diff --git a/-9E4T4oBgHgl3EQf4A3h/content/tmp_files/2301.05311v1.pdf.txt b/-9E4T4oBgHgl3EQf4A3h/content/tmp_files/2301.05311v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f62f0fd2aa7307faf1cd4be6bbe445554251ca6c --- /dev/null +++ b/-9E4T4oBgHgl3EQf4A3h/content/tmp_files/2301.05311v1.pdf.txt @@ -0,0 +1,895 @@ +A Framework for Active Haptic Guidance Using Robotic Haptic Proxies +Niall L. Williams1, Jiasheng Li1, and Ming C. Lin1 +https://gamma.umd.edu/active haptic guidance/ +Abstract— Haptic feedback is an important component of +creating an immersive virtual experience. Traditionally, haptic +forces are rendered in response to the user’s interactions +with the virtual environment. In this work, we explore the +idea of rendering haptic forces in a proactive manner, with +the explicit intention to influence the user’s behavior through +compelling haptic forces. To this end, we present a framework +for active haptic guidance in mixed reality, using one or more +robotic haptic proxies to influence user behavior and deliver +a safer and more immersive virtual experience. We provide +details on common challenges that need to be overcome when +implementing active haptic guidance, and discuss example +applications that show how active haptic guidance can be used +to influence the user’s behavior. Finally, we apply active haptic +guidance to a virtual reality navigation problem, and conduct a +user study that demonstrates how active haptic guidance creates +a safer and more immersive experience for users. +I. INTRODUCTION +In mixed reality (MR), the user is at least partially im- +mersed in a 3D, computer-generated environment. Included +within the mixed reality spectrum are augmented reality +and virtual reality (VR). A major factor that makes MR a +unique medium is that it is interactive—the user is able to +interact with the virtual environment (VE) through position- +tracking sensors that update the VE according to the user’s +movements in the physical environment (PE). For example, +when the user moves their head in the real world, the position +of the camera in the virtual world moves as well. Interactions +like these help to make users feel like they are really in the +VE that they see through the head-mounted display (HMD). +One key component to increasing the user’s sense of presence +in a VE is to improve the perceptual stimuli matching [8], +wherein the user is provided with perceptual information that +matches their actions (e.g. the viewing perspective updates +as the user moves their head). In this work, we focus on +the sense of touch provided by mechanical haptic feedback +and how we can use robots to provide more realistic haptic +sensations to improve the sense of immersion and safety in +mixed reality. +Robotic technology has in fact been used to provide haptic +feedback in MR to improve the sense of virtual touch and +virtual manipulation [10]. For example, MR can enhance +robotics via telepresence, wherein humans can remotely +operate robot to high precision using immersive controls +afforded by VR. +*This work is partially supported by National Science Foundation and +Lin’s professorship. +1Authors are with the Department of Computer Science, University of +Maryland, College Park. {niallw, jsli, lin}@umd.edu +Fig. 1. +An image of a user in the physical environment (left) and virtual +environment (right) in our implementation of active haptic guidance. The +user is tethered to a robot in the physical environment and to a virtual dog +companion in the virtual environment. The robot provides haptic feedback to +the user according to the virtual companion’s movements, which improves +the user’s sense of presence in the virtual world and encourages the user to +avoid the boundaries of the virtual reality system’s tracked space. +In this paper, we introduce the possibility of using robots +to enhance the virtual experience through haptic feedback. +Specifically, we use robots to guide the user as they navigate +through a VE, and reconfigure and virtually expand the +PE to align with the VE; we achieve this through manual +haptic feedback that directs the user’s locomotion behavior +in the VE, thereby making the virtual experience more +immersive and safer. To this end, we introduce the concept +of active haptic guidance, which describes the problem of +reconfiguring one or multiple robots in the PE in real time +such that they provide haptic feedback to guide the user +and influence their actions and motion in the VE, with the +ultimate goal of improving the user’s safety or level of +immersion in MR. One major challenge with robots for active +haptic feedback in MR is that the physical robots and their +virtual counterparts must be co-located relative to the user, +in order to provide the correct haptic feedback that aligns +with the virtual counterpart. This problem can be exacerbated +when the environments/interactions are dynamic (i.e. the +physical and virtual haptic proxy must move synchronously) +or when there is a decoupling between the user’s physical and +virtual locations (as is common with some VR interaction +techniques like redirected walking [24]). +Main contributions: We introduce the concept of ac- +tive haptic guidance for improved virtual locomotion, and +conduct a user study to show an example of how active +haptic guidance can be used to improve a user’s safety and +arXiv:2301.05311v1 [cs.RO] 12 Jan 2023 + +feelings of immersion in a virtual experience. Our framework +is general, so it can be applied to use cases other than +locomotion, and we provide examples of other possible use- +cases for active haptic guidance. Our main contributions +include: +• A formal description of the active haptic guidance +problem and details on common challenges that are +faced when implementing active haptic guidance. Ac- +tive haptic guidance involves using robots to provide +realistic haptic feedback to users in mixed reality, with +the goal of influencing users’ behaviors to improve +their safety and/or sense of presence in the virtual +environment. +• An prototype realization and user study showing the +benefits of active haptic guidance. In our study, par- +ticipants completed a virtual navigation task using real +walking, either with or without active haptic guidance. +Our results show that active haptic guidance can signif- +icantly improve the virtual experience by reducing the +number of “breaks in presence” and keeping them a safe +distance away from physical objects for longer. +II. BACKGROUND AND RELATED WORK +Haptic feedback can be utilized in any mixed reality +setting, but in this work we mainly discuss applications of +haptics to virtual reality (VR) settings, since our implementa- +tion was done in VR. In VR, the user wears a head-mounted +display (HMD) through which they view a 3D, computer- +generated virtual environment (VE) [15]. The user’s position +in the physical environment (PE) is tracked, so that whenever +the user updates their position in the PE, the position +of the virtual camera updates accordingly to provide an +accurate viewing perspective of the VE. VR is an interactive +experience, meaning that the user does not passively observe +the virtual content, but instead the environment changes in +response to the user’s actions and movements. When the +virtual experience feels sufficiently real, the user experiences +a sense of presence, which describes the subjective feeling of +really being in the environment [31]. Factors that contribute +to a user’s feelings of presence and immersion in a VE +include the HMD refresh rate [3], the environment realism +and visual quality [34], and perceptual stimuli matching +[8], [33] (the process of providing users with perceptual +information that matches their actions in the VE). In this +paper, we focus on providing haptic stimuli for perceptual +stimulus matching to improve the user’s experience in VR. +Haptic feedback can be provided in a passive or an active +manner. With passive haptics, objects are placed in the PE +such that they align with the locations of objects in the VE, +resulting in haptic feedback when the user tries to touch +objects in the VE [11]. Conversely, active haptics involves +a haptic proxy that dynamically alters its configuration in +real time to provide the appropriate haptic force feedback, +depending on the user’s interactions with the VE. It is +common to use robotic systems to render haptic forces. For +example, Zhang et al. [36] used a robotic arm to provide +haptic feedback during object assembly by aligning the arm’s +end effector with the handheld proxy. Siu et al. [28] used +an array of actuated pins to match the contours of virtual +objects. Similarly, Zhao et al. [37] used robotic assembly +to construct tangible representations of virtual objects, made +from magnetically attached blocks. To recreate the feelings of +grasping virtual objects, Kovacs et al. [18] used a wrist-worn +device to provide on-demand haptic feedback when users +grip virtual objects, while Sinclair et al. [27] used a force- +resisting, handheld controller to render haptic forces for rigid +and compliant objects. Suzuki et al. [32] used mobile robots +to rearrange physical furniture to align with virtual furniture +as the user moved through a virtual world. Robotic systems +have also been used to aid in navigation through VEs, via +handheld canes that use vibrations to provide information +about the VE [38], [19], [29], mechanical staircases [13] to +simulate uneven virtual terrain, or mobile tiles that simulate +infinite walking in any direction [12]. +The majority of prior work on active haptics for mixed +reality requires the user to initiate interactions with the VE +before the haptic forces are rendered. That is, the haptic +forces are triggered by the user’s interactions with the VE, +so it is the user’s actions that dictate when haptic forces +are rendered. In this work, we make the distinction of using +active haptics specifically to direct the user and influence +their behavior in the VE (in addition to providing a more +immersive experience, as all haptics aims to do). We define +this use of haptics as active haptic guidance, since it is the +haptic forces that direct the user’s behaviors, rather than +the other way around. We note that there already exists +research on “haptic guidance,” which Feygin et al. use to +refer to haptic feedback that is used to help people learn +motor skills [7]. The distinction between our work on active +haptic guidance and Feygin et al.’s work is that we use haptic +feedback to discreetly influence the user’s behavior in an +effort to enhance their feelings of presence and level of safety +in a mixed reality experience, while Feygin et al. use haptics +to teach people motor skills. +III. PROBLEM DESCRIPTION +Here we describe the active haptic guidance problem, as +well as constraints that need to be satisfied to effectively +utilize haptics to guide users in MR. +A. Definitions +In virtual reality, the user is located in a physical envi- +ronment (PE) and a virtual environment (VE) at the same +time. Each environment consists of objects (either physical +objects or virutal objects represented by textured meshes) +and agents (the users and robots). Note that it is common +to refer to virtual humans and animals as agents, but in this +work we will consider all components of the VE as generic +objects for simplicity, and we use “agents” to refer only to +humans and robots in the PE. +Let O = {o1, o2, ..., oi} be a set of polygonal objects, +where each object o is a mesh with vertices in R3. Let +U = {u1, u2, ..., uj} be the set of users in an environment. +Here, u represents the user’s state in an environment, and + +usually describes their position and orientation in said envi- +ronment. For example, we can define u = {p, θ}, where +p ∈ R2 represents their position in the 2D plane and +θ ∈ [0, 2π) represents their orientation in the environment. +Let R += +{r1, r2, ..., rk} be the set of robots in an +environment, and let A = {U ∪ R} be set of all agents. +Each of these sets O, U, R, and A may be empty. +We define an environment E as a set of obstacles and +agents; that is, E = {O, A}. To differentiate between the +PE and VE, we denote the PE as EP = {OP , AP } and the +VE as EV = {OV , AV }. For each user in virtual reality, +they will have a representation in both the PE and VE, so +|UP | = |UV | = n, where n is the number of users in virtual +reality. Since we only consider agents to be users and robots +in this work, |AV | = n (i.e., the only agents in the VE are +the users). In the VE, there are some objects that the user +is likely to interact with, which will improve their sense of +presence in the environment. We define this set of “objects +of interest” O ⊂ OV as the set of virtual objects for which +we render haptic forces when the user interacts with them. +With these definitions of the PE and VE, we can now de- +scribe the two main conditions that need to be met to provide +active haptic guidance to users in MR. First, the robots in the +physical environment need to provide the appropriate haptic +feedback to influence the user’s configuration. Second, we +need to ensure that the robots that provide haptic feedback +are co-located (relative to the user) with the virtual objects +of interest with which the haptic forces are associated. +B. Influential Haptics Constraint +The first condition that needs to be met in order to +implement active haptic guidance is that the rendered haptic +forces should influence the user’s behavior such that they +update their physical and virtual configurations. We dub +this constraint the influential haptics (IH) constraint. For +simplicity, we formalize this constraint using one user, one +robot, and one virtual object of interest, but this constraint +applies to any group of agents and virtual objects for which +we render haptic forces. +Given the user’s physical and virtual configurations uP +and uV , a virtual object of interest o, and a robot r that +provides haptic feedback for o, we wish to render a haptic +force F that compels the user to update uP and uV to +some goal configurations u∗ +P and u∗ +V . Thus, fulfilling the +IH constraint requires completing the following steps: +1) Compute the goal configurations u∗ +P and u∗ +V . +2) Detect or initiate an interaction I between o and uV . +3) Update the configuration of r to render a haptic force +F(I, uV , uP , u∗ +P , u∗ +V , r) that minimizes an objective +function f(uV , uP , u∗ +P , u∗ +V ). +In practice, computing F(I, uV , uP , u∗ +P , u∗ +V , r) depends +heavily on the mechanics of the haptic proxy r and the ob- +jective function f(uV , uP , u∗ +P , u∗ +V ). The objective function is +usually a distance function that measures the error between +uP and uV , and it depends on the user’s configuration +space. By rendering F, the user hopefully updates their +configuration such that they move closer to u∗ +P and u∗ +V . +Computing u∗ +P and u∗ +V is a matter of determining how we +want the user to behave. In mixed reality (MR), two main +reasons to influence the user’s behavior are to ensure their +safety and to deliver a more immersive experience. In MR +systems, the user tries to navigate through the PE and the VE +at the same time, but the PE is partially or fully occluded. +Thus, in order to prevent the user from bumping into physical +objects that they cannot see, locomotion interfaces for MR +usually display a notification that prompts them to reposition +themself to a safer position away from nearby objects. By +using haptics to warn users (either overtly or subtly), we can +decrease the likelihood that the user collides with unseen +physical obstacles or exits the designated tracking area. +In addition to ensuring user safety, influencing the user’s +behavior can be useful for improving the user’s sense of +presence in the VE. In MR, providing perceptual stimuli +that align with the content rendered on the visual display +enhances the user’s feeling that they are really in the VE that +they are seeing. To this end, haptic feedback can significantly +improve the user’s sense of presence in the VE [11]. In the +case of active haptic guidance, the haptic feedback can be +used as an additional narrative element that encourages users +to explore a particular area or interact with particular objects +in the VE (e.g. pairing visual distractors [21] with haptic +feedback to direct the user’s attention). +C. Relative Co-location Constraint +The second main constraint that should be met when +using active haptic guidance is that the physical robots +that render the haptic forces and their associated virtual +objects should be co-located relative to the user. That is, +the position of the robot and the virtual object should be the +same relative to the user’s configuration in the PE and VE. +This is done to ensure that the user perceives a congruent VE +that is augmented by haptic forces, rather than perceiving a +VE along with misaligned haptic forces, which may break +their sense of presence in the virtual experience. We call this +the relative co-location (RC) constraint. +Given the user’s physical and virtual configurations uP +and uV , a virtual object of interest o, and a robot r that +provides haptic feedback for o, we wish to update r such +that we minimize the error in the relative positions between +uV and o and uP and r. Fulfilling the RC constraint requires +completing the following steps: +1) Compute the configurations of o and r relative to uV +and uP , respectively. Usually, these are just positions +po and pr of o and r relative to the user in the +respective environment. +2) Compute a goal configuration r∗ for the haptic proxy +that minimizes an objective function f(po, pr). +3) Update the configuration of r to move it towards r∗. +In practice, updating the robot’s configuration in step #3 is +a motion planning problem where we aim to find a path +through the configuration space that brings r close to r∗, +and it depends on the mechanics of the haptic proxy. +Since MR is an interactive technology, the relative posi- +tions po and pr are constantly changing as the user explores + +and interacts with the VE. Thus, evaluating and fulfilling the +RC constraint must be done constantly to ensure that any per- +ceptual stimuli mismatch is minimized. Failure to adequately +meet this constraint can degrade the user experience, since +it increases the likelihood that the user notices a discrepancy +between visual stimuli and the haptic stimuli [14], [20]. +Furthermore, knowing how much error between their relative +positions the user will tolerate is a subjective measure [2], +[17], so it is usually not the case that the robot must reach +r∗ exactly. Note that this relative co-location constraint is +not unique to the active haptic guidance problem (unlike +subsection III-B); other work on active haptics for virtual +reality also has to deal with the problem of ensuring the +co-location of robotic agents and their virtual counterparts. +IV. PROTOTYPE REALIZATION EXAMPLES +In this section, we provide details on our prototype im- +plementation of an application of active haptic guidance. In +particular, we implement an active haptic-driven locomotion +application to provide a safer and more immersive virtual +navigation experience for users. We discuss other potential +use-cases for active haptic guidance in the supplementary +materials posted on our project page. +A. Natural Walking in Virtual Reality +In VR, it is common for the PE to be much smaller than +the VE. To enable users to explore large VEs, many different +locomotion interfaces such as teleportation, joystick naviga- +tion, and walking-in-place have been developed [6]. Ideally, +users explore the VE using natural, everyday walking since +it improves their sense of presence [33] and performance +in tasks [9], [22], [26]. One technique that enables natural +walking in VR is redirected walking (RDW) [24]. +RDW works by slowly rotating the VE around the user’s +virtual camera while they walk, which causes them to +unconsciously adjust their physical trajectory to counteract +the VE rotations and remain on their intended path in the +VE. It works because the human perceptual system tends to +believe the user’s visual stimuli over other stimuli (proprio- +ceptive, vestibular, etc.) when they conflict, as is the case in +RDW [23]. Using RDW, we can steer the user along paths in +the PE that direct them away from objects and edges of the +tracked space, resulting in a safer and more immersive virtual +experience. To help mask the VE rotations, researchers make +use of distractors which grab the user’s attention to decrease +the likelihood that they attend to the rotations of the VE [4], +[21], [35]. In our prototype implementation, we use a virtual +dog as a distractor in conjunction with a RDW algorithm +known as steer-to-center, which rotates the VE such that the +user is steered towards the center of the PE at all times [23]. +B. Virtual Experience and Equipment +For our implementation, a user u1 completed a navi- +gation task in a virtual city and had a virtual dog as a +companion (only a single user participated at a time, so +|UP | += +|UV | += +1). Additionally, u1 held a position- +tracked leash that was tethered to a differential wheeled robot +r1. The PE was an empty rectangular room with four walls +(represented by the boundaries of the VR tracking space). +Thus, EP += +{OP , AP }, where AP += +{u1, r1}. The +virtual dog served as a distractor and was the only object +of interest in EV (|O| = 1), meaning that the robot only +rendered haptic forces associated with the virtual dog. +Our application was implemented using one HTC VIVE +Cosmos VR HMD with two VIVE trackers, and one robot +car (ELEGOO UNO Robot Car kit). We attached one VIVE +tracker to the robot to track its location and orientation data, +and the other was attached to the leash handle to calculate +the distance between u1 and r1. The robot was equipped +with an HC-06 Bluetooth LE adapter, which connected to +the PC to transmit robot movement commands. The Unreal +4.22 game engine was used to render the VE. +C. Virtual Companion and Robot Behavior +Here we describe the behavior of the virtual dog com- +panion and how the robot matches the virtual companion’s +movements and provides haptic feedback. +1) Virtual Dog Companion Behavior: The virtual dog has +two main behavior states: following and distracting. When +the user walks around and is not at risk of leaving the +tracking space, the dog is in follow mode. In this mode, +the dog walks slightly ahead of the user as they walk, and +remains in one spot when the user stands still. +When the user reaches a boundary of the tracked space, the +VR system initiates what is called a reset, wherein the user +reorients themself such that they face towards the inside of +the tracking space in the PE. To ensure that their orientation +in the VE is not altered, the VR system applies redirection +that effectively cancels out their physical rotation in the +virtual space. When a reset is initiated, the virtual dog enters +distract mode. In distract mode, we compute a goal position +in the VE for the dog to move towards. The idea behind +distract mode is that the user is likely to pay attention to the +virtual dog as it runs to a goal position, which allows the +system to apply stronger redirection (away from the obstacles +in the PE) without interfering with the user’s experience [21]. +During a reset, the goal position is selected by first +computing the vector from the user towards the center of the +physical space. The goal position is then determined to be +either the endpoint of this vector in the VE, or a virtual object +near the vector’s endpoint that was labeled as a potential +goal position during development. Potential goal positions +are virtual objects that a dog would be likely to interact with, +such as a fire hydrant or a lamp post. If the vector intersects +with a virtual object (e.g. a virtual building) and there are no +potential goal objects nearby, the goal position is simply the +point furthest along the vector that does not intersect with +any objects. See Figure 2 for a visualization of this process. +2) Robot Haptic Proxy Behavior: The physical robot’s +main purpose is to provide haptic feedback to make the user’s +virtual experience feel more immersive and to encourage the +user to walk away from nearby objects or tracking space +boundaries in the PE. In both follow and distract mode, the +physical robot needs to update its position such that it is + +Fig. 2. +Our method of automatically choosing a suitable virtual goal position for the virtual companion. When the user gets close to a boundary of +the physical space, they need to be reoriented away from the boundary before they continue walking. In order to pick a goal destination for the virtual +companion and robotic haptic proxy, we cast a ray from the physical user to the center of the tracked space and then superimpose this vector onto the +user’s virtual position. If the endpoint of this vector is near a pre-defined potential goal position, that is chosen as the current goal position. Otherwise, we +choose the furthest point along the vector that does not intersect with any objects in the virtual environment. +aligned with the position of the virtual dog, relative to the +user in either environment. Checking if a position update is +necessary is easily achieved by computing the vector from +the virtual user to the virtual dog and comparing it to the +vector from the user’s HMD and the robot. +To compute the trajectory that the robot will follow, we +compute a circular arc path based on the robot’s position, +forward direction, and destination position (determined by +the relative position of the virtual dog and user). The ideal +path for a differential drive robot is a circular arc since it only +requires one set of wheel velocities [5]. The wheel velocities +are computed with the ratio +2rd +2r−d, where r is the arc radius +and d is the distance between the robot wheels. Note that we +do not use typical PID-based drift correction due to possible +unexpected complications that may arise from the tethering +to the user [1], [25], [30]. +D. Maintaining Active Haptic Guidance Constraints +This section describes how our active haptic-drive loco- +motion application satisfies the IH and RC constraints. +1) Directing Users With Haptic Feedback: +Since the +virtual object of interest is a dog, the user is attached to the +robot by an elastic tether that resembles a leash. When the +robot moves away from the user in the PE, it simulates the +sensation of a dog tugging on its leash, thereby improving the +realism of the virtual experience. Additionally, this tugging +encourages the user to follow the robot rather than “fight” it, +allowing us to further influence the user’s movement patterns +in the PE and VE. By triggering the robot to move away from +the user and towards the center of the PE when they get too +close to the tracking space boundaries, the tugging force on +the leash encourages the user to turn and walk towards the +robot and away from the tracked space boundaries. +2) Maintaining Co-location: Normally, maintaining rela- +tive co-location between a haptic proxy and a virtual object +is a matter of updating the position of the haptic proxy +whenever the virtual object’s position changes. We also do +this in our implementation by updating the position of the +robot to match the movements of the virtual dog. However, +our implementation requires additional work to maintain co- +location due to a new problem which we refer to as the haptic +proxy distortion (HPD) problem. +Virtual environment +before rotation. +Virtual environment after +rotation. +Physical environment +and superimposed +virtual relative positions. +Fig. 3. A visualization of the haptic proxy distortion problem. Left: Initially, +the virtual user and virtual companion have a particular relative position. +Middle: After rotating the virtual environment around the virtual user, the +relative position of the companion changes since the companion is rotated +along with the rest of the environment. Right: In the physical space, the +haptic proxy has not been updated, so its position coincides with the virtual +companion’s relative position before rotation (opaque robot and vector). The +new relative position of the virtual companion, which the haptic proxy needs +to match, is shown as the translucent robot and dashed-line vector. +In our implementation, we make use of a locomotion +interface called redirected walking (RDW) that enables nat- +ural walking in VR. RDW works by rotating the entire VE +around the virtual camera that represents the user’s viewpoint +in the VE. Consequently, the virtual dog companion may +change its position relative to the virtual user without the dog +actually moving to a new destination in the VE (see Figure 3). +Thus, as we apply redirection, the relative position of the +virtual dog changes constantly, while the relative position +of the physical robot does not. To resolve this discrepancy +in relative position, we check the relative positions of the +virtual dog and physical robot on each frame, and update +the robot’s destination in the PE to minimize the difference +in relative position. The user will perceive this as the haptic +proxy “sliding” across the floor around them, which might +result in unsmooth motion that may detract from the user +experience. In practice, this did not seem to be a major +problem for users, but we acknowledge that there may be +better solutions to the HPD problem, and leave that for future +work. This HPD problem adds onto the errors in relative co- +location between the haptic proxy and the virtual companion, +which makes it harder to satisfy the RC constraint. Note +that the HPD problem is not specific to our implementation; +this problem is present in any application that uses haptic +proxies and creates a mismatch between the user’s positions +in the physical and virtual environments, as is common for + +Physical Environment +Virtual Environment +Virtual Environment +Virtual Environment +Virtual Environment +UTU +User reached the tracked space boundary, so a +Superimpose the physical user-to-center +If there is a potential pre-defined goal +If there is no pre-defined goal position near +If the superimposed user-to-center vector +reorientation is required. Compute the vector +vector onto the virtual user to determine the +position near the endpoint of the user-to- +the endpoint of the user-to-center vector, +intersects with a virtual object, use the +from the user to the center of the physical +goal position of the virtual companion. +center vector (e.g., a fire hydrant), set that as +use the vector endpoint as the goal position. +furthest non-intersecting point along the +" vector). +the goal position. +vector as the goal position.locomotion interfaces for mixed reality. +V. EXPERIMENTS & RESULTS +A. Experiment Design and Procedure +To evaluate the effectiveness of our implementation of +active haptic-driven locomotion prototype, we conducted a +user study where participants completed a navigation task. +The study design was approved by our university’s Insti- +tutional Review Board. The goal of our user study was to +evaluate how effective the haptic guidance was at improving +users’ sense of presence in the VE and keeping users away +from the boundaries of the VR system’s tracked space. We +used a between participants design, where one group of +participants completed a navigation task with active haptic +guidance enabled, and the other group completed the same +task without any haptic guidance. The navigation task had +a time limit of 5 minutes and 30 seconds, after which the +experiment ended regardless of if the participant reached the +goal destination. Participants were unaware of this time limit +so that they did not rush to complete the task. We recruited 20 +participants (13 male, 5 female, 2 participants did not report) +through online advertising and oral recruitment. Participants’ +ages ranged from 18 to 28 (µ = 24.59, σ = 2.37). All +participants were able to walk without any assistance. +The study consisted of three sections, and lasted about 15 +minutes for each participant. First, we debriefed participants +on the experiment procedures and had them complete a pre- +study Simulator Sickness Questionnaire (SSQ) [16]. Next, +the user put on the HMD and completed the task in the +VE. The VE was a city environment with several streets and +blocks, and was populated with common objects such as bus +stops, stores, park squares, and virtual humans that roamed +around the environment (see Figure 1 and the supplementary +video). To mask any potentially distracting noises from the +robot as it moves, participants wore headphones and back- +ground music was played for the duration of the experiment +task. Participants started the task at one intersection in the +city, and their task was to reach a green question mark in +the environment that indicated their destination, which was +one block away from the their starting position. During the +experiment, we recorded how many times users reached the +bounds of the PE and the time taken to complete the task. +Once participants finished the task, they completed another +SSQ survey and a questionnaire with questions on a 7-point +Likert scale that measured their sense of presence in the VE +(7 = high presence, 1 = low presence). Finally, the experiment +was ended with open-ended questions where participants +could provide additional comments. +B. Results +The metrics we used to measure the effectiveness of our +active haptic-driven locomotion interface were the number of +breaks in presence (BiPs), the completion rate and time taken +to complete the task, and participants’ subjective feelings of +presence in the VE. A BiP is incurred when the user reaches +the boundaries of the tracking space and they are forced to +reorient away from the boundary before continuing to walk. +BiPs +Time (s) +Presence +Completed +Haptics +µ +σ +µ +σ +µ +σ +Total # +With +0.90 +0.74 +195.20 +22.25 +4.63 +1.77 +10 +Without +18.90 +5.17 +309.40 +65.14 +3.57 +1.64 +1 +TABLE I +Performance results from our user study. THE “WITH HAPTICS” +GROUP OF PARTICIPANTS INCURRED SIGNIFICANTLY fewer BREAKS IN +PRESENCE (“BIPS” COLUMN), COMPLETED THE EXPERIMENT MUCH +more quickly (“TIME” COLUMN) AND WITH MUCH higher SUCCESS +RATES (“COMPLETED” COLUMN), AND REPORTED A higher SENSE OF +PRESENCE IN THE VIRTUAL EXPERIENCE (“PRESENCE” COLUMN). +THESE RESULTS SHOW THAT HAPTIC GUIDANCE CAN BE EFFECTIVE FOR +IMPROVING USERS’ VIRTUAL EXPERIENCE. +Based on the results in Table I, the presence of our active +haptic guidance companion resulted in significantly fewer +BiPs, notably lower completion times and higher completion +rates, and slightly higher (and above-average) presence lev- +els. Meanwhile, participants who completed the navigation +task without any haptic guidance incurred a large number of +BiPs, did not finish the task in time, and reported below- +average levels of presence. These results support the notion +that active haptic guidance can be used to help keep users +safe and feel more immersed in mixed reality experiences. +VI. CONCLUSIONS & FUTURE WORK +In this work, we presented the active haptic guidance +problem for mixed reality (MR), which describes the use +of one or more robots to provide haptic feedback to users +in order to create a richer virtual experience for the user, +while also influencing the user’s behavior to improve their +safety and immersion in the virtual world. As a prototype +realization, we implemented active haptic guidance in a VR +locomotion application that enables the user to explore a +large VE while located in a much smaller PE. By combining +active haptic guidance and redirected walking, we increased +the effective area of the PE while also decreasing the +likelihood that the user exits the VR system’s tracked area. +The concept of active haptic guidance is general and can be +applied MR applications other than locomotion; we discuss +other potential use cases for active haptic guidance in the +supplementary materials on our project page. +Limitations and Future Work: One limitation of our +work is the haptic proxy distortion problem, in which the +haptic proxy and the associated virtual object can become +desynchronized due to mismatches between the user’s phys- +ical and virtual configurations. Solving this problem requires +continuously updating the position of the haptic proxy, and +our proposed solution in this work is likely not the most +optimized solution. Additionally, our system uses only a +rough estimation of drift to readjust the haptic proxy position, +instead of a more accurate method like PID-based drift +correction. Future work in this area should investigate the +use of more realistic companions and behavior models, and +should explore how active haptic guidance can be applied +to other types of VR experiences with different applications, +such as social mixed reality settings with other users. + +REFERENCES +[1] K.-E. ˚Aarz´en, “A simple event-based pid controller,” IFAC Proceedings +Volumes, vol. 32, no. 2, pp. 8687–8692, 1999. +[2] M. Azmandian, M. Hancock, H. Benko, E. Ofek, and A. D. Wilson, +“Haptic retargeting: Dynamic repurposing of passive haptics for en- +hanced virtual reality experiences,” in Proceedings of the 2016 chi +conference on human factors in computing systems, 2016, pp. 1968– +1979. +[3] W. Barfield and C. 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Billinghurst, “Robot-enabled tangible virtual assembly with coor- +dinated midair object placement,” Robotics and Computer-Integrated +Manufacturing, vol. 79, p. 102434, 2023. +[37] Y. Zhao, L. H. Kim, Y. Wang, M. Le Goc, and S. Follmer, “Robotic +assembly of haptic proxy objects for tangible interaction and virtual +reality,” in Proceedings of the 2017 ACM International Conference on +Interactive Surfaces and Spaces, 2017, pp. 82–91. +[38] Y. Zhao, C. L. Bennett, H. Benko, E. Cutrell, C. Holz, M. R. +Morris, and M. Sinclair, “Enabling people with visual impairments +to navigate virtual reality with a haptic and auditory cane simulation,” +in Proceedings of the 2018 CHI conference on human factors in +computing systems, 2018, pp. 1–14. + +APPENDIX +Additional Applications of Active Haptic Guidance: Here +we discuss other potential applications of active haptic guid- +ance for immersive applications: +• Wood Carving Application: In wood carving, the grain +of the wood will impact the direction in which the artist +carves the wood. That is, sometimes the artist will carve +“with the grain” and sometimes will carve “against the +grain.” Using active haptics, one could accurately render +the different resistance forces that arise from carving +with or against the grain of a virtual wooden block, +which will in turn influence the way in which the user +carves their virtual wooden sculpture. In addition to +providing a more realistic experience, this could be +used to guide the user to create a more appealing final +sculpture (e.g. by altering the direction of the grain to +subtly change their hand movements, which will change +the shape of the final carved surface). +• Immersive Cooperative Application: A major appeals +of mixed reality experiences is the ability to connect +with other users in shared virtual experiences. Important +to these shared experiences is the ability to touch the +other person, which can provide a greater sense of +companionship and connection between users. Haptic +forces can be used to encourage users to interact with +or follow other users who are also present in their virtual +experience, which may improve the users’ sense of +presence in the experience due to the enhanced realism. +• Virtual Cooking Training Application: Given a seated +VR experience where the user is practicing their cook- +ing skills in a virtual environment, a mobile, tabletop +robot can provide haptic feedback that represents feed- +back provided by cooking utensils. For example, when +spreading brownie batter in a baking pan, the user will +feel haptic forces when the virtual spreading utensil gets +too close to the edges of the virtual baking pan. These +forces could be rendered using a mobile robot with a +flat surface that serves as a wall that the user’s physical +hand will bump into. + diff --git a/-9E4T4oBgHgl3EQf4A3h/content/tmp_files/load_file.txt b/-9E4T4oBgHgl3EQf4A3h/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f1f54bb391c919f07e601988e7d18a7d3d6b74ef --- /dev/null +++ b/-9E4T4oBgHgl3EQf4A3h/content/tmp_files/load_file.txt @@ -0,0 +1,594 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf,len=593 +page_content='A Framework for Active Haptic Guidance Using Robotic Haptic Proxies Niall L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Williams1, Jiasheng Li1, and Ming C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Lin1 https://gamma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content='umd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content='edu/active haptic guidance/ Abstract— Haptic feedback is an important component of creating an immersive virtual experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Traditionally, haptic forces are rendered in response to the user’s interactions with the virtual environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' In this work, we explore the idea of rendering haptic forces in a proactive manner, with the explicit intention to influence the user’s behavior through compelling haptic forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' To this end, we present a framework for active haptic guidance in mixed reality, using one or more robotic haptic proxies to influence user behavior and deliver a safer and more immersive virtual experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' We provide details on common challenges that need to be overcome when implementing active haptic guidance, and discuss example applications that show how active haptic guidance can be used to influence the user’s behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Finally, we apply active haptic guidance to a virtual reality navigation problem, and conduct a user study that demonstrates how active haptic guidance creates a safer and more immersive experience for users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' INTRODUCTION In mixed reality (MR), the user is at least partially im- mersed in a 3D, computer-generated environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Included within the mixed reality spectrum are augmented reality and virtual reality (VR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' A major factor that makes MR a unique medium is that it is interactive—the user is able to interact with the virtual environment (VE) through position- tracking sensors that update the VE according to the user’s movements in the physical environment (PE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' For example, when the user moves their head in the real world, the position of the camera in the virtual world moves as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Interactions like these help to make users feel like they are really in the VE that they see through the head-mounted display (HMD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' One key component to increasing the user’s sense of presence in a VE is to improve the perceptual stimuli matching [8], wherein the user is provided with perceptual information that matches their actions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' the viewing perspective updates as the user moves their head).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' In this work, we focus on the sense of touch provided by mechanical haptic feedback and how we can use robots to provide more realistic haptic sensations to improve the sense of immersion and safety in mixed reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Robotic technology has in fact been used to provide haptic feedback in MR to improve the sense of virtual touch and virtual manipulation [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' For example, MR can enhance robotics via telepresence, wherein humans can remotely operate robot to high precision using immersive controls afforded by VR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' This work is partially supported by National Science Foundation and Lin’s professorship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' 1Authors are with the Department of Computer Science, University of Maryland, College Park.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' {niallw, jsli, lin}@umd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content='edu Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' An image of a user in the physical environment (left) and virtual environment (right) in our implementation of active haptic guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' The user is tethered to a robot in the physical environment and to a virtual dog companion in the virtual environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' The robot provides haptic feedback to the user according to the virtual companion’s movements, which improves the user’s sense of presence in the virtual world and encourages the user to avoid the boundaries of the virtual reality system’s tracked space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' In this paper, we introduce the possibility of using robots to enhance the virtual experience through haptic feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Specifically, we use robots to guide the user as they navigate through a VE, and reconfigure and virtually expand the PE to align with the VE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' we achieve this through manual haptic feedback that directs the user’s locomotion behavior in the VE, thereby making the virtual experience more immersive and safer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' To this end, we introduce the concept of active haptic guidance, which describes the problem of reconfiguring one or multiple robots in the PE in real time such that they provide haptic feedback to guide the user and influence their actions and motion in the VE, with the ultimate goal of improving the user’s safety or level of immersion in MR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' One major challenge with robots for active haptic feedback in MR is that the physical robots and their virtual counterparts must be co-located relative to the user, in order to provide the correct haptic feedback that aligns with the virtual counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' This problem can be exacerbated when the environments/interactions are dynamic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' the physical and virtual haptic proxy must move synchronously) or when there is a decoupling between the user’s physical and virtual locations (as is common with some VR interaction techniques like redirected walking [24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Main contributions: We introduce the concept of ac- tive haptic guidance for improved virtual locomotion, and conduct a user study to show an example of how active haptic guidance can be used to improve a user’s safety and arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content='05311v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content='RO] 12 Jan 2023 feelings of immersion in a virtual experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Our framework is general, so it can be applied to use cases other than locomotion, and we provide examples of other possible use- cases for active haptic guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Our main contributions include: A formal description of the active haptic guidance problem and details on common challenges that are faced when implementing active haptic guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Ac- tive haptic guidance involves using robots to provide realistic haptic feedback to users in mixed reality, with the goal of influencing users’ behaviors to improve their safety and/or sense of presence in the virtual environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' An prototype realization and user study showing the benefits of active haptic guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' In our study, par- ticipants completed a virtual navigation task using real walking, either with or without active haptic guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Our results show that active haptic guidance can signif- icantly improve the virtual experience by reducing the number of “breaks in presence” and keeping them a safe distance away from physical objects for longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' BACKGROUND AND RELATED WORK Haptic feedback can be utilized in any mixed reality setting, but in this work we mainly discuss applications of haptics to virtual reality (VR) settings, since our implementa- tion was done in VR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' In VR, the user wears a head-mounted display (HMD) through which they view a 3D, computer- generated virtual environment (VE) [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' The user’s position in the physical environment (PE) is tracked, so that whenever the user updates their position in the PE, the position of the virtual camera updates accordingly to provide an accurate viewing perspective of the VE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' VR is an interactive experience, meaning that the user does not passively observe the virtual content, but instead the environment changes in response to the user’s actions and movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' When the virtual experience feels sufficiently real, the user experiences a sense of presence, which describes the subjective feeling of really being in the environment [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Factors that contribute to a user’s feelings of presence and immersion in a VE include the HMD refresh rate [3], the environment realism and visual quality [34], and perceptual stimuli matching [8], [33] (the process of providing users with perceptual information that matches their actions in the VE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' In this paper, we focus on providing haptic stimuli for perceptual stimulus matching to improve the user’s experience in VR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Haptic feedback can be provided in a passive or an active manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' With passive haptics, objects are placed in the PE such that they align with the locations of objects in the VE, resulting in haptic feedback when the user tries to touch objects in the VE [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Conversely, active haptics involves a haptic proxy that dynamically alters its configuration in real time to provide the appropriate haptic force feedback, depending on the user’s interactions with the VE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' It is common to use robotic systems to render haptic forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' For example, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' [36] used a robotic arm to provide haptic feedback during object assembly by aligning the arm’s end effector with the handheld proxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Siu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' [28] used an array of actuated pins to match the contours of virtual objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Similarly, Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' [37] used robotic assembly to construct tangible representations of virtual objects, made from magnetically attached blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' To recreate the feelings of grasping virtual objects, Kovacs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' [18] used a wrist-worn device to provide on-demand haptic feedback when users grip virtual objects, while Sinclair et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' [27] used a force- resisting, handheld controller to render haptic forces for rigid and compliant objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Suzuki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' [32] used mobile robots to rearrange physical furniture to align with virtual furniture as the user moved through a virtual world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Robotic systems have also been used to aid in navigation through VEs, via handheld canes that use vibrations to provide information about the VE [38], [19], [29], mechanical staircases [13] to simulate uneven virtual terrain, or mobile tiles that simulate infinite walking in any direction [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' The majority of prior work on active haptics for mixed reality requires the user to initiate interactions with the VE before the haptic forces are rendered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' That is, the haptic forces are triggered by the user’s interactions with the VE, so it is the user’s actions that dictate when haptic forces are rendered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' In this work, we make the distinction of using active haptics specifically to direct the user and influence their behavior in the VE (in addition to providing a more immersive experience, as all haptics aims to do).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' We define this use of haptics as active haptic guidance, since it is the haptic forces that direct the user’s behaviors, rather than the other way around.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' We note that there already exists research on “haptic guidance,” which Feygin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' use to refer to haptic feedback that is used to help people learn motor skills [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' The distinction between our work on active haptic guidance and Feygin et al.’s work is that we use haptic feedback to discreetly influence the user’s behavior in an effort to enhance their feelings of presence and level of safety in a mixed reality experience, while Feygin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' use haptics to teach people motor skills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' PROBLEM DESCRIPTION Here we describe the active haptic guidance problem, as well as constraints that need to be satisfied to effectively utilize haptics to guide users in MR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Definitions In virtual reality, the user is located in a physical envi- ronment (PE) and a virtual environment (VE) at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Each environment consists of objects (either physical objects or virutal objects represented by textured meshes) and agents (the users and robots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Note that it is common to refer to virtual humans and animals as agents, but in this work we will consider all components of the VE as generic objects for simplicity, and we use “agents” to refer only to humans and robots in the PE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Let O = {o1, o2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=', oi} be a set of polygonal objects, where each object o is a mesh with vertices in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Let U = {u1, u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=', uj} be the set of users in an environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Here, u represents the user’s state in an environment, and usually describes their position and orientation in said envi- ronment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' For example, we can define u = {p, θ}, where p ∈ R2 represents their position in the 2D plane and θ ∈ [0, 2π) represents their orientation in the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Let R = {r1, r2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=', rk} be the set of robots in an environment, and let A = {U ∪ R} be set of all agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Each of these sets O, U, R, and A may be empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' We define an environment E as a set of obstacles and agents;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' that is, E = {O, A}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' To differentiate between the PE and VE, we denote the PE as EP = {OP , AP } and the VE as EV = {OV , AV }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' For each user in virtual reality, they will have a representation in both the PE and VE, so |UP | = |UV | = n, where n is the number of users in virtual reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Since we only consider agents to be users and robots in this work, |AV | = n (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=', the only agents in the VE are the users).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' In the VE, there are some objects that the user is likely to interact with, which will improve their sense of presence in the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' We define this set of “objects of interest” O ⊂ OV as the set of virtual objects for which we render haptic forces when the user interacts with them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' With these definitions of the PE and VE, we can now de- scribe the two main conditions that need to be met to provide active haptic guidance to users in MR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' First, the robots in the physical environment need to provide the appropriate haptic feedback to influence the user’s configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Second, we need to ensure that the robots that provide haptic feedback are co-located (relative to the user) with the virtual objects of interest with which the haptic forces are associated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Influential Haptics Constraint The first condition that needs to be met in order to implement active haptic guidance is that the rendered haptic forces should influence the user’s behavior such that they update their physical and virtual configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' We dub this constraint the influential haptics (IH) constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' For simplicity, we formalize this constraint using one user, one robot, and one virtual object of interest, but this constraint applies to any group of agents and virtual objects for which we render haptic forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Given the user’s physical and virtual configurations uP and uV , a virtual object of interest o, and a robot r that provides haptic feedback for o, we wish to render a haptic force F that compels the user to update uP and uV to some goal configurations u∗ P and u∗ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Thus, fulfilling the IH constraint requires completing the following steps: 1) Compute the goal configurations u∗ P and u∗ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' 2) Detect or initiate an interaction I between o and uV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' 3) Update the configuration of r to render a haptic force F(I, uV , uP , u∗ P , u∗ V , r) that minimizes an objective function f(uV , uP , u∗ P , u∗ V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' In practice, computing F(I, uV , uP , u∗ P , u∗ V , r) depends heavily on the mechanics of the haptic proxy r and the ob- jective function f(uV , uP , u∗ P , u∗ V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' The objective function is usually a distance function that measures the error between uP and uV , and it depends on the user’s configuration space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' By rendering F, the user hopefully updates their configuration such that they move closer to u∗ P and u∗ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Computing u∗ P and u∗ V is a matter of determining how we want the user to behave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' In mixed reality (MR), two main reasons to influence the user’s behavior are to ensure their safety and to deliver a more immersive experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' In MR systems, the user tries to navigate through the PE and the VE at the same time, but the PE is partially or fully occluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Thus, in order to prevent the user from bumping into physical objects that they cannot see, locomotion interfaces for MR usually display a notification that prompts them to reposition themself to a safer position away from nearby objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' By using haptics to warn users (either overtly or subtly), we can decrease the likelihood that the user collides with unseen physical obstacles or exits the designated tracking area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' In addition to ensuring user safety, influencing the user’s behavior can be useful for improving the user’s sense of presence in the VE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' In MR, providing perceptual stimuli that align with the content rendered on the visual display enhances the user’s feeling that they are really in the VE that they are seeing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' To this end, haptic feedback can significantly improve the user’s sense of presence in the VE [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' In the case of active haptic guidance, the haptic feedback can be used as an additional narrative element that encourages users to explore a particular area or interact with particular objects in the VE (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' pairing visual distractors [21] with haptic feedback to direct the user’s attention).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Relative Co-location Constraint The second main constraint that should be met when using active haptic guidance is that the physical robots that render the haptic forces and their associated virtual objects should be co-located relative to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' That is, the position of the robot and the virtual object should be the same relative to the user’s configuration in the PE and VE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' This is done to ensure that the user perceives a congruent VE that is augmented by haptic forces, rather than perceiving a VE along with misaligned haptic forces, which may break their sense of presence in the virtual experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' We call this the relative co-location (RC) constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Given the user’s physical and virtual configurations uP and uV , a virtual object of interest o, and a robot r that provides haptic feedback for o, we wish to update r such that we minimize the error in the relative positions between uV and o and uP and r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Fulfilling the RC constraint requires completing the following steps: 1) Compute the configurations of o and r relative to uV and uP , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Usually, these are just positions po and pr of o and r relative to the user in the respective environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' 2) Compute a goal configuration r∗ for the haptic proxy that minimizes an objective function f(po, pr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' 3) Update the configuration of r to move it towards r∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' In practice, updating the robot’s configuration in step #3 is a motion planning problem where we aim to find a path through the configuration space that brings r close to r∗, and it depends on the mechanics of the haptic proxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Since MR is an interactive technology, the relative posi- tions po and pr are constantly changing as the user explores and interacts with the VE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Thus, evaluating and fulfilling the RC constraint must be done constantly to ensure that any per- ceptual stimuli mismatch is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Failure to adequately meet this constraint can degrade the user experience, since it increases the likelihood that the user notices a discrepancy between visual stimuli and the haptic stimuli [14], [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Furthermore, knowing how much error between their relative positions the user will tolerate is a subjective measure [2], [17], so it is usually not the case that the robot must reach r∗ exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Note that this relative co-location constraint is not unique to the active haptic guidance problem (unlike subsection III-B);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' other work on active haptics for virtual reality also has to deal with the problem of ensuring the co-location of robotic agents and their virtual counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' PROTOTYPE REALIZATION EXAMPLES In this section, we provide details on our prototype im- plementation of an application of active haptic guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' In particular, we implement an active haptic-driven locomotion application to provide a safer and more immersive virtual navigation experience for users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' We discuss other potential use-cases for active haptic guidance in the supplementary materials posted on our project page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Natural Walking in Virtual Reality In VR, it is common for the PE to be much smaller than the VE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' To enable users to explore large VEs, many different locomotion interfaces such as teleportation, joystick naviga- tion, and walking-in-place have been developed [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Ideally, users explore the VE using natural, everyday walking since it improves their sense of presence [33] and performance in tasks [9], [22], [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' One technique that enables natural walking in VR is redirected walking (RDW) [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' RDW works by slowly rotating the VE around the user’s virtual camera while they walk, which causes them to unconsciously adjust their physical trajectory to counteract the VE rotations and remain on their intended path in the VE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' It works because the human perceptual system tends to believe the user’s visual stimuli over other stimuli (proprio- ceptive, vestibular, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=') when they conflict, as is the case in RDW [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Using RDW, we can steer the user along paths in the PE that direct them away from objects and edges of the tracked space, resulting in a safer and more immersive virtual experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' To help mask the VE rotations, researchers make use of distractors which grab the user’s attention to decrease the likelihood that they attend to the rotations of the VE [4], [21], [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' In our prototype implementation, we use a virtual dog as a distractor in conjunction with a RDW algorithm known as steer-to-center, which rotates the VE such that the user is steered towards the center of the PE at all times [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Virtual Experience and Equipment For our implementation, a user u1 completed a navi- gation task in a virtual city and had a virtual dog as a companion (only a single user participated at a time, so |UP | = |UV | = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Additionally, u1 held a position- tracked leash that was tethered to a differential wheeled robot r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' The PE was an empty rectangular room with four walls (represented by the boundaries of the VR tracking space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Thus, EP = {OP , AP }, where AP = {u1, r1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' The virtual dog served as a distractor and was the only object of interest in EV (|O| = 1), meaning that the robot only rendered haptic forces associated with the virtual dog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Our application was implemented using one HTC VIVE Cosmos VR HMD with two VIVE trackers, and one robot car (ELEGOO UNO Robot Car kit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' We attached one VIVE tracker to the robot to track its location and orientation data, and the other was attached to the leash handle to calculate the distance between u1 and r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' The robot was equipped with an HC-06 Bluetooth LE adapter, which connected to the PC to transmit robot movement commands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' The Unreal 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content='22 game engine was used to render the VE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Virtual Companion and Robot Behavior Here we describe the behavior of the virtual dog com- panion and how the robot matches the virtual companion’s movements and provides haptic feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' 1) Virtual Dog Companion Behavior: The virtual dog has two main behavior states: following and distracting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' When the user walks around and is not at risk of leaving the tracking space, the dog is in follow mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' In this mode, the dog walks slightly ahead of the user as they walk, and remains in one spot when the user stands still.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' When the user reaches a boundary of the tracked space, the VR system initiates what is called a reset, wherein the user reorients themself such that they face towards the inside of the tracking space in the PE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' To ensure that their orientation in the VE is not altered, the VR system applies redirection that effectively cancels out their physical rotation in the virtual space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' When a reset is initiated, the virtual dog enters distract mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' In distract mode, we compute a goal position in the VE for the dog to move towards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' The idea behind distract mode is that the user is likely to pay attention to the virtual dog as it runs to a goal position, which allows the system to apply stronger redirection (away from the obstacles in the PE) without interfering with the user’s experience [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' During a reset, the goal position is selected by first computing the vector from the user towards the center of the physical space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' The goal position is then determined to be either the endpoint of this vector in the VE, or a virtual object near the vector’s endpoint that was labeled as a potential goal position during development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Potential goal positions are virtual objects that a dog would be likely to interact with, such as a fire hydrant or a lamp post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' If the vector intersects with a virtual object (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' a virtual building) and there are no potential goal objects nearby, the goal position is simply the point furthest along the vector that does not intersect with any objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' See Figure 2 for a visualization of this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' 2) Robot Haptic Proxy Behavior: The physical robot’s main purpose is to provide haptic feedback to make the user’s virtual experience feel more immersive and to encourage the user to walk away from nearby objects or tracking space boundaries in the PE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' In both follow and distract mode, the physical robot needs to update its position such that it is Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Our method of automatically choosing a suitable virtual goal position for the virtual companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' When the user gets close to a boundary of the physical space, they need to be reoriented away from the boundary before they continue walking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' In order to pick a goal destination for the virtual companion and robotic haptic proxy, we cast a ray from the physical user to the center of the tracked space and then superimpose this vector onto the user’s virtual position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' If the endpoint of this vector is near a pre-defined potential goal position, that is chosen as the current goal position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Otherwise, we choose the furthest point along the vector that does not intersect with any objects in the virtual environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' aligned with the position of the virtual dog, relative to the user in either environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Checking if a position update is necessary is easily achieved by computing the vector from the virtual user to the virtual dog and comparing it to the vector from the user’s HMD and the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' To compute the trajectory that the robot will follow, we compute a circular arc path based on the robot’s position, forward direction, and destination position (determined by the relative position of the virtual dog and user).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' The ideal path for a differential drive robot is a circular arc since it only requires one set of wheel velocities [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' The wheel velocities are computed with the ratio 2rd 2r−d, where r is the arc radius and d is the distance between the robot wheels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Note that we do not use typical PID-based drift correction due to possible unexpected complications that may arise from the tethering to the user [1], [25], [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Maintaining Active Haptic Guidance Constraints This section describes how our active haptic-drive loco- motion application satisfies the IH and RC constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' 1) Directing Users With Haptic Feedback: Since the virtual object of interest is a dog, the user is attached to the robot by an elastic tether that resembles a leash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' When the robot moves away from the user in the PE, it simulates the sensation of a dog tugging on its leash, thereby improving the realism of the virtual experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Additionally, this tugging encourages the user to follow the robot rather than “fight” it, allowing us to further influence the user’s movement patterns in the PE and VE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' By triggering the robot to move away from the user and towards the center of the PE when they get too close to the tracking space boundaries, the tugging force on the leash encourages the user to turn and walk towards the robot and away from the tracked space boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' 2) Maintaining Co-location: Normally, maintaining rela- tive co-location between a haptic proxy and a virtual object is a matter of updating the position of the haptic proxy whenever the virtual object’s position changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' We also do this in our implementation by updating the position of the robot to match the movements of the virtual dog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' However, our implementation requires additional work to maintain co- location due to a new problem which we refer to as the haptic proxy distortion (HPD) problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Virtual environment before rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Virtual environment after rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Physical environment and superimposed virtual relative positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' A visualization of the haptic proxy distortion problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Left: Initially, the virtual user and virtual companion have a particular relative position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Middle: After rotating the virtual environment around the virtual user, the relative position of the companion changes since the companion is rotated along with the rest of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Right: In the physical space, the haptic proxy has not been updated, so its position coincides with the virtual companion’s relative position before rotation (opaque robot and vector).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' The new relative position of the virtual companion, which the haptic proxy needs to match, is shown as the translucent robot and dashed-line vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' In our implementation, we make use of a locomotion interface called redirected walking (RDW) that enables nat- ural walking in VR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' RDW works by rotating the entire VE around the virtual camera that represents the user’s viewpoint in the VE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Consequently, the virtual dog companion may change its position relative to the virtual user without the dog actually moving to a new destination in the VE (see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Thus, as we apply redirection, the relative position of the virtual dog changes constantly, while the relative position of the physical robot does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' To resolve this discrepancy in relative position, we check the relative positions of the virtual dog and physical robot on each frame, and update the robot’s destination in the PE to minimize the difference in relative position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' The user will perceive this as the haptic proxy “sliding” across the floor around them, which might result in unsmooth motion that may detract from the user experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' In practice, this did not seem to be a major problem for users, but we acknowledge that there may be better solutions to the HPD problem, and leave that for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' This HPD problem adds onto the errors in relative co- location between the haptic proxy and the virtual companion, which makes it harder to satisfy the RC constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Note that the HPD problem is not specific to our implementation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' this problem is present in any application that uses haptic proxies and creates a mismatch between the user’s positions in the physical and virtual environments,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' as is common for Physical Environment Virtual Environment Virtual Environment Virtual Environment Virtual Environment UTU User reached the tracked space boundary,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' so a Superimpose the physical user-to-center If there is a potential pre-defined goal If there is no pre-defined goal position near If the superimposed user-to-center vector reorientation is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Compute the vector vector onto the virtual user to determine the position near the endpoint of the user-to- the endpoint of the user-to-center vector, intersects with a virtual object, use the from the user to the center of the physical goal position of the virtual companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' center vector (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=', a fire hydrant), set that as use the vector endpoint as the goal position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' furthest non-intersecting point along the " vector).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' the goal position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' vector as the goal position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content='locomotion interfaces for mixed reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' EXPERIMENTS & RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Experiment Design and Procedure To evaluate the effectiveness of our implementation of active haptic-driven locomotion prototype, we conducted a user study where participants completed a navigation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' The study design was approved by our university’s Insti- tutional Review Board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' The goal of our user study was to evaluate how effective the haptic guidance was at improving users’ sense of presence in the VE and keeping users away from the boundaries of the VR system’s tracked space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' We used a between participants design, where one group of participants completed a navigation task with active haptic guidance enabled, and the other group completed the same task without any haptic guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' The navigation task had a time limit of 5 minutes and 30 seconds, after which the experiment ended regardless of if the participant reached the goal destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Participants were unaware of this time limit so that they did not rush to complete the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' We recruited 20 participants (13 male, 5 female, 2 participants did not report) through online advertising and oral recruitment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Participants’ ages ranged from 18 to 28 (µ = 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content='59, σ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content='37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' All participants were able to walk without any assistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' The study consisted of three sections, and lasted about 15 minutes for each participant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' First, we debriefed participants on the experiment procedures and had them complete a pre- study Simulator Sickness Questionnaire (SSQ) [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Next, the user put on the HMD and completed the task in the VE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' The VE was a city environment with several streets and blocks, and was populated with common objects such as bus stops, stores, park squares, and virtual humans that roamed around the environment (see Figure 1 and the supplementary video).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' To mask any potentially distracting noises from the robot as it moves, participants wore headphones and back- ground music was played for the duration of the experiment task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Participants started the task at one intersection in the city, and their task was to reach a green question mark in the environment that indicated their destination, which was one block away from the their starting position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' During the experiment, we recorded how many times users reached the bounds of the PE and the time taken to complete the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Once participants finished the task, they completed another SSQ survey and a questionnaire with questions on a 7-point Likert scale that measured their sense of presence in the VE (7 = high presence, 1 = low presence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Finally, the experiment was ended with open-ended questions where participants could provide additional comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Results The metrics we used to measure the effectiveness of our active haptic-driven locomotion interface were the number of breaks in presence (BiPs), the completion rate and time taken to complete the task, and participants’ subjective feelings of presence in the VE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' A BiP is incurred when the user reaches the boundaries of the tracking space and they are forced to reorient away from the boundary before continuing to walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' BiPs Time (s) Presence Completed Haptics µ σ µ σ µ σ Total # With 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content='74 195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content='20 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content='25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content='63 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content='77 10 Without 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content='90 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content='17 309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content='40 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content='14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content='57 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content='64 1 TABLE I Performance results from our user study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' THE “WITH HAPTICS” GROUP OF PARTICIPANTS INCURRED SIGNIFICANTLY fewer BREAKS IN PRESENCE (“BIPS” COLUMN), COMPLETED THE EXPERIMENT MUCH more quickly (“TIME” COLUMN) AND WITH MUCH higher SUCCESS RATES (“COMPLETED” COLUMN), AND REPORTED A higher SENSE OF PRESENCE IN THE VIRTUAL EXPERIENCE (“PRESENCE” COLUMN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' THESE RESULTS SHOW THAT HAPTIC GUIDANCE CAN BE EFFECTIVE FOR IMPROVING USERS’ VIRTUAL EXPERIENCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Based on the results in Table I, the presence of our active haptic guidance companion resulted in significantly fewer BiPs, notably lower completion times and higher completion rates, and slightly higher (and above-average) presence lev- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Meanwhile, participants who completed the navigation task without any haptic guidance incurred a large number of BiPs, did not finish the task in time, and reported below- average levels of presence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' These results support the notion that active haptic guidance can be used to help keep users safe and feel more immersed in mixed reality experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' CONCLUSIONS & FUTURE WORK In this work, we presented the active haptic guidance problem for mixed reality (MR), which describes the use of one or more robots to provide haptic feedback to users in order to create a richer virtual experience for the user, while also influencing the user’s behavior to improve their safety and immersion in the virtual world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' As a prototype realization, we implemented active haptic guidance in a VR locomotion application that enables the user to explore a large VE while located in a much smaller PE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' By combining active haptic guidance and redirected walking, we increased the effective area of the PE while also decreasing the likelihood that the user exits the VR system’s tracked area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' The concept of active haptic guidance is general and can be applied MR applications other than locomotion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' we discuss other potential use cases for active haptic guidance in the supplementary materials on our project page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Limitations and Future Work: One limitation of our work is the haptic proxy distortion problem, in which the haptic proxy and the associated virtual object can become desynchronized due to mismatches between the user’s phys- ical and virtual configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Solving this problem requires continuously updating the position of the haptic proxy, and our proposed solution in this work is likely not the most optimized solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Additionally, our system uses only a rough estimation of drift to readjust the haptic proxy position, instead of a more accurate method like PID-based drift correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Future work in this area should investigate the use of more realistic companions and behavior models, and should explore how active haptic guidance can be applied to other types of VR experiences with different applications, such as social mixed reality settings with other users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' REFERENCES [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content='-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' ˚Aarz´en, “A simple event-based pid controller,” IFAC Proceedings Volumes, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' 32, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' 2, pp.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Wang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Le Goc, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Follmer, “Robotic assembly of haptic proxy objects for tangible interaction and virtual reality,” in Proceedings of the 2017 ACM International Conference on Interactive Surfaces and Spaces, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' 82–91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' [38] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Zhao, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Bennett, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Benko, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Cutrell, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Holz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Morris, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Sinclair, “Enabling people with visual impairments to navigate virtual reality with a haptic and auditory cane simulation,” in Proceedings of the 2018 CHI conference on human factors in computing systems, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' 1–14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' APPENDIX Additional Applications of Active Haptic Guidance: Here we discuss other potential applications of active haptic guid- ance for immersive applications: Wood Carving Application: In wood carving, the grain of the wood will impact the direction in which the artist carves the wood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' That is, sometimes the artist will carve “with the grain” and sometimes will carve “against the grain.” Using active haptics, one could accurately render the different resistance forces that arise from carving with or against the grain of a virtual wooden block, which will in turn influence the way in which the user carves their virtual wooden sculpture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' In addition to providing a more realistic experience, this could be used to guide the user to create a more appealing final sculpture (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' by altering the direction of the grain to subtly change their hand movements, which will change the shape of the final carved surface).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Immersive Cooperative Application: A major appeals of mixed reality experiences is the ability to connect with other users in shared virtual experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Important to these shared experiences is the ability to touch the other person, which can provide a greater sense of companionship and connection between users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Haptic forces can be used to encourage users to interact with or follow other users who are also present in their virtual experience, which may improve the users’ sense of presence in the experience due to the enhanced realism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' Virtual Cooking Training Application: Given a seated VR experience where the user is practicing their cook- ing skills in a virtual environment, a mobile, tabletop robot can provide haptic feedback that represents feed- back provided by cooking utensils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' For example, when spreading brownie batter in a baking pan, the user will feel haptic forces when the virtual spreading utensil gets too close to the edges of the virtual baking pan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} +page_content=' These forces could be rendered using a mobile robot with a flat surface that serves as a wall that the user’s physical hand will bump into.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E4T4oBgHgl3EQf4A3h/content/2301.05311v1.pdf'} diff --git a/.gitattributes b/.gitattributes index 8b256dd9053642e33c1b3dfc3b38020ba11bfaa6..5ddcfada421b87143e2a32f67eb486d60b8ca365 100644 --- a/.gitattributes +++ b/.gitattributes @@ -4793,3 +4793,62 @@ ctE1T4oBgHgl3EQfxgX2/content/2301.03424v1.pdf filter=lfs diff=lfs merge=lfs -tex q9E2T4oBgHgl3EQfKwbV/content/2301.03707v1.pdf filter=lfs diff=lfs merge=lfs -text k9E0T4oBgHgl3EQf7wIz/content/2301.02779v1.pdf filter=lfs diff=lfs merge=lfs -text wNFST4oBgHgl3EQfQzi0/content/2301.13760v1.pdf filter=lfs diff=lfs merge=lfs -text +zNAyT4oBgHgl3EQfbPdo/content/2301.00258v1.pdf filter=lfs diff=lfs merge=lfs -text 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Modern Introduction +Harold Erbin1* +*Center for Theoretical Physics +Massachusetts Institute of Technology, Cambridge, MA 02139, Usa +*Cea, List, Gif-sur-Yvette, F-91191, France +7th January 2023 +1erbin@mit.edu +arXiv:2301.01686v1 [hep-th] 4 Jan 2023 + +Abstract +This book provides an introduction to string field theory (SFT). String theory is usually +formulated in the worldsheet formalism, which describes a single string (first-quantization). +While this approach is intuitive and could be pushed far due to the exceptional properties of +two-dimensional theories, it becomes cumbersome for some questions or even fails at a more +fundamental level. These motivations have led to the development of SFT, a description of +string theory using the field theory formalism (second-quantization). As a field theory, SFT +provides a rigorous and constructive formulation of string theory. +The main objective is to construct the closed bosonic SFT and to explain how to assess +the consistency of string theory with it. The accent is put on providing the reader with +the foundations, conceptual understanding and intuition of what SFT is. After reading this +book, they should be able to study the applications from the literature. +The book is organized in two parts. The first part reviews the topics of the worldsheet +theory that are necessary to build SFT (worldsheet path integral, CFT and BRST quant- +ization). The second part starts by introducing general concepts of SFT from the BRST +quantization. Then, it introduces off-shell string amplitudes before providing a Feynman +diagrams interpretation from which the building blocks of SFT are extracted. After con- +structing the closed SFT, it is used to outline the proofs of several important consistency +properties, such as background independence, unitarity and crossing symmetry. Finally, the +generalization to the superstring is also discussed. +This book grew up from lecture notes for a course given at the Ludwig-Maximilians- +Universität LMU (winter semesters 2017–2018 and 2018–2019). The current document is +the draft of the manuscript published by Springer. + +Preface +This book grew up from lectures delivered within the Elite Master Program “Theoretical +and Mathematical Physics” from the Ludwig-Maximilians-Universität during the winter +semesters 2017–2018 and 2018–2019. +The main focus of this book is the closed bosonic string field theory (SFT). While there +are many resources available for the open bosonic SFT, a single review [71] has been written +since the final construction of the bosonic closed SFT by Zwiebach [262]. For this reason, +it makes sense to provide a modern and extensive study. Moreover, the usual approach to +open SFT focuses on the cubic theory, which is so special that it is difficult to generalize the +techniques to other SFTs. Finally, closed strings are arguably more fundamental than open +strings because they are always present since they describe gravity, which further motivates +my choice. However, the reader should not take this focus as denying the major achievements +and the beauty of the open SFT; reading this book should provide most of the tools needed +to feel comfortable also with this theory. +While part of the original goal of SFT is to provide a non-perturbative definition of string +theory and to address important questions such as classifying consistent string backgrounds +or understanding dualities, no progress on this front has been achieved so far. Hence, there +is still much to understand and the recent surge of developments provide a new chance +to deepen our understanding of closed SFT. For example, several consistency properties of +string theory have been proven rigorously using SFT. Moreover, the recent construction of +the open-closed superstring field theory [165] together with earlier works [42, 218, 262, 264] +show that all types of string theories can be recast as a SFT. This is why, I believe, it is a +good time to provide a complete book on SFT. +The goal of this book is to offer a self-contained description of SFT and all the tools +necessary to build it. The emphasis is on describing the concepts behind SFT and to make +the reader build intuitions on what it means. +For this reason, there are relatively few +applications. +The reader is assumed to have some knowledge of QFT, and a basic knowledge of CFT +and string theory (classical string, Nambu–Goto action, light-cone and old-covariant quant- +izations). +Organization +The text is organized on three levels: the main content (augmented with examples), compu- +tations, and remarks. The latter two levels can be omitted in a first lecture. The examples, +computations and remarks are clearly separated from the text (respectively, by a half-box +on the left and bottom, by a vertical line on the left, and by italics) to help the navigation. +Many computations have been set aside from the main text to avoid breaking the flow +and to provide the reader with the opportunity to check by themself first. In some occasions, +computations are postponed well below the corresponding formula to gather similar compu- +tations or to avoid breaking an argument. While the derivations contain more details than +2 + +usual textbooks and may look pedantic to the expert, I think it is useful for students and +newcomers to have complete references where to check each step. This is even more the case +when there are many different conventions in the literature. The remarks are not directly +relevant to the core of the text but they make connections with other parts or topics. The +goal is to broaden the perspectives of the main text. +General references can be found at the end of each chapter to avoid overloading the +text. In-text references are reserved for specific points or explicit quotations (of a formula, a +discussion, a proof, etc.). I did not try to be exhaustive in the citations and I have certainly +missed important references: this should be imputed to my lack of familiarity with them +and not to their value. +This text is a preprint of the textbook [64] and is reproduced with permission of Springer. +My plan is to frequently update the draft of this book with new content. The last version +can be accessed on my professional webpage, currently located at: +http://www.lpthe.jussieu.fr/~erbin/ +Acknowledgements +I have started to learn string field theory at Hri by attending lectures from Ashoke Sen. +Since then, I have benefited from collaboration and many insightful discussions with him. +Following his lectures have been much helpful in building an intuition that cannot be found +in papers or reviews on the topic. Through this book, I hope being able to make some of +these insights more accessible. +I am particularly grateful to Ivo Sachs who proposed me to teach this course and to +Michael Haack for continuous support and help with the organization, and to both of them +for many interesting discussions during the two years I have spent at Lmu. Moreover, I have +been very lucky to be assigned an excellent tutor for this course, Christoph Chiaffrino. After +providing him with the topic and few references, Christoph has prepared all the tutorials +and the corrections autonomously. His help brought a lot to the course. +I am particularly obliged to all the students who have taken this course at Lmu for +many interesting discussions and comments: Enrico Andriolo, Hrólfur Ásmundsson, Daniel +Bockisch, Fabrizio Cordonnier, Julian Freigang, Wilfried Kaase, Andriana Makridou, Pouria +Mazloumi, Daniel Panea, Martin Rojo. +I am also grateful to all the string theory community for many exchanges. For discussions +related to the topics of this book, I would like to thank more particularly: Costas Bachas, +Adel Bilal, Subhroneel Chakrabarti, Atish Dabholkar, Benoit Douçot, Ted Erler, Dileep +Jatkar, Carlo Maccaferri, Juan Maldacena, Yuji Okawa, Sylvain Ribault, Raoul Santachiara, +Martin Schnabl, Dimitri Skliros, Jakub Vošmera. I have received a lot of feedback during +the different stages of writing this book, and I am obliged to all the colleagues who sent me +feedback. +I am thankful to my colleagues at Lmu for providing a warm and stimulating envir- +onment, with special thanks to Livia Ferro for many discussions around coffee. Moreover, +the encouragements and advice from Oleg Andreev and Erik Plauschinn have been strong +incentives for publishing this book. +The editorial process at Springer has been very smooth. +I would also like to thank +Christian Caron and Lisa Scalone for their help and efficiency during the publishing process. +I am also indebted to Stefan Theisen for having supported the publication at Springer and +for numerous comments and corrections on the draft. +Most of this book was written at the Ludwig–Maximilians–Universität (Lmu, Munich, +Germany) where I was supported by a Carl Friedrich von Siemens Research Fellowship of the +Alexander von Humboldt Foundation. The final stage has been completed at the University +3 + +of Turin (Italy). My research is currently funded by the European Union’s Horizon 2020 +research and innovation program under the Marie Skłodowska-Curie grant agreement No +891169. +Finally, writing this book would have been more difficult without the continuous and +loving support from Corinne. +November 2020 +Harold Erbin +4 + +Contents +Preface +2 +1 +Introduction +11 +1.1 +Strings, a distinguished theory +. . . . . . . . . . . . . . . . . . . . . . . . . . +11 +1.2 +String theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +14 +1.2.1 +Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +14 +1.2.2 +Classification of superstring theories . . . . . . . . . . . . . . . . . . . +18 +1.2.3 +Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +20 +1.3 +String field theory +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +23 +1.3.1 +From the worldsheet to field theory . . . . . . . . . . . . . . . . . . . . +23 +1.3.2 +String field action +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . +25 +1.3.3 +Expression with spacetime fields +. . . . . . . . . . . . . . . . . . . . . +25 +1.3.4 +Applications +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +26 +1.4 +Suggested readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +27 +I +Worldsheet theory +28 +2 +Worldsheet path integral: vacuum amplitudes +29 +2.1 +Worldsheet action and symmetries . . . . . . . . . . . . . . . . . . . . . . . . +29 +2.2 +Path integral +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +34 +2.3 +Faddeev–Popov gauge fixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . +36 +2.3.1 +Metrics on Riemann surfaces +. . . . . . . . . . . . . . . . . . . . . . . +37 +2.3.2 +Reparametrizations and analysis of P1 . . . . . . . . . . . . . . . . . . +42 +2.3.3 +Weyl transformations and quantum anomalies . . . . . . . . . . . . . . +47 +2.3.4 +Ambiguities, ultralocality and cosmological constant . . . . . . . . . . +48 +2.3.5 +Gauge-fixed path integral . . . . . . . . . . . . . . . . . . . . . . . . . +49 +2.4 +Ghost action +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +51 +2.4.1 +Actions and equations of motion . . . . . . . . . . . . . . . . . . . . . +51 +2.4.2 +Weyl ghost +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +52 +2.4.3 +Zero-modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +54 +2.5 +Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +55 +2.6 +Summary +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +56 +2.7 +Suggested readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +56 +3 +Worldsheet path integral: scattering amplitudes +58 +3.1 +Scattering amplitudes on moduli space . . . . . . . . . . . . . . . . . . . . . . +58 +3.1.1 +Vertex operators and path integral . . . . . . . . . . . . . . . . . . . . +58 +3.1.2 +Gauge fixing: general case . . . . . . . . . . . . . . . . . . . . . . . . . +61 +3.1.3 +Gauge fixing: 2-point amplitude +. . . . . . . . . . . . . . . . . . . . . +64 +5 + +3.2 +BRST quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +67 +3.2.1 +BRST symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +68 +3.2.2 +BRST cohomology and physical states . . . . . . . . . . . . . . . . . . +69 +3.3 +Summary +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +71 +3.4 +Suggested readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +71 +4 +Worldsheet path integral: complex coordinates +72 +4.1 +Geometry of complex manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . +72 +4.2 +Complex representation of path integral . . . . . . . . . . . . . . . . . . . . . +75 +4.3 +Summary +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +77 +4.4 +Suggested readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +77 +5 +Conformal symmetry in D dimensions +78 +5.1 +CFT on a general manifold +. . . . . . . . . . . . . . . . . . . . . . . . . . . . +78 +5.2 +CFT on Minkowski space +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . +79 +5.3 +Suggested readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +80 +6 +Conformal field theory on the plane +81 +6.1 +The Riemann sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +81 +6.1.1 +Map to the complex plane . . . . . . . . . . . . . . . . . . . . . . . . . +81 +6.1.2 +Relation to the cylinder – string theory +. . . . . . . . . . . . . . . . . +83 +6.2 +Classical CFTs +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +84 +6.2.1 +Witt conformal algebra +. . . . . . . . . . . . . . . . . . . . . . . . . . +85 +6.2.2 +PSL(2, C) conformal group +. . . . . . . . . . . . . . . . . . . . . . . . +86 +6.2.3 +Definition of a CFT +. . . . . . . . . . . . . . . . . . . . . . . . . . . . +88 +6.3 +Quantum CFTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +89 +6.3.1 +Virasoro algebra +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +90 +6.3.2 +Correlation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . +90 +6.4 +Operator formalism and radial quantization . . . . . . . . . . . . . . . . . . . +92 +6.4.1 +Radial ordering and commutators +. . . . . . . . . . . . . . . . . . . . +92 +6.4.2 +Operator product expansions . . . . . . . . . . . . . . . . . . . . . . . +94 +6.4.3 +Hermitian and BPZ conjugation +. . . . . . . . . . . . . . . . . . . . . +96 +6.4.4 +Mode expansion +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +97 +6.4.5 +Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 +6.4.6 +CFT on the cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 +6.5 +Suggested readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 +7 +CFT systems +107 +7.1 +Free scalar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 +7.1.1 +Covariant action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 +7.1.2 +Action on the complex plane +. . . . . . . . . . . . . . . . . . . . . . . 109 +7.1.3 +OPE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 +7.1.4 +Mode expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 +7.1.5 +Commutators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 +7.1.6 +Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 +7.1.7 +Euclidean and BPZ conjugates . . . . . . . . . . . . . . . . . . . . . . 118 +7.2 +First-order bc ghost system +. . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 +7.2.1 +Covariant action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 +7.2.2 +Action on the complex plane +. . . . . . . . . . . . . . . . . . . . . . . 119 +7.2.3 +OPE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 +7.2.4 +Mode expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 +7.2.5 +Commutators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 +6 + +7.2.6 +Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 +7.2.7 +Euclidean and BPZ conjugates . . . . . . . . . . . . . . . . . . . . . . 132 +7.2.8 +Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 +7.3 +Suggested readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 +8 +BRST quantization +134 +8.1 +BRST for reparametrization invariance . . . . . . . . . . . . . . . . . . . . . . 134 +8.2 +BRST in the CFT formalism +. . . . . . . . . . . . . . . . . . . . . . . . . . . 135 +8.2.1 +OPE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 +8.2.2 +Mode expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 +8.2.3 +Commutators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 +8.3 +BRST cohomology: two flat directions . . . . . . . . . . . . . . . . . . . . . . 138 +8.3.1 +Conditions on the states . . . . . . . . . . . . . . . . . . . . . . . . . . 139 +8.3.2 +Relative cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 +8.3.3 +Absolute cohomology, states and no-ghost theorem . . . . . . . . . . . 146 +8.3.4 +Cohomology for holomorphic and anti-holomorphic sectors . . . . . . . 147 +8.4 +Summary +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 +8.5 +Suggested readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 +II +String field theory +149 +9 +String field +150 +9.1 +Field functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 +9.2 +Field expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 +9.3 +Summary +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 +9.4 +Suggested readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 +10 Free BRST string field theory +154 +10.1 Classical action for the open string . . . . . . . . . . . . . . . . . . . . . . . . 154 +10.1.1 Warm-up: point-particle . . . . . . . . . . . . . . . . . . . . . . . . . . 154 +10.1.2 Open string action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 +10.1.3 Gauge invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 +10.1.4 Siegel gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 +10.2 Open string field expansion, parity and ghost number +. . . . . . . . . . . . . 160 +10.3 Path integral quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 +10.3.1 Tentative Faddeev–Popov gauge fixing . . . . . . . . . . . . . . . . . . 162 +10.3.2 Tower of ghosts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 +10.4 Spacetime action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 +10.5 Closed string +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 +10.6 Summary +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 +10.7 Suggested readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 +11 Introduction to off-shell string theory +171 +11.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 +11.1.1 3-point function +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 +11.1.2 4-point function +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 +11.2 Off-shell states +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 +11.3 Off-shell amplitudes +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 +11.3.1 Amplitudes from the marked moduli space +. . . . . . . . . . . . . . . 178 +11.3.2 Local coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 +11.4 Suggested readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 +7 + +12 Geometry of moduli spaces and Riemann surfaces +182 +12.1 Parametrization of Pg,n +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 +12.2 Tangent space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 +12.3 Plumbing fixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 +12.3.1 Separating case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 +12.3.2 Non-separating case +. . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 +12.3.3 Decomposition of moduli spaces and degeneration limit +. . . . . . . . 191 +12.3.4 Stubs +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 +12.4 Summary +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 +12.5 Suggested readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 +13 Off-shell amplitudes +197 +13.1 Cotangent spaces and amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . 197 +13.1.1 Construction of forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 +13.1.2 Amplitudes and surface states . . . . . . . . . . . . . . . . . . . . . . . 199 +13.2 Properties of forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 +13.2.1 Vanishing of forms with trivial vectors . . . . . . . . . . . . . . . . . . 202 +13.2.2 BRST identity +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 +13.3 Properties of amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 +13.3.1 Restriction to ˆPg,n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 +13.3.2 Consequences of the BRST identity +. . . . . . . . . . . . . . . . . . . 206 +13.4 Suggested readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 +14 Amplitude factorization and Feynman diagrams +208 +14.1 Amplitude factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 +14.1.1 Separating case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 +14.1.2 Non-separating case +. . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 +14.2 Feynman diagrams and Feynman rules . . . . . . . . . . . . . . . . . . . . . . 213 +14.2.1 Feynman graphs +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 +14.2.2 Propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 +14.2.3 Fundamental vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 +14.2.4 Stubs +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 +14.2.5 1PI vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 +14.3 Properties of fundamental vertices +. . . . . . . . . . . . . . . . . . . . . . . . 223 +14.3.1 String product +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 +14.3.2 Feynman graph interpretation . . . . . . . . . . . . . . . . . . . . . . . 224 +14.4 Suggested readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 +15 Closed string field theory +226 +15.1 Closed string field expansion +. . . . . . . . . . . . . . . . . . . . . . . . . . . 226 +15.2 Gauge fixed theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 +15.2.1 Kinetic term and propagator +. . . . . . . . . . . . . . . . . . . . . . . 227 +15.2.2 Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 +15.2.3 Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 +15.3 Classical gauge invariant theory . . . . . . . . . . . . . . . . . . . . . . . . . . 232 +15.4 BV theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 +15.5 1PI theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 +15.6 Suggested readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 +8 + +16 Background independence +238 +16.1 The concept of background independence +. . . . . . . . . . . . . . . . . . . . 238 +16.2 Problem setup +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 +16.3 Deformation of the CFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 +16.4 Expansion of the action +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 +16.5 Relating the equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . 242 +16.6 Idea of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 +16.7 Suggested readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 +17 Superstring +245 +17.1 Worldsheet superstring theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 +17.1.1 Heterotic worldsheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 +17.1.2 Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 +17.2 Off-shell superstring amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . 249 +17.2.1 Amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 +17.2.2 Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 +17.2.3 Spurious poles +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 +17.3 Superstring field theory +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 +17.3.1 String field and propagator . . . . . . . . . . . . . . . . . . . . . . . . 256 +17.3.2 Constraint approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 +17.3.3 Auxiliary field approach . . . . . . . . . . . . . . . . . . . . . . . . . . 257 +17.3.4 Large Hilbert space +. . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 +17.4 Suggested readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 +18 Momentum-space SFT +260 +18.1 General form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 +18.2 Generalized Wick rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 +18.3 Suggested readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 +A Conventions +266 +A.1 Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 +A.2 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 +A.3 QFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 +A.4 Curved space and gravity +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 +A.5 List of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 +B Summary of important formulas +273 +B.1 +Complex analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 +B.2 +QFT, curved spaces and gravity . . . . . . . . . . . . . . . . . . . . . . . . . . 273 +B.2.1 +Two dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 +B.3 +Conformal field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 +B.3.1 +Complex plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 +B.3.2 +General properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 +B.3.3 +Hermitian and BPZ conjugations . . . . . . . . . . . . . . . . . . . . . 276 +B.3.4 +Scalar field +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 +B.3.5 +Reparametrization ghosts . . . . . . . . . . . . . . . . . . . . . . . . . 277 +B.4 +Bosonic string . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 +9 + +C Quantum field theory +281 +C.1 Path integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 +C.1.1 +Integration measure +. . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 +C.1.2 +Field redefinitions +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 +C.1.3 +Zero-modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 +C.2 BRST quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 +C.3 BV formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 +C.3.1 +Properties of gauge algebra . . . . . . . . . . . . . . . . . . . . . . . . 289 +C.3.2 +Classical BV +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 +C.3.3 +Quantum BV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 +C.4 Suggested readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 +Bibliography +295 +Index +310 +10 + +Chapter 1 +Introduction +Abstract +In this chapter, we introduce the main motivations for studying string theory, +and why it is important to design a string field theory. After describing the central features +of string theory, we describe the most important concepts of the worldsheet formulation. +Then, we explain the reasons leading to string field theory (SFT) and outline the ideas +which will be discussed in the rest of the book. +1.1 +Strings, a distinguished theory +The first and simplest reason for considering theories of fundamental p-branes (fundamental +objects extended in p spatial dimensions) can be summarized by the following question: +“Why would Nature just make use of point-particles?” There is no a priori reason forbidding +the existence of fundamental extended objects and, according to Gell-Mann’s totalitarian +principle, “Everything not forbidden is compulsory.” If a consistent theory cannot be built +(after a reasonable amount of effort) or if it contradicts current theories (in their domains +of validity) and experiments, then one can support the claim that only point-particles exist. +On the other side, if such a theory can be built, it is of primary interest to understand it +deeper and to see if it can solve the current problems in high-energy theoretical physics. +The simplest case after the point particle is the string, so it makes sense to start with +it. It happens that a consistent theory of strings can be constructed, and that the latter +(in its supersymmetric version) contains all the necessary ingredients for a fully consistent +high-energy model:1 +• quantum gravity (quantization of general relativity plus higher-derivative corrections); +• grand unification (of matter, interactions and gravity); +• no divergences, UV finiteness (finite and renormalizable theory); +• fixed number of dimensions (26 = 25 + 1 for the bosonic string, 10 = 9 + 1 for the +supersymmetric version); +• existence of all possible branes; +• no dimensionless parameters and one dimensionful parameter (the string length ℓs). +It can be expected that a theory of fundamental strings (1-branes) occupies a distinguished +place among fundamental p-branes for the following reasons. +1There are also indications that a theory of membranes (2-branes) in 10+1 dimensions, called M-theory, +should exist. No direct and satisfactory description of the latter has been found and we will thus focus on +string theory in this book. +11 + +Figure 1.1: Locality of a particle interaction: two different observers always agree on the +interaction point and which parts of the worldline are 1- and 2-particle states. +Interaction non-locality +In a QFT of point particles, UV divergences arise because +interactions (defined as the place where the number and/or nature of the objects change) +are arbitrarily localized at a spacetime point. In Feynman graphs, such divergences can be +seen when the momentum of a loop becomes infinite (two vertices collide): this happens +when trying to concentrate an infinite amount of energy at a single point. However, these +divergences are expected to be reduced or absent in a field theory of extended objects: +whereas the interaction between particles is perfectly local in spacetime and agreed upon by +all observers (Figure 1.1), the spatial extension of branes makes the interactions non-local. +This means that two different observers will neither agree on the place of the interactions +(Figure 1.2), nor on the part of the diagram which describes one or two branes. +The string lies at the boundary between too much local and too much non-local: in any +given frame, the interaction is local in space, but not in spacetime. The reason is that a +string is one-dimensional and splits or joins along a point. For p > 1, the brane needs to +break/join along an extended spatial section, which looks non-local. +Another consequence of the non-locality is a drastic reduction of the possible interactions. +If an interaction is Lorentz invariant, Lorentz covariant objects can be attached at the vertex +(such as momentum or gamma matrices): this gives Lorentz invariants after contracting with +indices carried by the field. But, this is impossible if the interaction itself is non-local (and +thus not invariant): inserting a covariant object would break Lorentz invariance. +Brane degrees of freedom +The higher the number of spatial dimensions of a p-brane, the +more possibilities it has to fluctuate. As a consequence, it is expected that new divergences +appear as p increases due to the proliferations of the brane degrees of freedom. From the +worldvolume perspective, this is understood from the fact that the worldvolume theory +describes a field theory in (p + 1) dimensions, and UV divergences become worse as the +number of dimensions increase. +The limiting case happens for the string (p = 1) since +two-dimensional field theories are well-behaved in this respect (for example, any monomial +interaction for a scalar field is power-counting renormalizable). This can be explained by the +low-dimensionality of the momentum integration and by the enhancement of symmetries in +two dimensions. Hence, strings should display nice properties and are thus of special interest. +Worldvolume theory +The point-particle (0-brane) and the string (1-brane) are also re- +markable in another aspect: it is possible to construct a simple worldvolume field theory +(and the associated functional integral) in terms of a worldvolume metric. All components +of the latter are fixed by gauge symmetries (diffeomorphisms for the particle, diffeomorph- +isms and Weyl invariance for the string). This ensures the reparametrization invariance of +12 + +(a) Observers at rest and boosted. +(b) Observers close to the speed +of light moving in opposite dir- +ections. +The interactions are +widely separated in each case. +Figure 1.2: Non-locality of string interaction: two different observers see the interaction +happening at different places (denoted by the filled and empty circles) and they don’t agree +on which parts of the worldsheet are 1- and 2-string states (the litigation is denoted by the +grey zone). +the worldvolume without having to use a complicated action. Oppositely, the worldvolume +metric cannot be completely gauge fixed for p > 1. +Summary +As a conclusion, strings achieve an optimal balance between spacetime and +worldsheet divergences, as well as having a simple description with reparametrization in- +variance. +Since the construction of a field theory is difficult, it is natural to start with a worldsheet +theory and to study it in the first-quantization formalism, which will provide a guideline for +writing the field theory. In particular, this allows to access the physical states in a simple +way and to find other general properties of the theory. When it comes to the interactions +and scattering amplitudes, this approach may be hopeless in general since the topology of +the worldvolume needs to be specified by hand (describing the interaction process). In this +respect, the case of the string is again exceptional: because Riemann surfaces have been +classified and are well-understood, the arbitrariness is minimal. Combined with the tools of +conformal field theory, many computations can be performed. Moreover, since the modes +of vibrations of the strings provide all the necessary ingredients to describe the Standard +model, it is sufficient to consider only one string field (for one type of strings), instead +of the plethora found in point-particle field theory (one field for each particle). Similarly, +non-perturbative information (such as branes and dualities) could be found only due to the +specific properties of strings. +Coming back to the question which opened this section, higher-dimensional branes of all +the allowed dimensions naturally appear in string theory as bound states. Hence, even if +the worldvolume formulation of branes with p > 1 looks pathological2, string theory hints +towards another definition of these objects. +2Entering in the details would take us too far away from the main topic of this book. +Some of the +problems found when dealing with (p > 2)-branes are: how to define a Wick rotation for 3-manifolds, the +presence of Lorentz anomalies in target spacetime, problems with the spectrum, lack of renormalizability, +impossibility to gauge-fix the worldvolume metric [5–11, 41, 44, 45, 62, 127, 152, 156–158, 181, 193]. +13 + +1.2 +String theory +1.2.1 +Properties +The goal of this section is to give a general idea of string theory by introducing some concepts +and terminology. The reader not familiar with the points described in this section is advised +to follow in parallel some standard worldsheet string theory textbooks. +Worldsheet CFT +A string is characterized by its worldsheet field theory (Chapter 2).3 +The worldsheet is +parametrized by coordinates σa = (τ, σ). The simplest description is obtained by endowing +the worldsheet with a metric gab(σa) (a = 0, 1) and by adding a set of D scalar fields +Xµ(σa) living on the worldsheet (µ = 0, . . . , D − 1). The latter represents the position +of the string in the D-dimensional spacetime. From the classical equations of motion, the +metric gab is proportional to the metric induced on the worldsheet from its embedding in +spacetime. +More generally, one ensures that the worldsheet metric is non-dynamical by +imposing that the action is invariant under (worldsheet) diffeomorphisms and under Weyl +transformations (local rescalings of the metric). The consistency of these conditions at the +quantum level imposes that D = 26, and this number is called the critical dimension. Gauge +fixing the symmetries, and thus the metric, leads to the conformal invariance of the resulting +worldsheet field theory: a conformal field theory (CFT) is a field theory (possibly on a curved +background) in which only angles and not distances can be measured (Chapters 5 to 7). +This simplifies greatly the analysis since the two-dimensional conformal algebra (called the +Virasoro algebra) is infinite-dimensional. +CFTs more general than D free scalar fields can be considered: fields taking non-compact +values are interpreted as non-compact dimensions while compact or Grassmann-odd fields +are interpreted as compact dimensions or internal structure, like the spin. +While the light-cone quantization allows to find quickly the states of the theory, the +simplest covariant method is the BRST quantization (Chapter 8). It introduces ghosts (and +superghosts) associated to the gauge fixing of diffeomorphisms (and local supersymmetry). +These (super)ghosts form a CFT which is universal (independent of the matter CFT). +The trajectory of the string is denoted by xc(τ, σ). It begins and ends respectively at the +geometric shapes parametrized by xc(τi, σ) = xi(σ) and by xc(τf, σ) = xf(σ). Note that the +coordinate system on the worldsheet itself is arbitrary. The spatial section of a string can be +topologically closed (circle) or open (line) (Figure 1.3), leading to cylindrical or rectangular +worldsheets as illustrated in Figures 1.4 and 1.5. To each topology is associated different +boundary conditions and types of strings: +• closed: periodic and anti-periodic boundary conditions; +• open: Dirichlet and Neumann boundary conditions. +While a closed string theory is consistent by itself, an open string theory is not and requires +closed strings. +Spectrum +In order to gain some intuition for the states described by a closed string, one can write the +Fourier expansion of the fields Xµ (in the gauge gab = ηab and after imposing the equations +3We focus mainly on the bosonic string theory, leaving aside the superstring, except when differences +are important. +14 + +(a) Open string +(b) Closed string +Figure 1.3: Open and closed strings. +−−−−−−−−→ +Figure 1.4: Trajectory xµ +c (τ, σ) of a closed string in spacetime (worldsheet). It begins and +ends at the circles parametrized by xi(σ) and xf(σ). +The worldsheet is topologically a +cylinder and is parametrized by (τ, σ) ∈ [τi, τf] × [0, 2π). +−−−−−−−−→ +Figure 1.5: Trajectory xµ +c (τ, σ) of an open string in spacetime (worldsheet). +It begins +and ends at the lines parametrized by xi(σ) and xf(σ). The worldsheet is topologically a +rectangle and is parametrized by (τ, σ) ∈ [τi, τf] × [0, ℓ]. +15 + +of motion) +Xµ(τ, σ) ∼ xµ + pµτ + +i +√ +2 +� +n∈Z∗ +1 +n +� +αµ +ne−in(τ−σ) + ¯αµ +ne−in(τ+σ)� +, +(1.1) +where xµ is the centre-of-mass position of the string and pµ its momentum.4 +Canonical +quantization leads to the usual commutator: +[xµ, pν] = iηµν . +(1.2) +With respect to a point-particle for which only the first two terms are present, there are an +infinite number of oscillators αµ +n and ¯αµ +n which satisfy canonical commutation relations for +creation n < 0 and annihilation operators n > 0 +[αµ +m, αν +n] = m ηµνδm+n,0 . +(1.3) +The non-zero modes are the Fourier modes of the excitations of the embedded string. The +case of the open string is simply obtained by setting ¯αn = αn and p → 2p. The Hamiltonian +for the closed and open strings read respectively +Hclosed = −m2 +2 + N + ¯N − 2 , +(1.4a) +Hopen = −m2 + N − 1 +(1.4b) +where m2 = −pµpµ is the mass of the state (in Planck units), N and ¯N (level operators) +count the numbers Nn and ¯Nn of oscillators αn and ¯αn weighted by their mode index n: +N = +� +n∈N +nNn , +Nn = 1 +n α−n · αn , +¯N = +� +n∈N +n ¯Nn , +¯Nn = 1 +n ¯α−n · ¯αn . +(1.5) +With these elements, the Hilbert space of the string theory can be constructed. Invariance +under reparametrization leads to the on-shell condition, which says that the Hamiltonian +vanishes: +H |ψ⟩ = 0 +(1.6) +for any physical state |ψ⟩. Another constraint for the closed string is the level-matching +condition +(N − ¯N) |ψ⟩ = 0 . +(1.7) +It can be understood as fixing an origin on the string. +The ground state |k⟩ with momentum k is defined to be the eigenstate of the momentum +operator which does not contain any oscillator excitation: +pµ |k⟩ = kµ |k⟩ , +∀n > 0 : +αµ +n |k⟩ = 0 . +(1.8) +A general state can be built by applying successively creation operators +|ψ⟩ = +� +n>0 +D−1 +� +µ=0 +(αµ +−n)Nn,µ |k⟩ , +(1.9) +4In the introduction, we set α′ = 1. +16 + +where Nn,µ ∈ N counts excitation level of the oscillator αµ +−n. In the rest of this section, we +describe the first two levels of states. +The ground state is a tachyon (faster-than-light particle) because the Hamiltonian con- +straint shows that it has a negative mass (in the units where α′ = 1): +closed : +m2 = −4 , +open : +m2 = −1 . +(1.10) +The first excited state of the open string is found by applying α−1 on the vacuum |k⟩: +αµ +−1 |k⟩ . +(1.11) +This state is massless: +m2 = 0 +(1.12) +and since it transforms as a Lorentz vector (spin 1), it is identified with a U(1) gauge boson. +Writing a superposition of such states +|A⟩ = +� +dDk Aµ(k) αµ +−1 |k⟩ , +(1.13) +the coefficient Aµ(k) of the Fourier expansion is interpreted as the spacetime field for the +gauge boson. Reparametrization invariance is equivalent to the equation of motion +k2Aµ = 0 . +(1.14) +One can prove that the field obeys the Lorentz gauge condition +kµAµ = 0 , +(1.15) +which results from gauge fixing the U(1) gauge invariance +Aµ −→ Aµ + kµλ . +(1.16) +It can also be checked that the low-energy action reproduces the Maxwell action. +The first level of the closed string is obtained by applying both α−1 and ¯α−1 (this is the +only way to match N = ¯N at this level) +αµ +−1¯αν +−1 |k⟩ +(1.17) +and the corresponding states are massless +m2 = 0 . +(1.18) +These states can be decomposed into irreducible representations of the Lorentz group +� +αµ +−1¯αν +−1 + αν +−1¯αµ +−1 − 1 +D ηµνα−1 · ¯α−1 +� +|p⟩ , +� +αµ +−1¯αν +−1 − αν +−1¯αµ +−1 +� +|p⟩ , +1 +D ηµναµ +−1¯αν +−1 |p⟩ +(1.19) +which are respectively associated to the spacetime fields Gµν (metric, spin 2), Bµν (Kalb– +Ramond 2-form) and Φ (dilaton, spin 0). The appearance of a massless spin 2 particle (with +low-energy action being the Einstein–Hilbert action) is a key result and originally raised +interest for string theory. +17 + +Remark 1.1 (Reparametrization constraints) Reparametrization invariance leads to +other constraints than H = 0. They imply in particular that the massless fields have the +correct gauge invariance and hence the correct degrees of freedom. +Note that, after taking into account these constraints, the remaining modes correspond +to excitations of the string in the directions transverse to it. +Hence, each vibrational mode of the string corresponds to a spacetime field for a point- +particle (and linear superpositions of modes can describe several fields). This is how string +theory achieves unification since a single type of string (of each topology) is sufficient for +describing all the possible types of fields encountered in the standard model and in gravity. +They correspond to the lowest excitation modes, the higher massive modes being too heavy +to be observed at low energy. +Bosonic string theory includes tachyons and is thus unstable. While the instability of +the open string tachyon is well understood and indicates that open strings are unstable and +condense to closed strings, the status of the closed string tachyon is more worrisome (literally +interpreted, it indicates a decay of spacetime itself). In order to solve this problem, one can +introduce supersymmetry: in this case, the spectrum does not include the tachyon because +it cannot be paired with a supersymmetric partner. +Moreover, as its name indicates, the bosonic string possesses only bosons in its spectrum +(perturbatively), which is an important obstacle to reproduce the standard model. +By +introducing spacetime fermions, supersymmetry also solves this problem. The last direct +advantage of the superstring is that it reduces the number of dimensions from 26 to 10, +which makes the compactification easier. +1.2.2 +Classification of superstring theories +In this section, we describe the different superstring theories (Chapter 17). +In order to +proceed, we need to introduce some new elements. +The worldsheet field theory of the closed string is made of two sectors, called the left- and +right-moving sectors (the αn and ¯αn modes). While they are treated symmetrically in the +simplest models, they are in fact independent (up to the zero-mode) and the corresponding +CFT can be chosen to be distinct. +The second ingredient already evoked earlier is supersymmetry. This symmetry associ- +ates a fermion to each boson (and conversely) through the action of a supercharge Q +|boson⟩ = Q |fermion⟩ . +(1.20) +More generally, one can consider N supercharges which build up a family of several bosonic +and fermionic partners. Since each supercharge increases the spin by 1/2 (in D = 4), there +is an upper limit for the number of supersymmetries – for interacting theories with a finite +number of fields5 – in order to keep the spin of a family in the range where consistent actions +exist: +• Nmax = 4 without gravity (−1 ≤ spin ≤ 1); +• Nmax = 8 with gravity (−2 ≤ spin ≤ 2). +This counting serves as a basis to determine the maximal number of supersymmetries in +other dimensions (by relating them through dimensional reductions). +Let’s turn our attention to the case of the two-dimensional worldsheet theory. +The +number of supersymmetries of the closed left- and right-moving sectors can be chosen in- +dependently, and the number of charges is written as (NL, NR) (the index is omitted when +5These conditions exclude the cases of free theories and higher-spin theories. +18 + +statements are made at the level of the CFT). The critical dimension (absence of quantum +anomaly for the Weyl invariance) depends on the number of supersymmetry +D(N = 0) = 26, +D(N = 1) = 10. +(1.21) +Type II superstrings have (NL, NR) = (1, 1) and come in two flavours called IIA and IIB +according to the chiraly of the spacetime gravitini chiralities. A theory is called heterotic if +NL > NR; we will mostly be interested in the case NL = 1 and NR = 0.6 In such theories, +there cannot be open strings since both sectors must be equal in the latter. Since the critical +dimensions of the two sectors do not match, one needs to get rid of the additional dimensions +of the right-moving sector; this leads to the next topic – gauge groups. +Gauge groups associated with spacetime gauge bosons appear in two different places. In +heterotic models, the compactification of the unbalanced dimensions of the left sector leads +to the appearance of a gauge symmetry. The possibilities are scarce due to consistency +conditions which ensure a correct gluing with the right-sector. Another possibility is to +add degrees of freedom – known as Chan–Paton indices – at the ends of open strings: +one end transforms in the fundamental representation of a group G, while the other end +transforms in the anti-fundamental. The modes of the open string then reside in the adjoint +representation, and the massless spin-1 particles become the gauge bosons of the non-Abelian +gauge symmetry. +Finally, one can consider oriented or unoriented strings. An oriented string possesses an +internal direction, i.e. there is a distinction between going from the left to the right (for an +open string) or circling in clockwise or anti-clockwise direction (for a closed string). Such +an orientation can be attributed globally to the spacetime history of all strings (interacting +or not). The unoriented string is obtained by quotienting the theory by the Z2 worldsheet +parity symmetry which exchanges the left- and right-moving sectors. Applying this to the +type IIB gives the type I theory. +The tachyon-free superstring theories together with the bosonic string are summarized +in Table 1.1. +worldsheet +susy +D +spacetime +susy +gauge group +open string +oriented +tachyon +bosonic +(0, 0) +26 +0 +any* +yes +yes / no +yes +type I +(1, 1) +10 +(1, 0) +SO(32) +yes +no +no +type IIA +(1, 1) +10 +(1, 1) +U(1) +(yes)† +yes +no +type IIB +(1, 1) +10 +(2, 0) +none +(yes)† +yes +no +heterotic SO(32) +(1, 0) +10 +(1, 0) +SO(32) +no +yes +no +heterotic E8 +(1, 0) +10 +(1, 0) +E8 × E8 +no +yes +no +heterotic SO(16) +(1, 0) +10 +(0, 0) +SO(16) × SO(16) +no +yes +no +* UV divergences beyond the tachyon (interpreted as closed string dilaton tadpoles) cancel only for the unoriented +open plus closed strings with gauge group SO(213) = SO(8192). +† The parenthesis indicates that type II theories don’t have open strings in the vacuum: they require a D-brane +background. This is expected since there is no gauge multiplet in d = 10 (1, 1) or (2, 0) supergravities (the +D-brane breaks half of the supersymmetry). +Table 1.1: List of the consistent tachyon-free (super)string theories. The bosonic theory is +added for comparison. There are additional heterotic theories without spacetime supersym- +metry, but they contain a tachyon and are thus omitted. +6The case NL < NR is identical up to exchange of the left- and right-moving sectors. +19 + +(a) Closed strings +(b) Open strings +Figure 1.6: Graphs corresponding to 1-loop 4-point scattering after a conformal mapping. +1.2.3 +Interactions +Worldsheet and Riemann surfaces +After having described the spectrum and the general characteristics of string theory comes +the question of interactions. The worldsheets obtained in this way are Riemann surfaces, +i.e. 1-dimensional complex manifolds. They are classified by the numbers of handles (or +holes) g (called the genus) and external tubes n. In the presence of open strings, surfaces +have boundaries: in addition to the handles and tubes, they are classified by the numbers +of disks b and of strips m.7 A particularly important number associated to each surface is +the Euler characteristics +χ = 2 − 2g − b , +(1.22) +which is a topological invariant. It is remarkable that there is a single topology at every +loop order when one considers only closed strings, and just a few more in the presence of +open strings. The analysis is greatly simplified in contrast to QFT, for which the number +of Feynman graphs increases very rapidly with the number of loops and external particles. +Due to the topological equivalence between surfaces, a conformal map can be used in +order to work with simpler surfaces. In particular, the external tubes and strips are collapsed +to points called punctures (or marked points) on the corresponding surfaces or boundaries. +A general amplitude then looks like a sphere from which holes and disks have been removed +and to which marked points have been pierced (Figure 1.7). +Amplitudes +In order to compute an amplitude for the scattering of n strings (Chapters 3 and 4), one +must sum over all the inequivalent worldsheets through a path integral weighted by the CFT +action chosen to describe the theory.8 At fixed n, the sum runs over the genus g, such that +each term is described by a Riemann surface Σg,n of genus g with n punctures. +The interactions between strings follow from the graph topologies: since the latter are not +encoded into the action, the dependence in the coupling constant must be added by hand. +For closed strings, there is a unique cubic vertex with coupling gs. A direct inspection shows +that the correct factor is gn−2+2g +s +: +7We ignore unoriented strings in this discussion. +They would lead to an additional object called a +cross-cap, which is a place where the surface looses its orientation. +8For simplicity we focus on closed string amplitudes in this section. +20 + +Figure 1.7: General Riemann surfaces with boundaries and punctures. +• for n = 3 there is one factor gs, and every additional external string leads to the +addition of one vertex with factor gs, since this process can be obtained from the n−1 +process by splitting one of the external string in two by inserting a vertex; +• each loop comes with two vertices, so g-loops provide a factor g2g +s . +Remark 1.2 (Status of gs as a parameter) It was stated earlier that string theory has +no dimensionless parameter, but gs looks to be one. In reality it is determined by the expect- +ation value of the dilaton gs = e⟨Φ⟩. Hence the coupling constant is not a parameter defining +the theory but is rather determined by the dynamics of the theory. +Finally, the external states must be specified: this amounts to prescribe boundary condi- +tions for the path integral or to insert the corresponding wave functions. Under the conformal +mapping which brings the external legs to punctures located at zi, the states are mapped to +local operators Vi(ki, zi) inserted at the points zi. The latter are built from the CFT fields +and are called vertex operators: they are characterized by a momentum kµ which comes +from the Fourier transformation of the Xµ fields representing the non-compact dimensions. +These operators are inserted inside the path integral with integrals over the positions zi in +order to describe all possible conformal mappings. +Ultimately, the amplitude (amputated Green function) is computed as +An(k1, . . . , kn) = +� +g≥0 +gn−2+2g +s +Ag,n +(1.23) +where +Ag,n = +� +n +� +i=1 +d2zi +� +dgabdΨ e−Scft[gab,Ψ] +n +� +i=1 +Vi(ki, zi) +(1.24) +is the g-loop n-point amplitude (for simplicity we omit the dependence on the states beyond +the momentum). Ψ denotes collectively the CFT fields and gab is the metric on the surface. +The integration over the metrics and over the puncture locations contain a huge redund- +ancy due to the invariance under reparametrizations, which means that one integrates over +many equivalent surfaces. To avoid this, Faddeev–Popov ghosts must be introduced and the +integral is restricted to only finitely many (real) parameters tλ. They form the moduli space +Mg,n of the Riemann surfaces Σg,n whose dimension is +dimR Mg,n = 6g − 6 + 2n. +(1.25) +21 + +The computation of the amplitude Ag,n can be summarized as: +Ag,n = +� +Mg,n +6g−6+2n +� +λ=1 +dtλ F(t). +(1.26) +The function F(t) is a correlation function in the worldsheet CFT defined on the Riemann +surface Σg,n +F(t) = +� n +� +i=1 +Vi × ghosts × super-ghosts +� +Σg,n +. +(1.27) +Note that the (super)ghost part is independent of the choice of the matter CFT. +Divergences and Feynman graphs +Formally the moduli parameters are equivalent to Schwinger (proper-time) parameters si in +usual QFT: these are introduced in order to rewrite propagators as +1 +k2 + m2 = +� ∞ +0 +ds e−s(k2+m2), +(1.28) +such that the integration over the momentum k becomes a Gaussian times a polynomial. +This form of the propagator is useful to display the three types of divergences which can be +encountered: +1. IR: regions si → ∞ (for k2 + m2 ≤ 0). These divergences are artificial for k2 + m2 < 0 +and means that the parametrization is not appropriate. Divergences for k2 + m2 = 0 +are genuine and translates the fact that quantum effects shift the vacuum and the +masses. +Taking these effects into account necessitates a field theory framework in +which renormalization can be used. +2. UV: regions si → 0 (after integrating over k). Such divergences are absent in string +theories because these regions are excluded from the moduli space Mg,n (see Figure 1.8 +for the example of the torus).9 +3. Spurious: regions with finite si where the amplitude diverges. This happens typically +only in the presence of super-ghosts and it translates a breakdown of the gauge fixing +condition.10 Since these spurious singularities of the amplitudes are not physical, one +needs to ensure that they can be removed, which is indeed possible to achieve. +Hence, only IR divergences present a real challenge to string theory. Dealing with these +divergences requires renormalizing the amplitudes, but this is not possible in the standard +formulation of worldsheet string theory since the states are on-shell.11 +9There is a caveat to this statement: UV divergences reappear in string field theory in Lorentzian +signature due to the way the theory is formulated. +The solution requires a generalization of the Wick +rotation. +Moreover, this does not hold for open strings whose moduli spaces contains those regions: in this case, +the divergences are reinterpreted in terms of closed strings propagating. +10Such spurious singularities are also found in supergravity. +11The on-shell condition is a consequence of the BRST and conformal invariance. While the first will be +given up, the second will be maintained to facilitate the computations. +22 + +Figure 1.8: Moduli space of the torus: Re τ ∈ [−1/2, 1/2], Im τ > 0 and |τ| > 1. +1.3 +String field theory +1.3.1 +From the worldsheet to field theory +The first step is to solve the IR divergences problem is to go off-shell (Chapters 11 and 13). +This is made possible by introducing local coordinates around the punctures of the Riemann +surface (Chapter 12). +The IR divergences originate from Riemann surfaces close to degeneration, that is, sur- +faces with long tubes. The latter can be of separating and non-separating types, depending +on whether the Riemann surface splits in two pieces if the tube is cut (Figure 1.9). By +exploring the form of the amplitudes in this limit (Chapter 14), the expression naturally +separates into several pieces, to be interpreted as two amplitudes (of lower n and g) con- +nected by a propagator. The latter can be reinterpreted as a standard (k2 + m2)−1 term, +hence solving the divergence problem for k2 + m2 < 0. Taking this decomposition seriously +leads to identify each contribution with a Feynman graph. +Decomposing the amplitude recursively, the next step consists in finding the elementary +graphs, i.e. the interaction vertices from which all other graphs (and amplitudes) can be +built. These graphs are the building blocks of the field theory (Chapter 15), with the kinetic +term given by the inverse of the propagator. Having Feynman diagrams and a field theory +allows to use all the standard tools from QFT. +However, this field theory is gauge fixed because on-shell amplitudes are gauge invariant +and include only physical states. +For this reason, one needs to find how to re-establish +the gauge invariance. Due to the complicated structure of string theory, the full-fledged +Batalin–Vilkovisky (BV) formalism must be used (Chapter 15): it basically amounts to +introduce ghosts before the gauge fixing. The final stage is to obtain the 1PI effective action +from which the physics is more easily extracted. But, it is useful to study first the free +theory (Chapters 9 and 10) to gain some insights. The book ends with a discussion of the +momentum-space representation and of background independence (Chapters 16 and 18). +The procedure we will follow is a kind of reverse-engineering: we know what is the final +23 + +(a) Separating. +(b) Non-separating. +Figure 1.9: Degeneration of Riemann surfaces. +result and we want to study backwards how it is obtained: +on-shell amplitude → off-shell amplitude → Feynman graphs +→ gauge fixed field theory → BV field theory +In standard QFT, one follows the opposite process. +Remark 1.3 There are some prescriptions (using for example analytic continuation, the +optical theorem, some tricks. . . ) to address the problems mentioned above, but there is no +general and universally valid procedure. A field theory is much more satisfactory because it +provides a unique and complete framework. +We can now summarise the disadvantages of the worldsheet approach over the spacetime +field one: +• no natural description of (relativistic) multi-particle states; +• on-shell states: +– lack of renormalization, +– presence of infrared divergences, +– scattering amplitudes only for protected states; +• interactions added by hand; +• hard to check consistency (unitarity, causality. . . ); +• absence of non-perturbative processes. +Some of these problems can be addressed with various prescriptions, but it is desirable +to dispose of a unified and systematic procedure, which is to be found in the field theory +description. +24 + +1.3.2 +String field action +A string field theory (SFT) for open and closed strings is based on two fields Φ[X(σ)] (open +string field) and Ψ[X(σ)] (closed string field) governed by some action S[Φ, Ψ]. This action +is built from a diagonal kinetic term +S0 = 1 +2 KΨ(Ψ, Ψ) + 1 +2 KΦ(Φ, Φ) +(1.29) +and from an interaction polynomial in the fields +Sint = +� +m,n +Vm,n(Φm, Ψn) +(1.30) +where Vm,n is an appropriate product mapping m closed and n open string states to a number +(the power is with respect to the tensor product). In particular, it contains the coupling +constant. Contrary to the worldsheet approach where the cubic interaction looks sufficient, +higher-order elementary interactions with m, n ∈ N are typically needed. A second specific +feature is that the products also admit a loop (or genus g) expansion: a fundamental n- +point interaction is introduced at every loop order g. These terms are interpreted as (finite) +counter-terms needed to restore the gauge invariance of the measure. These two facts come +from the decomposition of the moduli spaces in pieces (Section 1.2.3). +Writing an action for a field Ψ[X(σ)] for which reparametrization invariance holds is +highly complicated. The most powerful method is to introduce a functional dependence in +ghost fields Ψ[X(σ), c(σ)] and to extend the BRST formalism to the string field, leading +ultimately to the BV formalism. +While the latter formalism is the most complete and +ensures that the theory is consistent at the quantum level, it is difficult to characterize the +interactions explicitly. Several constructions which exploit different properties of the theory +have been proposed: +• direct computation by reverse engineering of worldsheet amplitudes; +• specific parametrization of the Riemann surfaces (hyperbolic, minimal area); +• analogy with Chern–Simons and Wess–Zumino–Witten (WZW) theories; +• exploitation of the L∞ and A∞ algebra structures. +It can be shown that these constructions are all equivalent. For the superstring, the simplest +strategy is to dress the bosonic interactions with data from the super-ghost sector, which +motivates the study of the bosonic SFT by itself. The main difficulty in working with SFT is +that only the first few interactions have been constructed explicitly. Finally, the advantage +of the first formulation is that it provides a general formulation of SFT at the quantum +level, from which the general structure can be studied. +1.3.3 +Expression with spacetime fields +To obtain a more intuitive picture and to make contact with the spacetime fields, the field +is expanded in terms of 1-particle states in the momentum representation +|Ψ⟩ = +� +n +� +dDk +(2π)D ψα(k) |k, α⟩ , +(1.31) +where α denotes collectively the discrete labels of the CFT eigenstates. The coefficients +ψα(k) of the CFT states |k, α⟩ are spacetime fields, the first ones being the same as those +found in the first-quantized picture (Section 1.2.1) +ψα = {T, Gµν, Bµν, Φ, . . .}. +(1.32) +25 + +Then, inserting this expansion in the action gives an expression like S[T, Gµν, . . .]. The exact +expression of this action is out of reach and only the lowest terms are explicitly computable +for a given CFT background. Nonetheless, examining the string field action indicates what +is the generic form of the action in terms of the spacetime fields. One can then study the +properties of such a general QFT: since it is more general than the SFT (expanded) action, +any result derived for it will also be valid for SFT. This approach is very fruitful for studying +properties related to consistency of QFT (unitarity, soft theorems. . . ) and this can provide +helpful phenomenological models. +In conclusion, SFT can be seen as a regular QFT with the following properties: +• infinite number of fields; +• non-local interaction (proportional to e−k2#); +• the amplitudes agree with the worldsheet amplitudes (when the latter can be defined); +• genuine (IR) divergences agree but can be handled with the usual QFT tools. +1.3.4 +Applications +The first aspect is the possibility to use standard QFT techniques (such as renormalization) +to study – and to make sense of – string amplitudes. In this sense, SFT can be viewed as +providing recipes for computing quantities in the worldsheet theory which are otherwise not +defined. This program has been pushed quite far in the last years. +Another reason to use SFT is gauge invariance: it is always easier to describe a sys- +tem when its gauge invariance is manifest. We have explained that string theory contains +Yang–Mills and graviton fields with the corresponding (spacetime) gauge invariances (non- +Abelian gauge symmetry and diffeomorphisms). In fact, these symmetries are enhanced to +an enormous gauge invariance when taking into account the higher-spin fields. This invari- +ance is hidden in the standard formulation and cannot be exploited fully. On the other +hand, the full gauge symmetry is manifest in string field theory. +Finally, the worldvolume description of p-brane is difficult because there is no analogue +of the Polyakov action. If one could find a first-principle description of SFT which does not +rely on CFT and first-quantization, then one may hope to generalize it to build a brane field +theory. +We can summarize the general motivations for studying SFT: +• field theory (second-quantization); +• more rigorous and constructive formulation; +• make gauge invariance explicit (L∞ algebras et al.); +• use standard QFT techniques (renormalization, analyticity. . . ) +→ remove IR divergences, prove consistency (Cutkosky rules, unitarity, soft theorems, +background independence. . . ); +• worldvolume theory ill-defined for (p > 1)-branes. +Beyond these general ideas, SFT has been developed in order to address different questions: +• worldsheet scattering amplitudes; +• effective actions; +• map of the consistent backgrounds (classical solutions, marginal deformations, RR +fluxes. . . ); +26 + +• collective, non-perturbative, thermal, dynamical effects; +• symmetry breaking effects; +• dynamics of compactification; +• proof of dualities; +• proof of the AdS/CFT correspondence. +The last series of points is still out of reach within the current formulation of SFT. However, +the last two decades have seen many important develoments developments: +• construction of the open, closed and open-closed superstring field theories: +– 1PI and BV actions and general properties [73, 165, 166, 213, 214, 216, 218, 220, +222, 225, 226, 230], +– dressing of bosonic products using the WZW construction and homotopy al- +gebra [18, 19, 67–70, 74–76, 79, 80, 94, 111, 131, 134, 140–146, 180], +– light-cone super-SFT [114–117], +– supermoduli space [175, 241]; +• hyperbolic and minimal area constructions [38, 102, 103, 162–164, 183]; +• open string analytic solutions [77, 78]; +• level-truncation solutions [135–137]; +• field theory properties [34, 43, 150, 187, 221, 223, 224]; +• spacetime effective actions [65, 153, 154, 248]; +• defining worldsheet scattering amplitudes [184–186, 215, 216, 219, 227–229]; +• marginal and RR deformations [35, 229, 248]. +Recent reviews are [42, 71, 72]. +1.4 +Suggested readings +For references about different aspects in this chapter: +• Differences between the worldvolume and spacetime formalisms – and of the associated +first- and second-quantization – for the particle and string [124, chap. 1, 265, chap. 11]. +• General properties of relativistic strings [92, 265]. +• Divergences in string theory [42, 217, 256, sec. 7.2]. +• Motivations for building a string field theory [192, sec. 4]. +27 + +Part I +Worldsheet theory +28 + +Chapter 2 +Worldsheet path integral: +vacuum amplitudes +Abstract +In this chapter, we develop the path integral quantization for a generic closed +string theory in worldsheet Euclidean signature. We focus on the vacuum amplitudes, leaving +scattering amplitudes for the next chapter. This allows to focus on the definition and gauge +fixing of the path integral measure. +The exposition differs from most traditional textbooks in three ways: 1) we consider a +general matter CFT, 2) we consider the most general treatment (for any genus) and 3) we +don’t use complex coordinates but always a covariant parametrization. +The derivation is technical and the reader is encouraged to not stop at this chapter +in case of difficulties and to proceed forward: most concepts will be reintroduced from a +different point of view later in other chapters of the book. +2.1 +Worldsheet action and symmetries +The string worldsheet is a Riemann surface W = Σg of genus g: the genus counts the +number of holes or handles. Coordinates on the worldsheet are denoted by σa = (τ, σ). +When there is no risk of confusion, σ denotes collectively both coordinates. Since closed +strings are considered, the Riemann surface has locally the topology of a cylinder, with the +spatial section being circles S1 with radius taken to be 1, such that +σ ∈ [0, 2π), +σ ∼ σ + 2π. +(2.1) +The string is embedded in the D-dimensional spacetime M with metric Gµν through maps +Xµ(σa) : W → M with µ = 0, . . . , D − 1. +The Nambu–Goto action is the starting point of the worldsheet description: +SNG[Xµ] = +1 +2πα′ +� +d2σ +� +det Gµν(X)∂Xµ +∂σa +∂Xν +∂σb , +(2.2) +where α′ is the Regge slope (related to the string tension and string length). +However, +quantizing this action is difficult because it is highly non-linear. To solve this problem, a +Lagrange multiplier is introduced to remove the squareroot. This auxiliary field corresponds +to an intrinsic worldsheet metric gab(σ). +The worldsheet dynamics is described by the +Polyakov action: +SP[g, Xµ] = +1 +4πα′ +� +d2σ√g gabGµν(X)∂Xµ +∂σa +∂Xν +∂σb , +(2.3) +29 + +which is classically equivalent to the Nambu–Goto action (2.2). In this form, it is clear that +the scalar fields Xµ(σ) (µ = 0, . . . D − 1) characterize the string theory under consideration +in two ways. +First, by specifying some properties of the spacetime in which the string +propagates (for example, the number of dimensions is determined by the number of fields +Xµ), second, by describing the internal degrees of freedom (vibration modes).1 +But, nothing prevents to consider a more general matter content in order to describe a +different spacetime or different degrees of freedom. In Polyakov’s formalism, the worldsheet +geometry is endowed with a metric gab(σ) together with a set of matter fields living on it. +The scalar fields Xµ can be described by a general sigma model which encodes the embedding +of the string in the D non-compact spacetime dimensions, and other fields can be added, +for example to describe compactified dimensions or (spacetime) spin. Different sets of fields +(and actions) correspond to different string theories. +However, to describe precisely the +different possibilities, we first have to understand the constraints on the worldsheet theories +and to introduce conformal field theories (Part I). In this chapter (and in most of the book), +the precise matter content is not important and we will denote the fields collectively as Ψ(σ). +Before discussing the symmetries, let’s introduce a topological invariant which will be +needed throughout the text: the Euler characteristics. It is computed by integrating the +Riemann curvature R of the metric gab over the surface Σg: +χg := χ(Σg) := 2 − 2g = 1 +4π +� +Σg +d2σ√g R, +(2.4) +where g is the genus of the surface. Oriented Riemann surfaces without boundaries are +completely classified (topologically or as complex manifolds) by their Euler characteristics +χg, or equivalently by their genus g. +In order to describe a proper string theory, the worldsheet metric gab(σ) should not +be dynamical. +This means that the worldsheet has no intrinsic dynamics and that no +supplementary degrees of freedom are introduced when parametrizing the worldsheet with +a metric. A solution to remove these degrees of freedom is to introduce gauge symmetries +with as many gauge parameters as there are of degrees of freedom. The simplest symmetry +is invariance under diffeomorphisms: indeed, the worldsheet theory is effectively a QFT +coupled to gravity and it makes sense to require this invariance. Physically, this corresponds +to the fact that the worldsheet spatial coordinate σ used along the string and worldsheet +time are arbitrary. However, diffeomorphisms alone are not sufficient to completely fix the +metric. Another natural candidate is Weyl invariance (local rescalings of the metric). +A diffeomorphism f ∈ Diff(Σg) acts on the fields as +σ′a = f a(σb), +g′(σ′) = f ∗g(σ), +Ψ′(σ′) = f ∗Ψ(σ), +(2.5) +where the star denotes the pullback by f: this corresponds simply to the standard coordinate +transformation where each tensor index of the field receives a factor ∂σa/∂σ′b. In particular, +the metric and scalar fields transform explicitly as +g′ +ab(σ′) = ∂σc +∂σ′a +∂σd +∂σ′b gcd(σ), +X′µ(σ′) = Xµ(σ). +(2.6) +The index µ is inert since it is a target spacetime index: from the worldsheet point of view, +it just labels a collection of worldsheet scalar fields. Infinitesimal variations are generated +by vector fields on Σg: +δξσa = ξa, +δξΨ = LξΨ, +δξgab = Lξgab, +(2.7) +1Obviously, the vibrational modes are also constrained by the spacetime geometry. +30 + +where Lξ is the Lie derivative2 with respect to the vector field ξ ∈ diff(Σg) ≃ TΣg. The Lie +derivative of the metric is +Lξgab = ξc∂cgab + gac∂bξc + gbc∂aξc = ∇aξb + ∇bξa. +(2.8) +The Lie algebra generates only transformations in the connected component Diff0(Σg) of +the diffeomorphism group which contains the identity. +Transformations not contained in Diff0(Σg) are called large diffeomorphisms: this in- +cludes reflections, for example. The quotient of the two groups is called the modular group +Γg (also mapping class group or MCG): +Γg := π0 +� +Diff(Σg) +� += Diff(Σg) +Diff0(Σg). +(2.9) +It depends only on the genus g of the Riemann surface, but not on the metric. It is an +infinite discrete group for genus g ≥ 1 surfaces; in particular, Γ1 = SL(2, Z). +A Weyl transformation e2ω ∈ Weyl(Σg) corresponds to a local rescaling of the metric +and leaves the other fields unaffected3 +g′ +ab(σ) = e2ω(σ)gab(σ), +Ψ′(σ) = Ψ(σ). +(2.10) +The exponential parametrization is generally more useful, but one should remember that it +is e2ω and not ω which is an element of the group. The infinitesimal variation reads +δωgab = 2ω gab, +δωΨ = 0 +(2.11) +where ω ∈ weyl(Σ) ≃ F(Σg) is a function on the manifold. Two metrics related in this way +are said to be conformally equivalent. The conformal structure of the Riemann surface is +defined by +Conf(Σg) := Met(Σg) +Weyl(Σg), +(2.12) +where Met(Σg) denotes the space of all metrics on Σg. Each element is a class of conformally +equivalent metrics. +Diffeomorphisms have two parameters ξa (vector field) and Weyl invariance has one, +ω (function). +Hence, this is sufficient to locally fix the three components of the metric +(symmetric matrix) and the total gauge group of the theory is the semi-direct product +G := Diff(Σg) ⋉ Weyl(Σg). +(2.13) +Similarly, the component connected of the identity is written as +G0 := Diff0(Σg) ⋉ Weyl(Σg). +(2.14) +The semi-direct product arises because the Weyl parameter is not inert under diffeo- +morphisms. Indeed, the combination of two transformations is +g′ = f ∗� +e2ωg +� += e2f ∗ωf ∗g, +(2.15) +such that the diffeomorphism acts also on the conformal factor. +2For our purpose here, it is sufficient to accept the definition of the Lie derivative as corresponding to +the infinitesimal variation. +3For simplicity, we consider only fields which do not transform under Weyl transformations, which +excludes fermions. +31 + +The combination of transformations (2.15) can be chosen to fix the metric in a convenient +gauge. For example, the conformal gauge reads +gab(σ) = e2φ(σ)ˆgab(σ), +(2.16) +where ˆgab is some (fixed) background metric and φ(σ) is the conformal factor, also called +the Liouville field. Fixing only diffeomorphisms amount to keep φ arbitrary: the latter can +then be fixed with a Weyl transformation. For instance, one can adopt the conformally flat +gauge +ˆgab = δab, +φ arbitrary +(2.17) +with a diffeomorphism, and then reach the flat gauge +ˆgab = δab, +φ = 0 +(2.18) +with a Weyl transformation. Another common choice is the uniformization gauge where ˆg +is taken to be the metric of constant curvature on the sphere (g = 0), on the plane (g = 1) +or on the hyperbolic space (g > 1). All these gauges are covariant (both in spacetime and +worldsheet). +Remark 2.1 (Active and passive transformations) Usually, symmetries are described +by active transformations, which means that the field is seen to be changed by the transform- +ation. On the other hand, gauge fixing is seen as a passive transformation, where the field +is expressed in terms of other fields (i.e. a different parametrization). These are mathem- +atically equivalent since both cases correspond to inverse elements, and one can choose the +most convenient representation. We will use indifferently the same name for the parameters +to avoid introducing minus signs and inverse. +Remark 2.2 (Topology and gauge choices) While it is always possible to adopt locally +the flat gauge (2.18), it may not be possible to extend it globally. The can be seen intuitively +from the fact that the sign of the curvature is given by the one of 1 − g, but the curvature of +the flat metric is zero: curvature must then be localized somewhere and this prevents from +using a single coordinate patch. +The final step is to write an action Sm[g, Ψ] for the matter fields. According to the +previous discussion, it must have the following properties: +• local in the fields; +• renormalizable; +• non-linear sigma models for the scalar fields; +• periodicity conditions; +• invariant under diffeomorphisms (2.5); +• invariant under Weyl transformations (2.10). +The latter two conditions are summarized by +Sm[f ∗g, f ∗Ψ] = Sm[g, Ψ], +Sm[e2ωg, Ψ] = Sm[g, Ψ]. +(2.19) +The invariance under diffeomorphisms is straightforward to enforce by using only covariant +objects. Since the scalar fields represent embedding of the string in spacetime, the non- +linear sigma model condition means that spacetime is identified with the target space of the +sigma model, of which D dimensions are non-compact, and the spacetime metric appears +32 + +in the matter action as in (2.3). +The isometries of the target manifold metric become +global symmetries of Sm: while they are not needed in this chapter, they will have their +importances in other chapters. Finally, to make the action consistent with the topology of +the worldsheet, the fields must satisfy appropriate boundary conditions. For example, the +scalar fields Xµ must be periodic for the closed string: +Xµ(τ, σ) ∼ Xµ(τ, σ + 2π). +(2.20) +Remark 2.3 (2d gravity) The setup in two-dimensional gravity is exactly similar, except +that the system is, in general, not invariant under Weyl transformations. As a consequence, +one component of the metric (usually taken to be the Liouville mode) remains unconstrained: +in the conformal gauge, (2.16) only ˆg is fixed. +The symmetries (2.19) of the action have an important consequence: they imply that the +matter action is conformally invariant on flat space gab = δab. A two-dimensional conformal +field theory (CFT) is characterized by a central charge cm: roughly, it is a measure of the +quantum degrees of freedom. The central charge is additive for decoupled sectors. In partic- +ular, the scalar fields Xµ contribute as D, and it is useful to define the perpendicular CFT +with central charge c⊥ as the matter which does not describe the non-compact dimensions: +cm = D + c⊥. +(2.21) +This will be discussed in length in Part I. For this chapter and most of the book, it is +sufficient to know that the matter is a CFT of central charge cm and includes D scalar fields +Xµ: +matter CFT parameters: D, cm. +(2.22) +The energy–momentum is defined by +Tm,ab := − 4π +√g +δSm +δgab . +(2.23) +The variation of the action under the transformations (2.7) vanishes on-shell if the energy– +momentum tensor is conserved +∇aTm,ab = 0 +(on-shell). +(2.24) +On the other hand, the variation under (2.11) vanishes off-shell (i.e. without using the +equations of motion) if the energy–momentum tensor is traceless: +gabTm,ab = 0 +(off-shell). +(2.25) +The conserved charges associated to the energy–momentum tensor generate worldsheet +translations +P a := +� +dσ T 0a +m . +(2.26) +The first component is identified with the worldsheet Hamiltonian P 0 = H and generates +time translations, the second component generates spatial translations. +Remark 2.4 (Tracelessness of the energy–momentum tensor) In fact, the trace can +also be proportional to the curvature +gabTm,ab ∝ R. +(2.27) +Then, the equations of motion are invariant since the integral of R is topological. The theory +is invariant even if the action is not. Importantly, this happens for fields at the quantum +level (Weyl anomaly), for the Weyl ghost field (Section 2.4) and for the Liouville theory +(two-dimensional gravity coupled to conformal matter). +33 + +2.2 +Path integral +The quantization of the system is achieved by considering the path integral, which yields +the genus-g vacuum amplitude (or partition function): +Zg := +� +dggab +Ωgauge[g] Zm[g], +Zm[g] := +� +dgΨ e−Sm[g,Ψ] +(2.28) +at fixed genus g (not to be confused with the metric). The integration over gab is performed +over all metrics of the genus-g Riemann surface Σg: gab ∈ Met(Σg). The factor Ωgauge[g] +is a normalization inserted in order to make the integral finite: it depends on the metric +(but only through the moduli parameters, as we will show later) [53, p. 931], which explains +why it is included after the integral sign. Its value will be determined in the next section by +requiring the cancellation of the infinities due to the integration over the gauge parameters. +This partition function corresponds to the g-loop vacuum amplitude: interactions and their +associated scattering amplitudes are discussed in Section 3.1. +In order to perform the gauge fixing and to manipulate the path integral (2.28), it is +necessary to define the integration measure over the fields. Because the space is infinite- +dimensional, this is a difficult task. +One possibility is to define the measure implicitly +through Gaussian integration over the field tangent space (see also Appendix C.1). A Gaus- +sian integral involves a quadratic form, that is, an inner-product (or equivalently a metric) +on the field space. The explanation is that a metric also defines a volume form, and thus +a measure. To reduce the freedom in the definition of the inner-product, it is useful to +introduce three natural assumptions: +1. ultralocality: the measure is invariant under reparametrizations and defined point-wise, +which implies that it can depend on the fields but not on their derivatives; +2. invariant measure: the measure for the matter transforms trivially under any sym- +metry of the matter theory by contracting indices with appropriate tensors; +3. free-field measure: for fields other than the worldsheet metric and matter (like ghosts, +Killing vectors, etc.), the measure is the one of a free field. +This means that the inner-product is obtained by contracting the worldsheet indices of the +fields with a tensor built only from the worldsheet metric, by contracting other indices (like +spacetime) with some invariant tensor (like the spacetime metric), and finally by integrating +over the worldsheet. +We need to distinguish the matter fields from those appearing in the gauge fixing proced- +ure. The matter fields live in the representation of some group under which the inner-product +is invariant: this means that it is not possible to define each field measure independently +if the exponential of inner-products does not factorize. As an example, on a curved back- +ground: dX ̸= � +µ dXµ. However, we will not need to write explicitly the partition function +for performing the gauge fixing: it is sufficient to know that the matter is a CFT. In the +gauge fixing procedure, different types of fields (including the metric) appear which don’t +carry indices (beyond the worldsheet indices). Below, we focus on defining a measure for +each of those single fields (and use free-field measures according to the third condition). +Considering the finite elements δΦ1 and δΦ2 of tangent space at the point Φ of the state +of fields, the inner-product (·, ·)g and its associated norm | · |g read +(δΦ1, δΦ2)g := +� +d2σ√g γg(δΦ1, δΦ2), +|δΦ|2 +g := (δΦ, δΦ)g, +(2.29) +where γg is a metric on the δΦ space. It is taken to be flat for all fields except the metric itself, +that is, independent of Φ. The dependence in the metric ensures that the inner-product is +34 + +diffeomorphism invariant, which in turns will lead to a metric-dependent but diffeomorphism +invariant measure. The functional measure is then normalized by a Gaussian integral: +� +dgδΦ e− 1 +2 (δΦ,δΦ)g = +1 +� +det γg +. +(2.30) +This, in turn, induces a measure on the field space itself: +� +dΦ +� +det γg +(2.31) +The determinant can be absorbed in the measure, such that +� +dgδΦ e− 1 +2 (δΦ,δΦ)g = 1. +(2.32) +In fact, this normalization and the definition of the inner-product is ambiguous, but the +ultralocality condition allows to fix uniquely the final result (Section 2.3.4). Moreover, such +a free-field measure is invariant under field translations +Φ(σ) −→ Φ′(σ) = Φ(σ) + ε(σ). +(2.33) +The most natural inner-products for single scalar, vector and symmetric tensor fields are +(δf, δf)g := +� +d2σ√g δf 2 +(2.34a) +(δV a, δV a)g := +� +d2σ√g gabδV aδV b, +(2.34b) +(δTab, δTab)g := +� +d2σ√g GabcdδTabδTcd, +(2.34c) +where the (DeWitt) metric for the symmetric tensor is +Gabcd := Gabcd +⊥ ++ u gabgcd, +Gabcd +⊥ +:= gacgbd + gadgbc − gabgcd, +(2.35) +with u a constant. The first term G⊥ is the projector on the traceless component of the +tensor. Indeed, consider a traceless tensor gabTab = 0 and a pure trace tensor Λgab, then we +have: +GabcdTcd = Gabcd +⊥ +Tcd = 2Tab, +Gabcd(Λgcd) = 2u (Λgab). +(2.36) +While all measures are invariant under diffeomorphisms, only the vector measure is +invariant under Weyl transformations. This implies the existence of a quantum anomaly +(the Weyl or conformal anomaly): the classical symmetry is broken by quantum effects +because the path integral measure cannot respect all the classical symmetries. Hence, one +can expect difficulties for imposing it at the quantum level and ensuring that the Liouville +mode in (2.16) remains without dynamics. +The metric variation (symmetric tensor) is decomposed in its trace and traceless parts +δgab = gab δΛ + δg⊥ +ab, +δΛ = 1 +2 gabδgab, +gabδg⊥ +ab = 0. +(2.37) +In this decomposition, both terms are decoupled in the inner-product +|δgab|2 +g = 4u|δΛ|2 +g + |δg⊥ +µν| +2 +g, +(2.38) +35 + +where the norm of δΛ is the one of a scalar field (2.34a). The norm for δg⊥ +ab is equivalent +to (2.34c) with u = 0 (since it is traceless). Requiring positivity of the inner-product for a +non-traceless tensor imposes the following constraint on u: +u > 0. +(2.39) +One can absorb the coefficient with u in δΛ, which will just contribute as an overall factor: +its precise value has no physical meaning. The simple choice u = 1/4 sets the coefficient of +|δΛ|2 +g to 1 in (2.38) (another common choice is u = 1/2). Ultimately, this implies that the +measure factorizes as +dggab = dgΛ dgg⊥ +ab. +(2.40) +Computation – Equation (2.38) +Gabcd δgabδgcd = +� +Gabcd +⊥ ++ u gabgcd�� +gab δΛ + δg⊥ +ab +�� +gcd δΛ + δg⊥ +cd +� += +� +2u gcd δΛ + Gabcd +⊥ +δg⊥ +ab +�� +gcd δΛ + δg⊥ +cd +� += 4u (δΛ)2 + Gabcd +⊥ +δg⊥ +abδg⊥ +cd += 4u δΛ2 + 2gacgbdδg⊥ +abδg⊥ +cd. +Remark 2.5 Another common parametrization is +Gabcd = gacgbd + c gabgcd. +(2.41) +It corresponds to (2.35) up to a factor 1/2 and setting u = 1 + 2c. +Remark 2.6 (Matter and curved background measures) As explained previously, mat- +ter fields carry a representation and the inner-product must yield an invariant combination. +In particular, spacetime indices must be contracted with the spacetime metric Gµν(X) (which +is the non-linear sigma model metric appearing in front of the kinetic term) for a general +curved background. For example, the inner-product for the scalar fields Xµ is +(δXµ, δXµ)g = +� +d2σ√g Gµν(X)δXµδXν. +(2.42) +It is not possible to normalize anymore the measure to set det G(X) = 1 like in (2.32) since +it depends on the fields. On the other hand, this factor is not important for the manipu- +lations performed in this chapter. Any ambiguity in the measure will again corresponds to +a renormalization of the cosmological constant [53, p. 923]. Moreover, as explained above, +it is not necessary to write explicitly the matter partition function as long as it describes a +CFT. +2.3 +Faddeev–Popov gauge fixing +The naive integration over the space Met(Σg) of all metrics of Σg (note that the genus is +fixed) leads to a divergence of the functional integral since equivalent configurations +(f ∗g, f ∗Ψ) ∼ (g, Ψ), +(e2ωg, Ψ) ∼ (g, Ψ) +(2.43) +gives the same contribution to the integral. This infinite redundancy causes the integral +to diverge, and since the multiple counting is generated by the gauge group, the infinite +contribution corresponds to the volume of the latter. The Faddeev–Popov procedure is a +36 + +means to extract this volume by separating the integration over the gauge and physical +degrees of freedom +d(fields) = Jacobian × d(gauge) × d(physical). +(2.44) +The space of fields (g, Ψ) is divided into equivalence classes and one integrates over only one +representative of each class (gauge slice), see Figure 2.1. This change of variables introduces +a Jacobian which can be represented by a partition function with ghost fields (fields with +a wrong statistics). This program encounters some complications since G is a semi-direct +product and is non-connected. +Example 2.1 – Gauge redundancy +A finite-dimensional integral which mimics the problem is +Z = +� +R2 dx dy e−(x−y)2. +(2.45) +One can perform the change of variables +r = x − y, +y = a +(2.46) +such that +Z = +� +R +da +� ∞ +0 +e−r2 = +√π +2 Vol(R), +(2.47) +and Vol(R) is to be interpreted as the volume of the gauge group (translation by a real +number a). +Remark 2.7 Mathematically, the Faddeev–Popov procedure consists in identifying the or- +bits (class of equivalent metrics) under the gauge group G and to write the integral in terms +of G-invariant objects (orbits instead of individual metrics). This can be done by decompos- +ing the tangent space into variations generated by G and its complement. Then, one can +define a foliation of the field space which equips it with a fibre bundle structure: the base +is the push-forward of the complement and the fibre corresponds to the gauge orbits. The +integral is then defined by selecting a section of this bundle. +2.3.1 +Metrics on Riemann surfaces +According to the above procedure, each metric gab ∈ Met(Σg) has to be expressed in terms +of gauge parameters (ξ and ω) and of a metric ˆgab which contains the remaining gauge- +independent degrees of freedom. As there are as many gauge parameters as metric com- +ponents (Section 2.1), one could expect that there are no remaining physical parameters +and then that ˆg is totally fixed. But, this is not the case and the metric ˆg depends on a +finite number of parameters ti (moduli). The reason for this is topological: while locally it +is always possible to completely fix the metric, topological obstructions may prevent doing +it globally. This means that not all conformal classes in (2.12) can be (globally) related by +a diffeomorphism. +The quotient of the space of metrics by gauge transformations is called the moduli space +Mg := Met(Σg) +G +. +(2.48) +Accordingly, its coordinates ti with i = 1, . . . , dimR Mg are called moduli parameters. The +Teichmüller space Tg is obtained by taking the quotient of Met(Σg) with the component +37 + +Figure 2.1: The space of metrics decomposed in gauge orbits. Two metrics related by a +gauge transformation lie on the same orbit. Choosing a gauge slice amounts to pick one +metric in each orbit, and the projection gives the space of metric classes. +connected to the identity +Tg := Met(Σg) +G0 +. +(2.49) +The space Tg is the covering space of Mg: +Mg = Tg +Γg +, +(2.50) +where Γg is the modular group defined in (2.9). Both spaces can be endowed with a complex +structure and are finite-dimensional [172]: +Mg := dimR Mg = dimR Tg = +� +� +� +� +� +0 +g = 0, +2 +g = 1, +6g − 6 +g ≥ 2, +(2.51) +In particular, their volumes are related by +� +Mg +dMgt = +1 +ΩΓg +� +Tg +dMgt +(2.52) +where ΩΓg is the volume of Γg. +We will need to extract volumes of different groups, so it is useful to explain how they +are defined. A natural measure on a connected group G is the Haar measure dg, which is +the unique left-invariant measure on G. Integrating the measure gives the volume of the +group +ΩG := +� +G +dg = +� +G +d(hg), +(2.53) +for any h ∈ G. Given the Lie algebra g of the group, a general element of the algebra is a +linear combinations of the generators Ti with coefficients αi +α = αiTi. +(2.54) +38 + +Group elements can be parametrized in terms of α through the exponential map. Moreover, +since a Lie group is a manifold, it is locally isomorphic to Rn: this motivates the use of a +flat metric for the Lie algebra, such that +ΩG = +� +dα := +� � +i +dαi. +(2.55) +Finally, it is possible to perform a change of coordinates from the Lie parameters to co- +ordinates x on the group: the resulting Jacobian is the Haar measure for the coordinates +x. +Remark 2.8 While Tg is a manifold, this is not the case of Mg for g ≥ 2, which is an +orbifold: the quotient by the modular group introduces singularities [173]. +Remark 2.9 (Moduli space and fundamental domain) Given a group acting on a space, +a fundamental domain for a group is a subspace such that the full space is generated by act- +ing with the group on the fundamental domain. Hence, one can view the moduli space Mg +as a fundamental domain (sometimes denoted by Fg) for the group Γg and the space Tg. +In the conformal gauge (2.16), the metric gab can be parametrized by +gab = ˆg(f,φ) +ab +(t) := e2f ∗φf ∗ˆgab(t) = f ∗� +e2φˆgab(t) +� +(2.56) +where φ := ω and t denotes the dependence in the moduli parameters. To avoid surcharging +the notations, we will continue to write g when there is no ambiguity. In coordinates, this +is equivalent to: +gab(σ) = ˆg(f,φ) +ab +(σ; t) := e2φ(σ)ˆg′ +ab(σ; t), +ˆg′ +ab(σ; t) = ∂σ′c +∂σa +∂σ′d +∂σb ˆgcd(σ′; t). +(2.57) +Remark 2.10 Strictly speaking, the matter fields also transform and one should write Ψ = +Ψ(f) := f ∗ ˆΨ and include them in the change of integration measures of the following sections. +But, this does not bring any particular benefits since these changes are trivial because the +matter is decoupled from the metric. +Remark 2.11 Although the metric cannot be completely gauge fixed, having just a finite- +dimensional integral is much simpler than a functional integral. In higher dimensions, the +gauge fixing does not reduce that much the degrees of freedom and a functional integral over +ˆg remains (in similarity with Yang–Mills theories). +The corresponding infinitesimal transformations are parametrized by (φ, ξ, δti). +The +variation of the metric (2.56) can be expressed as +δgab = 2φ gab + ∇aξb + ∇bξa + δti∂igab, +(2.58) +which is decomposed in a reparametrization (2.7), a Weyl rescaling (2.11), and a contri- +bution from the variations of the moduli parameters. +The latter are called Teichmüller +deformations and describe changes in the metric which cannot be written as a combination +of diffeomorphism and Weyl transformation. Only the last term is written with a delta +because the parameters ξ and φ are already infinitesimal. There is an implicit sum over i +and we have defined +∂i := ∂ +∂ti +. +(2.59) +39 + +According to the formula (2.55), the volumes ΩDiff0[g] and ΩWeyl[g] of the diffeomorph- +isms connected to the identity and Weyl group are +ΩDiff0[g] := +� +dgξ, +(2.60a) +ΩWeyl[g] := +� +dgφ. +(2.60b) +The full diffeomorphism group has one connected component for each element of the modular +group Γg, according to (2.9): the volume ΩDiff[g] of the full group is the volume of the +component connected to the identity times the volume ΩΓg +ΩDiff[g] = ΩDiff0[g] ΩΓg. +(2.60c) +We have written that the volume depends on g: but, the metric itself is parametrized +in terms of the integration variables, and thus the LHS of (2.60) cannot depend on the +variable which is integrated over: ΩDiff0 can depend only on φ and ΩWeyl only on ξ. But, all +measures (2.34b) are invariant under diffeomorphisms, and thus the result cannot depend on +ξ. Moreover, the measure for vector is invariant under Weyl transformation, which means +that ΩDiff0 does not depend on φ. This implies that the volumes depend only on the moduli +parameters +ΩDiff0[g] := ΩDiff0[e2φˆg] = ΩDiff0[ˆg], +ΩWeyl[g] := ΩWeyl[Lξˆg] = ΩWeyl[ˆg]. +(2.61a) +For this reason, it is also sufficient to take the normalization factor Ωgauge to have the same +dependence: +Ωgauge[g] := Ωgauge[ˆg]. +(2.61b) +These volumes are also discussed in Section 2.3.4. +Computation – Equation (2.61) +ΩDiff0[e2φˆg] = +� +de2φLξˆgξ = +� +de2φˆgξ = +� +dˆgξ = ΩDiff0[ˆg], +ΩWeyl[Lξˆg] = +� +de2φLξˆgφ = +� +de2φˆgφ = ΩWeyl[ˆg]. +Remark 2.12 (Free-field measure for the Liouville mode) The explicit measure (2.60b) +of the Liouville mode is complicated since the inner-product contains an exponential of the +field: +|δφ|2 = +� +d2σ√g δφ2 = +� +d2σ +� +ˆg e2φδφ2. +(2.62) +It has been proposed by David–Distler–Kawai [40, 55], and later checked explicitly [50, 51, +160], how to rewrite the measure in terms of a free measure weighted by an effective action. +The latter is identified with the Liouville action (Section 2.3.3). +In principle, we could follow the standard Faddeev–Popov procedure by inserting a delta +function for the gauge fixing condition +Fab := gab − ˆg(f,φ) +ab +(t), +(2.63) +with ˆg(f,φ) +ab +(t) defined in (2.56). However, we will take a detour to take the opportunity to +study in details manipulations of path integrals and to understand several aspects of the +40 + +geometry of Riemann surfaces. In any case, several points are necessary even when going +the short way, but less apparent. +In order to make use of the factorization (2.40) of the integration measure, the variation +(2.58) is decomposed into its trace (first term) and traceless parts (last two terms) (2.37) +δgab = 2˜Λ gab + (P1ξ)ab + δti µiab, +(2.64) +where4 +(P1ξ)ab = ∇aξb + ∇bξa − gab∇cξc, +(2.65a) +µiab = ∂igab − 1 +2 gab gcd∂igcd, +(2.65b) +˜Λ = Λ + 1 +2 δti gab∂igab, +Λ = φ + 1 +2 ∇cξc. +(2.65c) +The objects µi are called Beltrami differentials and correspond to traceless Teichmüller +deformations (the factor of 1/2 comes from the symmetrization of the metric indices). The +decomposition emphasizes which variations are independent from each other. In particular, +changes to the trace of the metric due to a diffeomorphism generated by ξ or a modification +of the moduli parameters can be compensated by a Weyl rescaling. +One can use (2.40) to replace the integration over gab by one over the gauge parameters +ξ and φ and over the moduli ti since they contain all the information about the metric: +Zg = +� +dMgt dg ˜Λ dg(P1ξ) Ωgauge[g]−1 Zm[g]. +(2.66) +It is tempting to perform the change of variables +(P1ξ, ˜Λ) −→ (ξ, φ) +(2.67) +such that +dg(P1ξ) dg ˜Λ +?= dgξ dgφ ∆FP[g] +(2.68) +where ∆FP[g] is the Jacobian of the transformation +∆FP[g] = det ∂(P1ξ, ˜Λ) +∂(ξ, φ) += det +�P1 +0 +⋆ +1 +� += det P1. +(2.69) +But, one needs to be more careful: +1. The variations involving P1ξ and δti are not orthogonal and, as a consequence, the +measure does not factorize. +2. P1 has zero-modes, i.e. vectors such that P1ξ = 0, which causes the determinant to +vanish, det P1 = 0. +A rigorous analysis will be performed in Section 2.3.2 and will lead to additional factors in +the path integral. +Next, if the actions and measures were invariant under diffeomorphisms and Weyl trans- +formations (which amounts to replace g by ˆg everywhere), it would be possible to factor out +the integrations over the gauge parameters and to cancel the corresponding infinite factors +thanks to the normalization Ωgauge[g]. A new problem arises because the measures are not +Weyl invariant as explained above and one should be careful when replacing the metric +(Section 2.3.3). +4For comparison, Polchinski [193] defines P1 with an overall factor 1/2. +41 + +2.3.2 +Reparametrizations and analysis of P1 +The properties of the operator P1 are responsible for both problems preventing a direct +factorization of the measure; for this reason, it is useful to study it in more details. +The operator P1 is an object which takes a vector v to a symmetric traceless 2-tensor T, +see (2.65a). Conversely, its adjoint P † +1 can be defined from the scalar product (2.34c) +(T, P1v)g = (P † +1 T, v)g, +(2.70) +and takes symmetric traceless tensors to vectors. In components, one finds +(P † +1 T)a = −2∇bTab. +(2.71) +The Riemann–Roch theorem relates the dimension of the kernels of both operators [172]: +dim ker P † +1 − dim ker P1 = −3χg = 6g − 6. +(2.72) +Teichmüller deformations +We first need to characterize Teichmüller deformations, the variations of moduli parameters +which lead to transformations of the metric independent from diffeomorphisms and Weyl +rescalings. This means that the different variations must be orthogonal for the inner-product +(2.34). +First, the deformations must be traceless, otherwise they can be compensated by a Weyl +transformation. The traceless metric variations δg which cannot be generated by a vector +field ξ are perpendicular to P1ξ (otherwise, the former would a linear combination of the +latter): +(δg, P1ξ)g = 0 +=⇒ +(P † +1 δg, ξ)g = 0. +(2.73) +Since ξ is arbitrary, this means that the first argument vanishes +P † +1 δg = 0. +(2.74) +Metric variations induced by a change in the moduli ti are in the kernel of P † +1 +δg ∈ ker P † +1 . +(2.75) +Elements of ker P † +1 are called quadratic differentials and a basis (not necessarily orthonor- +mal) of ker P † +1 is denoted as: +ker P † +1 = Span{φi}, +i = 1, . . . , dim ker P † +1 +(2.76) +(these should not be confused with the Liouville field). The dimension of ker P † +1 is in fact +equal to the dimension of the moduli space (2.51): +dimR ker P † +1 = Mg = +� +� +� +� +� +0 +g = 0, +2 +g = 1, +6g − 6 +g > 1. +(2.77) +The last two terms in the variation (2.64) of δgab are not orthogonal. Let’s introduce +the projector on the complement space of ker P † +1 +Π := P1 +1 +P † +1 P1 +P † +1 . +(2.78) +42 + +The moduli variations can then be rewritten as +δti µi = δti (1 − Π)µi + δti Πµi = δti (1 − Π)µi + δti P1ζi. +(2.79) +The ζi exist because Πµi ∈ Im P1, and they read +ζi := +1 +P † +1 P1 +P † +1 µi. +(2.80) +The first term can be decomposed on the quadratic differential basis (2.76) +(1 − Π)µi = φj(M −1)jk(φk, µi)g +(2.81) +where +Mij := (φi, φj)g. +(2.82) +Ultimately, the variation (2.64) becomes +δgab = (P1 ˜ξ)ab + 2˜Λ gab + Qiab δti. +(2.83) +where +˜ξ = ξ + ζiδti, +Qiab = φjab (M −1)jk(φk, µi)g. +(2.84) +Correspondingly, the norm of the variation splits in three terms since each variation is +orthogonal to the others: +|δg|2 +g = |δ˜Λ| +2 +g + |P1 ˜ξ| +2 +g + |Qiδti|2 +g. +(2.85) +Since the norm is decomposed as a sum, the measure factorizes: +dggab = dg ˜Λ dg(P1 ˜ξ) dg(Qiδti). +(2.86) +One can then perform a change of coordinates +(˜ξ, ˜Λ, Qiδti) −→ (ξ, Λ, δti), +(2.87) +where Λ was defined in (2.65c). The goal of this transformation is to remove the dependence +in the moduli from the measures on the Weyl factor and vector fields, and to recover a finite- +dimensional integral over the moduli: +dg ˜Λ dg(P1 ˜ξ) dg(Qiδti) = dMgt dgΛ dg(P1ξ) det(φi, µj)g +� +det(φi, φj)g +, +(2.88) +where the determinants correspond to the Jacobian. The role of the determinant in the +denominator is to ensure a correct normalization when the basis is not orthonormal (in +particular, it ensures that the Jacobian is independent of the basis). Plugging this result in +(2.28) gives the partition function as +Zg = +� +Tg +dMgt +1 +Ωgauge[ˆg] +� +dgΛ dg(P1ξ) det(φi, µj)g +� +det(φi, φj)g +Zm[g]. +(2.89) +The ti are integrated over the Teichmüller space Tg defined by (2.49) because the vectors ξ +generate only reparametrizations connected to the identity, and thus the remaining freedom +lies in Met(Σg)/G0. Next, we study how to perform the changes of variables to remove P1 +from the measure. +43 + +Conformal Killing vectors +In this section, we focus on the dgΛ dg(P1ξ) part of the measure and we make contact with +the rest at the end. +Infinitesimal reparametrizations generated by a vector field ξa produce only transform- +ations close to the identity. For this reason, integrating over all possible vector fields yields +the volume (2.60a) of the component of the diffeomorphism group connected to the identity: +� +dgξ = ΩDiff0[ˆg]. +(2.90) +Remember that the volume depends only on the moduli, but obviously not on ξ (integrated +over) nor φ (the inner-product (2.34b) is invariant). But, due to the existence of zero-modes, +one gets an integration over a subset of all vector fields, and this complicates the program, +as we discuss now. +Zero-modes ξ(0) of P1 are called conformal Killing vectors (CKV) +ξ(0) ∈ Kg := ker P1 +(2.91) +and satisfy the conformal Killing equation (see also Section 5.1): +(P1ξ(0))ab = ∇aξ(0) +b ++ ∇bξ(0) +a +− gab∇cξ(0)c = 0. +(2.92) +CKVs correspond to reparametrizations which can be absorbed by a change of the con- +formal factor. They should be removed from the ξ integration in order to not double-count +the corresponding metrics. The dimension of the zero-modes CKV space depends on the +genus [172]: +Kg := dimR Kg = dimR ker P1 = +� +� +� +� +� +6 +g = 0, +2 +g = 1, +0 +g > 1. +(2.93) +The associated transformations will be interpreted later (Chapter 5). The groups generated +by the CKVs are +g = 0 : +K0 = SL(2, C), +g = 1 : +K1 = U(1) × U(1). +(2.94) +Note that the first group is non-compact while the second is compact. +A general vector ξ can be separated into a zero-mode part and its orthogonal complement +ξ′: +ξ = ξ(0) + ξ′, +(2.95) +such that +(ξ(0), ξ′)g = 0 +(2.96) +for the inner-product (2.34b). Because zero-modes are annihilated by P1, the correct change +of variables in the partition function (2.66) maps to ξ′ only: +(P1ξ, Λ) −→ (ξ′, φ). +(2.97) +Integrating over ξ at this stage would double count the CKV (since they are already described +by the φ integration). The appropriate Jacobian reads +dgΛ dg(P1ξ) = dgφ dgξ′ ∆FP[g], +(2.98) +where the Faddeev–Popov determinant is +∆FP[g] = det′ ∂(P1ξ, Λ) +∂(ξ′, φ) += det′ P1 = +� +det′ P1P † +1 , +(2.99) +44 + +the prime on the determinant indicating that the zero-modes are excluded. This brings the +partition function (2.89) to the form +Zg = +� +Tg +dMgt Ωgauge[ˆg]−1 +� +dgφ dgξ′ +det(φi, µj)g +� +det(φi, φj)g +∆FP[g]Zm[g]. +(2.100) +Computation – Equation (2.98) +The Jacobian can be evaluated directly: +∆FP[g] = det′ ∂(P1ξ, Λ) +∂(ξ′, φ) += det′ +� P1 +0 +1 +2∇ +1 +� += det′ P1. +(2.101) +As a consequence of det′ P † +1 = det′ P1, the Jacobian can be rewritten as: +� +det′ P † +1 P1 = det′ P1. +(2.102) +It is instructive to derive this result also by manipulating the path integral. Con- +sidering small variations of the fields, one has: +1 = +� +dgδΛ dg(P1δξ) e−|δΛ|2 +g−|P1δξ′|2 +g += ∆FP[g] +� +dgδφ dgδξ′ e−|δφ+ 1 +2 ∇cδξc|2 +g−|P1δξ′|2 +g += ∆FP[g] +� +dgδφ dgδξ′ e−|δφ|2 +g−(δξ′,P † +1 P1δξ′)g += ∆FP[g] +� +det′ P † +1 P1 +�−1/2 +. +That the expression is equal to 1 follows from the normalization of symmetric tensors +and scalars (2.34) (the measures appearing in the path integral (2.89) arises without any +factor). The third equality holds because the measure is invariant under translations +of the fields, and we used the definition of the adjoint. +The volume of the group generated by the vectors orthogonal to the CKV is denoted as +Ω′ +Diff0[g] := Ω′ +Diff0[ˆg] = +� +dgξ′. +(2.103) +As explained in the beginning of this section, one should extract the volume of the full Diff0 +group, not only the volume Ω′ +Diff0[g]. Since the two sets of vectors are orthogonal, we can +expect the measures, and thus the volumes, to factorize. However, a Jacobian can and does +arise: its role it to take into account the normalization of the zero-modes. Denoting by ψi +a basis (not necessarily orthonormal) for the zero-modes +ker P1 = Span{ψi}, +i = 1, . . . , Kg, +(2.104) +the change of variables +ξ′ −→ ξ +(2.105) +reads +dgξ′ = +1 +� +det(ψi, ψj)g +dgξ +Ωckv[g], +(2.106) +45 + +where Ωckv[g] is the volume of the CKV group. The determinant is necessary when the basis +is not orthonormal. The relation between the gauge volumes is then +ΩDiff0[g] = +� +det(ψi, ψj)g Ωckv[g] Ω′ +Diff0[g]. +(2.107) +Note that the CKV volume is given in (2.111) and depends only on the topology but not on +the metric. By using arguments similar to the ones which lead to (2.61), one can expect that +each term is independently invariant under Weyl rescaling: this is indeed true (Section 2.3.3). +Computation – Equation (2.106) +Let’s expand ξ(0) on the zero-mode basis +ξ(0) = αiψi, +(2.108) +where the αi are real numbers, such that one can write the changes of variables +ξ −→ (ξ′, αi). +(2.109) +The Jacobian is computed from +1 = +� +dξ e−|ξ|2 +g = J +� +dξ(0) dξ′ e−|ξ′|2 +g−|ξ(0)| +2 +g += J +� � +i +dαi e−αiαj(ψi,ψj)g +� +dξ′ e−|ξ′|2 +g += J (det(ψi, ψj)g)−1/2 . +Note that the integration over the αi is a standard finite-dimensional integral. This +gives +dξ = +� +det(ψi, ψj)g dξ′ � +i +dαi. +(2.110) +Since nothing depends on the αi, they can be integrated over as in (2.53), giving the +volume of the CKV group +Ωckv[g] = +� � +i +dαi. +(2.111) +Replacing the integration over ξ′ thanks to (2.106), the path integral becomes +Zg = +� +Tg +dMgt Ωgauge[ˆg]−1 +� +dgφ dgξ +det(φi, µj)g +� +det(φi, φj)g +Ωckv[g]−1 +� +det(ψi, ψj)g +∆FP[g] Zm[g]. +(2.112) +Since the matter action and measure, and the Liouville measure are invariant under +reparametrizations, one can perform a change of variables +(f ∗ˆg, f ∗φ, f ∗Ψ) −→ (ˆg, φ, Ψ) +(2.113) +such that everything becomes independent of f (or equivalently ξ). Since the measure for ξ +is Weyl invariant, it is possible to separate it from the rest of the expression, which yields +an overall factor of ΩDiff0[g]. This brings the partition function to the form +Zg = +� +Tg +dMgt ΩDiff0[ˆg] +Ωgauge[ˆg] +� +dgφ det(φi, µj)g +� +det(φi, φj)g +Ωckv[g]−1 +� +det(ψi, ψj)g +∆FP[g] Zm[g] +(2.114) +46 + +where the same symbol is used for the metric +gab := g(φ) +ab = e2φˆgab. +(2.115) +Since the expression is invariant under the full diffeomorphism group Diff(Σg) and not +just under its component Diff0(Σg), one needs to extract the volume of the full diffeomorph- +ism group before cancelling it with the normalization factor. Otherwise, there is still an +over-counting the configurations. Using the relation (2.60c) leads to: +Zg = +1 +ΩΓg +� +Tg +dMgt ΩDiff[ˆg] +Ωgauge[ˆg] +� +dgφ det(φi, µj)g +� +det(φi, φj)g +Ωckv[g]−1 +� +det(ψi, ψj)g +∆FP[g] Zm[g]. +(2.116) +The volume ΩΓg can be factorized outside the integral because it depends only on the genus +and not on the metric. Finally, using the relation (2.52), one can replace the integration +over the Teichmüller space by an integration over the moduli space +Zg = +� +Mg +dMgt ΩDiff[ˆg] +Ωgauge[ˆg] +� +dgφ det(φi, µj)g +� +det(φi, φj)g +Ωckv[g]−1 +� +det(ψi, ψj)g +∆FP[g] Zm[g]. +(2.117) +2.3.3 +Weyl transformations and quantum anomalies +The next question is whether the integrand depends on the Liouville mode φ such that +the Weyl volume can be factorized out. While the matter action has been chosen to be +Weyl invariant – see the condition (2.19) – the measures cannot be defined to be Weyl +invariant. This means that there is a Weyl (or conformal) anomaly, i.e. a violation of the +Weyl invariance due to quantum effects. Since the techniques needed to derive the results +of this section are outside the scope of this book, we simply state the results. +It is possible to show that the Weyl anomaly reads [53, p. 929]5 +∆FP[e2φˆg] +� +det(φi, φj)e2φˆg += e +cgh +6 SL[ˆg,φ] +∆FP[ˆg] +� +det(ˆφi, ˆφj)ˆg +(2.118a) +Zm[e2φˆg] = e +cm +6 SL[ˆg,φ]Zm[ˆg], +(2.118b) +where SL is the Liouville action +SL[ˆg, φ] := 1 +4π +� +d2σ +� +ˆg +� +ˆgab∂aφ∂bφ + ˆRφ +� +, +(2.119) +where ˆR is the Ricci scalar of the metric ˆgab. These relations require to introduce counter- +terms, discussed further in Section 2.3.4. The coefficients cm and cgh are the central charges +respectively of the matter and ghost systems, with: +cgh = −26. +(2.120) +This value will be derived in Section 7.2. +The inner-products between φi and µj, and between the ψi, and the CKV volume are +independent of φ [172, sec. 14.2.2, 53, p. 931] +det(φi, µj)e2φˆg = det(ˆφi, ˆµj)ˆg, +det(ψi, ψj)e2φˆg = det(ψi, ψj)ˆg, +Ωckv[e2φˆg] = Ωckv[ˆg]. +(2.121) +5The relation is written for Zm since the action is invariant and is not affected by the anomaly. +47 + +Remark 2.13 (Weyl and gravitational anomalies) The Weyl anomaly translates into +a non-zero trace of the quantum energy–momentum tensor +⟨gµνTµν⟩ = c +12 R, +(2.122) +where c is the central charge of the theory. The Weyl anomaly can be traded for a gravita- +tional anomaly, which means that diffeomorphisms are broken at the quantum level [122]. +Inserting (2.118) in (2.117) yields +Zg = +� +Mg +dMgt ΩDiff[ˆg] +Ωgauge[ˆg] +det(φi, ˆµj)ˆg +� +det(φi, φj)ˆg +Ωckv[ˆg]−1 +� +det(ψi, ψj)ˆg +∆FP[ˆg] Zm[ˆg] +� +dgφ e− +cL +6 SL[ˆg,φ], +(2.123) +with the Liouville central charge +cL := 26 − cm. +(2.124) +The critical “dimension” is defined to be the value of the matter central charge cm such that +the Liouville central charge cancels +cL = 0 +=⇒ +cm = 26. +(2.125) +If the number of non-compact dimensions is D, it means that the central charge (2.21) of +the transverse CFT satisfies +c⊥ = 26 − D. +(2.126) +In this case, the integrand does not depend on the Liouville mode (because ΩDiff is +invariant under Weyl transformations) and the integration over φ can be factored out and +yields the volume of the Weyl group (2.60b) +� +dgφ = ΩWeyl[ˆg]. +(2.127) +Then, taking +Ωgauge[ˆg] = ΩDiff[ˆg] × ΩWeyl[ˆg] +(2.128) +removes the infinite gauge contributions and gives the partition function +Zg = +� +Mg +dMgt det(φi, ˆµj)ˆg +� +det(φi, φj)ˆg +Ωckv[ˆg]−1 +� +det(ψi, ψj)ˆg +∆FP[ˆg] Zm[ˆg]. +(2.129) +2.3.4 +Ambiguities, ultralocality and cosmological constant +Different ambiguities remain in the previous computations, starting with the definitions of +the measures (2.32) and (2.34), then in obtaining the volume of the diffeomorphism (2.60a) +and Weyl (2.60b) groups, and finally in deriving the conformal anomaly (2.118). +These different ambiguities can be removed by renormalizing the worldsheet cosmological +constant. This implies that the action +Sµ[g] = +� +d2σ√g +(2.130) +must be added to the classical Lagrangian, where µ0 is the bare cosmological constant. This +means that Weyl invariance is explicitly broken at the classical level. After performing all +the manipulations, µ0 is determined by removing all ambiguities and enforcing invariance +under the Weyl symmetry at the quantum level. +This amounts to set the renormalized +cosmological constant to zero (since it breaks the Weyl symmetry). +The possibility to +48 + +introduce a counter-term violating a classical symmetry arises because the symmetry itself +is broken by a quantum anomaly, so there is no reason to enforce it in the classical action. +We now review each issue separately. First, consider the inner-product of a single tensor +(2.32): the determinant det γg depends on the metric and one should be more careful when +fixing the gauge or integrating over all metrics. +However, ultralocality implies that the +determinant can only be of the form [53, pp. 923] +� +det γg = e−µγ Sµ[g], +(2.131) +for some µγ ∈ R, since Sµ is the only renormalizable covariant functional depending on the +metric but not on its derivatives. The effect is just to redefine the cosmological constant. +Second, the volume of the field space can be defined as the limit λ → 0 of a Gaussian +integral [53, pp. 931]: +ΩΦ = lim +λ→0 +� +dgΦ e−λ (Φ,Φ)g. +(2.132) +Due to ultralocality, the Gaussian integral should again be of the form +� +dgΦ e−λ (Φ,Φ)g = e−µ(λ) Sµ[g], +(2.133) +for some constant µ(λ). Hence, the limit λ → 0 gives +ΩΦ = +� +dgΦ = e−µ(0) Sµ[g], +(2.134) +which can be absorbed in the cosmological constant. However, the situation is more com- +plicated if Φ = ξ, φ since the integration variables also appear in the measure, as it was +also discussed before (2.61). But, in that case, it cannot appear in the expression of the +volume in the LHS. Moreover, invariances under diffeomorphisms for both measures, and +under Weyl rescalings for the vector measure, imply that the LHS can only depend on the +moduli through the background metric ˆg. The diffeomorphism and Weyl volumes can be +written in terms of e−ˆµ Sµ[ˆg]: since there is no counter-term left (the cosmological constant +counter-term is already fixed to cancel the coefficient of Sµ[g]), it is necessary to divide by +Ωgauge to cancel the volumes. +Finally, the computation of the Weyl anomaly (2.118) yields divergent terms of the form +lim +ϵ→0 +1 +ϵ +� +d2σ√g. +(2.135) +These divergences are canceled by the cosmological constant counter-term, see [54, app. 5.A] +for more details. +2.3.5 +Gauge-fixed path integral +As a conclusion of this section, we found that the partition function (2.28) can be written +as +Zg = +� +Mg +dMgt det(φi, ˆµj)ˆg +� +det(φi, φj)ˆg +Ωckv[ˆg]−1 +� +det(ψi, ψj)ˆg +∆FP[ˆg] Zm[ˆg], +(2.136a) += +� +Mg +dMgt +� +det(φi, ˆµj)2 +ˆg +det(φi, φj)ˆg +det′ ˆP † +1 ˆP1 +det(ψi, ψj)ˆg +Zm[ˆg] +Ωckv[ˆg]. +(2.136b) +after gauge fixing of the worldsheet diffeomorphisms and Weyl rescalings. It is implicit that +the factors for the CKV and moduli are respectively absent for g > 1 and g < 1. For g = 0 +the CKV group is non-compact and its volume is infinite. It looks like the partition vanishes, +but there are subtleties which will be discussed in Section 3.1.3. +49 + +Remark 2.14 (Weil–Petersson metric) When the metric is chosen to be of constant +curvature ˆR = −1, the moduli measure together with the determinants form the Weil– +Petersson measure +d(WP) = +� +Mg +dMgt det(φi, ˆµj)ˆg +� +det(φi, φj)ˆg +. +(2.137) +In (2.136), the background metric ˆgab is fixed. However, the derivation holds for any +choice of ˆgab: as a consequence, it makes sense to relax the gauge fixing and allow it to vary +while adding gauge symmetries. The first symmetry is background diffeomorphisms: +σ′a = ˆf a(σb), +ˆg′(σ′) = f ∗ˆg(σ), +φ′(σ′) = f ∗φ(σ), +Ψ′(σ′) = f ∗Ψ(σ). +(2.138) +This symmetry is automatic for Sm[ˆg, Ψ] since Sm[g, Ψ] was invariant under (2.5). Similarly, +the integration measures are also invariant. A second symmetry is found by inspecting the +decomposition (2.56) +gab = f ∗� +e2φˆgab(t) +� +, +(2.139) +which is left invariant under a background Weyl symmetry (also called emergent): +g′ +ab(σ) = e2ω(σ)gab(σ), +φ′(σ) = φ(σ) − ω(σ), +Ψ′(σ) = Ψ(σ). +(2.140) +Let us stress that it is not related to the Weyl rescaling (2.10) of the metric gab. +The +background Weyl rescaling (2.140) is a symmetry even when the physical Weyl rescaling +(2.10) is not. Together, the background diffeomorphisms and Weyl symmetry have three +gauge parameters, which is sufficient to completely fix the background metric ˆg up to moduli. +In fact, the combination of both symmetries is equivalent to invariance under the physical +diffeomorphisms. +To prove this statement, consider two metrics g and g′ related by a +diffeomorphism F and both gauge fixed to pairs (f, φ, ˆg) and (f ′, φ′, ˆg′): +g′ +ab = F ∗gab, +g′ +ab = f ′∗� +e2φ′ˆg′ +ab +� +, +gab = f ∗� +e2φˆgab +� +. +(2.141) +Then, the gauge fixing parametrizations are related by background symmetries ( ˆF, ω) as +ˆF = f ′−1 ◦ F ◦ f, +φ′ = ˆF ∗(φ − ω), +ˆg′ +ab = ˆF ∗(e2ωˆgab). +(2.142) +Moreover, this also implies that there is a diffeomorphism ˜f = F ◦ f such that g′ is gauge +fixed in terms of (φ, ˆg): +g′ +ab = ˜f ∗� +e2φˆgab +� +. +(2.143) +Computation – Equation (2.142) +The functions F, f, f ′, φ, φ′ and the metrics gab, g′ +ab, ˆgab and ˆg′ +ab are all fixed and one +must find ˆF and ω such that the relations (2.141) are compatible. First, one rewrites +g′ +ab in terms of ˆgab and compare with the expression with ˆg′ +ab: +g′ +ab = F ∗gab = F ∗� +f ∗� +e2φˆgab +�� += F ∗� +f ∗� +e2(φ−ω)e2ωˆgab +�� += f ′∗� +e2φ′ˆg′ +ab +� +. +In the third equality, we have introduced ω because ˆg′ +ab = ˆF ∗ˆgab is not true in general +since there are 3 independent components but ˆF has only 2 parameters, so we cannot +just define f ′ = F ◦ f and φ′ = φ. This explains the importance of the emergent Weyl +symmetry. +50 + +Remark 2.15 (Gauge fixing and field redefinition) Although it looks like we are un- +doing the gauge fixing, this is not exactly the case since the original metric is not used any- +more. One can understand the procedure of this section as a field redefinition: the degrees of +freedom in gab are repackaged into two fields (φ, ˆgab) adapted to make some properties of the +system more salient. A new gauge symmetry is introduced to maintain the number of degrees +of freedom. The latter helps to understand the structure of the action on the background. +Finally, in this context, the Liouville action is understood as a Wess–Zumino action, which +is defined as the difference between the effective actions evaluated in each metric. Another +typical use of this point of view is to rewrite a massive vector field as a massless gauge field +together with an axion [195]. +Remark 2.16 (Two-dimensional gravity) In 2d gravity, one does not work in the crit- +ical dimension (2.125) and cL ̸= 0. Thus, the Liouville mode does not decouple: the con- +formal anomaly breaks the Weyl symmetry at the quantum level which gives dynamics to +gravity, even if it has no degree of freedom classically. +As a consequence, one chooses +Ωgauge = ΩDiff. +Since the role of the classical Weyl symmetry is not as important as for string theory, it is +even not necessary to impose it classically. This leads to consider non-conformal matter [21, +22, 31, 82, 83]. Following the arguments from Section 2.1, the existence of the emergent +Weyl symmetry (2.140) implies that the total action Sgrav[ˆg, φ] + Sm[ˆg, Ψ] must be a CFT +for a flat background ˆg = δ, even if the two actions are not independently CFTs. +2.4 +Ghost action +2.4.1 +Actions and equations of motion +It is well-known that a determinant can be represented with two anticommuting fields, called +ghosts. The fields carry indices dictated by the map induced by the operator of the Faddeev– +Popov determinant: one needs a symmetric and traceless anti-ghost bab and a vector ghost +ca fields: +∆FP[g] = +� +d′ +gb d′ +gc e−Sgh[g,b,c], +(2.144) +where the prime indicates that the ghost zero-modes are omitted. The ghost action is +Sgh[g, b, c] := 1 +4π +� +d2σ√g gabgcdbac(P1c)bd +(2.145a) += 1 +4π +� +d2σ√g gab� +bac∇bcc + bbc∇acc − bab∇ccc� +. +(2.145b) +The ghosts ca and anti-ghosts bab are associated respectively to the variations due to the +diffeomorphisms ξa and to the variations perpendicular to the gauge slice. The normalization +of 1/4π is conventional. In Minkowski signature, the action is multiplied by a factor i. +Since bab is traceless, the last term of the action vanishes and could be removed. However, +this implies to consider traceless variations of the bab when varying the action (to compute +the equations of motion, the energy–momentum tensor, etc.). On the other hand, one can +keep the term and consider unconstrained variation of bab (since the structure of the action +will force the variation to have the correct symmetry), which is simpler. A last possibility is +to introduce a Lagrange multiplier. These aspects are related to the question of introducing +a ghost for the Weyl symmetry, which is described in Section 2.4.2. +The equations of motion are +(P1c)ab = ∇acb + ∇bca − gab∇ccc = 0, +(P † +1 b)a = −2∇bbab = 0. +(2.146) +51 + +Hence, the classical solutions of b and c are respectively mapped to the zero-modes of the +operators P † +1 and P1, and they are thus associated to the CKV and Teichmüller parameters. +The energy–momentum tensor is +T gh +ab = −bac∇bcc − bbc∇acc + cc∇cbab + gabbcd∇ccd. +(2.147) +Its trace vanishes off-shell (i.e. without using the b and c equations of motion) +gabT gh +ab = 0, +(2.148) +which shows that the action (2.145) is invariant under Weyl transformations +Sgh[e2ωg, b, c] = Sgh[g, b, c]. +(2.149) +The action (2.145) also has a U(1) global symmetry. The associated conserved charge is +called the ghost number and counts the number of c ghosts minus the number of b ghosts, +i.e. +Ngh(b) = −1, +Ngh(c) = 1. +(2.150a) +The matter fields are inert under this symmetry: +Ngh(Ψ) = 0. +(2.150b) +In terms of actions, the path integral (2.136) can be rewritten as +Zg = +� +Mg +dMgt det(φi, ˆµj)ˆg +� +det(φi, φj)ˆg +Ωckv[ˆg]−1 +� +det(ψi, ψj)ˆg +� +dˆgΨ d′ +ˆgb d′ +ˆgc e−Sm[ˆg,Ψ]−Sgh[ˆg,b,c]. +(2.151) +One can use (2.136) or (2.151) indifferently: the first is more appropriate when using spectral +analysis to compute the determinant explicitly, while the second is more natural in the +context of CFTs. +2.4.2 +Weyl ghost +Ghosts have been introduced for the reparametrizations (generated by ξa) and the traceless +part of the metric (the gauge field associated to the transformation): one may wonder why +there is not a ghost cw associated to the Weyl symmetry along with an antighost for the trace +of the metric (i.e. the conformal factor). This can be understood from several viewpoints. +First, the relation between a metric and its transformation – and the corresponding gauge +fixing condition – does not involve any derivative: as such, the Jacobian is trivial. Second, +one could choose +F ⊥ +ab = √ggab − +� +ˆgˆgab = 0 +(2.152) +as a gauge fixing condition instead of (2.63), and the trace component does not appear +anywhere. Finally, a local Weyl symmetry is not independent from the diffeomorphisms. +Remark 2.17 (Local Weyl symmetry) The topic of obtaining a local Weyl symmetry by +gauging a global Weyl symmetry (dilatation) is very interesting [86, chap. 15, 113]. Under +general conditions, one can express the new action in terms of the Ricci tensor (or of the +curvature): this means that the Weyl gauge field and its curvature are composite fields. +Moreover, one finds that local Weyl invariance leads to an off-shell condition while diffeo- +morphisms give on-shell conditions. This explains why one imposes only Virasoro constraints +(associated to reparametrizations) and no constraints for the Weyl symmetry in the covariant +quantization. +52 + +However, it can be useful to introduce a ghost field cw for the Weyl symmetry nonetheless. +In view of the previous discussion, this field should appear as a Lagrange multiplier which +ensures that bab is traceless. Starting from the action (2.145), one finds +S′ +gh[g, b, c, cw] = 1 +4π +� +d2σ√g gab� +bac∇bcc + bbc∇acc + 2babcw +� +, +(2.153) +where bab is not traceless anymore. The ghost cw is not dynamical since the action does not +contain derivatives of it, and it can be integrated out of the path integral to recover (2.145). +The equations of motion for this modified action are +∇acb + ∇bca + 2gabcw = 0, +∇abab = 0, +gabbab = 0. +(2.154) +Contracting the first equation with the metric gives +cw = −1 +2 ∇aca, +(2.155) +and thus cw is nothing else than the divergence of the ca field: the Weyl ghost is a composite +field (this makes connection with Remark 2.17) – see also (2.65c). The energy–momentum +tensor of the ghosts with action (2.153) is +T ′gh +ab = − +� +bac∇bcc + bbc∇acc + 2babcw +� +− ∇c(babcc) ++ 1 +2 gabgcd� +bce∇dce + bde∇cce + 2bcdcw +� +. +(2.156) +The trace of this tensor +gabT ′gh +ab = −gab∇c(babcc) +(2.157) +does not vanish off-shell, but it does on-shell since gabbab = 0. This implies that the theory +is Weyl invariant even if the action is not. It is interesting to contrast this with the trace +(2.148) when the Weyl ghost has been integrated out. +The equations of motion (2.146) and energy–momentum tensor (2.147) for the action +(2.145) can be easily derived by replacing cw by its solution in the previous formulas. +Computation – Equation (2.156) +The first parenthesis comes from varying gab, the second from the covariant derivatives, +the last from the √g. The second term comes from +gab� +bacδ∇bcc + bbcδ∇acc� += 2gabbacδ∇bcc = 2gabbacδΓc +bdcd += gabbacgce� +∇bδgde + ∇dδgbe − ∇eδgbd +� +cd += bab� +∇aδgbc + ∇cδgab − ∇bδgac +� +cc += bab∇cδgabcc, +where two terms have cancelled due to the symmetry of bab. Integrating by part gives +the term in the previous equation. +Note that the integration on the Weyl ghost yields a delta function +� +dgcw e−(cw,gabbab)g = δ +� +gabbab +� +. +(2.158) +53 + +2.4.3 +Zero-modes +The path integral (2.151) excludes the zero-modes of the ghosts. One can expect them to +be related to the determinants of elements of ker P1 and ker P † +1 with Grassmann coefficients. +They can be included after few simple manipulations (see also Appendix C.1.3). +It is simpler to first focus on the b ghost (to avoid the problems related to the CKV). +The path integral (2.151) can be rewritten as +Zg = +� +Mg +dMgt +Ωckv[ˆg]−1 +� +det(ψi, ψj)ˆg +� +dˆgΨ dˆgb d′ +ˆgc +Mg +� +i=1 +(b, ˆµi)ˆg e−Sm[ˆg,Ψ]−Sgh[ˆg,b,c]. +(2.159) +In this expression, c zero-modes are not integrated over, only the b zero-modes are. This is +the standard starting point on Riemann surfaces with genus g ≥ 1. The inner-product reads +explicitly +(b, ˆµi)ˆg = +� +d2σ +� +ˆg Gabcd +⊥ +babˆµi,cd = +� +d2σ +� +ˆg gacgbdbabˆµi,cd. +(2.160) +Computation – Equation (2.159) +Since the zero-modes of b are in the kernel of P † +1 , it means that the quadratic differentials +(2.76) also provide a suitable basis: +b = b0 + b′, +b0 = b0iφi, +where the b0i are Grassmann-odd coefficients. The first step is to find the Jacobian for +the changes of variables b → (b′, b0i): +1 = +� +dˆgb e−|b|2 +ˆg = J +� +dˆgb′ � +i +db0i e−|b′|2 +ˆg−|b0iφi|2 = J +� +det(φi, φj). +Next, (2.151) has no zero-modes, so one must insert Mg of them at arbitrary positions +σ0 +j to get a non-vanishing result when integrating over dMgb0i. The result of the integral +is: +� +dMgb0i +� +j +b0(σ0 +j ) = +� +dMgb0i +� +j +� +b0iφi(σ0 +j ) +� += det φi(σ0 +j ). +The only combination of the φi which does not vanish is the determinant due to the +anti-symmetry of the Grassmann numbers. Combining both results leads to: +dˆgb′ +� +det(φi, φj)ˆg += +dˆgb +det φi(σ0 +j ) +Mg +� +j=1 +b(σ0 +j ). +(2.161) +The locations positions σ0 +j are arbitrary (in particular, the RHS does not depend on +them since the LHS does not either). +Note that more details are provided in Ap- +pendix C.1.3. +An even simpler result can be obtained by combining the previous formula with the +factor det(φi, ˆµj)ˆg: +dˆgb′ +det(φi, ˆµj)ˆg +� +det(φi, φj)ˆg += dˆgb +Mg +� +j=1 +(b, ˆµj)ˆg. +(2.162) +54 + +This follows from +Mg +� +j=1 +b(σ0 +j ) = +Mg +� +j=1 +� +b0iφi(σ0 +j ) +� += det φi(σ0 +j ) +Mg +� +j=1 +b0i, +det(φi, ˆµj)ˆg +Mg +� +j=1 +b0i = +Mg +� +j=1 +� +b0i(φi, ˆµj)ˆg +� += +Mg +� +j=1 +(b0iφi, ˆµj)ˆg = +Mg +� +j=1 +(b, ˆµj)ˆg. +Note that the previous manipulations are slightly formal: the symmetric traceless fields +bab and φi,ab carry indices and there should be a product over the (two) independent +components. This is a trivial extension and would just make the notations heavier. +Similar manipulations lead to a new expression which includes also the c zero-mode (but +which is not very illuminating): +Zg = +� +Mg +dMgt Ωckv[ˆg]−1 +det ψi(σ0 +j ) +� +dˆgΨ dˆgb dˆgc +Kc +g +� +j=1 +ϵab +2 ca(σ0 +j )cb(σ0 +j ) +× +Mg +� +i=1 +(ˆµi, b)ˆg e−Sm[ˆg,Ψ]−Sgh[ˆg,b,c]. +(2.163) +The σ0a +j +are Kc +g = Kg/2 fixed positions and the integral does not depend on their values. +Note that only Kc +g positions are needed because the coordinate is 2-dimensional: fixing 3 +points with 2 components correctly gives 6 constraints. Then, ψi(σ0a +j ) is a 6-dimensional +matrix, with the rows indexed by i and the columns by the pair (a, j). +The expression cannot be simplified further because the CKV factor is infinite for g = 0. +This is connected to a fact mentioned previously: there is a remaining gauge symmetry +which is not taken into account +c −→ c + c0, +P1c0 = 0. +(2.164) +A proper account requires to gauge fix this symmetry: the simplest possibility is to insert +three or more vertex operators – this topic is discussed in Section 3.1. +Finally, note that the same question arises for the b-ghost since one has the symmetry +b −→ b + b0, +P † +1 b0 = 0. +(2.165) +That there is no problem in this case is related to the presence of the moduli. +2.5 +Normalization +In the previous sections, the closed string coupling constant gs did not appear in the ex- +pressions. Loops in vacuum amplitudes are generated by splitting of closed strings. By +inspecting the amplitudes, it seems that there are 2g such splittings (Figure 2.2), which +would lead to a factor g2g +s . However, this is not quite correct: this result holds for a 2-point +function. Gluing the two external legs to get a partition function (that is, taking the trace) +leads to an additional factor g−2 +s +(to be determined later), such that the overall factor is +g2g−2 +s +. The fact that it is the appropriate power of the coupling constant can be more easily +understood by considering n-point amplitudes (Section 3.1). The normalization of the path +integral can be completely fixed by unitarity [193]. +The above factor has a nice geometrical interpretation. Defining +Φ0 = ln gs +(2.166) +55 + +and remembering the expression (2.4) of the Euler characteristics χg = 2 − 2g, the coupling +factor can be rewritten as +g2g−2 +s += e−Φ0χg = exp +� +−Φ0 +4π +� +d2σ√gR +� += e−Φ0SEH[g], +(2.167) +where SEH is the Einstein–Hilbert action. +This action is topological in two dimensions. +Hence, the coupling constant can be inserted in the path integral simply by shifting the +action by the above term. +This shows that string theory on a flat target spacetime is +completely equivalent to matter minimally coupled to Einstein–Hilbert gravity with a cos- +mological constant (tuned to impose Weyl invariance at the quantum level). The advantage +of describing the coupling power in this fashion is that it directly generalizes to scattering +amplitudes and to open strings. The parameter Φ0 is interpreted as the expectation value of +the dilaton. Replacing it by a general field Φ(Xµ) is a generalization of the matter non-linear +sigma model, but this topic is beyond the scope of this book. +Figure 2.2: g-loop partition function. +2.6 +Summary +In this chapter, we started with a fairly general matter CFT – containing at least D scalar +fields Xµ – and explained under which condition it describes a string theory. The most +important consequence is that the matter 2d QFT must in fact be a 2d CFT. We then +continued by describing how to gauge fix the integration over the surfaces and we identified +the remaining degrees of freedom – the moduli space Mg – up to some residual redundancy +– the conformal Killing vector (CKV). Then, we showed how to rewrite the result in terms +of ghosts and proved that they are also a CFT. This means that a string theory can be com- +pletely described by two decoupled CFTs: a universal ghost CFT and a theory-dependent +matter CFT describing the string spacetime embedding and the internal structure. The +advantage is that one can forget the path integral formalism altogether and employ only +CFT techniques to perform the computations. This point of view will be developed for off- +shell amplitudes (Chapter 11) in order to provide an alternative description of how to build +amplitudes. It is particularly fruitful because one can also consider matter CFTs which do +not have a Lagrangian description. In the next chapter, we describe scattering amplitudes. +2.7 +Suggested readings +Numerous books have been published on the worldsheet string theory. +Useful (but not +required) complements to this chapter and subsequent ones are [151, 265] for introductory +texts and [24, 47, 48, 128, 193] for more advanced aspects. +• The definition of a field measure from a Gaussian integral and manipulations thereof +can be found in [100, sec. 15.1, 22.1, 172, chap. 14, 191, 53]. +56 + +• The most complete explanations of the gauge fixing procedure are [100, sec. 15.1, 22.1, +24, sec. 3.4, 6.2, 193, chap. 5, 48, 124, chap. 5]. The original derivation can be found +in [52, 161]. +• For the geometry of the moduli space, see [172, 173]. +• Ultralocality and its consequences are described in [53, 191] (see also [98, sec. 2.4]). +• The use of a Weyl ghost is shown in [240, sec. 8, 258, sec. 9.2]. +57 + +Chapter 3 +Worldsheet path integral: +scattering amplitudes +Abstract +In this chapter, we generalize the worldsheet path integral to compute scattering +amplitudes, which corresponds to insert vertex operators. The gauge fixing from the previous +chapter is generalized to this case. In particular, we discuss the 2-point amplitude on the +sphere. Finally, we introduce the BRST symmetry and motivate some properties of the +BRST quantization, which will be performed in details later. The formulas in this chapter +are all covariant: they will be rewritten in complex coordinates in the next chapter. +3.1 +Scattering amplitudes on moduli space +In this section, we describe the scattering of n strings. The momentum representation is +more natural for describing interactions, especially in string theory. Therefore, each string is +characterized by a state Vαi(ki) with momentum ki and some additional quantum numbers +αi (i = 1, . . . , n). We start from the worldsheet path integral (2.28) before gauge fixing: +Zg = +� +dggab +Ωgauge[g] Zm[g], +Zm[g] = +� +dgΨ e−Sm[g,Ψ]. +(3.1) +3.1.1 +Vertex operators and path integral +The external states are represented by infinite semi-tubes attached to the surfaces. Under a +conformal mapping, the tubes can be mapped to points called punctures on the worldsheet. +At g loops, the resulting space is a Riemann surface Σg,n of genus g with n punctures (or +marked points). The external states are represented by integrated vertex operators +Vα(ki) := +� +d2σ +� +g(σ) Vα(k; σ). +(3.2) +The vertex operators Vα(k; σ) are built from the matter CFT operators and from the world- +sheet metric gab. The functional dependence is omitted to not overload the notation, but +one should read Vα(k; σ) := Vα[g, Ψ](k; σ). The integration over the state positions is neces- +sary because the mapping of the tube to a point is arbitrary. Another viewpoint is that it +is needed to obtain an expression invariant under worldsheet diffeomorphisms. The vertex +operators described general states which not necessarily on-shell: this restriction will be +found later when discussing the BRST invariance of scattering amplitudes (Section 3.2.2). +58 + +Following Section 2.3.5, the Einstein–Hilbert action with boundary term +SEH[g] := 1 +4π +� +d2σ√g R + 1 +2π +� +ds k = χg,n. +(3.3) +is inserted in the path integral equals the Euler characteristics χg,n (the g in χg,n denotes +the genus). On a surface with punctures, the latter is shifted by the number of punctures +(which are equivalent to boundaries or disks) with respect to (2.4): +χg,n := χ(Σg,n) = 2 − 2g − n. +(3.4) +This gives the normalization factor: +g−χg,n +s += e−Φ0SEH[g], +Φ0 := ln gs. +(3.5) +The correctness factor can be verified by inspection of the Riemann surface for the scat- +tering of n string at g loops. In particular, the string coupling constant is by definition +the interaction strength for the scattering of 3 strings at tree-level. Moreover, the tree-level +2-point amplitude contains no interaction and should have no power of gs. This factor can +also be obtained by unitarity [193]. +By inserting these factors in (2.28), the g-loop n-point scattering amplitude is described +by: +Ag,n({ki}){αi} := +� +dggab +Ωgauge[g] dgΨ e−Sm[g,Ψ]−Φ0SEH[g] +n +� +i=1 +�� +d2σi +� +g(σi) Vαi(ki; σi) +� +. +(3.6) +The σi dependence of each √g will be omitted from now on since no confusion is possible. +The following equivalent notations will be used: +Ag,n({ki}){αi} := Ag,n(k1, . . . , kn)α1,...,αn := Ag,n +� +Vα1(k1), . . . , Vαn(kn) +� +. +(3.7) +The complete (perturbative) amplitude is found by summing over all genus: +An(k1, . . . , kn)α1,...,αn = +∞ +� +g=0 +Ag,n(k1, . . . , kn)α1,...,αn. +(3.8) +We omit a genus-dependent normalization which can be determined from unitarity [193]. +Sometimes, it is convenient to extract the factor e−Φ0χg,n of the amplitude Ag,n to display +explicitly the genus expansion, but we will not follow this convention here. Since each term of +the sum scales as Ag,n ∝ g2g+n−2 +s +, this expression clearly shows that worldsheet amplitudes +are perturbative by definition: this motivates the construction of a string field theory from +which the full non-perturbative S-matrix can theoretically be computed. +Finally, the amplitude (3.6) can be rewritten in terms of correlation functions of the +matter QFT integrated over worldsheet metrics: +Ag,n({ki}){αi} = +� +dggab +Ωgauge[g] e−Φ0SEH[g] +� +n +� +i=1 +d2σi +√g +� n +� +i=1 +Vαi(ki; σi) +� +m,g +. +(3.9) +The correlation function plays the same role as the partition function in (2.28). This shows +that string expressions are integrals of CFT expressions over the space of worldsheet metrics +(to be reduced to the moduli space). +We address a last question before performing the gauge fixing: what does (3.6) computes +exactly: on-shell or off-shell? Green functions or amplitudes? if amplitudes, the S-matrix +59 + +or just the interacting part T (amputated Green functions)? The first point is that a path +integral over connected worldsheets will compute connected processes. We will prove later, +when discussing the BRST quantization, that string states must be on-shell (Sections 3.2 +and 3.2.2) and that it corresponds to setting the Hamiltonian (2.26) to zero: +H = 0. +(3.10) +From this fact, it follows that (2.28) must compute amplitudes since non-amputated Green +functions diverge on-shell (due to external propagators). Finally, the question of whether it +computes the S-matrix S = 1 + iT, or just the interacting part T is subtler. At tree-level, +they agree for n ≥ 3, while T = 0 for n = 2 and S reduces to the identity. This difficulty +(discussed further in Section 3.1.2) is thus related to the question of gauge-fixing tree-level +2-point amplitude (Section 3.1.3). It has long been believed that (2.28) computes only the +interacting part (amputated Green functions), but it has been understood recently that this +is not correct and that (2.28) computes the S-matrix. +Remark 3.1 (Scattering amplitudes in QFT) Remember that the S-matrix is separ- +ated as: +S = 1 + iT, +(3.11) +where 1 denotes the contribution where all particles propagate without interaction. +The +connected components of S and T are denoted by Sc and T c. +The n-point (connected) +scattering amplitudes An for n ≥ 3 can be computed from the Green functions Gn through +the LSZ prescription (amputation of the external propagators): +An(k1, . . . , kn) = Gn(k1, . . . , kn) +n +� +i=1 +(k2 +i + m2 +i ). +(3.12) +The path integral computes the Green functions Gn; perturbatively, they are obtained from +the Feynman rules. They include a D-dimensional delta function +Gn(k1, . . . , kn) ∝ δ(D)(k1 + · · · + kn). +(3.13) +The 2-point amputated Green function T2 computed from the LSZ prescription vanishes +on-shell. For example, considering a scalar field at tree-level, one finds: +T2 = G2(k, k′) (k2 + m2)2 ∼ (k2 + m2) δ(D)(k + k′) −−−−−−→ +k2→−m2 0 +(3.14) +since +G2(k, k′) = δ(D)(k + k′) +k2 + m2 +. +(3.15) +Hence, T2 = 0 and the S-matrix (3.11) reduces to the identity component Sc +2 = 12 (which is +a connected process). There are several way to understand this result: +1. The recursive definition of the connected S-matrix Sc from the cluster decomposition +principle requires a non-vanishing 2-point amplitude [121, sec. 5.1.5, 251, sec. 4.3, 63, +sec. 6.1]. +2. The 2-point amplitude corresponds to the normalization of the 1-particle states (overlap +of a particle state with itself, which is non-trivial) [250, eq. 4.1.4, 239, chap. 5]. +3. A single particle in the far past propagating to the far future without interacting is a +connected and physical process [63, p. 133]. +4. It is required by the unitarity of the 2-point amplitude [66]. +60 + +These points indicate that the 2-point amplitude is proportional to the identity in the mo- +mentum representation [121, p. 212, 250, eq. 4.3.3 and 4.1.5] +A2(k, k′) = 2k0 (2π)D−1δ(D−1)(k − k′). +(3.16) +The absence of interactions implies that the spatial momentum does not change (the on- +shell condition implies that the energy is also conserved). This relation is consistent with +the commutation relation of the operators with the Lorentz invariant measure1 +[a(k), a†(k′)] = 2k0 (2π)D−1δ(D−1)(k − k′). +(3.17) +That this holds for all particles at all loops can be proven using the Källen–Lehman repres- +entation [121, p. 212]. +On the other hand, the identity part in (3.11) is absent for n ≥ 3 for connected amp- +litudes: Sc +n = T c +n for n ≥ 3. This shows that the Feynman rules and the LSZ prescription +compute only the interacting part T of the on-shell scattering amplitudes. The reason is that +the derivation of the LSZ formula assumes that the incoming and outgoing states have no +overlap, which is not the case for the 2-point function. A complete derivation of the S-matrix +from the path integral is more involved [121, sec. 5.1.5, 260, sec. 6.7, 81] (see also [37]). +The main idea is to consider a superposition of momentum states (here, in the holomorphic +representation [260, sec. 5.1, 6.4]) +φ(α) = +� +dD−1k α(k)∗a†(k). +(3.18) +They contribute a quadratic piece to the connected S-matrix and, setting them to delta func- +tions, one recovers the above result. +3.1.2 +Gauge fixing: general case +The Faddeev–Popov gauge fixing of the worldsheet diffeomorphisms and Weyl rescaling +(2.15) goes through also in this case if the integrated vertex operators are diffeomorphism +and Weyl invariant: +δξVαi(ki) = δξ +� +d2σ√g Vαi(ki; σ) = 0, +(3.19a) +δωVαi(ki) = δω +� +d2σ√g Vαi(ki; σ) = 0, +(3.19b) +with the variations defined in (2.7) and (2.11). Diffeomorphism invariance is straightforward +if the states are integrated worldsheet scalars. However, if the states are classically Weyl +invariant, they are not necessary so at the quantum level: vertex operators are composite +operators, which need to be renormalized to be well-defined at the quantum level. Renor- +malization introduces a scale which breaks Weyl invariance. Enforcing it to be a symmetry +of the vertex operators leads to constraints on the latter. We will not enter in the details +since it depends on the matter CFT and we will assume that the operators Vαi(ki) are indeed +Weyl invariant (see [193, sec. 3.6] for more details). In the rest of this book, we will use +CFT techniques developed in Chapter 6. The Einstein–Hilbert action is clearly invariant +under both symmetries since it is a topological quantity. +1If the modes are defined as ˜a(k) = a(k)/ +√ +2k0 such that [˜a(k), ˜a†(k′)] = (2π)D−1δ(D−1)(k − k′), then +one finds ˜ +A2(k, k′) = (2π)D−1δ(D−1)(k − k′). +61 + +Following the computations from Section 2.3 leads to a generalization of (2.136) with +the vertex operators inserted for the amplitude (3.6): +Ag,n({ki}){αi} = g−χg,n +s +� +Mg +dMgt det(φi, ˆµj)ˆg +� +det(φi, φj)ˆg +Ωckv[ˆg]−1 +� +det(ψi, ψj)ˆg +× +� +n +� +i=1 +d2σi +� +ˆg +� n +� +i=1 +ˆVαi(ki; σi) +� +m,ˆg +. +(3.20) +The hat on the vertex operators indicates that they are evaluated in the background metric +ˆg. +The next step is to introduce the ghosts: following Section 2.4, the generalization of +(2.159) is +Ag,n({ki}){αi} = g−χg,n +s +� +Mg +dMgt +Ωckv[ˆg]−1 +� +det(ψi, ψj)ˆg +� +dˆgb d′ +ˆgc +Mg +� +i=1 +(b, ˆµi)ˆg e−Sgh[ˆg,b,c] +× +� +n +� +i=1 +d2σi +� +ˆg +� n +� +i=1 +ˆVαi(ki; σi) +� +m,ˆg +. +(3.21) +For the moment, only the b ghosts come with zero-modes. +Then, c zero-modes can be +introduced in (3.21) +Ag,n = g−χg,n +s +� +Mg +dMgt Ωckv[ˆg]−1 +det ψi(σ0 +j ) +� +dˆgb dˆgc +Kc +g +� +j=1 +ϵab +2 ca(σ0 +j )cb(σ0 +j ) +Mg +� +i=1 +(ˆµi, b)ˆg e−Sgh[ˆg,b,c] +× +� +n +� +i=1 +d2σi +� +ˆg +� n +� +i=1 +ˆVαi(ki; σi) +� +m,ˆg +, +(3.22) +by following the same derivation as (2.163). The formulas (3.21) and (3.22) are the correct +starting point for all g and n. In particular, the c ghosts are not paired with any vertex +(a condition often assumed or presented as mandatory). This fact will help resolve some +difficulties for the 2-point function on the sphere. +Remember that there is no CKV and no c zero-mode for g ≥ 2. For the sphere g = 0 +and the torus g = 1, there are CKVs, indicating that there is a residual symmetry in (3.21) +and (3.22), which is the global conformal group of the worldsheet. It can be gauge fixed by +imposing conditions on the vertex operators.2 The simplest gauge fixing condition amounts +to fix the positions of Kc +g vertex operators through the Faddeev–Popov trick: +1 = ∆(σ0 +j ) +� +dξ +Kc +g +� +j=1 +δ(2)(σj − σ0(ξ) +j +), +σ0(ξ) +j += σ0 +j + δξσ0 +j , +δξσ0 +j = ξ(σ0 +j ), +(3.23) +where ξ is a conformal Killing vector, and the variation of σ was given in (2.7). We find +that +∆(σ0 +j ) = det ψi(σ0 +j ). +(3.24) +A priori, the positions σ0 +j are not the same as the one appearing in (2.163) (since both sets +are arbitrary): however, considering the same positions allows to cancel the factor (3.24) +with the same one in (2.163). +2In fact, it is only important to gauge fix for the sphere because the volume of the group is infinite. On +the other hand, the volume of the CKV group for the torus is finite-dimensional such that dividing by Ωckv +is not ambiguous. +62 + +Computation – Equation (3.24) +The first step is to compute ∆ in (3.23). For this, we decompose the CKV ξ on the +basis (2.104) +ξ(σ0 +j ) = αiψi(σ0 +j ) +and write the Gaussian integral: +1 = +� +Kc +g +� +j=1 +d2δσj e +−� +j(δσj,δσj) = ∆ +� +Kg +� +j=1 +dαi e +−� +j,i,i′(αiψi(σj),αi′ψi′(σj)) += ∆ +� +det ψi(σj) +�−1. +Again, we have reduced rigour in order to simplify the manipulations. +After inserting the identity (3.23) into (3.22), one can integrate over Kc +g vertex operator +positions to remove the delta functions – at the condition that there are at least Kc +g operators. +As a consequence, we learn that the proposed gauge fixing works only for n ≥ 1 if g = 1 or +n ≥ 3 if g = 0. This condition is equivalent to +χg,n = 2 − 2g − n < 0. +(3.25) +In this case, the factors det ψi(σ0 +j ) cancel and (3.21) becomes +Ag,n({ki}){αi} = g−χg,n +s +� +Mg +dMgt +� +dˆgb dˆgc +Kc +g +� +j=1 +ϵab +2 ca(σ0 +j )cb(σ0 +j ) +Mg +� +i=1 +(ˆµi, b)ˆg e−Sgh[ˆg,b,c] +× +� +n +� +i=Kc +g+1 +d2σi +� +ˆg +� Kc +g +� +j=1 +ˆVαj(kj; σ0 +j ) +n +� +i=Kcg+1 +ˆVαi(ki; σi) +� +m,ˆg +. +(3.26) +The result may be divided by a symmetry factor if the delta functions have solutions for +several points [193, sec. 5.3]. Performing the gauge fixing for the other cases (in particular, +g = 0, n = 2 and g = 1, n = 0) is more subtle (Section 3.1.3 and [193]). +The amplitude can be rewritten in two different ways. First, the ghost insertions can be +rewritten in terms of a ghost correlation functions +Ag,n({ki}){αi} = g−χg,n +s +� +Mg +dMgt +� +n +� +i=Kc +g+1 +d2σi +� +ˆg +� Kc +g +� +j=1 +ϵab +2 ca(σ0 +j )cb(σ0 +j ) +Mg +� +i=1 +(ˆµi, b)ˆg +� +gh,ˆg +× +� Kc +g +� +j=1 +ˆVαj(kj; σ0 +j ) +n +� +i=Kc +g+1 +ˆVαi(ki; σi) +� +m,ˆg +. +(3.27) +This form is particularly interesting because it shows that, before integration over the mod- +uli, the amplitudes factorize. This is one of the main advantage of the conformal gauge, since +the original complicated amplitude (3.6) for a QFT on a dynamical spacetime reduces to +the product of two correlation functions of QFTs on a fixed curved background. In fact, the +situation is even simpler when taking a flat background ˆg = δ since both the ghost and mat- +ter sectors are CFTs and one can employ all the tools from two-dimensional CFT (Part I) to +perform the computations and mostly forget about the path integral origin of these formulas. +This approach is particularly fruitful for off-shell (Chapter 11) and superstring amplitudes +(Chapter 17). +63 + +Remark 3.2 (Amplitudes in 2d gravity) The derivation of amplitudes for 2d gravity +follows the same procedure, up to two differences: 1) there is an additional decoupled (before +moduli and position integrations) gravitational sector described by the Liouville field, 2) the +matter and gravitational action are not CFTs if the original matter was not. +A second formula can be obtained by bringing the c-ghost on top of the matter vertex +operators which are at the same positions +Ag,n({ki}){αi} = g−χg,n +s +� +Mg +dMgt +� +n +� +i=Kc +g+1 +d2σi +� +ˆg +� Mg +� +i=1 +ˆBi +Kc +g +� +j=1 +ˆ +Vαj(kj; σ0 +j ) +n +� +i=Kc +g+1 +ˆVαi(ki; σi) +� +ˆg +, +(3.28) +and where +ˆ +Vαj(kj; σ0 +j ) := ϵab +2 ca(σ0 +j )cb(σ0 +j ) ˆVαj(kj; σ0 +j ), +ˆBi := (ˆµi, b)ˆg. +(3.29) +The operators Vαi(ki; σ0 +j ) (a priori off-shell) are called unintegrated operators, by opposition +to the integrated operators Vαi(ki). We will see that both are natural elements of the BRST +cohomology. +To stress that the ˆBi insertions are really an element of the measure, it is finally possible +to rewrite the previous expression as +Ag,n({ki}){αi} = g−χg,n +s +� +Mg×Cn−Kcg +� Mg +� +i=1 +ˆBi dti +Kc +g +� +j=1 +ˆ +Vαi(ki; σ0 +j ) +n +� +i=Kc +g+1 +ˆVαi(ki; σi) d2σi +� +ˆg +� +ˆg +. +(3.30) +The result (3.28) suggests a last possibility for improving the expression of the amplitude. +Indeed, the different vertex operators don’t appear symmetrically: some are integrated over +and other come with c ghosts. Similarly, the two types of integrals have different roles: the +moduli are related to geometry while the positions look like external data (vertex operators). +However, punctures can obviously be interpreted as part of the geometry, and one may +wonder if it is possible to unify the moduli and positions integrals. It is, in fact, possible to +put all vertex operators and integrals on the same footing by considering the amplitude to +be defined on the moduli space Mg,n of genus-g Riemann surfaces with n punctures instead +of just Mg [193] (see also Section 11.3.1). +3.1.3 +Gauge fixing: 2-point amplitude +As discussed at the end of Section 3.1.1, it has long been believed that the tree-level 2-point +amplitude vanishes. There were two main arguments: there are not sufficiently many vertex +operators 1) to fix completely the SL(2, C) invariance or 2) to saturate the number of c- +ghost zero-modes. Let’s review both points and then explain why they are incorrect. We +will provide the simplest arguments, referring the reader to the literature [66, 208] for more +general approaches. +For simplicity, we consider the flat metric ˆg = δ and an orthonormal basis of CKV. The +two weight-(1, 1) matter vertex operators are denoted as Vk(z, ¯z) and Vk′(z′, ¯z′) such that +the 2-point correlation function on the sphere reads (see Chapters 6 and 7 for more details): +⟨Vk(z, ¯z)Vk′(z′, ¯z′)⟩S2 = i (2π)Dδ(D)(k + k′) +|z − z′|4 +. +(3.31) +The numerator comes from the zero-modes ei(k+k′)·x for a target spacetime with a Lorentzian +signature [48, p. 866, 193] (required to make use of the on-shell condition). +64 + +Review of the problem +The tree-level amplitude (3.20) for n = 2 reads: +A0,2(k, k′) = +CS2 +Vol K0,0 +� +d2zd2z′ ⟨Vk(z, ¯z)Vk′(z′, ¯z′)⟩S2 , +(3.32) +where K0,n is the CKV group of Σ0,n, the sphere with n punctures. In particular, the group +of the sphere without puncture is K0,0 = PSL(2, C). The normalization of the amplitude is +CS2 = 8πα′−1 for gs = 1 [193, 249]. Since there are two insertions, the symmetry can be +partially gauge fixed by fixing the positions of the two punctures to z = 0 and z′ = ∞. In +this case, the amplitude (3.32) becomes: +A0,2(k, k′) = +CS2 +Vol K0,2 +⟨Vk(∞, ∞)Vk′(0, 0)⟩S2 , +(3.33) +where K0,2 = R∗ ++×U(1) is the CKV group of the 2-punctured sphere – containing dilatations +and rotations.3 Since the volume of this group is infinite Vol K0,2 = ∞, it looks like A0,2 = 0. +However, this forgets that the 2-point correlation function (3.31) contains a D-dimensional +delta function. +The on-shell condition implies that the conservation of the momentum +k + k′ = 0 is automatic for one component, such that the numerator in (3.33) contains a +divergent factor δ(0): +A0,2(k, k′) = (2π)D−1δ(D−1)(k + k′) CS2 2πi δ(0) +Vol K0,2 +. +(3.34) +Hence, (3.33) is of the form A0,2 = ∞/∞ and one should be careful when evaluating it. +The second argument relies on a loophole in the understanding of the gauge fixed amp- +litude (3.28). The result (3.28) is often summarized by saying that one can go from (3.20) +to (3.28) by replacing Kc +g integrated vertices +� +V by unintegrated vertices c¯cV in order to +saturate the ghost zero-modes and to obtain a non-zero result. For g = 0, this requires 3 +unintegrated vertices. But, since there are only two operators in (3.32), this is impossible +and the result must be zero. However, this is also incorrect because it is always possible +to insert 6 c zero-modes, as show the formulas (2.163) and (3.27). Indeed, they are part +of how the path integral measure is defined and do not care of the matter operators. The +question is whether they can be attached to vertex operators (for aesthetic reasons or more +pragmatically to get natural states of the BRST cohomology). To find the correct result +with ghosts requires to start with (3.27) and to see how this can be simplified when there +are only two operators. +Computation of the amplitude +In this section, we compute the 2-point amplitude from (3.33): +A0,2(k, k′) = +CS2 +Vol K0,2 +⟨Vk(∞, ∞)Vk′(0, 0)⟩S2 . +(3.35) +The volume of K0,2 reads (by writing a measure invariant under rotations and dilatations, +but not translations nor special conformal transformations) [53, 60]: +Vol K0,2 = +� d2z +|z|2 = 2 +� 2π +0 +dσ +� ∞ +0 +dr +r , +(3.36) +3The subgroup and the associated measure depend on the locations of the two punctures. +65 + +by doing the change of variables z = reiσ. Since the volume is infinite, it must be regu- +larized. A first possibility is to cut-off a small circle of radius ϵ around r = 0 and r = ∞ +(corresponding to removing the two punctures at z = 0, ∞). A second possibility consists +in performing the change of variables r = eτ and to add an imaginary exponential: +Vol K0,2 = 4π +� ∞ +0 +dr +r = 4π +� ∞ +−∞ +dτ = 4π lim +ε→0 +� ∞ +−∞ +dτ eiετ = 4π × 2π lim +ε→0 δ(ε), +(3.37) +such that the regularized volume reads +Volε K0,2 = 8π2 δ(ε). +(3.38) +In fact, τ can be interpreted as the Euclidean worldsheet time on the cylinder since r +corresponds to the radial direction of the complex plane. +Since the worldsheet is an embedding into the target spacetime, both must have the +same signature. As a consequence, for the worldsheet to be also Lorentzian, the formula +(3.37) must be analytically continued as ε = −iE and τ = it such that +VolM,E K0,2 = 8π2i δ(E), +(3.39) +where the subscript M reminds that one considers the Lorentzian signature. Inserting this +expression in (3.34) and taking the limit E → 0, it looks like the two δ(0) will cancel. +However, we need to be careful about the dimensions. Indeed, the worldsheet time τ and +energy E are dimensionless, while the spacetime time and energy are not. Thus, it is not +quite correct to cancel directly both δ(0) since they don’t have the same dimensions. In order +to find the correct relation between the integrals in (3.37) and of the zero-mode in (3.31), +we can look at the mode expansion for the scalar field (removing the useless oscillators): +X0(z, ¯z) = x0 + i +2 α′k0 ln |z|2 = x0 + iα′k0τ, +(3.40) +where the second equality follows by setting z = eτ. After analytic continuation k0 = −ik0 +M, +X0 = iX0 +M, x0 = ix0 +M and τ = it, we find [265, p. 186]: +X0 +M = x0 +M + α′k0 +Mt. +(3.41) +This indicates that the measure of the worldsheet time in (3.39) must be rescaled by 1/α′k0 +M +such that: +VolM K0,2 −→ 8π2i δ(0) +α′k0 +M += CS2 2πi δ(0) +2k0 +M +. +(3.42) +This is equivalent to rescale E by α′k0 and to use δ(ax) = a−1δ(x). +Ultimately, the 2-point amplitude becomes (removing the subscript on k0): +A0,2(k, k′) = 2k0(2π)D−1δ(D−1)(k + k′) +(3.43) +and matches the QFT formula (3.16). We see that taking into account the scale of the +coordinates is important to reproduce this result. +The computation displayed here presents some ambiguities because of the regularization. +However, this ambiguity can be fixed from unitarity of the scattering amplitudes. A more +general version of the Faddeev–Popov gauge fixing has been introduced in [66] to avoid +dealing altogether with infinities. +It is an interesting question whether these techniques +can be extended to the compute the tree-level 1- and 0-point amplitudes on the sphere. In +most cases, the 1-point amplitude is expected to vanish since 1-point correlation functions +66 + +of primary operators other than the identity vanish in unitary CFTs.4 The 0-point function +corresponds to the sphere partition function: the saddle point approximation to leading +order allows to relate it to the spacetime action evaluated on the classical solution φ0, +Z0 ∼ e−S[φ0]/ℏ. Since the normalization is not known and because S[φ0] is expected to +be infinite, only comparison between two spacetimes should be meaningful (à la Gibbons– +Hawking–York [190, sec. 4.1]). In particular, for Minkowski spacetime we find naively +Z0 ∼ δ(D)(0) +Vol K0 +, +(3.44) +which is not well-defined. This question has no yet been investigated. +Expression with ghosts +There are different ways to rewrite the 2-point amplitude in terms of ghosts. In all cases, one +correctly finds the 6 insertions necessary to get a non-vanishing result since, by definition, +it is always possible to rewrite the Faddeev–Popov determinant in terms of ghosts. A first +approach is to insert 1 = +� +d2z δ(2)(z) inside (3.32) to mimic the presence of a third operator. +This is equivalent to use the identity +⟨0| c−1¯c−1c0¯c0c1¯c1 |0⟩ = 1 +(3.45) +inside (3.33), leading to: +A0,2(k, k′) = +CS2 +Vol K0,2 +⟨Vk(∞, ∞)c0¯c0 Vk′(0, 0)⟩S2 , +(3.46) +where Vk(z, ¯z) = c¯cVk(z, ¯z). This shows that (3.16) can also be recovered using the correct +insertions of ghosts. The presence of c0¯c0 can be expected from string field theory since they +appear in the kinetic term (10.115). +The disadvantage of this formula is to still contain the infinite volume of the dilatation +group. It is also possible to introduce ghosts for the more general gauge fixing presented +in [66]. An alternative approach has been proposed in [208]. +3.2 +BRST quantization +The symmetries of a Lagrangian dictate the possible terms which can be considered. This +continues to hold at the quantum level and the counter-terms introduced by renormalization +are constrained by the symmetries. However, if the path integral is gauge fixed, the original +symmetry is no more available for this purpose. Fortunately, one can show that there is +a global symmetry (with anticommuting parameters) remnant of the local symmetry: the +BRST symmetry. It ensures consistency of the quantum theory. It also provides a direct +access to the physical spectrum. +The goal of this section is to provide a general idea of the BRST quantization for the +worldsheet path integral. A more detailed CFT analysis and the consequence for string +theory are given in Chapter 8. The reader is assumed to have some familiarity with the +BRST quantization in field theory – a summary is given in Appendix C.2. +4The integral over the zero-mode gives a factor δ(D)(k) which implies k = 0. +At zero momentum, +the time scalar X0 is effectively described by unitary CFT. However, there can be some subtleties when +considering marginal operator. +67 + +3.2.1 +BRST symmetry +The partition function (2.159) is not the most suitable to display the BRST symmetry. +The first step is to restore the dependence in the original metric gab by introducing a delta +function +Zg = +� +Mg +dMgt +Ωckv[g] +� +dggab dgΨ dgb d′ +gc δ +�√ggab − +� +ˆgˆgab +� Mg +� +i=1 +(φi, b)g e−Sm[g,Ψ]−Sgh[g,b,c]. +(3.47) +Note that it is necessary to use the traceless gauge fixing condition (2.152) as it will become +clear. The delta function is Fourier transformed in an exponential thanks to an auxiliary +bosonic field: +Zg = +� +Mg +dMgt +Ωckv[g] +� +dggab dgBab dgΨ dgb d′ +gc +Mg +� +i=1 +(φi, b)g e−Sm[g,Ψ]−Sgf[g,ˆg,B]−Sgh[g,b,c] +(3.48) +where the gauge-fixing action reads: +Sgf[g, ˆg, B] = − i +4π +� +d2σ Bab�√ggab − +� +ˆgˆgab +� +. +(3.49) +Varying the action with respect to the auxiliary field Bab, called the Nakanish–Lautrup field, +produces the gauge-fixing condition. +The BRST transformations are +δϵgab = iϵ Lcgab, +δϵΨ = iϵ LcΨ, +δϵca = iϵ Lcca, +δϵbab = ϵ Bab, +δϵBab = 0, +(3.50) +where ϵ is a Grassmann parameter (anticommuting number) independent of the position. +If the traceless gauge fixing (2.152) is not used, then Bab is not traceless: in that case, the +variation δϵbab will generate a trace, which is not consistent. Since the transformations act +on the matter action Sm as a diffeomorphism with vector ϵca, it is obvious that it is invariant +by itself. It is easy to show that the transformations (3.50) leave the total action invariant +in (3.48). The invariance of the measure is given in [193]. +Remark 3.3 (BRST transformations with Weyl ghost) One can also consider the ac- +tion (2.153) with the Weyl ghost. +In this case, the transformation law of the metric is +modified and the Weyl ghost transforms as a scalar: +δϵgab = iϵ Lcgab + iϵ gabcw, +δϵcw = iϵ Lccw. +(3.51) +The second term in δϵgab is a Weyl transformation with parameter ϵcw. Moreover, bab and +Bab are not symmetric traceless. +The equation of motion for the auxiliary field is +Bab = i Tab := i +� +T m +ab + T gh +ab +� +, +(3.52) +where the RHS is the total energy–momentum tensor (matter plus ghosts). Integrating it +out imposes the gauge condition gab = ˆgab and yields the modified BRST transformations +δϵΨ = iϵ LcΨ, +δϵca = iϵ Lcca, +δϵbab = iϵ Tab. +(3.53) +Without starting with the path integral (3.48) with auxiliary field, it would have been +difficult to guess the transformation of the b ghost. Since ca is a vector, one can also write +δϵca = ϵ cb∂bca. +(3.54) +68 + +Associated to this symmetry is the BRST current ja +B and the associated conserved BRST +charge QB +QB = +� +dσ j0 +B. +(3.55) +The charge is nilpotent +Q2 +B = 0, +(3.56) +and, through the presence of the c-ghost in the BRST transformation, the BRST charge has +ghost number one +Ngh(QB) = 1. +(3.57) +Variations of the matter fields can be written as +δϵΨ = i [ϵQB, Ψ]±. +(3.58) +Note that the energy–momentum tensor is BRST exact +Tab = [QB, bab]. +(3.59) +3.2.2 +BRST cohomology and physical states +Physical state |ψ⟩ are elements of the absolute cohomology of the BRST operator: +|ψ⟩ ∈ H(QB) := ker QB +Im QB +, +(3.60) +or, more explicitly, closed but non-exact states: +QB |ψ⟩ = 0, +∄ |χ⟩ : |ψ⟩ = QB |χ⟩ . +(3.61) +The adjective “absolute” is used to distinguish it from two other cohomologies (relative +and semi-relative) defined below. Two states of the cohomology differing by an exact state +represent identical physical states: +|ψ⟩ ∼ |ψ⟩ + QB |Λ⟩ . +(3.62) +This equivalence relation, translated in terms of spacetime fields, correspond to spacetime +gauge transformations. In particular, it contains the (linearized) reparametrization invari- +ance of the spacetime metric in the closed string sector, and, for the open string sector, it +contains Yang–Mills symmetries. We will find that it corresponds to the gauge invariance +of free string field theory (Chapter 10). +However, physical states satisfy two additional constraints (remember that bab is traceless +symmetric): +� +dσ bab |ψ⟩ = 0. +(3.63) +These conditions are central to string (field) theory, so they will appear regularly in this +book. For this reason, it is useful to provide first some general motivations, and to refine +the analysis later since the CFT language will be more appropriate. Moreover, these two +conditions will naturally emerge in string field theory. +In order to introduce some additional terminology, let’s define the following quantities:5 +b+ := +� +dσ b00, +b− := +� +dσ b01. +(3.64) +5The objects b± are zero-modes of the b ghost fields. They correspond (up to a possible irrelevant factor) +to the modes b± +0 in the CFT formulation of the ghost system (7.132). +69 + +The semi-relative and relative cohomologies H−(QB) and H0(QB) are defined as6 +H−(QB) = H(QB) ∩ ker b−, +H0(QB) = H−(QB) ∩ ker b+. +(3.65) +The first constraint arises as a consequence of the topology of the closed string worldsheet: +the spatial direction is a circle, which implies that the theory must be invariant under +translations along the σ direction (the circle is invariant under rotation). However, choosing +a parametrization implies to fix an origin for the spatial direction: this is equivalent to +a gauge fixing condition. +As usual, this implies that the corresponding generator Pσ of +worldsheet spatial translations (2.26) must annihilate the states: +Pσ |ψ⟩ = 0. +(3.66) +This is called the level-matching condition. Using (3.59), this can be rewritten as +Pσ |ψ⟩ = +� +dσ T01 |ψ⟩ = +� +dσ {QB, b01} |ψ⟩ = QB +� +dσ b01 |ψ⟩ , +(3.67) +since QB |ψ⟩ = 0 for a state |ψ⟩ in the cohomology. +The simplest way to enforce this +condition is to set the state on which QB acts to zero:7 +b− |ψ⟩ = 0, +(3.68) +which is equivalent to one of the conditions in (3.63). +The second condition does not follow as simply. The Hilbert space can be decomposed +according to b+ as +H− := H↓ ⊕ H↑, +H↓ := H0 := H− ∩ ker b+. +(3.69) +Indeed, b+ is a Grassmann variable and generates a 2-state system. In the ghost sector, the +two Hilbert spaces are generated from the ghost vacua | ↓⟩ and | ↑⟩ obeying +b+ | ↓⟩ = 0, +b+ | ↑⟩ = | ↓⟩ . +(3.70) +The action of the BRST charge on states |ψ↓⟩ ∈ H↓ and |ψ↑⟩ ∈ H↑ follow from these relations +and from the commutation relation (3.59): +QB |ψ↓⟩ = H |ψ↑⟩ , +QB |ψ↑⟩ = 0, +(3.71) +where H is the worldsheet Hamiltonian defined in (2.26). To prove this relation, start first +with H |ψ↑⟩, then use (3.59)) to get the LHS of the first condition; then apply QB to get +the second condition (using that QB commutes with H, and b+ with any other operators +building the states). For H ̸= 0, the state |ψ↓⟩ is not in the cohomology and |ψ↑⟩ is exact. +Thus, the exact and closed states are +Im QB = +� +|ψ↑⟩ ∈ H↑ | H |ψ↑⟩ ̸= 0 +� +, +(3.72a) +ker QB = +� +|ψ↑⟩ ∈ H↑ +� +∪ +� +|ψ↓⟩ ∈ H↓ | H |ψ↓⟩ = 0 +� +. +(3.72b) +This implies that eigenstates of H in the cohomology satisfy the on-shell condition: +H |ψ⟩ = 0. +(3.73) +6The BRST cohomologies described in this section are slightly different from the ones used in the rest +of this book. +To distinguish them, indices are written as superscripts in this section, and as subscripts +otherwise. +7The reverse is not true. We will see in Section 3.2.2 the relation between the two conditions in more +details. +70 + +This is consistent with the fact that scattering amplitudes involve on-shell states. In this +case, |ψ↑⟩ is not exact and is thus a member of the cohomology H(QB), as well as |ψ↓⟩ since +it becomes close. But, the Hilbert space H↑ must be rejected for two reasons: there would +be an apparent doubling of states and scattering amplitudes would behave badly. The first +problem arises because one can show that the cohomological subspaces of each space are +isomorphic: H↓(QB) ≃ H↑(QB). Hence, keeping both subspaces would lead to a doubling +of the physical states. For the second problem, consider an amplitude where one of the +external state is built from |ψ↑⟩: the amplitude vanishes if the states are off-shell since the +state |ψ↑⟩ is exact, but it does not vanish on-shell [193, ch. 4]. This means that it must +be proportional to δ(H). But, general properties in QFT forbid such dependence in the +amplitude (only poles and cuts are allowed, except if D = 2). Projecting out the states in +H↑ is equivalent to require +b+ |ψ⟩ = 0 +(3.74) +for physical states, which is the second condition in (3.63). +In fact, this condition can be obtained very similarly as the b− = 0 condition: using the +expression of H (2.26) and the commutation relation (3.59), (3.73) is equivalent to +QB +� +dσ b00 |ψ⟩ = 0. +(3.75) +Hence, imposing (3.74) allows to automatically ensure that (3.73) holds. +Since the on-shell character (3.73) of the BRST states and of the BRST symmetry are +intimately related to the construction of the worldsheet integral, one can expect difficulty +for going off-shell. +3.3 +Summary +In this chapter, we derived general formulas for string scattering amplitudes. The general +BRST formalism has been summarized. Moreover, we gave general motivations for restrict- +ing the absolute cohomology to the smaller relative cohomology. +In Chapter 8, a more +precise derivation of the BRST cohomology is worked out. It includes also a proof of the +no-ghost theorem: the ghosts and the negative norm states (in Minkowski signature) are +unphysical particles and should not be part of the physical states. This theorem asserts that +it is indeed the case. It will also be the occasion to recover the details of the spectrum in +various cases. +3.4 +Suggested readings +• The delta function approach to the gauge fixing is described in [193, sec. 3.3, 151, +sec. 15.3.2], with a more direct computation is in [128]. +• The most complete references for scattering amplitudes in the path integral formalism +are [53, 193]. +• Computation of the tree-level 2-point amplitude [66, 208] (for discussions of 2-point +function, see [53, p. 936–7, 207, 60, 61, 48, p. 863–4]). +• The BRST quantization of string theory is discussed in [155, 39, 193, chap. 4]. For +a general discussion see [105, 247, 251]. The use of an auxiliary field is considered +in [252, sec. 3.2]. +71 + +Chapter 4 +Worldsheet path integral: +complex coordinates +Abstract +In the two previous chapters, the amplitudes computed from the worldsheet +path integrals have been written covariantly for a generic curved background metric. In +this chapter, we start to use complex coordinates and finally take the background metric to +be flat. This is the usual starting point for computing amplitudes since it allows to make +contact with CFTs and to employ tools from complex analysis. We first recall few facts on +2d complex manifolds before briefly describing how to rewrite the scattering amplitudes in +complex coordinates. +4.1 +Geometry of complex manifolds +Choosing a flat background metric simplifies the computations. However, we have seen in +Section 2.3 that there is a topological obstruction to get a globally flat metric. The solution +is to work with coordinate patches (σ0, σ1) = (τ, σ) such that the background metric ˆgab is +flat in each patch (conformally flat gauge): +ds2 = gabdσadσb = e2φ(τ,σ)� +dτ 2 + dσ2� +, +(4.1) +or +gab = e2φδab, +ˆgab = δab. +(4.2) +To simplify the notations, we remove the dependence in the flat metric and the hat for +quantities (like the vertex operators) expressed in the background metric when no confusion +is possible. +Introducing complex coordinates +z = τ + iσ, +¯z = τ − iσ, +(4.3a) +τ = z + ¯z +2 +, +σ = z − ¯z +2i +, +(4.3b) +the metric reads1 +ds2 = 2gz¯zdzd¯z = e2φ(z,¯z)|dz|2. +(4.4) +1In Section 6.1, we provide more details on the relation between the worldsheet (viewed as a cylinder or +a sphere) and the complex plane. +72 + +The metric and its inverse can also be written in components: +gz¯z = e2φ +2 , +gzz = g¯z¯z = 0, +(4.5a) +gz¯z = 2e−2φ, +gzz = g¯z¯z = 0. +(4.5b) +Equivalently, the non-zero components of the background metric are +ˆgz¯z = 1 +2, +ˆgz¯z = 2. +(4.6) +An oriented two-dimensional manifold is a complex manifold: this means that there exists +a complex structure, such that the transition functions and changes of coordinates between +different patches are holomorphic at the intersection of the two patches: +w = w(z), +¯w = ¯w(¯z). +(4.7) +For such a transformation, the Liouville mode transforms as +e2φ(z,¯z) = +���� +∂w +∂z +���� +2 +e2φ(w, ¯ +w) +(4.8) +such that +ds2 = e2φ(w, ¯ +w)|dw|2. +(4.9) +This shows also that a conformal structure (2.12) induces a complex structure since the +transformation law of φ is equivalent to a Weyl rescaling. +The integration measures are related as +d2σ := dτdσ = 1 +2 d2z, +d2z := dzd¯z. +(4.10) +Due to the factor of 2 in the expression, the delta function δ(2)(z) also gets a factor of 2 +with respect to δ(2)(σ) +δ(2)(z) = 1 +2 δ(2)(σ). +(4.11) +Then, one can check that +� +d2z δ(2)(z) = +� +d2σ δ(2)(σ) = 1. +(4.12) +The basis vectors (derivatives) and one-forms can be found using the chain rule: +∂z = 1 +2 (∂τ − i∂σ), +∂¯z = 1 +2 (∂τ + i∂σ), +(4.13a) +dz = dτ + idσ, +d¯z = dτ − idσ. +(4.13b) +The Levi–Civita (completely antisymmetric) tensor is normalized by +ϵ01 = ϵ01 = 1. +(4.14a) +ϵz¯z = i +2, +ϵz¯z = −2i, +(4.14b) +remembering that it transforms as a density. Integer indices run over local frame coordinates. +The different tensors can be found from the tensor transformation law. For example, the +components of a vector V a in both systems are related by +V z = V 0 + iV 1, +V ¯z = V 0 − iV 1 +(4.15) +73 + +such that +V = V 0∂0 + V 1∂1 = V z∂z + V ¯z∂¯z. +(4.16) +For holomorphic coordinate transformations (4.7), the components of the vector do not mix: +V w = ∂w +∂z V z, +V ¯ +w = ∂ ¯w +∂¯z V ¯z. +(4.17) +This implies that the tangent space of the Riemann surface is decomposed into holomorphic +and anti-holomorphic vectors:2 +TΣg ≃ TΣ+ +g ⊕ TΣ− +g , +(4.18a) +V z∂z ∈ TΣ+ +g , +V ¯z∂¯z ∈ TΣ− +g , +(4.18b) +as a consequence of the existence of a complex structure. Similarly, the components of a +1-form ω – which is the only non-trivial form on Σg – can be written in terms of the real +coordinates as: +ωz = 1 +2 (ω0 − iω1), +ω¯z = 1 +2 (ω0 + iω1) +(4.19) +such that +ω = ω0dσ0 + ω1dσ1 = ωzdz + ω¯zd¯z. +(4.20) +Hence, a 1-form is decomposed into complex (1, 0)- and (0, 1)-forms: +T ∗Σg ≃ Ω1,0(Σg) ⊕ Ω0,1(Σg), +(4.21a) +ωzdz ∈ Ω1,0(Σg), +ω¯zd¯z ∈ Ω0,1(Σg), +(4.21b) +since both components will not mixed under holomorphic changes of coordinates (4.7). Fi- +nally, the metric provides an isomorphism between TΣ+ +g and Ω0,1(Σg), and between TΣ− +g +and Ω1,0(Σg), since it can be used to lower/raise an index while converting it from holo- +morphic to anti-holomorphic, or conversely: +Vz = gz¯zV ¯z, +V¯z = gz¯zV z. +(4.22) +This can be generalized further by considering components with more indices: all anti- +holomorphic indices can be converted to holomorphic indices thanks to the metric: +T +q++p− +���� +z···z +z···z +���� +p++q− += (gz¯z)p−(gz¯z)q−T +q+ +���� +z···z +q− +���� +¯z···¯z +z···z +���� +p+ +¯z···¯z +���� +p− +. +(4.23) +Hence, it is sufficient to study (p, q)-tensors with p upper and q lower holomorphic indices. +In this case, the transformation rule under (4.7) reads +T +q +���� +w···w +w···w +���� +p += +�∂w +∂z +�n +T +q +���� +z···z +z···z +���� +p +, +n := q − p. +(4.24) +The number n ∈ Z is called the helicity or rank.3 The set of helicity-n tensors is denoted +by T n. +The first example is vectors (or equivalently 1-forms): V z ∈ T 1, Vz ∈ T −1. The second +most useful case is traceless symmetric tensors, which are elements of T ±2. Consider a +2However, at this stage, each component can still depend on both z and ¯z: V z = V z(z, ¯z) and V ¯z = +V ¯z(z, ¯z). +3In fact, it is even possible to consider n ∈ Z + 1/2 to describe spinors. +74 + +traceless symmetric tensor T ab = T ba and gabT ab = 0: this implies T 01 = T 10 and T 00 = +−T 11 in real coordinates. The components in complex coordinates are: +T zz = 2(T 00 + iT 01) ∈ T 2, +T ¯z¯z = 2(T 00 − iT 01) ∈ T −2, +T z +z = 0. +(4.25) +Note that +Tzz = gz¯zgz¯zT ¯z¯z = 1 +2(T 00 − iT 01), +(4.26) +and T z +z = gz¯zT z¯z ∈ T 0 corresponds to the trace. +Computation – Equation (4.25) +T zz = +�∂z +∂τ +�2 +T 00 + +� ∂z +∂σ +�2 +T 11 + 2 ∂z +∂τ +∂z +∂σ T 01 = T 00 − T 11 + 2i T 01. +Stokes’ theorem in complex coordinates follows directly from (B.10): +� +d2z (∂zvz + ∂¯zv¯z) = −i +� � +dz v¯z − d¯zvz� += −2i +� +∂R +(vzdz − v¯zd¯z), +(4.27) +where the integration contour is anti-clockwise. To obtain this formula, note that d2x = 1 +2d2z +and ϵz¯z = i/2, such that the factor 1/2 cancels between both sides. +4.2 +Complex representation of path integral +In the previous section, we have found that tensors of a given rank are naturally decomposed +into different subspaces thanks to the complex structure of the manifold. +Accordingly, +complex coordinates are natural and one can expect most objects in string theory to split +similarly into holomorphic and anti-holomorphic sectors (or left- and right-moving). This +will be particularly clear using the CFT language (Chapter 6). The main difficulty for this +program is due to the matter zero-modes. In this section, we focus on the path integral +measure and expression of the ghosts. +There is, however, a subtlety in displaying explicitly the factorization: the notion of +“holomorphicity” depends on the metric (because the complex structure must be compatible +with the metric for an Hermitian manifold). Since the metric depends on the moduli which +are integrated over in the path integral, it is not clear that there is a consistent holomorphic +factorization. We will not push the question of achieving a global factorization further (but +see Remark 4.1) to focus instead on the integrand. The latter is local (in moduli space) and +there is no ambiguity. +The results of the previous section indicate that the basis of Killing vectors (2.104) and +quadratic differentials (2.76) split into holomorphic and anti-holomorphic components: +ψi(z, ¯z) = ψz +i ∂z + ψ¯z +i ∂¯z, +φi(z, ¯z) = φi,zz(dz)2 + φi,¯z¯z(d¯z)2. +(4.28) +Similarly, the operators P1 (2.65a) and P † +1 (2.71) also split: +(P1ξ)zz = 2∇zξz = ∂zξ¯z, +(P1ξ)¯z¯z = 2∇¯zξ¯z = ∂¯zξz, +(4.29a) +(P † +1 T)z = −2∇zTzz = −4 ∂¯zTzz, +(P † +1 T)¯z = −2∇¯zT¯z¯z = −4 ∂zT¯z¯z +(4.29b) +for arbitrary vector ξ and traceless symmetric tensor T (in the background metric). As a +consequence, the components of Killing vectors and quadratic differentials are holomorphic +or anti-holomorphic as a function of z: +ψz = ψz(z), +ψ¯z = ψ¯z(¯z), +φzz = φzz(z), +φ¯z¯z = φ¯z¯z(¯z), +(4.30) +75 + +such that it makes sense to consider a complex basis instead of the previous real basis: +ker P1 = Span{ψK(z)} ⊕ Span{ ¯ψK(¯z)}, +K = 1, . . . , Kc +g, +(4.31a) +ker P † +1 = Span{φI(z)} ⊕ Span{¯φI(¯z)}, +I = 1, . . . , Mc +g. +(4.31b) +The last equation can inspire to search for a similar rewriting of the moduli parameters. +In fact, the moduli space itself is a complex manifold and can be endowed with complex +coordinates [173, 193]: +mI = t2I−1 + it2I, +¯mI = t2I−1 − it2I, +I = 1, . . . , Mc +g +(4.32) +with the integration measure +dMgt = d2Mc +gm. +(4.33) +The last ingredient to rewrite the vacuum amplitudes (2.136) is to obtain the determin- +ants. The inner-products of vector and traceless symmetric fields also factorize: +(T1, T2) = 2 +� +d2σ +� +ˆg ˆgacgbdT1,abT2,cd = 4 +� +d2z +� +T1,zzT2,¯z¯z + T1,¯z¯zT2,zz +� +, +(4.34a) +(ξ1, ξ2) = +� +d2σ +� +ˆg ˆgabξaξb = 1 +4 +� +d2z +� +ξz +1ξ¯z +2 + ξ¯z +1ξz +2 +� +. +(4.34b) +All inner-products are evaluated in the flat background metric. For (anti-)holomorphic fields, +only one term survives in each integral: since each field appears twice in the determinants +(φi, φj) and (φi, φj), the final expression is a square, which cancels against the squareroot +in (2.136). The remaining determinant involves the Beltrami differential (2.65b): +µizz = ∂i¯gzz, +µi¯z¯z = ∂i¯g¯z¯z +(4.35) +(¯gzz = 0 in our coordinates system, but its variation under a shift of moduli is not zero). +The basis can be changed to a complex basis such that the determinant of inner-products +between Beltrami and quadratic differentials is a modulus squared. All together, the different +formulas lead to the following rewriting of the vacuum amplitude : +Zg = +� +Mg +d2Mc +gm | det(φI, µJ)|2 +| det(φI, ¯φJ)| +det′ P † +1 P1 +| det(ψI, ¯ψJ)| +Zm[δ] +Ωckv[δ], +(4.36) +where the absolute values are to be understood with respect to the basis of P1 and P † +1 , for +example |f(mI)|2 := f(mI)f( ¯mI). +The same reasoning can be applied to the ghosts. The c and b ghosts are respectively +a vector and a symmetric traceless tensor, both with two independent components: it is +customary to define +c := cz, +¯c := c¯z, +b := bzz, +¯b := b¯z¯z. +(4.37) +In that case, the action (2.145) reads +Sgh[g, b, c] = 1 +2π +� +d2z +� +b∂¯zc + ¯b∂z¯c +� +. +(4.38) +The action is the sum of two holomorphic and anti-holomorphic contributions and it is +independent of φ(z, ¯z) as expected. In fact, the equations of motion are +∂zc = 0, +∂zb = 0, +∂¯z¯c = 0, +∂¯z¯b = 0, +(4.39) +76 + +such that b and c (resp. ¯b and ¯c) are holomorphic (anti-holomorphic) functions. Then, the +integration measure is simply +Mg +� +i=1 +Bi dti = +Mc +g +� +I=1 +BI ¯BI dmI ∧ ¯mI, +BI := (µI, b). +(4.40) +Note that BI does not contain ¯b(¯z), it is built only from b(z). +Finally, the vacuum amplitude (2.163) reads +Zg = +� +Mg +d2Mc +gm +Ωckv[δ]−1 +| det ψI(z0 +j )|2 +� +d(b,¯b) d(c, ¯c) +Kc +g +� +j=1 +c(z0 +j )¯c(¯z0 +j ) +Mc +g +� +I=1 +|(µI, b)|2 e−Sgh[b,c] Zm[δ]. +(4.41) +The c insertions are separated in holomorphic and anti-holomorphic components because, +at the end, only the zero-modes contribute. The measures are written as d(b,¯b) and d(c, ¯c) +because proving that they factorize is difficult (Remark 4.1). +Remark 4.1 (Holomorphic factorization) It was proven in [15, 27, 33] (see [173, sec. 9, +53, sec. VII, 237, sec. 3] for reviews) that the ghost and matter path integrals can be globally +factorized, up to a factor due to zero-modes. Such a result is suggested by the factorization +of the inner-products, which imply a factorization of the measures: the caveat is due to the +zero-mode determinants and matter measure. Interestingly, the factorization is possible only +in the critical dimension (2.125). +4.3 +Summary +In this chapter, we have introduced complex notations for the fields, path integral and +moduli space. +4.4 +Suggested readings +• Good references for this chapter are [24, 53, 172, 173, 193]. +• Geometry of complex manifolds is discussed in [24, sec. 6.2, 172, chap. 14, 53]. +77 + +Chapter 5 +Conformal symmetry in D +dimensions +Abstract +Starting with this chapter, we discuss general properties of conformal field the- +ories (CFT). The goal is not to be exhaustive, but to provide a short introduction and to +gather the concepts and formulas that are needed for string theory. However, the subject is +presented as a standalone topic such that it can be of interest for a more general public. +The conformal group in any dimension is introduced in this chapter. The specific case +D = 2, which is the most relevant for the current book, is developed in the following chapters. +5.1 +CFT on a general manifold +In this chapter and in the next one, we discuss CFTs as QFTs living on a spacetime M, +independently from string theory (there is no reference to a target spacetime). As such, we +will use spacetime notations together with some simplifications: coordinates are written as +xµ with µ = 0, . . . , D − 1 and time is written as x0 = t (x0 = τ) in Lorentzian (Euclidean) +signature. +Given a metric gµν on a D-dimensional manifold M, the conformal group CISO(M) is +the set of coordinate transformations (called conformal symmetries or isometries) +xµ −→ x′µ = x′µ(x) +(5.1) +which leaves the metric invariant up to an overall scaling factor: +gµν(x) −→ g′ +µν(x′) = ∂xρ +∂x′µ +∂xσ +∂x′ν gρσ(x) = Ω(x′)2gµν(x′). +(5.2) +This means that angles between two vectors u and v are left invariant under the transform- +ation: +u · v +|u| |v| = u′ · v′ +|u′| |v′|. +(5.3) +It is often convenient to parametrize the scale factor by an exponential +Ω := eω. +(5.4) +Considering an infinitesimal transformation +δxµ = ξµ, +(5.5) +78 + +the condition (5.2) becomes the conformal Killing equation +δgµν = Lξgµν = ∇µξν + ∇νξµ = 2 +d gµν∇ρξρ, +(5.6) +such that the scale factor is +Ω2 = 1 + 2 +d ∇ρξρ. +(5.7) +The vector fields ξ satisfying this equation are called conformal Killing vectors (CKV). Con- +formal transformations form a global subgroup of the diffeomorphism group: the generators +of the transformations do depend on the coordinates, but the parameters do not (for an +internal global symmetry, both the generators and the parameters don’t depend on the +coordinates). +The conformal group contains the isometry group ISO(M) of M as a subgroup, corres- +ponding to the case Ω = 1: +ISO(M) ⊂ CISO(M). +(5.8) +These transformations also preserve distances between points. The corresponding generators +of infinitesimal transformations are called Killing vectors and satisfies the Killing equation +δgµν = Lξgµν = ∇µξν + ∇νξµ = 0. +(5.9) +They form a subalgebra of the CKV algebra. +An important point is to be made for the relation between infinitesimal and finite trans- +formations: with spacetime symmetries it often happens that the first cannot be exponenti- +ated into the second. The reason is that the (conformal) Killing vectors may be defined only +locally, i.e. they are well-defined in a given domain but have singularities outside. When +this happens, they do not lead to an invertible transformation, which cannot be an element +of the group. These notions are sometimes confused in physics and the term of “group” is +used instead of “algebra”. We shall be careful in distinguishing both concepts. +Remark 5.1 (Isometries of M ⊂ Rp,q) In order to find the conformal isometries of a +manifold M which is a subset of Rp,q defined in (5.10), it is sufficient to restrict the trans- +formations of Rp,q to the subset M [206]. In the process, not all global transformations +generically survive. On the other hand, the algebra of local (infinitesimal) transformations +for M and Rp,q are identical since M is locally like Rp,q. +5.2 +CFT on Minkowski space +In this section, we consider the case where M = Rp,q (D = p + q) and where g = η is the +flat metric with signature (p, q): +η = diag(−1, . . . , −1 +� +�� +� +q +, 1, . . . , 1 +� �� � +p +). +(5.10) +The conformal Killing equation becomes +� +ηµν∆ + (D − 2)∂µ∂ν +� +∂ · ϵ = 0, +(5.11) +where ∆ is the D-dimensional Beltrami–Laplace operator for the metric ηµν. The case D = 2 +is relegated to the next chapter. For D > 2, one finds the following transformations: +translation: +ξµ = aµ, +(5.12a) +rotation & boost: +ξµ = ωµ +νxν, +(5.12b) +dilatation: +ξµ = λ xµ, +(5.12c) +SCT: +ξµ = bµx2 − 2b · x xµ, +(5.12d) +79 + +where ωµν is antisymmetric. The rotations include Lorentz transformations and SCT means +“special conformal transformation”. +All parameters {aµ, ωµν, λ, bµ} are constant. The generators are respectively denoted by +{Pµ, Jµν, D, Kµ}. The finite translations and rotations form the Poincaré group SO(p, q), +while the conformal group can be shown to be SO(p + 1, q + 1): +ISO(Rp,q) = SO(p, q), +CISO(Rp,q) = SO(p + 1, q + 1). +(5.13) +The dimension of this group is +dim SO(p + 1, q + 1) = 1 +2 (p + q + 2)(p + q + 1). +(5.14) +5.3 +Suggested readings +• References on higher-dimensional CFTs are [54, 196, 203, 206, 236]. +80 + +Chapter 6 +Conformal field theory on the +plane +Abstract +Starting with this chapter, we focus on two-dimensional Euclidean CFTs on the +complex plane (or equivalently the sphere). We start by describing the geometry of the +sphere and the relation to the complex plane and to the cylinder, in order to make contact +with the string worldsheet. Then, we discuss classical CFTs and the Witt algebra obtained +by classifying the conformal isometries of the complex plane. Then, we describe quantum +CFTs and introduce the operator formalism. This last section is the most important for this +book as it includes information on the operator product expansion, Hilbert space, Hermitian +and BPZ conjugations. +As described at the beginning of Chapter 5, we use spacetime notations for the coordin- +ates, but follow otherwise the normalization for the worldsheet. In particular, integrals are +normalized by 2π. However, the spatial coordinate on the cylinder is still written as σ to +avoid confusions: xµ = (τ, σ). +6.1 +The Riemann sphere +6.1.1 +Map to the complex plane +The Riemann sphere Σ0, which is diffeomorphic to the unit sphere S2, has genus g = 0 and +is thus the simplest Riemann surface. Its most straightforward description is obtained by +mapping it to the extended1 complex plane ¯C (also denoted ˆC), which is the complex plane +z ∈ C to which the point at infinity z = ∞ is added: +¯C = C ∪ {∞}. +(6.1) +One speaks about “the point at infinity” because all the points at infinity (i.e. the points z +such that |z| → ∞) +lim +r→∞ r eiθ := ∞ +(6.2) +are identified (the limit is independent of θ). +The identification can be understood by mapping (say) the south pole to the origin of +the plane and the north pole to infinity2 (Figure 6.1) through the stereographic projection +z = eiφ cot θ +2, +(6.3) +1This qualification will often be omitted. +2Note that the points are distinguished in order to write the map, but they have nothing special by +themselves (i.e. they are not punctures). +81 + +−−−−−−−−→ +Figure 6.1: Map from the Riemann sphere to the complex plane. The south and north poles +are denoted by the letter S and N, and the equatorial circle by E. +where (θ, φ) are angles on the sphere. Any circle on the sphere is mapped to a circle in the +complex plane. Conversely, the Riemann sphere can be viewed as a compactification of the +complex plane. +Introducing Cartesian coordinates (x, y) related to the complex coordinates by3 +z = x + iy, +¯z = x − iy, +(6.4a) +x = z + ¯z +2 +, +y = z − ¯z +2i +, +(6.4b) +the metric reads +ds2 = dx2 + dy2 = dzd¯z. +(6.5) +The relations between the derivatives in the two coordinate systems are easily found: +∂ := ∂z = 1 +2 (∂x − i∂y), +¯∂ := ∂¯z = 1 +2 (∂x + i∂y). +(6.6) +The indexed form will be used when there is a risk of confusion. If the index is omitted then +the derivative acts directly to the field next to it, for example +∂φ(z1)∂φ(z2) := ∂z1∂z2φ(z1)φ(z2). +(6.7) +Generically, the meromorphic and anti-meromorphic parts of a object will be denoted +without and with a bar, see (6.55) for an example. +The extended complex plane ¯C can be covered by two coordinate patches z ∈ C and +w ∈ C. In the first, the point at infinity (north pole) is removed, in the second, the origin +(south pole) is removed. On the overlap, the transition function is +w = 1 +z . +(6.8) +This description avoids to work with the infinity: studying the behaviour of f(z) at z = ∞ +is equivalent to study f(1/w) at w = 0. +Since any two-dimensional metric is locally conformally equivalent to the flat metric, it +is sufficient to work with this metric in each patch. This is particularly convenient for the +Riemann sphere since one patch covers it completely except for one point. +3General formulas can be found in Section 4.1 by replacing (τ, σ) with (x, y). In most cases, the conformal +factor is set to zero (φ = 0) in this chapter. +82 + +6.1.2 +Relation to the cylinder – string theory +The worldsheet of a closed string propagating in spacetime is locally topologically a cylinder +R × S1 of circumference L. In this section, we show that the cylinder can also be mapped +to the complex plane – and thus to the Riemann sphere – after removing two points. Since +the cylinder has a clear physical interpretation in string theory, it is useful to know how to +translate the results from the plane to the cylinder. +It makes also sense to define two-dimensional models on the cylinder independently of +a string theory interpretation since the compactification of the spatial direction from R to +S1 regulates the infrared divergences. Moreover, it leads to a natural definition of a “time” +and of an Hamiltonian on the Euclidean plane. +Denoting the worldsheet coordinates in Lorentzian signature by (t, σ) with4 +t ∈ R, +σ ∈ [0, L), +σ ∼ σ + L, +(6.9) +the metric reads +ds2 = −dt2 + dσ2 = −dσ+dσ−, +(6.10) +where the light-cone coordinates +dσ± = dt ± dσ +(6.11) +have been introduced. It is natural to perform a Wick rotation from the Lorentzian time t +to the Euclidean time +τ = it, +(6.12) +and the metric becomes +ds2 = dτ 2 + dσ2. +(6.13) +It is convenient to introduce the complex coordinates +w = τ + iσ, +¯w = τ − iσ +(6.14) +for which the metric is +ds2 = dwd ¯w. +(6.15) +Note that the relation to Lorentzian light-cone coordinates are +w = i(t + σ) = iσ+, +¯w = i(t − σ) = iσ−. +(6.16) +Hence, an (anti-)holomorphic function of w ( ¯w) depends only on σ+ (σ−) before the Wick +rotation: this leads to the identification of the left- and right-moving sectors with the holo- +morphic and anti-holomorphic sectors of the theory. +The cylinder can be mapped to the complex plane through +z = e2πw/L, +¯z = e2π ¯ +w/L, +(6.17) +and the corresponding metric is +ds2 = +� L +2π +�2 dzd¯z +|z|2 . +(6.18) +A conformal transformation brings this metric to the flat metric (6.5). The conventions +for the various coordinates and maps vary in the different textbooks. We have gathered in +Table A.1 the three main conventions and which references use which. +4Consistently with the comments at the beginning of Chapter 5, the Lorentzian worldsheet time is +denoted by t instead of τM. +83 + +−−−−−→ +−−−−−→ +Figure 6.2: Map from the cylinder to the sphere with two tubes, to the 2-punctured sphere +Σ0,2. +The map from the cylinder to the plane is found by sending the bottom end (corres- +ponding to the infinite past t → −∞) to the origin of the plane, and the top end (infinite +future t → ∞) to the infinity. Since the cylinder has two boundaries (its two ends) the map +excludes the point z = 0 and z = ∞ and one really obtains the space ¯C − {0, ∞} = C∗. +This space can, in turn, be mapped to the 2-punctured Riemann sphere Σ0,2. +The physical interpretation for the difference between Σ0 and Σ0,2 is simple: since one +considers the propagation of a string, it means that the worldsheet corresponds to an amp- +litude with two external states, which are the mapped to the sphere as punctures (Figure 6.2, +Section 3.1.1). Removing the external states (yielding the tree-level vacuum amplitude) cor- +responds to gluing half-sphere (caps) at each end of the cylinder (Figure 6.3). Then, it can +be mapped to the Riemann sphere without punctures. As a consequence, the properties of +tree-level string theory are found by studying the matter and ghost CFTs on the Riemann +sphere. Scattering amplitudes are computed through correlation functions of appropriate op- +erators on the sphere. This picture generalizes to higher-genus Riemann surfaces. Moreover, +since local properties of the CFT (e.g. the spectrum of operators) are determined by the +conformal algebra, they will be common to all surfaces. +Mathematically, a difference between Σ0 and Σ0,2 had to be expected since the sphere +has a positive curvature (and χ = −2) but the cylinder is flat (with χ = 0). Punctures +contribute negatively to the curvature (and thus positively to the Euler characteristics). +Remark 6.1 The coordinate z is always used as a coordinate on the complex plane, but the +corresponding metric may be different – compare (6.5) and (6.18). As explained previously, +this does not matter since the theory is insensitive to the conformal factor. +6.2 +Classical CFTs +In this section, we consider an action S[Ψ] which is conformally invariant. We first identify +and discuss the properties of the conformal algebra and group, before explaining how a CFT +is defined. +84 + +−−−−−−−−→ +Figure 6.3: Map from the cylinder with two caps (half-spheres) to the Riemann sphere Σ0. +6.2.1 +Witt conformal algebra +Since the Riemann sphere is identified with the complex plane, they share the same conformal +group and algebra. Consider the metric (6.5) +ds2 = dzd¯z, +(6.19) +then, any meromorphic change of coordinates +z −→ z′ = f(z), +¯z −→ ¯z′ = ¯f(¯z) +(6.20) +is a conformal transformation since the metric becomes +ds2 = dz′d¯z′ = +���� +df +dz +���� +2 +dzd¯z. +(6.21) +However, only holomorphic functions which are globally defined on ¯C are elements of +the group. At the algebra level, any holomorphic function f(z) regular in a domain D gives +a well-defined transformation in this domain D. Hence, the algebra is infinite-dimensional. +On the other hand, f(z) is only meromorphic on C generically: it cannot be exponentiated +to a group element. We first characterize the algebra and then obtain the conditions to +promote the local transformations to global ones. +Since the transformations are defined only locally, it is sufficient to consider an infinites- +imal transformation +δz = v(z), +δ¯z = ¯v(¯z), +(6.22) +where v(z) is a meromorphic vector field on the Riemann sphere. Indeed, the conformal +Killing equation (5.6) in D = 2 is equivalent to the Cauchy–Riemann equations: +¯∂v = 0, +∂¯v = 0. +(6.23) +The vector field admits a Laurent series +v(z) = +� +n∈Z +vnzn+1, +¯v(¯z) = +� +n∈Z +¯vn¯zn+1, +(6.24) +and the vn and ¯vn are to be interpreted as the parameters of the transformation. A basis of +vectors (generators) is: +ℓn = −zn+1∂z, +¯ℓn = −¯zn+1∂¯z, +n ∈ Z. +(6.25) +One can check that each set of generators satisfies the Witt algebra +[ℓm, ℓn] = (m − n)ℓm+n, +[¯ℓm, ¯ℓn] = (m − n)¯ℓm+n, +[ℓm, ¯ℓn] = 0. +(6.26) +85 + +Since there are two commuting copies of the Witt algebra, it is natural to extend the +ranges of the coordinates from C to C2 and to consider z and ¯z as independent variables. +In particular, this gives a natural action of the product algebra over C2. This procedure +will be further motivated when studying CFTs since the holomorphic and anti-holomorphic +parts will generally split, and it makes sense to study them separately. Ultimately, phys- +ical quantities can be extracted by imposing the condition ¯z = z∗ at the end (the star is +always reserved for the complex conjugation, the bar will generically denote an independent +variable). In that case, the two algebras are also related by complex conjugation. +Note that the variation of the metric (B.6) under a meromorphic change of coordinates +(6.22) becomes +δgz¯z = ∂v + ¯∂¯v, +δgzz = δg¯z¯z = 0. +(6.27) +6.2.2 +PSL(2, C) conformal group +The next step is to determine the globally defined vectors and to study the associated group. +First, the conditions for a vector v(z) to be well-defined at z = 0 are +lim +|z|→0 v(z) < ∞ +=⇒ +∀n < −1 : +vn = 0. +(6.28) +The behaviour at z = ∞ can be investigated thanks to the map z = 1/w +v(1/w) = dz +dw +� +n +vnw−n−1, +(6.29) +where the additional derivative arises because v is a vector. Then, the regularity conditions +at z = ∞ are +lim +|z|→∞ v(z) = lim +|w|→0 +dz +dw v(1/w) = − lim +|w|→0 +v(1/w) +w2 +< ∞ +=⇒ +∀n > 1 : +vn = 0. +(6.30) +As a result, the globally defined generators are +{ℓ−1, ℓ0, ℓ1} ∪ {¯ℓ−1, ¯ℓ0, ¯ℓ1} +(6.31) +where +ℓ−1 = −∂z, +ℓ0 = −z∂z, +ℓ1 = −z2∂z. +(6.32) +It is straightforward to check that they form two copies of the sl(2, C) algebra +[ℓ0, ℓ±1] = ∓ℓ±1, +[ℓ1, ℓ−1] = 2ℓ0. +(6.33) +The global conformal group is sometimes called Möbius group: +PSL(2, C) := SL(2, C)/Z2 ∼ SO(3, 1), +(6.34) +where the additional division by Z2 is clearer when studying an explicit representation. It +corresponds with ker P1 defined in (2.91): +K0 = PSL(2, C). +(6.35) +A matrix representation of SL(2, C) is +g = +�a +b +c +d +� +, +a, b, c, d ∈ C, +det g = ad − bc = 1, +(6.36) +86 + +which shows that this group has six real parameters +K0 := dim SL(2, C) = 6. +(6.37) +The associated transformation on the complex plane reads +fg(z) = az + b +cz + d. +(6.38) +The quotient by Z2 is required since changing the sign of all parameters does not change the +transformation. These transformations have received different names: Möbius, projective, +homographic, linear fractional transformations. . . +Holomorphic vector fields are then of the form +v(z) = β + 2αz + γz2, +¯v(¯z) = ¯β + 2¯α¯z + ¯γ¯z2, +(6.39) +where +a = 1 + α, +b = β, +c = −γ, +d = 1 − α. +(6.40) +The finite transformations associated to (5.12) are: +translation: +fg(z) = z + a, +a ∈ C, +(6.41a) +rotation: +fg(z) = ζ z, +|ζ| = 1, +(6.41b) +dilatation: +fg(z) = λ z, +λ ∈ R, +(6.41c) +SCT: +fg(z) = +z +cz + 1, +c ∈ C. +(6.41d) +Investigation leads to the following association between the generators and transformations: +• translation: ℓ−1 and ¯ℓ−1; +• dilatation (or radial translation): (ℓ0 + ¯ℓ0); +• rotation (or angular translation): i(ℓ0 − ¯ℓ0); +• special conformal transformation: ℓ1 and ¯ℓ1. +The inversion defined by +inversion: +I+(z) := I(z) := 1 +z +(6.42a) +is not an element of SL(2, C). However, the inversion with a minus sign +I−(z) := −I(z) = I(−z) = −1 +z +(6.42b) +is a SL(2, C) transformation. +A useful transformation is the circular permutation of (0, 1, ∞): +g∞,0,1(z) = +1 +1 − z . +(6.43) +87 + +6.2.3 +Definition of a CFT +A CFT is characterized by its set of (composite) fields (also called operators) O(z, ¯z) which +correspond to any local expression constructed from the fields Ψ appearing in the Lagrangian +and of their derivatives.5 For example, in a scalar field theory, the simplest operators are of +the form ∂mφn. +Among the operators, two particular categories are distinguished according to their trans- +formation laws: +• primary operator: +∀f meromorphic : +O(z, ¯z) = +�df +dz +�h �d ¯f +d¯z +�¯h +O′� +f(z), ¯f(¯z) +� +, +(6.44) +• quasi-primary (or SL(2, C) primary) operator: +∀f ∈ PSL(2, C) : +O(z, ¯z) = +�df +dz +�h �d ¯f +d¯z +�¯h +O′� +f(z), ¯f(¯z) +� +. +(6.45) +The parameters (h, ¯h) are the conformal weights of the operator O (both are independent +from each other), and combinations of them give the conformal dimension ∆ and spin s: +∆ := h + ¯h, +s := h − ¯h. +(6.46) +The conformal weights correspond to the charges of the operator under ℓ0 and ¯ℓ0. We will +use “(h, ¯h) (quasi-)primary” as a synonym of “(quasi-)primary field with conformal weight +(h, ¯h)”. +Remark 6.2 (Complex conformal weights) While we consider h, ¯h ∈ R, and more spe- +cifically h, ¯h ≥ 0 for a unitary theory (which is the case of string theory except for the re- +parametrization ghosts), theories with h, ¯h ∈ C make perfectly sense. One example is the +Liouville theory with complex central charge c ∈ C [200, 202] (central charges are defined +below, see (6.58)). +Primaries and quasi-primaries are hence operators which have nice transformations re- +spectively under the algebra and group. Obviously, a primary is also a quasi-primary. These +transformations are similar to those of a tensor with h holomorphic and ¯h anti-holomorphic +indices (Section 4.1). Another point of view is that the object +O(z, ¯z) dzhd¯z +¯h +(6.47) +is invariant under local / global conformal transformations. +The notation f ◦ O indicates the complete change of coordinates, including the tensor +transformation law and the possible corrections if the operator is not primary.6 For a primary +field, we have: +f ◦ O(z, ¯z) := f ′(z)h ¯f ′(¯z) +¯h O′� +f(z), ¯f(¯z) +� +. +(6.48) +We stress that it does not correspond to function composition. +Under an infinitesimal transformations +δz = v(z), +δ¯z = ¯v(¯z), +(6.49) +5Not all CFTs admit a Lagrangian description. But, since we are mostly interested in string theories +defined from Polyakov’s path integral, it is sufficient to study CFTs with a Lagrangian. +6In fact, one has f ◦ O := f∗O in the notations of Chapter 2. +88 + +a primary operator changes as +δO(z, ¯z) = (h ∂v + v ∂)O(z, ¯z) + (¯h ¯∂¯v + ¯v ¯∂)O(z, ¯z). +(6.50) +The transformation of a non-primary field contains additional terms, see for example (6.89). +Remark 6.3 (Higher-genus Riemann surfaces) According to Remark 5.1, all Riemann +surfaces Σg share the same conformal algebra since locally they are all subsets of R2. On +the other hand, one finds that no global transformations are defined for g > 1, and only the +subgroup U(1) × U(1) survives for the torus. +The most important operator in a CFT is the energy–momentum tensor Tµν, if it exists +as a local operator. According to Section 2.1, this tensor is conserved and traceless +∇νTµν = 0, +gµνTµν = 0. +(6.51) +The traceless equation in components reads +gµνTµν = 4 Tz¯z = Txx + Tyy = 0 +(6.52) +which implies that the off-diagonal component vanishes in complex coordinates +Tz¯z = 0. +(6.53) +Then, the conservation equation yields +∂zT¯z¯z = 0, +∂¯zTzz = 0, +(6.54) +such that the non-vanishing components Tzz and T¯z¯z are respectively holomorphic and anti- +holomorphic. This motivates the introduction of the notations: +T(z) := Tzz(z), +¯T(¯z) := T¯z¯z(¯z). +(6.55) +This is an example of the factorization between the holomorphic and anti-holomorphic sec- +tors. +Currents are local objects and thus one expects to be able to write an infinite number of +such currents associated to the Witt algebra. Applying the Noether procedure gives +Jv(z) := J ¯z +v (z) = −T(z)v(z), +¯Jv(¯z) := Jz +v (¯z) = − ¯T(¯z)¯v(¯z). +(6.56) +6.3 +Quantum CFTs +The previous section was purely classical. The quantum theory is first defined through the +path integral +Z = +� +dΨ e−S[Ψ]. +(6.57) +We will also develop an operator formalism. The latter is more general than the path integral +and allows to work without reference to path integrals and Lagrangians. This is particularly +fruitful as it extends the class of theories and parameter ranges (e.g. Remark 6.2) which can +be studied. +89 + +6.3.1 +Virasoro algebra +As discussed in Section 2.3.3, field measures in path integrals display a conformal anom- +aly, meaning that they cannot be defined without introducing a scale. This anomaly can +be traded for a gravitational anomaly by introducing counter-terms in the action [87, 95, +sec. 3.2, 96, 101, 122, 129]. As a consequence, the Witt algebra (6.26) is modified to its +central extension, the Virasoro algebra.7 +The generators in both sectors are denoted by +{Ln} and {¯Ln} and are called Virasoro operators (or modes). The algebra is given by: +[Lm, Ln] = (m − n)Lm+n + c +12 m(m − 1)(m + 1)δm+n, +(6.58a) +[¯Lm, ¯Ln] = (m − n)¯Lm+n + ¯c +12 m(m − 1)(m + 1)δm+n, +(6.58b) +[Lm, ¯Ln] = 0, +[c, Lm] = 0, +[¯c, ¯Lm] = 0, +(6.58c) +where c, ¯c ∈ C are the holomorphic and anti-holomorphic central charges. Consistency of +the theory on a curved space implies ¯c = c, but there is otherwise no constraint on the +plane [95, 246]. +The sl(2, C) subalgebra is not modified by the central extension. This means that states +are still classified by eigenvalues of (h, ¯h) of (L0, ¯L0). +Remark 6.4 In most models relevant for string theory, one finds that the central charges +are real, c, ¯c ∈ R. +Moreover, unitarity requires them to be positive c, ¯c > 0, and only +reparametrization ghosts do not satisfy this condition. On the other hand, it makes perfect +sense to discuss general CFTs for c, ¯c ∈ C (the Liouville theory is such an example [200, +202]). +6.3.2 +Correlation functions +A n-point correlation function is defined by +� n +� +i=1 +Oi(zi, ¯zi) +� += +� +dΨ e−S[Ψ] +n +� +i=1 +Oi(zi, ¯zi), +(6.59) +choosing a normalization such that ⟨1⟩ = 1. The path integral defines the time-ordered +product (on the cylinder) of the corresponding operators. +Invariance under global transformations leads to strong constraints on the correlation +functions. For quasi-primary fields, they transform under SL(2, C) as +� n +� +i=1 +Oi(zi, ¯zi) +� += +n +� +i=1 +�df +dz (zi) +�hi �df +d¯z (¯zi) +�¯hi +× +� n +� +i=1 +Oi +� +f(zi), ¯f(¯zi) +� +� +. +(6.60) +Considering an infinitesimal variation (6.50) yields a differential equation for the n-point +function +δ +� n +� +i=1 +Oi(zi, ¯zi) +� += +n +� +i=1 +� +hi∂iv(zi) + v(zi)∂i + c.c. +� +� n +� +i=1 +Oi(zi, ¯zi) +� += 0, +(6.61) +where ∂i := ∂zi and v is a vector (6.39) of sl(2, C). These equations are sufficient to determine +7That the central charge in the Virasoro algebra indicates a diffeomorphism anomaly can be understood +from the fact that +90 + +completely the forms of the 1-, 2- and 3-point functions of quasi-primaries: +⟨Oi(zi, ¯zi)⟩ = δhi,0δ¯hi,0, +(6.62a) +⟨Oi(zi, ¯zi)Oj(zj, ¯zj)⟩ = δhi,hjδ¯hi,¯hj +gij +z2hi +ij ¯z2¯hi +ij +, +(6.62b) +⟨Oi(zi, ¯zi)Oj(zj, ¯zj)Ok(zk, ¯zk)⟩ = +Cijk +zhi+hj−hk +ij +zhj+hk−hi +jk +zhi+hk−hj +ki +× +1 +¯z +¯hi+¯hj−¯hk +ij +¯z +¯hj+¯hk−¯hi +jk +¯z +¯hi+¯hk−¯hj +ki +, +(6.62c) +where we have defined +zij = zi − zj. +(6.63) +The coefficients Cijk are called structure constants and the matrix gij defines a metric +(Zamolodchikov metric) on the space of fields. The metric is often taken to be diagonal +gij = δij, which amounts to use an orthonormal eigenbasis of L0 and ¯L0. The vanishing of +the 1-point function of a non-primary quasi-primary holds only on the plane: for example +the value on the cylinder can be non-zero since the map is not globally defined – see in +particular (6.167). +Remark 6.5 (Logarithmic CFTs) Logarithmic CFTs display a set of unusual proper- +ties [84, 85, 90, 97, 130]. +In particular, the correlation functions are not of the form +displayed above. The most striking feature of those theories is that the L0 operator is non- +diagonalisable (but it can be set in the Jordan normal form). +Remark 6.6 (Fake identity) Usually, the only primary operator with h = ¯h = 0 is the +identity 1. While this is always true for unitary theories, there are non-unitary theories +(c ≤ 1 Liouville theory, SLE, loop models) where there is another field (called the indicator, +marking operator, or also fake identity) with h = ¯h = 0 [13, 46, 99, 112, 182, 200, 202]. +The main difference between both fields is that the identity is a degenerate field (it has a +null descendant), whereas the other operator with h = ¯h = 0 is not. Such theories will not +be considered in this book. Operators with h = ℏ = 0 can also be built by comining several +CFTs, and they play a very important role in string theory since they describe on-shell states. +Finally, the 4-point function is determined up to a function of a single variable x and its +complex conjugate: +� 4 +� +i=1 +Oi(zi, ¯zi) +� += f(x, ¯x) +� +i |w|, +(−1)F B(w)A(z) +|w| > |z|, +(6.72) +where F = 0 (F = 1) for bosonic (fermionic) operators. Radial ordering will often be kept +implicit. +The equal-time (anti-)commutator becomes an equal radius commutator defined by +point-splitting: +[A(z), B(w)]±,|z|=|w| = lim +δ→0 +� +A(z)B(w)||z|=|w|+δ ± B(w)A(z)||z|=|w|−δ +� +. +(6.73) +If A and B are two operators which can be written as the contour integrals of a(z) and b(z) +(corresponding to integral over closed curves on the cylinder) +A = +� +C0 +dz +2πi a(z), +B = +� +C0 +dz +2πi b(z), +(6.74) +then one finds the following commutators: +[A, B]± = +� +C0 +dw +2πi +� +Cw +dz +2πi a(z)b(w), +(6.75a) +[A, b(w)]± = +� +Cw +dz +2πi a(z)b(w). +(6.75b) +92 + +Figure 6.4: Graphical proof of (6.75). +The contours C0 and Cw are respectively centered around the points 0 and w. For a proof, +see Figure 6.4. Since these are contour integrals in the complex plane, the Cauchy–Riemann +formula (B.1) can be used to write the result as soon as one knows the poles of the above +expression (ultimately, this amounts to pick the sum of residues). In CFTs, the poles of such +expressions are given by operator product expansions (OPE), defined below (Section 6.4.2). +Given a conserved current jµ +∂µjµ = ∂jz + ¯∂j ¯z = 2(∂j¯z + ¯∂jz) = 0, +(6.76) +the associated conserved charge is defined by +Q = +1 +2πi +� +C0 +(jzdz − j¯zd¯z), +(6.77) +where C0 denotes the anti-clockwise contour around z = 0 (equivalently the interior of the +contour is located to the left). The difference of sign in the second term follows directly +from Stokes’ theorem (B.14g) (and can be understood as a conjugation of the contour). +The additional factor of 1/2π is consistent with the normalization of spatial integrals in two +dimensions. The current components are not necessarily holomorphic and anti-holomorphic +at this level, but in practice this will often be the case (and each component is independently +conserved), and one writes +j(z) := jz(z), +¯ȷ(¯z) := j¯z(¯z). +(6.78) +In this case, the charge also splits into a holomorphic and an anti-holomorphic (left- and +right-moving8) contributions +Q = QL + QR, +QL := +1 +2πi +� +C0 +j(z)dz, +QR := − 1 +2πi +� +C0 +¯ȷ(¯z)d¯z. +(6.79) +The infinitesimal variation of a field under the symmetry generated by Q reads +δϵO(z, ¯z) = −[ϵQ, O(z, ¯z)] = −ϵ +� +Cz +dw +2πi j(w)O(z, ¯z) + ϵ +� +Cz +d ¯w +2πi ¯ȷ( ¯w)O(z, ¯z). +(6.80) +The contour integrals are easily evaluated once the OPE between the current and the oper- +ator is known. This formula gives the infinitesimal variation under the transformation for +any field, not only for primaries. +8For charges, we use subscript L and R to distinguish both sectors to avoid introducing a new symbol for +the total charge. However, since Q = QL in the holomorphic sector, it is often not necessary to distinguish +between the two symbols when acting on an operator or a state (however, this is useful for writing mode +expansions). We do not write a bar on QR because the charges don’t depend on the position. +93 + +Computation – Equation (6.77) +In real coordinates, the charge is defined by integrating the time component of the +current jµ over space for fixed time (A.23): +Q = 1 +2π +� +dσ j0. +The first step is to rewrite this formula covariantly. Since the time is fixed on the slice, +dτ = 0 and one can write +Q = 1 +2π +� +(dσ j0 − dτ j1) = − 1 +2π +� +ϵµνjµ dxν. +The last formula is valid for any contour. Moreover, it can be evaluated for complex +coordinates: +Q = − 1 +2π +� +ϵz¯z +� +jz d¯z − j ¯z dz +� += − i +4π +� � +jz d¯z − j ¯z dz +� += − 1 +2πi +� � +jz dz − j¯z d¯z +� +. +One finds a contour integral because τ = cst circles of the cylinder are mapped to +|z| = cst contours. +6.4.2 +Operator product expansions +The operator product expansion (OPE) is a tool used frequently in CFT: it means that +when two local operators come close to each other, it is possible to replace their product by +a sum of local operators +Oi(zi, ¯zi)Oj(zj, ¯zj) = +� +k +ck +ij +zhi+hj−hk +ij +¯z +¯hi+¯hj−¯hk +ij +Ok(zj, ¯zj), +(6.81) +where the OPE coefficients ck +ij are some constants and the sum runs over all operators. +When Ok is primary, the coefficients ck +ij are related to the structure constants and the field +metric by +Cijk = gkℓcℓ +ij. +(6.82) +The radius of convergence for the OPE is given by the distance to the nearest operators in +the correlation function. The OPE defines an associative algebra (commutative for bosonic +operators), and the holomorphic sector forms a subalgebra (called the chiral algebra). +Example 6.1 – OPE with the identity +The OPE of a field φ(z) with the identity 1 is found by a direct series expansion +φ(z)1 = +� +n∈N +(z − w)n +n! +∂nφ(w). +(6.83) +Obviously there are no singular terms. +Starting from this point we consider only the holomorphic sector except when stated +otherwise. The formula for the OPE (6.81) can be rewritten as +A(z)B(w) := +N +� +n=−∞ +{AB}n(z) +(z − w)n +(6.84) +94 + +to simplify the manipulations. +N is an integer and there are singular terms if N > 0. +Generally, only the terms singular as w → z are necessary in the computations (for example, +to use the Cauchy–Riemann formula (B.1)): equality up to non-singular terms is denoted +by a tilde +A(z)B(w) ∼ +N +� +n=1 +{AB}n(z) +(z − w)n =: A(z)B(w). +(6.85) +The RHS of this expression defines the contraction of the operators A and B. +While, most of the time, only singular terms are kept +φi(zi)φj(zj) ∼ +� +k +θ(hi + hj − hk) +ck +ij +(z − w)hi+hj−hk φk(w) +(6.86) +(with θ(x) the Heaviside step function), it can happen that one keeps also non-singular terms +(the product of two OPE have singular terms coming from non-singular terms multiplying +singular terms). Explicit contractions of operators through the OPE is also denoted by a +bracket when there are other operators. +For a primary field φ(z), one finds the OPE with the energy–momentum tensor to be +T(z)φ(w) ∼ +h φ(w) +(z − w)2 + ∂φ(w) +z − w , +(6.87) +where h is the conformal weight of the field. This OPE together with (6.80) for j(z) = +−v(z)T(z) correctly reproduces (6.50). +Computation – Equation (6.50) +δφ(z) = +� +Cz +dw +2πi v(w)T(w)φ(z) ∼ +� +Cz +dw +2πi v(w) +� h φ(z) +(w − z)2 + ∂φ(z) +w − z +� += h ∂v(z) φ(z) + v(z)∂φ(z). +For a non-primary operator, the OPE becomes more complicated (as it is reflected by +the transformation property), but the conformal weight can still be identified at the term +in z−2. The most important example is the energy–momentum tensor: the central charge is +found as the coefficient of the z−4 term its OPE with itself: +T(z)T(w) ∼ +c/2 +(z − w)4 + +2T(w) +(z − w)2 + ∂T(w) +z − w . +(6.88) +The OPE indicates that the conformal weight of T is h = 2. +Using (6.80) for j(z) = +−v(z)T(z), one finds the infinitesimal variation +δT = 2 ∂v T + v ∂T + c +12 ∂3v, +(6.89) +The last term vanishes for global transformations: this translates the fact that T is only a +quasi-primary. The finite form of this transformation is +T ′(w) = +� dz +dw +�−2 � +T(z) − c +12 S(w, z) +� += +� dz +dw +�−2 +T(z) + c +12 S(z, w) +(6.90) +where S(w, z) is the Schwarzian derivative +S(w, z) = w(3) +w′ − 3 +2 +�w′′ +w′ +�2 +, +(6.91) +95 + +where the derivatives of w are with respect to z. This vanishes if the transformation is in +SL(2, C), and it transforms as +S(u, z) = S(w, z) + +�dw +dz +�2 +S(u, w) +(6.92) +under successive changes of coordinates. +Computation – Equation (6.89) +δT(z) = +� +Cz +dw +2πi v(w)T(w)T(z) ∼ +� +Cz +dw +2πi v(w) +� +c/2 +(z − w)4 + +2T(w) +(z − w)2 + ∂T(w) +z − w +� += +c +2 × 3! ∂3v(z) + 2∂v(z) T(z) + v(z)∂T(z). +6.4.3 +Hermitian and BPZ conjugation +In this section, we introduce two different notions of conjugations: one is adapted for amp- +litudes because it defines a unitary Euclidean time evolution, while the second is more +natural as an inner product of CFT states. Both can be interpreted as providing a map +from in-states to out-states on the cylinder. +Given an operator O, we need to define an operation O‡ – called Euclidean adjoint (or +simply adjoint) – which, after Wick rotation from Euclidean to Lorentzian signature, can +be interpreted as the Hermitian adjoint.9 This is necessary in order to define a Hermitian +inner-product and to impose reality conditions. +To motivate the definition, consider first the cylinder in Lorentzian signature. +Since +Hermitian conjugation does not affect the Lorentzian coordinates, the Euclidean time must +reverse its sign: +t† = −iτ † = t +=⇒ +τ † = −τ. +(6.93) +Hence, an appropriate definition of the Euclidean adjoint is an Hermitian conjugation to- +gether with time reversal.10 +Another point of view is that the time evolution operator +U(τ) := e−τH is not unitary when H is Hermitian H† = H: the solution is to define a new +Euclidean adjoint U(τ)‡ := U(−τ)† such that U(τ) is unitary for it. +Time reversal on the cylinder corresponds to inversion and complex conjugation on the +complex plane: +z +τ→−τ +−−−−→ e−τ+iσ = 1 +z∗ = I(¯z), +(6.94) +where I(z) = 1/z is the inversion (6.42).11 On the real surface12 ¯z = z∗, which leads to the +definition of the Euclidean adjoint as follows: +O(z, ¯z)‡ := +�¯I ◦ O(z, ¯z) +�†, +(6.95) +9In [193], it is denoted by a bar on top of the operator: we avoid this notation since the bar already +denotes the anti-holomorphic sector. In [262], it is indicated by a subscript hc. Otherwise, in most of the +literature, it has no specific symbol since one directly works with the modes. +10The Euclidean adjoint can be used to define an inner product: positive-definiteness of the latter is +called reflection positivity or OS-positive and is a central axiom of constructive QFT. +11We do not write “z†” because this notation is confusing as one should not complex conjugate the factor +of i in the exponential (Section 6.2.1). +12Remember that ¯z is not the complex conjugate of z but an independent variable. +96 + +where ¯I(z) := 1/¯z. If O is quasi-primary, we have: +O(z, ¯z)‡ = +� +1 +¯z2hz2¯h O +�1 +¯z , 1 +z +��† += +1 +z2h¯z2¯h O† +�1 +z , 1 +¯z +� +, +(6.96) +The last equality shows that Euclidean conjugation is equivalent to take the conjugate of all +factors of i but otherwise leaves z and ¯z unaffected. The Euclidean adjoint acts by complex +conjugation of any c-number and reverses the order of the operators (acting as a transpose): +(λ O1 · · · On)‡ = λ∗ O‡ +n · · · O‡ +1, +λ ∈ C, +(6.97) +without any sign. +A second operation, called the BPZ conjugation, is useful. +It can be defined in two +different ways: +O(z, ¯z)t := I± ◦ O(z, ¯z) = (∓1)h+¯h +z2h¯z2¯h O +� +±1 +z , ±1 +¯z +� +, +(6.98) +where I±(z) = ±1/z is the inversion (6.42). The minus and plus signs are respectively more +convenient when working with the open and closed strings.13 The BPZ conjugation does +not complex conjugate c-number nor changes the order of the operators:14 +(λ O1 · · · On)t = λ Ot +1 · · · Ot +n, +λ ∈ C. +(6.99) +The identity is invariant under both conjugation +1‡ = 1t = 1. +(6.100) +6.4.4 +Mode expansion +Any field of weight (h, ¯h) can be expanded in terms of modes Om,n +O(z, ¯z) = +� +m,n +Om,n +zm+h¯zn+¯h . +(6.101) +Note that the modes Om,n themselves are operators. The ranges of the two indices are such +that +m + h ∈ Z + ν, +n + ¯h ∈ Z + ¯ν, +ν, ¯ν = +� +0 +periodic, +1/2 +anti-periodic. +(6.102) +The values of ν and ¯ν depend on whether the fields satisfies periodic or anti-periodic bound- +ary conditions on the plane (for half-integer weights, the periodicity is reversed on the +cylinder): +O(e2πiz, ¯z) = e2πiνO(z, ¯z), +O(z, e2πi¯z) = e2πi¯νO(z, ¯z). +(6.103) +Depending on whether the weights are integers or half-integers, additional terminology is +introduced: +13The index t should not be confused with the matrix transpose: it is used in opposition with ‡ and † to +indicate that no complex conjugation is involved. +14However, the fields become anti-radially ordered after a BPZ conjugation since it sends z to 1/z. The +radial ordering can be restored by (anti-)commuting the fields, which can introduce additional signs [220]. +This problem does not arise when working in terms of the modes. +97 + +• If h ∈ Z + 1/2, then one can choose anti-periodic (Neveu–Schwarz or NS) or periodic +(Ramond or R) boundary conditions on the cylinder (reversed for the plane): +ν, ¯ν = +� +0 +NS +1/2 +R +(6.104) +The indices are half-integers (resp. integers) for the NS (R) sector. +• If h ∈ Z, periodic (or untwisted) boundary conditions are more natural, but anti- +periodic boundary conditions may also be considered: +ν, ¯ν = +� +0 +untwisted +1/2 +twisted +(6.105) +The modes of untwisted (resp. twisted) fields have integer (half-integers) indices. +The mode expansions have no branch cut (fractional power of z or ¯z) for periodic fields +(bosonic untwisted or fermionic twisted). We will see explicit examples of such operators +later in this book. +Under Euclidean conjugation (6.95), the modes are related by +(O‡)−m,−n = (Om,n)†. +(6.106) +In particular, if the operator is Hermitian (under the Euclidean adjoint), the reality condition +on the modes relates the negative modes with the conjugated positive modes +O‡ = O +=⇒ +(Om,n)† = O−m,−n. +(6.107) +When no confusion is possible (for Hermitian operators), we will write O† +m,n instead of +(Om,n)†. +For a holomorphic field φ(z), the above expansion becomes +φ(z) = +� +n∈Z+h+ν +φn +zn+h . +(6.108) +Conversely, the modes are recovered from the field through +φn = +� +C0 +dz +2πi zn+h−1φ(z), +(6.109) +where the integration is counter-clockwise around the origin. +If the field is Hermitian, then +φ‡ = φ +=⇒ +(φn)† = φ−n. +(6.110) +The operators φn have a conformal weight of −n (since the weight of z is −1). The BPZ +conjugate of the modes is +φt +n = (I± ◦ φ)n = (−1)h(±1)nφ−n. +(6.111) +98 + +Computation – Equation (6.111) +φt +n = (I± ◦ φ)n = +� +dz +2πi zn+h−1I± ◦ φ(z) += +� +dz +2πi zn+h−1 +� +∓ 1 +z2 +�h +φ +� +±1 +z +� += (∓1)h +� +dz +2πi zn−h−1φ +� +±1 +z +� += (∓1)h +� dw +2πi +� +± 1 +w +�n−h +w−1φ(w) += (∓1)h(±1)n−h +� dw +2πi w−n+h−1φ(w), +where we have set w = ±1/z such that +dz +z = ∓ dw +w2z = −dw +w , +(6.112) +and the minus sign disappears upon reversing the contour orientation. +The mode expansion of the energy–momentum tensor is +T(z) = +� +n∈Z +Ln +zn+2 , +Ln = +� +dz +2πi T(z)zn+1, +(6.113) +where one recognizes the Virasoro operators as the modes. In most situations, the Virasoro +operators are Hermitian +L† +n = L−n. +(6.114) +The OPE (6.88) and (6.87) together with (6.75a) help to reconstruct the Virasoro algebra +(6.58) and the commutation relations between the Lm and the modes φn of a weight h +primary: +[Lm, φn] = +� +m(h − 1) − n +� +φm+n. +(6.115) +This easily gives the commutation relation for the complete field: +[Lm, φ(z)] = zm� +z∂ + (n + 1)h +� +φ(z). +(6.116) +We will often use (6.58) and (6.115) for m = 0: +[L0, L−n] = nL−n, +[L0, φ−n] = nφ−n. +(6.117) +This means that both φn and Ln act as raising operators for L0 if n < 0, and as lowering +operators if n > 0 (remember that L0 is the Hamiltonian in the holomorphic sector). When +both the holomorphic and anti-holomorphic sectors enter, it is convenient to introduce the +combinations +L± +n = Ln ± ¯Ln, +(6.118) +such that L+ +0 is the Hamiltonian. +Finally, every holomorphic current j(z) has a conformal weight h = 1 and can be expan- +ded as +j(z) = +� +n +jn +zn+1 . +(6.119) +By definition, the zero-mode is equal to the holomorphic charge +QL = j0. +(6.120) +99 + +6.4.5 +Hilbert space +The Hilbert space of the CFT is denoted by H. The SL(2, C) (or conformal) vacuum15 |0⟩ +is defined by the state which is invariant under the global conformal transformations: +L0 |0⟩ = 0, +L±1 |0⟩ = 0. +(6.121) +Expectation value of an operator O in the SL(2, C) vacuum is denoted as: +⟨O⟩ :=⟨0| O |0⟩ . +(6.122) +If the fields are expressed in terms of creation and annihilation operators (which happens +e.g. for free scalars, free fermions and ghosts), then the Hilbert space has the structure of a +Fock space. +State–operator correspondence +The state–operator correspondence identifies every state |O⟩ of the CFT Hilbert space with +an operator O(z, ¯z) through +|O⟩ = lim +z,¯z→0 O(z, ¯z) |0⟩ = O(0, 0) |0⟩ . +(6.123) +Such a state can be interpreted as an “in” state since it is located at τ → −∞ on the +cylinder. Focusing now on a holomorphic field φ(z), the state is defined as +|φ⟩ = lim +z→0 φ(z) |0⟩ = φ(0) |0⟩ . +(6.124) +For this to make sense, the modes which diverge as z → 0 must annihilate the vacuum. In +particular, for a weight h field φ(z), one finds: +∀n ≥ −h + 1 : +φn |0⟩ = 0. +(6.125) +Thus, the φn for n ≥ −h + 1 are annihilation operators for the vacuum |0⟩, and conversely +the states φn with n < −h + 1 are creation operators. As a consequence, the state |φ⟩ is +found by applying the mode n = −h to the vacuum: +|φ⟩ = φ−h |0⟩ = +� +dz +2πi +φ(z) +z +|0⟩ . +(6.126) +Since L−1 is the generator of translations on the plane, one finds +φ(z) |0⟩ = ezL−1φ(0)e−zL−1 |0⟩ = ezL−1 |φ⟩ . +(6.127) +The vacuum |0⟩ is the state associated to the identity 1. Translating the conditions (6.125) +to the energy–momentum tensor gives +∀n ≥ −1 : +Ln |0⟩ = 0. +(6.128) +This is consistent with the definition (6.121) since it includes the sl(2, C) subalgebra. +If h < 0, some of the modes with n > 0 do not annihilate the vacuum: (6.117) implies that +some states have an energy lower than the one of |0⟩. The state |Ω⟩ (possibly degenerate) +with the lowest energy is called the energy vacuum +∀ |φ⟩ ∈ H : +⟨Ω| L0 |Ω⟩ ≤⟨φ| L0 |φ⟩ . +(6.129) +15There are different notions of “vacuum”, see (6.129). However, the SL(2, C) vacuum is unique. Indeed, +it is mapped to the unique identity operator under the state–operator correspondence (however, there can +be other states of weight 0, see Remark 6.6). +100 + +It is obtained by acting repetitively with the modes φn>0. +This vacuum defines a new +partition of the non-zero-modes operators into annihilation and creation operators. If there +are zero-modes, i.e. n = 0 modes, then the vacuum is degenerate since they commute +with the Hamiltonian, [L0, φ0] = 0 according to (6.117). The partition of the zero-modes +into creation and annihilation operators depends on the specific state chosen among the +degenerate vacua. +The energy aΩ of |Ω⟩, which is also its L0 eigenvalue +L0 |Ω⟩ := aΩ |Ω⟩ , +(6.130) +is called zero-point energy. Bosonic operators with negative h are dangerous because they +lead to an infinite negative energy together with an infinite degeneracy (from the zero-mode). +The conjugate vacuum is defined by BPZ or Hermitian conjugation +⟨0| = |0⟩‡ = |0⟩t +(6.131) +since both leave the identity invariant. It is also annihilated by the sl(2, C) subalgebra: +⟨0| L0 = 0, +⟨0| L±1 = 0. +(6.132) +Since there are two kinds of conjugation, two different conjugated states can be defined. They +are also called “out” states since they are located at τ → ∞ on the cylinder (Figure 6.2). +Euclidean and BPZ conjugations and inner products +The Euclidean adjoint ⟨O‡| of the state |O⟩ is defined as +⟨O‡| = +lim +w, ¯ +w→0⟨0| O(w, ¯w)‡ = +lim +w, ¯ +w→0 +1 +w2h ¯w2¯h ⟨0| O +� 1 +¯w, 1 +w +�† +(6.133a) += +lim +z,¯z→∞ z2h¯z2¯h⟨0| O†(z, ¯z) +(6.133b) +=⟨0| I ◦ O†(0, 0), +(6.133c) +where the two coordinate systems are related by w = 1/¯z. From this formula, the definition +of the adjoint of a holomorphic operator φ follows +⟨φ‡| = lim +¯ +w→0⟨0| φ(w)‡ = lim +¯ +w→0 +1 +w2h ⟨0| φ† +� 1 +w +� +(6.134a) += lim +z→∞ z2h⟨0| φ†(z) +(6.134b) +=⟨0| I ◦ φ†(0). +(6.134c) +Then, expanding the field in terms of the modes gives +⟨φ‡| =⟨0| (φ†)h. +(6.135) +The BPZ conjugated state is +⟨φ| := lim +w→0⟨0| φ(w)t +(6.136a) += (±1)h lim +z→∞ z2h⟨0| φ(z) +(6.136b) +=⟨0| I± ◦ φ(0). +(6.136c) +In terms of the modes, one has +⟨φ| = (±1)h⟨0| φh. +(6.137) +101 + +If φ is Hermitian, then the relation between both conjugated states corresponds to a reality +condition: +⟨φ‡| = (±1)h⟨φ| . +(6.138) +Taking the BPZ conjugation of the conditions (6.125) tells which modes must annihilate +the conjugate vacuum: +∀n ≤ h − 1 : +⟨0| φn = 0, +(6.139) +and one finds more particularly for the Virasoro operators +∀n ≤ 1 : +⟨0| Ln = 0. +(6.140) +This can also be derived directly from (6.136) by requiring that applying an operator on the +conjugate vacuum ⟨0| is well-defined. +All conditions taken together mean that the expectation value of the energy–momentum +tensor in the conformal vacuum vanishes: +⟨0| T(z) |0⟩ = 0. +(6.141) +In particular, this means that the energy vacuum |Ω⟩, if different from |0⟩, has a negative +energy. +The Hermitian16 and BPZ inner products are respectively defined by: +⟨φ‡ +i|φj⟩ =⟨0| ¯I ◦ φj(0)φi(0) |0⟩ = lim +z→∞ +w→0 +z2hi⟨0| φ† +i(z)φj(w) |0⟩ , +(6.142a) +⟨φi|φj⟩ =⟨0| I ◦ φj(0)φi(0) |0⟩ = (±1)hi lim +z→∞ +w→0 +z2hi⟨0| φi(z)φj(w) |0⟩ . +(6.142b) +These products can be recast as 2-point correlation functions (6.62b) on the sphere: +⟨φi|φj⟩ = ⟨I ◦ φi(0)φj(0)⟩, +⟨φ‡ +i|φj⟩ = ⟨I ◦ φ† +i(0)φj(0)⟩. +(6.143) +From the state–operator correspondence, the action of one operator on the in-state can be +reinterpreted as the matrix element of this operator using the two external states, or also as +a 3-point function: +⟨φi| φj(z) |φk⟩ = (±1)hi lim +w→∞ w2hi⟨φi(w)φj(z)φk(0)⟩. +(6.144) +Given a basis of states {φi} (i can run over both discrete and continuous indices), the +conjugate or dual states {φc +i} are defined by: +⟨φc +i|φj⟩ = δij +(6.145) +(the delta function is discrete and/or continuous according to the indices). +Verma modules +If φ(z) is a weight h primary, then the associated state |φ⟩ satisfies: +L0 |φ⟩ = h |φ⟩ , +∀n ≥ 1 : +Ln |φ⟩ = 0. +(6.146) +Such a state is also called a highest-weight state. The descendant states are defined by all +possible states of the form +|φ{ni}⟩ := +� +i +L−ni |φ⟩ , +(6.147) +where the same L−ni can appear multiple times and ni > 0. The set of states φ{ni} is called +a Verma module V (h, c). One finds that the L0 eigenvalues of this state is +L0 = h + +� +i +ni. +(6.148) +16Depending on the normalization, it can also be anti-Hermitian. +102 + +Normal ordering +The normal ordering of an operator with respect to a vacuum corresponds to placing all cre- +ation (resp. annihilation) operators of this vacuum on the left (resp. right). From this defin- +ition, the expectation value of a normal ordered operator in the vacuum vanishes identically. +The main reason for normal ordering is to remove singularities in expectation values. +Given an operator φ(z), we define two normal orderings: +• The conformal normal order (CNO) :O: is defined with respect to the conformal va- +cuum (6.121): +⟨0| :O: |0⟩ = 0. +(6.149) +• The energy normal order (ENO) +⋆ +⋆ O +⋆ +⋆ is defined with respect to the energy vacuum +(6.129): +⟨Ω| +⋆ +⋆O +⋆ +⋆ |Ω⟩ = 0. +(6.150) +We first discuss the conformal normal ordering before explaining how to relate it to the +energy normal ordering. +Given two operators A and B, the simplest normal ordering amounts to subtract the +expectation value: +:A(z)B(w): +?= A(z)B(w) − ⟨A(z)B(w)⟩. +(6.151) +This is equivalent to defining the products of two operators at coincident points via point- +splitting: +:A(z)B(z): +?= lim +w→z +� +A(z)B(w) − ⟨A(z)B(w)⟩ +� +. +(6.152) +While this works well for free fields, this does not generalize for composite or interacting +fields. +The reason is that this procedure removes only the highest singularity in the product: it +does not work if the OPE has more than one singular term. An appropriate definition is +:A(z)B(w): := A(z)B(w) − A(z)B(w) = +� +n∈N +(z − w)n{AB}−n(z), +(6.153) +where the contraction between A and B is defined in (6.85), and the second equality comes +from (6.84). +Then, the product evaluated at coincident points is found by taking the limit (in this +case the argument is often indicated only at the end of the product) +:AB(z): := :A(z)B(z): := lim +w→z :A(z)B(w): = {AB}0(z). +(6.154) +Indeed, since all powers of (z − w) are positive in the RHS of (6.153), all terms but the first +one disappear. The form of (6.154) shows that the normal order can also be computed with +the contour integral +:AB(z): = +� +Cz +dw +2πi +A(z)B(w) +z − w +. +(6.155) +It is common to remove the colons of normal ordering when there is no ambiguity and, in +particular, to write: +AB(z) := :AB(z):. +(6.156) +In terms of modes, one has +:AB(z): = +� +m +:AB:m +zm+hA+hB , +(6.157a) +:AB:m = +� +n≤−hA +AnBm−n + +� +n>−hA +Bm−nAn. +(6.157b) +103 + +This expression makes explicit that normal ordering is non-commutative and non-associative: +:AB(z): ̸= :BA(z):, +:A(BC)(z): ̸= :(AB)C(z):. +(6.158) +The product of normal ordered operators can then be computed using Wick theorem. In +fact, one is more interested in the contraction of two such operators in order to recover the +OPE between these operators: the product is then derived with (6.153). +If Ai (i = 1, 2, 3) are free fields, one has +A1(z) :A2A3(w): = :A1(z)A2A3(w): + A1(z) :A2 A3(w):, +A1(z) :A2 A3(w): = A1(z)A2(w) :A3(w): + A1(z)A3(w) :A2(w):. +(6.159) +If the fields are not free, then the contraction cannot be extracted from the normal ordering. +Similarly if there are more fields, then one needs to perform all the possible contractions. +Given two free fields A and B, one has the following identities: +A(z) :B(w)n: = n A(z)B(w) :B(w)n−1:, +(6.160a) +A(z) :eB(w): = A(z)B(w) :eB(w):, +(6.160b) +:eA(z): :eB(w): = exp +� +A(z)B(w) +� +:eA(z)eB(w):. +(6.160c) +The last relation generalizes for a set of n fields Ai: +n +� +i=1 +:eAi: = : exp +� n +� +i=1 +Ai +� +: exp +� +i0 +Bm−nAn. +(6.162b) +To simplify the definition we assume that A0 is a creation operator and it is thus included +in the first sum (this must be adapted in function of which vacuum state is chosen if the +latter is degenerate). +The relation between the normal ordered modes is +:AB:m = +⋆ +⋆AB +⋆ +⋆m + +hA−1 +� +n=0 +[Bm+n, A−n]. +(6.163) +Computation – Equation (6.163) +:AB:m = +� +n≤−hA +AnBm−n + +� +n>−hA +Bm−nAn += +� +n≥hA +A−nBm+n + +� +n>0 +Bm−nAn + +hA−1 +� +n=0 +Bm+nA−n += +� +n≥0 +A−nBm+n + +� +n>0 +Bm−nAn + +hA−1 +� +n=0 +[Bm+n, A−n] += +⋆ +⋆AB +⋆ +⋆m + +hA−1 +� +n=0 +[Bm+n, A−n]. +The choice of the normal ordering for the operators is related to the ordering ambiguity +when quantizing the system: when the product of two non-commuting modes appears in the +classical composite field, the corresponding quantum operator is ambiguous (generally up to +a constant). In practice, one starts with the conformal ordering since it is invariant under +conformal transformations and because one can compute with contour integrals. Then, the +expression can be translated in the energy ordering using (6.163). But, knowing how the +conformal and energy vacua are related, it is often simpler to find the difference between +the two orderings by applying the operator on the vacua. +6.4.6 +CFT on the cylinder +According to (6.44), the relation between the field on the cylinder and on the plane is +φ(z) = +� L +2π +�h +z−hφcyl(w) +(6.164) +(quantities without indices are on the plane by definition). The mode expansion on the +cylinder is +φcyl = +�2π +L +�h � +n∈Z +φne− 2π +L w = +�2π +L +�h � +n∈Z +φn +zn . +(6.165) +105 + +Using the finite transformation (6.90) for the energy–momentum tensor T, one finds the +relation +Tcyl(w) = +�2π +L +�2 � +T(z)z2 − c +24 +� +. +(6.166) +For the L0 mode, one finds +(L0)cyl = L0 − c +24, +(6.167) +and thus the Hamiltonian is +H = (L0)cyl + (¯L0)cyl = L0 + ¯L0 − c + ¯c +24 . +(6.168) +6.5 +Suggested readings +• The most complete reference on CFTs is [54] but it lacks some recent developments. +Two excellent complementary books are [25, 206]. +String theory books generally dedicate a fair amount of pages to CFTs: particularly +good summaries can be found in [24, 128, 193, 194]. +Finally, a modern and fully algebraic approach can be found in [200, 201]. Other good +reviews are [196, 259]. +• There are various other books [104, 120, 126, 171] and reviews [32, 89, 91, 204, 243]. +• The maps from the sphere and the cylinder to the complex plane are discussed in [193, +sec. 2.6, 6.1]. +• Normal ordering is discussed in details in [54, chap. 6] (see also [24, sec. 4.2, 193, +sec. 2.2]). +• Euclidean conjugation is discussed in [54, sec. 6.1.1, 193, p. 202–3]. For a comparison +of Euclidean and BPZ conjugations, see [262, sec. 2.2, 220, p. 11]. +• Normal ordering and difference between the different definitions are described in [193, +chap. 2, 54, sec. 6.5]. +106 + +Chapter 7 +CFT systems +Abstract +This chapter summarizes the properties of some CFT systems. We focus on +the free scalar field and on the first-order bc system (which generalizes the reparametriz- +ation ghosts). For the different systems, we first provide an analysis on a general curved +background before focusing on the complex plane. This is sufficient to describe the local +properties on all Riemann surfaces g ≥ 0. +7.1 +Free scalar +7.1.1 +Covariant action +The Euclidean action of a free scalar X on a curved background gµν is +S = +ϵ +4πℓ2 +� +d2x√g gµν∂µX∂νX, +(7.1) +where ℓ is a length scale1 and +ϵ := +� ++1 +spacelike +−1 +timelike , +√ϵ := +� ++1 +spacelike +i +timelike +(7.2) +denotes the signature of the kinetic term. The field is periodic along σ +X(τ, σ) ∼ X(τ, σ + 2π). +(7.3) +The energy–momentum tensor reads +Tµν = − ϵ +ℓ2 +� +∂µX∂νX − 1 +2 gµν(∂X)2 +� +, +(7.4) +and it is traceless +T µ +µ = 0. +(7.5) +The equation of motion is +∆X = 0, +(7.6) +where ∆ is the Laplacian (A.28). +1To be identified with the string scale, such that α′ = ℓ2. +107 + +The simplest method for finding the propagator in flat space is by using the identity +(assuming that there is no boundary term) +0 = +� +dX +δ +δX(σ) +� +e−S[X]X(σ′) +� +, +(7.7) +which yields a differential equation for the propagator: +⟨∂2X(σ)X(σ′)⟩ = −2πϵℓ2 δ(2)(σ − σ′). +(7.8) +This is easily integrated to +⟨X(σ)X(σ′)⟩ = −ϵℓ2 +2 +ln |σ − σ′|2. +(7.9) +Computation – Equation (7.9) +By translation and rotation invariance, one has +⟨X(σ)X(σ′)⟩ = G(r), +r = |σ − σ′|. +(7.10) +In polar coordinates, the Laplacian reads +∆G(r) = 1 +r ∂r(rG′(r)). +(7.11) +Integrating the differential equation (7.8) over d2σ = rdrdθ yields +−2πϵℓ2 = 2π +� r +0 +dr′ r′ × 1 +r′ ∂r′(r′G′(r′)) = 2πrG′(r). +(7.12) +The solution is +G′(r) = −ϵℓ2 ln r +(7.13) +and the form (7.9) follows by writing +ln r = 1 +2 ln r2 = 1 +2 ln |σ − σ′|2. +(7.14) +The action (7.1) is obviously invariant under constant translations of X: +X −→ X + a, +a ∈ R. +(7.15) +The associated U(1) current2 is conserved and reads +Jµ := 2πiϵ +∂L +∂(∂µX) = i +ℓ2 gµν∂νX, +∇µJµ = 0. +(7.16) +On flat space, the charge follows from (A.23): +p = 1 +2π +� +dσ J0 = +i +2πℓ2 +� +dσ ∂0X. +(7.17) +This charge is called momentum because it corresponds to the spacetime momentum in +string theory. +2The group is R but the algebra is u(1) (since locally there is no difference between the real line and the +circle). +108 + +Moreover, there is a another topological current +�Jµ := −i ϵµνJν = 1 +ℓ2 ϵµν∂νX, +(7.18) +which is identically conserved: +∇µ �Jµ ∝ ϵµν[∇µ, ∇ν]X = 0 +(7.19) +since [∇µ, ∇ν] = 0 when acting on a scalar field. Note that ˜Jµ is the Hodge dual of Jµ. The +conserved charge is called the winding number and reads on flat space: +w = 1 +2π +� +dσ �J0 = +1 +2πℓ2 +� 2π +0 +dσ ∂1X = +1 +2πℓ2 +� +X(τ, 2π) − X(τ, 0) +� +. +(7.20) +Remark 7.1 (Normalization of the current) The definition of the current (7.16) may +look confusing. The factor of i is due to the Euclidean signature, see (A.25a), and the factor +of 2π comes from the normalization of the spatial integral. We have inserted ϵ in order to +interpret the conserved charge p as a component of the momentum contravariant vector in +string theory. +To make contact with string theory, consider D scalar fields Xa(xµ). Then, the current +becomes +Jµ +a = +i +2πℓ2 ηab∂µXb, +(7.21) +where the position of the indices is in agreement with the standard form of Noether’s formula +(A.25a) (a current has indices in opposite locations as the parameters and fields). Since we +have η00 = −1 = ϵX0, we find that J0µ = ϵX0Jµ +0 has no epsilon after replacing the expression +(7.17) of Jµ +0 . +The transformation Xa → Xa +ca is a global translation in target spacetime: the charge +pa is identified with the spacetime momentum. +The factor of i indicates that pa is the +Euclidean contravariant momentum vector by comparison with (A.7). +The convention of this section is to always work with quantities which will become con- +travariant vector to avoid ambiguity. +7.1.2 +Action on the complex plane +In complex coordinates, the action on flat space reads +S = +ϵ +2πℓ2 +� +dzd¯z ∂zX∂¯zX, +(7.22) +giving the equation of motion: +∂z∂¯zX = 0. +(7.23) +This indicates that ∂zX and ∂¯zX are respectively holomorphic and anti-holomorphic such +that +X(z, ¯z) = XL(z) + XR(¯z), +(7.24) +and we will remove the subscripts when there is no ambiguity (for example, when the position +dependence is written): +X(z) := XL(z), +X(¯z) := XR(¯z). +(7.25) +It looks like XL(z) and XR(¯z) are unrelated, but this is not the case because of the zero- +mode, as we will see below. +109 + +The U(1) current is written as +J := Jz = i +ℓ2 ∂zX, +¯J := J¯z = i +ℓ2 ∂¯zX, +(7.26) +where we used the relations Jz = J ¯z/2 and J¯z = Jz/2. The equation of motion implies that +the current J is holomorphic, and ¯J is anti-holomorphic: +¯∂J = 0, +∂ ¯J = 0. +(7.27) +The momentum splits into left- and right-moving parts: +p = pL + pR, +pL = +1 +2πi +� +dz J, +pR = − 1 +2πi +� +d¯z ¯J. +(7.28) +The components of the topological current (7.18) are related to the ones of the U(1) +current: +�Jz = i +ℓ2 ∂zX = J, +�J¯z = − i +ℓ2 ∂¯zX = − ¯J. +(7.29) +As a consequence, the winding number is +w = pL − pR. +(7.30) +Note that we have the relations +pL = p + w +2 +, +pR = p − w +2 +, +(7.31a) +p2 + w2 = p2 +L + p2 +R, +2pw = p2 +L − p2 +R. +(7.31b) +The energy–momentum tensor is +T := Tzz = − ϵ +ℓ2 ∂zX∂zX, +¯T := T¯z¯z = − ϵ +ℓ2 ∂¯zX∂¯zX, +Tz¯z = 0. +(7.32) +Since the ∂zX (∂¯zX) is (anti-)holomorphic, so is T(z) ( ¯T(¯z)). Since the energy–momentum +tensor, the current and the field itself (up to zero-modes) split in holomorphic and anti- +holomorphic components in a symmetric way, it is sufficient to focus on one of the sectors, +say the holomorphic one. +The other primary operators of the theory are given by the vertex operators Vk(z):3 +Vk(z, ¯z) := :eiϵkX(z,¯z):. +(7.33) +Remark 7.2 In fact, it is possible to introduce more general vertex operators +VkL,kR(z, ¯z) := :e2iϵ +� +kLX(z)+kRX(¯z) +� +:, +(7.34) +but we will not consider them in this book. +Remark 7.3 (Plane and cylinder coordinates) The action in w-coordinate (cylinder) +takes the same form as a result of the conformal invariance of the scalar field, which in +practice results from the cancellation between the determinant and inverse metric. As a +consequence, every quantity derived from the classical action (equation of motion, energy– +momentum tensor. . . ) will have the same form in both coordinate systems: we will focus +on the z-coordinate, writing the w-coordinate expression when it is insightful to compare. +This is not anymore the case at the quantum level: anomalies may translate into differences +between quantities: to differentiate between the plane and cylinder quantities an index “cyl” +will be added when necessary (by convention, all quantities without qualification are on the +plane). +3The ϵ in the exponential is consistent with interpreting X and k as a contravariant vector. +110 + +7.1.3 +OPE +The OPE between X and itself is directly found from the propagator: +X(z)X(w) ∼ −ϵℓ2 +2 +ln(z − w). +(7.35) +By successive derivations, one finds the OPE between X and ∂X +∂X(z)X(w) ∼ −ϵℓ2 +2 +1 +z − w, +(7.36) +and between ∂X with itself +∂X(z)∂X(w) ∼ −ϵℓ2 +2 +1 +(z − w)2 . +(7.37) +The invariance under the permutation of z and w reflects the fact that X is bosonic and +that both operators in (7.37) are identical. +The OPE between ∂X and T allows to verify that the field ∂X is primary with h = 1: +T(z)∂X(w) ∼ ∂X(w) +(z − w)2 + ∂ +� +∂X(w) +� +z − w +. +(7.38) +The OPE of T with itself gives +T(z)T(w) ∼ 1 +2 +1 +(z − w)4 + +2T(w) +(z − w)2 + ∂T(w) +z − w +(7.39) +which shows that the central charge is +c = 1. +(7.40) +One finds that the operator ∂nX has conformal weight +h = n +(7.41) +since the OPE with T is +T(z)∂nX(w) ∼ · · · + n ∂nX(w) +(z − w)2 + ∂(∂nX(w)) +z − w +(7.42) +where the dots indicate higher negative powers of (z − w). These states are not primary for +n ≥ 2. Explicitly, for n = 2, one finds +T(z)∂2X(w) ∼ 2 ∂X(w) +(z − w)3 + +2 ∂2X +(z − w)2 + ∂(∂2X(w)) +z − w +. +(7.43) +The OPE of a vertex operator with the current J is +J(z)Vk(w, ¯w) ∼ ℓ2k +2 +Vk(w, ¯w) +z − w +. +(7.44) +This shows that the vertex operators Vk are eigenstates of the U(1) holomorphic current +with the eigenvalue given by the momentum (with a normalization of ℓ2). Then, the OPE +with T: +T(z)Vk(w, ¯w) ∼ hk Vk(w, ¯w) +(z − w)2 ++ ∂Vk(w, ¯w) +z − w +(7.45) +111 + +together with its anti-holomorphic counterpart show that the Vk are primary operators with +weight +(hk, ¯hk) = +�ϵℓ2k2 +4 +, ϵℓ2k2 +4 +� +, +∆k = ϵℓ2k2 +2 +, +sk = 0. +(7.46) +Note that classically hk = 0 since ℓ ∼ ℏ [246, p. 81]. The weight is invariant under k → −k. +Finally, the OPE between two vertex operators is +Vk(z, ¯z)Vk′(w, , ¯w) ∼ +Vk+k′(w, ¯w) +(z − w)−ϵkk′ℓ2/2 , +(7.47) +where only the leading term (non-necessarily singular) is displayed. In particular, correlation +functions should be computed for ϵkk′ < 0 in order to avoid exponential growth. +Computation – Equation (7.38) +T(z)∂X(w) = − ϵ +ℓ2 :∂X(z)∂X(z): ∂X(w) ∼ −2ϵ +ℓ2 :∂X(z)∂X(z): ∂X(w) ∼ +∂X(z) +(z − w)2 . +The result (7.38) follows by Taylor expanding the numerator. +Computation – Equation (7.39) +T(z)∂X(w) = 1 +ℓ4 :∂X(z)∂X(z): :∂X(w)∂X(w): +∼ 1 +ℓ4 +� +:∂X(z)∂X(z): :∂X(w)∂X(w): + :∂X(z)∂X(z): :∂X(w)∂X(w): ++ :∂X(z)∂X(z): :∂X(w)∂X(w): + perms +� +∼ 2 × 1 +4 +1 +(z − w)4 − 4 × 1 +2ℓ2 +1 +(z − w)2 :∂X(z)∂X(w): +∼ 1 +2 +1 +(z − w)4 − 2 +ℓ2 +1 +(z − w)2 +� +:∂X(w)∂X(w): + (z − w) :∂2X(w)∂X(w): +� +. +Computation – Equation (7.42) +T(z)∂nX(w) ∼ ∂n−1 +w +∂X(z) +(z − w)2 +∼ n! +∂X(z) +(z − w)n+1 +∼ +n! +(z − w)n+1 +� +· · · + +1 +(n − 1)! (z − w)n−1∂n−1(∂X(w)) ++ 1 +n! (z − w)n∂n(∂X(w)) +� +. +112 + +Computation – Equation (7.44) +Using (6.160b), one has: +∂X(z)Vk(w, ¯w) ∼ iϵk ∂X(z)X(w) Vk(w, ¯w) ∼ iϵk +� +−ϵℓ2 +2 +1 +z − w +� +Vk(w, ¯w). +Computation – Equation (7.45) +T(z)Vk(w, ¯w) ∼ − ϵ +ℓ2 :∂X(z)∂X(z): :eiϵkX(w, ¯ +w): +∼ iϵk +2 +1 +z − w ∂X(z) :eiϵkX(w, ¯ +w): − ϵ +ℓ2 ∂X(z) :∂X(z)eiϵkX(w, ¯ +w): +∼ iϵk +2 +1 +z − w +� +:∂X(z) eiϵkX(w, ¯ +w): + ∂X(z) :eiϵkX(w, ¯ +w): +� ++ iϵk +2 +:∂X(z)eiϵkX(w, ¯ +w): +z − w +∼ ϵk2ℓ2 +4 +Vk(w, ¯w) +(z − w)2 + iϵk :∂X(w)eiϵkX(w, ¯ +w): +z − w +. +In the first line, we consider a single contraction (hence, there is no factor of 2): the +reason is that considering the contractions symmetrically and not successively counts +twice the first term of the last line. Indeed, there is only one way to generate this term. +It is also possible to achieve the same result by expanding the exponential. +Computation – Equation (7.47) +Using (6.160c) and keeping only the leading term, one has: +Vk(z, ¯z)Vk′(w, ¯w) ∼ exp +� +− kk′ X(z, ¯z)X(w, ¯w) +� +:eiϵkX(z,¯z)eiϵk′X(w, ¯ +w): +∼ (z − w)ϵkk′ℓ2/2 Vk+k′(w, ¯w). +7.1.4 +Mode expansions +Since ∂X is holomorphic and of weight h = 1, it can be expanded as:4 +∂X = −i +� +ℓ2 +2 +� +n∈Z +αn z−n−1, +¯∂X = −i +� +ℓ2 +2 +� +n∈Z +¯αn ¯z−n−1, +(7.48) +where an individual mode can be extracted with a contour integral: +αn = i +� +dz +2πi zn−1∂X(z), +¯αn = i +� +dz +2πi zn−1 ¯∂X(z). +(7.49) +4The Fourier expansion is taken to be identical for ϵ = ±1 fields since ∂X is contravariant in target +space. The difference between the two cases will appear in the commutators. +113 + +Integrating this formula gives: +X(z) = xL +2 − i +� +ℓ2 +2 α0 ln z + i +� +ℓ2 +2 +� +n̸=0 +αn +n z−n, +X(¯z) = xR +2 − i +� +ℓ2 +2 ¯α0 ln ¯z + i +� +ℓ2 +2 +� +n̸=0 +¯αn +n ¯z−n. +(7.50) +The zero-modes are respectively α0 and ¯α0 for ∂X and ¯∂X, and xL and xR for XL and +XR. The meaning of the modes will become clearer in Section 7.1.5 where we study the +commutation relations. +First, we relate the zero-modes α0 and ¯α0 to the conserved charges pL and pR (7.28) of +the U(1) current: +pL = +α0 +√ +2ℓ2 , +pR = +¯α0 +√ +2ℓ2 +(7.51) +such that +X(z) = xL +2 − iℓ2 pL ln z + i +� +ℓ2 +2 +� +n̸=0 +αn +n z−n, +(7.52) +Then, the relations (7.28) and (7.30) allow to rewrite this result in terms of the momentum +p and winding w: +p = +1 +√ +2ℓ2 +� +α0 + ¯α0 +� +, +w = +1 +√ +2ℓ2 +� +α0 − ¯α0 +� +. +(7.53) +These relations can be inverted as +α0 = +� +ℓ2 +2 (p + w), +¯α0 = +� +ℓ2 +2 (p − w). +(7.54) +In the same sense that there are two momenta pL and pR conjugated to xL and xR, +it makes sense to introduce two coordinates x and q conjugated to p and w. From string +theory, the operator x is called the center of mass. The expression (7.54) suggests to write: +xL = x + q, +xR = x − q, +(7.55) +and conversely: +x = 1 +2 (xL + xR), +q = 1 +2 (xL − xR). +(7.56) +In terms of these new variables, the expansion of the full X(z, ¯z) reads: +X(z, ¯z) = x − i ℓ2 +2 +� +p ln |z|2 + w ln z +¯z +� ++ i +� +ℓ2 +2 +� +n̸=0 +1 +n +� +αn z−n + ¯αn ¯z−n� +. +(7.57) +In terms of the coordinates on the cylinder, the part without oscillations becomes: +X(τ, σ) = x − i ℓ2 pτ + ℓ2 wσ + · · · +(7.58) +Note how the presence of ℓ2 gives the correct scale to the second term. The mode q does not +appear at all, and x is the zero-mode of the complete field X(z, ¯z). As it is well-known, the +physical interpretation of x and p is as the position and momentum of the centre-of-mass +of the string.5 If there is a compact dimension, then w counts the number of times the +string winds around it, and q can be understood as the position of the centre-of-mass after +a T-duality.6 +5In worldsheet Lorentzian signature, this becomes X(τ, σ) = x + ℓ2 pt + ℓ2 wσ as expected. +6T-duality and compact bosons fall outside the scope of this book and we refer the reader to [265, +chap. 17, 193, chap. 8] for more details. +114 + +Computation – Equation (7.51) +pL = +1 +2πi +� +dz J = i +ℓ2 +1 +2πi +� +dz ∂X = i +ℓ2 +1 +2πi +� +dz ∂X += +1 +√ +2ℓ2 +1 +2πi +� +dz +� +n +αn z−n−1 = +1 +√ +2ℓ2 α0. +The computation gives pR after replacing α0 by ¯α0. +If the scalar field is non-compact but periodic on the cylinder, the periodicity condition +X(τ, σ + 2π) ∼ X(τ, σ) +(7.59) +translates as +X(e2πiz, e−2πi¯z) ∼ X(z, ¯z). +(7.60) +Evaluating the LHS from (7.50) gives a constraint on the zero-modes: +X(e2πiz, e−2πi¯z) = X(z, ¯z) − i +� +ℓ2 +2 (α0 − ¯α0), +(7.61) +which implies +α0 = ¯α0 +=⇒ +pL = pR = p +2, +w = 0. +(7.62) +The other cases will not be discussed in this book, but we still use the general notation to +make the contact with the literature easier. This also implies that XL and XR cannot be +periodic independently. Hence, the zero-mode couples the holomorphic and anti-holomorphic +sectors together. +The number operators Nn ¯Nn at level n > 0 are defined by: +Nn = ϵ +n α−nαn, +¯Nn = ϵ +n ¯α−n¯αn. +(7.63) +The modes have been normal ordered. They count the number of excitations at the level n: +the factor n−1 is necessary because the modes are not canonically normalized. Then, one +can build the level operators +N = +� +n>0 +n Nn. +(7.64) +They count the number of excitations at level n weighted by the level itself. This corresponds +to the total energy due to the oscillations (the higher the level, the more energy it needs to +be excited). +The Virasoro operators are +Lm = ϵ +2 +� +n +:αnαm−n: +(7.65) +For m ̸= 0, we have +m ̸= 0 : +Lm = ϵ +2 +� +n̸=0,m +:αnαm−n: + ϵ α0αm, +(7.66) +there is no ordering ambiguity and the normal order can be removed. In the case of the +zero-mode, one finds +L0 = ϵ +2 +� +n +:αnα−n: = N + ϵ +2 α2 +0 = N + ϵℓ2 p2 +L, +(7.67) +115 + +using (7.64) and (7.51). It is also useful to define �L0 which corresponds to L0 stripped from +the zero-mode contribution: +�L0 := N. +(7.68) +Similarly, the anti-holomorphic zero-mode is +¯L0 = ¯N + ϵℓ2 p2 +R, +�¯L0 := ¯N, +(7.69) +such that +L+ +0 = N + ¯N + ϵℓ2 (p2 +L + p2 +R) = N + ¯N + ϵℓ2 +2 (p2 + w2), +(7.70a) +L− +0 = N − ¯N + ϵℓ2 (p2 +L − p2 +R) = N − ¯N + ϵℓ2 wp, +(7.70b) +where L± +0 := L0 ± ¯L0 as defined in (6.118). The last equality of each line follows from +(7.31b). The expression of L+ +0 for N = ¯N = 0 matches the weights (7.46) of the vertex +operators for pL = pR = p/2 (no winding), which will be interpreted below. It is a good +place to stress that pL, pR, p and w are operators, while k is a number. +7.1.5 +Commutators +The commutators can be computed from (6.75a) knowing the OPE (7.37). The modes of +∂X and ¯∂X satisfy +[αm, αn] = ϵ m δm+n,0, +[¯αm, ¯αn] = ϵ m δm+n,0, +[αm, ¯αn] = 0 +(7.71) +for all m, n ∈ Z (including the zero-modes). The appearance of the factor m in the RHS +explains the normalization of the number operator (7.63). +From the commutators of the zero-modes, we directly find the ones for the momentum +and winding: +[p, w] = [p, p] = [w, w] = 0, +[p, αn] = [p, ¯αn] = [w, αn] = [w, ¯αn] = 0. +(7.72) +The OPE (7.36) yields +[xL, pL] = iϵ, +[xR, pR] = iϵ, +(7.73) +which can be used to determine the commutators of x and q: +[x, p] = [q, w] = iϵ, +[x, w] = [q, p] = 0. +(7.74) +This shows that (x, p) and (q, w) are pairs of conjugate variables. Interestingly, the winding +number w commutes will all other modes except q, but the latter disappears from the +description. Hence, it can be interpreted as a number which labels different representations: +if no other principle (like periodicity) forbids w ̸= 0, then one can except to have states with +all possible w in the spectrum, each value of w forming a different sector. There are other +interpretations from the point of view of T-duality and double field theory [107, 110, 189, +265]. +The commutator of the modes with the Virasoro operators is +[Lm, αn] = −n αm+n. +(7.75) +as expected from (6.115). For m = 0, this reduces to +[L0, α−n] = n α−n, +(7.76) +which shows that negative modes increase the energy. +The commutator of the creation +modes α−n with the number operators is +[Nm, α−n] = α−mδm,n. +(7.77) +116 + +7.1.6 +Hilbert space +The Hilbert space of the free scalar has the structure of a Fock space. +From (7.76), the momentum p commutes with the Hamiltonian L+ +0 such that it is a +good quantum number to label the states:7 this translates the fact the action (7.1) does not +depend on the conjugate variable x. As a consequence, there exists a family of vacua |k⟩. +The vacua |k⟩ are the states related to the vertex operators (7.33) through the state- +operator correspondence: +|k⟩ := lim +z,¯z→0 Vk(z, ¯z) |0⟩ = eiϵkx |0⟩ , +(7.78) +where |0⟩ is the SL(2, C) vacuum and x is the zero-mode of X(z, ¯z). That this identification +is correct follows by applying the operator p: +p |k⟩ = k |k⟩ . +(7.79) +The notation is consistent with the one of the SL(2, C) vacuum since p |0⟩ = 0. +The vacuum is annihilated by the action of the positive-frequency modes: +∀n > 0 : +αn |k⟩ = 0, +(7.80) +which is equivalent to +Nn |k⟩ = 0. +(7.81) +The different vacua are each ground state of a Fock space (they are all equivalent), but they +are not ground states of the Hamiltonian since they have different energies: +L+ +0 |k⟩ = 2ϵℓ2 k2 |k⟩ , +L− +0 |k⟩ = 0, +(7.82) +using (7.70). The SL(2, C) vacuum is the lowest (highest) energy state if ϵ = 1 (ϵ = −1). +The Fock space F(k) built from the vacuum at momentum k is found by acting repet- +itively with the negative-frequency modes. A convenient basis, the oscillator basis, is given +by the states: +F(k) = Span +� +|k; {Nn}⟩ +� +, +(7.83a) +|k; {Nn}⟩ := +� +n≥1 +(α−n)Nn +� +nNnNn! +|k⟩ , +Nn ∈ N∗ +(7.83b) +(we don’t distinguish the notations between the number operators and their eigenvalues). +The full Hilbert space is given by: +H = +� +R +dk F(k). +(7.84) +Computation – Equation (7.78) +We provide a quick argument to justify the second form of (7.78). Take the limit of +(7.57) with w = 0: +lim +z,¯z→0 eiϵkX(z,¯z) |0⟩ = lim +z,¯z→0 exp iϵk +� +�x − i ℓ2 +2 p ln |z|2 + i +� +ℓ2 +2 +� +n̸=0 +1 +n +� +αn z−n + ¯αn ¯z−n� +� +� |0⟩ += lim +z,¯z→0 exp +� +�iϵkx − ϵk +� +ℓ2 +2 +� +n̸=0 +1 +n +� +αn z−n + ¯αn ¯z−n� +� +� |0⟩ . +7To simplify the discussion, we do not consider winding but only vertex operators of the form (7.33). +117 + +The second term from the first line disappears because p |0⟩ = 0. +For ϵk > 0, as +z, ¯z → 0, the terms with αn and ¯αn for n < 0 disappear since they are accompanied +with a positive power of zn and ¯zn. The modes with n > 0 diverge but the minus sign +makes the exponential to vanish. A more rigorous argument requires to normal order +the exponential and then to use (7.80). +Computation – Equation (7.79) +p |k⟩ = 1 +ℓ2 +1 +2πi +� � +dz i∂X(z) + d¯z i¯∂X(¯z) +� +Vk(0, 0) |0⟩ += 1 +ℓ2 +1 +2πi +� �dz +z +ℓ2k +2 ++ d¯z +¯z +ℓ2k +2 +� +Vk(0, 0) |0⟩ += k Vk(0, 0) |0⟩ +using (7.44). +Remark 7.4 (Fock space and Verma module isomorphism) Note that, in the absence +of the so-called null states, there is a one-to-one map between states in the α−n oscillator +basis and in the L−n Virasoro basis. This translates an isomorphism between the Fock space +and the Verma module of Vk. +One hint for this relation is that applying α−n and L−n +changes the weight (eigenvalue of L0) by the same amount, and there are as many operators +in both basis. +7.1.7 +Euclidean and BPZ conjugates +Since X is a real scalar field, it is self-adjoint (6.95) such that +x† = x +p† = p, +α† +n = α−n. +(7.85) +This implies that the Virasoro operators (7.65) are Hermitian: +L† +n = L−n, +(7.86) +as expected since T(z) is self-adjoint for a free scalar field. +As a consequence of (7.85), the adjoint of the vacuum |k⟩ follows from (7.78): +⟨k| = |k⟩‡ =⟨0| e−iϵkx, +⟨k| p =⟨k| k. +(7.87) +The BPZ conjugate (6.111) of the mode αn is: +αt +n = −(±1)nα−n, +(7.88) +where the sign depends on the choice of I± in (6.111). Using (7.53), this implies that the +momentum operator gets a minus sign:8 +pt = −p, +⟨−k| = |k⟩t . +(7.89) +The inner product between two vacua |k⟩ and |k′⟩ is normalized as: +⟨k|k′⟩ = 2π δ(k − k′) +(7.90) +8Be careful that |k⟩ is not the state associated to the operator p through the state–operator correspond- +ence. Instead, they are associated to Vk, see (7.78). This explains why ⟨k| ̸= (|k⟩)t as in (6.136). +118 + +such that the conjugate state (6.145) of the vacuum reads +⟨kc| = 1 +2π ⟨k| . +(7.91) +The Hermitian and BPZ conjugate states are related as: +|k⟩‡ = − |k⟩t , +(7.92) +which can be interpreted as a reality condition on |k⟩. +7.2 +First-order bc ghost system +First-order systems describe two free fields called ghosts which have a first-order action +and whose conformal weights sum to 1. Commuting (resp. anti-commuting) fields are often +denoted by β and γ (resp. b and c) and correspondingly first-order systems are also called +βγ or bc systems. We will introduce a sign ϵ = ±1 to denote the Grassmann parity of the +fields and always write them as b and c. In string theory, first-order systems describe the +Faddeev–Popov ghosts associated to reparametrizations and supersymmetries (Sections 2.4 +and 17.1). +7.2.1 +Covariant action +A first-order system is defined by two symmetric and traceless fields bµ1···µλ and cµ1···µλ−1 +called ghosts. For fields of integer spins, the dynamics is governed by the first-order action +S = 1 +4π +� +d2x√g gµν bµµ1···µλ−1∇νcµ1···µλ−1 +(7.93) +after taking into account the symmetries of the field indices. Obviously, for λ = 2, one +recovers the reparametrization ghost action (2.145). The action (7.93) is invariant under +Weyl transformations (the fields and covariant derivatives are inert) such that it describes +a CFT on flat space. +When the fields have half-integer spins (and often denoted as β and γ in this case), they +carry a spinor index. In this case, the action contains a Dirac matrix, and the covariant +derivative a spin connection. +The ghost action (7.93) is invariant under a global U(1) symmetry +bµ1···µn −→ e−iθbµ1···µn, +cµ1···µn−1 −→ eiθcµ1···µn−1. +(7.94) +7.2.2 +Action on the complex plane +The simplest description of the system is on the complex plane. Due to the conditions im- +posed on the fields, they have only two independent components for all n, and the equations +of motion imply that one is holomorphic, and the other anti-holomorphic: +b(z) := bz···z(z), +¯b(¯z) := b¯z···¯z(¯z), +c(z) := cz···z(¯z), +¯c(¯z) := c¯z···¯z(z). +(7.95) +In this language, the action becomes +S = 1 +2π +� +d2z +� +b¯∂c + ¯b∂¯c). +(7.96) +This action gives the correct equations of motion +∂¯b = 0, +¯∂b = 0, +∂¯c = 0, +¯∂c = 0. +(7.97) +119 + +Since the fields split into holomorphic and anti-holomorphic sectors, it is convenient to study +only the holomorphic sector as usual. +This system is even simpler than the scalar field +because the zero-modes don’t couple both sectors.9 All formulas for the anti-holomorphic +sector are directly obtained from the holomorphic one by adding bars on quantities, except +for conserved charges which have an index L or R and are both written explicitly. +The action describes a CFT, and the weight of the fields are given by +h(b) = λ, +h(c) = 1 − λ, +h(¯b) = λ, +h(¯c) = 1 − λ, +(7.98) +where λ = n if the fields are in a tensor representation, and λ = n + 1/2 if they are in a +spinor-tensor representation. The holomorphic energy–momentum reads +T = −λ :b∂c: + (1 − λ) :∂b c: +(7.99a) += −λ :∂(bc): + :∂b c: +(7.99b) += (1 − λ) :∂(bc): − :b ∂c:. +(7.99c) +Normal ordering is taken with respect to the SL(2, C) vacuum (6.121). +Finally, both fields can be classically commuting or anticommuting (see below for the +quantum commutators): +b(z)c(w) = −ϵ c(w)b(z), +b(z)b(w) = −ϵ b(w)b(z), +c(z)c(w) = −ϵ c(w)c(z), (7.100) +where ϵ denotes the Grassmann parity +ϵ = +� ++1 +anticommuting, +−1 +commuting. +(7.101) +Sometimes, if ϵ = +1, one denotes b and c respectively by β and γ. If b and c are ghosts +arising from Faddeev–Popov gauge fixing, then ϵ = 1 if λ is integer; and ϵ = −1 if λ is +half-integer (“wrong” spin–statistics assignment). +The U(1) global symmetry (7.94) reads infinitesimally +δb = −ib, +δc = ic, +δ¯b = −i¯b, +δ¯c = i¯c. +(7.102) +It is generated by the conserved ghost current with components: +j(z) = −:b(z)c(z):, +¯ȷ(¯z) = −:¯b(¯z)¯c(¯z): +(7.103) +and the associated charge is called the ghost number +Ngh = Ngh,L + Ngh,R, +Ngh,L = +� +dz +2πi j(z), +Ngh,R = − +� +d¯z +2πi ¯ȷ(¯z). +(7.104) +This charge counts the number of c ghosts minus the number of b ghosts, such that +Ngh(c) = 1, +Ngh(b) = −1, +Ngh(¯c) = 1, +Ngh(¯b) = −1. +(7.105) +The propagator can be derived from the path integral +� +d′b d′c +δ +δb(z) +� +b(w)e−S[b,c]� += 0 +(7.106) +which gives the differential equation +δ(2)(z − w) + 1 +2π ⟨b(w)¯∂c(z)⟩ = 0. +(7.107) +Using (B.2), the solution is easily found to be +⟨c(z)b(w)⟩ = +1 +z − w. +(7.108) +9For the scalar field, the coupling of both sectors happened because of the periodicity condition (7.62). +120 + +Remark 7.5 The propagator is constructed with the path integral. For convenience, the +zero-modes are removed from the measure: reintroducing them, one finds that the propag- +ator is computed not in the conformal vacuum (which has no operator insertion), but in +a state with ghost insertions. This explains why the propagator (7.108) is not of the form +(6.62b). However, this form is sufficient to extract the OPE as changing the vacuum does +not introduce singular terms. +7.2.3 +OPE +The OPEs between the b and c fields are found from the propagator (7.108): +c(z)b(w) ∼ +1 +z − w, +b(z)c(w) ∼ +ϵ +z − w, +(7.109a) +b(z)b(w) ∼ 0, +c(z)c(w) ∼ 0. +(7.109b) +The OPE of each ghost with T confirms the conformal weights in (7.98): +T(z)b(w) ∼ λ +b(w) +(z − w)2 + ∂b(w) +z − w , +(7.110a) +T(z)c(w) ∼ (1 − λ) +c(w) +(z − w)2 + ∂c(w) +z − w . +(7.110b) +The OPE of T with itself is +T(z)T(w) ∼ +cλ/2 +(z − w)4 + +2T(w) +(z − w)2 + ∂T(w) +z − w , +(7.111) +where the central charge is: +cλ = 2ϵ(−1 + 6λ − 6λ2) = −2ϵ +� +1 + 6λ(λ − 1) +� +. +(7.112) +Introducing the ghost charge: +qλ = ϵ(1 − 2λ), +(7.113) +the central charge can also be written as +cλ = ϵ(1 − 3q2 +λ). +(7.114) +This parameter will appear many times in this section and its meaning will become clearer +as we proceed. +The OPE between the ghost current (7.103) and the b and c ghosts read +j(z)b(w) ∼ − b(w) +z − w, +(7.115a) +j(z)c(w) ∼ c(w) +z − w. +(7.115b) +The coefficients of the (z − w)−1 terms correspond to the ghost number of the b and c fields +(7.105). More generally, the ghost number Ngh(O) of any operator O(z) is defined by +j(z)O(w) ∼ Ngh(O) O(w) +z − w. +(7.116) +The OPE for j with itself is +j(z)j(w) ∼ +ϵ +(z − w)2 . +(7.117) +121 + +Finally, the OPE of the current with T reads: +T(z)j(w) ∼ +qλ +(z − w)3 + +j(w) +(z − w)2 + ∂j(w) +z − w . +(7.118) +Due to the presence of the z−3 term, the current j(z) is not a primary field if qλ ̸= 0, that is, +if λ ̸= 1/2. In that case, its transformation under changes of coordinates gets an anomalous +contribution: +j(z) = dw +dz j′(w) + qλ +2 +d +dz ln dw +dz = dw +dz j′(w) + qλ +2 +∂2 +zw +∂zw . +(7.119) +This implies in particular that the currents on the plane and on the cylinder (w = ln z) are +related by: +j(z) = dw +dz +� +jcyl(w) − qλ +2 +� +, +(7.120) +which leads to the following relation between the ghost numbers on the plane and on the +cylinder: +Ngh = N cyl +gh − qλ, +Ngh,L = N cyl +gh,L − qλ +2 , +Ngh,R = N cyl +gh,R − qλ +2 . +(7.121) +For this reason, it is important to make clear the space with respect to which is given the +ghost number: if not explicitly stated, ghost numbers in this book are given on the plane.10 +Due to this anomaly, one finds that the ghost number is not conserved on a curved space: +N c − N b = −ϵ qλ +2 +χg = (1 − 2λ)(g − 1), +(7.122) +where χg is the Euler characteristics (2.4), N b and N c are the numbers of b and c operators. +In string theory, where the only ghost insertions are zero-modes, this translates into a +statement on the number of zero-modes to be inserted. Hence, this can be interpreted as a +generalization of (2.72). For a proof, see for example [24, p. 397]. +Computation – Equation (7.110a) +T(z)b(w) = +� +− λ :b(z)∂c(z): + (1 − λ) :∂b(z) c(z): +� +b(w) +∼ −λ :b(z)∂c(z): b(w) + (1 − λ) :∂b(z) c(z): b(w) +∼ −λ b(z)∂z +1 +z − w + (1 − λ) ∂b(z) +1 +z − w +∼ λ +� +b(w) +(((((( +(z − w)∂b(w) +� +1 +(z − w)2 + (1 − �λ) ∂b(w) +z − w . +10Other references, especially old ones, give it on the cylinder. This can be easily recognized if some ghost +numbers in the holomorphic sector are half-integers: for the reparametrization ghosts, qλ is an integer such +that the shift in (7.121) is a half-integer. +122 + +Computation – Equation (7.110b) +T(z)c(w) = +� +− λ :b(z)∂c(z): + (1 − λ) :∂b(z)c(z): +� +c(w) +∼ ϵλ :∂c(z)b(z): c(w) − ϵ(1 − λ) :c(z)∂b(z): c(w) +∼ λ ∂c(z) +z − w − (1 − λ) c(z) ∂z +1 +z − w +∼ λ ∂c(w) +z − w + (1 − λ) +� +c(w) + (z − w)∂c(w) +� +1 +(z − w)2 +∼ (1 − λ) +c(w) +(z − w)2 + +∂c(w) +(z − w)2 . +Computation – Equation (7.115a) +j(z)b(w) = −:b(z)c(z): b(w) ∼ −:b(z)c(z): b(w) ∼ − b(z) +z − w ∼ − b(w) +z − w. +Computation – Equation (7.115b) +j(z)c(w) = −:b(z)c(z): c(w) ∼ ϵ :c(z)b(z): c(w) ∼ c(z) +z − w ∼ c(w) +z − w. +Computation – Equation (7.117) +j(z)j(w) = :b(z)c(z): :b(w)c(w): +∼ :b(z)c(z): :b(w)c(w): + :b(z)c(z): :b(w)c(w): + :b(z)c(z): :b(w)c(w): +∼ +ϵ +(z − w)2 + ϵ :c(z)b(w): +z − w ++ :b(z)c(w): +z − w +∼ +ϵ +(z − w)2 . +7.2.4 +Mode expansions +The b and c ghosts are expanded as +b(z) = +� +n∈Z+λ+ν +bn +zn+λ , +c(z) = +� +n∈Z+λ+ν +cn +zn+1−λ , +(7.123) +where ν = 0, 1/2 depends on ϵ and on the periodicity of the fields, see (6.102). The modes +are extracted with the contour formulas +bn = +� +dz +2πi zn+λ−1b(z), +cn = +� +dz +2πi zn−λc(z). +(7.124) +Ghosts with λ ∈ Z have integer indices and ν = 0 (we don’t consider ghosts with twisted +boundary conditions). On the other hand, ghosts with λ ∈ Z + 1/2 have integer indices +123 + +and ν = 1/2 in the R sector, and half-integer indices and ν = 0 in the NS sector (see +Section 6.4.4). The choices in the boundary conditions arise from the Z2 symmetry of the +action: +b −→ −b, +c −→ −c. +(7.125) +The number operators N b +n and N c +n are defined to count the numbers of excitations above +the SL(2, C) vacuum of b and c ghosts at level n: +N b +n = :b−ncn:, +N c +n = ϵ :c−nbn:. +(7.126) +The definitions follow from the commutators (7.134). Then, the level operators N b and N c +are obtained by summing over n: +N b = +� +n>0 +n N b +n, +N c = +� +n>0 +n N c +n. +(7.127) +The Virasoro operators are +Lm = +� +n +� +n − (1 − λ)m +� +:bm−ncn: = +� +n +(λm − n) :bncm−n:. +(7.128) +Of particular importance is the zero-mode +L0 = − +� +n +n :bnc−n: = +� +n +n :b−ncn:. +(7.129) +We will give the expression of L0 in terms of the level operators below, see (7.159). To +do this, we will first need to change the normal ordering, which first requires to study the +Hilbert space. +The modes of the ghost current are +jm = − +� +n +:bm−ncn: = − +� +n +:bncm−n:. +(7.130) +Note that the zero-mode of the current also equals the ghost number +Ngh,L = j0 = − +� +n +:b−ncn:. +(7.131) +When both the holomorphic and anti-holomorphic sectors enter, it is convenient to in- +troduce the combinations +b± +n = bn ± ¯bn, +c± +n = 1 +2 (cn ± ¯cn). +(7.132) +The normalization of b± +m is chosen to match the one of L± +m (6.118), and the one of c± +m such +that (7.135) holds. Note the following useful identities: +b− +n b+ +n = 2bn¯bn, +c− +n c+ +n = 1 +2 cn¯cn. +(7.133) +124 + +Computation – Equation (7.128) +T = −λ :b∂c: + (1 − λ) :∂bc: += +� +m,n +� +λ :bmcn: +n + 1 − λ +zm+λzm+2−λ − (1 − λ) :bmcn: +m + λ +zm+λ+1zm+1−λ +� += +� +m,n +� +λ (n + 1 − λ) − (1 − λ)(m + λ) +� :bmcn: +zm−n+2 += +� +m,n +� +λ (n + 1 − λ) − (1 − λ)(m − n + λ) +� :bm−ncn: +zm+2 += +� +m,n +(n − m + λm) :bm−ncn: +zm+2 += +� +m +Lm +zm+2 . +The fourth line follows from shifting m → m−n. The second equality in (7.128) follows +by shifting n → m − n. +Computation – Equation (7.130) +j = −:bc: = +� +m,n +:bmcn: +zm+λzn+1−λ = +� +m,n +:bm−ncn: +zm+1 += +� +m +jm +zm+1 . +7.2.5 +Commutators +The (anti)commutators between the modes bn and cn read: +[bm, cn]ϵ = δm+n,0, +[bm, bn]ϵ = 0, +[cm, cn]ϵ = 0. +(7.134) +Therefore, the modes with n < 0 are creation operators and the modes with n > 0 are +annihilation operators: +• a b ghost excitation at level n > 0 is created by b−n and annihilated by cn; +• a c ghost excitation at level n > 0 is created by c−n and annihilated by bn. +In terms of b± +m and c± +m (7.132), we have: +[b+ +m, c+ +n ]ϵ = δm+n, +[b− +m, c− +n ]ϵ = δm+n. +(7.135) +The commutators of the number operators with the modes are: +[N b +m, b−n] = b−nδm,n, +[N c +m, c−n] = c−nδm,n, +(7.136) +while those between the Ln and the ghost modes are: +[Lm, bn] = +� +m(λ − 1) − n +� +bm+n, +[Lm, cn] = −(mλ + n)cm+n, +(7.137) +in agreement with (6.115). +If n ∈ Z, each ghost field has zero-modes b0 and c0 which +commutes with L0 +[L0, b0] = 0, +[L0, c0] = 0. +(7.138) +125 + +The commutator of the current modes reads +[jm, jn] = m δm+n,0. +(7.139) +Then, the commutator with the Virasoro operators are +[Lm, jn] = −njm+n + qλ +2 m(m + 1)δm+n,0. +(7.140) +Finally, the commutators of the ghost number operator with the ghosts are: +[Ngh, b(w)] = −b(w), +[Ngh, c(w)] = c(w). +(7.141) +Computation – Equation (7.134) +[bm, cn]ϵ = ϵ +� +C0 +dw +2πi w−1 +� +Cw +dz +2πi z−1 wn+λzm−λ+1b(z)c(w) +∼ ϵ +� +C0 +dw +2πi w−1 +� +Cw +dz +2πi z−1 wn+λzm−λ+1 +ϵ +z − w += +� +C0 +dw +2πi wm+n−1 = δm+n,0. +Computation – Equation (7.141) +[Ngh, b(w)] = +� +dz +2πi j(z)b(w) ∼ − +� +dz +2πi +b(w) +z − w = −b(w). +The computation for c is similar. +7.2.6 +Hilbert space +The SL(2, C) vacuum |0⟩ (6.121) is defined by: +∀n > −λ : +bn |0⟩ = 0, +∀n > λ − 1 : +cn |0⟩ = 0. +(7.142) +If λ > 1, there are positive modes which do not annihilate the vacuum. +To simplify the notation, we consider the case λ ∈ Z, the half-integer case following by +shifting the indices by 1/2. Since the modes {c1, . . . , cλ−1} do not annihilate |0⟩, one can +create states +|n1, . . . , nλ−1⟩ = cn1 +1 · · · cnλ−1 +λ−1 |0⟩ +(7.143) +which have negative energies: +L0 |n1, . . . , nλ−1⟩ = − +� +� +λ−1 +� +j=1 +j nj +� +� |n1, . . . , nλ−1⟩ , +(7.144) +where (7.137) has been used. Moreover, this state is degenerate due to the existence of +zero-modes since they commute with the Hamiltonian – see (7.138). As a consequence, it +must be in a representation of the zero-mode algebra. +If the ghosts are commuting (ϵ = −1), then it seems hard to make sense of the theory +since one can find a state of arbitrarily negative energy since ni ∈ N. The zero-modes make +the problem even worse. The appropriate interpretation of these states will be discussed in +the context of the superstring theory for λ = 3/2 (superconformal ghosts). +In the rest of this section, we focus on the Grassmann odd case ϵ = 1. +126 + +Energy vacuum (Grassmann odd) +Since ni = 0 or ni = 1 for anticommuting ghosts (ϵ = 1), there is a state of lowest energy. +This is the energy vacuum (6.129). Since the zero-modes b0 and c0 commute with L0, it is +doubly degenerate. A convenient basis is +� +| ↓⟩ , | ↑⟩ +� +, +(7.145) +where +| ↓⟩ := c1 · · · cλ−1 |0⟩ , +| ↑⟩ := c0c1 · · · cλ−1 |0⟩ . +(7.146) +A general vacuum is a linear combination of the two basis vacua: +|Ω⟩ = ω↓ | ↓⟩ + ω↑ | ↑⟩ , +ω↓, ω↑ ∈ C. +(7.147) +The algebra of these vacua is the one of a two-state system: +b0 | ↑⟩ = | ↓⟩ , +c0 | ↓⟩ = | ↑⟩ , +b0 | ↓⟩ = 0, +c0 | ↑⟩ = 0. +(7.148) +Hence, for the vacuum | ↓⟩ (resp. | ↑⟩), b0 (resp. c0) acts as an annihilation operator, and +conversely c0 (resp. b0) acts as a creation operator. Finally, both states are annihilated by +all positive modes: +∀n > 0 : +bn | ↓⟩ = bn | ↑⟩ = 0, +cn | ↓⟩ = bn | ↓⟩ = 0. +(7.149) +Note that the SL(2, C) vacuum can be recovered by acting with b−n with n < λ: +|0⟩ = b1−λ · · · b−1 | ↓⟩ = b1−λ · · · b−1b0 | ↑⟩ . +(7.150) +The zero-point energy (6.130) of these states is the conformal weight of the vacuum: +L0 | ↓⟩ = aλ | ↓⟩ , +L0 | ↑⟩ = aλ | ↑⟩ , +(7.151) +where aλ can be written in various forms: +aλ = − +λ−1 +� +n=1 +n = −λ(λ − 1) +2 += cλ +24 + 2 +24. +(7.152) +Taking into account the anti-holomorphic sector leads to a four-fold degeneracy. The +basis +� +| ↓↓⟩ , | ↑↓⟩ , | ↓↑⟩ , | ↑↑⟩ +� +, +(7.153) +is built as follows: +| ↓↓⟩ := c1¯c1 · · · cλ−1¯cλ−1 |0⟩ , +| ↑↓⟩ := c0 | ↓↓⟩ , +| ↓↑⟩ := ¯c0 | ↓↓⟩ , +| ↑↑⟩ := c0¯c0 | ↓↓⟩ . +(7.154) +The modes b0 and ¯b0 can be used to flip the arrows downward, leading to the following +algebra: +c0 | ↓↓⟩ = | ↑↓⟩ , +¯c0 | ↓↓⟩ = | ↓↑⟩ , +c0 | ↓↑⟩ = −¯c0 | ↑↓⟩ = | ↑↑⟩ , +b0 | ↑↑⟩ = | ↓↑⟩ , +¯b0 | ↑↑⟩ = − | ↑↓⟩ , +b0 | ↑↓⟩ = ¯b0 | ↓↑⟩ = | ↓↓⟩ , +(7.155a) +The vacua are annihilated by different combinations of the zero-modes: +b0 | ↓↓⟩ = ¯b0 | ↓↓⟩ = 0, +c0 | ↑↓⟩ = ¯b0 | ↑↓⟩ = 0, +b0 | ↓↑⟩ = ¯c0 | ↓↑⟩ = 0, +c0 | ↑↑⟩ = ¯c0 | ↑↑⟩ = 0. +(7.155b) +127 + +In these manipulations, one has to be careful to correctly anti-commute the modes with the +ones hidden in the definitions of the vacua. +There is a second basis which is more natural when using the zero-modes c± +0 and b± +0 +(7.132): +� +| ↓↓⟩ , |+⟩ , |−⟩ , | ↑↑⟩ +� +, +(7.156) +where the two vacua |±⟩ are combinations of the | ↓↑⟩ and | ↑↓⟩ vacua: +|±⟩ = | ↑↓⟩ ± | ↓↑⟩ . +(7.157) +The different vacua are naturally related by acting with c± +0 and b± +0 which act as raising and +lowering operators: +c± +0 | ↓↓⟩ = 1 +2 |±⟩ , +c∓ +0 |±⟩ = ± | ↑↑⟩ , +b± +0 |±⟩ = ±2 | ↓↓⟩ , +b∓ +0 | ↑↑⟩ = ± |±⟩ . +(7.158a) +From the previous relations, it follows that the different vacua are annihilated by the zero- +modes as follow: +b+ +0 | ↓↓⟩ = b− +0 | ↓↓⟩ = 0, +c− +0 |−⟩ = b+ +0 |−⟩ = 0, +c+ +0 |+⟩ = b− +0 |+⟩ = 0 +c+ +0 | ↑↑⟩ = c− +0 | ↑↑⟩ = 0, +(7.158b) +This also means that we have +c− +0 c+ +0 | ↓↓⟩ = 1 +2 | ↑↑⟩ , +b+ +0 b− +0 | ↑↑⟩ = 2 | ↓↓⟩ . +(7.158c) +Computation – Equation (7.158) +2 c+ +0 |±⟩ = (c0 + ¯c0) | ↑↓⟩ ± (c0 + ¯c0) | ↓↑⟩ = ¯c0 | ↑↓⟩ ± c0 | ↓↑⟩ = (−1 ± 1) | ↑↑⟩ +b+ +0 |±⟩ = (b0 + ¯b0) | ↑↓⟩ ± (b0 + ¯b0) | ↓↑⟩ = b0 | ↑↓⟩ ± ¯b0 | ↓↑⟩ = (1 ± 1) | ↓↓⟩ +2 c± +0 | ↓↓⟩ = (c0 ± ¯c0) | ↓↓⟩ = c0 | ↓↓⟩ ± ¯c0 | ↓↓⟩ = | ↑↓⟩ ± | ↓↑⟩ = |±⟩ +b± +0 | ↑↑⟩ = (b0 ± ¯b0) | ↑↑⟩ = b0 | ↑↑⟩ ± ¯b0 | ↑↑⟩ = | ↓↑⟩ ∓ | ↑↓⟩ = ∓ |∓⟩ +Energy normal ordering (Grassmann odd) +We now turn towards the definition of the energy normal ordering (6.150). Ultimately, it +will be found that | ↓⟩ is the physical vacuum in string theory. For this reason, the energy +normal ordering +⋆ +⋆ · · · +⋆ +⋆ is associated to the vacuum | ↓⟩ in order to resolve the ambiguity +of the zero-modes. In particular, b0 is an annihilation operator in this case, while c0 is a +creation operator. In the rest of this section, we translate the normal ordering of expressions +from the conformal vacuum to the energy vacuum. +The Virasoro operators Ln for n ̸= 0 have no ordering problems since the modes which +compose them commute. The expression of L0 (7.129) in the energy ordering becomes +L0 = +� +n +n +⋆ +⋆b−ncn +⋆ +⋆ + aλ = N b + N c + aλ +(7.159) +where aλ is the zero-point energy (7.152) and N b and N c are the ghost mode numbers +(7.127). The contribution of the non-zero modes is denoted by: +�L0 = N b + N c. +(7.160) +128 + +The expression can be rewritten to encompass all modes: +Lm = +� +n +� +n − (1 − λ)m +� ⋆ +⋆bm−ncn +⋆ +⋆ + aλ δm,0 +(7.161) +Similarly, the expression of the ghost number is +Ngh,L = j0 = +� +n +⋆ +⋆b−ncn +⋆ +⋆ − +�qλ +2 + 1 +2 +� +(7.162a) += +� +n>0 +� +N c +n − N b +n +� ++ 1 +2 +� +N c +0 − N b +0 +� +− qλ +2 , +(7.162b) +and thus: +jm = +� +n +⋆ +⋆bm−ncn +⋆ +⋆ − +�qλ +2 + 1 +2 +� +δm,0. +(7.163) +It is useful to define the ghost number without ghost zero-modes: +� +Ngh,L := +� +n>0 +� +N c +n − N b +n +� +. +(7.164) +One can straightforwardly compute the ghost number of the vacua: +j0 | ↓⟩ = (λ − 1) | ↓⟩ = +� +−qλ +2 − 1 +2 +� +| ↓⟩ , +(7.165a) +j0 | ↑⟩ = λ | ↑⟩ = +� +−qλ +2 + 1 +2 +� +| ↑⟩ . +(7.165b) +This confirms that the SL(2, C) vacuum has vanishing ghost number since | ↓⟩ contains +exactly λ − 1 ghosts: +j0 |0⟩ = 0. +(7.166) +Using (7.121) allows to write the ghost numbers on the cylinder: +jcyl +0 +| ↓⟩ = −1 +2 | ↓⟩ , +jcyl +0 +| ↑⟩ = 1 +2 | ↑⟩ . +(7.167) +That both ghost numbers have same magnitude but opposite signs could be expected: since +the ghost number changes as Ngh → −Ngh when b ↔ c, the mean value of the ghost number +should be zero. +Remark 7.6 (Ghost number conventions) Since the ghost number is an additive quan- +tum number, it is always possible to shift its definition by a constant. This can be used to +set the ghost numbers of the vacua to some other values. For example, [24, p. 116] adds +qλ/2 to the ghost number in order to get Ngh = ±1/2 on the plane (instead of the cylinder). +We do not follow this convention in order to keep the symmetry between the vacuum ghost +numbers on the cylinder. +129 + +Computation – Equation (7.159) +Start with (7.129) and use (6.157): +L0 = − +� +n +n :bnc−n: = − +� +n≤−λ +n bnc−n + ϵ +� +n>−λ +n c−nbn += +� +n≥λ +n b−ncn + ϵ +� +n>−λ +n c−nbn += +� +n≥λ +n b−ncn + ϵ +� +n>0 +n c−nbn + ϵ +0 +� +n=−λ+1 +n c−nbn += +� +n≥λ +n b−ncn + ϵ +� +n>0 +n c−nbn + ϵ +λ−1 +� +n=0 +n b−ncn + aλ += +� +n>0 +n b−ncn + ϵ +� +n>0 +n c−nbn + aλ, += +� +n +n +⋆ +⋆b−ncn +⋆ +⋆ + aλ, +using that +0 +� +n=−λ+1 +c−nbn = − +λ−1 +� +n=0 +n cnb−n = − +λ−1 +� +n=0 +n (−ϵ b−ncn + 1) = ϵ +λ−1 +� +n=0 +n b−ncn + aλ. +The result also follows from (6.163). +Computation – Equation (7.162) +j0 = − +� +n +:b−ncn: = − +� +n≥λ +b−ncn + ϵ +� +n>−λ +c−nbn += − +� +n≥λ +b−ncn + ϵ +� +n>0 +c−nbn + ϵ +λ−1 +� +n=1 +cnb−n + ϵ c0b0 += − +� +n≥λ +b−ncn + ϵ +� +n>0 +c−nbn − +λ−1 +� +n=1 +b−ncn + ϵ(λ − 1) + ϵ c0b0 += − +� +n>0 +b−ncn + ϵ +� +n>0 +c−nbn + ϵ(λ − 1) + ϵ c0b0. +Finally, one can write +ϵ(λ − 1) = −qλ +2 − ϵ +2. +(7.168) +The result also follows from (6.163). The second expression is obtained by symmetrizing +the last term such that +ϵ c0b0 + ϵ(λ − 1) = ϵ +2 c0b0 + 1 +2(−b0c0 + ϵ) + ϵ(λ − 1) += 1 +2 (ϵ c0b0 − b0c0) + ϵ +� +λ − 1 +2 +� +. +130 + +Structure of the Hilbert space (Grassmann odd) +Since the zero-modes commute with the Hamiltonian and with all other negative- and +positive-frequency modes, the Hilbert space is decomposed in several subspaces, each as- +sociated to a zero-mode.11 +Starting with the holomorphic sector only, the Hilbert space Hgh is: +Hgh = Hgh,0 ⊕ c0Hgh,0, +Hgh,0 := Hgh ∩ ker b0, +(7.169) +which follows from the 2-state algebra (7.148). Obviously, one has c0Hgh,0 = Hgh ∩ ker c0. +The oscillator basis of the Hilbert space Hgh,0 is generated by applying the negative- +frequency modes and has the structure of a fermionic Fock space without zero-modes: +Hgh,0 = Span +� ��↓; {N b +n}; {N c +n} +� � +, +(7.170a) +��↓; {N b +n}; {N c +n} +� += +� +n≥1 +(b−n)Nb +n(c−n)Nc +n | ↓⟩ , +N b +n, N c +n ∈ N∗ +(7.170b) +(again, number operators and their eigenvalues are not distinguished). This means that +Hgh,0 can also be regarded as a Fock space built on the vacuum | ↓⟩, for which c0 and b0 +are respectively creation and annihilation operators. Conversely, c0 and b0 are respectively +annihilation and creation operators for c0Hgh,0. +In particular, this means that any state can be written as the sum of two states +ψ = ψ↓ + ψ↑, +ψ↓ ∈ Hgh,0, +ψ↑ ∈ c0Hgh,0, +(7.171) +with ψ↓ and ψ↑ built respectively on top of the | ↓⟩ and | ↑⟩ vacua. +This pattern generalizes when considering both the holomorphic and anti-holomorphic +sectors. In that case, the Hilbert space is decomposed in four subspaces:12 +Hgh = Hgh,0 ⊕ c0Hgh,0 ⊕ ¯c0Hgh,0 ⊕ c0¯c0Hgh,0, +Hgh,0 := Hgh ∩ ker b0 ∩ ker¯b0. +(7.172) +Basis states of the Hilbert space Hgh,0 are: +��↓↓; {N b +n}; {N c +n}; { ¯N b +n}; { ¯N c +n} +� += +� +n≥1 +(b−n)N b +n(¯b−n) +¯ +Nb +n(c−n)Nc +n(¯c−n) +¯ +Nc +n | ↓↓⟩ , +N b +n, ¯N b +n, N c +n, ¯N c +n ∈ N∗. +(7.173) +A general state of Hgh can be decomposed as +ψ = ψ↓↓ + ψ↑↓ + ψ↓↑ + ψ↑↑, +(7.174) +where each state is built by acting with negative-frequency modes on the corresponding +vacuum. +In terms of the second basis (7.156), the Hilbert space admits a second decomposition: +Hgh = Hgh,0 ⊕ c+ +0 Hgh,0 ⊕ c− +0 Hgh,0 ⊕ c− +0 c+ +0 Hgh,0, +Hgh,0 := Hgh ∩ ker b− +0 ∩ ker b+ +0 . +(7.175) +11Due to the specific structure of the inner product defined below, these subspaces are not orthonormal +to each other. +12The reader should not get confused by the same symbol Hgh,0 as in the case of the holomorphic sector. +131 + +In view of applications to string theory, it is useful to introduce two more subspaces: +Hgh,± := Hgh ∩ ker b± +0 = Hgh,0 ⊕ c∓ +0 Hgh,0, +(7.176) +and the associated decomposition +Hgh = Hgh,± ⊕ c± +0 Hgh,±. +(7.177) +In off-shell closed string theory, the principal Hilbert space will be H− +gh due to the level- +matching condition. In this case, H− +gh has the same structure as Hgh in the pure holomorphic +sector, and c+ +0 plays the same role as c0. A state in H− +gh is built on top of the vacua | ↓↓⟩ +and |+⟩. +7.2.7 +Euclidean and BPZ conjugates +In order for the Virasoro operators to be Hermitian, the bn and cn must satisfy the following +conditions: +b† +n = ϵb−n, +c† +n = c−n. +(7.178) +Hence, bn is anti-Hermitian if ϵ = −1. The BPZ conjugates of the modes are: +bt +n = (−1)λ b−n, +ct +n = (−1)1−λ c−n, +(7.179) +using I+(z) with (6.111). +In the rest of this section, we consider only the case ϵ = 1 and λ ∈ N. The adjoints of +the vacuum read: +| ↓⟩‡ =⟨0| c1−λ · · · c−1, +| ↑⟩‡ =⟨0| c1−λ · · · c−1c0. +(7.180) +The BPZ conjugates of the vacua are: +⟨↓ | := | ↓⟩t = (−1)(1−λ)2⟨0| c−1 · · · c1−λ, +⟨↑ | := | ↑⟩t = (−1)λ(1−λ)⟨0| c0c−1 · · · c1−λ. +(7.181) +The signs are inconvenient but will disappear when considering both the left and right vacua +together as in (7.154). We have the following relations: +⟨↓ | = (−1)aλ+(1−λ)(2−λ) | ↓⟩‡ , +⟨↑ | = (−1)aλ | ↑⟩‡ , +(7.182) +where aλ is the zero-point energy (7.152). +Computation – Equation (7.182) +To prove the relation, we can start from the BPZ conjugate ⟨↓ | and reorder the modes +to bring them in the same order as the adjoint: +⟨↓ | = (−1)(1−λ)2+ 1 +2 (2−λ)(1−λ) | ↓⟩‡ = (−1)−aλ+(1−λ)(2−λ) | ↓⟩‡ +The reordering gives a factor (−1) to the power: +λ−2 +� +i=1 +i = 1 +2(2 − λ)(1 − λ) = −aλ + 1 − λ. +Similarly, for the second vacuum: +⟨↑ | = (−1)λ(1−λ)− 1 +2 λ(1−λ) | ↑⟩‡ = (−1) +1 +2 λ(1−λ) | ↑⟩‡ . +132 + +We can identify the power with (7.152). +Then, we have the following relations: +⟨↑ | b0 =⟨↓ | , +⟨↓ | c0 =⟨↑ | , +⟨↓ | b0 = 0, +⟨↑ | c0 = 0. +(7.183) +There is a subtlety in defining the inner product because the vacuum is degenerate. If +we write the two vacua as vectors +| ↓⟩ = +� +0 +1 +� +, +| ↑⟩ = +� +1 +0 +� +, +(7.184) +then the zero-modes have the following matrix representation: +b0 = +�0 +0 +1 +0 +� +, +c0 = +�0 +1 +0 +0 +� +. +(7.185) +These matrices are not Hermitian as required by (7.178): since Hermiticity follows from the +choice of an inner product, it means that the vacua cannot form an orthonormal basis. An +appropriate choice for the inner products is:13 +⟨↓ | ↓⟩ = ⟨↑ | ↑⟩ = 0, +⟨↑ | ↓⟩ =⟨↓ | c0 | ↓⟩ =⟨0| c1−λ · · · c−1c0c1 · · · cλ−1 |0⟩ = 1. +(7.186) +The effect of changing the definition of the inner product or to consider a non-orthonormal +basis is represented by the insertion of c0. The last condition implies that the conjugate +state (6.145) to the SL(2, C) vacuum is: +⟨0c| =⟨0| c1−λ · · · c−1c0c1 · · · cλ−1, +⟨↓c | =⟨↑ | . +(7.187) +7.2.8 +Summary +In this section we summarize the values of the parameters for different theories of interest +(Table 7.1). The (η, ξ) system will be introduced in Chapter 17 in the bosonization of the +super-reparametrization (β, γ) ghosts. +The ψ± system can be used to describe spin-1/2 +fermions. +ϵ +λ +qλ +cλ +aλ +b, c (diff.) +1 +2 +−3 +−26 +−1 +β, γ (susy.) +−1 +3/2 +2 +11 +3/8 +ψ± +1 +1/2 +0 +1 +0 +η, ξ +1 +1 +−1 +−2 +0 +Table 7.1: Summary of the first-order systems. Remember that h(b) = λ and h(c) = 1 − λ. +7.3 +Suggested readings +• Free scalar: general references [246, sec. 4.1.3, 4.3, 4.6.2, 54, sec. 5.3.1, 6.3, 24, sec. 4.2, +193, 128], topological current and winding [109, 265, sec. 17.2–3]. +• First-order system: general references [24, chap. 5, sec. 13.1, 128, sec. 4.15, 193, +sec. 2.5], ghost vacua [151, sec. 15.3]. +13To avoid confusions, let us note that the adjoint in (7.182) are defined only through the adjoint of +the modes (6.110) but not with respect to the inner product given here, which would lead to exchanging +| ↓⟩‡ ∼⟨↑ | and | ↑⟩‡ ∼⟨↓ |. +133 + +Chapter 8 +BRST quantization +Abstract +The BRST quantization can be introduced either by following the standard +QFT treatment (outlined in Section 3.2), or by translating it in the CFT language. One +can then use all the CFT techniques to extract information on the spectrum, which makes +this approach more powerful. Moreover, this also provides an elegant description of states +and string fields. In this chapter, we set the stage of the BRST quantization using the CFT +language and we apply it to string theory. The main results of this chapter are a proof of +the no-ghost theorem and a characterization of the BRST cohomology (physical states). +8.1 +BRST for reparametrization invariance +The BRST symmetry we are interested in results from gauge fixing the reparametrization +invariance. +In this chapter, we focus on the holomorphic sector: since both sectors are +independent, most results follow directly, except those concerning the zero-modes. +We +consider a generic matter CFT coupled to reparametrization ghosts: +1. matter: central charge cm, energy–momentum tensor Tm and Hilbert space Hm; +2. reparametrization ghosts: bc ghost system (Sections 2.3 and 7.2) with ϵ = +1 and +λ = 2, cgh = −26, energy–momentum tensor T gh and Hilbert space Hgh. +The formulas for the reparametrization ghosts are summarized in Appendix B.3.5. +For +modes, the system (m, gh, b or c) is indicated as a superscript to not confuse it with the +mode index. +The total central charge, energy–momentum tensor and Hilbert space are +denoted by: +c = cm + cgh = cm − 26, +T(z) = T m(z) + T gh(z), +H = Hm ⊗ Hgh. +(8.1) +The goal is to find the physical states in the cohomology, that is, which are BRST closed +QB |ψ⟩ = 0 +(8.2) +but non exact (Section 3.2): the latter statement can be understood as an equivalence +between closed states under shift by exact states: +|ψ⟩ ∼ |ψ⟩ + QB |Λ⟩ . +(8.3) +We introduce the BRST current and study its CFT properties. Then, we give a compu- +tation of the BRST cohomology when the matter CFT contains at least two scalar fields. +134 + +8.2 +BRST in the CFT formalism +The BRST current can be found from (3.50) to be [193]: +jB(z) = :c(z) +� +T m(z) + 1 +2 T gh(z) +� +: + κ ∂2c(z) +(8.4a) += c(z)T m(z) + :b(z)c(z)∂c(z): + κ ∂2c(z), +(8.4b) +and similarly for the anti-holomorphic sector. This can be derived from (3.53): the generator +of infinitesimal changes of coordinates (given by the Lie derivative) is the energy–momentum +tensor. The factor of 1/2 comes from the expression (7.99) of the ghost energy–momentum +tensor: the second term does not contribute while the first has a factor of 2. Since the +transformation of c in (3.53) has no factor, the 1/2 is necessary to recover the correct +normalization. Finally, one finds that the transformation of b is reproduced. The different +computations can be checked using the OPEs given below. The last piece is a total derivative +and does not contribute to the charge: for this reason, it cannot be derived from (3.53), its +coefficient will be determined below. Note that it is the only total derivative of dimension +1 and of ghost number 1. +The BRST charge is then obtained by the contour integral: +QB = QB,L + QB,R, +QB,L = +� +dz +2πi jB(z), +QB,R = +� +d¯z +2πi ¯ȷB(¯z). +(8.5) +As usual, QB ∼ QB,L when considering only the holomorphic sectors such that we generally +omit the index. +8.2.1 +OPE +The OPE of the BRST current with T is +T(z)jB(w) ∼ +�cm +2 − 4 − 6κ +� +c(w) +(z − w)4 + (3 − 2κ) ∂c(w) +(z − w)3 + +jB(w) +(z − w)2 + ∂jB(w) +z − w . +(8.6) +Hence, the BRST current is a primary operator only if +cm = 26, +κ = 3 +2. +(8.7) +The BRST current must be primary, otherwise, the BRST symmetry is anomalous, which +means that the theory is not consistent. This provides another derivation of the critical +dimension. In this case, the OPE becomes +T(z)jB(w) ∼ +jB(w) +(z − w)2 + ∂jB(w) +z − w . +(8.8) +Remark 8.1 (Critical dimension in 2d gravity) The value cm = 26 (critical dimen- +sion) was obtained in Section 2.3 by requiring that the Liouville field decouples from the +path integral. In 2d gravity, where this condition is not necessary, (nor even desirable) the +Liouville field is effectively part of the matter, such that cL + cm = 26. One can also study +the BRST cohomology in this case. +The OPE of jB(z) with the ghosts are +jB(z)b(w) ∼ +2κ +(z − w)3 + +j(w) +(z − w)2 + T(w) +z − w, +(8.9a) +jB(z)c(w) ∼ :c(w)∂c(w): +z − w +. +(8.9b) +135 + +Similarly, the OPE with any matter weight h primary field φ is +jB(z)φ(w) ∼ h c(w)φ(w) +(z − w)2 + :h ∂c(w)φ(w) + c(w)∂φ(w): +z − w +, +(8.9c) +using that c(w)2 = 0 to cancel one term. +The OPE with the ghost current is +jB(z)j(w) ∼ +2κ + 1 +(z − w)3 − 2∂c(w) +(z − w)2 − jB(w) +z − w , +(8.10) +while the OPE with itself is (for κ = 3/2) +jB(z)jB(w) ∼ −cm − 18 +2 +:c(w)∂c(w): +(z − w)3 +− cm − 18 +4 +:c(w)∂2c(w): +(z − w)2 +− cm − 26 +12 +:c(w)∂3c(w): +z − w +. +(8.11) +There is no first order pole if cm = 26: as we will see shortly, this implies that the BRST +charge is nilpotent. +8.2.2 +Mode expansions +The mode expansion of the BRST charge can be written equivalently +QB = +� +m +:cm +� +Lm +−m + 1 +2 Lgh +−m +� +: +(8.12a) += +� +m +c−mLm +m + 1 +2 +� +m,n +(n − m) :c−mc−nbm+n: +(8.12b) +In the energy ordering, this expression becomes +QB = +� +m +⋆ +⋆cm +� +Lm +−m + 1 +2 Lgh +−m +� +⋆ +⋆ − c0 +2 +(8.13a) += +� +n +cmLm +−m + 1 +2 +� +m,n +(n − m) +⋆ +⋆c−mc−nbm+n +⋆ +⋆ − c0, +(8.13b) +where the ordering constant is the same as in Lgh +0 +(as can be checked by comparing both +sides of the anticommutator). The simplest derivation of this term is to use the algebra +and to ensure that it is consistent. The only ambiguity is in the second term, when one c +does not commute with the b: this happens for −n + (m + n) = 0, such that the ordering +ambiguity is proportional to c0. Then, one finds that it is equal to agh = −1. +The BRST operator can be decomposed on the ghost zero-modes as +QB = c0L0 − b0M + �QB +(8.14a) +where +�QB = +� +m̸=0 +c−mLm +m − 1 +2 +� +m,n̸=0 +m+n̸=0 +(m − n) +⋆ +⋆c−mc−nbm+n +⋆ +⋆ , +(8.14b) +M = +� +m̸=0 +m c−mcm +(8.14c) +136 + +The interest of this decomposition is that L0, M and �Q do not contain b0 or c0, which +make it very useful to act on states decomposed according to the zero-modes (7.169). The +nilpotency of the BRST operator implies the relations +[L0, M] = [ �QB, M] = [ �QB, L0] = 0, +�Q2 +B = L0M. +(8.15) +Moreover, one has Ngh( �QB) = 1 and Ngh(M) = 2. +8.2.3 +Commutators +From the various OPEs, one can compute the (anti-)commutators of the BRST charge with +the other operators. For the ghosts and a weight h primary field φ, one finds +{QB, b(z)} = T(z), +(8.16a) +{QB, c(z)} = c(z)∂c(z), +(8.16b) +[QB, φ(z)] = h ∂c(z)φ(z) + c(z)∂φ(z). +(8.16c) +This reproduces correctly (3.53). +Two facts will be useful in string theory. First, (8.16c) is a total derivative for h = 1: +[QB, φ(z)] = ∂ +� +c(z)φ(z) +� +. +(8.17) +Second, c(z)φ(z) is closed if h = 1 +{QB, c(z)φ(z)} = (1 − h)c(z)∂c(z)φ(z). +(8.18) +The commutator with the ghost current is +[QB, j(z)] = −jB(z), +(8.19) +which confirms that the BRST charge increases the ghost number by 1 +[Ngh, QB] = QB. +(8.20) +One finds that the BRST charge is nilpotent +{QB, QB} = 0 +(8.21) +and commutes with the energy–momentum tensor +[QB, T(z)] = 0 +(8.22) +only if the matter central charge corresponds to the critical dimension: +cm = 26. +(8.23) +The most important commutator for the modes is +Ln = {QB, bn}. +(8.24) +Nilpotency of QB then implies that QB commutes with Ln: +[QB, Ln] = 0. +(8.25) +137 + +8.3 +BRST cohomology: two flat directions +The simplest case for studying the BRST cohomology is when the target spacetime has at +least two non-compact flat directions represented by two free scalar fields (X0, X1) (Sec- +tion 7.1). The remaining matter fields are arbitrary as long as the critical dimension cm = 26 +is reached. The reason for introducing two flat directions is that the cohomology is easily +worked out by introducing light-cone (or complex) coordinates in target spacetime. +The field X0 can be spacelike or timelike ϵ0 = ±1, while we consider X1 to be always +spacelike, ϵ1 = 1. The oscillators are denoted by α0 +m and α1 +m, and the momenta of the Fock +vacua by k∥ = (k0, k1) such that +k2 +∥ = ϵ0(k0)2 + (k1)2. +(8.26) +The rest of the matter sector, called the transverse sector ⊥, is an arbitrary CFT with +energy–momentum tensor T ⊥, central charge c⊥ = 24 and Hilbert space H⊥. The ghost +together with the two scalar fields form the longitudinal sector ∥. The motivation for the +names longitudinal and transverse will become clear later: they will be identified with the +light-cone and perpendicular directions in the target spacetime (and, correspondingly, with +unphysical and physical states). +The Hilbert space of the theory is decomposed as +H := H∥ ⊗ H⊥, +H∥ := +� +dk0 F0(k0) ⊗ +� +dk1 F1(k1) ⊗ Hgh, +(8.27) +where F0(k0) and F1(k1) are the Fock spaces (7.83a) of the scalar fields X0 and X1, and +Hgh is the ghost Hilbert space (7.169). As a consequence, a generic state of H reads +|ψ⟩ = |ψ∥⟩ ⊗ |ψ⊥⟩ , +(8.28) +where ψ⊥ is a generic state of the transverse matter CFT H⊥ and ψ∥ is built by acting with +oscillators on the Fock vacuum of H∥: +|ψ∥⟩ = cNc +0 +0 +� +m>0 +(α0 +−m)N0 +m(α1 +−m)N1 +m (b−m)Nb +m(c−m)Nc +m |k0, k1, ↓⟩ +|k0, k1, ↓⟩ := |k0⟩ ⊗ |k1⟩ ⊗ | ↓⟩ , +N 0 +m, N 1 +m ∈ N, +N b +m, N c +m = 0, 1. +(8.29) +Since the Virasoro modes commute with the ghost number, eigenstates of the Virasoro +operators without zero-modes �L0, given by the sum of (7.68) and (7.160), can also be taken +to be eigenstates of Ngh. It is also useful to define the Hilbert space of states lying in the +kernel of b0: +H0 = H ∩ ker b0 +(8.30) +such that +H = H0 ⊕ c0H0. +(8.31) +The full L0 operator reads +L0 = Lm +0 + Lgh +0 = (Lm +0 − 1) + N b + N c, +(8.32) +using (7.129) for Lgh +0 . A more useful expression is obtained by separating the two sectors +and by extracting the zero-modes using (7.67): +L0 = +� +L⊥ +0 − m2 +∥,Lℓ2 − 1 +� ++ �L∥ +0, +(8.33) +where the longitudinal mass and total level operator are: +m2 +∥,L = −p2 +∥,L, +�L∥ +0 = N 0 + N 1 + N b + N c ∈ N. +(8.34) +138 + +A state |ψ⟩ is said to be on-shell if it is annihilated by L0: +on-shell: +L0 |ψ⟩ = 0. +(8.35) +The absolute BRST cohomology Habs(QB) defines the physical states (Section 3.2) and +is given by the states ψ ∈ H that are QB-closed but not exact: +Habs(QB) := +� +|ψ⟩ ∈ H +�� QB |ψ⟩ = 0, ∄ |χ⟩ ∈ H +�� |ψ⟩ = QB |χ⟩ +� +. +(8.36) +Since QB commutes with L0, (8.25), the cohomology subspace is preserved under time +evolution. +Before continuing, it is useful to outline the general strategy for studying the cohomology +of a BRST operator Q in the CFT language. The idea is to find an operator ∆ – called +contracting homotopy operator – which, if it exists, trivializes the cohomology. Conversely, +this implies that the cohomology is to be found within states which are annihilated by ∆ +or for which ∆ is not defined. Then, it is possible to restrict Q on these subspaces: this is +advantageous when the restriction of the BRST charge on these subspaces is a simpler. In +fact, we will find that the reduced operator is itself a BRST operator, for which one can +search for another contracting homotopy operator.1 +Given a BRST operator Q, a contracting homotopy operator ∆ for Q is an operator such +that +{Q, ∆} = 1. +(8.37) +Interpreting Q as a derivative operator, ∆ corresponds to the Green function or propagator. +The existence of a well-defined ∆ with empty kernel implies that the cohomology is empty +because all closed states are exact. Indeed, consider a state |ψ⟩ ∈ H which is an eigenstate +of ∆ and closed QB |ψ⟩ = 0. Inserting (8.37) in front of the state gives: +|ψ⟩ = {QB, ∆} |ψ⟩ = QB +� +∆ |ψ⟩ +� +. +(8.38) +If ∆ is well-defined on |ψ⟩ and |ψ⟩ /∈ ker ∆, then ∆ |ψ⟩ is another state in H, which implies +that |ψ⟩ is exact. Hence, the BRST cohomology has to be found inside the subspaces ker ∆ +or on which ∆ is not defined. +8.3.1 +Conditions on the states +In this subsection, we apply explicitly the strategy just discussed to get conditions on the +states. A candidate contracting homotopy operator for QB is +∆ := b0 +L0 +(8.39) +thanks to (8.24): +L0 = {QB, b0}. +(8.40) +Indeed, suppose that |ψ⟩ is an eigenstate of L0, and that it is closed but not on-shell: +QB |ψ⟩ = 0, +L0 |ψ⟩ ̸= 0. +(8.41) +One can use (8.40) in order to write: +|ψ⟩ = QB +� b0 +L0 +|ψ⟩ +� +. +(8.42) +1A similar strategy shows that there is no open string excitation for the open SFT in the tachyon vacuum. +139 + +The operator inside the parenthesis is ∆ defined above in (8.39). The formula (8.42) breaks +down if ψ is in the kernel of L0 since the inverse is not defined. This implies that a necessary +condition for a L0-eigenstate |ψ⟩ to be in the BRST cohomology is to be on-shell (8.35). +Considering explicitly the subset of states annihilated by b0 is not needed at this stage since +ker b0 ⊂ ker L0 for QB-closed states, according to (8.24). Hence, we conclude: +Habs(QB) ⊂ ker L0. +(8.43) +Note that this statement holds only at the level of vector spaces, i.e. when considering +equivalence classes of states |ψ⟩ ∼ |ψ⟩+Q |Λ⟩. This means that there exists a representative +state of each equivalence class inside ker L0, but a generic state is not necessarily in ker L0. +For example, consider a state |ψ⟩ ∈ ker L0 and closed. Then, |ψ′⟩ = |ψ⟩ + QB |Λ⟩ with +|Λ⟩ /∈ ker L0 is still in Habs(QB) but |ψ′⟩ /∈ ker L0 since [L0, QB] = 0. +Computation – Equation (8.42) +For L0 |ψ⟩ ̸= 0, one has: +|ψ⟩ = L0 +L0 +|ψ⟩ = 1 +L0 +{QB, b0} |ψ⟩ = 1 +L0 +QB +� +b0 |ψ⟩ +� +where the fact that |ψ⟩ is closed has been used to cancel the second term of the anti- +commutator. Note that L0 commutes with both QB and b0 such that it can be moved +freely. +This shows that ∆ = b0/L0 given by (8.39) is not a contracting homotopy operator. A +proper definition involves the projector P0 on the kernel of L0: +|ψ⟩ ∈ ker L0 : +P0 |ψ⟩ = |ψ⟩ , +|ψ⟩ ∈ (ker L0)⊥ : +P0 |ψ⟩ = 0. +(8.44) +Then, the appropriate contracting homotopy operator reads ∆(1−P0) and (8.37) is changed +to: +{QB, ∆(1 − P0)} = (1 − P0). +(8.45) +This parallels completely the definition of the Green function in presence of zero-modes, see +(B.3). By abuse of language, we will also say that ∆ is a contracting homotopy operator, +remembering that this statement is correct only when multiplying with (1 − P0). +We will revisit these aspects later from the SFT perspective. In fact, we will find that QB +is the kinetic operator of the gauge invariant theory, while ∆ is the gauge fixed propagator +in the Siegel gauge. This is expected from experience with standard gauge theories: the +inverse of the kinetic operator (Green function) is not defined when the gauge invariance is +not fixed. +The on-shell condition (8.35) is already a good starting point. In order to simplify the +analysis further, one can restrict the question of computing the cohomology on the subspace: +H0 := H ∩ ker b0 = Hm ⊗ Hgh,0, +(8.46) +where Hgh,0 = Hgh ∩ker b0 was defined in (7.2.6). This subspace contains all states |ψ⟩ such +that: +|ψ⟩ ∈ H0 +=⇒ +b0 |ψ⟩ = 0. +(8.47) +In this subspace, there is no exact state |ψ⟩ with L0 |ψ⟩ ̸= 0 such that b0 |ψ⟩ = QB |ψ⟩ = 0. +Indeed, assuming these conditions, (8.42) leads to a contraction: +b0 |ψ⟩ = QB |ψ⟩ = 0, +L0 |ψ⟩ ̸= 0 +=⇒ +|ψ⟩ = 0. +(8.48) +140 + +Note that the converse statement is not true: there are on-shell states such that b0 |ψ⟩ ̸= 0. +This also makes sense because the ghost Hilbert space can be decomposed with respect to +the ghost zero-modes. The cohomology of QB in the subspace H0 is called the relative +cohomology: +Hrel(QB) := H0(QB) = +� +|ψ⟩ ∈ H0 +�� QB |ψ⟩ = 0, ∄ |χ⟩ ∈ H +�� |ψ⟩ = QB |χ⟩ +� +. +(8.49) +The advantage of the subspace b0 = 0 is to precisely pick the representative of Habs which +lies in ker L0. In particular, the operator L0 is simple and has a direct physical interpretation +as the worldsheet Hamiltonian. This condition is also meaningful in string theory because +these states are also mass eigenstates, which have a nice spacetime interpretation, and it +will later be interpreted in SFT as fixing the Siegel gauge. Moreover, it is implied by the +choice of ∆ in (8.39) as the contracting homotopy operator, which is particularly convenient +to work with to derive the cohomology. However, there are other possible choices, which are +interpreted as different gauge fixings. +After having built this cohomology, we can look for the full cohomology by relaxing the +condition b0 = 0. In view of the structure of the ghost Hilbert space (7.169), one can expect +that Habs(QB) = Hrel(QB) ⊕ c0Hrel(QB), which is indeed the correct answer. But, we will +see (building on Section 3.2.2) that, in fact, it is this cohomology which contains the physical +states in string theory, instead of the absolute cohomology. +As a summary, we are looking for QB-closed non-exact states annihilated by b0 and L0: +QB |ψ⟩ = 0, +L0 |ψ⟩ = 0, +b0 |ψ⟩ = 0. +(8.50) +8.3.2 +Relative cohomology +In (8.14a), the BRST operator was decomposed as: +QB = c0L0 − b0M + �QB, +�Q2 +B = L0M. +(8.51) +This shows that, on the subspace L0 = b0 = 0, �QB is nilpotent and equivalent to QB: +|ψ⟩ ∈ H0 ∩ ker L0 +=⇒ +QB |ψ⟩ = �QB |ψ⟩ , +�Q2 +B |ψ⟩ = 0. +(8.52) +Hence, this implies that �QB is a proper BRST operator and the relative cohomology of QB +is isomorphic to the cohomology of �QB: +H0(QB) = H0( �QB). +(8.53) +Next, we introduce light-cone coordinates in the target spacetime. While it does not allow +to write Lorentz covariant expressions, it is helpful mathematically because it introduces a +grading of the Hilbert space, for which powerful theorems exist (even if we will need only +basic facts for our purpose). +Light-cone parametrization +The two scalar fields X0 and X1 are combined in a light-cone (if ϵ0 = −1) or complex (if +ϵ0 = 1) fashion: +X± +L = +1 +√ +2 +� +X0 +L ± +i +√ϵ0 +X1 +L +� +. +(8.54) +141 + +The modes of X± are found by following (7.50):2 +α± +n = +1 +√ +2 +� +α0 +n ± +i +√ϵ0 +α1 +n +� +, +n ̸= 0, +(8.55a) +x± +L = +1 +√ +2 +� +x0 +L ± +i +√ϵ0 +x1 +L +� +, +p± +L = +1 +√ +2 +� +p0 +L ± +i +√ϵ0 +p1 +L +� +, +(8.55b) +The non-zero commutation relations are: +[α+ +m, α− +n ] = ϵ0 m δm+n,0, +[x± +L, p∓ +L] = iϵ0. +(8.56) +This implies that negative-frequency (creation) modes α± +−n are canonically conjugate to +positive-frequency (annihilation) modes α∓ +n . Note the similarity with the first-order system +(7.134). +For later purposes, it is useful to note the following relations: +2 p+ +Lp− +L = (p0 +L)2 + ϵ0(p1 +L)2 = ϵ0 p2 +∥,L, +(8.57a) +x+p− + x−p+ = x0p0 + ϵ0 x1p1, +(8.57b) +� +n +α+ +n α− +m−n = 1 +2 +� +n +� +α0 +nα0 +m−n + ϵ0 α1 +nα1 +m−n +� +. +(8.57c) +In view of the commutators (8.56), the appropriate definitions of the light-cone number +N ± +n and level operators N ± are: +N ± +n = ϵ0 +n α± +−nα∓ +n , +N ± = +� +n>0 +n N ± +n . +(8.58) +The insertion of ϵ0 follows (7.63). Then, one finds the following relation: +N + + N − = N 0 + N 1. +(8.59) +Using these definitions, the variables appearing in L0 (8.33) +L0 = +� +L⊥ +0 − m2 +∥,Lℓ2 − 1 +� ++ �L∥ +0 +(8.60) +can be rewritten as: +m2 +∥,L = −2ϵ0 p+ +Lp− +L, +�L∥ +0 = N + + N − + N b + N c. +(8.61) +The expression for the sum of the Virasoro operators (7.65) easily follows from (8.57): +L0 +m + L1 +m = ϵ0 +� +n +:α+ +n α− +m−n: = ϵ0 +� +n̸=0,m +:α+ +n α− +m−n: + ϵ0 +� +α− +0 α+ +m + α+ +mα− +m +� +. +(8.62) +Computation – Equation (8.56) +For the modes α± +m, we have: +[α+ +m, α± +n ] = 1 +2 +�� +α0 +m + +i +√ϵ0 +α1 +m +� +, +� +α0 +n ± +i +√ϵ0 +α1 +n +�� += 1 +2 +� +[α0 +m, α0 +n] ∓ 1 +ϵ0 +[α1 +m, α1 +n] +� += ϵ0 +2 m δm+n,0(1 ∓ 1), +2For ϵ0 = 1, this convention matches the ones from [29] for X0 = X and X1 = φ. For ϵ = −1, this +convention matches [193]. +142 + +where we used (7.72). The other commutators follow similarly from (7.73), for example: +[x− +L, p± +L] = 1 +2 +�� +x0 +L − +i +√ϵ0 +x1 +L +� +, +� +p0 +L ± +i +√ϵ0 +p1 +L +�� += 1 +2 +� +[x0 +L, p0 +L] ± ϵ0[x1 +L, p1 +L] +� += ϵ0 +2 (1 ± 1). +Computation – Equation (8.57) +For the modes α± +m, we have: +� +n +α+ +n α− +m−n = 1 +2 +� +n +� +α0 +n + +i +√ϵ0 +α1 +n +� � +α0 +m−n − +i +√ϵ0 +α1 +m−n +� += 1 +2 +� +n +� +α0 +nα0 +m−n + ϵ0 α1 +nα1 +m−n + +i +√ϵ0 +(α0 +m−nα1 +n − α0 +nα1 +m−n) +� +. +The last two terms in parenthesis cancel as can be seen by shifting the sum n → m − n +in one of the term. Note that, for m ̸= 2n, there is no cross-term only after summing +over n. +The relations for the zero-modes follow simply by observing that expressions in both +coordinates can be rewritten in terms of the 2-dimensional (spacetime) flat metric. +Computation – Equation (8.59) +Using (8.57), one finds: +N 0 + N 1 = +� +n +n +� +N 0 +n + N 1 +n +� += +� +n +n +� +N + +n + N − +n +� += N + + N −. +Reduced cohomology +In terms of the light-cone variables, the reduced BRST operator �QB reads: +�QB = +� +m̸=0 +c−m +� +L⊥ +m + ϵ0 +� +n +α+ +n α− +m−n +� ++ 1 +2 +� +m,n +(n − m) :c−mc−nbm+n:. +(8.63) +This operator can be further decomposed. Introducing the degree +deg := N + − N − + � +N c − � +N b +(8.64) +such that +∀m ̸= 0 : +deg(α+ +m) = deg(cm) = 1, +deg(α− +m) = deg(bm) = −1, +(8.65) +and deg = 0 for the other variables, the operator �QB is decomposed as:3 +�QB = Q0 + Q1 + Q2, +deg(Qj) = j, +(8.66a) +where +Q1 = +� +m̸=0 +c−mL⊥ +m + +� +m,n̸=0 +m+n̸=0 +⋆ +⋆c−m +� +ϵ0 α+ +n α− +m−n + 1 +2 (m − n) c−mbm+n +� +⋆ +⋆, +Q0 = +� +n̸=0 +α+ +0 c−nα− +n , +Q2 = +� +n̸=0 +α− +0 c−nα+ +n . +(8.66b) +3The general idea behind this decomposition is the notion of filtration, nicely explained in [3, sec. 3, 36]. +143 + +The nilpotency of �QB implies the following conditions on the Qj: +Q2 +0 = Q2 +2 = 0, +{Q0, Q1} = {Q1, Q2} = 0, +Q2 +1 + {Q0, Q2} = 0. +(8.67) +Hence, Q0 and Q2 are both nilpotent and define a cohomology. +One can show that the cohomologies of �QB and Q0 are isomorphic4 +H0( �QB) ≃ H0(Q0) +(8.68) +under general conditions [29], in particular, if the cohomology is ghost-free (i.e. all states +have Ngh = 1). +The contracting homotopy operator for Q0 is +�∆ := B +�L∥ +0 +, +B := ϵ0 +� +n̸=0 +1 +α+ +0 +α+ +−nbn. +(8.69) +Indeed, it is straightforward to check that +�L∥ +0 = {Q0, B} +=⇒ +{Q0, �∆} = 1. +(8.70) +As a consequence, a necessary condition for a closed �L∥ +0-eigenstate |ψ⟩ to be in the +cohomology of Q0 is to be annihilated by �L∥ +0: +�L∥ +0 |ψ⟩ = 0, +=⇒ +N ± |ψ⟩ = N c |ψ⟩ = N b |ψ⟩ = 0, +(8.71) +since �L∥ +0 is a sum of positive integers. This means that the state ψ contains no ghost or +light-cone excitations α± +−n, b−n and c−n, and lies in the ground state of the Fock space H∥,0. +Then, we need to prove that this condition is sufficient: states with �L∥ +0 = 0 are closed. +First, note that a state |ψ⟩ ∈ H0 with �L∥ +0 has ghost number 1 since there are no ghost +excitations on top of the vacuum | ↓⟩, which has Ngh = 1. Second, �L0 and Q0 commute, +such that: +0 = Q0�L∥ +0 |ψ⟩ = �L∥ +0Q0 |ψ⟩ . +(8.72) +Since Q0 increases the ghost number by 1, one can invert �L∥ +0 = N b + N c + · · · in the last +term since �L∥ +0 ̸= 0 in this subspace. This gives: +Q0 |ψ⟩ = 0. +(8.73) +Hence, the condition �L∥ +0 |ψ⟩ = 0 is sufficient for |ψ⟩ to be in the cohomology. This has to be +contrasted with Section 8.3.1 where the condition L∥ +0 = 0 is necessary but not sufficient. +In this case, the on-shell condition (8.33) reduces to +L0 = L⊥ +0 − m2 +∥,Lℓ2 − 1 = 0. +(8.74) +But, additional states can be found in ker B or in a subspace of H on which B is singular. +We have ker B = ker �L∥ +0 such that nothing new can be found there. However, the operator B +is not defined for states with vanishing momentum α+ +0 ∝ p+ +L = 0. In fact, one must also have +α− +0 ∝ p− +L = 0 (otherwise, the contracting operator for Q2 is well-defined and can be used +instead). But, these states do not satisfy the on-shell condition (except for massless states +with L⊥ +0 = 1), as it will be clear later (see [245, sec. 2.2] for more details). For this reason, +we assume that states have a generic non-zero momentum and that there is no pathology. +4The role of Q0 and Q2 can be reversed by changing the sign in the definition of the degree and the role +of P ± +n . +144 + +Full relative cohomology +This section aims to construct states in H0( �QB) from states in H(Q0). +We follow the +construction from [29]. +Given a state |ψ0⟩ ∈ H0(Q0), the state Q1 |ψ0⟩ is Q0-closed since Q0 and Q1 anticommute +(8.67): +{Q0, Q1} |ψ0⟩ = 0 +=⇒ +Q0 +� +Q1 |ψ0⟩ +� += 0. +(8.75) +Since Q1 |ψ0⟩ is not in ker �L∥ +0 (because Q1 increases the ghost number by 1), the state Q1 |ψ0⟩ +is Q0-exact and can be written as Q0 of another state |ψ1⟩: +Q1 |ψ0⟩ =: −Q0 |ψ1⟩ +=⇒ +|ψ1⟩ = − B +�L∥ +0 +Q1 |ψ0⟩ . +(8.76) +Computation – Equation (8.76) +Start from the definition and insert (8.70) since �L0 is invertible: +Q1 |ψ0⟩ = +� +Q0, B +�L∥ +0 +� +Q1 |ψ0⟩ = Q0 +� +B +�L∥ +0 +Q1 |ψ0⟩ +� +. +The state |ψ1⟩ is identified with minus the state inside the parenthesis (up to a BRST +exact state). +As for |ψ0⟩, apply {Q0, Q1} on ψ1: +{Q0, Q1} |ψ1⟩ = Q0 +� +Q1 |ψ1⟩ + Q2 |ψ0⟩ +� +. +(8.77) +This implies that the combination in parenthesis is Q0-closed and, for the same reason as +above, it is exact: +Q1 |ψ1⟩ + Q2 |ψ0⟩ = Q0 |ψ2⟩ , +|ψ2⟩ = − B +�L∥ +0 +� +Q1 |ψ1⟩ + Q2 |ψ0⟩ +� +. +(8.78) +Computation – Equation (8.77) +{Q0, Q1} |ψ1⟩ = Q0Q1 |ψ1⟩ − Q2 +1 |ψ0⟩ = Q0Q1 |ψ1⟩ + {Q0, Q2} |ψ0⟩ . +The first equality follows from (8.76), the second by using (8.67). The final result is +obtained after using that |ψ0⟩ is Q0-closed. +Iterating this procedure leads to a series of states: +|ψk+1⟩ = − B +�L∥ +0 +� +Q1 |ψk⟩ + Q2 |ψk−1⟩ +� +. +(8.79) +We claim that a state in the relative cohomology |ψ⟩ ∈ H0( �QB) is built by summing all +these states: +|ψ⟩ = +� +k∈N +|ψk⟩ . +(8.80) +Indeed, it is easy to check that |ψ⟩ is �QB-closed: +�QB |ψ⟩ = 0. +(8.81) +145 + +We leave aside the proof that ψ is not exact (see [29]). Note that ψ and ψ0 have the same +ghost numbers +Ngh(ψ) = Ngh(ψ0) = 1 +(8.82) +since Ngh(BQj) = 0. +In fact, since ψ0 does not contain longitudinal modes, it is annihilated by Q1 and Q2 +(these operators contain either a ghost creation operator together with a light-cone annihil- +ation operator, or the reverse): +Q1 |ψ0⟩ = Q2 |ψ0⟩ = 0. +(8.83) +As a consequence, one has ψk = 0 for k ≥ 1 and ψ = ψ0. +Computation – Equation (8.81) +�QB |ψ⟩ = +� +k∈N +�QB |ψk⟩ += Q0 |ψ0⟩ + Q1 |ψ0⟩ + Q0 |ψ1⟩ +� +�� +� +=0 ++ Q2 |ψ0⟩ + Q1 |ψ1⟩ + Q0 |ψ2⟩ +� +�� +� +=0 ++ · · · += 0. +8.3.3 +Absolute cohomology, states and no-ghost theorem +The absolute cohomology is constructed from the relative cohomology: +Habs(QB) = Hrel(QB) ⊕ c0 Hrel(QB). +(8.84) +The interested reader is refereed to [29] for the proof. A simple motivation is that the Hilbert +space is decomposed in terms of the ghost zero-modes as in (7.169). Since the zero-modes +commute with �Q0, linear combination of states in Hrel(QB) and c0Hrel(QB) are expected +to be in the cohomology. Obviously, one has to work out the other terms of QB and prove +that there are no other states. +It looks like there is a doubling of the physical states, one built on | ↓⟩ and one on | ↑⟩. +The remedy is to impose the condition b0 = 0 on the states (see also Section 3.2.2 and [245, +sec. 2.2] for more details). As already pointed out, states in Habs form equivalence class +under |ψ⟩ ∼ |ψ⟩ + QB |Λ⟩, and it is necessary to select a single representative. This is what +the condition b0 = 0 achieves. Obviously, it is always possible to add BRST exact states to +write another representative (for example, to restore the Lorentz covariance). +The last step is to discuss the no-ghost theorem: the latter states that there is no +negative-norm states in the BRST cohomology of string theory. This follows straightfor- +wardly from the condition �L0 = 0: it implies that there are no ghost and no light-cone +excitations. The ghosts and the time direction (if X0 is timelike) are responsible for negative- +norm states. Hence, the cohomology has no negative-norm states if the transverse CFT is +unitary (which implies that all states in H⊥ have a positive-definite inner-product). +Physical states |ψ⟩ ∈ Hrel(QB) are thus of the form: +|ψ⟩ = |k0, k1, ↓⟩ ⊗ |ψ⊥⟩ , +|ψ⊥⟩ ∈ H⊥, +(8.85a) +� +L⊥ +0 − m2 +∥,Lℓ2 − 1 +� +|ψ⟩ = 0, +p2 +L,∥ = −m2 +∥,Lℓ2. +(8.85b) +This form can be made covariant: taking a state of the form |ψ⟩⊗| ↓⟩ with |ψ⟩ ∈ Hm, acting +with QB implies the equivalence with the old covariant quantization: +(Lm +0 − 1) |ψ⟩ = 0, +∀n > 0 : +Lm +n |ψ⟩ = 0. +(8.86) +146 + +This means that ψ must be a weight 1 primary field of the matter CFT. +Remark 8.2 (Open string) The results of this section provide, in fact, the cohomology +for the open string after taking pL = p (instead of pL = p/2 for the closed string). +8.3.4 +Cohomology for holomorphic and anti-holomorphic sectors +It remains to generalize the computation of the cohomology when considering both the +holomorphic and anti-holomorphic sectors. +In this case, the BRST operator is +QB = c0L0 − b0M + �QB + ¯c0 ¯L0 − ¯b0 ¯ +M + �QB. +(8.87) +It is useful to rewrite this expression in terms of L± +0 , b± +0 and c± +0 : +QB = c+ +0 L+ +0 − b+ +0 M + + c− +0 L− +0 − b− +0 M − + �Q+ +B, +(8.88) +where +L+ +0 = +� +L⊥+ +0 +− +m2 +∥ℓ2 +2 +− 2 +� ++ �L∥+ +0 , +L− +0 = L⊥− +0 ++ �L∥− +0 +(8.89) +and +M ± := 1 +2(M ± ¯ +M). +(8.90) +Because of the relations L± +0 = {QB, b± +0 }, we find that states in the cohomology must be +on-shell L+ +0 = 0 and must satisfy the level-matching condition L− +0 = 0:5 +L+ +0 |ψ⟩ = L− +0 |ψ⟩ = 0. +(8.91) +Again, it is possible to reduce the cohomology by imposing conditions on the zero-modes +such that the above conditions are automatically satisfied (see also Section 3.2.2). Imposing +first the condition b− +0 = 0 defines the semi-relative cohomology. The relative cohomology is +found by imposing b± +0 = 0 and in fact corresponds to the physical space (see [245, sec. 2.3] for +more details). The rest of the derivation follows straightforwardly because the two sectors +commute: we find that the cohomology is ghost-free and has no light-cone excitations: +�L∥± +0 += N 0 ± ¯N 0 + N 1 ± ¯N 1 + N b ± ¯N b + N c ± ¯N c = 0. +(8.92) +In general, it is simpler to work with a covariant expression and to impose the necessary +conditions. Taking a state |ψ⟩ ⊗ | ↓↓⟩ with |ψ⟩ ∈ Hm, we find that ψ is a weight (1, 1) +primary field of the matter CFT: +(Lm +0 + ¯Lm +0 − 2) |ψ⟩ = 0, +(Lm +0 − ¯Lm +0 ) |ψ⟩ = 0, +∀n > 0 : +Lm +n |ψ⟩ = ¯Lm +n |ψ⟩ = 0. +(8.93) +An important point is that the usual mass-shell condition k2 = −m2 is provided by the first +condition only. This also shows that states in the cohomology naturally appears with c¯c +insertion since +| ↓↓⟩ = c(0)¯c(0) |0⟩ = c1¯c1 |0⟩ . +(8.94) +This hints at rewriting of scattering amplitudes in terms of unintegrated states (3.29) only. +A state is said to be of level (ℓ, ¯ℓ) and denoted as ψℓ,¯ℓ if it satisfies: +�L0 |ψℓ,¯ℓ⟩ = ℓ |ψℓ,¯ℓ⟩ , +�¯L0 |ψℓ,¯ℓ⟩ = ¯ℓ |ψℓ,¯ℓ⟩ . +(8.95) +5In the current case, the propagator is less easily identified. We will come back on its definition later. +147 + +Example 8.1 – Closed string tachyon +As an example, let’s construct the state ψ0,0 with level zero for a spacetime with D +non-compact dimensions. In this case, the transverse CFT contains D − 2 free scalars +which combine with X0 and X1 into D scalars Xµ. The Fock space is built on the +vacuum |k⟩ and we define the mass such that on-shell condition reduces to the standard +QFT expression: +k2 = −m2, +m2 := 2 +ℓ2 (N + ¯N − 2), +(8.96) +where N and ¯N are the matter level operators. The state in the remaining transverse +CFT (without the D − 2 scalars) is the SL(2, C) vacuum with L⊥ +0 = ¯L⊥ +0 = 0 (this is +the state with the lowest energy for a unitary CFT). In this case, the on-shell condition +reads +m2ℓ2 = −4 < 0. +(8.97) +Since the mass is negative, this state is a tachyon. The vertex operator associated to +the state reads: +V (k, z, ¯z) = c(z)¯c(¯z)eik·X(z,¯z). +(8.98) +8.4 +Summary +In this chapter, we have described the BRST quantization from the CFT point of view. We +have first considered only the holomorphic sector (equivalently, the open string). We proved +that the cohomology does not contain negative-norm states and we provided an explicit way +to construct the states. Finally, we glued together both sectors and characterized the BRST +cohomology of the closed string. +What is the next step? We could move to computations of on-shell string amplitudes, +but this falls outside the scope of this book. We can also start to consider string field theory. +Indeed, the BRST equation QB |ψ⟩ = 0 and the equivalence |ψ⟩ ∼ |ψ⟩ + QB |Λ⟩ completely +characterize the states. In QFT, states are solutions of the linearized equations of motion: +hence, the BRST equation can provide a starting point for building the action. This is the +topic of Chapter 10. +8.5 +Suggested readings +• The general method to construct the absolute cohomology follows [29, 193]. Other +works and reviews include [20, 28, 57, 118, 119, 170, 177]. +• String states are discussed in [24, sec. 3.3, 193, sec. 4.1]. +148 + +Part II +String field theory +149 + +Chapter 9 +String field +In this chapter, we introduce general concepts about the string field. The goal is to give an +idea of which type of object it is and of the different possibilities for describing it. We will +see that the string field is a functional and, for this reason, it is more convenient to work +with the associated ket field, which can itself be represented in momentum space. We focus +on what to expect from a free field, taking inspiration from the worldsheet theory. The +interpretation becomes more difficult when taking into account the interactions. +9.1 +Field functional +A string field, after quantization, is an operator which creates or destroys a string at a given +time. Since a string is a 1-dimension extended object, the string field Ψ must depend on the +spatial positions of each point of the string denoted collectively as Xµ. Hence, the string +field is a functional Ψ[Xµ]. The fact that it is a functional rather than a function makes the +construction of a field theory much more challenging: it asks for revisiting all concepts we +know in point-particle QFT without any prior experience with a simple model.1 +It is important that the dependence is only on the shape and not on the parametrization. +However, it is simpler to first work with a specific parametrization X(σ) and make sure that +nothing depends on it at the end (equivalent to imposing the invariance under reparamet- +rization of the worldsheet). This leads to work with a functional Ψ[X(σ)] of fields on the +worldsheet (at fixed time). To proceed, one should first determine the degrees of freedom of +the string, and then to find the interactions. The simplest way to achieve the first step is to +perform a second-quantization of the string wave-functional: the string field is written as a +linear combination of first-quantized states with spacetime wave functionals as coefficients.2 +This provides a free Hamiltonian; trying to add interactions perturbatively does not work +well. +It is not possible to go very far with this approach and one is lead to choose a specific +gauge, breaking the manifest invariance under reparametrizations. The simplest is the light- +cone gauge since one works only with the physical degrees of freedom of the string. While +this approach is interesting to gain some intuitions and to show that, in principle, it is +possible to build a string field theory, it requires making various assumptions and ends up +1The problem is not in working with the wordline formalism and writing a BRST field theory, but really +to take into account the spatial extension of the objects. +In fact, generalizing further to functionals of +extended (p > 1)-dimensional objects – branes – shows that SFT is the simplest of such field theories. +2The description of the first-quantized states depends on the CFT used to describe the theory. This +explains the lack of manifest background independence of SFT. Unfortunately, no better approach has been +found until now. +150 + +with problems (especially for superstrings).3 +Since worldsheet reparametrization invariance is just a kind of gauge symmetry – maybe +less familiar than the non-Abelian gauge symmetries in Yang–Mills, but still a gauge sym- +metry –, one may surmise that it should be possible to gauge fix this symmetry and to +introduce a BRST symmetry in its place. This is the program of the BRST (or covariant) +string field theory in which the string field depends not only of the worldsheet (at fixed +time), but also on the ghosts: Ψ[X(σ), c(σ)]. There is no dependence on the b ghost because +the latter is the conjugate momentum of the c ghost: in the operator language, b(σ) ∼ +δ +δc(σ). +The BRST formalism has the major advantage to allow to move easily from D = 26 +dimensions – described by Xµ scalars (µ = 0, . . . , 25) – to a (possibly curved) D-dimensional +spacetime and a string with some internal structure – described by a more general CFT, +in which D scalars Xµ represent the non-compact dimensions and the remaining system +with central charge 26 − D describes the compactification and structure. It is sufficient to +consider the string field as a general functional of all the worldsheet fields. For simplicity, +we will continue to write X in the functional dependence, keeping the other matter fields +implicit. +It is complicated to find an explicit expression for the string field as a functional of X(σ) +and c(σ). In fact, the field written in this way is in the position representation and, as usual +in quantum mechanics, one can choose to work with the representation independent ket |Ψ⟩: +Ψ[X(σ), c(σ)] := ⟨X(σ), c(σ)|Ψ⟩ . +(9.1) +It is often more convenient to work with |Ψ⟩ (which we will also denote simply as Ψ, not +distinguishing between states and operators). The latter will be the basic object of SFT in +most of this book. +Writing a field theory in terms of |Ψ⟩ may not be intuitive since in point-particle QFT, +one is used to work with the position or momentum representation. In fact, there is a very +simple way to recover a formulation in terms of spacetime point-particle fields, which can be +used almost whenever there is a doubt about what is going on. Indeed, as is well-known from +standard worldsheet string theory, the string states behave like a collection of particles. This +is because the modes of the CFT fields (like αµ +n) carry spacetime indices (Lorentz, group +representation. . . ) such that the states themselves carries indices. Indeed, these quantum +numbers classify eigenstates of the operators L0 and ¯L0. On the other hand, positions and +shapes are not eigenstates of any simple CFT operator. +9.2 +Field expansion +It follows that the second-quantized string field can be written as a linear combination +of first-quantized off-shell states |φα(k)⟩ = Vα(k; 0, 0) |0⟩ (which form a basis of the CFT +Hilbert space H): +|Ψ⟩ = +� +α +� +dDk +(2π)D ψα(k) |φα(k)⟩ , +(9.2) +where k is the D-dimensional momentum of the string (conjugated to the position of the +centre-of-mass) and α is a collection of discrete quantum numbers (Lorentz indices, group +representation. . . ). When inserting this expansion inside the action, we find that it reduces +to a standard field theory with an infinite number of particles described by the spacetime +3While this approach has been mostly abandoned, recent results show that it can still be used when +defined with a proper regularization [4, 114–117]. +151 + +fields ψα(k) (momentum representation). +The fields can also be written in the position +representation by Fourier transforming only the momentum k to the centre-of-mass x: +ψα(x) = +� +dDk +(2π)D eik·xψα(k). +(9.3) +However, we will see that it is often not convenient because the action is non-local in position +space (including for example exponentials of derivatives). +The physical intuition is that the string is a non-local object in spacetime. It can be +expressed in momentum space through a Fourier transformation: variables dual to non- +compact (resp. compact) dimensions are continuous (discrete). As a consequence, the mo- +mentum is continuous since the centre-of-mass move in the non-compact spacetime, while +the string itself has a finite extension and the associated modes are discrete but still not +bounded (and similarly for compact dimensions). This indicates that the spectrum is the +collection of a set of continuous and discrete modes. Hence, the non-locality of the string +(due to the spatial extension) is traded for an infinite number of modes which behave like +standard particles. In this description, the non-locality arises: 1) in the infinite number of +fields, 2) in the coupling between the modes, 3) as a complicated momentum-dependence of +the action. +When we are not interested in the spacetime properties, we will write a generic basis of +the Hilbert space H as {φr}: +|Ψ⟩ = +� +r +ψr |φr⟩ . +(9.4) +The sum over r includes discrete and continuous labels. +Example 9.1 – Scalar field +In order to illustrate the notations for a point-particle, consider a scalar field φ(x). It +can be expanded in Fourier modes as: +φ(x) = +� +dDk +(2π)D φ(k)eik·x. +(9.5) +The corresponding ket |φ⟩ is found by expanding on a basis {|k⟩}: +|φ⟩ = +� +dDk +(2π)D φ(k) |k⟩ , +φ(k) = ⟨k|φ⟩ . +(9.6) +Similarly, the position space field is defined from the basis {|x⟩} such that: +φ(x) = ⟨x|φ⟩ = +� +dDk +(2π)D ⟨x|k⟩ ⟨k|φ⟩ , +⟨x|k⟩ = eik·x. +(9.7) +9.3 +Summary +In this chapter, we introduced general ideas about what a string field is. We now need to +write an action. In general, one proceeds in two steps: +1. build the kinetic term (free theory): +(a) equations of motion → physical states +(b) equivalence relation → gauge symmetry +2. add interactions and deform the gauge transformation +152 + +We consider the first point in the next chapter, but we will have to introduce more machinery +in order to discuss interactions. +9.4 +Suggested readings +• General discussions of the string field and of the ideas of string field theory can be +found in [192, sec. 4, 261]. +• Light-cone SFT is reviewed in [245, 124, chap. 6, 125, chap. 9]. +153 + +Chapter 10 +Free BRST string field theory +Abstract +In this chapter, we construct the BRST (or covariant) free bosonic string field +theories. It is useful to first ignore the interactions in order to introduce some general tools +and structures in a simpler setting. Moreover, the free SFT is easily constructed and does +not require as much input as the interactions. In this chapter, we discuss mostly the open +string, keeping the closed string for the last section. We start by describing the classical +theory: equations of motion, action, gauge invariance and gauge fixing. Then, we perform +the path integral quantization and compute the action in terms of spacetime fields for the +first two levels (tachyon and gauge field). +10.1 +Classical action for the open string +Contrary to most of this book, we will exemplify the discussion with the open string. The +reason is that most computations are the same in both the open and closed string theories, +but the latter requires twice more writing. There are also a few subtleties which can be more +easily explained once the general structure is understood. Everything needed for the open +string for this chapter can be found in Chapter 8: in fact, describing the open string (at this +level) is equivalent to consider only the holomorphic sector of the CFT and to set pL = p +(instead of p/2). We consider a generic matter CFT in addition to the ghost system and we +denote as H the space of states. The open and closed string fields are denoted respectively +by Φ and Ψ, such that it is clear which theory is studied. +An action can be either constructed from first principles, or it can be derived from the +equations of motion. Since the fundamental structure of string field theory is not (really) +known, one needs to rely on the second approach. +But do we already know the (free) +equations of motion for the string field? The answer is yes. But, before showing how these +can be found from the worldsheet formalism, we will study the case of the point-particle to +fix ideas and notations. +10.1.1 +Warm-up: point-particle +The free (or linearized) equation of motion for a scalar particle reads: +(−∆ + m2)φ(x) = 0. +(10.1) +Solutions to this equation provides one-particle state of the free theory: a convenient basis +is {eikx}, where each state satisfies the on-shell condition +k2 = −m2. +(10.2) +154 + +The field φ(x) is decomposed on the basis as +φ(x) = +� +dk φ(k)eikx, +(10.3) +where φ(k) are the coefficients of the expansion. Since the field is off-shell, the condition +k2 = −m2 is not imposed. Following Chapter 9, the field can also be represented as a ket: +φ(x) = ⟨x|φ⟩ , +φ(k) = ⟨k|φ⟩ , +(10.4) +or, conversely: +|φ⟩ = +� +dx φ(x) |x⟩ = +� +dk φ(k) |k⟩ . +(10.5) +Writing the kinetic operator as a kernel: +K(x, x′) :=⟨x| K |x′⟩ = δ(x − x′) (−∆x + m2), +(10.6) +the equations of motion reads +� +dx′ K(x, x′)φ(x′) = 0 +⇐⇒ +K |φ⟩ = 0. +(10.7) +An action can easily be found from the equation of motion by multiplying with φ(x) and +integrating: +S = 1 +2 +� +dx φ(x)(−∆ + m2)φ(x) = 1 +2 +� +dxdx′ φ(x)K(x, x′)φ(x′). +(10.8) +It is straightforward to write the action in terms of the ket: +S = 1 +2 ⟨φ| K |φ⟩ . +(10.9) +There is one hidden assumption in the previous lines: the definition of a scalar product. +A natural inner product is provided in the usual quantum mechanics by associating a bra to +a ket. Similarly, integration provides another definition of the inner product when working +with functions. We will find that the definition of the inner product requires more care in +closed SFT. To summarize, to write the kinetic term of the action, one needs the linearized +equation of motion and an appropriate inner product on the space of states. +10.1.2 +Open string action +The worldsheet equation which yields precisely all the string physical states |ψ⟩ is the BRST +condition: +QB |ψ⟩ = 0. +(10.10) +Considering the open string field Φ to be a linear combination of all possible one-string +states |ψ⟩ +Φ ∈ H, +(10.11) +the equation of motion is: +QB |Φ⟩ = 0. +(10.12) +Moving away from the physical state condition, the string field Φ is off-shell and is expanded +on a general basis {φr} of H. This presents a first difficulty because the worldsheet approach +– and the description of amplitudes – looks ill-defined for off-shell states: extending the usual +155 + +formalism will be the topic of Chapter 11. However, this is not necessary for the free theory +and we can directly proceed. +Next, we need to find an inner product ⟨·, ·⟩ on the Hilbert space H. A natural candidate +is the BPZ inner product since it is not degenerate +⟨A, B⟩ := ⟨A|B⟩ , +(10.13) +where ⟨A| = |A⟩t is the BPZ conjugate (6.98) of |A⟩, using I−. This leads to the action: +S = 1 +2 ⟨Φ, QBΦ⟩ = 1 +2 ⟨Φ| QB |Φ⟩ . +(10.14) +Due to the definition of the BPZ product, the action is equivalent to a 2-point correlation +function on the disk. +The inner product satisfies the following identities: +⟨A, B⟩ = (−1)|A||B|⟨B, A⟩, +⟨QBA, B⟩ = −(−1)|A|⟨A, QBB⟩, +(10.15) +where |A| denotes the Grassmann parity of the operator A. +A first consistency check is to verify that the ghost number of the string can be defined +such that the action is not vanishing. Indeed, the ghost number anomaly on the disk implies +that the total ghost number must be Ngh = 3. Since physical states have Ngh = 1, it is +reasonable to take the string field to satisfy the same condition, even off-shell: +Ngh(Φ) = 1. +(10.16) +This condition means that there is no ghost at the classical level beyond the one of the +energy vacuum | ↓⟩, which has Ngh = 1. Moreover, the BRST charge has Ngh(QB) = 1, +such that the action has ghost number 3. +One needs to find the Grassmann parity of the string field. Using the properties of the +BPZ inner product, the string field should be Grassmann odd +|Φ| = 1 +(10.17) +for the action to be even. This is in agreement with the fact that the string field has ghost +number 1 and that the ghosts are Grassmann odd. One must impose a reality condition on +the string field (a complex field would behave like two real fields and have too many states). +The appropriate reality condition identifies the Euclidean and BPZ conjugates: +|Φ⟩‡ = |Φ⟩t . +(10.18) +That this relation is correct will be checked a posteriori for the tachyon field in Section 10.4. +Computation – Equation (10.17) +⟨Φ, QBΦ⟩ = (−1)|Φ|(|QBΦ|)⟨QBΦ, Φ⟩ = (−1)|Φ|(1+|Φ|)⟨QBΦ, Φ⟩ += ⟨QBΦ, Φ⟩ = −(−1)|Φ|⟨Φ, QBΦ⟩, +where both properties (10.15), together with the fact that |Φ|(1 + |Φ|) is necessarily +even. In order for the bracket to be non-zero, one must have |Φ| = 1. +Since the Hilbert space splits as H = H0 ⊕ c0H0 with H0 = H ∩ ker b0, see (8.31), it is +natural to split the field as (this is discussed further in Section 10.2): +|Φ⟩ = |Φ↓⟩ + c0 |�Φ↓⟩ , +(10.19) +156 + +where +Φ↓, �Φ↓ ∈ H0 +=⇒ +b0 |Φ↓⟩ = b0 |�Φ↓⟩ = 0. +(10.20) +The ghost number of each component is +Ngh(Φ↓) = 1, +Ngh(�Φ↓) = 0. +(10.21) +Remembering the decomposition (8.14a) of the BRST operator +QB = c0L0 − b0M + �QB, +(10.22) +inserting the decomposition (10.19) in the action (10.14) gives: +S = 1 +2⟨Φ↓| c0L0 |Φ↓⟩ + 1 +2⟨�Φ↓| c0M |�Φ↓⟩ +⟨�Φ↓| c0 �QB |Φ↓⟩ . +(10.23) +The equations of motion are obtained by varying the different fields: +0 = −M |�Φ↓⟩ + �QB |Φ↓⟩ , +0 = c0L0 |Φ↓⟩ + c0 �QB |�Φ↓⟩ . +(10.24) +Computation – Equation (10.23) +Let’s introduce the projector Πs = b0c0 on the space H0 = H∩ker b0 and the orthogonal +projector ¯Πs = c0b0 such that +|Φ⟩ = |Φ↓⟩ + |Φ↑⟩ , +|Φ↓⟩ = Πs |Φ⟩ , +|Φ↑⟩ = ¯Πs |Φ⟩ . +(10.25) +We then have: +ΠsQB |Φ⟩ = −b0M |Φ↑⟩ + �QB |Φ↓⟩ , +¯ΠsQB |Φ⟩ = c0L0 |Φ↓⟩ + �QB |Φ↑⟩ , +(10.26) +using +[Πs, �QB] = [Πs, M] = [Πs, L0] = 0. +(10.27) +Then, we need the fact that Π† +s = ¯Πs. to compute the action: +S = 1 +2 ⟨Φ, QBΦ⟩ += 1 +2 ⟨ΠsΦ + ¯ΠsΦ, QBΦ⟩ += 1 +2 ⟨ΠsΦ, ¯ΠsQBΦ⟩ + 1 +2 ⟨¯ΠsΦ, ΠsQBΦ⟩ += 1 +2 ⟨Φ↓, c0L0Φ↓ + �QBΦ↑⟩ + 1 +2 ⟨Φ↑, −b0MΦ↑ + �QBΦ↓⟩ += 1 +2 ⟨Φ↓, c0L0Φ↓⟩ + 1 +2 ⟨Φ↓, �QBΦ↑⟩ − 1 +2 ⟨Φ↑, b0MΦ↑⟩ + 1 +2 ⟨Φ↑, �QBΦ↓⟩. +The result follows by setting |Φ↑⟩ = c0 |�Φ⟩, using (10.15) and that the BPZ conjugate +of c0 is −c0. +10.1.3 +Gauge invariance +In writing the action, only the condition that the states are BRST closed has been used. One +needs to interpret the condition that the state are not BRST-exact, or phrased differently +that two states differing by a BRST exact state are equivalent: +|φ⟩ ∼ |ψ⟩ + QB |λ⟩ . +(10.28) +157 + +Uplifting this condition to the string field, the most direct interpretation is that it corres- +ponds to a gauge invariance: +|Φ⟩ −→ |Φ′⟩ = |Φ⟩ + δΛ |Φ⟩ , +δΛ |Φ⟩ = QB |Λ⟩ +Ngh(Λ) = 0. +(10.29) +In order for the ghost numbers to match, the gauge parameter has vanishing ghost number. +The action (10.14) is obviously invariant since the BRST charge is nilpotent. +10.1.4 +Siegel gauge +In writing the action (10.14), the condition b0 |ψ⟩ = 0 has not been imposed on the string +field. In Section 3.2.2, this condition was found by restricting the BRST cohomology, pro- +jecting out states built on the ghost vacuum | ↑⟩, as required by the behaviour of the on-shell +scattering amplitudes. In Chapter 8, we obtained it by finding that the absolute cohomology +contains twice more states as necessary. This was also understood as a way to work with +a specific representative of the BRST cohomology. Since the field is off-shell and since the +action computes off-shell Green functions, these arguments cannot be used, which explains +why we did not use this condition earlier. +On the other hand, the condition +b0 |Φ⟩ = 0 +(10.30) +can be interpreted as a gauge fixing condition, called Siegel gauge. It can be reached from +any field through a gauge transformation (10.29) with +|Λ⟩ = −∆ |Φ⟩ , +∆ = b0 +L0 +, +(10.31) +where ∆ was defined in (8.39) and will be identified with the propagator. Note that b0 = 0 +does not imply L0 = 0 since the string field is not BRST closed. +This gauge choice is well-defined and completely fixes the gauge symmetry off-shell, +meaning that no solution of the equation of motion is pure gauge after the gauge fixing. +This is shown as follows: assume that |ψ⟩ = QB |χ⟩ is an off-shell pure-gauge state with +L0 ̸= 0, then, because it is also annihilated by b0, one finds: +0 = {QB, b0} |ψ⟩ = L0 |ψ⟩ +(10.32) +which yields a contradiction. +The gauge fixing condition breaks down for L0 = 0, but this does not pose any problem +when working with Feynman diagrams since they are not physical by themselves (nor are +the off-shell and on-shell Green functions). Only the sum giving the scattering amplitudes +(truncated on-shell Green functions) is physical: in this case, the singularity L0 = 0 cor- +responds to the on-shell condition and it is well-known how such infrared divergences for +intermediate states are removed (through the LSZ prescription, mass renormalization and +tadpole cancellation). +Computation – Equation (10.31) +Performing a gauge transformation gives +b0 |Φ′⟩ = b0 |Φ⟩ + b0QB |Λ⟩ = 0. +(10.33) +Then, one writes +b0 |Φ⟩ = b0{QB, ∆} |Φ⟩ = b0QB∆ |Φ⟩ , +(10.34) +using the relation (8.37), the expression (8.39) and the fact that b2 +0 = 0. Plugging this +158 + +back in the first equation gives: +b0QB (∆ |Φ⟩ + |Λ⟩) = 0. +(10.35) +The factor of b0 can be removed by multiplying with c0, and the parenthesis should +vanish (since it is not identically closed), which means that (10.31) holds up to a BRST +exact state. +Example 10.1 – Gauge fixing and singularity +In Maxwell theory, the gauge transformation +A′ +µ = Aµ + ∂µλ, +(10.36) +is used to impose the Lorentz condition +∂µA′ +µ = 0 +=⇒ +∆λ = −∂µAµ. +(10.37) +In momentum space, the parameter reads +λ = −kµ +k2 Aµ. +(10.38) +It is singular when k is on-shell, k2 = 0. However, this does not prevent from computing +Feynman diagrams. +To understand the effect of the gauge fixing on the string field components, decompose +the field as (10.19) |Φ⟩ = |Φ↓⟩ + c0 |�Φ↓⟩. Then, imposing the condition (10.30) yields +|�Φ↓⟩ = 0 +=⇒ +|Φ⟩ = |Φ↓⟩ . +(10.39) +This has the expected effect of dividing by two the number of states and show that they are +not physical. +Plugging this condition in the action (10.23) leads to gauge fixed action +S = 1 +2 ⟨Φ| c0L0 |Φ⟩ , +(10.40) +for which the equation of motion is +L0 |Φ⟩ = 0. +(10.41) +But, note that this equation contains much less information than the original (10.12): as |�Φ↓⟩ +is truncated from (10.40), a part of the equations of motion is lost. The missing equation +can be found by setting |�Φ⟩ = 0 in (10.24) and must be imposed on top of the action: +�QB |Φ⟩ = 0. +(10.42) +It is called out-of-Siegel gauge constraint and is equivalent to the Gauss constraint in elec- +tromagnetism: the equations of motion for pure gauge states contain also the physical fields, +thus, when one fixes a gauge, these relations are lost and must be imposed on the side of the +action. This procedure mimics what happens in the old covariant theory, where the Virasoro +constraints are imposed after choosing the flat gauge (if Φ contains no ghost on top of | ↓⟩, +then �QB = 0 implies Ln = 0, see Section 8.3.3). Moreover, the states which do not satisfy +the condition b0 = 0 do not propagate: this restricts the external states to be considered in +amplitudes. +159 + +Remark 10.1 Another way to derive (10.40) is to insert {b0, c0} = 1 in the action: +S = 1 +2 ⟨Φ| QB{c0, b0} |Φ⟩ = 1 +2 ⟨Φ| QBb0c0 |Φ⟩ += 1 +2 ⟨Φ| {b0, QB}c0 |Φ⟩ − 1 +2 ⟨Φ| b0QBc0 |Φ⟩ += 1 +2 ⟨Φ| c0L0 |Φ⟩ . +The drawback of this computation is that it does not show directly how the constraints (10.42) +arise. +Remark 10.2 (Generalized gauge fixing) It is possible to generalize the Siegel gauge, +in the same way that the Feynman gauge generalizes the Lorentz gauge. This has been studied +in [1, 2]. +In this section, we have motivated different properties and adopted some normalizations. +The simplest way to check that they are consistent is to derive the action in terms of the +spacetime fields and to check that it has the expected properties from standard QFT. This +will be the topic of Section 10.4. +10.2 +Open string field expansion, parity and ghost num- +ber +A basis for the off-shell Hilbert space H is denoted by {φr}, where the ghost numbers and +parity of the states are written as: +nr := Ngh(φr), +|φr| = nr +mod 2. +(10.43) +The corresponding basis of dual (or conjugate) states {φc +r} is defined by (6.145): +⟨φc +r|φs⟩ = δrs. +(10.44) +The basis states can be decomposed according to the ghost zero-modes +|φr⟩ = |φ↓,r⟩ + |φ↑,r⟩ , +b0 |φ↓,r⟩ = c0 |φ↑,r⟩ = 0. +(10.45) +Finally, each state ψ↑ ∈ c0H can be associated to a state �ψ: +|ψ↑⟩ = c0 | �ψ↓⟩ , +b0 | �ψ↓⟩ = 0, +Ngh(ψ↑) = Ngh( �ψ↓) + 1. +(10.46) +More details can be found in Section 11.2. +Any field Φ can be expanded as +|Φ⟩ = +� +r +ψr |φr⟩ , +(10.47) +where the ψr are spacetime fields (remembering that r denotes collectively the continuous +and discrete quantum numbers).1 +1The notation is slightly ambiguous: from (10.45), it looks like both components of φr have the same +coefficient ψr. But, in fact, one sums over all linearly independent states: in terms of the components of φr, +different basis can be considered; for example {φ↓,r, φ↑,r}, or {φ↓,r ± φ↑,r}. A more precise expression can +be found in (10.54) and (10.56). +160 + +Obviously, the coefficients do not carry a ghost number since they are not worldsheet +operators. However, they can be Grassmann even or odd such that each term of the sum +has the same parity, so that the field has a definite parity: +∀r : +|Φ| = |ψr| |φr|. +(10.48) +If the field is Grassmann odd (resp. even) then the coefficients ψr and the basis states must +have opposite (resp. identical) parities, such that |Φ| = 1. +Since the parity results from worldsheet ghosts and since there would be Grassmann odd +states even in a purely bosonic theory, it suggests that the parity of the coefficients ψr is +also related to a spacetime ghost number G defined as: +G(ψr) = 1 − nr. +(10.49) +The normalization is chosen such that the component of a classical string field (Ngh = 1) +are classical spacetime fields with G = 0 (no ghost). We will see later that this definition +makes sense. +A quantum string field Φ generally contains components Φn of all worldsheet ghost +numbers n: +Φ = +� +n∈Z +Φn, +Ngh(Φn) = n. +(10.50) +The projections on the positive and negative (cylinder) ghost numbers are denoted by Φ±: +Φ = Φ+ + Φ−, +Φ+ = +� +n>1 +Φn, +Φ− = +� +n≤1 +Φn. +(10.51) +The shift in the indices is explained by the relation (B.56) between the cylinder and plane +ghost numbers. +For a field Φn of fixed ghost number, coefficients of the expansion vanish whenever the +ghost number of the basis state does not match the one of the field: +∀nr ̸= n : +ψr = 0. +(10.52) +Another possibility to define the field Φn is to insert a delta function: +|Φn⟩ = δ(Ngh − n) |Ψ⟩ = +� +r +δ(nr − n) ψr |φr⟩ . +(10.53) +According to (10.45), a string field Φ can also be separated in terms of the ghost zero- +modes: +|Φ⟩ = |Φ↓⟩ + |Φ↑⟩ = |Φ↓⟩ + c0 |�Φ↓⟩ , +(10.54a) +|Φ↑⟩ = c0 |�Φ↓⟩ , +|�Φ↓⟩ = b0 |Φ↑⟩ , +(10.54b) +where the components satisfy the constraints +b0 |Φ↓⟩ = 0, +c0 |Φ↑⟩ = 0, +b0 |�Φ↓⟩ = 0. +(10.55) +The fields |Φ↓⟩ and |Φ↑⟩ (or |�Φ↓⟩) are called the down and top components and they can be +expanded as: +|Φ↓⟩ = +� +r +ψ↓,r |φ↓,r⟩ , +|Φ↑⟩ = +� +r +ψ↑,r |φ↑,r⟩ . +(10.56) +161 + +10.3 +Path integral quantization +The string field theory can be quantized with a path integral: +Z = +� +dΦcl e−S[Φcl] = +� +dΦcl e− 1 +2⟨Φcl|QB|Φcl⟩. +(10.57) +An index has been added to the field to emphasize that it is the classical field (no spacetime +ghosts). The simplest way to define the measure is to use the expansion (9.4) such that +Z = +� � +s +dψs e−S[{ψr}]. +(10.58) +10.3.1 +Tentative Faddeev–Popov gauge fixing +The action can be gauge fixed using the Faddeev–Popov formalism. The gauge fixing con- +dition is +F(Φcl) := b0 |Φcl⟩ = 0. +(10.59) +Its variation under a gauge transformation (10.29) reads +δF = b0QB |Λcl⟩ , +(10.60) +which implies that the Faddeev–Popov determinant is +det δF +δΛcl += det b0QB. +(10.61) +This determinant is rewritten as a path integral by introducing a ghost C and an antighost +B′ string fields (the prime on B′ will become clear below): +det b0QB = +� +dB′dC e−SFP, +SFP = −⟨B′| b0QB |C⟩ . +(10.62) +The ghost numbers are attributed by selecting the same ghost number for the C ghost and for +the gauge parameter, and then requiring that the Faddeev–Popov action is non-vanishing: +Ngh(B′) = 3, +Ngh(C) = 0. +(10.63) +The ghosts can be expanded as +|B′⟩ = δ(Ngh − 3) +� +r +b′ +r |φr⟩ , +|C⟩ = δ(Ngh) +� +r +cr |φr⟩ , +(10.64) +where the coefficients br and cr are Grassmann odd in order for the determinant formula to +make sense: +|br| = |cr| = 1. +(10.65) +Then, since the basis states appearing in B′ and C are respectively odd and even, this +implies +|B′| = 0, +|C| = 1. +(10.66) +However, there is a redundancy in the gauge fixing because the Faddeev–Popov action +is itself invariant under two independent transformations: +δ |C⟩ = QB |Λ−1⟩ , +Ngh(Λ−1) = −1, +(10.67a) +δ |B′⟩ = b0 |Λ′⟩ , +Ngh(Λ′) = 4. +(10.67b) +162 + +This residual invariance arises because not all |Λcl⟩ generate a gauge transformation. Indeed, +if +|Λ⟩ = |Λ0⟩ + QB |Λ−1⟩ , +(10.68) +the field transforms as +|Φ′ +cl⟩ −→ |Φcl⟩ + QB |Λ0⟩ +(10.69) +and there is no trace left of |Λ−1⟩, so it should not be counted. +The second invariance (10.67b) is not problematic because b0 is an algebraic operator (the +Faddeev–Popov action associated to the determinant has no dynamics). The decompositions +of the gauge parameter Λ′ and the B′ field into components (10.54) read: +|B′⟩ = |B′ +↓⟩ + c0 |B⟩ , +|B⟩ := | �B′ +↓⟩ , +(10.70a) +|Λ′⟩ = |Λ′ +↓⟩ + c0 |�Λ′ +↓⟩ . +(10.70b) +The gauge transformations act on the components as: +δ |B′ +↓⟩ = |�Λ′ +↓⟩ , +δ |B⟩ = 0. +(10.71) +This shows that B is gauge invariant and B′ +↓ can be completely removed by the gauge +transformation. This makes sense because B′ +↓ does not appear in the action (10.62). The +gauge transformation (10.67b) can be used to fix the gauge: +|F ′⟩ = c0 |B′⟩ = 0 +=⇒ +|B′ +↓⟩ = 0. +(10.72) +This fixes completely the gauge invariance since the field B is restricted to satisfy b0 |B⟩ = 0, +and the component form (10.71) of the gauge transformation shows that no transformation +is allowed. Moreover, there is no need to introduce a Faddeev–Popov determinant for this +gauge fixing because the corresponding ghosts would not couple to the other fields (and this +would continue to hold even in the presence of interactions, see Remark 10.4). Indeed, from +the absence of derivatives in the gauge transformation, one finds that the determinant is +constant and thus a ghost-representation is not necessary: +det δF ′ +δΛ′ = det c0b0 = det c0 det b0 = 1 +2 det{b0, c0} = 1 +2. +(10.73) +Then, redefining the measure, the partition function and action reduce to +∆FP = +� +dB dC e−SFP[B,C], +SFP =⟨B| QB |C⟩ . +(10.74) +Note that the field B satisfies +b0 |B⟩ = 0, +Ngh(B) = 2, +|B| = 1. +(10.75) +Since both fields are Grassmann odd, the action can be rewritten in a symmetric way: +SFP = 1 +2 +� +⟨B| QB |C⟩ +⟨C| QB |B⟩ +� +. +(10.76) +Remark 10.3 (Ghost and anti-ghost definitions) The definition of the anti-ghost B +and ghost C is appropriate because the worldsheet and spacetime ghost numbers are related +by a minus sign (and a shift of one unit). In the BV formalism, we will see that the fields +contain the matter and ghost fields, while the antifields contain the anti-ghosts. These two +sets are respectively defined with Ngh ≤ 1 and Ngh > 1. +163 + +The constraint b0 |B⟩ = 0 can be lifted by adding a top component: +|B⟩ = |B↓⟩ + c0 | �B↓⟩ +(10.77) +together with the gauge invariance +δ |B⟩ = QB |Λ1⟩ . +(10.78) +Note the difference with (10.70): while B = �B′ +↓ was the top component of the B′ field, here, +it is defined to be the down component, such that |B↓⟩ = | �B′ +↓⟩. However, for the moment, +we keep B to satisfy b0 |B⟩ = 0. +Remark 10.4 (Decoupling of the ghosts) Since the theory is free the Faddeev–Popov +action (10.74) could be ignored and absorbed in the normalization because it does not couple +to the field. On the other hand, when interactions are included, the gauge transformation +is modified and the ghosts couple to the matter fields. +But this is true only for the C +transformation (10.67a), not for (10.67b). Then it means that ghosts introduced for gauge +fixing (10.67b) will never couple to the matter and other ghosts. +The invariance (10.67a) is a gauge invariance for C and must be treated in the same +way as (10.29). Then, following the Faddeev–Popov procedure, one is lead to introduce new +ghosts for the ghosts. But, the same structure appears again. This leads to a residual gauge +invariance, which has the same form. This process continues recursively and one finds an +infinite tower of ghosts. +10.3.2 +Tower of ghosts +In order to simplify the notations, all the fields are denoted by Φn where n gives the ghost +number: +• Φ1 := Φcl is the original physical field +• Φ0 := C and, more generally, Φn with n < 1 are ghosts +• Φ2 := B and, more generally, Φn with n > 1 are anti-ghosts +The recipe is that each pair of ghost fields (Φn+2, Φ−n) is associated to a gauge parameter +Λ−n−1 with n ≥ 0. It is then natural to gather all the fields in a single field +|Φ⟩ = +� +n +|Φn⟩ +(10.79) +satisfying the gauge fixing constraint: +b0 |Φ⟩ = 0 +=⇒ +b0 |Φn⟩ = 0. +(10.80) +For n ≤ 1, these constraints are gauge fixing conditions for the invariance δ |Φn⟩ = QBΛn. +For n > 1, they arise by considering only the top component of the B field. +Finally, the gauge fixing condition can be incorporated inside the action by using a +Lagrange multiplier β, which is an auxiliary string field containing also components of all +ghost numbers: +|β⟩ = +� +n∈Z +|βn⟩ . +(10.81) +The path integral then reads +Z = +� +dΦdβ e−S[Φ,β], +(10.82) +164 + +where +S[Φ, β] = 1 +2⟨Φ| QB |Φ⟩ +⟨β| b0 |Φ⟩ +(10.83a) += +� +n∈Z +�1 +2⟨Φ2−n| QB |Φn⟩ +⟨β4−n| b0 |Φn⟩ +� +. +(10.83b) +The first term of the action has the same form as the classical action (10.14), but now includes +fields at every ghost number. The complete BV analysis is relegated to the interacting theory. +Removing the auxiliary field β = 0, one finds that the action is invariant under the +extended gauge transformation +δ |Φ⟩ = QB |Λ⟩ , +(10.84) +where the gauge parameter has also components of all ghost numbers: +|Λ⟩ = +� +n∈Z +|Λn⟩ . +(10.85) +10.4 +Spacetime action +In order to make the string field action more concrete, and as emphasized in Chapter 9, +it is useful to expand the string field in spacetime fields and to write the action for the +lowest modes. This also helps to check that the normalization chosen until here correctly +reproduces the standard QFT normalizations. For simplicity we focus on the open bosonic +string in D = 26. +We build the string from the vacuum |k, ↓⟩ (Chapter 8) by acting with the ghost positive- +frequency modes b−n and c−n, the zero-mode c0, and from the scalar oscillators iαµ +−n. +Up to level ℓ = 1, the classical open string field can be expanded as +|Φ⟩ = +1 +√ +α′ +� +dDk +(2π)D +� +T(k) + Aµ(k)αµ +−1 + i +� +α′ +2 B(k)b−1c0 + · · · +� +|k, ↓⟩ +(10.86) +before gauge fixing. The spacetime fields are T(k), Aµ(k) and B(k): their roles will be +interpreted below. The first two terms are part of the |Φ↓⟩ component, while the last term +is part of the |Φ↑⟩ component. +All terms are correctly Grassmann even and they have +vanishing spacetime ghost numbers. The normalizations are chosen in order to retrieve the +canonical normalization in QFT. The factor of i in front of B is needed for the field B to +be real (as can be seen below, this leads to the expected factor ikµ which maps to ∂µ in +position space). +The equation (10.12) leads to the following equations of motion of the spacetime fields: +(α′k2 − 1)T(k) = 0, +k2Aµ(k) + ikµB(k) = 0, +kµAµ(k) + iB(k) = 0. +(10.87) +Moreover, plugging the last equation into the second one gives +k2Aµ(k) − kµk · A(k) = 0. +(10.88) +After Fourier transformation, the equations in position space read: +(α′∆ + 1) T = 0, +B = ∂µAµ, +∆Aµ = ∂µB. +(10.89) +This shows that T(k) is a tachyon with mass m2 = −1/α′ and Aµ(k) is a massless gauge +field. The field B(k) is the Nakanishi–Lautrup auxiliary field: it is completely fixed once +Aµ is known since its equation has no derivative. Siegel gauge imposes B = 0 which shows +that it generalizes the Feynman gauge to the string field. +165 + +Computation – Equation (10.87) +Keeping only the levels 0 and 1 terms in the string field, it is sufficient to truncate the +BRST operator as +QB = c0L0 − b0M + �QB, +M ∼ 2c−1c1, +�QB ∼ c1Lm +−1 + c−1Lm +1 , +Lm +1 ∼ α0 · α1, +Lm +−1 ∼ α0 · α−1. +(10.90) +Acting on the string field gives +QB |Φ⟩ = +1 +√ +α′ +� +dDk +(2π)D +� +T(k)c0L0 |k, ↓⟩ + Aµ(k) +� +c0L0 + ηνρc−1αν +1αρ +0 +� +αµ +−1 |k, ↓⟩ ++ i +� +α′ +2 B(k) +� +− 2b0c−1c1 + ηνρc1αν +−1αρ +0 +� +b−1c0 |k, ↓⟩ +� += +1 +√ +α′ +� +dDk +(2π)D +� +T(k)(α′k2 − 1)c0 |k, ↓⟩ ++ Aµ(k) +� +α′k2c0αµ +−1 + +√ +2α′ηνρηµν kρ c−1 +� +|k, ↓⟩ ++ i +� +α′ +2 B(k) +� +2c−1 + +√ +2α′ηνρkραν +−1c0 +� +|k, ↓⟩ +� += +1 +√ +α′ +� +dDk +(2π)D +� +T(k)(α′k2 − 1)c0 |k, ↓⟩ ++ α′� +Aµ(k)k2 + ikµB(k +� +c0αµ +−1 |k, ↓⟩ ++ +√ +2α′ +� +kµAµ(k) + iB(k) +� +c−1 |k, ↓⟩ +� +. +One needs to be careful when anticommuting the ghosts and we used that pL = k and +α0 = +√ +2α′k for the open string. It remains to require that the coefficient of each state +vanishes. +In order to confirm that Aµ is indeed a gauge field, we must study the gauge transform- +ation. The gauge parameter is expanded at the first level: +|Λ⟩ = +i +√ +2α′ +� +dDk +(2π)D +� +λ(k) b−1 |k, ↓⟩ + · · · +� +. +(10.91) +Note that b−1 | ↓⟩ is the SL(2, C) ghost vacuum. Since +QB |Λ⟩ = +i +√ +α′ +� +dDk +(2π)D λ(k) +� +− +� +α′ +2 k2 b−1c0 + kµαµ +−1 +� +|k, ↓⟩ , +(10.92) +matching the coefficients in (10.29) gives +δAµ = −ikµλ, +δB = k2λ. +(10.93) +This is the appropriate transformation for a U(1) gauge field. +Finally, one can derive the action; for simplicity, we work in the Siegel gauge. We consider +only the tachyon component: +|T⟩ = +� +dDk +(2π)D T(k)c1 |k, 0⟩ , +(10.94) +166 + +with c1 |0⟩ = | ↓⟩. The BPZ conjugate and Hermitian conjugates are respectively: +⟨T| = +� +dDk +(2π)D T(k)⟨−k, 0| c−1, +(10.95a) +⟨T ‡| = +� +dDk +(2π)D T(k)∗⟨k, 0| c−1. +(10.95b) +since ct +1 = c−1 when using the operator I− in (6.111). Imposing equality of both leads to +the reality condition +T(k)∗ = T(−k), +(10.96) +which agrees with the fact that the tachyon is real (the integration measure changes as +dDk → −dDk, but the contour is reversed). +Then, the action reads: +S[T] = 1 +2 +� +dDk +(2π)D T(−k) +� +k2 − 1 +α′ +� +T(k). +(10.97) +This shows that the action is canonically normalized as it should for a real scalar field. +Similarly, one can compute the action for the gauge field: +S[A] = 1 +2 +� +dDk +(2π)D Aµ(−k)k2Aµ(k). +(10.98) +The correct normalization of the tachyon (real scalar field of negative mass) gives a jus- +tification a posteriori for the normalization of the action (10.14). +Typically, string field +actions are normalized in this way, by requiring that the first physical spacetime fields has +the correct normalization. Note how this implies the correct normalization for all the others +physical fields. Generalizing this computation for higher-levels, one always find the kinetic +term to be: +L+ +0 +2 += 1 +2(k2 + m2), +(10.99) +which is the canonical normalization. +Computation – Equation (10.97) +⟨T| c0L0 |T⟩ = 1 +α′ +� +dDk +(2π)D +dDk′ +(2π)D T(k)T(k′)⟨−k′, 0| c−1c0L0c1 |k, 0⟩ += 1 +α′ +� +dDk +(2π)D +dDk′ +(2π)D T(k)T(k′)(α′k2 − 1)⟨−k′, 0| c−1c0c1 |k, 0⟩ += 1 +α′ +� +dDk +(2π)D dDk′ T(k)T(k′)(α′k2 − 1) δ(D)(k + k′), +where we used ⟨0| c−1c0c1 |0⟩ = 1 and ⟨k′|k⟩ = (2π)Dδ(D)(k + k′). +10.5 +Closed string +The derivation of the BRST free action for the closed string is very similar. The starting +point is the equation of motion +QB |Ψ⟩ = 0 +(10.100) +167 + +for the closed string field |Ψ⟩. The difference with (10.12) is that the BRST charge QB now +includes both the left- and right-moving sectors. In the case of the open string, the field Φ +was free of any constraint: we will see shortly that this is not the case for the closed string. +The next step is to find an inner product ⟨·, ·⟩ to write the action: +S = 1 +2 ⟨Ψ, QBΨ⟩. +(10.101) +Following the open string, it seems logical to give the string field Ψ the same ghost number +as the states in the cohomology: +Ngh(Ψ) = 2. +(10.102) +In this case, the ghost number of the arguments of ⟨·, ·⟩ in (10.101) is Ngh = 5. The ghost +number anomaly requires the total ghost number to be 6, that is: +Ngh(⟨·, ·⟩) = 1. +(10.103) +There is no other choice because Ngh(Ψ) must be integer. +The simplest solution is to +insert one c zero-mode c0 or ¯c0, or a linear combination. The BRST operator QB contains +both L± +0 (see the decomposition (8.88)): the natural expectation (and by analogy with the +open string) is that the gauge fixed equation of motion (to be discussed below) should be +equivalent to the on-shell equation L+ +0 = 0 (see also Section 8.3.4). This is possible only if +the insertion is c− +0 . With this insertion, ⟨·, ·⟩ can be formed from the BPZ product: +⟨A, B⟩ =⟨A| c− +0 |B⟩ . +(10.104) +Then, the action reads: +S = 1 +2 ⟨Ψ| c− +0 QB |Ψ⟩ . +(10.105) +However, the presence of c− +0 has a drastic effect because it annihilates part of the string +field. Decomposing the Hilbert space as in (7.175) +H = H− ⊕ c− +0 H−, +H− := H ∩ ker b− +0 , +(10.106) +the string field reads: +|Ψ⟩ = |Ψ−⟩ + c− +0 |�Ψ−⟩ , +Ψ−, �Ψ− ∈ H−, +(10.107) +such that +c− +0 |Ψ⟩ = c− +0 |Ψ−⟩ . +(10.108) +The problem in such cases is that the kinetic term may become non-invertible. This motiv- +ates to project out the component �Ψ− by imposing the following constraint on the string +field: +b− +0 |Ψ⟩ = 0. +(10.109) +The constraint (10.109) is stronger than the constraint L− +0 = 0 for states in the cohomo- +logy (Section 8.3.1), so there is no information lost on-shell by imposing it. For this reason, +we will also impose the level-matching condition: +L− +0 |Ψ⟩ = 0, +(10.110) +such that +Ψ ∈ H− ∩ ker L− +0 . +(10.111) +This will later be motivated by studying the propagator and the off-shell scattering amp- +litudes. To avoid introducing more notations, we will not use a new symbol for this space +and keep implicit that Ψ ∈ ker L− +0 . +168 + +The necessity of this condition can be understood differently. We had found that it is +necessary to ensure that the closed string parametrization is invariant under translations +along the string (Section 3.2.2). Since there is no BRST symmetry associated to this sym- +metry, one needs to keep the constraint.2 This suggests that one may enlarge further the +gauge symmetry and interpret (10.109) as a gauge fixing condition. This would be quite +desirable: one could argue that a fundamental field should be completely described by the +Lagrangian (if such a description exists) and that it should not be necessary to supplement +it with constraints imposed by hand. While this can be achieved at the free level, this idea +runs into problems in the presence of interactions (Section 13.3.1) and the interpretation is +not clear.3 +The action (10.105) is gauge invariant under: +|Ψ⟩ −→ |Ψ′⟩ = |Ψ⟩ + δΛ |Ψ⟩ , +δΛ |Ψ⟩ = QB |Λ⟩ , +(10.112) +where the gauge parameter has ghost number 1 and also lives in H− ∩ ker L− +0 : +Ngh(Λ) = 1, +L− +0 |Λ⟩ = 0, +b− +0 |Λ⟩ = 0. +(10.113) +As for the open string, the gauge invariance (10.112) can be gauge fixed in the Siegel +gauge: +b+ +0 |Ψ⟩ = 0. +(10.114) +Then, the action reduces to: +S = 1 +2 ⟨Ψ| c− +0 c+ +0 L+ +0 |Ψ⟩ = 1 +4 ⟨Ψ| c0¯c0L+ +0 |Ψ⟩ . +(10.115) +The equation of motion is equivalent to the on-shell condition as expected: +L+ +0 |Ψ⟩ = 0. +(10.116) +Additional constraints must be imposed to ensure that only the physical degrees of freedom +propagate. +Computation – Equation (10.115) +c− +0 QB = (c0 − ¯c0)(c0L0 + ¯c0 ¯L0) = c0¯c0(L0 + ¯L0). +10.6 +Summary +In this chapter, we have shown how the BRST conditions defining the cohomology can be +interpreted as an equation of motion for a string field together with a gauge invariance. We +found a subtlety for the closed string due to the ghost number anomaly and because of the +level-matching condition. Then, we studied several basic properties in order to prove that +the free action has the expected properties. +The next step is to add the interactions to the action, but we don’t know first principles +to write them. For this reason, we need to take a detour and to consider off-shell amplitudes. +By introducing a factorization of the amplitudes, it is possible to rewrite them as Feynman +diagrams, where fundamental interactions are connected by propagators (which we will find +to match the one in the Siegel gauge). This can be used to extract the interacting terms of +the action. +2Yet another reason can be found in Section 3.2.2 (see also Section 8.3.4): to motivate the need of the +b+ +0 condition, we could take the on-shell limit from off-shell states because L+ +0 is continuous. However, the +L− +0 operator is discrete and there is no such limit we can consider [245]. So we must always impose this +condition, both off- and on-shell. +3A recent proposal can be found in [179]. +169 + +10.7 +Suggested readings +• The free BRST string field theory is discussed in details in [245] (see also [124, chap. 7, +125, chap. 9, 235, chap. 11]). Shorter discussions can be found in [261, 192, sec. 4, +253, 1, 244]. +• Spacetime fields and actions are discussed in [242, 192, sec. 4]. +• Gauge fixing [147, 242, sec. 6.5, 7.2, 7.4, 1, 134, sec. 2.1, 2, 26]. +• General properties of string field (reality, parity, etc.) [1, 262]. +170 + +Chapter 11 +Introduction to off-shell string +theory +Abstract +In this chapter, we introduce a framework to describe off-shell amplitudes in +string theory. We first start by motivating various concepts – in particular, local coordinates +and factorization – by focusing on the 3- and 4-point amplitudes. We then prepare the stage +for a general description of off-shell amplitudes. We focus again on the closed bosonic string +only. +11.1 +Motivations +11.1.1 +3-point function +The tree-level 3-point amplitude of 3 weight hi vertex operators1 Vi is given by +A0,3 = +� 3 +� +i=1 +Vi(zi) +� +S2 +∝ (z1 − z2)h3−h1−h2 × perms × c.c. +(11.1) +There is no integration since dim M0,3 = 0. +The amplitude is independent of the zi only if the matter state is on-shell, hi = 0, for +example if Vi = c¯cVi with h(Vi) = 1. Indeed, if hi ̸= 0, then A0,3 is not invariant under +conformal transformations (6.38): +z −→ fg(z) = az + b +cz + d ∈ SL(2, C) +(11.2) +(it transforms covariantly). This is a consequence of the punctures: the presence of the latter +modifies locally the metric, since they act as sources of negative curvature. When performing +a conformal transformation, the metric around the punctures changes in a different way as +away from them. This implies that the final result depends on the metric chosen around +the punctures. This looks puzzling because the original path integral derivation (Chapter 3) +indicates that the 3-point amplitude should not depend on the locations of the operators +because its moduli space is empty (hence, all choices of zi should be equivalent). +1The quantum number (k, j) of the vertex operator is mostly irrelevant for the discussion of the current +and next chapters, and they are omitted. +We will distinguish them by a number and reintroduce the +momentum k when necessary. We also omit the overall normalization of the amplitudes. +171 + +The solution is to introduce local coordinates wi with a flat metric |dwi|2 around each +puncture conventionally located at wi = 0. The local coordinates are defined by the maps: +z = fi(wi), +zi = fi(0). +(11.3) +This is also useful to characterize in a simpler way the dependence of off-shell amplitudes +rather than using the metric around the punctures (computations may be more difficult with +a general metric). +The expression of a local operator in the local coordinate system is found by applying +the corresponding change of coordinates (6.48): +f ◦ V (w) = f ′(w)hf ′(w) +¯h V +� +f(w) +� +. +(11.4) +The amplitude reads then +A0,3 = +� 3 +� +i=1 +fi ◦ Vi(0) +� +S2 += +� 3 +� +i=1 +f ′ +i(0)hif ′ +i(0) +¯hi +� � 3 +� +i=1 +Vi +� +fi(0) +� +� +S2 +(11.5a) +∝ +� 3 +� +i=1 +f ′ +i(0)hif ′ +i(0) +¯hi +� +� +f1(0) − f2(0) +�h3−h1−h2 × perms × c.c. +(11.5b) +The amplitude depends on the local coordinate choice fi, but not on the metric around the +punctures. It is also invariant under SL(2, C): the transformation (11.2) written in terms of +the local coordinates is +fi −→ afi + b +cfi + d +(11.6) +from which we get: +f ′ +i −→ +f ′ +i +(cfi + d)2 , +fi − fj −→ +fi − fj +(cfi + d)(cfj + d). +(11.7) +All together, this implies the invariance of the 3-point amplitude since the factors in the +denominator cancel. +When the states are on-shell hi = 0, the dependence in the local +coordinate cancels, showing that the latter is non-physical. +One can ask how Feynman graphs can be constructed in string theory. By definition, an +amplitude is the sum of Feynman graphs contributing at that order in the loop expansion +and for the given number of external legs. The Feynman graphs are themselves built from a +set of Feynman rules. These correspond to the data of the fundamental interactions together +with the definition of a propagator. Since a tree-level cubic interaction is the interaction of +the lowest order, it makes sense to promote it to a fundamental cubic vertex2 V0,3: +V0,3(V1, V2, V3) := += A0,3(V1, V2, V3). +(11.8) +The index 0 reminds that it is a tree-level interaction. +2The notation will become clear later, and should not be confused with the vertex operators. +172 + +11.1.2 +4-point function +The tree-level 4-point amplitude is expressed as +A0,4 = +� +d2z4 +� 3 +� +i=1 +c¯cVi(zi) V4(z4) +� +S2 +. +(11.9) +The conformal weights are denoted by h(Vi) = hi. For on-shell states, hi = 1: while there +is no dependence on the positions z1, z2 and z3, there are divergences for +z4 −→ z1, z2, z3, +(11.10) +corresponding to collisions of punctures in the integration process. Moreover, the expression +does not look symmetric: it would me more satisfactory if all the insertions were accompanied +by ghost insertions and if all the puncture locations were treated on an equal footing. +Example 11.1 – Tachyons +Given tachyon states Vi = eiki·X, the amplitude reads: +A0,4 ∝ +3 +� +i,j=1 +i |q|1/2, +(12.32) +and conversely. The idea is that the disk D(1) +q +of Σg1,n1 is removed and replaced by the +complement of D(2) +q +in Σg2,n2, i.e. the full surface Σg2,n2 −D(2) +q +is glued inside D(1) +q . While it +is clear geometrically, this statement may look confusing from the coordinate point of view +because the local coordinates w(1) +n1 and w(2) +n2 do not cover completely the Riemann surfaces, +but their relation still encodes information about the complete surface. The reason is that +one can always use transition functions to relate the coordinates on the two surfaces. +Example 12.1 +Denote by S(1) +a +and S(2) +b +the spheres sharing a boundary with D(1) +n1 and D(2) +n2 , and write +the corresponding coordinates by z(1) +a +and z(2) +b +such that the transition functions are +z(1) +a += f (1) +an1(w(1) +n1 ), +z(2) +b += f (2) +bn2(w(2) +n2 ). +(12.33) +188 + +(a) Direct gluing of circles. +(b) Connection by a long tube. +(c) Insertion of the second surface into the first one. +(d) Insertion of the first surface +into the second one. +Figure 12.5: Different representations of the surface Σ1,2 obtained after gluing Σ1,1 and Σ0,3 +through the plumbing fixture. +Figure 12.6: Smoothed connection between both surfaces. +189 + +Then the coordinates za and zb are related by +z(1) +a += f (1) +an1 +� +w(1) +n1 +� += f (1) +an1 +� +q +w(2) +n2 +� += f (1) +an1 +� +q +f (2)−1 +bn2 +� +z(2) +b +� +� +(12.34) +such that the new transition function reads +za = Fab(zb), +Fab = f (1) +an1 ◦ (q · I) ◦ f (2)−1 +bn2 +, +(12.35) +where I is the inversion (the superscript on the coordinates za and zb has been removed +to indicate that they are now seen as coordinates on the same surface Σg,n). +The Riemann surface Σg,n is a point of Mg,n. By varying the moduli parameters of +Σg1,n1 and Σg2,n2, one obtains other surfaces in Mg,n. +But the number of parameters +furnished by Σg1,n1 and Σg2,n2 does not match the dimension (11.53) of Mg,n: +Mg1,n1 + Mg2,n2 = 6g1 − 6 + 2n1 + 6g2 − 6 + 2n2 = Mg,n − 2. +(12.36) +This means that the subspace of Mg,n obtained by gluing all the possible surfaces in Mg1,n1 +and Mg2,n2 is of codimension 2. +The missing complex parameter is q: in writing the +plumbing fixture, it was taken to be fixed, but it can be varied to generate a 2-parameter +family of Riemann surfaces in Mg,n, with the moduli of the original surfaces held fixed. +The surface Σg,n is equipped with local coordinates inherited from the original surfaces +Σg1,n1 and Σg2,n2. Hence, the plumbing fixture of points in Pg1,n1 and Pg2,n2 automatically +leads to a point of Pg,n. The fact that the local coordinates are inherited from lower-order +surfaces is called gluing compatibility. It is also not necessary to add parameters to describe +the fibre direction. +12.3.2 +Non-separating case +In the previous section, the plumbing fixture was used to glue punctures on two different +surfaces. In fact, one can also glue two punctures on the same surface to get a new surface +with an additional handle: +Σg,n = #Σg1,n1, +� +g = g1 + 1, +n = n1 − 2, +(12.37) +defining # as a unary operator. This gluing is called non-separating because there is a single +surface before the identification of the disks. +In terms of the local coordinates, the gluing relation reads +w(1) +n1−1w(1) +n1 = q, +(12.38) +where we consider the last two punctures for definiteness. +The dimensions of both moduli spaces are related by +Mg1,n1 = Mg,n − 2. +(12.39) +Again, the two missing parameters are provided by varying q and we obtain a Mg,n- +dimensional subspace of Mg,n. +Example 12.2 +Here are some examples of surfaces obtained by gluing: +190 + +• Σ0,4 = Σ0,3#Σ0,3 +• Σ0,5 = Σ0,3#Σ0,3#Σ0,3, Σ0,3#Σ0,4 +• Σ1,1 = #Σ0,3 +• Σ1,2 = #Σ0,4, Σ1,1#Σ0,3 +Note that the moduli on the LHS and RHS are fixed (we will see later that not all +surfaces can be obtained by gluing). +12.3.3 +Decomposition of moduli spaces and degeneration limit +We have seen that the separating and non-separating plumbing fixtures yield a family of +surfaces in Mg,n described in terms of lower-dimensional moduli spaces. The question is +whether all points in Mg,n can be obtained in this way by looking at all the possible gluing +(varying g1, n1, g2 and n2). It turns out that this is not possible, which is at the core of the +difficulties to construct a string field theory. +Which surfaces are obtained from this construction? In order to interpret the regions +of Mg,n covered by the plumbing fixture, the parametrization (12.31) is the most useful. +Previously, we explained that s gives the size of the tube connecting the two surfaces. Since +the latter is like a sphere with two punctures, it corresponds to a cylinder (interpreted as an +intermediate closed string propagating). The angle θ in (12.31) is the twist of the cylinder +connecting both components. This amounts to start with θ = 0, then to cut the cylinder, +to twist it by an angle θ and to glue again. +The limit s → ∞ (|q| → 0) is called the degeneration limit: the degenerate surface +Σg,n reduces to Σg1,n1 and Σg2,n2 connected by a very long tube attached to two punctures +(separating case), or to Σg−1,n+2 with a very long handle (non-separating case). So it means +that the family of surfaces described by the plumbing fixture are “close” to degeneration. +Another characterization (for the separating case) is that the punctures on Σg1,n1 are closer +(according to some distance, possibly after a conformal transformation) to each other than +to the punctures on Σg2,n2. +Conversely, there are surfaces which cannot be described in this way: the plumbing +fixture does not cover all the possible values of the moduli. For a given Mg,n, we denote +the surfaces which cannot be obtained by the plumbing fixture by Vg,n. This space does +not contain any surface arbitrarily close to degeneration (i.e. with long handles or tubes). +In terms of punctures, it also means that there is no conformal frame where the punctures +split in two sets. +In the previous subsection, we considered two specific punctures, but any other punctures +could be chosen. Hence, there are many ways to split Σg,n in two surfaces Σg1,n1 and Σg2,n2 +(with fixed g1, g2, n1 and n2): every partition of the punctures and holes in two sets lead to +different degeneration limits (because they are associated to different moduli – Figure 12.7). +Since each puncture is described by a modulus, choosing different punctures for gluing give +different set of moduli for Σg,n, such that each possibility covers a different subspace of +Mg,n. The part of the moduli space Mg,n covered by the plumbing fixture of all surfaces +Σg1,n1 and Σg2,n2 (with fixed g1, g2, n1, n2) is denoted by Mg1,n1#Mg2,n2: +Mg1,n1#Mg2,n2 ⊂ Mg,n, +(12.40) +where the operation # includes the plumbing fixture for all values of q and all pairs of +punctures. Similarly, the part covered by the non-separating plumbing fixture is written as +#Mg1,n1: +#Mg1,n1 ⊂ Mg,n. +(12.41) +Importantly, the regions covered by the plumbing fixture depend on the choice of the +local coordinates because (12.30) is written in terms of local coordinates. The subspaces +Mg1,n1#Mg2,n2 and #Mg1,n1 are not necessarily connected (in the topological sense). +191 + +(a) Degeneration 12 → 34 +(b) Degeneration 13 → 24 +(c) Degeneration 14 → 23 +Figure 12.7: Permutations of punctures while gluing two spheres: they correspond to differ- +ent (disconnected) parts of M0,4. +The moduli space Mg,n cannot be completely covered by the plumbing fixture of lower- +dimensional surfaces. We define the propagator and fundamental vertex regions Fg,n and +Vg,n as the subspaces which can and cannot be described by the plumbing fixture: +Fg,n := #Mg−1,n+2 +� +� +� +n1+n2=n+2 +g1+g2=g +Mg1,n1#Mg2,n2 +� +, +(12.42a) +Vg,n := Mg,n − Fg,n, +(12.42b) +In the RHS, it is not necessary to consider multiple non-separating plumbing fixtures for +the first term because #Mg−2,n+4 ⊂ Mg−1,n+2, etc. For the same reason, it is sufficient to +consider a single separating plumbing fixture. Note that Vg,n and Fg,n are in general not +connected subspaces. A simple illustration is given in Figure 12.9. The actual decomposition +of M0,4 is given in Figure 12.8. Importantly, Fg,n and Vg,n depend on the choice of the +local coordinates for all Vg′,n′ appearing in the RHS. +It is also useful to define the subspaces F1PR +g,n +and V1PI +g,n of Mg,n which can and cannot +be described with the separating plumbing fixture only: +F1PR +g,n := +� +n1+n2=n+2 +g1+g2=g +Mg1,n1#Mg2,n2, +(12.43a) +V1PI +g,n := Mg,n − F1PR +g,n . +(12.43b) +1PR (1PI) stands for 1-particle (ir)reducible, a terminology which will become clear later. +Note the relation: +V1PI +g,n = Vg,n +� +� � +g′ +#Mg−g′,n+g′ +� +. +(12.44) +The two plumbing fixtures behave as follow: +• separating: increases both n and g (if both surfaces have a non-vanishing g); +192 + +Figure 12.8: In white are the subspaces of the moduli space M0,4 covered by the plumb- +ing fixture. The three different regions correspond to the three different ways to pair the +punctures (see Figure 12.7). In grey is the fundamental vertex region V0,4. +Figure 12.9: Schematic illustration of the covering of Mg,n from the plumbing fixture of +lower-dimensional spaces. The fundamental region Vg,n (usually disconnected) is not covered +by the plumbing fixture. +193 + +• non-separating plumbing: increases g but decreases n. +The construction is obviously recursive: starting from the lowest-dimensional moduli space, +which is M0,3 (no moduli), one has: +V0,3 = M0,3, +F0,3 = ∅. +(12.45) +Next, the subspace of M0,4 obtained from the plumbing fixture is: +F0,4 = V0,3#V0,3, +(12.46) +and V0,4 is characterized as the remaining region. Then, one has: +F0,5 = M0,4#M0,3 += F0,4#V0,3 + V0,4#V0,3 = V0,3#V0,3#V0,3 + V0,4#V0,3, +(12.47) +and V0,5 is what remains of M0,5. The pattern continues for g = 0. The same story holds +for g ≥ 1: the first such space is +F1,1 = #V0,3, +(12.48) +and V1,1 = M1,1 − F1,1. The gluing of a 3-punctured sphere and the addition of a handle +are the two most elementary operations. +To keep track of which moduli spaces can contribute, it is useful to find a function of +Σg,n, called the index, which increases by 1 for each of the two elementary operations: +r(Σg1,n1#Σ0,3) = r(Σg1,n1) + 1, +r(#Σg1,n1) = r(Σg1,n1) + 1. +(12.49) +An appropriate function is +r(Σg,n) = 3g + n − 2 ∈ N∗. +(12.50) +which is normalized such that: +r(Σ0,3) = 1. +(12.51) +For a generic separating plumbing fixture, we find: +r(Σg1,n1#Σg2,n2) = r(Σg1,n1) + r(Σg2,n2). +(12.52) +Since the index increases, surfaces with a given r can be obtained by considering all the +gluings of surfaces with r′ < r. +12.3.4 +Stubs +To conclude this chapter, we introduce the concept of stubs. Previously in (12.31), the range +of the parameter s was the complete line of positive numbers, s ∈ R+. This means that +tubes of all lengths were considered to glue surfaces. But, we could also introduce a minimal +length s0 > 0, called the stub parameter, for the tube. In this case, the plumbing fixture +parameter is generalized to: +q = e−s+iθ, +s ∈ [s0, ∞), +θ ∈ [0, 2π), +s0 ≥ 0. +(12.53) +What is the effect on the subspaces Fg,n(s0) and Vg,n(s0)? Obviously, less surfaces can be +described by the plumbing fixture if s0 > 0 than if s0 = 0, since the plumbing fixture cannot +describe anymore surfaces which contain a tube of length less than s0. Equivalently, the +values of the moduli described by the plumbing fixture is more restricted when s0 > 0. More +generally, one has: +s0 < s′ +0 : +Fg,n(s′ +0) ⊂ Fg,n(s0) +Vg,n(s0) ⊂ Vg,n(s′ +0). +(12.54) +194 + +Figure 12.10: In light grey is the subspace covered by the V0,4(s0) as in Figure 12.8. In dark +grey is the difference δV0,4 = V0,4(s0 + δs0) − V0,4(s0) with δs0 > 0. +This is illustrated on Figure 12.10. Even if s0 is very large, Vg,n still does not include surfaces +arbitrarily close to degeneracy. In general, we omit the dependence in s0 except when it is +necessary. +To interpret the stub parameter, consider two local coordinates w1 and w2 and rescale +them by λ ∈ C with Re λ > 0: +w1 = λ ˜w1, +w2 = λ ˜w2. +(12.55) +Then, the plumbing fixture (12.30) becomes +˜w1 ˜w2 = e−˜s+i˜θ. +(12.56) +with +˜s = s + 2 ln |λ|, +˜θ = θ + i ln λ +¯λ. +(12.57) +If s ∈ R+, the corresponding range of ˜s is +˜s ∈ [s0, ∞), +s0 := 2 ln |λ|. +(12.58) +This shows that rescaling the local coordinates by a constant parameter is equivalent to +change the stub parameter. +Note also how performing a global phase rotation in (12.57) is equivalent to shift the +twist parameter. Working in ˆPg,n forces to take λ ∈ R+. +12.4 +Summary +In this chapter, we have explained how to parametrize the fibre bundle Pg,n, that is, appro- +priate coordinates for the moduli space and the local coordinate systems. This was realized +195 + +by introducing different coordinate patches and encoding all the informations of Pg,n in +the transition functions. Then, this description lead to a simple description of the tangent +vectors through the Schiffer variation. +In the next chapter, we will continue the program by building the p-forms required to +describe off-shell amplitudes. +12.5 +Suggested readings +• Plumbing fixture [193, sec. 9.3]. +196 + +Chapter 13 +Off-shell amplitudes +Abstract +While the previous chapter was purely geometrical, this one makes contact +with string theory through the worldsheet CFT. We continue the description of Pg,n by +constructing p-forms. +The reason why we need to consider the CFT is that ghosts are +necessary to build the p-forms: this can be understood from Chapter 2, where we found +that the ghosts must be interpreted as part of the measure on the moduli space. Then, we +build the off-shell amplitudes and discuss some properties. +13.1 +Cotangent spaces and amplitudes +In this section, we construct the p-forms on Pg,n which are needed for the amplitudes. We +first motivate the expressions from general ideas, and check later that they have the correct +properties. +13.1.1 +Construction of forms +A p-form ω(g,n) +p +∈ �p T ∗Pg,n is a multilinear antisymmetric map from �p TPg,n to a function +of the moduli parameters. The superscript on the form is omitted when there is no ambiguity +about the space considered. The components ωi1···ip of the p-form are defined by inserting +p basis vectors ∂s1, . . . , ∂sp +ωi1···ip := ωp(∂s1, . . . , ∂sp), +(13.1) +where ∂s = +∂ +∂xs and xs are the coordinates (12.13). It is antisymmetric in any pair of two +indices +ωi1i2···ip = −ωi2i1···ip, +(13.2) +and multilinearity implies that +ωp +� +V (1), . . . , V (p)� += ωp +� +V (1) +s1 ∂s1, . . . , V (p) +sp ∂sp +� += ωi1···ipV (1) +s1 · · · V (p) +sp , +(13.3) +given vectors V (α) = V (α) +s +∂s. +The p-forms which are needed to define off-shell amplitudes depend on the external states +Vi (i = 1, . . . , n) inserted at the punctures zi. They are maps from �p TPg,n × Hn to a +function on Pg,n. The dependence on the states is denoted equivalently as +ωp(V1, . . . , Vn) := ωp(⊗iVi). +(13.4) +The simplest way to get a function on Pg,n from the states Vi is to compute a CFT correlation +function of the operators inserted at the points zi = fi(0) on the surface Σg,n described by +the point in Mg,n. +197 + +The 0-form is just a function and is defined by: +ω0 = (2πi)−Mc +g,n +� n +� +i=1 +fi ◦ Vi(0) +� +Σg,n +. +(13.5) +For simplicity, the dependence in the local coordinates fi is kept implicit in the rest of the +chapter. +A natural approach for constructing p-forms is to build them from elementary 1-forms +and to use ghosts to enforce the antisymmetry. +Remembering the Beltrami differentials +found in Chapter 2, the contour integral of ghosts b(z) weighted by some vector field is a +good starting point. In the current language, it is defined by its contraction with a vector +V = (v, C) ∈ TPg,n defined in (12.23): +B(V ) := +� +C +dz +2πi b(z)v(z) + +� +C +d¯z +2πi +¯b(¯z)¯v(¯z), +(13.6) +where b(z) and ¯b(¯z) are the b-ghost components, and v is the vector field on Σg,n defining +V . The contours run anti-clockwise. If the contour C includes several circles (C = ∪αCα), +B(V ) is defined as the sum of the contour integral on each circle: +B(V ) := +� +α +� +Cα +dz +2πi b(z)v(z) + c.c. +(13.7) +It is also useful to define another object built from the energy–momentum tensor: +T(V ) := +� +C +dz +2πi T(z)v(z) + +� +C +d¯z +2πi +¯T(¯z)¯v(¯z), +(13.8) +where T and ¯T are the components of the energy–momentum tensor. It is defined such that +T(V ) = {QB, B(V )}. +(13.9) +Considering the coordinate system (12.13), the Beltrami form can be decomposed as: +B = Bsdxs, +Bs := B(∂s), +(13.10a) +Bs = +� +α +� +Cα +dσα +2πi b(σα) ∂Fα +∂xs +� +F −1 +α (σα) +� ++ +� +α +� +Cα +d¯σα +2πi +¯b(¯σα) ∂ ¯Fα +∂xs +� ¯F −1 +α (¯σα) +� +, +(13.10b) +where the contour orientations are defined by having the σα coordinate system on the left. +We define the p-form contracted with a set of vectors V (1), . . . , V (p) by +ωp +� +V (1), . . . , V (p)� +(V1, . . . , Vn) := (2πi)−Mc +g,n +� +B(V (1)) · · · B(V (p)) +n +� +i=1 +Vi +� +Σg,n +, +(13.11) +and the corresponding p-form reads +ωp = ωp,s1···sp dxs1 ∧ · · · ∧ dxsp +(13.12a) += (2πi)−Mc +g,n +� +Bs1dxs1 ∧ · · · ∧ Bspdxsp +n +� +i=1 +Vi +� +Σg,n +. +(13.12b) +In this expression, the form contains an infinite numbers of components ωp,s1···sp since there +is an infinite number of coordinates. Note that the normalization is independent of p. +198 + +In practice, one is not interested in Pg,n, but rather in a subspace of it. Given a q- +dimensional subspace S of Pg,n parametrized by q real coordinates t1, . . . , tq +xs = xs(t1, . . . , tq), +(13.13) +the restriction of a p-form to this subspace is obtained by the chain rule: +∀p ≤ q : +ωp|S = (2πi)−Mc +g,n +� +Br1 +∂xs1 +∂tr1 +dtr1 ∧ · · · ∧ Brp +∂xsp +∂trp +dtrp +n +� +i=1 +Vi +� +Σg,n +, +∀p > q : +ωp|S = 0. +(13.14) +We will often write the expression directly in terms of the coordinates of S and abbreviate +the notation as: +Br := ∂xs +∂tr +Bs. +(13.15) +13.1.2 +Amplitudes and surface states +It is now possible to write the amplitude more explicitly. An on-shell amplitude is defined +as an integral over Mg,n. Off-shell, one needs to consider local coordinates around each +puncture, that is, a point of the fibre for each point of the base Mg,n. This defines a Mg,n- +dimensional section Sg,n of Pg,n (Figure 11.2). The g-loop n-point off-shell amplitude of the +states V1, . . . , Vn reads: +Ag,n(V1, . . . , Vn)Sg,n := +� +Sg,n +ωg,n +Mg,n(V1, . . . , Vn) +�� +Sg,n, +(13.16a) +ωg,n +Mg,n(V1, . . . , Vn) +�� +Sg,n = (2πi)−Mc +g,n +�Mg,n +� +λ=1 +Bs +∂xs +∂tλ +dtλ +n +� +i=1 +fi ◦ Vi(0) +� +Σg,n +, +(13.16b) +where the choice of the fi is dictated by the section Sg,n. From now on, we stop to write the +restriction of the form to the section. We also restrict to the cases where χg,n = 2−2g−n < 0. +The complete (perturbative) n-point amplitude is the sum of contributions from all loops: +An(V1, . . . , Vn) := +� +g≥0 +Ag,n(V1, . . . , Vn). +(13.17) +More generally, we define the integral over a section Rg,n which projection on the base +is a subspace of Mg,n (and not the full space as for the amplitude) as: +Rg,n(V1, . . . , Vn) := +� +Rg,n +ωg,n +Mg,n(V1, . . . , Vn), +(13.18) +For simplicity, we will sometimes use the same notation for the section of Pg,n and its +projection on the base Mg,n. For this reason, the reader should assume that some choice of +local coordinates around the punctures is made except otherwise stated. +Given sections Rg,n, the sum over all genus contribution is written formally as +Rn := +� +g≥0 +Rg,n, +(13.19) +such that +Rn(V1, . . . , Vn) := +� +g≥0 +Rg,n(V1, . . . , Vn) = +� +g≥0 +� +Rg,n +ωg,n +Mg,n(V1, . . . , Vn). +(13.20) +199 + +A surface state is defined as a n-fold bra which reproduces the expression of a given +function when contracted with n states Ai. The surface ⟨Σg,n|, form ⟨ωg,n|, section ⟨Rg,n| +and amplitude ⟨Ag,n| n-fold states are defined by the following expressions: +⟨Σg,n| Bs1 · · · Bsp | ⊗i Vi⟩ := ωs1···sp(V1, . . . , Vn), +(13.21a) +⟨ωg,n +p +| ⊗i Vi⟩ := ωp(V1, . . . , Vn), +(13.21b) +⟨Ag,n| ⊗i Vi⟩ := Ag,n(V1, . . . , Vn). +(13.21c) +The last relation is generalized to any section Rg,n: +⟨Rg,n| ⊗i Vi⟩ := Rg,n(V1, . . . , Vn). +(13.21d) +The reason for introducing these objects is that the form (13.12) is a linear map from H⊗n +to a form on Mg,n – see (13.4). Thus, there is always a state ⟨Σg,n| such that its BPZ +product with the states reproduces the form. In particular, the state ⟨Σg,n| contains all the +information about the local coordinates and the moduli (the dependence is kept implicit). +The definition of the other states follow similarly. These states are defined as bras, but they +can be mapped to kets. +One finds the obvious relations: +⟨ωg,n +p +| =⟨Σg,n| Bs1dxs1 · · · Bspdxsp, +⟨Ag,n| = +� +Mg,n +⟨ωg,n +p +| . +(13.22) +The surface states don’t contain information about the matter CFT: they collect the +universal data (like local coordinates) needed to describe amplitudes. Hence, it is an im- +portant step in the description of off-shell string theory to characterize this data. However, +note that the relation between a surface state and the corresponding form does depend on +the CFT. +Example 13.1 – On-shell amplitude A0,4 +The transition functions are given by (see Figure 12.1): +C1 : w1 = z1 − y1, +C3 : w3 = z2 − y3, +C5 : z1 = z2, +C2 : w2 = z1 − y2, +C4 : w4 = z2 − y4. +(13.23) +Three of the parameters (y1, y2 and y3) are fixed while the single complex modulus +of M0,4 is taken to be y4. Since we are interested in the on-shell amplitude, it is not +necessary to introduce local coordinates and the associated parameters. +A variation of the modulus +y4 −→ y4 + δy4, +¯y4 −→ ¯y4 + δ¯y4 +(13.24) +is equivalent to a change in the transition function of C4. This translates in turn into +a transformation of z2: +z′ +2 = z2 + δy4, +¯z′ +2 = ¯z2 + δ¯y4. +(13.25) +Then, the tangent vector V = ∂y4 is associated to the vector field +v = 1, +¯v = 0, +(13.26) +with support on C4. For V = ∂¯y4, one finds +v = 0, +¯v = 1. +(13.27) +200 + +The Beltrami 1-form for the unit vectors are +B(∂y4) = +� +C4 +dz2 b(z2)(+1), +B(∂¯y4) = +� +C4 +d¯z2 ¯b(¯z2)(+1), +(13.28) +with both contours running anti-clockwise. +The components of the 2-form reads +ω2(∂y4, ∂¯y4) = +1 +2πi +� +B(∂y4)B(∂¯y4) +4 +� +i=1 +Vi +� +Σ0,4 += +1 +2πi +�� +C4 +dz2 b(z2) +� +C4 +d¯z2 ¯b(¯z2) +4 +� +i=1 +Vi +� +Σ0,4 +. +For on-shell states Vi = c¯cVi(yi, ¯yi), this becomes +ω2(∂y4, ∂¯y4) = +1 +2πi +� 3 +� +i=1 +c¯cVi(yi, ¯yi) +� +C4 +dz2 b(z2) +� +C4 +d¯z2 ¯b(¯z2)¯c(¯y4)c(y4)V4(y4, ¯y4) +� +Σ0,4 +. +The first three operators could be moved to the left because they are not encircled by the +integration contour. Note the difference with the example discussed in Section 11.1.2: +here, the contour encircles z3, while it was encircling y3 for the s-channel. +Using the OPE +� +C4 +dz2 b(z2)c(y4) ∼ +� +C4 +dz2 +1 +z2 − y4 +(13.29) +to simplify the product of b and c gives the amplitude +A0,4 = +1 +2πi +� +dy4 ∧ d¯y4 +� 3 +� +i=1 +c¯cVi(yi) V4(y4) +� +Σ0,4 +. +(13.30) +This is the standard formula for the 4-point function derived from the Polyakov path +integral. +13.2 +Properties of forms +In this section, we check that the form (13.12) has the correct properties: +• antisymmetry under exchange of two vectors; +• given a trivial vector of (a subspace of) Pg,n (Section 12.1), its contraction with the +form vanishes: ωp(V (1), . . . , V (p)) = 0 if any of the V (i) generates: +– reparametrizations of za for V (i) ∈ TPg,n, +– rotation wi → (1 + iαi)wi for V (i) ∈ T ˆPg,n, +– reparametrizations of wi keeping wi = 0 if the states are on-shell for V (i) ∈ +TMg,n; +• BRST identity, which is necessary to prove several properties of the amplitudes. +The first property is obvious. Indeed, the form is correctly antisymmetric under the +exchange of two vectors V (i) and V (j) due to the ghost insertions. +201 + +13.2.1 +Vanishing of forms with trivial vectors +Reparametrization of za +Consider the sphere Sa with coordinate za, and denote by C1, +C2 and C3 the three boundaries. Then, a reparametrization +za −→ za + φ(za) +(13.31) +is generated by a vector field φ(z) which is regular on Sa. This transformation modifies +the transition functions on the three circles and is thus associated to a tangent vector V +described by a vector field v with support on the three circles: +Ci : +v(i) = φ|Ci. +(13.32) +The Beltrami form then reads +B(V ) = +3 +� +i=1 +� +Ci +dza b(za)φ(za) + c.c. +(13.33) +where the orientations of the contours are such that Sa is on the left. Since the vector +field φ is regular in Sa, two of the contours can be deformed until they merge together. +The resulting orientation is opposite to the one of the last contour (Figure 13.1). As a +consequence, both cancel and the integral vanishes. +Figure 13.1: Deformation of the contour of integration defining the Beltrami form for a +reparametrization of za. +The figure is drawn for two circles at a hole, but the proof is +identical for other types of circles. +Rotation of wi +Consider an infinitesimal phase rotation of the local coordinate wi in the +disk Di: +wi −→ (1 + iαi)wi, +¯wi −→ (1 − iαi) ¯wi, +(13.34) +with αi ∈ R. The tangent vector is defined by the circle Ci and the vector field by +v = iwi, +¯v = −i ¯wi. +(13.35) +The Beltrami form for this vector is +B(V ) = i +� +Ci +dwi wi b(wi) − i +� +Ci +d ¯wi ¯wi ¯b( ¯wi), +(13.36) +where Di is kept to the left. +In the p-form (13.11), the ith operator Vi is inserted in Di and encircled by Ci. Because +there is no other operator inside Di, the contribution of this disk to the form is +B(V )Vi(0) = i +� +Ci +dwi wi b(wi)Vi(0) − i +� +Ci +d ¯wi ¯wi ¯b( ¯wi)Vi(0), +(13.37) +202 + +The state–operator correspondence allows to rewrite this result as +i(b0 − ¯b0) |Vi⟩ , +(13.38) +since the contour integral picks the zero-modes of b and of ¯b. +Requiring that the form +vanishes implies the ghost counter-part of the level-matching condition: +b− +0 |Vi⟩ = 0. +(13.39) +Hence, consistency of off-shell amplitudes imply that +Vi ∈ H−, +(13.40) +where H− is defined in (11.38). +Reparametrization of wi +A reparametrization of the local coordinate wi keeping the +origin of Di fixed reads: +wi −→ f(wi), +f(0) = 0. +(13.41) +The function can be expanded in series: +f(wi) = +� +m≥0 +pmwm+1 +i +. +(13.42) +Because the transformation is holomorphic, it can be extended on Ci. Each parameter pm +provides a coordinate of Pg,n and whose deformation corresponds to a vector field: +vm = wm+1 +i +, +¯vm = 0. +(13.43) +The corresponding Beltrami differential is +B(∂pm) = +� +Ci +dwi b(wi)wm+1 +i +. +(13.44) +Since only the operator Vi is inserted in the disk, the state–operator correspondence gives +bm |Vi⟩. Requiring that the form vanishes on Mg,n for all m and also for the anti-holomorphic +vectors gives the conditions: +∀m ≥ 0 : +bm |Vi⟩ = 0, +¯bm |Vi⟩ = 0. +(13.45) +This holds automatically for on-shell states Vi = c¯cVi. +13.2.2 +BRST identity +The BRST identity for the p-form (13.12) reads +ωp +� � +i +Q(i) +B ⊗i Vi +� += (−1)pdωp−1(⊗Vi), +(13.46) +using the notation (13.4). The BRST operator acting on the ith Hilbert space is written as +Q(i) +B = 1i−1 ⊗ QB ⊗ 1n−i +(13.47) +and acts as +QBVi(z, ¯z) = +1 +2πi +� +dw jB(w)Vi(z, ¯z) + c.c. +(13.48) +203 + +More explicitly, the LHS corresponds to +ωp +� � +i +Q(i) +B ⊗i Vi +� += ωp(QBV1, V2, . . . , Vn) + (−1)|V1|ωp(V1, QBV2, . . . , Vn) ++ · · · + (−1)|V1|+···+|Vn−1|ωp(V1, V2, . . . , QBVn). +(13.49) +We give just an hint of this identity, the complete proof can be found in [262, pp. 85–89, +215, sec. 2.5]. +The contour of the BRST current around each puncture can be deformed, picking singu- +larities due to the presence of the Beltrami forms. Using (13.9), we find that anti-commuting +the BRST charge with the Beltrami form Bs leads to an insertion of +Ts = {QB, Bs}. +(13.50) +The energy–momentum tensor generates changes of coordinates. Hence, Ts = T∂s is precisely +the generator associated to an infinitesimal change of the coordinate xs on Pg,n. The latter +is given by the vector ∂s. For this reason, one can write: +dxs {QB, Bs} = dxs Ts = dxs ∂s = d, +(13.51) +where d is the exterior derivative on Pg,n. The minus signs arise if the states Vi are Grass- +mann odd. +13.3 +Properties of amplitudes +In order for the p-form (13.12) to be non-vanishing, its total ghost number should match +the ghost number anomaly: +Ngh +� +ωp(V1, . . . , Vn) +� += +n +� +i=1 +Ngh(Vi) − p = 6 − 6g, +(13.52) +using Ngh(B) = −1. For an amplitude, one has p = Mg,n = 6g − 6 + 2n and thus: +Ngh(ωMg,n) = 6 − 6g +=⇒ +n +� +i=1 +Ngh(Vi) = 2n. +(13.53) +This condition holds automatically for on-shell states since Ngh(c¯cVi) = 2. +13.3.1 +Restriction to ˆPg,n +The goal of this section is to explain why amplitudes must be described in terms of a section +of ˆPg,n (12.10) instead of Pg,n. +This means that one should identify local coordinates +differing by a global phase rotation. +The off-shell amplitudes (13.16) are multi-valued on Pg,n. Indeed, the amplitude depends +on the local coordinates1 and changes by a factor under a global phase rotation of any local +coordinate wi → eiαwi. However, such a global rotation leaves the surface unchanged, since +the flat metric |dwi|2 is invariant. This means that the same surface leads to different values +for the amplitude. To prevent this multi-valuedness of the amplitudes, it is necessary to +identify local coordinates differing by a constant phase. +A second way to obtain this condition is to require that the section Sg,n is globally +defined: every point of the section should correspond to a single point of the moduli space +1The current argument does not apply for on-shell amplitudes. +204 + +Mg,n. However, there is a topological obstruction which prevents finding a global section in +Pg,n in general. One hint [56, sec. 2, 71, sec. 3] is to exhibit a nowhere vanishing 1-form if +Sg,n is globally defined: this leads to a contradiction since such a 1-form does not generally +exist (see for example [59, sec. 6.3.2, ch. 7]). Then, consider a closed curve in the moduli +space (such curves exist since Mg,n is compact). Starting at a given point Σ of the curve, +one finds that the local coordinates typically change by a global phase when coming back +to the point Σ (Figure 13.2), since this describes the same surface and there is no reason +to expect the phase to be invariant. Up to this identification, it is possible to find a global +section. The latter corresponds to a section of ˆPg,n. +(a) Closed curve in Mg,n. +(b) Change in the phase of wi. +Figure 13.2: Schematic plot of the change in the phase of the local coordinate wi as one +follows a closed curve in Mg,n. If the original phase at Σ is α0 and if the phase varies +continuously along the path, then α1 ̸= α0 when returning back to Σ by continuity. +Remark 13.1 (Degeneracy of the antibracket) It is possible to define a BV structure +on Riemann surfaces [233, 234]. The antibracket is degenerate in Pg,n but not in ˆPg,n [234]. +Global phase rotations of the local coordinates are generated by L− +0 . Hence, identifying +the local coordinates wi → eiαiwi amounts to require that the amplitude is invariant under +L− +0 . This is equivalent to imposing the level-matching condition +L− +0 |Vi⟩ = 0 +(13.54) +on the off-shell states. This condition was interpreted in Section 3.2.2 as a gauge-fixing +condition for translations along the S1 of the string. This shows, in agreement with earlier +comments, that the level-matching condition should also be imposed off-shell because no +gauge symmetry is introduced for the corresponding transformation. +If the generator L− +0 is trivial, this means that the ghost associated to the corresponding +tangent vector must be decoupled. According to Section 13.2.1, this corresponds to the +constraint: +b− +0 |Vi⟩ = 0. +(13.55) +This can be interpreted as a gauge fixing condition (Section 10.5), which could in principle +be relaxed. However, the decoupling of physical states (equivalent to gauge invariance in +SFT) happens only after integrating over the moduli space. This requires having a globally +defined section. +As a consequence, off-shell states are elements of the semi-relative Hilbert space +Vi ∈ H− ∩ ker L− +0 , +(13.56) +and the amplitudes are defined by integrating the form ωMg,n over a section Sg,n ⊂ ˆPg,n. +205 + +Computation – Equation (13.54) +The operator associated to the state through |Ai⟩ = Ai(0) |0⟩ transforms as +Vi(0) −→ (eiαi)h(e−iαi) +¯hVi(0) +(13.57) +which translates into +|Vi⟩ −→ eiαi(L0−¯L0) |Vi⟩ +(13.58) +for the state, using the fact that the vacuum is invariant under L0 and ¯L0. +Then, +requiring the invariance of the state leads to (13.54). +13.3.2 +Consequences of the BRST identity +Two important properties of the on-shell amplitudes can be deduced from the BRST identity +(13.46): the independence of physical results on the choice of local coordinates and the +decoupling of pure gauge states. +Given BRST closed states, the LHS of (13.46) vanishes identically +∀i : +QB |Vi⟩ = 0 +=⇒ +dωp−1(V1, . . . , Vn) = 0. +(13.59) +Using this result, one can compare the on-shell amplitudes computed for two different sec- +tions S and S′: +� +S +ωMg,n − +� +S′ ωMg,n = +� +∂T +ωMg,n−1 = +� +T +dωMg,n−1 = 0, +(13.60) +using Stokes’ theorem and where T is the surface delimited by the two sections (Figure 13.3). +This implies that on-shell amplitudes do not depend on the section, and thus on the local +coordinates. In obtaining the result, one needs to assume that the vertical segments do +not contribute. The latter correspond to boundary contributions of the moduli space. In +general, many statements hold up to this condition, which we will not comment more in this +book. +Figure 13.3: Two sections S and S′ of Pg,n delimiting a surface T . +Next, we consider a pure gauge state together with BRST closed states: +|V1⟩ = QB |Λ⟩ , +QB |Vi⟩ = 0. +(13.61) +The BRST identity (13.46) reads: +ωMg,n(QBΛ, V2, . . . , Vn) = dωMg,n−1(Λ, V2, . . . , Vn), +(13.62) +206 + +which gives the amplitude +� +S +ωMg,n(QBΛ, V2, . . . , Vn) = +� +S +dωMg,n−1(Λ, V2, . . . , Vn) = +� +∂S +ωMg,n−1(Λ, V2, . . . , Vn) +(13.63) +where the last equality follows from Stokes’ theorem. +Assuming again that there is no +boundary contribution, this vanishes: +� +S +ωMg,n(QBΛ, V2, . . . , Vn) = 0. +(13.64) +This implies that pure gauge states decouple from the physical states. +13.4 +Suggested readings +• Definition of the forms [71, 73, 215, 262]. +• Global phase rotation of local coordinates [56, sec. 2, 71, sec. 3, 174, 262, p. 54]. +207 + +Chapter 14 +Amplitude factorization and +Feynman diagrams +Abstract +In the previous chapter, we built the off-shell amplitudes by integrating forms +on sections of Pg,n. Studying their factorizations lead to rewrite them in terms of Feynman +diagrams, which allows to identify the fundamental interactions vertices. We will then be +able to write the SFT action in the next chapter. +14.1 +Amplitude factorization +We have seen how to write off-shell amplitudes. The next step is to rewrite them as a sum +of Feynman diagrams through factorization of amplitudes. +Factorization consists in writing a g-loop n-point amplitude in terms of lower-order +amplitudes in both g and n connected by propagators. Since an amplitude corresponds +to a sum over all possible processes, which corresponds to integrating over the moduli space, +it is natural to associate Feynman diagrams to different subspaces of the moduli space. One +can expect that the plumbing fixture (Section 12.3) is the appropriate translation of the +factorization at the level of Riemann surfaces. We will assume that it is the case and check +that it is correct a posteriori. +To proceed, we consider the contribution to the amplitude Ag,n of the family of surfaces +obtained by the plumbing fixture of two surfaces (separating case) or a surface with itself +(non-separating case). +14.1.1 +Separating case +In this section, we consider the separating plumbing fixture where part of the moduli space +Mg,n is covered by Mg1,n1#Mg2,n2 with g = g1 + g2 and n = n1 + n2 − 2 (Section 12.3.1). +The local coordinates read w(1) +i +and w(2) +j +for i = 1, . . . , n1 and j = 1, . . . , n2. By convention, +the last coordinate of each set is used for the plumbing fixture: +w(1) +n1 w(2) +n2 = q. +(14.1) +The g-loop n-point amplitude with external states {V (1) +1 +, . . . , V (1) +n1−1, V (2) +1 +, . . . , V (2) +n2−1} is +denoted as: +Ag,n = +� +Sg,n +ωg,n +Mg,n +� +V (1) +1 +, . . . , V (1) +n1−1, V (2) +1 +, . . . , V (2) +n2−1 +� +. +(14.2) +208 + +We need to study the form ωg,n +Mg,n on Mg1,n1#Mg2,n2, which means to rewrite it in terms +of the data from Mg1,n1 and from Mg2,n2. This corresponds to the degeneration limit where +the two groups of punctures denoted by V (1) +i +and V (2) +j +(i = 1, . . . , n1 − 1, j = 1, . . . , n2 − 1) +together with g1 and g2 holes move apart from each other (Figure 14.1). +Figure 14.1: Degeneration limit of Σg,n where the punctures V (1) +i +and V (2) +j +move apart from +each other. +Since q is a coordinate of Pg,n, its variation is associated with a tangent vector and a +Beltrami 1-form. The latter has to be inserted inside ωg,n +Mg,n. A change q → q + δq translates +into a change of coordinate +w′(1) +n1 = w(1) +n1 + w(1) +n1 +q +δq, +(14.3) +where w(2) +n2 is kept fixed (obviously, this choice is conventional as explained in Section 12.2). +Thus, the vector field and the Beltrami form are +vq = w(1) +n1 +q +, +Bq = 1 +q +� +Cq +dw(1) +n1 b +� +w(1) +n1 +� +w(1) +n1 . +(14.4) +Computation – Equation (14.3) +Starting from (14.1), vary q → q + δq while keeping w(2) +n2 fixed: +w′(1) +n1 w(2) +n2 = q + δq +w′(1) +n1 = w(1) +n1 +q +(q + δq) = w(1) +n1 + +q +w(1) +n1 +δq. +The second line follows by replacing w(2) +n2 using (14.1). +The Mg,n-form for the moduli described by the plumbing fixture can be expressed as: +ωMg,n +� +V (1) +1 +, . . . , V (1) +Mg1,n1, ∂q, ∂¯q, V (2) +1 +, . . . , V (2) +Mg2,n2 +� +(14.5) += (2πi)−Mc +g,n +�Mg1,n1 +� +λ=1 +B +� +V (1) +λ +� +B(∂q)B(∂¯q) +Mg2,n2 +� +κ=1 +B +� +V (2) +κ +�n1−1 +� +i=1 +V (1) +i +n2−1 +� +j=1 +V (2) +j +� +Σg,n +. +We introduce the surface states Σn1 and Σn2 such that the BPZ inner product with the +209 + +new states V (1) +n1 +and V (2) +n2 +reproduce the Mg1,n1- and Mg2,n2-forms: +⟨Σn1|V (1) +n1 ⟩ := ωMg1,n1(V (1) +1 +, . . . , V (1) +n1 ) = (2πi)−Mc +g1,n1 +�Mg1,n1 +� +λ=1 +B +� +V (1) +λ +� n1−1 +� +i=1 +V (1) +i +� +Σg1,n1 +, +(14.6a) +⟨Σn2|V (2) +n2 ⟩ := ωMg2,n2(V (2) +1 +, . . . , V (2) +n2 ) = (2πi)−Mc +g2,n2 +�Mg1,n1 +� +λ=1 +B +� +V (2) +λ +� n2−1 +� +j=1 +V (2) +j +� +Σg2,n2 +. +(14.6b) +As described in Section 13.1.2, these states exist since the p-form are linear in each of the +external state and the BPZ inner-product is non-degenerate. Each of the surface states +corresponds to an operator +⟨Σn1| =⟨0| I ◦ Σn1(0), +⟨Σn2| =⟨0| I ◦ Σn2(0), +(14.7) +defined from (6.136). Then, the forms can be interpreted as 2-point functions on the complex +plane: +⟨Σn1|V (1) +n1 ⟩ = ⟨I ◦ Σn1(0)Vn1(0)⟩w(1) +n1 , +⟨Σn2|V (2) +n2 ⟩ = ⟨I ◦ Σn2(0)Vn2(0)⟩w(2) +n2 . +(14.8) +All the complexity of the amplitudes has been lumped into the definitions of the surface +states which contain information about the surface moduli (including the ghost insertions) +and about the n1 − 1 remaining states (including the local coordinate systems). The local +coordinates around V (1) +n1 +and V (2) +n2 +are denoted respectively as w(1) +n1 and w(2) +n2 . Correspond- +ingly, the surface operators are inserted in the local coordinates w1 and w2 which are related +to w(1) +n1 and w(2) +n2 through the inversion: +w1 = I +� +w(1) +n1 +� +, +w2 = I +� +w(2) +n2 +� +. +(14.9) +In order to rewrite (14.5) in terms of Σ1 and Σ2, it is first necessary to express all +operators in one coordinate system, for example w(1) +n1 . Hence, we need to find its relation to +w2. Using the plumbing fixture (14.1), the relation between w(1) +n1 and w2 is: +w(1) +n1 = +q +w(2) +n2 += +q +I(w2) = qw2 := f(w2). +(14.10) +Then, the form (14.5) becomes +ωMg,n = +1 +2πi ⟨I ◦ Σn1(0)BqB¯q f ◦ Σn2(0)⟩w(1) +n1 = +1 +2πi ⟨Σn1| BqB¯q qL0 ¯q +¯L0 |Σ2⟩ , +(14.11) +using that Σ2 has a well-defined scaling dimension. The factor of 2πi arises by comparing the +contribution from Σn1 and Σn2 with the factor in (14.5). The expression can be simplified +by using the relation +⟨Σn1| BqB¯q |V (1) +n1 ⟩ = 1 +q¯q ⟨Σn1| b0¯b0 |V (1) +n1 ⟩ +(14.12) +using the expression (14.4) for Bq and the state–operator correspondence: +BqV (1) +n1 (z, ¯z) = 1 +q +� +Cq +dw(1) +n1 b +� +w(1) +n1 +� +w(1) +n1 V (1) +n1 (z, ¯z) −→ 1 +q b0 |V (1) +n1 ⟩ . +(14.13) +210 + +Ultimately, the form (14.5) reads +ωMg,n = +1 +2πi +1 +q¯q ⟨Σn1| b0¯b0 qL0 ¯q +¯L0 |Σn2⟩ . +(14.14) +It is important to remember that the plumbing fixture describes only a patch of the +moduli space, and the form defined in this way is valid only locally. As a consequence, the +integration over all moduli of Mg1,n1#Mg2,n2 does not describe Mg,n, but only a part of +it (Section 12.3.3). Every degeneration limit with a different puncture distribution in two +different groups contributes to a different part of the amplitude. +We denote the contribution to the total amplitude (14.2) from the region of the moduli +space connected to this degeneration limit as: +Fg,n +� +V (1) +i +|V (2) +j +� +:= +1 +2πi +� +Mg1,n1 +� +λ=1 +dt(1) +λ +Mg2,n2 +� +κ=1 +dt(2) +κ ∧ dq +q ∧ d¯q +¯q ⟨Σn1| b0¯b0 qL0 ¯q +¯L0 |Σn2⟩ . (14.15) +To proceed, we introduce a basis {φα(k)} of eigenstates of L0 and ¯L0, where kµ is the D- +dimensional momentum and α denotes the remaining quantum number. Then, introducing +twice the resolution of the identity (11.36) gives: +Fg,n +� +V (1) +i +|V (2) +j +� += +1 +2πi +� +dDk +(2π)D +dDk′ +(2π)D (−1)|φα| +× +� dq +q ∧ d¯q +¯q ⟨φα(k)c| b0¯b0 qL0 ¯q +¯L0 |φβ(k′)c⟩ +(14.16) +× +� +Mg1,n1 +� +λ=1 +dt(1) +λ ⟨Σn1|φα(k)⟩ +� +Mg2,n2 +� +κ=1 +dt(2) +κ ⟨φβ(k′)|Σn2⟩ +(with implicit sums over α and β). In the last line, one recognizes the expressions of the +g1-loop n1-point amplitude with external states {V (1) +1 +, . . . , V (1) +n1−1, φα} and of the g2-loop +and n2-point amplitudes with external states {V (2) +1 +, . . . , V (2) +n2−1, φβ}: +Ag1,n1 +� +V (1) +1 +, . . . , V (1) +n1−1, φα(k) +� += +� +Sg1,n1 +ωMg1,n1 +� +V (1) +1 +, . . . , V (1) +n1−1, φα(k) +� += +� +Sg1,n1 +Mg1,n1 +� +λ=1 +dt(1) +λ ⟨Σn1|φα(k)⟩ , +(14.17a) +Ag2,n2 +� +V (2) +1 +, . . . , V (2) +n2−1, φβ(k′) +� += +� +Sg2,n2 +ωMg2,n2 +� +V (2) +1 +, . . . , V (2) +n2−2, φβ(k′) +� += +� +Sg2,n2 +Mg2,n2 +� +λ=1 +dt(2) +λ ⟨Σn2|φβ(k′)⟩ . +(14.17b) +The property (B.27) has been used to reverse the order of the BPZ product for the second +Riemann surface, and this cancels the factor (−1)|φα|. +Defining the second line of (14.16) as +∆αβ(k, k′) := ∆ +� +φα(k)c, φβ(k′)c� +:= +1 +2πi +� dq +q ∧ d¯q +¯q ⟨φα(k)c| b0¯b0 qL0 ¯q +¯L0 |φβ(k′)c⟩ , (14.18) +one has: +Fg,n +� +V (1) +i +|V (2) +j +� += +� +dDk +(2π)D +dDk′ +(2π)D Ag1,n1 +� +V (1) +1 +, . . . , V (1) +n1−1, φα(k) +� +∆αβ(k, k′) +× Ag2,n2 +� +V (2) +1 +, . . . , V (2) +n2−1, φβ(k′) +� +. +(14.19) +211 + +We recover the expressions from Section 11.1.2, but for a more general amplitude. +We +had found that ∆ corresponds to the propagator: its properties are studied further in +Section 14.2.2. +Hence, the object (14.16) corresponds to the product of two amplitudes +connected by a propagator (Figure 14.2). +There are several points to mention about this amplitude: +• We will find that the propagator depends only on one momentum because ⟨k|k′⟩ ∼ +δ(D)(k + k′), which removes one of the integral. Then, both amplitudes Ag1,n1 and +Ag2,n2 contain a delta function for the momenta: +Ag1,n1 ∼ δ(D)� +k(1) +1 +· · ·+k(1) +n1−1+k +� +, +Ag2,n2 ∼ δ(D)� +k(2) +1 +· · ·+k(2) +n2−1+k′� +. (14.20) +As a consequence, the second momentum integral can be performed and yields a delta +function: +Fg,n ∼ δ(D)� +k(1) +1 ++ · · · + k(1) +n1−1 + k(2) +1 ++ · · · + k(2) +n2−1 +� +. +(14.21) +Hence, the momentum flowing in the internal line is fixed and this ensures the overall +momentum conservation as expected. +• The ghost numbers of the states φα and φβ are also fixed (in terms of the external +states). Indeed, because of the ghost number anomaly, the amplitudes on Mg1,n1 and +Mg2,n2 are non-vanishing only if the ghost numbers of these states satisfy: +Ngh(φα) = 2n1 − +n1−1 +� +i=1 +Ngh +� +V (1) +i +� +, +Ngh(φβ) = 2n2 − +n2−1 +� +j=1 +Ngh +� +V (2) +j +� +. +(14.22) +The non-vanishing of Fg,n also gives another relation: +Ngh(φα) + Ngh(φβ) = 4. +(14.23) +In particular, if the external states are on-shell with Ngh = 2, we find: +Ngh(φα) = Ngh(φβ) = 2. +(14.24) +As indicated in Chapter 10, such states are appropriate at the classical level since they +do not contain spacetime ghosts. +• The sum over α and β is over an infinite number of states and could diverge. In fact, +the sum can be made convergent by tuning the stub parameter (Section 14.2.4). +Properties of Feynman graphs and amplitudes in the momentum space will be discussed +further in Chapter 18. +14.1.2 +Non-separating case +Next, we consider the non-separating plumbing fixture (Section 12.3.2). The computations +are almost identical to the separating case, thus we outline only the general steps. +Part of the moduli space Mg,n is covered by #Mg1,n1, with g = g1 + 1 and n = n1 − 2. +The local coordinates are denoted as wi for i = 1, . . . , n1 and the plumbing fixture reads: +wn1−1wn1 = q. +(14.25) +The g-loop n-point amplitude with external states {V (1) +1 +, . . . , V (1) +n1−2} is denoted as: +Ag,n = +� +Sg,n +ωg,n +Mg,n +� +V (1) +1 +, . . . , V (1) +n1−2 +� +. +(14.26) +212 + +Figure 14.2: Factorization of the amplitude into two sub-amplitudes connected by a propag- +ator (dashed line). +When the n1 − 2 punctures and g1 = g − 1 holes move lose to each other, the form can +be written as: +ωMg,n +� +V (1) +1 +, . . . , V (1) +Mg1,n1, ∂q, ∂¯q +� += (2πi)−Mc +g,n +�Mg1,n1 +� +λ=1 +B +� +V (1) +λ +� +B(∂q)B(∂¯q) +n1−2 +� +i=1 +V (1) +i +� +Σg,n +. +(14.27) +To proceed, one needs to introduce the surface state Σn1−1,n1: +⟨Σn1−1,n1|V (1) +n1−1 ⊗ V (1) +n1 ⟩ := ωMg1,n1(V (1) +1 +, . . . , V (1) +n1 ). +(14.28) +Following the same step as in the previous section leads to: +Fg,n +� +V (1) +i +| +� += +� +dDk +(2π)D +dDk′ +(2π)D Ag1,n1 +� +V (1) +1 +, . . . , V (1) +n1−2, φα(k), φβ(k′) +� +∆αβ(k, k′), +(14.29) +where the propagator is given in (14.18). This is equivalent to an amplitude for which two +external legs are glued together with a propagator, giving a loop (Figure 14.3). +Since both states φα and φβ are inserted on the same surface, their ghost numbers are +not fixed, even if the external states are physical. The non-vanishing of Fg,n only leads to +the constraint: +Ngh(φα) + Ngh(φβ) = 2n1 − +n1−2 +� +i=1 +Ngh +� +V (1) +i +� += 4. +(14.30) +As a consequence, loop diagrams force to introduce states of every ghost number. Internal +states with Ngh ̸= 2 correspond to spacetime ghosts. +Since the propagator contains a delta function δ(D)(k − k′), the integral over k′ can be +removed by setting k′ = −k. However, the integral over k remains since +Ag1,n1 +� +V (1) +1 +, . . . , V (1) +n1−2, φα(k), φβ(−k) +� +∼ δ(D)� +k(1) +1 ++ · · · + k(1) +n1−2 +� +. +(14.31) +Hence, the loop momentum k is not fixed, as expected in QFT. +Remark 14.1 Not all values of the moduli associated to the holes can be associated to loops +in Feynman diagrams. Only the values close to the degeneration limit can be interpreted in +this way, the other being just standard (quantum) vertices. +14.2 +Feynman diagrams and Feynman rules +In the standard QFT approach, Feynman graphs compute Green functions, and scattering +amplitudes are obtained by amputating the external propagators through the LSZ prescrip- +tion. For connected tree-level processes, this requires n ≥ 3 (corresponding to χ0,n < 0). +213 + +Figure 14.3: Factorization of the amplitude into two sub-amplitudes connected by a propag- +ator (dashed line). The propagator connects two punctures of the same surface, which is +equivalent to a loop. +Given a theory, there is a minimal set of Feynman diagrams – the Feynman rules – from +which every other diagram can be constructed. These rules include the definitions of the +fundamental vertices – the fundamental interactions – and of the propagator – how states +propagate between two interactions (or, how to glue vertices together). In this section, we +describe these different elements. +14.2.1 +Feynman graphs +The amplitude factorization described in Section 14.1 gives a natural separation of amp- +litudes into several contributions. Considering all the possible degeneration limits lead to +a set of diagrams with amplitudes of lower order connected by propagators (Figure 14.2, +Figure 14.3). This corresponds exactly to the idea behind Feynman graphs. Then, the goal +is to find the Feynman rules of the theory: since the propagator has already been identified +(further studied in Section 14.2.2), it is sufficient to find the interaction vertices. +Let’s make this more precise by considering an amplitude Ag,n(V1, . . . , Vn). The index +of an amplitude is defined to be the index (12.50) of the corresponding Riemann surfaces +r(Ag,n) := r(Σg,n) = 3g + n − 2. +(14.32) +Contributions to an amplitude with a given r(Ag,n) can be described in terms of amp- +litudes Ag′,n′ with r(Ag′,n′) < r(Ag,n). But, the moduli space Mg,n cannot (generically) be +completely covered with the plumbing fixture of lower-dimensional moduli spaces, i.e. with +r(Mg′,n′) < r(Mg,n) (Section 12.3.3). Then, the same must be true for the amplitudes, +such that Ag,n cannot be uniquely expressed in terms of amplitudes Ag′,n′. +The g-loop n-point fundamental vertex is defined by: +Vg,n(V1, . . . , Vn) := +:= +� +Rg,n +ωg,n +Mg,n(V1, . . . , Vn), +(14.33) +The form defined in (13.16) is integrated over a sub-section Rg,n ⊂ Sg,n of ˆPg,n. Its pro- +jection on the base is the region Vg,n ⊂ Mg,n which cannot be described by the plumbing +214 + +fixture, see (12.42b). In general, we will keep the choice of local coordinates implicit and +always write Vg,n to avoid surcharging the notations. +It corresponds to the remaining contribution of the amplitude once all graphs containing +propagators have been taken into account: += +� +0≤h≤g +0≤m 0. As argued in the introduction, divergences +for L+ +0 ≤ 0 are either non-physical or IR divergences which can be cured by renormalization. +For this reason, we take the RHS as a definition of the integral, which would be the correct +result if one starts with a field theory action instead of a first-quantized formalism. +In this case, the propagator becomes +∆ = b+ +0 +L+ +0 +b− +0 δL− +0 ,0. +(14.41) +This is the standard expression for the propagator. For completeness, the form in terms of +the holomorphic and anti-holomorphic components is: +∆ = −2b0¯b0 +1 +L0 + ¯L0 +δL0,¯L0. +(14.42) +The delta function restricts the amplitude to states satisfying the level-matching condition, +that is, annihilated by L− +0 . +Considering a basis {φα(k)} of eigenstates of both L0 and ¯L0: +L+ +0 |φα(k)⟩ = α′ +2 (k2 + m2 +α) |φα(k)⟩ , +L− +0 |φα(k)⟩ = 0 +(14.43) +leads to the following momentum-space kernel for the propagator: +∆αβ(k, k′) :=⟨φα(k)c| ∆ |φβ(k′)c⟩ := (2π)Dδ(D)(k + k′) ∆αβ(k), +(14.44a) +∆αβ(k) := Mαβ(k) +k2 + m2α +, +Mαβ(k) := 2 +α′ ⟨φc +α(k)| b+ +0 b− +0 |φc +β(−k)⟩ , +(14.44b) +with Mαβ a finite-dimensional matrix giving the overlap of states of identical masses (because +the number of states at a given level is finite). +For the propagator to be well-defined, it must be invertible (in particular, to define a +kinetic term). The propagator (14.41) is non-vanishing if the states it acts on satisfy: +b+ +0 |φc +α⟩ ̸= 0, +b− +0 |φc +α⟩ ̸= 0. +(14.45) +216 + +Necessary and sufficient conditions for this to be true are +c+ +0 |φc +α⟩ = 0, +c− +0 |φc +α⟩ = 0. +(14.46) +Indeed, decomposing the state on the ghost zero-modes +|φc +α⟩ = |φ1⟩ + b± +0 |φ2⟩ , +c± +0 |φ1⟩ = c± +0 |φ2⟩ = 0 +(14.47) +gives +c± +0 |φc +α⟩ = 0 +=⇒ +|φ2⟩ = 0, +(14.48) +and one has correctly b± +0 |φ1⟩ ̸= 0. +These conditions are given for the dual states: translating them on the normal states +reverses the roles of b0 and c0. Hence, the states must satisfy the conditions: +b+ +0 |φα⟩ = 0, +b− +0 |φα⟩ = 0. +(14.49) +The second condition is satisfied automatically because the Hilbert space is H− when work- +ing with ˆPg,n (Section 13.3.1). However, the first condition further restricts the states which +propagate in internal lines. This leads to postulate that the external states should also be +taken to satisfy this condition +b+ +0 |Vi⟩ = 0, +(14.50) +since external states are usually a subset of the internal states. This provides another mo- +tivation of the statement in Section 3.2.2 that scattering amplitudes for the states not anni- +hilated by b+ +0 must be trivial. A field interpretation of this condition is given in Chapters 10 +and 15. +Under these constraints on the states, the propagator can be inverted: +∆−1 = c+ +0 c− +0 L+ +0 δL− +0 ,0. +(14.51) +14.2.3 +Fundamental vertices +The vertices (14.33) can be constructed recursively assuming that all amplitudes are known. +The starting point is the tree-level cubic amplitude A0,3: since it does not contain any +internal propagator, it is equal to the fundamental vertex V0,3. +The fist thing to extract from the recursion relations are the background independent +data. This amounts to find local coordinates and a characterization of the subspaces Vg,n ⊂ +Mg,n, starting with P0,3 and iterating. +In the rest of this section, we show how this works schematically. +Recursive definition: tree-level vertices +The description of tree-level amplitudes A0,n is the simplest since only the separating plumb- +ing fixture is used and Feynman graphs are trees. The possible factorizations of the amp- +litude correspond basically to all the partitions of the set {Vi} into subsets. +Tree-level cubic vertex +Since M0,3 = 0, the moduli space of the 3-punctured sphere +Σ0,3 reduces to a point, and so does the section S0,3 of P0,3 (Figure 14.4a): +V0,3(V1, V2, V3) := A0,3(V1, V2, V3) = ω0,3 +0 (V1, V2, V3). +(14.52) +The corresponding graph is indicated in Figure 14.4b. +217 + +(a) A section S0,3 over P0,3 reduces to +a point. +(b) Fundamental cubic vertex. +Figure 14.4: Section of P0,3 and cubic vertex. +Tree-level quartic vertex +Part of the contributions to the 4-point amplitude A0,4 with +external states Vi (i = 1, . . . , 4) comes from gluing two cubic vertices. Because there are four +external states, there are three different partitions 2|2 which are described in Figure 14.5 +(see also Figure 12.7). The sum of these three diagrams does not reproduce A0,4: the moduli +space M0,4 is not completely covered by the three amplitudes. Equivalently, the projection +of the section over P0,4 does not cover all of M0,4. The missing contribution is defined by +the quartic vertex (Figure 14.7) +V0,4(V1, V2, V3, V4) := +� +R0,4 +ω0,4 +2 (V1, . . . , V4), +(14.53) +and the corresponding section is denoted by R0,4 (Figure 14.6). Denoting by F(s,t,u) +0,4 +the +graphs 14.5 in the s-, t- and u-channels, one has the relation +A0,4 = F(s) +0,4 + F(t) +0,4 + F(u) +0,4 + V0,4. +(14.54) +Tree-level quintic vertex +The amplitude A0,5 can be factorized in a greater number +of channels, the two types being 2|2|1 and 2|3. +The possible Feynman graphs are built +either from three cubic vertices and two propagators (Figure 14.8a and permutations), or +from one cubic and one quartic vertices together with one propagator (Figure 14.8b and +permutations). The remaining contribution is the fundamental vertex (Figure 14.8c): +V0,5(V1, . . . , V5) := +� +R0,5 +ω0,5 +4 (V1, . . . , V5). +(14.55) +The construction to higher-order follows exactly this scheme. +Recursive definition: general vertices +Next, one needs to consider Feynman diagrams with loops. The first amplitude which can +be considered is the one-loop tadpole A1,1(V1). +The factorization region corresponds to +the graph obtained by gluing two legs of the cubic vertex (Section 14.2.3). The remaining +contribution is the fundamental tadpole vertex V1,1(V1) (Section 14.2.3) – note the index +g = 1 on the vertex, indicating that it is a 1-loop effect. +Next, the 1-loop 2-point amplitude can be obtained using the cubic and quartic tree-level +vertices V0,3 and V0,4, but also the one-loop tadpole V1,1. Iterating, the number of loops +218 + +(a) s-channel. +(b) t-channel. +(c) u-channel. +Figure 14.5: Factorization of the quartic amplitude A0,4 in the s-, t- and u-channels. +Figure 14.6: A section S0,4 over P0,4, the contribution from the s-, t- and u-channels (Fig- +ure 14.5) are indicated by the corresponding indices. The fundamental vertex is defined by +the section V0,4. +219 + +Figure 14.7: Fundamental quartic vertex. +(a) Factorization 12|3|45. +(b) Factorization 12|345. +(c) Fundamental vertex. +Figure 14.8: Factorization of the amplitude G0,5 in channels and fundamental quintic vertex. +Only the cases where V1 and V2 factorize on one side is indicated, the other cases follow by +permutations of the external states. +220 + +can be increased either by gluing together two external legs of a graph, or by gluing two +different graphs with loops together. +For g ≥ 2, the recursion implies the existence of vertices with no external states Vg,0: +they should be interpreted as loop corrections to the vacuum energy density. +It is important to realize that, in this language, a handle in the Riemann surface is +not necessarily mapped to a loop in the Feynman graph: only handles described by the +region Fg,n = Mg,n − Vg,n do. The higher-order vertices – corresponding to surfaces with +small handles only and described by Vg,n – should be regarded as quantum fundamental +interactions. +In Chapter 15, it will be explained that they really correspond to (finite) +counter-terms: the measure is not invariant under the gauge symmetry of the theory and +these terms must be introduced to restore it. +(a) Internal loop. +(b) Fundamental vertex. +Figure 14.9: Factorization of the amplitude G1,1 and fundamental tadpole at 1-loop. +Other vertices +The definition given at the end of (14.2.1) suggests to introduce additional vertices. The +previous recursive definition gives only vertices with χg,n = 2 − 2g − n < 0, but, in fact, it +makes sense to consider the additional cases: g = 0 and n = 0, 1, 2, and g = 1, n = 0. +The definition of the vertices as amputated Green function without internal propagators +provides a hint for the tree-level quadratic vertex V0,2. We define the latter as the amputated +tree-level 2-point Green function: +V0,2 := ∆−1∆∆−1 = ∆−1. +(14.56) +Hence, we have +V0,2(V1, V2) :=⟨V1| c+ +0 c− +0 L+ +0 δL− +0 ,0 |V2⟩ . +(14.57) +Note that V0,2 is not the 2-point scattering amplitude. +We denote the tree-level 1-point and 0-point vertices as V0,1(V1) and V0,0. The first can +be interpreted as a classical source in the action, while the second is a classical vacuum +energy. They are set to zero in most applications and can be safely ignored. However, they +appear when formulating the theory on a background which does not solve the equation of +motion [263]. +Finally, the 1-loop vacuum energy V1,0 can also be defined as the partition function of +the worldsheet CFT integrated over the torus modulus. +This allows to define the vertices Vg,n for all g, n ∈ N. We define the sum of all loop +contributions for a fixed n as: +Vn(V1, . . . , Vn) := +� +g≥0 +(ℏg2 +s)g Vg,n(V1, . . . , Vn). +(14.58) +221 + +14.2.4 +Stubs +In Section 12.3.4, we have indicated that the plumbing fixture can be modified by adding +stubs or, equivalently, by rescaling the local coordinates. This amounts to introduce a cut-off +(12.53) on the variable s such that +q = e−s+iθ, +s ∈ [s0, ∞), +θ ∈ [0, 2π). +(14.59) +instead of (14.37). In this case, the s-integral in the propagator (14.36) is modified to +� ∞ +s0 +ds e−sL+ +0 = e−s0L+ +0 +L+ +0 +. +(14.60) +This leads to a new expression for the propagator: +∆(s0) = b+ +0 +e−s0L+ +0 +L+ +0 +b− +0 δL− +0 ,0. +(14.61) +In momentum space, this reads +∆αβ(k) := e− α′s0 +2 +(k2+m2 +α) +k2 + m2α +Mαβ(k). +(14.62) +It is more convenient to work with the canonical propagator (14.41). This can be achieved +by absorbing e− s0 +2 L+ +0 in the interaction vertex: a n-point interaction will get n such factors.1 +Since s0 changes the local coordinates, this means that it also changes the region Vg,n +(Figure 12.10). The freedom in the choice of s0 translates into a freedom to choose which part +of the amplitude is described by propagator graphs Fg,n(s0), and which part is described by +a fundamental vertex Vg,n(s0). The amplitude Ag,n is independent of s0 since it is described +in terms of the complete moduli space Mg,n. This also means that the parameter s0 must +disappear when summing over the contributions from Vg,n(s0) and Fg,n(s0). This indicates +that the value of s0 is not relevant, even off-shell: it can be taken to any convenient value. +The possibility of adding stubs solves the problem that the sum over all states could +diverge (see Section 14.1.1). Indeed, the expression (14.62) in momentum space shows that +the propagator includes an exponential suppression for very massive particle propagating as +intermediate states. Since the mass of a particle increases with the level, this shows that the +sum converges for a sufficiently large value of s0 thanks to the factor e−α′s0m2. A second +interesting aspect is the exponential momentum suppression e−α′s0k2: this is responsible for +the nice UV behaviour of string theory. Since the value of s0 is not physical, this means +that all Feynman graphs must share these properties. These two points will be made more +precise in Chapter 18. +14.2.5 +1PI vertices +We can follow the same procedure as before, but considering only the separating plumb- +ing fixture. In this case, the Feynman diagrams are all 1PR (1-particle reducible): if the +propagator line is cut, then the graphs split in two disconnected components. The region +of the moduli space covered by these graphs is written as F1PR +g,n +(12.43a). The complement +1To make this identification precise for vertices involving external states, one has to consider the non- +amputated Green functions. +222 + +defines the 1PI region V1PI +g,n (12.43b). Then, the 1PI g-loop n-point fundamental vertices are +defined as: +V1PI +g,n (V1, . . . , Vn) := +:= +� +R1PI +g,n +ωg,n +Mg,n(V1, . . . , Vn), +(14.63) +where R1PI +g,n is a section of Pg,n which projection on the base is V1PI +g,n . +14.3 +Properties of fundamental vertices +14.3.1 +String product +Following the definition of surfaces states (Section 13.1.2), the vertex state is defined as: +⟨Vg,n| ⊗i Vi⟩ := Vg,n(⊗iVi). +(14.64) +The vertex is a map Vg,n : H⊗n → C where C ≃ H⊗0. We will find very useful to +introduce the string products ℓg,n : H⊗n → H through the closed string inner product: +Vg,n+1(V0, V1, . . . , Vn) :=⟨V0| c− +0 |ℓg,n(V1, . . . , Vn)⟩ . +(14.65) +An alternative notation is: +ℓg,n(V1, . . . , Vn) := [V1, . . . , Vn]g +(14.66) +The advantage of the second notation is to show that the products with n ≥ 3 are direct +generalization of the 2-product, which is very similar to a super-Lie bracket. These products +play a central role in SFT – in fact, the description of SFT is more natural using the ℓg,n +rather than the Vg,n. +Note that the products with n = 0 are maps C → H, which means that they correspond +to a particular fixed state. +ℓg,0 := [·]g ∈ H. +(14.67) +The ghost number of the product (14.65) is +Ngh +� +ℓg,n(V1, . . . , Vn) +� += 3 − 2n + +n +� +i=1 +Ngh(Vi) = 3 + +n +� +i=1 +� +Ngh(Vi) − 2 +� +, +(14.68) +and it is independent of the genus g. As a consequence, the parity of the product is +|ℓg,n(V1, . . . , Vn)| = 1 + +n +� +i=1 +|Vi| +mod 2, +(14.69) +and the string product itself is always odd. +The vertices satisfy the following identity for g ≥ 0 and n ≥ 1 [262, pp. 41–42] +0 = +� +g1,g2≥0 +g1+g2=g +� +n1,n2≥0 +n1+n2=n +n! +n1! n2!Vg1,n1+1 +� +Ψn1, ℓg2,n2(Ψn2) +� ++ (−1)|φs|Vg−1,n+2 +� +φs, b− +0 φc +s, Ψn� +. +(14.70) +The last term is absent for g = 0. It is a consequence of the definition of the vertices as the +missing region from gluing lower-order vertices. +223 + +14.3.2 +Feynman graph interpretation +The vertices must satisfy a certain number of conditions to be interpreted as Feynman +diagrams. The first is that they must be symmetric under permutations of the states. Not +every choice of local coordinates satisfies this requirement: this can be solved by defining +the vertex over a generalized section. In this case, the vertex is defined as the average of +the integrals over N sections S(a) +g,n of Pg,n: +Vg,n(V1, . . . , Vn) = 1 +N +N +� +a=1 +� +R(a) +g,n +ωg,n +Mg,n(V1, . . . , Vn). +(14.71) +Example 14.2 – 3-point vertex +The cubic vertex must be symmetric under permutations +V0,3(V1, V2, V3) = V0,3(V3, V1, V2) + · · · +(14.72) +Taking the vertex to be given by a section S0,3 with local coordinates fi +V0,3(V1, V2, V3) = ω0,3 +0 (V1, V2, V3)|S0,3 = ⟨f1 ◦ V1(0)f2 ◦ V2(0)f3 ◦ V3(0)⟩, +(14.73) +one finds that a permutation looks different +V0,3(V3, V1, V2) = ⟨f1 ◦ V3(0)f2 ◦ V1(0)f3 ◦ V2(0)⟩ ̸= V0,3(V1, V2, V3), +(14.74) +unless the local coordinates satisfy special properties (remember that the local co- +ordinates are specified by the vertex state V and not by the external states Vi, so a +permutation of them does not permute the local maps). Obviously, both amplitudes +agree on-shell since the dependence in the local coordinates cancel (equivalently one +can rotate the punctures using SL(2, C)). +Writing zi = fi(0), there is a SL(2, C) transformation g(z) such that +g(z1) = z2, +g(z2) = z3, +g(z3) = z1 +(14.75) +such that +V0,3(V3, V1, V2) = ⟨g ◦ f1 ◦ V3(0)g ◦ f2 ◦ V1(0)g ◦ f3 ◦ V2(0)⟩. +(14.76) +While the state Vi is correctly inserted at the puncture zi in this expression, this is not +sufficient to guarantee the equality of the amplitudes. Indeed the fibre is defined by the +complete functions fi(w) and not only by their values at w = 0. For this reason the +amplitudes can be equal only if +g ◦ f1 = f2, +g ◦ f2 = f3, +g ◦ f3 = f1. +(14.77) +This provides constraints on the functions fi, but it is often not possible to solve them. +If the constraints cannot be solved, then one must introduce a general section. In +this case a generalized section will be made of 6 sections S(a) (a = 1, . . . , 6) because +there are 6 permutations. Then the amplitude reads +V0,3(V1, V2, V3) = 1 +6 +6 +� +a=1 +ω0,3 +0 (V1, V2, V3)|S(a) +0,3 . +(14.78) +224 + +Figure 14.10: A generalized section {S(a) +0,3} (a = 1, . . . , 6) of P0,3 for the 3-point vertex. This +is to be compared with Figure 14.4a. +When computing the Feynman graphs by gluing lower-dimensional amplitudes, it is +possible that parts of the section overlap, meaning that several graphs cover the same part +of the moduli space. In this case, the fundamental vertex should be defined as a negative +contribution in the overlap region. This procedure is perfectly well-defined since all graphs +are finite and there is no ambiguity. In practice, it is always simpler to work with non- +overlapping sections (i.e. a single covering of the moduli space). A simple way to prevent +overlaps is to tune the stub parameter s0 to a large value. +By construction, the integral over Vg,n should be finite. If this is not the case, it means +that the propagator graphs also diverge and that the parametrization is not good. This can +also be solved by considering a sufficiently large value of the stub parameter s0. +14.4 +Suggested readings +• Plumbing fixture and amplitude factorization [193, sec. 9.3, 9.4, 256, sec. 6]. +225 + +Chapter 15 +Closed string field theory +Abstract +We bring together the elements from the previous chapters in order to write +the closed string field action. We first study the gauge fixed theory before reintroducing the +gauge invariance. We then prove that the action satisfies the BV master equation meaning +that closed SFT is completely consistent at the quantum level. Finally, we describe the 1PI +effective action. +15.1 +Closed string field expansion +In Chapters 11, 13 and 14, constraints on the external and internal states were found to +be necessary. But, to provide another perspective and decouples the properties of the field +from the ones of the state, we assume that the string field does not obey any constraint. +They will be derived later in order to reproduce the scattering amplitudes from the action +and to make the latter well-defined. +The string field is expanded on a basis {φr} of the CFT Hilbert space H (see Section 11.2 +for more details) +|Ψ⟩ = +� +r +ψr |φr⟩ . +(15.1) +Using the decomposition (11.40) of the Hilbert space according to the ghost zero-modes, the +string field can also be expanded as +|Ψ⟩ = +� +r +� +ψ↓↓,r |φ↓↓,r⟩ + ψ↓↑,r |φ↓↑,r⟩ + ψ↑↓,r |φ↑↓,r⟩ + ψ↑↑,r |φ↑↑,r⟩ +� +, +(15.2) +where we recall that the basis states satisfy +b0 |φ↓↓,r⟩ = ¯b0 |φ↓↓,r⟩ = 0, +b0 |φ↓↑,r⟩ = ¯c0 |φ↓↑,r⟩ = 0, +c0 |φ↑↓,r⟩ = ¯b0 |φ↑↓,r⟩ = 0, +c0 |φ↑↑,r⟩ = ¯c0 |φ↑↑,r⟩ = 0. +(15.3) +We recall the definition of the dual basis {φc +r} through the BPZ inner product +⟨φc +r|φs⟩ = δrs. +(15.4) +In terms of the ghost decomposition, the components of the dual states satisfy: +⟨φc +↓↓,r| c0 =⟨φc +↓↓,r| ¯c0 = 0, +⟨φc +↓↑,r| c0 =⟨φc +↓↑,r|¯b0 = 0, +⟨φc +↑↓,r| b0 =⟨φc +↑↓,r| ¯c0 = 0, +⟨φc +↑↑,r| b0 =⟨φc +↑↑,r|¯b0 = 0, +⟨φc +x,r|φy,s⟩ = δxyδrs, +(15.5) +226 + +where x, y =↓↓, ↑↓, ↓↑, ↑↑. The spacetime ghost number of the fields ψr is defined by +G(ψr) = 2 − nr. +(15.6) +Remember that the ghost number of the basis states are denoted by +nr = Ngh(φr), +nc +r = Ngh(φc +r) = 6 − nr. +(15.7) +15.2 +Gauge fixed theory +Having built the kinetic term (Chapter 9), one needs to construct the interactions. For +the same reason – our ignorance of SFT first principles – that forced us to start with the +free equation of motion to derive the quadratic action (Chapter 10), we also need to infer +the interactions from the scattering amplitudes. Preparing the stage for this analysis was +the goal of Chapter 14, where we introduced the factorization of amplitudes to derive the +fundamental interactions. +Scattering amplitudes are expressed in terms of gauge fixed states since only them are +physical. This allows to give an alternative derivation of the kinetic term by defining it as +the inverse of the propagator, which is well-defined for gauge fixed states.1 The price to +pay by constructing interactions in this way is that the SFT action itself is gauge fixed. To +undercover its deeper structure it is necessary to release the gauge fixing condition. In view +of the analysis of the quadratic action in Chapter 10, we can expect that the BV formalism +is required. Another possibility is to consider directly the 1PI action. +In this section, we first derive the kinetic term by inverting the propagator. For this to +be possible, the string field must obey some constraints: we will find that they correspond +to the level-matching and Siegel gauge conditions. Then, we introduce the interactions into +the action. +15.2.1 +Kinetic term and propagator +In Chapter 14, it was found that the propagator reads (14.41): +∆ = b+ +0 b− +0 +1 +L+ +0 +δL− +0 ,0, +∆rs =⟨φc +r| b+ +0 b− +0 +1 +L+ +0 +δL− +0 ,0 |φc +s⟩ . +(15.8) +The most natural guess for the kinetic term is +S0,2 = 1 +2 ⟨Ψ| K |Ψ⟩ = 1 +2 ψrKrsψs +(15.9) +where +K = c− +0 c+ +0 L+ +0 δL− +0 ,0 +Krs =⟨φr| c− +0 c+ +0 L+ +0 δL− +0 ,0 |φs⟩ . +(15.10) +Indeed, it looks like K∆ = 1 using the identities c± +0 b± +0 ∼ 1 and it matches (10.115). In +terms of the holomorphic and anti-holomorphic modes, we have +K = 1 +2 c0¯c0L+ +0 δL− +0 ,0. +(15.11) +But, when writing c± +0 b± +0 ∼ 1, the second part of the anti-commutator {b± +0 , c± +0 } = 1 is +missing. The relation c± +0 b± +0 ∼ 1 is correct only when acting on basis dual states annihilated +1This step is not necessary because the propagator corresponding to the plumbing fixture (Section 14.2.2) +matches the one found in Section 10.5 by considering the simplest gauge fixing. However, this would have +been necessary if the factorization had given another propagator, or if the structure of the theory was more +complicated, for example for the superstring. +227 + +by c± +0 . +The problem stems from the fact that Ψ is not yet subject to any constraint. +Moreover, some of the string field components will not appear in the expression since they +are annihilated by the ghost zero-mode. As a consequence, the kinetic operator in (15.10) +(or equivalently the propagator) is not invertible in the Hilbert space H because its kernel +is not empty: +ker K|H ̸= ∅. +(15.12) +This can be seen by writing φr as a 4-vector and Krs as a 4 × 4-matrix: +Krs = 1 +2 +� +� +� +� +⟨φ↓↓,r| +⟨φ↓↑,r| +⟨φ↑↓,r| +⟨φ↑↑,r| +� +� +� +� +t � +� +� +� +c0¯c0L+ +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +� +� +� +� +� +� +� +� +|φ↓↓,s⟩ +|φ↓↑,s⟩ +|φ↑↓,s⟩ +|φ↑↑,s⟩ +� +� +� +� . +(15.13) +The matrix is mostly empty because the states φx,r with different x =↓↓, ↑↓, ↓↑, ↑↑ are +orthogonal (no non-diagonal terms) and the states with x ̸=↓↓ are annihilated by c0 or ¯c0. +The same consideration applies for the delta-function: if the field does not satisfy L− +0 = 0, +then the kinetic operator is non-invertible. +To summarize the string field must satisfy three conditions in order to have an invertible +kinetic term +L− +0 |Ψ⟩ = 0, +b− +0 |Ψ⟩ = 0, +b+ +0 |Ψ⟩ = 0. +(15.14) +This means that the string field is expanded on the H0 ∩ ker L− +0 Hilbert space: +|Ψ⟩ = +� +r +ψ↓↓,r |φ↓↓,r⟩ . +(15.15) +Ill-defined kinetic terms are expected in the presence of a gauge symmetry: this was already +discussed in Sections 10.1.4 and 10.5 for the free theory, and this will be discussed further +later in this chapter for the interacting case. +Computation +Let’s check that Krs is correctly the inverse of ∆rs when Ψ is restricted to H0: +Krs∆st =⟨φr| c− +0 c+ +0 L+ +0 δL− +0 ,0 |φs⟩⟨φc +s| b+ +0 b− +0 +1 +L+ +0 +δL− +0 ,0 |φc +t⟩ +=⟨φr| c− +0 c+ +0 L+ +0 δL− +0 ,0b+ +0 b− +0 +1 +L+ +0 +δL− +0 ,0 |φc +t⟩ +=⟨φr| {c− +0 , b− +0 }{c+ +0 , b+ +0 } |φc +t⟩ += ⟨φr|φc +t⟩ = δrt. +The second equality follows from the resolution of the identity (11.36): due to the zero- +mode insertions, the resolution of the identity collapses to a sum over the ↓↓ states +1 = +� +r +|φr⟩⟨φc +r| = +� +r +|φ↓↓,r⟩⟨φc +↓↓,r| . +(15.16) +The third equality uses that L+ +0 commutes with the ghost modes, that φr is annihilated +by b± +0 , and that (δL− +0 ,0)2 = δL− +0 ,0 = 1 on states with L− +0 = 0. +Finally, we find that the kinetic term matches the classical quadratic vertex V0,2 defined +in (14.56) such that +S0,2 = 1 +2 V0,2(Ψ2) = 1 +2 ⟨Ψ| c− +0 c+ +0 L+ +0 δL− +0 ,0 |Ψ⟩ . +(15.17) +228 + +15.2.2 +Interactions +The second step to build the action is to write the interaction terms from the Feynman rules. +Before proceeding to SFT, it is useful to remember how this works for a standard QFT. +Example 15.1 – Feynman rules for a scalar field +Consider a scalar field with a standard kinetic term and a n-point interaction: +S = +� +dDx +�1 +2 φ(x)(−∂2 + m2)φ(x) + λ +n! φ(x)n +� +. +(15.18) +First, one needs to find the physical states, which correspond to solutions of the lin- +earised equation of motion. In the current case, they are plane-waves (in momentum +representation): +φk(x) = eik·x. +(15.19) +Then, the vertex (in momentum representation) Vn(k1, . . . , kn) is found by replacing +in the interaction each occurrence of the field by a different state, and summing over +all the different contributions. Here, this means that one considers states φki(x) with +different momenta: +Vn(k1, . . . , kn) = λ +n! +� +dDx n! +n +� +i=1 +φki(x) = λ +� +dDx ei(k1+···+kn)x += λ(2π)D δ(D)(k1 + · · · + kn). +(15.20) +The factor n! comes from all the permutations of the n states in the monomial of order +n. Reversing the argument, one sees how to move from the vertex Vn(k1, . . . , kn) written +in terms of states to the interaction in the action in terms of the field. +Obviously, if the field has more states (for example if it has a spin or if it is in a +representation of a group), then one needs to consider all the different possibilities. The +above prescription also yields directly the insertion of the momentum necessary if the +interaction contains derivatives. +In Section 14.2, the Feynman rule for a g-loop n-point fundamental vertex of states +(V1, . . . , Vn) was found to be given by (14.33): +Vg,n(V1, . . . , Vn) = +� +Rg,n +ωg,n +Mg,n(V1, . . . , Vn) = +(15.21) +where Rg,n is a section over the fundamental region Vg,n ⊂ Mg,n (12.42b) which cannot be +covered from the plumbing fixture of lower-dimensional surfaces. +From the example Example 15.1, it should be clear that the g-loop n-point contribution +to the action can be obtained simply by replacing every state with a string field in Vg,n: +Sg,n = ℏg g2g−2+n +s +n! +Vg,n(Ψn). +(15.22) +where Ψn := Ψ⊗n. +The power of the coupling constant has been reinstated: it can be +motivated by the fact that it should have the same power as the corresponding amplitude +(Section 3.1.1). Note that the interactions are defined only when the power of gs is positive: +229 + +χg,n = 2 − 2g − n < 0. We have also written explicitly the power of ℏ, which counts the +number of loops. +Before closing this section, we need to comment on the effect of the constraints (15.14) +on the interactions. Building a Feynman graph by gluing two m- and n-point interactions +with a propagator, one finds that the states proportional to φx,r for x ̸=↓↓ do not propagate +inside internal legs +Vg,m(V1, . . . , Vm−1, φr)⟨φc +r| b+ +0 b− +0 +1 +L+ +0 +|φc +s⟩ Vg′,n(W1, . . . , Wn−1, φs) += Vg,m(V1, . . . , Vm−1, φ↓↓,r)⟨φc +↓↓,r| b+ +0 b− +0 +1 +L+ +0 +|φc +↓↓,s⟩ Vg′,n(W1, . . . , Wn−1, φ↓↓,s). +(15.23) +Thus, they do not contribute to the final result even if the interactions contain them. While +the conditions L− +0 = b− +0 = 0 were found to be necessary for defining off-shell amplitudes, the +condition b+ +0 = 0 does not arise from any consistency requirement. But, it is also consistent +with the interactions, since only fundamental vertices have a chance to give a non-vanishing +result for states which do not satisfy (15.14). Hence, the interactions (15.22) are compatible +with the definition of the kinetic term and the restriction of the string field. +15.2.3 +Action +The interacting gauge-fixed action is built from the kinetic term V0,2 (15.17) and from the +interactions Vg,n (15.22) with χg,n < 0. However, this is not sufficient: we have seen in +Section 14.3 that it makes sense to consider the vertices with χg,n ≥ 0. First, we should +consider the 1-loop cosmological constant V1,0. Then, we can also add the classical source +V0,1 and the tree-level cosmological constant V0,0. With all the terms together, the action +reads: +S = +� +g,n≥0 +ℏg g2g−2+n +s +n! +Vg,n(Ψn) +:= 1 +2 ⟨Ψ| c− +0 c+ +0 L+ +0 δL− +0 ,0 |Ψ⟩ + +�′ +g,n≥0 +ℏg g2g−2+n +s +n! +Vg,n(Ψn). +(15.24) +where Vn was defined in (14.58). A prime on the sum indicates that the term g = 0, n = 2 +is removed, such that one can single out the kinetic term. We will often drop the delta +function imposing L− +0 = 0 because the field are taken to satisfy this constraint. +Rewriting the vertices in terms of the products ℓg,n defined in (14.65) +Vg,n(Ψn) :=⟨Ψ| c− +0 +��ℓg,n−1(Ψn−1) +� +(15.25) +leads to the alternative form +S = +� +g,n≥0 +ℏg g2g−2+n +s +n! +⟨Ψ| c− +0 +��ℓg,n−1(Ψn−1) +� +. +(15.26) +The definition (14.56) leads to the following explicit expression for ℓ0,1: +ℓ0,1(Ψcl) = c+ +0 L+ +0 |Ψcl⟩ . +(15.27) +In most cases, the terms g = 0, n = 0, 1 vanish such that the action reads: +S = +� +g,n≥0 +χg,n≤0 +ℏg g2g−2+n +s +n! +Vg,n(Ψn). +(15.28) +230 + +However, we will often omit the condition χg,n ≤ 0 to simplify the notation, except when the +distinction is important, and the reader can safely assumes V0,0 = V0,1 = 0 if not otherwise +stated. The classical action is obtained by setting ℏ = 0: +Scl = 1 +2 ⟨Ψcl| c− +0 c+ +0 L+ +0 |Ψcl⟩ + +� +n≥3 +gn +s +n! V0,n(Ψn +cl). +(15.29) +Rescaling the string field by g−1 +s +gives the more canonical form of the action (using the +same symbol): +S = +� +g,n≥0 +ℏgg2g−2 +s +1 +n! Vg,n(Ψn) +:= +1 +2g2s +⟨Ψ| c− +0 c+ +0 L+ +0 δL− +0 ,0 |Ψ⟩ + 1 +g2s +�′ +g,n≥0 +(ℏg2 +s)g +n! +Vg,n(Ψn). +(15.30) +In the path integral, the action is divided by ℏ such that +S +ℏ = +� +g,n≥0 +(ℏg2 +s)g−1 1 +n! Vg,n(Ψn). +(15.31) +This shows that there is a single coupling constant ℏg2 +s, instead of two (ℏ and gs separately) +as it looks at the first sight. This makes sense because gs is in fact the expectation value +of the dilaton field (2.166) and its value can be changed by deforming the background with +dilatons [16, 17, 197]. +The previous remark also allows to easily change the normalization of the action, for +example, to perform a Wick rotation, to normalize canonically the action in terms of space- +time fields, or reintroduce ℏ. Rescaling the action by α is equivalent to rescale g2 +s by α−1: +S → α S +=⇒ +g2 +s → g2 +s +α . +(15.32) +The linearized equation of motion is: +L+ +0 |Ψ⟩ = 0, +(15.33) +which corresponds to the Siegel gauge equation of motion of the free theory (10.116). Hence, +this equation is not sufficient to determine the physical states (cohomology of the BRST op- +erator, Chapter 8), as discussed in Chapter 10, and additional constraints must be imposed. +One can interpret this by saying that the action (15.24) provides only the Feynman rules, +not the physical states. Removing the gauge fixing will be done in Sections 15.3 and 15.4. +The action (15.24) looks overly more complicated than a typical QFT theory: instead +of few interaction terms for low n (n ≤ 4 in d = 4 renormalizable theories), it has contact +interactions of all orders n ∈ N. The terms with g ≥ 1 are associated to quantum corrections +as indicate the power of ℏ, which means that they can be interpreted as counter-terms. But, +how is it that one needs counter-terms despite the claim that every Feynman graphs (includ- +ing the fundamental vertices) in SFT are finite? The role of renormalization is not only to +cure UV divergences, but also IR divergences (due to vacuum shift and mass renormaliza- +tion). Equivalently, this can be understood by the necessity to correct the asymptotic states +of the theory, or to consider renormalized instead of bare quantities. Indeed, the asymptotic +states obtained from the linearized classical equations of motion are idealization: turning +on interactions modify the states. In typical QFTs, these corrections are infinite and renor- +malization is crucial to extract a number; however, even if the effect is finite, it is needed to +describe correctly the physical quantities [250, p. 411]. There is a second reason for these +231 + +additional terms: when relaxing the gauge fixing condition, the path integral is anomalous +under the gauge symmetry, and the terms with g > 0 are necessary to cancel the anomaly +(this will be discussed more precisely in Section 15.4). It may thus seem that SFT cannot +be predictive because of the infinite number of counter-terms. Fortunately, this is not the +case: the main reason for the loss of predictability in non-renormalizable theory is that +the renormalization procedure introduces an infinite number2 of arbitrary parameters (and +thus making a prediction would require to have already made an infinite number of observa- +tions to determine all the parameters). These parameters come from the subtraction of two +infinities: there is no unique way to perform it and thus one needs to introduce a new para- +meter. The case of SFT is different: since every quantity is finite, the renormalization has +no ambiguity because one subtracts two finite numbers, and the result is unambiguous. As +a consequence, renormalization does not introduce any new parameter and there is a unique +coupling constant gs in the theory, which is determined by the tree-level cubic interaction. +The coupling constants of higher-order and higher-loop interactions are all determined by +powers of gs, and thus a unique measurement is sufficient to make predictions. +Another important point is that the action (15.24) is not uniquely defined. The definition +of the vertices depends on the choice of the local coordinates and of the stub parameter s0. +Changing them modifies the vertices, and thus the action. +But, one can show that the +different theories are related by field redefinitions and are thus equivalent. +15.3 +Classical gauge invariant theory +In the previous section, we have found the gauge fixed action (15.24). Since the complete +gauge invariant quantum action has a complicated structure, it is instructive to first focus +on the classical action (15.29). The full action is discussed in Section 15.4. +The gauge fixing is removed by relaxing the b+ +0 = 0 constraint on the field (the other +constraints must be kept in order to have well-defined the interactions). The classical field +Ψcl is then defined by: +Ψcl ∈ H− ∩ ker L− +0 , +Ngh(Ψcl) = 2. +(15.34) +The restriction on the ghost number translates the condition that the field is classical, i.e. +that there are no spacetime ghosts at the classical level. The relation (15.6) implies that all +components have vanishing spacetime ghost number. +In the free limit, the gauge invariant action should match (10.105) +S0,2 = 1 +2 ⟨Ψ| c− +0 QB |Ψ⟩ . +(15.35) +and lead to the results from Section 10.5. A natural guess is that the form of the interactions +is not affected by the gauge fixing (the latter usually modifies the propagator but not the +interactions). This leads to the gauge invariant classical action: +Scl = 1 +2 ⟨Ψcl| c− +0 QB |Ψcl⟩ + 1 +g2s +� +n≥3 +gn +s +n! V0,n(Ψn +cl), +(15.36) +where the vertices V0,n with n ≥ 3 are the ones defined in (14.33) (we consider the case +where V0,0 = V0,1 = 0). It is natural to generalize the definition of V0,2 as: +V0,2(Ψ2 +cl) :=⟨Ψcl| c− +0 QB |Ψcl⟩ +(15.37) +2In practice, this number does not need to be infinite to wreck predictability, it is sufficient that it is +very large. +232 + +such that +Scl = 1 +g2s +� +n≥2 +gn +s +n! V0,n(Ψn +cl) = 1 +g2s +� +n≥2 +gn +s +n! ⟨Ψcl| c− +0 +��ℓ0,n−1(Ψn−1 +cl +) +� +, +(15.38) +where (15.37) implies: +ℓ0,1(Ψcl) = QB |Ψcl⟩ . +(15.39) +The equation of motion is +Fcl(Ψcl) := +� +n≥1 +gn−1 +s +n! +ℓ0,n(Ψn +cl) = QB |Ψcl⟩ + +� +n≥2 +gn−1 +s +n! +ℓ0,n(Ψn +cl) = 0. +(15.40) +Computation – Equation (15.40) +δScl = 1 +g2s +� +n≥2 +gn +s +n! n{δΨcl, Ψn−1 +cl +}0 = 1 +g2s +� +n≥2 +gn +s +(n − 1)! ⟨δΨcl| c− +0 +��ℓ0,n−1(Ψn−1 +cl +� +. +(15.41) +The first equality follows because the vertex is completely symmetric. Simplifying and +shifting n, one obtains c− +0 |Fcl⟩. The factor c− +0 is invertible because of the constraint +b− +0 = 0 imposed on the field. +The action is invariant +δΛScl = 0 +(15.42) +under the gauge transformation +δΛΨcl = +� +n≥0 +gn +s +n! ℓ0,n+1(Ψn +cl, Λ) = QB |Λ⟩ + +� +n≥1 +gn +s +n! ℓ0,n+1(Ψn +cl, Λ). +(15.43) +The gauge algebra is [262, sec. 4]: +[δΛ2, δΛ1]Ψcl = δΛ(Λ1,Λ2,Ψcl) |Ψcl⟩ + +� +n≥0 +gn+2 +s +n! +ℓ0,n+3 +� +Ψn +cl, Λ2, Λ1, Fcl(Ψcl) +� +, +(15.44a) +where Fcl is the equation of motion (15.40), and Λ(Λ1, Λ2, Ψcl) is a field-dependent gauge +parameter: +Λ(Λ1, Λ2, Ψcl) = +� +n≥0 +gn+1 +s +n! +ℓ0,n+2(Λ1, Λ2, Ψn +cl) += gs ℓ0,2(Λ1, Λ2) + +� +n≥1 +gn+1 +s +n! +ℓ0,n+2(Λ1, Λ2, Ψn +cl). +(15.44b) +The classical gauge algebra is complicated which explains why a direct quantization (for +example through the Faddeev–Popov procedure) cannot work: the second term in (15.44a) +indicates that the algebra is open (it closes only on-shell), while the first term is a gauge +transformation with a field-dependent parameter. As reviewed in Appendix C.3, both prop- +erties require using the BV formalism for the quantization, and the latter is performed in +Section 15.4. An important point is that if the theory had only cubic interactions, i.e. if +∀n ≥ 4 : +V0,4(V1, . . . , Vn) = 0, +ℓg,n−1(V1, . . . , Vn−1) = 0, +(cubic theory), (15.45) +then the algebra closes off-shell and Λ(Λ1, Λ2, Ψcl) becomes field-independent. +233 + +Computation – Equation (15.42) +δΛScl = +� +n≥2 +gn−2 +s +n! +nV0,n(δΨcl, Ψn−1 +cl +) = +� +m,n≥0 +gm+n−1 +s +m! n! +V0,n+1 +� +ℓ0,m+1(Ψm +cl , Λ), Ψn +cl +� += +� +m≥0 +m +� +n=0 +gm−1 +s +(m − n)! n! +� +ℓm−n+1(Ψm−n +cl +, Λ) +�� c− +0 |ℓ0,n(Ψn +cl)⟩ . +For simplicity we have extended the sum up to n = 0 and m = 0 by using the fact that +lower-order vertices vanish. The bracket can be rewritten as += ⟨ℓ0,n(Ψn +cl)| c− +0 +��ℓ0,m−n+1(Ψm−n +cl +, Λ) +� += V0,m−n+2 +� +ℓ0,n(Ψn +cl), Ψm−n +cl +, Λ +� += −V0,m−n+2 +� +Λ, ℓ0,n(Ψn +cl), Ψm−n +cl +� += ⟨Λ| c− +0 +��ℓ0,m−n+1 +� +ℓ0,n(Ψn +cl), Ψm−n +cl +�� +. +Then, one needs to use the identity (defined for all m ≥ 0) +0 = +m +� +n=0 +m! +(m − n)! n! ℓ0,m−n+1 +� +ℓ0,n(Ψn +cl), Ψm−n +cl +� +, +(15.46) +which comes from (14.70). Multiplying this by gm−1 +s +/m! and summing over m ≥ 0 +proves (15.42). +Remark 15.1 (L∞ algebra) The identities satisfied by the products ℓ0,n from the gauge +invariance of the action implies that they form a L∞ homotopy algebra [74, 169, 262] (for +more general references, see [106, 108, 148, 149]). The latter can also be mapped to a BV +structure, which explains why the BV quantization Section 15.4 is straightforward. This +interplay between gauge invariance, covering of the moduli space, BV and homotopy algebra +is particularly beautiful. It has also been fruitful in constructing super-SFT. +15.4 +BV theory +As indicated in the previous section (Section 15.3), the classical gauge algebra is open and +has field-dependent structure constants. The BV formalism (Appendix C.3) is necessary to +define the theory. +In the BV formalism, the classical action for the physical fields is extended to the +quantum master action by solving the quantum master equation (C.114). +It is generic- +ally difficult to build this action exactly, but the discussion of Section 10.3 can serve as a +guide: it was found that the free quantum action (with the tower of ghosts) has exactly the +same form as the free classical action (without ghosts). Hence, this motivates the ansatz +that it should be of the same form as the classical action (15.36) to which are added the +234 + +counter-terms from (15.24): +S = 1 +g2s +� +g≥0 +ℏgg2g +s +� +n≥0 +gn +s +n! Vg,n(Ψn) +(15.47a) += 1 +2 ⟨Ψ| c− +0 QB |Ψ⟩ + +�′ +g,n≥0 +ℏgg2g−2+n +s +n! +Vg,n(Ψn) +(15.47b) += 1 +g2s +� +g,n≥0 +ℏgg2g−2+n +s +n! +⟨Ψ| c− +0 +��ℓg,n−1(Ψn−1) +� +, +(15.47c) +but without any constraint on the ghost number of Ψ: +Ψ ∈ H− ∩ ker L− +0 . +(15.48) +In order to show that (15.47) is a consistent quantum master action, it is necessary to +show that it solves the master BV equation (C.114): +(S, S) − 2ℏ∆S = 0. +(15.49) +The first step is to introduce the fields and antifields. In fact, because the CFT ghost number +induces a spacetime ghost number, there is a natural candidate set. +The string field is expanded as (15.1) +|Ψ⟩ = +� +r +ψr |φr⟩ , +(15.50) +where the {φr} forms a basis of H−. The string field can be further separated as: +Ψ = Ψ+ + Ψ−, +(15.51) +where Ψ− (Ψ+) contains only states which have negative (positive) cylinder ghost numbers +(this gives an offset of 3 when using the plane ghost number): +Ψ− = +� +r +� +nr≤2 +|φr⟩ ψr, +Ψ+ = +� +r +� +ncr>2 +b− +0 |φc +r⟩ ψ∗ +r. +(15.52) +The order of the basis states and coefficients matter if they anti-commute. The sum in Ψ+ +can be rewritten as a sum over nr ≤ 2 like the first term since nr +nc +r = 6. Correspondingly, +the spacetime ghost numbers (15.6) for the coefficients in Ψ− (Ψ+) are positive (negative) +G(ψr) ≥ 0, +G(ψ∗ +r) < 0. +(15.53) +Moreover, one finds that the ghost numbers of ψr and ψ∗ +r are related as: +G(ψ∗ +r) = −1 − G(ψr), +(15.54) +which also implies that they have opposite parity. Comparing with Appendix C.3, this shows +that the ψr (ψ∗ +r) contained in Ψ− (Ψ+) can be identified with the fields (antifields). +Computation – Equation (15.54) +G(ψ∗ +r) = 2 − Ngh(b− +0 φc +r) = 2 + 1 − nc +r += 3 − (6 − nr) = −3 + (2 − G(ψr)) = −1 − G(ψr). +235 + +In terms of fields and antifields, the master action is +∂RS +∂ψr +∂LS +∂ψ∗r ++ ℏ ∂R∂LS +∂ψr∂ψ∗r += 0. +(15.55) +Plugging the expression (15.47) of S inside and requiring that the expression vanishes order +by order in g and n give the set of equations: +� +g1,g2≥0 +g1+g2=g +� +n1,n2≥0 +n1+n2=n +∂RSg1,n1 +∂ψr +∂LSg2,n2 +∂ψ∗r ++ ℏ ∂R∂LSg−1,n +∂ψr∂ψ∗r += 0, +(15.56) +where Sg,n was defined in (15.22). This holds true due to the identity (14.70) (the complete +proof can be found in [262, pp. 42–45]). The fact that the second term is not identically zero +means that the measure is not invariant under the classical gauge symmetry (anomalous +symmetry): corrections need to be introduced to cancel the anomaly. It is a remarkable +fact that one can construct directly the quantum master action in SFT and that it takes +the same form as the classical action. +15.5 +1PI theory +The BV action is complicated: instead, it is often simpler and sufficient to work with the +1PI effective action. The latter incorporates all the quantum corrections in 1PI vertices +such that scattering amplitudes are expressed only in terms of tree Feynman graphs (there +are no loops in diagrams since they correspond to quantum effects, already included in the +definitions of the vertices). +A 1PI graph is a Feynman graph which stays connected if one cuts any single internal +line. On the other hand, a 1PR graph splits in two disconnected by cutting one of the line. +The scattering amplitudes Ag,n are built by summing all the different ways to connect two +1PI vertices with a propagator: diagrams connecting two legs of the same 1PI vertex are +forbidden by definition. +The g-loop n-point 1PR and 1PI Feynman diagrams are associated to some regions of +the moduli space Mg,n. Comparing the previous definitions with the gluing of Riemann +surfaces (Section 12.3), 1PR diagrams are obtained by gluing surfaces with the separating +plumbing fixture (Section 14.1.1). Thus, the 1PR and 1PI regions F1PR +g,n +and V1PI +g,n can be +identified with the regions defined in (12.43a) and (12.43b). In particular, the n-point 1PI +interaction is the sum over g of the g-loop n-point 1PI interactions (14.63): +V1PI +n +(V1, . . . , Vn) := +:= +� +g≥0 +(ℏg2 +s)g V1PI +g,n (V1, . . . , Vn), +V1PI +g,n (V1, . . . , Vn) := +� +R1PI +g,n +ωg,n +Mg,n(V1, . . . , Vn), +(15.57) +where R1PI +g,n is a section of Pg,n over V1PI +g,n . +Given the interactions vertices, it is possible to follow the same reasoning as in Sec- +tions 15.2 and 15.3. +236 + +The gauge fixed 1PI effective action reads: +S1PI = 1 +g2s +� +n≥0 +gn +s +n! V1PI +n +(Ψn) := 1 +2 ⟨Ψ| c− +0 c+ +0 L+ +0 |Ψ⟩ + 1 +g2s +�′ +n≥0 +gn +s +n! V1PI +n +(Ψn). +(15.58) +Here, the prime means again that the term g = 0, n = 2 is excluded from the definition of +V1PI +2 +. The action has the same form as the classical gauge fixed action (15.29), which is +logical since it generates only tree-level Feynman graphs. For this reason the vertices V1PI +n +have exactly the same properties as the brackets V0,n. This fact can be used to write the +1PI gauge invariant action: +S1PI = 1 +2 ⟨Ψ| c− +0 QB |Ψ⟩ + 1 +g2s +�′ +n≥0 +gn +s +n! V1PI +n +(Ψn), +(15.59) +which mirrors the classical gauge invariant action (15.36). Then it is straightforward to see +that it enjoys the same gauge symmetry upon replacing the tree-level vertices by the 1PI +vertices. But, since this action incorporates all quantum corrections this also proves that +the quantum theory is correctly invariant under a quantum gauge symmetry. +Remark 15.2 The 1PI action (15.59) can also be directly constructed from the BV action +(15.47). +15.6 +Suggested readings +• Gauge fixed and classical gauge invariant closed SFT [262] (see also [138, 139]). +• BV closed SFT [262] (see also [245]). +• Construction of the open–closed BV SFT [264]. +• 1PI SFT [213, 214, 42, sec. 4.1, 5.2]. +237 + +Chapter 16 +Background independence +Abstract +Spacetime background independence is a fundamental property of any candidate +quantum gravity theory. In this chapter, we outline the proof of background independence +for the closed SFT by proving that the equations of motion of two background related by a +marginal deformation are equivalent after a field redefinition. +16.1 +The concept of background independence +Background independence means that the formalism does not depend on the background +– if any – used to write the theory. A dependence in the background would imply that +there is a distinguished background among all possibilities, which seems in tension with the +dynamics of spacetime and the superposition principle from quantum mechanics. Moreover, +one would expect a fundamental theory to tell which backgrounds are consistent and that +they could be derived instead of postulated. Background independence allows spacetime to +emerge as a consequence of the dynamics of the theory and of its defining fundamental laws. +Background independence can be manifest or not. In the second case, one needs to fix a +background to define the theory, but the dynamics on different backgrounds are physically +equivalent.1 This implies that two theories with different backgrounds can be related, for +example by a field redefinition. +While fields other than the metric can also be expanded around a background, no diffi- +culty is expected in this case. Indeed, the topic of background independence is particularly +sensible only for the metric because it provides the frame for all other computations – and +in particular for the questions of dynamics and quantization. Generally, these questions are +subsumed into the problem of the emergence of time in a generally covariant theory. In +the previous language, QFTs without gravity are (generically) manifestly background inde- +pendent after minimal coupling.2 For example, a classical field theory is defined on a fixed +Minkowski background and a well-defined time is necessary to perform its quantization and +to obtain a QFT, but it is not needed to choose a background for the other fields. For this +reason, the extension of a QFT on a curved background is generally possible if the space- +time is hyperbolic, implying that there is a distinguished time direction. But the coupling +to gravity is difficult and restricted to a (semi-)classical description. +What is the status of background independence in string theory? The worldsheet for- +mulation requires to fix a background (usually Minkowski) to quantize the theory and to +compute scattering amplitudes. Thus, the quantum theory is at least not manifestly back- +1This does not mean that the physics in all backgrounds are identical, but that the laws are. Hence, a +computation made in one specific background can be translated into another background. +2However, non-minimal coupling terms may be necessary to make the theory physical. +238 + +ground independent. On the other hand, the worldsheet action can be modified to a generic +CFT including a generic non-linear sigma model describing an arbitrary target spacetime. +Conformal invariance reproduces (at leading order) Einstein equations coupled to various +matter and gauge field equations of motion. From this point of view, the classical theory can +be written as a manifestly background independent theory, and this provides hopes that the +quantum theory may be also background independent, even if non-manifestly. This idea is +supported by other definitions of string theory (e.g. through the AdS/CFT conjecture – and +other holographic realizations – or through matrix models) which provide, at least partially, +background independent formulations. +Ultimately, the greatest avenue to establish the background independence is string field +theory. Indeed, the form of the SFT action and of its properties (gauge invariance, equa- +tion of motion. . . ) are identical irrespective of the background [234]. This provides a good +starting point. The background dependence enters in the precise definition of the string +products (BRST operator and vertices). The origin of this dependence lies in the derivation +of the action (Chapters 14 and 15): one begins with a particular CFT describing a given +background (spacetime compactifications, fluxes, etc.) and defines the vertices from correl- +ation functions of vertex operators, and the Hilbert space from the CFT operators. As a +consequence, even though it is clear that no specific property of the background has been +used in the derivation – and that the final action describes SFT for any background –, this +is not sufficient to establish background independence. Since the theory assumes implicitly +a background choice, one cannot guarantee that the physical quantities have no residual +dependence in the background, even if the action looks superficially background independ- +ent. Background independence in SFT is thus the statement that theories characterized by +different CFTs can be related by a field redefinition. +In this chapter, we will sketch the proof of background independence for backgrounds +related by marginal deformations.3 It is possible to prove it at the level of the action [232, +233], or at the level of the equations of motion [226]. The advantage of the second approach +is that one can use the 1PI theory, which simplifies vastly the analysis. It also generalizes +directly to the super-SFT. +Remark 16.1 (Field theory on the CFT space) As mentioned earlier, the string field +is defined as a functional on the state space of a given CFT and not as a functional on the +field theory space (off-shell states would correspond to general QFTs, only on-shell states are +CFTs). In this case, background independence would amount to reparametrization invariance +of the action in the theory space, and would thus almost automatically hold. A complete +formulation of SFT following this line is currently out of reach, but some ideas can be found +in [255]. +16.2 +Problem setup +Given a SFT on a background, there are two ways to describe it on another background: +• deform the worldsheet CFT and express the SFT on the new background; +• expand the original action around the infinitesimal classical solution (to the linearised +equations of motion) corresponding to the deformation. +Background independence amounts to the equivalence of both theories up to a field redefin- +ition. The derivation can be performed at the level of the action or of the equations of +motion. To prove the background independence at the quantum level, one needs to take +into account the changes in the path integral measure or to work with the 1PI action. +3An alternative approach based on morphism of L∞ algebra is followed in [169]. +239 + +The simplest case is when the two CFTs are related by an infinitesimal marginal deform- +ation +δScft = λ +2π +� +d2z ϕ(z, ¯z), +(16.1) +with ϕ a (1, 1)-primary operator and λ infinitesimal. The two CFTs are denoted by CFT1 +and CFT2, and quantities associated to each CFTs is indexed with the appropriate number. +Establishing background independence in this case also implies it for finite marginal +deformation since they can be built from a series of successive deformations. In the latter +case, the field redefinition may be singular, which reflects that the parametrization of one +CFT is not adapted for the other (equivalently, the coordinate systems for the string field +breaks down), which is expected if both CFTs are far in the field theory space. +Remember the form of the 1PI action (15.59): +S1[Ψ1] = 1 +g2s +� +�1 +2 ⟨Ψ1| c− +0 QB |Ψ1⟩ + +�′ +n≥0 +1 +n! V1PI +n +(Ψn +1) +� +� , +(16.2) +where the prime indicates that vertices with n < 3 do not include contributions from the +sphere. In all this chapter, we remove the index 1PI to lighten the notations. The equation +of motion is: +F1(Ψ1) = QB |Ψ1⟩ + +� +n +1 +n! ℓn(Ψn +1) = 0. +(16.3) +16.3 +Deformation of the CFT +Consider the case where the theory CFT1 is described by an action Scft,1[ψ1] given in terms +of fields ψ1. Then, the deformation of this action by (16.1) gives an action for CFT2: +Scft,2[ψ1] = Scft,1[ψ1] + λ +2π +� +d2z ϕ(z, ¯z). +(16.4) +Correlation functions on a Riemann surface Σ in both theories can be related by expanding +the action to first order in λ in the path integral: +�� +i +Oi(zi, ¯zi) +� +2 += +� +exp +� +− λ +2π +� +d2z ϕ(z, ¯z) +� � +i +Oi(zi, ¯zi) +� +1 +(16.5a) +≈ +�� +i +Oi(zi, ¯zi) +� +1 +− λ +2π +� +Σ +d2z +� +ϕ(z, ¯z) +� +i +Oi(zi, ¯zi) +� +1 +, +(16.5b) +where the Oi are operators built from the matter fields ψ1. This expression presents two +obvious problems. First, the correlation function may diverge when ϕ collides with one of the +insertions, i.e. when z = zi in the integration. Second, there is an inherent ambiguity: the +correlation functions are written in terms of operators in the Hilbert space of CFT1, which +is different from the CFT2 Hilbert space, and there is no canonical isomorphism between +both spaces. +Seeing the Hilbert space as a vector bundle over the CFT theory space, the second +problem can be solved by introducing a connection on this bundle. This allows to relate +Hilbert spaces of neighbouring CFTs. In fact, the choice of a non-singular connection also +regularizes the divergences. +The simplest definition of a connection corresponds to cut unit disks around each operator +insertions [30, 198, 199, 210, 238]. This amounts to define the variation between the two +240 + +correlation functions as: +δ +�� +i +Oi(zi, ¯zi) +� +1 += − λ +2π +� +Σ−∪iDi +d2z +� +ϕ(z, ¯z) +� +i +Oi(zi, ¯zi) +� +1 +. +(16.6) +The integration is over Σ minus the disks Di = {|wi| ≤ 1} where wi is the local coordinate +for the insertion Oi. The divergences are cured because ϕ never approaches another operator +since the corresponding regions have been removed. The changes in the correlation functions +induce a change in the string vertices denoted by δVn(V1, . . . , Vn). +The next step consists in computing the deformations of the operator modes. Since it +involves only a matter operator, the modes in the ghost sector are left unchanged. The +Virasoro generators change as: +δLn = λ +� +|z|=1 +d¯z +2πi zn+1ϕ(z, ¯z), +δ ¯Ln = λ +� +|z|=1 +dz +2πi ¯zn+1ϕ(z, ¯z). +(16.7) +As a consequence, the BRST operator changes as +δQB = λ +� +|z|=1 +d¯z +2πi c(z)ϕ(z, ¯z) + λ +� +|z|=1 +d¯z +2πi ¯c(¯z)ϕ(z, ¯z). +(16.8) +One can prove that +{QB, δQB} = O(λ2) +(16.9) +such that the BRST charge QB + δQB in CFT2 is correctly nilpotent if QB is nilpotent in +CFT1. +For the deformation to provide a consistent SFT, the conditions b− +0 = 0 and L− +0 = 0 must +be preserved. The first is automatically satisfied since the ghost modes are not modified. +Considering an weight-(h, h) operator O, one finds +δL− +0 |O⟩ = λ +� +|z|=1 +d¯z +2πi z +� +p,q +zp−1¯zq−1 |Op,q⟩ − λ +� +|z|=1 +d¯z +2πi +� +p,q +zp−1¯zq−1 |Op,q⟩ , +(16.10) +where Op,q are the fields appearing in the OPE with ϕ: +ϕ(z, ¯z)O(0, 0) = +� +p,q +zp−1¯zq−1Op,q(0, 0). +(16.11) +The terms with p ̸= q vanish because the contour integrals are performed around circles of +unit radius centred at the origin. Moreover, the terms p = q are identical and cancel with +each other, showing that δL− +0 = 0 when acting on states satisfying L− +0 = 0. +The SFT action S2[Ψ1] in the new background reads +S2[Ψ1] = S1[Ψ1] + δS1[Ψ1] +(16.12) +where the change δS1 in the action is induced by the changes in the string vertices: +δS1[Ψ1] = 1 +g2s +� +�1 +2 ⟨Ψ1| c− +0 δQB |Ψ1⟩ + +� +n≥0 +1 +n! δVn(Ψn +1) +� +� . +(16.13) +The equation of motion is: +F2(Ψ1) = F1(Ψ1) + λ δF1(Ψ1) = 0, +(16.14) +where F1 is given in (16.3) and +λ δF1(Ψ1) = δQB |Ψ1⟩ + +� +n +1 +n! δℓn(Ψn +1). +(16.15) +241 + +16.4 +Expansion of the action +Given a (1, 1) primary ϕ, a BRST invariant operator is c¯cϕ. Hence the field +|Ψ1⟩ = λ |Ψ0⟩ , +|Ψ0⟩ = c1¯c1(0) |ϕ⟩ +(16.16) +is a classical solution to first order in λ since the interactions on the sphere are at least +cubic. +Separating the string field as the contribution from the (fixed) background and a fluctu- +ation Ψ′ +|Ψ1⟩ = λ |Ψ0⟩ + |Ψ′⟩ , +(16.17) +the action expanded to first order in λ reads: +S1[Ψ1] = S1[Ψ0] + S′[Ψ′], +(16.18) +where +S′[Ψ′] = 1 +g2s +� +1 +2 ⟨Ψ′| c− +0 QB |Ψ′⟩ + +� +n +1 +n! +� +Vn(Ψ′n) + λ Vn+1(Ψ0, Ψ′n) +� +� +. +(16.19) +The equation of motion is: +F′(Ψ′) := F1(Ψ′) + λ δF′(Ψ′) = 0, +(16.20) +where F1 is given in (16.3) and +δF′(Ψ′) = +� +n +1 +n! ℓn+1(Ψ0, Ψ′n). +(16.21) +16.5 +Relating the equations of motion +In the previous section, we have derived the equations of motion for two different descriptions +of a SFT obtained after shifting the background: (16.14) arises by deforming the CFT and +computing the changes in the BRST operator and string products, while (16.20) arises by +expanding the SFT action around the new background. The theory is background independ- +ent if both sets of equations (16.14) and (16.20) are related by a (possibly field-dependent) +linear transformation M(Ψ′) after a field redefinition of Ψ1 = Ψ1(Ψ′): +F1(Ψ1) + λ δF1(Ψ1) = +� +1 + λM(Ψ′) +�� +F1(Ψ′) + λ δF′(Ψ′) +� +, +(16.22a) +|Ψ1⟩ = |Ψ′⟩ + λ |δΨ′⟩ . +(16.22b) +The zero-order equation is automatically satisfied. To first order, this becomes +d +dλF1(Ψ′ + λδΨ′) +���� +λ=0 ++ δF1(Ψ1) − δF′(Ψ′) = M(Ψ′)F1(Ψ′). +(16.23) +Taking Ψ′ to be a solution of the original action removes the RHS, such that: +λ QB |δΨ′⟩ + λ +� +n +1 +n! ℓn+1(δΨ′, Ψ′n) + δQB |Ψ′⟩ ++ +� +n +1 +n! δℓn(Ψ′n) − λ +� +n +1 +n! ℓn+1(Ψ0, Ψ′n) = 0. +(16.24) +242 + +To simplify the computations, it is simpler to consider the inner product of this quantity +with an arbitrary state A (assumed to be even): +∆ := λ ⟨A| c− +0 QB |δΨ′⟩ + λ +� +n +1 +n! Vn+2(A, δΨ′, Ψ′n) +⟨A| c− +0 δQB |Ψ′⟩ ++ +� +n +1 +n! δVn+1(A, Ψ′n) − λ +� +n +1 +n! Vn+2(A, Ψ0, Ψ′n). +(16.25) +The goal is to prove the existence of δΨ′ such that ∆ = 0 up to the zero-order equation of +motion F1(Ψ′) = 0. +16.6 +Idea of the proof +In this section, we give an idea of how the proof ends, referring to [226] for the details. +The first step is to introduce new vertices V′ +0,3 and V′ +n parametrizing the variations of +the string vertices: +⟨A| c− +0 δQB |B⟩ = λ V′ +0,3(Ψ0, B, A), +δVn(Ψ′n) = λ V′ +n+1(Ψ0, Ψ′n), +(16.26) +where the notation (13.19) has been used. Each subspace V′ +g,n is defined such that the LHS +is recovered upon integrating the appropriate ωg,n over this section segment. Next, the field +redefinition δΨ′ is parametrized as: +⟨A| c− +0 |δΨ′⟩ = +� +n +1 +n! Bn+2(Ψ0, Ψ′n, A). +(16.27) +The objective is to prove the existence (and if possible the form) of the subspaces Bn+2. +Both the vertices V′ +n and Bn admit a genus expansion: +V′ +n = +� +g≥0 +V′ +g,n, +Bn = +� +g≥0 +Bg,n. +(16.28) +Plugging the new expressions in (16.25) give: +∆ = − +� +n +1 +n! Bn+2(Ψ0, Ψ′n, QBA) + +� +m,n +1 +m!n! Bn+2(Ψ0, Ψ′m, ℓn+1(A, Ψ′n)) ++ +� +n +1 +n! V′ +n+2(A, Ψ0, Ψ′n) − +� +n +1 +n! Vn+2(A, Ψ0, Ψ′n). +(16.29) +Next, the BRST identity (13.46) and the equation of motion F1(Ψ′) = 0 allow to rewrite +the first term as: +Bn+2(Ψ0, Ψ′n, QBA) = ∂Bn+2(Ψ0, Ψ′n, A) + n Bn+2(Ψ0, Ψ′n−1, QBΨ′, A) +(16.30a) += ∂Bn+2(Ψ0, Ψ′n, A) − +� +m +n +m! Bn+2(Ψ0, Ψ′n−1, ℓm(Ψ′m), A). +(16.30b) +In the second term, the sum over n is shifted. Combining everything together gives: +∆ = +� +n +1 +n! ∂Bn+2(Ψ0, Ψ′n, A) − +� +m,n +1 +m!n! Bn+3(Ψ0, Ψ′n, ℓm(Ψ′m), A) ++ +� +m,n +1 +m!n! Bn+2(Ψ0, Ψ′m, ℓn+1(A, Ψ′n)) + +� +n +1 +n! V′ +n+2(A, Ψ0, Ψ′n) +− +� +n +1 +n! Vn+2(A, Ψ0, Ψ′n). +(16.31) +243 + +Solving for ∆ = 0 requires that each term with a different power of Ψ′ vanishes independ- +ently: +∂Bn+2(Ψ0, Ψ′n, A) = − V′ +n+2(A, Ψ0, Ψ′n) + Vn+2(A, Ψ0, Ψ′n) ++ +� +m1,m2 +m1+m2=n +n! +m1!m2! Bm1+3(Ψ0, Ψ′m1, ℓm2(Ψ′m2), A) +− +� +m1,m2 +m1+m2=n +n! +m1!m2! Bm1+2 +� +Ψ0, Ψ′m1, ℓm2+1(A, Ψ′m2) +� +. +(16.32) +In order to proceed, one needs to perform a genus expansion of the various spaces: +this allows to solve recursively for all Bg,n starting from B0,3. One can then build |δΨ′⟩ +recursively, which provides the field redefinition. Indeed, the RHS of this equation contains +only Bg′,n′ for g′ < g or n′ < n and the equation for B0,3 contains no Bg,n in the RHS. +It should be noted that the field redefinition is not unique, but there is the freedom of +performing (infinite-dimensional) gauge transformations. Finding an obstruction to solve +these equations mean that the field redefinition does not exist, and thus that the theory is +not background independent +The form of the equation +∂B0,3 = V0,3 − V′ +0,3 +(16.33) +suggests to use homology theory. The interpretation of B0,3 is that it is a space interpolating +between V0,3 and V′ +0,3. A preliminary step is to check that there is no obstruction: since the +LHS is already a boundary one has ∂2B0,3 = 0 and one should check that ∂(RHS) = 0 as +well. It can be shown that it is indeed true. It was proved in [226] that this equation admits +a solution and that the equations for higher g and n can all be solved. Hence, there exists +a field redefinition and SFT is background independent. +16.7 +Suggested readings +• Proof of the background independence under marginal deformations [226, 232, 233] +(see also [209–211] for earlier results laying foundations for the complete proof). +• L∞ perspective [169, sec. 4] (see also [168, 167, sec. III.B]. +• Connection on the space of CFTs [30, 198, 199, 210, 238]. +244 + +Chapter 17 +Superstring +Abstract +Superstring theory is generally the starting point for physical model building. +It has indeed several advantages over the bosonic string, most importantly, the removal +of the tachyon and the inclusion of fermions in the spectrum. The goal of this chapter is +to introduce the most important concepts needed to generalize the bosonic string to the +superstring, both for off-shell amplitudes and string field theory. We refer to the review [42] +for more details. +17.1 +Worldsheet superstring theory +There are five different superstring theories with spacetime supersymmetry: the types I, IIA +and IIB, and the E8 × E8 and SO(32) heterotic models. +In the Ramond–Neveu–Schwarz formalism (RNS), the left- and right-moving sectors of +the superstring worldsheet are described by a two-dimensional super-conformal field theory +(SCFT), possibly with different numbers of supersymmetries. The prototypical example is +the heterotic string with N = (1, 0) and we will focus on this case: only the left-moving +sector is supersymmetric, while the right-moving is given by the same bosonic theory as in +the other chapters. Up to minor modifications, the type II theory follows by duplicating the +formulas of the left-moving sector to the right-moving one. +17.1.1 +Heterotic worldsheet +The ghost super-CFT is characterized by anti-commuting ghosts (b, c) (left-moving) and +(¯b, ¯c) (right-moving) with central charge c = (−26, −26), associated to diffeomorphisms, and +by commuting ghosts (β, γ) with central charge c = (11, 0), associated to local supersym- +metry. As a consequence the matter SCFT must have a central charge c = (15, 26). If +spacetime has D non-compact dimensions, then the matter CFT is made of: +• a free theory of D scalars Xµ and D left-moving fermions ψµ (µ = 0, . . . , D − 1) such +that cfree = 3D/2 and ¯cfree = D; +• an internal theory with cint = 15 − 3D/2 and ¯cint = 26 − D. +The critical dimension is reached when cint = 0 which corresponds to D = 10. +The diffeomorphisms are generated by the energy–momentum tensor T(z); correspond- +ingly, supersymmetry is generated by its super-partner G(z) (sometimes also denoted by +245 + +TF ). The OPEs of the algebra formed by T(z) and G(z) is: +T(z)T(w) ∼ +c/2 +(z − w)4 + +2T(w) +(z − w)2 + ∂T(w) +z − w , +(17.1a) +G(z)G(w) ∼ +2c/3 +(z − w)3 + 2T(w) +(z − w), +(17.1b) +T(z)G(w) ∼ 3 +2 +G(w) +(z − w)2 + ∂G(w) +(z − w). +(17.1c) +The superconformal ghosts form a first-order system (see Section 7.2) with ϵ = −1 and +λ = 3/2. Hence, they have conformal weights +h(β) = +�3 +2, 0 +� +, +h(γ) = +� +−1 +2, 0 +� +(17.2) +and OPEs +γ(z)β(w) ∼ +1 +z − w, +β(z)γ(w) ∼ − +1 +z − w. +(17.3) +The expressions of the ghost energy–momentum tensors are +T gh = −2b ∂c + c∂b, +T βγ = 3 +2 β∂γ + 1 +2 γ ∂β. +(17.4) +The ghost numbers of the different fields are +Ngh(b) = Ngh(β) = −1, +Ngh(c) = Ngh(γ) = 1. +(17.5) +The worldsheet scalars satisfy periodic boundary conditions. On the other hand, fermions +can satisfy anti-periodic or periodic conditions: this leads to two different sectors, called +Neveu–Schwarz (NS) and Ramond (R) respectively. +βγ system +The βγ system can be bosonized as +γ = η eφ, +β = ∂ξ e−φ, +(17.6) +where (ξ, η) are fermions with conformal weights 0 and 1 (this is a first-order system with +ϵ = 1 and λ = 1), and φ is a scalar field with a background charge (Coulomb gas). This +provides an alternative representation of the delta functions: +δ(γ) = e−φ, +δ(β) = eφ. +(17.7) +Introducing these operators is necessary to properly define the path integral with bosonic +zero-modes. They play the same role as the zero-modes insertions for fermionic fields needed +to obtain a finite result (see also Appendix C.1.3): +� +dc0 = 0 +=⇒ +� +dc0 c0 = 1, +(17.8) +because c0 = δ(c0). For a bosonic path integral, one needs a delta function: +� +dγ0 = ∞ +=⇒ +� +dγ0 δ(γ0) = 1. +(17.9) +246 + +By definition of the bosonization, one has: +T βγ = T ηξ + T φ, +(17.10) +where +T ηξ = −η ∂ξ, +T φ = −1 +2 (∂φ)2 − ∂2φ. +(17.11) +The OPE between the new fields are: +ξ(z)η(w) ∼ +1 +z − w, +eq1φ(z)eq2φ(w) ∼ e(q1+q2)φ(w) +(z − w)q1q2 , +∂φ(z)∂φ(w) ∼ − +1 +(z − w)2 . +(17.12) +The simplest attribution of ghost numbers to the new fields is: +Ngh(η) = 1, +Ngh(ξ) = −1, +Ngh(φ) = 0. +(17.13) +To the scalar field φ is associated another U(1) symmetry whose quantum number is +called the picture number Npic. The picture number of η and ξ are assigned1 such that β +and γ have Npic = 0: +Npic(eqφ) = q, +Npic(ξ) = 1, +Npic(η) = −1. +(17.14) +Because of the background charge, this symmetry is anomalous and correlation functions +are non-vanishing if the total picture number (equivalently the number of φ zero-modes) is: +Npic = 2(g − 1) = −χg. +(17.15) +For the same reason, the vertex operators eqφ are the only primary operators: +h(eqφ) = −q +2(q + 2), +(17.16) +and the Grassmann parity of these operators is (−1)q. Special values are +h(eφ) = 3 +2, +h(e−φ) = 1 +2. +(17.17) +The superstring theory features a Z2 symmetry called the GSO symmetry. All fields are +taken to be GSO even, except β and γ which are GSO odd and eqφ whose parity is (−1)q. +Physical states in the NS sector are restricted to be GSO even: it is required to remove +the tachyon of the spectrum and to get a spacetime with supersymmetry. In type II, the +Ramond sector can be projected in two different ways, leading to the type IIA and type IIB +theories. +The components of the BRST current are: +jB = c(T m + T βγ) + γG + bc∂c − 1 +4 γ2b, +(17.18a) +¯ȷB = ¯c ¯T m + ¯b¯c¯∂¯c. +(17.18b) +From there, it is useful to define the picture changing operator (PCO): +X(z) = {QB, ξ(z)} = c∂ξ + eφG − 1 +4 ∂η e2φ b − 1 +4 ∂(η e2φb), +(17.19) +which is a weight-(0, 0) primary operator which carries a unit picture number. It is obviously +BRST exact. This operator will be necessary to saturate the picture number condition: the +1Any linear combination of both U(1) could have been used. The one given here is conventional, but +also the most convenient. +247 + +naive insertion of eφ ∼ δ(β) breaks the BRST invariance. The PCO zero-mode is obtained +from the contour integral: +X0 = +1 +2πi +� dz +z X(z). +(17.20) +It can be interpreted as delocalizing a PCO insertion from a point to a circle, which decreases +the risk of divergence. +17.1.2 +Hilbert spaces +The description in terms of the (η, ξ, φ) fields leads to a subtlety: the bosonization involves +only the derivative ∂ξ and not the field ξ itself, meaning that the zero-mode ξ0 is absent from +the original Hilbert space defined from (β, γ). In the bosonized language, the Hilbert space +without the ξ zero-mode is called the small Hilbert space and is made of state annihilated +by η0 (the η zero-mode) +Hsmall = +� +|ψ⟩ | η0 |ψ⟩ = 0 +� +. +(17.21) +Removing this condition leads to the large Hilbert space:2 +Hsmall = Hlarge ∩ ker η0. +(17.22) +A state in Hsmall contains ξ with at least one derivative acting on it. +A correlation function defined in terms of the (η, ξ, φ) system is in the large Hilbert space +and will vanish since there is no ξ factor to absorb the zero-mode of the path integral. As +a consequence, correlation functions (and the inner product) are defined with a ξ0 insertion +(by convention at the extreme left) or, equivalently, ξ(z). The position does not matter since +only the zero-mode contribution survives, and the correlation function is independent of z. +Sometimes it is more convenient to work in the large Hilbert space and to restrict later to +the small Hilbert space. +The SL(2, C) invariant vacuum is normalized as +⟨k| c−1¯c−1c0¯c0c1¯c1 e−2φ(z) |k′⟩ = (2π)Dδ(D)(k + k′). +(17.23) +Remark 17.1 (Normalization in type II) In type II theory, the SL(2, C) is normalised +as: +⟨k| c−1¯c−1c0¯c0c1¯c1 e−2φ(z)e− ¯φ( ¯ +w) |k′⟩ = −(2π)Dδ(D)(k + k′). +(17.24) +The sign difference allows to avoid sign differences between type II and heterotic string +theories in most formulas [42]. +The Hilbert space of GSO even states satisfying the b− +0 = 0 and L− +0 = 0 conditions is +denoted by HT (ghost and picture numbers are arbitrary). This Hilbert space is the direct +sum of the NS and R Hilbert spaces: +HT = HNS ⊕ HR. +(17.25) +The subspace of states with picture number Npic = n is written Hn. The picture number +of NS and R states are respectively integer and half-integer. Two special subspaces of HT +play a distinguished role: +� +HT = H−1 ⊕ H−1/2, +� +HT = H−1 ⊕ H−3/2. +(17.26) +2The relation between the small and large Hilbert spaces is similar to the one between the H and +H0 = b0H Hilbert space from the open string since the (b, c) and (η, ξ) are both fermionic first-order +systems. +248 + +To understand this, consider the vacuum |p⟩ of the φ field with picture number p: +|p⟩ = epφ(0) |0⟩ . +(17.27) +Then, acting on the vacuum with the βn and γn modes implies +∀n ≥ −p − 1 +2 : +βn |p⟩ = 0, +∀n ≥ p + 3 +2 : +γn |p⟩ = 0. +(17.28) +For p = −1, all positive modes (starting with n = 1/2) annihilate the vacuum in the NS +sector. This is a positive asset because positive modes which do not annihilate the vacuum +can create states with arbitrary negative energy (since it is bosonic).3 For p = −1/2 or +p = −3/2, the vacuum is annihilated by all positive modes, but not by one of the zero-mode +γ0 or β0. Nonetheless, one can show that the propagator in the R sector allows to propagate +only a finite number of states if one chooses H−1/2; the role of H−3/2 will become apparent +when discussing how to build the superstring field theory. +Basis states are introduced as in the bosonic case: +� +HT = Span{|φr⟩}, +� +HT = Span{|φc +r⟩} +(17.29) +such that +⟨φc +r|φs⟩ = δrs. +(17.30) +The completeness relations are +1 = +� +r +|φr⟩⟨φc +r| +(17.31) +� +HT , and +1 = +� +r +(−1)|φr| |φc +r⟩⟨φr| +(17.32) +on � +HT . +Finally, the operator G is defined as: +G = +� +1 +NS sector, +X0 +R sector. +(17.33) +Note the following properties +[G, L± +0 ] = [G, b± +0 ] = [G, QB] = 0. +(17.34) +It will be appear in the propagator and kinetic term of the superstring field theory. +17.2 +Off-shell superstring amplitudes +In this section, we are going to build the scattering amplitudes. +The procedure is very +similar to the bosonic case, except for the PCO insertions and of the Ramond sector. For +this reason, we will simply state the result and motivate the modifications with respect to +the bosonic case. +3This is not a problem on-shell since the BRST cohomology is independent of the picture number. +However, this matters off-shell since such states would propagate in loops and make the theory inconsistent. +249 + +17.2.1 +Amplitudes +External states can be either NS or R: the Riemann surface corresponding to the g-loop +scattering of m external NS states and n external R states is denoted by Σg,m,n. R states +must come in pairs because they correspond to fermions. As in the bosonic case, the amp- +litude is written as the integration of an appropriate p-form Ω(g,m,n) +p +over the moduli space +Mg,m,n (or, more precisely, of a section of a fibre bundle with this moduli space as a basis). +From the geometric point of view, nothing distinguishes the punctures and thus: +Mg,m,n := dim Mg,m,n = 6g − 6 + 2m + 2n. +(17.35) +The form ΩMg,m,n is defined as a SCFT correlation function of the physical vertex operators +together with ghost and PCO insertions. +Remark 17.2 A simple way to avoid making errors with signs is to multiply every Grass- +mann odd external state with a Grassmann odd number. These can be removed at the end +to read the sign. +The two conditions from the U(1) anomalies on the scattering amplitude are: +Ngh = 6 − 6g, +Npic = 2g − 2. +(17.36) +Given an amplitude with m NS states V NS +i +∈ H−1 and n R states V R +j +∈ H−1/2, the above +picture number can be reached by introducing a certain number of PCO X(yA): +npco := 2g − 2 + m + n +2 . +(17.37) +These PCO are inserted at various positions: while the amplitude does not depend on these +locations on-shell, off-shell it will (because the vertex operators are not BRST invariant). +The choices of PCO locations are arbitrary except for several consistency conditions: +1. avoid spurious poles (Section 17.2.3); +2. consistent with factorization (each component of the surface in the degeneration limits +must saturate the picture number condition). +This parallels the discussion of the choices of local coordinates: as a consequence, the +natural object is a fibre bundle �Pg,m,n with the local coordinate choices (up to global phase +rotations) and the PCO locations as fibre, and the moduli space Mg,m,n as base. Forgetting +about the PCO locations leads to a fibre bundle �Pg,m,n which is a generalization of the +one found in the bosonic case. The coordinate system of the fibre bundle presented in the +bosonic case is extended by including the PCO locations {yA}. +With these information, the amplitude can be written as: +Ag,m,n(V NS +i +, V R +j ) = +� +Sg,m,n +ΩMg,m,n(V NS +i +, V R +j ), +(17.38a) +where +ΩMg,m,n = (−2πi)−Mc +g,m,n +�Mg,m,n +� +λ=1 +Bλ dtλ +npco +� +A=1 +X(yA) +m +� +i=1 +V NS +i +n +� +j=1 +V R +j +� +Σg,n +. +(17.38b) +where Sg,m,n is a Mg,m,n-dimensional section of �Pg,m,n parametrized by coordinates tλ. The +1-form B corresponds to a generalization of the bosonic 1-form. +It has ghost number 1 +250 + +and includes a correction to compensate the variation of the PCO locations in terms of the +moduli parameters: +Bλ = +� +α +� +Cα +dσα +2πi b(σα) ∂Fα +∂tλ +� +F −1 +α (σα) +� ++ +� +α +� +Cα +d¯σα +2πi +¯b(¯σα) ∂ ¯Fα +∂tλ +� ¯F −1 +α (¯σα) +� +− +� +A +1 +X(yA) +∂yA +∂tλ +∂ξ(yA). +(17.39) +The last factor amounts to consider the combination +X(yA) − ∂ξ(yA) dyA +(17.40) +for each PCO insertion:4 the correction is necessary to ensure that the BRST identity +(13.46) holds. This can be understood as follows: the derivative acting on the PCO gives a +term dX(z) = ∂X(z)dz which must be cancelled. This is achieved by the second term since +{QB, ∂ξ(z)} = ∂X(z). +Remark 17.3 While it is sufficient to work with Mg,n for on-shell bosonic amplitudes, +on-shell superstring amplitudes are naturally expressed in �Pg,m,n (with the local coordinate +removed) since the positions of the PCO must be specified even on-shell. +Remark 17.4 (Amplitudes on the supermoduli space) Following Polyakov’s appro- +ach from Chapters 2 and 3 to the superstring would lead to replace the moduli space by the +supermoduli space. The latter includes Grassmann-odd moduli parameters in addition to the +moduli parameters from Mg,m+n (in the same way the superspace includes odd coordinates +θ along with spacetime coordinates x). The natural question is whether it is possible to split +the integration over the even and odd moduli, and to integrate over the latter such that only +an integral over Mg,m+n remains. In view of (17.38a), the answer seems positive. However, +this is incorrect: it was proven in [58] that there is no global holomorphic projection of the +supermoduli space to the moduli space. +This is related to the problem of spurious poles +described below. But, this does not prevent to do it locally: in that case, implementing the +procedure carefully should give the rules of vertical integration [73, 215, 230]. +17.2.2 +Factorization +The plumbing fixture of two Riemann surfaces Σg1,m1,n1 and Σg2,m2,n2 can be performed in +two different ways since two NS or two R punctures can be glued. +If two NS punctures are glued, the resulting Riemann surface is Σ(NS) +g1+g2,m1+m2−2,n1+n2. +The number of PCO inherited from the two original surfaces is +n(1) +pco + n(2) +pco = 2(g1 + g2) − 2 + (m1 + m2 − 2) + n1 + n2 +2 += n(NS) +pco , +(17.41) +which is the required number for a non-vanishing amplitude. As a consequence, the propag- +ator is the same as in the bosonic case: +∆NS = b+ +0 b− +0 +1 +L0 + ¯L0 +δ(L− +0 ). +(17.42) +If two R punctures are glued, the numbers of PCO do not match by one unit: +n(1) +pco + n(2) +pco = 2(g1 + g2) − 2 + (m1 + m2) + n1 + n2 − 2 +2 +− 1 = n(R) +pco − 1. +(17.43) +4The sum is formal since it is composed of 0- and 1-forms. +251 + +This means that an additional PCO must be inserted in the plumbing fixture procedure: +the natural place for it is in the propagator since this is the only way to keep both vertices +symmetric as required for a field theory interpretation. Another way to see the need of +this modification is to study the propagator (17.42) for Ramond states: since Ramond +states carry a picture number −1/2, the conjugate states have Npic = −3/2 and thus the +propagator has a total picture number −3 instead of −2 (the propagator graph is equivalent +to a sphere). Then, to avoid localizing the PCO at a point of the propagator, one inserts +the zero-mode which corresponds to smear the PCO: +∆R = b+ +0 b− +0 +X0 +L0 + ¯L0 +δ(L− +0 ). +(17.44) +Delocalizing the PCO amounts to average the amplitude over an infinite number of points +(i.e. to consider a generalized section): this is necessary to preserve the L− +0 eigenvalue since +X0 is rotationally invariant while X(z) is not. +Note that the zero-mode can be written +equivalently as a contour integral around one of the two glued punctures: +X0 = +1 +2πi +� dw(1) +n +w(1) +n +X +� +w(1) +n +� += +1 +2πi +� dw(2) +n +w(2) +n +X +� +w(2) +n +� +. +(17.45) +The equality of both expressions holds because X(z) has conformal weight 0. +Using the operator G (17.33), the propagator can be written generically as +∆ = b+ +0 b− +0 +G +L0 + ¯L0 +δ(L− +0 ). +(17.46) +Remark 17.5 (Propagators) NS and R states correspond respectively to bosonic and fer- +mionic fields: the operators L+ +0 and X0 can be interpreted as the (massive) Laplacian and +Dirac operators, such that both propagators can be written +∆NS ∼ +1 +k2 + m2 , +∆R ∼ i/∂ + m +k2 + m2 . +(17.47) +To motivate the identification of X0 with the Dirac operator, remember that X(z) contains +a term eφ(z)G(z) (this is the only term which contributes on-shell), where G(z) in turn +contains ψµ∂Xµ. But, the zero-modes of ψµ and ∂Xµ correspond respectively to the gamma +matrix γµ and momentum kµ when acting on a state. +The PCO zero-mode insertion inside the propagator has another virtue. It was noted +previously that states with Npic = −3/2 are infinitely degenerate since one can apply β0 +an arbitrary number of time. These states have large negative ghost numbers. Considering +a loop amplitude, all these states would appear in the sum over the states and lead to a +divergence. The problem is present only for loops because the ghost number is not fixed: in +a tree propagator, the ghost number is fixed and only a finite number of β0 can be applied. +But, the PCO insertion turns these states into Npic = −1/2 states. In this picture number, +one cannot create an arbitrarily large negative ghost number since γn +0 can only increase the +ghost number. +17.2.3 +Spurious poles +A spurious pole corresponds to a singularity of the amplitude which cannot be interpreted +as the degeneration limit of Riemann surfaces. As a consequence, they do not correspond to +infrared divergences and don’t have any physical meaning; they must be avoided in order to +define a consistent theory. To achieve this, the section Sg,m,n must be chosen such that it +252 + +avoids all spurious poles. However, while it is always possible to avoid these poles locally, it +is not possible globally (this is related to the results from [58]). Poles can be avoided using +vertical integration: two methods have been proposed, in the small (Sen–Witten) [215, 230] +and large (Erler–Konopka) [73] Hilbert spaces respectively. Before describing the essence of +both approaches, we review the origin of spurious poles. +Origin +Spurious poles arise in three different ways: +• two PCOs collide; +• one PCO and one matter vertex collide; +• other singularities of the correlation functions. +The last source is the less intuitive one and we focus on it. +A general correlation function of (η, ξ, φ) on the torus5 (satisfying the ghost number +condition) reads +C(xi, yj, zq) = +�n+1 +� +i=1 +ξ(xi) +n +� +j=1 +η(yj) +m +� +k=1 +eqkφ(zk) +� += +n� +j′=1 +ϑδ +� +− yj′ + � +i +xi − � +j +yj + � +k +qkzk +� +n+1 +� +i′=1 +ϑδ +� +− xi′ + � +i +xi − � +j +yj + � +k +qkzk +� × +� +i 0 is a number. There is an implicit sum over the momentum +indices. +The terms quadratic in the momenta inside the exponential arise from two sources: +• The correlation functions of the vertex operators ⟨� +i eiki·X(zi)⟩ is proportional to +e−ki·kjG(zi,zj), where G is the Green function. +Additional factors like ∂X contrib- +ute to the polynomial Pα1,...,αn. +• It is possible to add stubs to the vertices. The effect is to multiply each leg by a factor +e−λ(k2 +i +m2 +i ) with λ > 0 (we take λ to be the same for all vertices for simplicity). The +first term of the exponential contributes to the diagonal of the matrix gij. By taking +λ sufficiently large, one can enforce that all eigenvalues are positive. +Finally, the exponential term with the masses m2 +α ensures that the sum over all intermediate +states converge despite an infinite number of states. Indeed, the number of states of mass +261 + +mα grows as ecmα, which is dominated by e−λm2 +α for sufficiently large λ. Hence, the addition +of stubs make explicit the absence of divergences in SFT.1 +The vertices have no singularity for ki ∈ C finite. As the energy becomes infinite |k0 +i | → +∞, they behave as: +lim +k0→±i∞ V (n) = 0, +lim +k0→±∞ V (n) = ∞. +(18.5) +The first property is responsible for the soft UV behaviour of string theory in Euclidean +signature, while the second prevents from performing the Wick rotation (indeed, the pole +at infinity implies that the arcs closing the contour contribute). +The g-loop n-point amputated Green functions are sums of Feynman diagrams, each of +the form: +Fg,n(p1, . . . , pn) ∼ +� +dT +� +s +dDℓs e−Grs(T ) ℓr·ℓs−2Hri(T ) ℓr·pi−Fij(T ) pi·pj +× +� +a +1 +k2a + m2a +P(pi, ℓr; T), +(18.6) +where {pi} are the external momenta, {ℓr} the loop momenta and {ki} the internal mo- +menta, with the latter given by a linear combination of the other. Moreover, T denotes the +dependence in the moduli parameters of all the internal vertices, and P is a polynomial in +(pi, ℓr). The matrix Grs is positive definite, which implies that: +• integrations over spatial loop momenta ℓr converge; +• integrations over loop energies ℓ0 +r diverge. +As a consequence, the Feynman diagrams in Lorentzian signature are ill-defined: we will +explain in the next section how to fix this problem. +18.2 +Generalized Wick rotation +We have seen that loop integrals in Lorentzian signature are divergent because of the large +energy behaviour of the interactions. But, this is not different from the usual QFT, where the +loop integrals are also ill-defined in Lorentzian signature. Indeed, poles of the propagators +sit on the real axis and also give divergent loop integrals (note that the same problem arise +also here). In that case, the strategy is to define the Feynman diagrams in Euclidean space +and to perform a Wick rotation: the latter matches the expressions in Lorentzian signature +up to the iε-prescription. The goal of the latter is to move slightly the poles away from the +real axis. +Example 18.1 – Scalar field +Consider a scalar field of mass m with a quartic interaction. The 1-loop 4-point Feyn- +man diagram is given in Figure 18.1. The external momenta are pi, i = 1, . . . , 4. There +are one loop momentum ℓ and two internal momenta k1 = ℓ and k2 = p − ℓ, where +p = p1 + p2. The poles in the loop energy ℓ0 are located at: +p± = ± +� +ℓ2 + m2, +q± = p0 ± +� +(p − ℓ)2 + m2. +(18.7) +The graph is first defined in Euclidean signature, where the external and loop en- +ergies are pure imaginary, p0 +i , ℓ0 ∈ iR. The poles are shown in Figure 18.2. Then, +the external momenta are analytically continued to real values, p0 +i ∈ R. At the same +1Remember that λ is not a physical parameter and disappears on-shell. This means that the cancellation +of the divergences is independent of λ and must always happen on-shell. +262 + +time, the integration contour is also analytically continued thanks to the Wick rotation +(Figure 18.3). The contour is closed with arcs, but they don’t contribute since there is +no poles in the upper-right and lower-left quadrants, and no poles at infinity. However, +one cannot continue the contour such that ℓ0 ∈ R because of the poles on the real axis. +The Wick rotation is possible for ℓ0 in the upper-right quadrant, Re ℓ0 ≥ 0, Im ℓ0 > 0, +which leads to the iε-prescription ℓ0 ∈ R + iε. +Figure 18.1: 1-loop 4-point function for a scalar field theory. +Figure 18.2: Integration contour for external Euclidean momenta. +Since the Feynman diagram (18.6) is not defined in Lorentzian signature because of the +poles at ℓ0 +r → ±∞, it is also necessary to start with Euclidean momenta. However, the +same behaviour at infinity prevents from using the Wick rotation since the contribution +from the arcs does not vanish. It is then necessary to find another prescription for defining +the Feynman diagrams in SFT starting from the Euclidean Green functions. This is given +by the following generalized Wick rotation (Pius–Sen [187]): +1. Define the Green functions for Euclidean internal and external momenta. +2. Perform an analytic continuation of the external energies and of the integration contour +such that: +• keep poles on the same side; +• keep the contour ends fixed at ±i∞. +One can show [187] that the Green functions are analytic in the upper-right quadrant Im p0 +a > +0, Re p0 +a ≥ 0, for pa ∈ R, p0 +a. Moreover, the result is independent of the contour chosen as +long as it satisfies the conditions described above. In fact, this generalized Wick rotation is +263 + +Figure 18.3: Integration contour for external Lorentzian momenta after Wick rotation (reg- +ular vertices). +valid even for normal QFT, which raises interesting questions. For example, it seems that +the internal and external set of states have no intersection, which can be puzzling when +trying to interpret the Cutkosky rules. Nonetheless, everything works as expected. +Remark 18.1 (Timelike Liouville theory) It has been shown in [13] that this general- +ized Wick rotation is also the correct way for defining the timelike Liouville theory. +The fact that the amplitude is analytic only when the imaginary parts of the momenta +are not zero, Im p0 +a > 0, is equivalent to the usual iε-prescription for QFT. Moreover, it has +been shown [222] to be equivalent to the moduli space iε-prescription from [257]. Then, it +has also been used to prove several important properties of string theory shared by local +QFTs: Cutkosky rules [187, 188], unitarity [220, 221], analyticity in a subset of the primitive +domain and crossing symmetry [43]. Finally, general soft theorems for string theory (and, in +fact, any theory of quantum gravity) have been proven in [34, 150, 223, 224]. All together, +these properties establish string theory as a very strong candidate for a consistent theory +of everything. +The next main question is how to obtain an expression of SFT which is +amenable to explicit computations. This will certainly require to understand even better +the deep structure of SFT, a goal which this book will hopefully help the reader to achieve. +18.3 +Suggested readings +• SFT momentum space action [42, 187, 225]. +• Consistency properties of string theory [42]: +– generalized Wick rotation, Cutkosky rules and unitarity [187, 188, 219–222]. +– analyticity and crossing symmetry [43]. +– soft theorems [34, 150, 223, 224]. +264 + +(a) +(b) +(c) +Figure 18.4: Integration contour after analytic continuation to external Lorentzian momenta. +Depending on the values of the external momenta, different cases can happen. +265 + +Appendix A +Conventions +Most of the book uses natural units where c = ℏ = 1, but the string length ℓs (or Regge +slope α′) are kept. +A bar is used to denote both complex conjugation and the anti-holomorphic operators. +The symbol := (resp. =:) means that the LHS (RHS) is defined by the expression in the +RHS (LHS). +A.1 +Coordinates +The number of spacetime (target-space) dimensions is denoted by D = d + 1, where d is +the number of spatial dimensions. The corresponding spacetime and spatial coordinates are +written with Greek and Latin indices: +xµ = (x0, xi), +µ = 0, . . . , D − 1 = d +i = 1, . . . , d +(A.1) +When time is singled out, one writes x0 = t in Lorentzian signature and x0 = tE in Euclidean +signature (or x0 = τ when there is no ambiguity with the worldsheet time). +A p-brane is a (p+1)-dimensional object whose worldvolume is parametrized by coordin- +ates: +σa = (σ0, σα), +a = 0, . . . , p − 1, +α = 1, . . . , p. +(A.2) +The time coordinate can also be singled out as σ0 = τM in Lorentzian signature and as +σ0 = τ in Euclidean signature. For the string, the index α is omitted since it takes only one +value. +The Lorentzian signature is taken to be mostly plus and the flat Minkowski metric reads +ηµν = diag(−1, 1, . . . , 1 +� �� � +d +). +(A.3) +The flat Euclidean metric is +δµν = diag(1, . . . , 1 +� �� � +D +). +(A.4) +Similar notations hold for the worldvolume metrics ηab and δab. The Levi–Civita (completely +antisymmetric) tensor is normalized by +ϵ01 = −ϵ01 = 1. +(A.5) +Wick rotation from Lorentzian time t to Euclidean time τ (either worldsheet or target +spacetime) is defined by +t = −iτ. +(A.6) +266 + +Accordingly, contravariant (covariant) vector transforms with the same (opposite) factor: +V 0 +M = −iV 0 +E, +VM,0 = iVE,0. +(A.7) +Most computations are performed with both spacetime and worldsheet Euclidean signatures. +Expressions are Wick rotated when needed. +Light-cone coordinates are defined by +x± = x0 ± x1. +(A.8) +A function depending only on x+ (x−) is said to be left-moving (right-moving) by ana- +logy with the displacement of a wave. Under analytic continuation, the left-moving (right- +moving) coordinate is mapped to the holomorphic1 (anti-holomorphic) coordinate z (¯z). In +chiral theories, the left-moving value is written first. +The worldsheet coordinates (τ, σ) on the cylinder are defined by +τ ∈ R, +σ ∈ [0, L), +σ ∼ σ + L, +(A.9) +where typically L = 2π. The integration over the spatial coordinate is normalized such that +the perimeter of spatial slice is normalized to 1 if L = 2π: +L = 1 +2π +� L +0 +dσ = L +2π . +(A.10) +This implies that 2d action, conserved charges, etc. are divided by an extra factor of 2π. +The coordinates can be written in terms of complex coordinates +w = τ + iσ, +¯w = τ − iσ +(A.11) +such that the flat metric is +ds2 = dτ 2 + dσ2 = dwd ¯w. +(A.12) +Under Wick rotation, the complex coordinates are mapped to light-cone coordinates as +follows: +w = iσ+, +¯w = iσ−. +(A.13) +The cylinder can be mapped to the complex plane through +z = e2πw/L, +¯z = e2π ¯ +w/L. +(A.14) +The definition of the Levi–Civita tensor includes the √g factor, such that +ϵz¯z = i +2, +ϵz¯z = −2i +(A.15) +on the complex plane with flat metric. +1The terms of holomorphic is simply used to indicate that the object depends only on z, but not on ¯z. +Typically, the objects have singularities and are really meromorphic in z. +2In fact, the terms of “left”- and “right”-moving are interchanged in [193, p. 34] to get agreement with +the literature. But, it means that the spatial axis orientation is reversed. +Moreover, concerning [54], the first definition agrees with (6.1) but not with (6.53) since the definition of +ξ (our w) is modified in-between. This explains why the definitions of left- and right-moving [54, p. 161] do +not agree with the one given in the table. +267 + +refs +cylinder +plane +light-cone +left-moving +here, Di Francesco et al. +[25, 54, 124, 126, 128, 212, 252, 265] +w = τ + iσ +z = ew +w = iσ+, ¯w = iσ− +holomorphic +Blumenhagen et al. +[14, 24, 123, 205] +w = τ − iσ +z = ew +w = iσ−, ¯w = iσ+ +anti-holomorphic +Polchinski [176, 193, 246] +w = σ + iτ +z = e−iw +w = −σ−, ¯w = σ+ +anti-holomorphic2 +Table A.1: Conventions for the coordinates.The notations are the following (they can slightly +vary depending on the references): the Euclidean time is obtained by the analytic continu- +ation τ = it (denoted also by τ = σ0 = σ2) the spatial direction is σ = σ1, and the light-cone +coordinates are σ± = t ± σ. +A.2 +Operators +Commutators and anti-commutators are denoted by +[A, B] := [A, B]− = AB − BA, +{A, B} := [A, B]+ = AB + BA. +(A.16) +The Grassmann parity of a field A is denoted by |A| +|A| = +� ++1 +Grassmann odd, +0 +Grassmann even. +(A.17) +Two (anti-)commuting operators satisfy +AB = (−1)|A||B|BA. +(A.18) +A.3 +QFT +Energy is defined as the first component of the momentum vector +pµ := (E, pi). +(A.19) +The following notations are used to denote the number of supersymmetries: +(NL, NR), +N = NL + NR, +(A.20) +where NL and NR are the numbers of left- and right-chirality supersymmetries. The last +form is used when it is not important to know the chirality of the supercharges. +The variation of a field φ(x) is defined by +δφ(x) = φ′(x) − φ(x). +(A.21) +Given an internal symmetry with parameters αa, the Noether current in Lorentzian signature +is given by: +Jµ +a = λ +∂L +∂(∂µφ) +δφ +δαa , +∇µJµ +a = 0, +(A.22) +where L is the Lagrangian which does not include the factor √g for curved spaces and λ is +some normalization.3 The conserved charges Qa associated to the currents Jµ +a for a fixed +spatial slice t = cst are +Qa = 1 +λ +� +Σ +dD−1x +√ +h J0 +a, +(A.23) +3Including the √g would give the current density √gJµ +a . The simple derivative of the latter vanishes +∂µ(√gJµ +a ) = 0 in view of the identity (B.4). +268 + +where Σ is a spatial slice and h is the induced metric. One sets λ = 2π in two dimensions, +otherwise λ = 1. The variation of a field under a transformation generated by Q is +δαaφ(x) = iαa[Qa, φ(x)] +(A.24) +In Euclidean signature, the current and variation are: +Jµ +a = iλ +∂L +∂(∂µφ) +δφ +δαa , +(A.25a) +δαaφ(x) = −αa[Qa, φ(x)]. +(A.25b) +Note that the charge is still given by (A.23). The factor of i in (A.25a) can be understood +as follows.4 +First, the time component J0 +a of the current transforms like time such that +J0 +a → iJ0 +a, which implies that the charge also gets a factor i, Qa → iQa. This explains the +minus sign in (A.25b). Then, one needs to make this consistent with the formula (B.9) for +the charge associated to a general surface. Given a spacelike nµ, the integration measure +includes the time which transforms with a factor of i: one can interpret it as coming from the +spatial components of the current, Ji +a → iJi +a, while working with a Euclidean region. Another +way to understand this factor for the spatial vector is by considering the electromagnetic +case, where J contains a time derivative. +The term “zero-mode” has two (related) meanings: +1. given an operator D acting on a space of fields ψ(z), zero-modes ψ0,i(z) of the operator +are all fields with zero eigenvalue Dψ0,i(z) = 0, i = 1, . . . , dim ker D +2. the zero-mode of a field expansion ψ = � +n ψnz−n−h is the mode ψ0 for n = 0: on the +cylinder, it corresponds to the constant term of the Fourier expansion on the cylinder +(hence, a zero-mode of ∂z according to the previous definition) +A prime indicates that the zero-modes are excluded. For example, det′ D is the product of +non-zero eigenvalues, φ′ is a field without zero-mode and d′φ the corresponding integration +measure. +A.4 +Curved space and gravity +The covariant derivative is defined by +∇µ = ∂µ + Γµ +(A.26) +where Γµ is the connection. For example, one has for a vector field +∇µAν = ∂µAν + Γ +ν +µρ Aρ. +(A.27) +The negative-definite Laplacian (or Laplace–Beltrami operator) is defined by +∆ = gµν∇µ∇ν = +1 +√g ∇µ +�√ggµν∇ν). +(A.28) +Note that ∇µ does not contain the Christoffel symbol for the index ν because of the identity +(B.4) (but it contains a connection for any other index of the field). For a scalar field, both +derivatives become simple derivatives. +The energy–momentum tensor is defined by +Tµν = − 2λ +√g +δS +δgµν , +(A.29) +where λ = 2π for D = 2 and λ = 1 otherwise. +4We stress that these formulas and arguments do not apply to the energy–momentum tensor. +269 + +A.5 +List of symbols +General: +• D: number of non-compact spacetime dimensions +• g: loop order for a scattering amplitude +• n: number of external closed string states +• xµ: spacetime non-compact coordinates +• σa = (t, σ): worldsheet coordinates +• gs: closed string coupling +• Zg = Ag,0: genus-g vacuum amplitude +• Ag,n(k1, . . . , kn)α1,...,αn := Ag,n({ki}){αi}: g-loop n-point scattering amplitude for +states with quantum numbers {ki, αi} (if connected, amputated Green functions for +n ≥ 3) +• Gg,n(k1, . . . , kn)α1,...,αn: g-loop n-point Green function for states with quantum num- +bers {ki, αi} +• T ⊥ +ab: traceless symmetric tensor or traceless component of the tensor Tab +• Ψ: generic (set of) matter field(s) +Hilbert spaces: +• H: generic Hilbert space (in general, Hilbert space of the matter plus ghost CFT) +• H± = H ∩ ker b± +0 +• H0 = H ∩ ker b− +0 ∩ ker b+ +0 +• H(QB): absolute cohomology of the operator QB inside the space H +• H−(QB) = H(QB) ∩ H−: semi-relative cohomology of the operator QB inside the +space H +• H0(QB) = H(QB) ∩ H0: relative cohomology of the operator QB inside the space H +• A: Grassmann parity of the operator or state A +Riemann surfaces: +• g: Riemann surface genus (number of holes / handles) +• n: number of bulk punctures / marked points +• Σg,n: genus-g Riemann surface with n punctures +• Σg = Σg,0: genus-g Riemann surface +• Mg,n: moduli space of genus-g Riemann surfaces with n punctures +• Mg = Mg,0: moduli space of genus-g Riemann surfaces +• Mg,n = dim Mg,n +270 + +• Mc +g,n = dimC Mg,n +• Mg = Mg,0 = dim Mg = dim ker P † +1 +• Mc +g = Mc +g,0 = dimC Mg +• Kg,n: conformal Killing vector group of genus-g Riemann surfaces with n punctures +• Kg = Kg,0 = ker P1: conformal Killing vector group of genus-g Riemann surfaces +• Kg,n = dim Kg,n +• Kc +g,n = dimC Kg,n = dimC ker P1 +• Kg = Kg,0 = dim ker P1 +• Kc +gKc +g,0 = dimC ker P1 +• ψi: real basis of ker P1, CKV +• φi: real basis of ker P † +1 , real quadratic differentials +• (ψK, ¯ψK): complex basis of ker P1, (anti-)holomorphic CKV +• (φI, ¯φI): complex basis of ker P † +1 , (anti-)holomorphic quadratic differentials +• tλ ∈ Mg,n: real moduli of Mg,n +• mΛ ∈ Mg,n: complex moduli of Mg,n +• ti ∈ Mg: real moduli of Mg +• mI ∈ Mg: complex moduli of Mg +• z: coordinate on the Riemann surface +• wi: local coordinates around punctures +• za: local coordinates away from punctures +• fi(wi): transition functions from wi to z +• σα: coordinate system on the left of the contour Cα +• τα: coordinate system on the right of the contour Cα +CFT: +• Vα(k; σa) := Vk,α(σa): matter vertex operator with5 momentum k and quantum num- +bers α inserted at position σa = (z, ¯z) +• Vα(k; σa): unintegrated vertex operator with momentum k and quantum numbers α +inserted at position σa +• Vα(k) = +� +d2σ√g Vα(k; σ): integrated vertex operator +• on-shell (closed bosonic string): Vα(k; σa) = c¯cVα(k; σa) is a (0, 0)-primary, with +Vα(k; σa) a (1, 1)-primary matter operator +5When the momentum and/or quantum numbers are not relevant, we remove them or simply index the +operators by a number. +271 + +• � +O: operator O with zero-modes removed +• O†: Hermitian adjoint +• O‡: Euclidean adjoint +• Ot: BPZ conjugation +• ⟨O1|O2⟩: BPZ inner-product +• ⟨O‡ +1|O2⟩: Hermitian inner-product +• |0⟩: SL(2, C) (conformal) vacuum +• |Ω⟩: energy vacuum (lowest energy state) +• :O: : conformal normal ordering (with respect to SL(2, C) vacuum |0⟩) +• +⋆ +⋆O +⋆ +⋆ : energy normal ordering (with respect to energy vacuum |Ω⟩) +SFT: +• Ψ: closed string field +• {φr} = {φα(k)}: basis of H (or some subspace) +Indices: +• µ = 0, . . . , D − 1: non-compact spacetime dimensions +• a = 0, . . . , p: worldvolume coordinates (p = 1: worldsheet) +• i = 1, . . . , n: external states, local coordinates +• λ = 1, . . . , Mg,n: real moduli of Mg,n +• Λ = 1, . . . , Mc +g,n: complex moduli of Mg,n +• i = 1, . . . , Mg: real moduli of Mg +• I = 1, . . . , Mc +g: complex moduli of Mg +• i = 1, . . . , Kg: real CKV of Kg +• K = 1, . . . , Kc +g: complex CKV of Kg +• r = (k, α): index for basis state of H (or some subspaces), α: non-momentum indices +272 + +Appendix B +Summary of important formulas +This appendix summarizes formulas which appear in the book or which are needed but +assumed to be known to the reader (such as formulas from QFT and general relativity). +B.1 +Complex analysis +The Cauchy–Riemann formula is +� +Cz +dw +2πi +f(w) +(w − z)n = f (n−1)(z) +(n − 1)! , +(B.1) +where f(z) is an holomorphic function. +One has +¯∂ 1 +z = 2π δ(2)(z). +(B.2) +B.2 +QFT, curved spaces and gravity +The Green function G of a differential operator D is defined by +DxG(x, y) = δ(x − y) +√g +− P(x, y), +(B.3) +where P is the projector on the zero-modes of D. +The covariant divergence of a vector can be rewritten in terms of a simple derivative: +∇µvµ = +1 +√g ∂µ(√gvµ). +(B.4) +Under an infinitesimal change of coordinates +δxµ = ξµ, +(B.5) +the metric transforms as +δgµν = Lξgµν = ∇µξν + ∇νξµ. +(B.6) +Stokes’ theorem reads +� +V +dDx ∇µvµ = +� +∂V +dΣµvµ, +dΣµ := ϵ nµ dD−1Σ, +(B.7) +273 + +where V is a spacetime region, S = ∂V its boundary and dD−1Σ the induced integration +measure. The vector nµ normal to S points outward and ϵ := nµnµ = 1 (−1) if S is timelike +(spacelike). If the surface is defined by x0 = cst, then +dD−1Σ = √g dD−1x, +nµ = δ0 +µ. +(B.8) +We can write a generalization of (A.23) for a charge associated to a general surface S: +QS = 1 +λ +� +S +dΣµ Jµ +a . +(B.9) +If the current Jµ +a is conserved, ∇µJµ +a = 0 (no source), Stokes’ theorem (B.7) shows that the +charge vanishes QS = 0 if S is a closed surface and that it is conserved QS1 = −QS2 for +two spacelike surfaces S1 and S2 extending to infinity (if Jµ +a vanishes at infinity) (see [190, +chap. 3, 265, sec. 8.4] for more details). +B.2.1 +Two dimensions +Stokes’ theorem (B.7) on flat space reads +� +d2x ∂µvµ = +� +ϵµν dxνvµ = +� +(v0dσ − v1dτ), +(B.10) +since dΣµ = ϵµνdxν. +The integral of the curvature is a topological invariant +χg;b := 1 +4π +� +d2σ√g R + 1 +2π +� +ds k += 2 − 2g − b, +(B.11) +called the Euler characteristics and where g is the number of holes and b the number of +boundaries. +B.3 +Conformal field theory +In two dimensions, the energy–momentum tensor is defined by +Tab = − 4π +√g +δS +δgab . +(B.12) +B.3.1 +Complex plane +Defining the real coordinates (x, y) from the complex coordinate on the complex plane +z = x + iy, +z = x − iy, +(B.13) +274 + +we have the formulas: +ds2 = dx2 + dy2 = dzd¯z, +gz¯z = 1 +2, +gzz = g¯z¯z = 0, +(B.14a) +ϵz¯z = i +2, +ϵz¯z = −2i, +(B.14b) +∂ := ∂z = 1 +2 (∂x − i∂y), +¯∂ := ∂¯z = 1 +2 (∂x + i∂y), +(B.14c) +V z = V x + iV y, +V ¯z = V x − iV y, +(B.14d) +d2x = dxdy = 1 +2 d2z, +d2z = dzd¯z, +(B.14e) +δ(z) = 1 +2 δ(2)(x), +1 = +� +d2z δ(2)(z) = +� +d2x δ(2)(x), +(B.14f) +� +R +d2z (∂zvz + ∂¯zv¯z) = −i +� +∂R +� +dz v¯z − d¯zvz� += −2i +� +∂R +(vzdz − v¯zd¯z). +(B.14g) +B.3.2 +General properties +A primary holomorphic field φ(z) of weight h transforms as +f ◦ φ(z) = +�df +dz +�h +φ +� +f(z) +� +(B.15) +for any local change of coordinates f. A quasi-primary operator transforms like this only +for f ∈ SL(2, C). Its mode expansion reads +φ(z) = +� +n +φn +zn+h , +φn = +� +C0 +dz +2πi zn+h−1φ(z), +(B.16) +where the integration is counter-clockwise around the origin. +The SL(2, C) vacuum |0⟩ is defined by +∀n ≥ −h + 1 : +φn |0⟩ = 0. +(B.17) +Its BPZ conjugate ⟨0| satisfies: +∀n ≤ h − 1 : +⟨0| φn = 0. +(B.18) +The state–operator correspondence associates a state |φ⟩ to each operator φ(z): +|φ⟩ := φ(0) |0⟩ = φ−h |0⟩ . +(B.19) +The operator corresponding to the vacuum is the identity 1.1 +The Hermitian and BPZ +conjugated states are +⟨φ‡| :=⟨0| I ◦ φ†(0) = lim +z→∞ z2h⟨0| φ†(z), +⟨φ| :=⟨0| I± ◦ φ(0) = (±1)h lim +z→∞ z2h⟨0| φ(z). +(B.20) +The energy–momentum tensor is a quasi-primary operator of weight h = 2 +T(z) = +� +n +Ln +zn+2 . +(B.21) +1Exceptionally, the state |0⟩ and the operator 1 does not have the same symbol. +275 + +The OPE between T and a primary operator h of weight h is +T(z)φ(w) ∼ +h φ(w) +(z − w)2 + ∂φ(w) +z − w . +(B.22) +The OPE of T with itself defines the central charge c +T(z)T(w) ∼ +c/2 +(z − w)4 + +2T(w) +(z − w)2 + ∂T(w) +z − w . +(B.23) +B.3.3 +Hermitian and BPZ conjugations +Both conjugations do not change the ghost number of a state. +Hermitian +The Hermitian conjugate of a general state built from n operators Ai and a complex number +λ is +(λ A1 · · · An |0⟩)† = λ∗ ⟨0| A† +n · · · A† +1. +(B.24) +BPZ +The BPZ conjugate of modes is +φt +n = (I± ◦ φ)n = (−1)h(±1)nφ−n, +(B.25) +where I±(z) = ±1/z. The plus sign is usually used for the closed string, and the minus sign +for the open string. Given a general state built from n operators n and a complex number λ, +the conjugation does not change the order of the operators and does not conjugate complex +numbers: +(λ A1 · · · An |0⟩)t = λ ⟨0| (A1)t · · · (An)t. +(B.26) +However, it reverses radial ordering such that operators must be (anti-)commuted in radial +ordered expressions. +The BPZ product satisfies +⟨A, B⟩ = (−1)|A||B|⟨B, A⟩. +(B.27) +Moreover the inner product is non-degenerate, so +∀A : +⟨A|B⟩ = 0 +=⇒ +|B⟩ = 0. +(B.28) +Denoting by {|φr⟩} a complete basis of states, then the conjugate basis {⟨φc +r|} is defined +by the BPZ product as +⟨φc +r|φs⟩ = δrs. +(B.29) +We have +⟨φr|φc +s⟩ = (−1)|φr|δrs. +(B.30) +B.3.4 +Scalar field +The simplest matter CFT is a set of D scalar field Xµ(z, ¯z) such that the i∂Xµ and i¯∂Xµ +are of weight h = (1, 0) and h = (0, 1) +i∂Xµ = +� +n +αµ +n +zn+1 , +i¯∂Xµ = +� +n +¯αµ +n +¯zn+1 . +(B.31) +276 + +The commutation relations between the modes are: +[αµ +m, αν +n] = mδm+n,0ηµν, +[¯αµ +m, ¯αν +n] = mδm+n,0ηµν, +[αµ +m, ¯αν +n] = 0. +(B.32) +The zero-modes of both operators are equal and correspond to the (centre-of-mass) mo- +mentum +αµ +0 = ¯αµ +0 = +� +α′ +2 pµ. +(B.33) +The conjugate of pµ is the centre-of-mass position xµ: +[xµ, pν] = ηµν. +(B.34) +Vertex operators are defined by +Vk(z, ¯z) = :eik·X(z,¯z):, +h = ¯h = α′2k2 +4 +. +(B.35) +The scalar vacuum |k⟩ is annihilated by all positive-frequency oscillators and it is char- +acterized by its eigenvalue for the zero-mode operator +pµ |k⟩ = kµ |k⟩ , +∀n > 0 : +αµ +n |k⟩ = 0, +¯αµ +n |k⟩ = 0. +(B.36) +The vacuum is associated to the vertex operator Vk: +|k⟩ = Vk(0, 0) |0⟩ = eik·x |0⟩ . +(B.37) +The conjugate vacuum is +⟨k| pµ =⟨k| kµ, +⟨k| = |k⟩† , +⟨−k| = |k⟩t . +(B.38) +B.3.5 +Reparametrization ghosts +The reparametrization ghosts are described by an anti-commuting first-order system with +the parameters (Chapter 7 and table 7.1): +ϵ = 1, +λ = 2, +cgh = −26, +qgh = −3, +agh = −1. +(B.39) +We focus on the holomorphic sector. +The b and c ghosts have weights: +h(b) = 2, +h(c) = −1 +(B.40) +such that the mode expansions are: +b(z) = +� +n∈Z +bn +zn+2 , +c(z) = +� +n∈Z +cn +zn−1 , +(B.41a) +bn = +� +dz +2πi zn+1b(z), +cn = +� +dz +2πi zn−2c(z). +(B.41b) +The anti-commutators between the modes bn and cn read: +{bm, cn} = δm+n,0, +{bm, bn} = 0, +{cm, cn} = 0. +(B.42) +The energy–momentum tensor and the Virasoro modes are respectively: +T = −2 :b∂c: − :∂b c:, +(B.43a) +Lm = +� +n +� +n + m +� +:bm−ncn: = +� +n +(2m − n) :bncm−n:. +(B.43b) +277 + +The expression of the zero-mode is: +L0 = − +� +n +n :bnc−n: = +� +n +n :b−ncn:. +(B.44) +The commutators between the Ln and the ghost modes are: +[Lm, bn] = +� +m − n +� +bm+n, +[Lm, cn] = −(2m + n)cm+n. +(B.45) +In particular, L0 commutes with the zero-modes: +[L0, b0] = 0, +[L0, c0] = 0. +(B.46) +The anomalous global U(1) symmetry for the ghost number Ngh is generated by the +ghost current: +j = −:bc:, +Ngh,L = +� +dz +2πi j(z), +(B.47) +such that +Ngh(c) = 1, +Ngh(b) = −1. +(B.48) +Remember that Ngh = Ngh,L in the left sector, such that we omit the index L. The modes +of the ghost current are +jm = − +� +n +:bm−ncn: = − +� +n +:bncm−n:, +Ngh,L = j0 = − +� +n +:b−ncn:. +(B.49) +The commutator of the current modes with itself and with the Virasoro modes are: +[jm, jn] = m δm+n,0, +[Lm, jn] = −njm+n − 3 +2 m(m + 1)δm+n,0. +(B.50) +Finally, the commutators of the ghost number operator are: +[Ngh, b(w)] = −b(w), +[Ngh, c(w)] = c(w). +(B.51) +The level operators N b and N c and number operators N b +n and N c +n are defined as: +N b = +� +n>0 +n N b +n, +N c = +� +n>0 +n N c +n, +(B.52a) +N b +n = :b−ncn:, +N c +n = :c−nbn:. +(B.52b) +The commutator of the number operators with the modes are: +[N b +m, b−n] = b−nδm,n, +[N c +m, c−n] = c−nδm,n. +(B.53) +The OPE between the ghosts and different currents are: +c(z)b(w) ∼ +1 +z − w, +b(z)c(w) ∼ +1 +z − w, +b(z)b(w) ∼ 0, +c(z)c(w) ∼ 0, +(B.54a) +T(z)b(w) ∼ +2b(w) +(z − w)2 + ∂b(w) +z − w , +T(z)c(w) ∼ +−c(w) +(z − w)2 + ∂c(w) +z − w . +(B.54b) +j(z)b(w) ∼ − b(w) +z − w, +j(z)c(w) ∼ c(w) +z − w. +j(z)O(w) ∼ Ngh(O) O(w) +z − w, +(B.54c) +j(z)j(w) ∼ +1 +(z − w)2 . +(B.54d) +T(z)j(w) ∼ +−3 +(z − w)3 + +j(w) +(z − w)2 + ∂j(w) +z − w . +(B.54e) +278 + +any operator O(z) is defined by +The OPE (B.54e) implies that the ghost number is not conserved on a curved space: +N c − N b = 3 − 3g, +(B.55) +and leads to a shift between the ghost numbers on the plane and on the cylinder: +Ngh,L = N cyl +gh,L + 3 +2. +(B.56) +The SL(2, C) vacuum |0⟩ is defined by +∀n > −2 : +bn |0⟩ = 0, +∀n > 1 : +cn |0⟩ = 0. +(B.57) +The mode c1 does not annihilate the vacuum and the two degenerate energy vacua are: +| ↓⟩ := c1 |0⟩ , +| ↑⟩ := c0c1 |0⟩ . +(B.58) +The zero-point energy of these states is: +L0 | ↓⟩ = agh | ↓⟩ , +L0 | ↑⟩ = agh | ↑⟩ , +agh = −1. +(B.59) +The energy for the normal ordering of the different currents is: +Lm = +� +n +� +n − (1 − λ)m +� ⋆ +⋆bm−ncn +⋆ +⋆ + agh δm,0, +(B.60a) +jm = +� +n +⋆ +⋆bm−ncn +⋆ +⋆ + δm,0. +(B.60b) +The energy–momentum and ghost current zero-modes are explicitly: +L0 = +� +n +n +⋆ +⋆b−ncn +⋆ +⋆ + agh = �L0 − 1, +(B.61a) +Ngh,L = j0 = +� +n +⋆ +⋆b−ncn +⋆ +⋆ + 1 = � +Ngh,L + 1 +2 +� +N c +0 − N b +0 +� +− 3 +2, +(B.61b) +�L0 = N b + N c, +� +Ngh,L := +� +n>0 +� +N c +n − N b +n +� +. +(B.61c) +Then, one can straightforwardly compute the ghost number of the vacua: +Ngh |0⟩ = 0, +Ngh | ↓⟩ = | ↓⟩ , +Ngh | ↑⟩ = 2 | ↑⟩ . +(B.62) +Using (B.56) allows to write the ghost numbers on the cylinder: +N cyl +gh | ↓⟩ = −1 +2 | ↓⟩ , +N cyl +gh | ↑⟩ = 1 +2 | ↑⟩ . +(B.63) +The bn and cn are Hermitian: +b† +n = b−n, +c† +n = c−n. +(B.64) +The BPZ conjugates of the modes are: +bt +n = (±1)nb−n, +ct +n = −(±1)nc−n, +(B.65) +using I±(z) with (6.111). +279 + +The conjugates of the vacuum read: +| ↓⟩‡ =⟨0| c−1, +| ↑⟩‡ =⟨0| c−1c0. +(B.66) +The BPZ conjugates of the vacua are: +⟨↓ | := | ↓⟩t = ∓⟨0| c−1, +⟨↑ | := | ↑⟩t = ±⟨0| c0c−1. +(B.67) +We have the following relations: +⟨↓ | = ∓ | ↓⟩‡ , +⟨↑ | = ∓ | ↑⟩‡ . +(B.68) +The ghost are normalized with +⟨↑ | ↓⟩ =⟨↓ | c0 | ↓⟩ =⟨0| c−1c0c1 |0⟩ = 1, +(B.69) +which selects the minus sign in the BPZ conjugation. The conjugate of the ghost vacuum is +⟨0c| =⟨0| c−1c0c1. +(B.70) +Considering both the holomorphic and anti-holomorphic sectors, we introduce the com- +binations: +b± +n = bn ± ¯bn, +c± +n = 1 +2 (cn ± ¯cn). +(B.71) +The normalization of b± +m is chosen to match the one of L± +m (B.75), and the one of c± +m such +that +{b+ +m, c+ +n } = δm+n, +{b− +m, c− +n } = δm+n. +(B.72) +We have the following useful identities: +b− +n b+ +n = 2bn¯bn, +c− +n c+ +n = 1 +2 cn¯cn. +(B.73) +B.4 +Bosonic string +The BPZ conjugates of the scalar and ghost modes are +(αn)t = −(±1)n α−n, +(bn)t = (±1)n b−n, +(cn)t = −(±1)n c−n. +(B.74) +Combinations of holomorphic and anti-holomorphic modes: +L± +n = Ln ± ¯Ln, +b± +n = bn ± ¯bn, +c± +n = 1 +2 (cn ± ¯cn). +(B.75) +The closed string inner product is defined from the BPZ product by an additional inser- +tion of c− +0 +⟨A, B⟩ =⟨A| c− +0 |B⟩ , +(B.76) +while the open string inner product is equal to the BPZ product +⟨A, B⟩ = ⟨A|B⟩ . +(B.77) +The vacuum for the matter and ghosts is +|k, 0⟩ := |k⟩ ⊗ |0⟩ , +|k, ↓⟩ := |k⟩ ⊗ | ↓⟩ . +(B.78) +The vacuum is normalized as +open: +⟨k, ↓ | c0 |k, ↓⟩ =⟨k′, 0| c−1c0c1 |k, 0⟩ = (2π)Dδ(D)(k + k′), +(B.79a) +closed: +⟨k, ↓↓ | c0¯c0 |k, ↓↓⟩ =⟨k′, 0| c−1¯c−1c0¯c0c1¯c1 |k, 0⟩ = (2π)Dδ(D)(k + k′), +(B.79b) +280 + +Appendix C +Quantum field theory +In this appendix, we gather useful information on quantum field theories. The first section +describes how to compute with path integral with non-trivial measures, generalizing tech- +niques from finite-dimensional integrals. Then, we summarize the important concepts from +the BRST and BV formalisms. +C.1 +Path integrals +In this section, we explain how analysis, algebra and differential geometry are generalized +to infinite-dimensional vector spaces (fields). +C.1.1 +Integration measure +In order to construct a path integral for the field Φ, one needs to define a notion of distance +on the space of fields. The distance between a field Φ and a neighbouring field Φ + δΦ is +|δΦ|2 = G(Φ)(δΦ, δΦ), +(C.1) +where G is the (field-dependent) metric on the field tangent space (the field dependence will +be omitted when no confusion is possible). This induces a metric on the field space itself +|Φ|2 = G(Φ)(Φ, Φ), +(C.2) +from which the integration measure over the field space can be defined as +dΦ +� +det G(Φ). +(C.3) +Moreover, the field metric also defines an inner-product between two different elements of +the tangent space or field space: +(δΦ1, δΦ2) = G(Φ)(δΦ1, δΦ2), +(Φ1, Φ2) = G(Φ)(Φ1, Φ2). +(C.4) +Remark C.1 (Metric in component form) If one has a set of spacetime fields Φa(x), +then a local norm is defined by +|δΦa|2 = +� +dx ρ(x)γab +� +Φ(x) +� +δΦa(x)δΦb(x), +(C.5) +which means that the metric in component form is +Gab(x, y)(Φ) = δ(x − y)ρ(x)γab +� +Φ(x) +� +. +(C.6) +281 + +Locality means that all fields are evaluated at the same point. +On a curved space, it is +natural to write γ only in terms of the metric g and to set ρ(x) = +� +det g(x), such that the +inner-product is diffeomorphism invariant. +Since a Gaussian integral is proportional to the squareroot of the operator determinant, +the integration measure can be determined by considering the Gaussian integral over the +tangent space: +� +dδΦ e−G(Φ)(δΦ,δΦ) = +1 +� +det G(Φ) +. +(C.7) +Note that one needs to work on the tangent space because G(Φ) can depend on the field, +which means that the integral +� +dΦ e−G(Φ)(Φ,Φ). +(C.8) +is not Gaussian. +Having constructed the Gaussian measure with respect to the metric G(Φ), it is now +possible to consider the path integral of general functional F of the fields: +� +dΦ +� +det G(Φ) F(Φ). +(C.9) +The (effective) action S(Φ) provides a natural metric on the field space by defining +√ +det G = +e−S, or +S = −1 +2 tr ln G(Φ). +(C.10) +However, it can be simpler to work with a Gaussian measure by considering only the quad- +ratic terms in S, and expanding the rest in a power series. +In particular, the partition +function is defined from the classical action Scl by +Z = +� +dΦ e−Scl(Φ). +(C.11) +Given an operator D, its adjoint D† is defined with respect to the metric as +G(δΦ, DδΦ) = G(D†δΦ, δΦ). +(C.12) +The free-field measure is such that the metric on the field space is independent from the +field itself: G(X) = G0. In particular, this implies that the metric is flat and its determinant +can be absorbed in the measure, setting det G0 = 1. In this case, the measure is invariant +under shift of the field: +Φ → Φ + ε +(C.13) +such that +� +dΦ e− 1 +2 |Φ+ε|2 = +� +dΦ e− 1 +2 |Φ|2. +(C.14) +This property allows to complete squares and shift integration variables (for example to +generate a perturbative expansion and to derive the propagator). +Computation – Equation (C.14) +� +dΦ e− 1 +2 |Φ+ε|2 = +� +d�Φ det δΦ +δ�Φ +e− 1 +2 |� +Φ| +2 += +� +d�Φ e− 1 +2 |� +Φ| +2 +(C.15) +The first equality follows by setting �Φ = Φ + ε, and the result (C.14) follows by the +redefinition �Φ = Φ. +282 + +C.1.2 +Field redefinitions +Under a field redefinition Φ → Φ′, the norm and the measure are invariant: +dΦ +� +det G(Φ) = d�Φ +� +det �G(�Φ), +G(Φ)(δΦ, δΦ) = �G(�Φ)(δ�Φ, δ�Φ). +(C.16) +Conversely, one can find the Jacobian J(Φ, �Φ) between two coordinate systems by writing +dΦ = J(Φ, �Φ)d�Φ, +J(Φ, �Φ) = +����det ∂Φ +∂�Φ +���� = +� +det �G(�Φ) +det G(Φ). +(C.17) +If the measure of the initial field coordinate is normalized such that det G = 1, or equivalently +� +dδΦ e−|δΦ|2 = 1, +(C.18) +one can determine the Jacobian by performing explicitly the integral +J(�Φ)−1 = +� +dδ�Φ e−� +G(δ� +Φ,δ� +Φ). +(C.19) +Remark C.2 (Identity of the Jacobian for Φ and δΦ) The Jacobian agrees on the space +of fields and on its tangent space. This is most simply seen by using a finite-dimensional +notation: considering the coordinates xµ and a vector v = vµ∂µ, the Jacobian for changing +the coordinates to ˜xµ is equivalently +J = det ∂˜xµ +∂xµ = det ∂˜vµ +∂vµ +(C.20) +since the vector transforms as +˜vµ = vν ∂˜xµ +∂xν . +(C.21) +C.1.3 +Zero-modes +A zero-mode Φ0 of an operator D is a field such that +DΦ0 = 0. +(C.22) +In the definition of the path integral over the space of fields Φ, the measure is defined +over the complete space. However, this will lead respectively to a divergent or vanishing +integral if the field is bosonic or fermionic, because the integration over the zero-modes can +be factorized from the rest of the integral. Writing the field as +Φ = Φ0 + Φ′, +(Φ0, Φ′) = 0, +(C.23) +where Φ′ is orthogonal to the zero-mode Φ0, a Gaussian integral of an operator D reads: +Z[D] = +� +dΦ +√ +det G e− 1 +2 (Φ,DΦ) = +�� +dΦ0 +� � +dΦ′ e− 1 +2 (Φ′,DΦ′) +(C.24) +A first solution could be to simply strip the first factor (for example, by absorbing it in the +normalization), but this is not satisfactory. In particular, the partition function with source +Z[D, J] = +� +dΦ +√ +det G e− 1 +2 (Φ,DΦ)−(J,Φ) +(C.25) +283 + +will depend on the zero-modes through the sources. +But, since the zero-modes are still +singled out, it is interesting to factorize the integration +Z[D, J] = +� +dΦ0 e−(J,Φ0) +� +dΦ′ e− 1 +2 (Φ′,DΦ′)−(J,Φ′) +(C.26) +and to understand what makes it finite. Ensuring that zero-modes are correctly inserted +is an important consistency and leads to powerful arguments. Especially, this can help to +guess an expression when it cannot be derived easily from first principles. +To exemplify the problem, consider the cases where there is a single constant zero-mode +denoted as x (bosonic) or θ (fermionic). The integral over x is infinite: +� +dx = ∞. +(C.27) +Oppositely, the integral of a Grassmann variable θ vanishes: +� +dθ = 0. +(C.28) +A Grassmann integral satisfies also +� +dθ θ = +� +dθ δ(θ) = 1, +(C.29) +such that an integral over a zero-mode does not vanish if there one zero-mode in the integrand +(due to the Grassmann nature of θ, the integrand can be at most linear). By analogy with +the fermionic case, a possibility for getting a finite bosonic integral is to insert a delta +function: +� +dx δ(x) = 1. +(C.30) +We will see that this is exactly what happens for the ghosts and super-ghosts in (super)string +theories. +Since ker D is generally finite-dimensional, it is interesting to decompose the zero-mode +on a basis and to integrate over the coefficients in order to obtain a finite-dimensional +integral. Writing the zero-mode as +θ0(x) = θ0iψi(x), +ker D = Span{ψi} +(C.31) +where the coefficients θ0i are constant Grassmann numbers, the change of variables θ → +(θ0i, θ′) implies: +dθ = +1 +� +det(ψi, ψj) +dθ′ +n +� +i=1 +dθ0i, +(C.32) +where n = dim ker D. +Next, according to the discussion above, one can ask if it is possible to rewrite an integ- +ration over dθ′ in terms of an integration over dθ together with zero-mode insertions. This +is indeed possible and one finds: +dθ +n +� +i=1 +θ(xi) = +det ψi(xj) +� +det(ψi, ψj) +dθ′. +(C.33) +284 + +Computation – Equation (C.32) +1 = +� +dθ e−|θ|2 = +� +dθ′dθ0 e−|θ|2−|θ0|2 += J +� +dθ′ � +i +dθ0i e−|θ′|2−|θ0iψi|2 = J +� +det(ψi, ψj) +Computation – Equation (C.33) +The simplest approach is to start with the LHS. This formula is motivated from the +previous discussion: if the integration measure contains n zero-modes, it will vanish +unless there are n zero-mode insertions. Moreover, one can replace each of them by the +complete field since only the zero-mode part can contribute: +� +dθ0 +n +� +j=1 +θ(xj) = +� +dθ0 +n +� +j=1 +θ0(xj) = +1 +� +det(ψi, ψj) +� +dnθ0i +n +� +j=1 +� +θ0iψi(xj) +� += +det ψi(xj) +� +det(ψi, ψj) +� � +i +dθ0i θ0i = +det ψi(xj) +� +det(ψi, ψj) +. +The third equality follows by developing the product and ordering the θ0i: minus signs +result from anticommuting the θ0i such that one gets the determinant of the basis +elements. +C.2 +BRST quantization +Consider an action Sm[φi] which depends on some fields φi subject to a gauge symmetry: +δφi = ϵaδaφi = ϵaRi +a(φ), +(C.34) +where ϵa are the (local) bosonic parameters, such that the action is invariant +ϵaδaSm = 0. +(C.35) +The gauge transformations form a Lie algebra with structure coefficients f c +ab +[δa, δb] = f c +abδc. +(C.36) +It is important 1) that the algebra closes off-shell (without using the equations of motion), +2) that the structure coefficients are field independent and 3) that the gauge symmetry is +irreducible (each gauge parameter is independent). +Remark C.3 (Interpretation of the Ri +a matrices) If the φi transforms in a represent- +ation R of the gauge group, then the transformation is linear in the field +Ri +a(φ) = (T R +a )i +jφj, +(C.37) +with T R +a the generators in the representation R. But, in full generality, this is not the case: +for example the gauge fields Aa +µ do not transform in the adjoint representation even if they +carry an adjoint index (only the field strength does), and in this case +Rb +aµ = δb +a∂µ + f c +abAb +µ. +(C.38) +When the fields φi form a non-linear sigma models, the Ri +a(φ) correspond to Killing +vectors of the target manifold. +285 + +In order to fix the gauge symmetry in the path integral +Z = Ω−1 +gauge +� +dφi e−Sm, +(C.39) +gauge fixing conditions must be imposed: +F A(φi) = 0. +(C.40) +Indeed, without gauge fixing, the integration is performed over multiple identical configura- +tions and the result diverges. The index A is different from the gauge index a because they +can refer to different representations, but for the gauge fixing to be possible they should run +over as many values. +Next, ghost fields ca (fermionic) are introduced for every gauge parameter, anti-ghosts +bA (fermionic) and auxiliary (Nakanishi–Laudrup) fields BA (bosonic) for every gauge con- +dition. The gauge-fixing and ghost actions are then defined by +Sgh = bAca δaF A(φi), +(C.41a) +Sgf = −i BAF A(φi) +(C.41b) +such that the original partition function is equivalent to +Z = +� +dφi dbA dca dBA e−Stot +(C.42) +where +Stot = Sm + Sgf + Sgh. +(C.43) +The total action is invariant +δϵStot = 0. +(C.44) +under the (global) BRST transformations +δϵφi = iϵ caδaφi, +δϵca = − i +2 ϵ f a +bccbcc, +δϵbA = ϵ BA, +δϵBA = 0, +(C.45) +where ϵ is an anti-commuting constant parameter. +Note that the original action Sm is +invariant by itself since the transformation acts like a gauge transformation with parameter +ϵca. The transformation of ca follows because it transforms in the adjoint representation of +the gauge group. Direct computations show that this transformation is nilpotent +δϵδϵ′ = 0. +(C.46) +These transformations are generated by a (fermionic) charge QB called the BRST charge +δϵφi = i [ϵQB, φi] +(C.47) +and similarly for the other fields (stripping the ϵ outside the commutator turns it to an +anticommutator if the field is fermionic). Taking the ghosts to be Hermitian leads to an +Hermitian charge. +An important consequence is that the two additional terms of the action can be rewritten +as a BRST exact terms +Sgf + Sgh = {QB, bAF A}. +(C.48) +A small change in the gauge-fixing condition δF leads to a variation of the action +δS = {QB, bAδF A}. +(C.49) +286 + +The BRST charge should commute with the Hamiltonian in order to be conserved: this +should hold in particular when changing the gauge fixing condition +[QB, {QB, bAδF A}] = 0 +=⇒ +Q2 +B = 0. +(C.50) +Some vocabulary is needed before proceeding further. A state |ψ⟩ is said to be BRST +closed if it is annihilated by the BRST charge +|ψ⟩ closed +⇐⇒ +|ψ⟩ ∈ ker QB +⇐⇒ +QB |ψ⟩ = 0. +(C.51) +States which are in the image of QB (i.e. they can be written as QB applied on some other +states) are said to be exact +|ψ⟩ exact +⇐⇒ +|ψ⟩ ∈ Im QB +⇐⇒ +∃ |χ⟩ : |ψ⟩ = QB |χ⟩ . +(C.52) +The cohomology H(QB) of QB is the set of closed states which are not exact +|ψ⟩ ∈ H(QB) +⇐⇒ +|ψ⟩ ∈ ker QB, +∄ |χ⟩ : |ψ⟩ = QB |χ⟩ . +(C.53) +Hence the cohomology corresponds to +H(QB) = ker QB +Im QB +. +(C.54) +Two elements of the cohomology differing by an exact state are in the same equivalence class +|ψ⟩ ≃ |ψ⟩ + QB |χ⟩ . +(C.55) +Considering the S-matrix ⟨ψf|ψi⟩ between a set of physical initial states ψi and final +states ψf, a small change in the gauge-fixing condition leads to +δF ⟨ψf|ψi⟩ =⟨ψf| {QB, bAδF A} |ψi⟩ +(C.56) +after expanding the exponential to first order. Since the S-matrix should not depend on the +gauge this implies that a physical state ψ must be BRST closed (i.e. invariant) +QB |ψ⟩ = 0. +(C.57) +Conversely, this implies that any state of the form QB |χ⟩ cannot be physical because it is +orthogonal to every physical state |ψ⟩ +⟨ψ| QB |χ⟩ = 0. +(C.58) +This implies in particular that the amplitudes involving |ψ⟩ and |ψ⟩ + QB |χ⟩ are identical, +and any amplitude for which an external state is exact vanishes. As a conclusion, physical +states are in the BRST cohomology +|ψ⟩ physical +⇐⇒ +|ψ⟩ ∈ H(QB). +(C.59) +If there is a gauge where the ghosts decouple from the matter field, then the invariance +of the action and of the S-matrix under changes of the gauge fixing ensures that this state- +ment holds in any gauge (but, one still need to check that the gauge preserves the other +symmetries). If such a gauge does not exist, then one needs to employ other methods to +show the desired result. +Note that BA can be integrated out by using its equations of motion +δF A +δφi BA = −δSm +δφi , +(C.60) +287 + +and this modifies the BRST transformation of the anti-ghost to +δϵbA = −ϵ +�δF A +δφi +�−1 δSm +δφi . +(C.61) +It is also possible to introduce a term +{QB, bABBM AB} = i BAM ABBB +(C.62) +for any constant matrix M AB. Since this is also a BRST exact term, the amplitudes are +not affected. Integrating over BA produces a Gaussian average instead of a delta function +to fix the gauge. +In the previous discussion, the BRST symmetry was assumed to originate from the +Faddeev–Popov gauge fixing. But, in fact, it is possible to start directly with an action of +the form +S[φ, b, c, B] = S0[φ] + QBΨ[φ, b, c, B] +(C.63) +where Ψ has ghost number −1. It can be proven that this is the most general action invariant +under the BRST transformations (C.45). This can describe gauge fixed action which cannot +be described by the Faddeev–Popov procedure: in particular, the latter yields actions which +are quadratic in the ghost fields (by definition of the Gaussian integral representation of the +determinant), but this does not exhaust all the possibilities. For example, the background +field method applied to Yang–Mills theory requires using an action quartic in the ghosts. +In this section, several hypothesis have been implicit (off-shell closure, irreducibility and +constant structure coefficients). If one of them breaks, then it is necessary to employ the +more general BV formalism. +C.3 +BV formalism +The Batalin–Vilkovisky (BV, or also field–antifield) formalism is the most general framework +to quantize theories with a gauge symmetry. While the BRST formalism (Appendix C.2) +is sufficient to describe simple systems, it breaks down when the structure of the gauge +symmetry is more complicated, for example in systems implying gravity. The BV formalism +is required in the three following cases (which can occur simultaneously): +1. the gauge algebra is open (on-shell closure); +2. the structure coefficients depend on the fields; +3. the gauge symmetry is reducible (not all transformations are independent). +The BV formalism is also useful for standard gauge symmetries to demonstrate renormaliz- +ability and to deal with anomalies. +As explained in the previous section, the ghosts and the BRST symmetry are crucial to +ensure the consistency of the gauge theory. The idea of the BV formalism is to put on an +equal footing the physical fields and all the required auxiliary and ghost fields (before gauge +fixing). The introduction of antifields – one for each of the fields – and the description of the +full quantum dynamics in terms of a quantum action (constrained by the quantum master +equation) ensure the consistency of the system. Additional benefits are the presence of a +(generalized) BRST symmetry, the existence of a Poisson structure (which allows to bring +concepts from the Hamiltonian formalism), the covariance of the formalism and the simple +interpretation of counter-terms as corrections to the classical action. +For giving a short intuition, the BV formalism can be interpreted as providing a (anti)ca- +nonical structure in the Lagrangian formalism, the role of the Hamiltonian being played by +the action. +288 + +C.3.1 +Properties of gauge algebra +Before explaining the BV formalism, we review the situations listed above. The classical +action for the physical fields φi is denoted by S0[φ] and the associated equations of motion +by +Fi(φ) = ∂S0 +∂φi . +(C.64) +Then, a gauge algebra is open and has field-dependent structure coefficients F c +ab(φ) if: +[Ta, Tb] = F c +ab(φ)Tc + λi +abFi(φ). +(C.65) +On-shell, Fi = 0 and the second term is absent, such that the algebra closes. The fields +themselves are constants from the point of view of the gauge algebra, but their presences in +the structure coefficients complicate the analysis of the theory. Moreover, the path integral +is off-shell and for this reason one needs to take into account the last term. +Finally, the gauge algebra can be reducible: in brief, it means that there are gauge +invariances associated to gauge parameters – and correspondingly ghosts for ghosts –, and +this recursively. Since there is one independent ghost for each generator, there are too many +ghosts if the generators are not all independent, and there is a remnant gauge symmetry for +the ghost fields (in the standard Faddeev–Popov formalism, the ghosts are not subject to +any gauge invariance). This originates from relations between the generators Ri +a: denoting +by m0 the number of level-0 gauge transformations, the number of independent generators +is rank Ri +a. Then, the +m1 = m0 − rank Ri +a +(C.66) +relations between the generators translate into a level-1 gauge invariance of the ghosts. This +symmetry can be gauge fixed by performing a second time the Faddeev–Popov procedure, +yielding commuting ghosts. This symmetry can also be reducible, and the procedure can +continue without end. If one finds that the gauge invariance at level n = ℓ is irreducible, one +says that the gauge invariance is ℓ-reducible. If this does not happen, one defines ℓ = ∞. +The number of generators at level n is denoted by mn. +Example C.1 – p-form gauge theory +A p-form gauge theory is written in terms of a gauge field Ap with a a gauge invariance +δAp = dλp−1. +(C.67) +But, due to the nilpotency of the derivative, deformations of the gauge parameter +satisfying +δλp−1 = dλp−2 +(C.68) +does not translate into a gauge invariance of Ap. Similarly from this should be excluded +the transformation +δλp−2 = dλp−3, +(C.69) +and so on until one reaches the case p = 0. Hence, a p-form field has a p-reducible +gauge invariance. +C.3.2 +Classical BV +Denoting the fields collectively as +ψr = {φi, BA, bA, ca}, +(C.70) +289 + +the simplest BV action reads +S[ψr, ψ∗ +r] = S0[φ] + QBψr ψ∗ +r +(C.71) +with the antifields +ψ∗ +r = {φ∗ +i , BA∗, bA∗, c∗ +a}. +(C.72) +The action (C.63) is recovered by writing +ψ∗ +r = ∂Ψ +∂ψr . +(C.73) +This indicates that the general BRST formalism could be rephrased in the BV language. +But, in the same way that the BRST formalism generalizes the Faddeev–Popov formalism, +it is in turn generalized by the BV formalism. Indeed, the above action is linear in the +antifields: this constraint is not required and one can write more general actions. In the rest +of this section, we explain how this works at the level of the action (classical level) and how +the sets of fields and antifields are defined. +Consider a set of physical fields φi with the gauge invariance +δφi = ϵa0 +0 Ri +a0(φi). +(C.74) +Then, associate a ghost field ca0 to each of the gauge parameters ϵa0. If the gauge symmetry +is reducible, a new gauge invariance is associated to the ghosts +δca0 +0 = ϵa1 +1 Ra0 +a1(φi, ca0). +(C.75) +This structure is recurring and the ghosts of the level-n gauge invariance are denoted by can +and they satisfy +δcan +n = ϵan+1 +n+1 Ran +an+1(φi, ca0 +0 , . . . , can +n ). +(C.76) +Thus, the set of fields is +ψr = {can +n }n=−1,...,ℓ, +c−1 := φ. +(C.77) +A ghost number is introduced +Ngh(φi) = 0, +Ngh(can +n ) = n + 1, +(C.78) +and the Grassmann parity of the ghosts is defined to be opposite (resp. identical) of the +parity of the associated gauge parameter for even (resp. odd) n +|cn| = |ϵan +n | + n + 1. +(C.79) +To each of these fields is associated an antifield ψ∗ +r of opposite parity as ψr and such that +their ghost numbers sum to −1 +Ngh(ψ∗ +r) = −1 − Ngh(ψr), +|ψ∗ +r| = −|ψr|. +(C.80) +The fields and antifields together are taken to define a graded symplectic structure +ω = +� +r +dψr ∧ dψ∗ +r +(C.81) +with respect to which they are conjugated to each other +(ψr, ψ∗ +s) = δrs, +(ψr, ψs) = 0, +(ψ∗ +r, ψ∗ +s) = 0. +(C.82) +290 + +The antibracket (graded Poisson bracket) (·, ·) reads +(A, B) = ∂RA +∂ψr +∂LB +∂ψ∗r +− ∂RA +∂ψ∗r +∂LB +∂ψr , +(C.83) +where the L and R indices indicate left and right derivatives. It is graded symmetric, which +means +(A, B) = −(−1)(|A|+1)(|B|+1)(B, A). +(C.84) +It also satisfies a graded Jacobi identity and the property +Ngh((A, B)) = Ngh(A) + Ngh(B) + 1, +|(A, B)| = |A| + |B| + 1 +mod 2. +(C.85) +Moreover, the antibracket acts as a derivative +(A, BC) = (A, B)C + (−1)|B|C(A, C)B. +(C.86) +The dynamics of the theory is described by the (classical) master action S[ψr, ψ∗ +r] which +satisfies +Ngh(S) = 0, +|S| = 0. +(C.87) +In order to reproduce correctly the dynamics of the classical system without ghosts, this +action is required to satisfy the boundary condition +S[ψr, ψ∗ +r = 0] = S0[φi], +∂L∂RS +∂c∗ +n−1,an−1∂can +n +���� +ψ∗=0 += Ran−1 +an +. +(C.88) +Indeed, if the antifields are set to zero, the ghost fields cannot appear because they all have +positive ghost numbers and it is not possible to build terms with vanishing ghost numbers +from them. +In analogy with the Hamiltonian formalism, the master action can be used as the gen- +erator of a global fermionic symmetry, and inspection will show that it corresponds to a +generalization of the BRST symmetry. Writing the generalized and classical BRST operator +as s, the transformations of the fields and antifields read +δθψr = θ sψr = −θ (S, ψr) = θ ∂RS +∂ψ∗r +, +(C.89a) +δθψ∗ +r = θ sψ∗ +r = −θ (S, ψ∗ +r) = −θ ∂RS +∂ψr , +(C.89b) +where θ is a constant Grassmann parameter. The variation of a generic functional F[ψr, ψ∗ +r] +is +δθF = θ sF = −θ (S, F). +(C.90) +For the BRST transformation to be a symmetry of the action, the action must satisfy the +classical master equation +(S, S) = 0. +(C.91) +This equation can easily be solved by expanding S in the ghosts: the various terms can be +interpreted in terms of properties of the gauge algebra. Then, the Jacobi identity used with +two S and an arbitrary functional gives +(S, (S, F)) = 0 +(C.92) +and this implies that the transformation is nilpotent +s2 = 0. +(C.93) +291 + +A classical observable O satisfies +sO = 0. +(C.94) +Due to the BRST symmetry, the action is not uniquely defined and the action +S′ = S + (S, δF) +(C.95) +also satisfies the master equation, where δF is arbitrary up to the condition Ngh(δF) = −1. +This can be interpreted as the action S in a new coordinate system (ψ′r, ψ′∗ +r ) with +ψ′ = ψ − δF +δψ∗ , +ψ′∗ = ψ∗ + δF +δψ +(C.96) +such that +S′[ψ, ψ∗] = S +� +ψ − δF +δψ∗ , ψ∗ + δF +δψ +� +. +(C.97) +Indeed, for F = F[ψ, ψ∗], one has +S′[ψ, ψ∗] = S[ψ, ψ∗] + (S, ψ)δF +δψ + (S, ψ∗) δF +δψ∗ = S[ψ, ψ∗] − ∂RS +∂ψ∗ +δF +δψ + ∂RS +∂ψ +δF +δψ∗ . +(C.98) +It can be shown that this transformation preserves the antibracket and the master equation +(ψ′r, ψ′∗ +s ) = δrs, +(S′, S′) = 0. +(C.99) +More generally, any transformation preserving the antibracket is called an (anti)canonical +transformation. One can also consider generating functions depending on both the old and +new coordinates, as is standard in the Hamiltonian formalism. Under a transformation, any +object depending on the coordinates changes as +G′ = G + (δF, G). +(C.100) +One can consider finite transformation without problems. +In order to perform the gauge fixing, one needs to eliminate the antifields. A convenient +condition is +SΨ[ψr] = S +� +ψr, ∂Ψ +∂ψr +� +, +ψ∗ +r = ∂Ψ +∂ψr , +(C.101) +where Ψ[ψr] is called the gauge fixing fermion and satisfies +Ngh(Ψ) = −1, +|Ψ| = 1. +(C.102) +From the discussion on coordinate transformations this amounts to work in new coordinates +where ψ′∗ +r = 0. But such a function Ψ cannot be built from the fields because they all have +positive ghost numbers. One needs to introduce trivial pairs of fields. +A trivial pair (B, ¯c) is defined by the properties +|B| = −|¯c|, +Ngh(B) = Ngh(¯c) + 1, +(C.103a) +s¯c = B, +sB = 0 +(C.103b) +and the new action reads +¯S = S[ψr, ψ∗ +r] − B¯c∗ +(C.104) +(the position dependence is kept implicit). In this context ψr and ψ∗ +r are sometimes called +minimal variables. From this, one learns that +( ¯S, ¯S) = (S, S) = 0. +(C.105) +292 + +At level-0, one introduces the pair +(B0a0, ¯c0a0) := (B0 +0a0, ¯c0 +0a0) +(C.106) +and the associated antifields. The field ¯c0 := b is the Faddeev–Popov anti-ghost associated +to c0 and the trivial pair satisfies +|B0| = |ϵ0|, +|¯c0| = −|ϵ0|, +Ngh(B0) = 0, +Ngh(c0) = −1. +(C.107) +For the level 1, two additional pairs are introduced: +(B0 +1a1, ¯c0 +1a1), +( ¯B1a1 +1 +, c1a1 +1 +) +(C.108) +and the corresponding antifields. The motivation for adding an additional pair is that the +level-0 pair only fixes m0 − m1 of the generators: the additional m1 extra-ghosts c1a1 +1 +can +be fixed by the residual level-0 symmetry. The first level-1 pair fixes the level-1 symmetry. +Then, the gauge fixed action enjoys a BRST symmetry acting only on the fields +δθψr = θ sψr = θ ∂RS +∂ψ∗r +���� +ψ∗ +r =∂rΨ +. +(C.109) +Note that this BRST operator is generically nilpotent only on-shell +s2 ∝ eom. +(C.110) +C.3.3 +Quantum BV +At the quantum level, one considers the path integral +Z = +� +dψrdψ∗ +r e−W [ψr,ψ∗ +r ]/ℏ +(C.111) +where W is called the quantum master action. The reason for distinguishing it from the +classical master action S is that the measure is not necessarily invariant by itself under +the generalized BRST transformation – this translates into a non-gauge invariance of the +measure of the physical fields, i.e. a gauge anomaly. +Quantum BRST transformation are generated by the quantum BRST operator σ +δθF = θ σF = (W, F) − ℏ ∆F, +(C.112) +where +∆ = ∂R +∂ψ∗r +∂L +∂ψr . +(C.113) +Then, the path integral is invariant if W satisfies the quantum master equation +(W, W) − 2ℏ∆W = 0, +(C.114) +which can also be written as +∆e−W/ℏ = 0. +(C.115) +This can be interpreted as the invariance of Z under changes of coordinates: indeed one +finds that +δW = 1 +2(W, W), +(C.116) +and the integration measure picks a Jacobian +sdet J ∼ 1 + ∆W. +(C.117) +293 + +In the limit ℏ → 0, one recovers the classical master equation. More generally, the action +can be expanded in powers of ℏ +W = S + +� +p≥1 +ℏpWp. +(C.118) +Observables are given by operators O[ψ, ψ∗] invariant under σ: +σO = 0, +(C.119) +which ensures that the expectation value is invariant under changes of Ψ +δ⟨O⟩ = 0. +(C.120) +Note that if O depends just on ψ the condition reduces to sO = 0, but generically there is +no such operators (except constants) satisfying this condition for open algebra. +Consider the gauge fixed integral +Z = +� +dψr e−WΨ[ψr], +WΨ[ψr] = W +� +ψr, ∂Ψ +∂ψr +� +. +(C.121) +Varying the gauge fixing fermion by δΨ gives +Z = +� +dψr e−WΨ[ψr] +�∂RS +∂ψ∗r +� +ψ∗=∂ψΨ +∂(δΨ) +∂ψr . +(C.122) +Integrating by part gives the quantum master equation. +C.4 +Suggested readings +• Manipulations of functional integral are given in [100, sec. 15.1, 22.1, 172, chap. 14, +191, 53]. +• Zero-modes are discussed in [23]. +• A general summary of path integrals for bosonic and fermionic fields can be found +in [193, app. A]. +• BRST formalism: most QFT books contain an introduction, more complete references +are [251, chap. 15, 247, 105]; +• BV formalism [251, chap. 15, 88, 93, 247, 105, 49] (several explicit examples are given +in [93, sec. 3], see [12, 231, 254] for more specific details). +294 + +Bibliography +[1] +M. 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Cambridge University +Press, Jan. 2009. +309 + +Index +#, 187, 190 +Σ0, see Riemann sphere +Σg, Σg,n, see Riemann surface +χg, χg,n, see Euler characteristics +| ↓⟩, see first-order system, vacuum +| ↑⟩, see first-order system, vacuum +⟨·⟩, 100 +ωg,n +p +, see Pg,n space, p-form +∼, 95 +†, see Hermitian adjoint +‡, see Euclidean adjoint +c, see conjugate state +t, see BPZ conjugation +[· · · ]g, see string product +{· · · }g, see fundamental vertex +1PI vertex, 222, 236 +region, 192 +1PR region, 192 +2d gravity, 51, 135 +adjoint, see Euclidean, Hermitian adjoint +AdS/CFT, 239 +Ag,n, see (off-shell) string amplitude +αn, see scalar field CFT, mode expansion +background independence, 238 +background metric, 32 +Batalin–Vilkovisky formalism, 23, 234, 288 +antibracket, 291 +BRST transformation, 291 +classical master equation, 291 +field redefinition, 292 +fields and antifields, 289 +gauge fixing, 292 +quantum BRST transformation, 293 +quantum master equation, 293 +bc CFT, see first-order CFT, reparametriza- +tion bc ghosts +Beltrami differential, 41 +Berkovits’ super-SFT, 256 +βγ CFT, see first-order CFT, superconformal +βγ ghosts +bosonic string CFT, 138 +L0, 138, 142 +complex parametrization, 141 +Hilbert space, 138 +level operator, 138, 142 +light-cone parametrization, 141 +boundary condition, 97 +Neveu–Schwarz (NS), 98 +Ramond (R), 98 +twisted, 98 +untwisted, 98 +BPZ conjugation, 97 +mode, 98 +state, 101 +BRST cohomology, see also string states, 69, +134 +absolute, 69, 139, 146 +relative, 70, 141, 144, 145, 147 +semi-relative, 70, 147 +two flat directions, 138–148 +BRST current, 135 +OPE, 135–136 +BRST operator, 135 +commutator, 137 +full +zero-mode decomposition, 147 +mode expansion, 136 +zero-mode decomposition, 136 +BRST quantization +change of gauge fixing condition, 287 +charge nilpotency, 287 +cohomology, 287 +Nakanishi–Lautrup auxiliary field, 286 +physical states, 287 +transformations, 286 +BRST SFT, see covariant SFT +BV formalism, see Batalin–Vilkovisky form- +alism +central charge, 33, 90, 95 +CFT, see conformal field theory +310 + +CISO, see conformal isometry group +CKV, see conformal Killing vector +classical solution +marginal deformation, 242 +closed string amplitude +Feynman diagram decomposition, 215 +on punctured moduli space, 178 +tree-level +3-point, 171 +4-point, 173, 200 +closed string fundamental vertex, 214 +g-loop +0-point, 221 +1-loop +0-point, 221 +1-point, 218 +properties, 223 +recursive construction, 217 +special vertices, 221 +tree-level +0-point, 221 +1-point, 221 +2-point, 221, 232 +3-point, 172 +4-point, 175 +closed string product, 223 +g = 0, n = 1, 230, 233 +ghost number, 223 +closed string states +dilaton Φ, 17 +Kalb–Ramond Bµν, 17 +level-matching, see level-matching +massless field, 17 +metric Gµν, 17 +tachyon T, 17 +physical, 148 +zero-mode decomposition, 176 +closed string vertex +fundamental identity, 223 +commutator +[Lm, Ln], 90 +[Lm, φn], 99 +CFT, 92–93 +complex coordinates, 72, 82 +cylinder, 83 +conformal +anomaly, see Weyl anomaly +dimension, 88 +factor, see Liouville field +spin, 88 +vacuum, see SL(2, C) vacuum +weight, 88 +conformal algebra, 79 +2d, see Witt algebra, Virasoro algebra +conformal field theory +classical, 84 +cylinder, 105 +definition, 88 +finite transformation, 88 +conformal isometry group, see also conformal +Killing vector, 78 +sphere, see SL(2, C) +conformal Killing +equation, 44, 79 +group volume, 46 +vector, 44, 62, 79 +conformal structure, 31 +conjugate state, 102 +conserved charge, 99, 268, 274 +CFT, 93 +conserved current, 268 +CFT, 93 +mode expansion, 99 +contracting homotopy operator, 139, 144 +contraction, 95 +conventions +complex coordinates, 268 +ghost number, 129 +spacetime momentum current, 109 +correlation function, 90 +quasi-primary operator, 90 +sphere 1-point (-), 91 +sphere 2-point (-), 91 +sphere 3-point (-), 91 +covariant SFT, see also classical (-), free (-), +quantum (-) +background independence, 239 +closed bosonic +1PI action, 236 +1PI gauge symmetry, 237 +classical action, 232 +classical equation of motion, 233 +classical gauge algebra, 233 +classical gauge transformation, 233 +fields and antifields, 235 +gauge fixed action, 230, 231 +inner product, 168 +normalization, 231 +quantum action, 235 +quantum BV master equation, 235 +open bosonic +inner product, 156 +parameters, 231 +renormalization, 231 +311 + +critical dimension, 14, 19, 48, 77, 135, 137 +superstring, 245 +cross-ratio, 91 +degeneration limit, 191 +diffeomorphism, 30 +group volume, 39, 44, 46 +infinitesimal (-), 30 +large (-), 31 +dual state, see conjugate state +ϵ (scalar action sign), 107 +ηξ ghosts, 246 +energy vacuum, 100 +energy–momentum tensor +finite transformation, 95 +mode expansion, 99 +Euclidean adjoint, 96–97 +mode, 98 +state, 101 +Euler characteristics, 30, 59 +extended complex plane ¯C, 81 +f◦, see CFT, finite transformation +factorization, see string amplitude +Faddeev–Popov +determinant, 44 +gauge fixing, see path integral +Fg,n, see propagator region +F1PR +g,n , see 1PR region +field space, see also path integral +DeWitt metric, 35 +inner-product, 34, 281 +norm, 34, 281 +field theory space, 239 +connection, 240 +Hilbert space bundle, 240 +first-order CFT, 119 +L0, 124, 128 +U(1) ghost current, 120, 129 +mode expansion, 124 +transformation law, 122 +U(1) symmetry, 119, 120 +action, 119 +boundary condition, 123 +BPZ conjugate +modes, 132 +vacuum, 132 +central charge, 121 +commutator, 125 +complex components, 119 +cylinder, 122 +energy normal ordering, 128 +energy–momentum tensor, 120 +mode expansion, 124 +equation of motion, 119 +Euclidean adjoint +modes, 132 +vacuum, 132 +Fock space, see Hilbert space, 131 +ghost charge, 121 +ghost number, 120, 129 +cylinder, 122 +Hilbert space +full, 131 +holomorphic, 131 +inner product, 133 +level operator, 124 +mode expansion, 123 +number operator, 124 +OPE, 121–123 +propagator, 120 +summary, 133 +vacuum +SL(2, C), 126 +conjugate, 133 +energy, 127–128 +Virasoro operators, 124, 129 +weight, 120 +zero-mode decomposition, 131 +zero-point energy, 127 +free covariant SFT +closed bosonic, 168 +action, 232 +classical action, 168 +equation of motion, 167 +gauge fixed action, 169, 228 +gauge fixed equation of motion, 159, +169, 231 +gauge transformation, 169 +open bosonic +BV action, 164 +classical action, 156, 157 +equation of motion, 155 +gauge fixed action, 159 +gauge transformation, 157, 165 +zero-mode decomposition, 156 +path integral, see string field path integ- +ral +free SFT, 140 +free super-SFT +action, 257, 258 +gauge transformation, 257, 258 +fundamental vertex, see also closed string (-) +312 + +interpretation, 231 +region, 191 +G, see spacetime ghost number +ˆgab, see background metric +gab, see worldsheet metric +ghost number, 52 +anomaly, 204, 250 +ghosts, see first-order CFT, reparametriza- +tion bc ghosts, superconformal βγ +ghosts +gluing compatibility, 190 +Green function, 59–61, 215 +tree-level 2-point, 221 +group +fundamental domain, 39 +volume, 38 +GSO symmetry, 247 +Hermitian adjoint, 96 +higher-genus Riemann surface +conformal group, 89 +Hilbert space +CFT, 100 +holomorphic factorization, 77, 251 +holomorphic/anti-holomorphic sectors, 83, 267 +I(z), I±(z), see inversion +iε-prescription, 262 +index +amplitude, 214 +Riemann surface, 194, 214 +inversion map, 87, 97 +ISO, see isometry group +isometry group, 79 +ker P1, see conformal Killing vector +ker P † +1 , see quadratic differential +Kg, see conformal Killing vector +Killing vector, 79 +L±, 99 +L∞ algebra, 234, 255 +left/right-moving sectors, 83, 267 +level-matching condition, 16, 70, 147, 168, +176, 184, 203, 205, 216, 228, 257 +ℓg,n, see string product +light-cone coordinates, 83 +Liouville +action, 47 +central charge, 48 +field, 32, 47 +free-field measure, 40 +theory, 88, 90, 264 +local coordinates, 23, 171, 179 +constraints, 184, 190 +global phase, 184, 202, 204 +global rescaling, 195 +reparametrization, 184, 202, 203 +transition function, 184, 185 +logarithmic CFT, 91 +mapping class group (MCG), see modular +group +marginal deformation +action, 239, 240 +correlation function, 240 +marked Riemann surface, see punctured Riemann +surface +metric +gauge +conformal (-), 32 +conformally flat (-), 32, 72 +flat (-), 32 +gauge decomposition, 39, 41, 43, 50 +gauge fixing, 31, 37 +uniformization gauge, 32 +local rescaling, see Weyl transformation +Mg, see moduli space +Möbius group, see SL(2, C) +mode expansion, 97 +Hermiticity, 98 +mode range, 97 +modular group, 31, 38 +moduli space, 21, 37 +complex coordinates, 76 +plumbing fixture decomposition, 190–194 +with punctures, 178 +momentum-space SFT +action, 261 +consistency, 264 +Feynman rules, 261 +finiteness +infinite number of states, 222, 261 +UV divergence, 222, 261 +Green function, 262 +interaction vertex, 261 +properties, 260 +string field expansion, 260 +Ngh, see ghost number +non-locality, 12, 260 +normal ordering, 103 +conformal (-), 103 +energy (-), 103 +313 + +mode relation, 105 +off-shell closed string amplitude, 179, 199 +contribution from subspace, 199 +tree-level +3-point, 172 +4-point, 218 +off-shell string amplitude, 23, 177 +conformal invariance, 172 +off-shell superstring amplitude, 250 +consistency, 250, 252 +factorization, 251–252 +PCO insertions, 250 +supermoduli space, 251 +old covariant quantization (OCQ), 146 +on-shell condition, 16, 60, 70, 92, 140, 144, +146, 147 +OPE, see operator product expansion +open string states +gauge field Aµ, 17 +gauge invariance, 17 +momentum expansion, 17 +tachyon T, 17 +operator product expansion, 94 +GG, 246 +T-primary, 95 +TG, 246 +TT, 95 +identity-primary, 94 +out-of-Siegel gauge constraint, 159 +p-brane, 11 +P1, 41, 42 +path integral +Faddeev–Popov gauge fixing, 36 +field redefinition, 283 +measure, 34, 282 +free-field, 34, 282 +ultralocality, 34, 49 +PCO, see picture changing operator +ˆPg,n, 184 +Pg,n space, 179 +p-form, 197, 198 +BRST identity, 203 +properties, 201–204 +0-form, 197 +1-form, 198 +coordinates, 185 +section, 179, 214 +vector, 185 +�Pg,m,n space, 250 +p-form, 250 +1-form, 250 +picture changing operator, 256 +picture number, 247 +anomaly, 247, 250 +plumbing fixture, 187, 208, 251 +p-form, 209 +ghost 1-form, 209 +moduli, 187 +non-separating, 190, 212 +separating, 187–190, 208, 222 +vector field, 209 +Polyakov path integral, 29 +complex representation, 75 +Faddeev–Popov gauge fixing, 36, 61 +primary operator/state, 88 +finite transformation, 88 +weight-0, 91 +projector +on-shell, ker L0, 140 +propagator, 139 +closed bosonic, 212, 215–217 +with stub, 222 +closed string, 174 +NS sector, 251 +open bosonic, 158 +R sector, 252 +Schwinger parametrization, 22, 216 +superstring, 256 +propagator region, 191 +puncture, see Riemann surface +punctured Riemann surface, 178 +Euler characteristics, 178 +parametrization, 182–183 +quadratic differential, 42 +quasi-primary operator/state, 88 +r(Σg,n), see index +radial quantization, 92 +reparametrization bc ghost +zero-mode, 69 +reparametrization bc ghosts, see also first- +order CFT, 51 +action, 51, 76 +central charge, 47 +equation of motion, 51 +zero-mode, 54 +Rg,n, see off-shell string amplitude contribu- +tion +Riemann sphere S2, 81 +complex plane map, 81 +cylinder map, 83 +314 + +Riemann surface +degeneration, 23 +g = 0, see Riemann sphere +genus, 29 +puncture, 20, 58 +scalar field CFT +L0, 115–116 +U(1) current, 108, 109 +U(1) symmetry, 108 +action, 107, 109 +boundary condition +periodic, 115 +BPZ conjugate +modes, 118 +vacuum, 118 +central charge, 111 +commutator, 116 +complex components, 109 +complex plane, 109 +cylinder, 110 +dual position, 114 +energy–momentum tensor, 107, 110 +equation of motion, 107, 109 +Euclidean adjoint +modes, 118 +vacuum, 118 +Fock space, 117 +Hilbert space, 117 +inner product, 118 +level operator, 115 +mode expansion, 16, 113–115 +momentum, 108, 110, 114, 115 +normal ordering, 113 +number operator, 115 +OPE, 111–113 +periodic boundary condition, 107 +position (center of mass), 114 +propagator, 107 +topological current, 108, 110 +vacuum, 117 +conjugate, 118 +vertex operator, 110 +Virasoro operators, 115 +winding number, 109, 110, 114, 115 +zero-mode, 114 +scattering amplitude, 59–61 +tree-level 2-point (-), 60 +Schwarzian derivative, 95 +Schwinger parameter, see propagator +section of Pg,n +generalized, 224 +overlap, 224 +SFT, see string field theory +Siegel gauge, 141, 158, 165, 169 +SL(2, C) group, 86 +SL(2, C) vacuum, 100 +S-matrix, see scattering amplitude +spacetime ghost number +closed string, 227 +open string, 161 +spacetime level-truncated action +open bosonic, 165 +gauge transformation, 166 +spacetime momentum, see also scalar field +CFT, 109 +spurious pole, 250, 252–255 +state (CFT) +bra, 101 +ket, 100 +state–operator correspondence, 100 +Stokes’ theorem, 75 +string +gauge group, 19 +interactions, 12, 20 +orientation, 19 +parameters, 21 +properties, 14 +string amplitude, 20 +CKV gauge fixing, 62 +closed string, see closed string amplitude +conformal invariance, 171 +divergence, 22 +factorization, 174, 208, 214 +ghost number, 204 +g-loop vacuum (-), 34, 49, 52, 54, 55, 76 +matching QFT, 59–60, 64 +normalization, 55, 58 +properties, 204–207 +pure gauge states decoupling, 206 +section independence, 206 +tree-level 2-point (-), 64–67 +string amplitudegn +g-loop n-point (-), 59, 63, 64 +string coupling constant, 21 +string Feynman diagram, see also momentum- +space SFT, 172 +1PR diagram, 211, 222 +change of stub parameter, 222 +Feynman rules, 213 +intermediate states +ghost number, 212, 213 +momentum, 212, 213 +IR divergence, 231 +315 + +loop diagram, 213 +string field, see also string states +closed bosonic, 168 +classical, 232 +expansion, 226, 235 +quantum, 235 +expansion, 152 +functional, 150 +gauge fixing, see Siegel gauge +ket representation, 151 +momentum expansion, 25, 151, 260 +open bosonic, 155 +classical, 156 +expansion, 160 +Nakanishi–Lautrup auxiliary field, 165 +parity, 156 +quantum, 161 +reality condition, 156 +position representation, 151 +string field path integral +Faddeev–Popov gauge fixing, 162 +free covariant open bosonic string, 162 +string field theory, 25 +construction, 23 +string states, see also closed, open (-), 14, +175–177 +dual, 176 +off-shell, 175 +on-shell, see on-shell condition +physical, see BRST cohomology +resolution of identity, 176 +string theory +background independence, 238 +consistency, 64, 264 +motivations, 11–13 +structure constant (CFT), 91 +stub, 194–195 +Feynman diagram, 222 +parameter, 194 +stub parameter, 232, 261 +super-SFT, 255 +superconformal βγ ghosts, see also first-order +CFT, 246 +bosonization, 246 +conformal weights, 246 +energy–momentum tensor, 246 +OPE, 246 +superstring, 18, 245 +BRST current, 247 +heterotic (-), 19, 245 +Hilbert space, 248 +large, 248 +picture number, 248 +small, 248 +motivations, 18 +normalization, 248 +type I (-), 19 +type II (-), 19 +superstring field +auxiliary field, 257 +constrained Ramond field, 256 +large Hilbert space, 258 +Ramond field, 255 +small Hilbert space, 256 +supersymmetry, 18 +surface state, 200, 209 +T-duality, 114 +tachyon, 17 +instability, 18 +Teichmüller deformation, 39, 41, 42 +Teichmüller space, 37 +ultralocality, see path integral measure +Verma module, 102 +vertex operator, see also scalar field CFT, see +also string states +integrated, 58 +unintegrated, 64 +Weyl invariance, 61 +vertex state, 223 +vertical integration, 253 +Vg,n, see fundamental vertex +V1PI +g,n , see 1PI vertex +Virasoro algebra, 90 +Virasoro operators, 90, 99 +Hermiticity, 99 +Weyl +anomaly, 35, 47, 49 +ghost, 52, 68 +group volume, 39 +symmetry, 31, 52 +Wick rotation, 262, 266 +generalized, 263 +Wick theorem, 103 +winding number, see also scalar field CFT +Witt algebra, 85 +worldsheet +action +Einstein–Hilbert (-), 56 +gauge-fixing (-), 68 +Nambu–Goto (-), 29 +316 + +Polyakov (-), 29 +sigma model (-), 30 +boundary conditions, 14 +CFT, 14, 18 +classification, 14 +cosmological constant, 48 +cylinder, 83 +energy–momentum tensor, 33 +trace, 33 +ghosts, see reparametrization ghosts +Nakanishi–Lautrup auxiliary field, 68 +path integral, see Polyakov path integral +Riemann surface, 20, 29 +symmetry, 30 +background diffeomorphisms, 50 +background Weyl, 50 +BRST, 68 +diffeomorphisms, 30 +Weyl, 30 +worldsheet metric, 29 +worldvolume description, 12 +X, see picture changing operator +Zamolodchikov metric, 91 +zero-mode, 269, 283 +zero-point energy, 101 +317 + diff --git a/19AzT4oBgHgl3EQfuP0B/content/tmp_files/load_file.txt b/19AzT4oBgHgl3EQfuP0B/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f62e0beae1ae3b746cd5ec9a2da2e3d425f862cc --- /dev/null +++ b/19AzT4oBgHgl3EQfuP0B/content/tmp_files/load_file.txt @@ -0,0 +1,18250 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf,len=18249 +page_content='String Field Theory – A Modern Introduction Harold Erbin1* Center for Theoretical Physics Massachusetts Institute of Technology, Cambridge, MA 02139, Usa Cea, List, Gif-sur-Yvette, F-91191, France 7th January 2023 1erbin@mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='edu arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='01686v1 [hep-th] 4 Jan 2023 Abstract This book provides an introduction to string field theory (SFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' String theory is usually formulated in the worldsheet formalism, which describes a single string (first-quantization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' While this approach is intuitive and could be pushed far due to the exceptional properties of two-dimensional theories, it becomes cumbersome for some questions or even fails at a more fundamental level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' These motivations have led to the development of SFT, a description of string theory using the field theory formalism (second-quantization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' As a field theory, SFT provides a rigorous and constructive formulation of string theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The main objective is to construct the closed bosonic SFT and to explain how to assess the consistency of string theory with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The accent is put on providing the reader with the foundations, conceptual understanding and intuition of what SFT is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' After reading this book, they should be able to study the applications from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The book is organized in two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The first part reviews the topics of the worldsheet theory that are necessary to build SFT (worldsheet path integral, CFT and BRST quant- ization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The second part starts by introducing general concepts of SFT from the BRST quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Then, it introduces off-shell string amplitudes before providing a Feynman diagrams interpretation from which the building blocks of SFT are extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' After con- structing the closed SFT, it is used to outline the proofs of several important consistency properties, such as background independence, unitarity and crossing symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Finally, the generalization to the superstring is also discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This book grew up from lecture notes for a course given at the Ludwig-Maximilians- Universität LMU (winter semesters 2017–2018 and 2018–2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The current document is the draft of the manuscript published by Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Preface This book grew up from lectures delivered within the Elite Master Program “Theoretical and Mathematical Physics” from the Ludwig-Maximilians-Universität during the winter semesters 2017–2018 and 2018–2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The main focus of this book is the closed bosonic string field theory (SFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' While there are many resources available for the open bosonic SFT, a single review [71] has been written since the final construction of the bosonic closed SFT by Zwiebach [262].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' For this reason, it makes sense to provide a modern and extensive study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Moreover, the usual approach to open SFT focuses on the cubic theory, which is so special that it is difficult to generalize the techniques to other SFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Finally, closed strings are arguably more fundamental than open strings because they are always present since they describe gravity, which further motivates my choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' However, the reader should not take this focus as denying the major achievements and the beauty of the open SFT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' reading this book should provide most of the tools needed to feel comfortable also with this theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' While part of the original goal of SFT is to provide a non-perturbative definition of string theory and to address important questions such as classifying consistent string backgrounds or understanding dualities, no progress on this front has been achieved so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Hence, there is still much to understand and the recent surge of developments provide a new chance to deepen our understanding of closed SFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' For example, several consistency properties of string theory have been proven rigorously using SFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Moreover, the recent construction of the open-closed superstring field theory [165] together with earlier works [42, 218, 262, 264] show that all types of string theories can be recast as a SFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This is why, I believe, it is a good time to provide a complete book on SFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The goal of this book is to offer a self-contained description of SFT and all the tools necessary to build it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The emphasis is on describing the concepts behind SFT and to make the reader build intuitions on what it means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' For this reason, there are relatively few applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The reader is assumed to have some knowledge of QFT, and a basic knowledge of CFT and string theory (classical string, Nambu–Goto action, light-cone and old-covariant quant- izations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Organization The text is organized on three levels: the main content (augmented with examples), compu- tations, and remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The latter two levels can be omitted in a first lecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The examples, computations and remarks are clearly separated from the text (respectively, by a half-box on the left and bottom, by a vertical line on the left, and by italics) to help the navigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Many computations have been set aside from the main text to avoid breaking the flow and to provide the reader with the opportunity to check by themself first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In some occasions, computations are postponed well below the corresponding formula to gather similar compu- tations or to avoid breaking an argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' While the derivations contain more details than 2 usual textbooks and may look pedantic to the expert, I think it is useful for students and newcomers to have complete references where to check each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This is even more the case when there are many different conventions in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The remarks are not directly relevant to the core of the text but they make connections with other parts or topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The goal is to broaden the perspectives of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' General references can be found at the end of each chapter to avoid overloading the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In-text references are reserved for specific points or explicit quotations (of a formula, a discussion, a proof, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' I did not try to be exhaustive in the citations and I have certainly missed important references: this should be imputed to my lack of familiarity with them and not to their value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This text is a preprint of the textbook [64] and is reproduced with permission of Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' My plan is to frequently update the draft of this book with new content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The last version can be accessed on my professional webpage, currently located at: http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='lpthe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='jussieu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='fr/~erbin/ Acknowledgements I have started to learn string field theory at Hri by attending lectures from Ashoke Sen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Since then, I have benefited from collaboration and many insightful discussions with him.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Following his lectures have been much helpful in building an intuition that cannot be found in papers or reviews on the topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Through this book, I hope being able to make some of these insights more accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' I am particularly grateful to Ivo Sachs who proposed me to teach this course and to Michael Haack for continuous support and help with the organization, and to both of them for many interesting discussions during the two years I have spent at Lmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Moreover, I have been very lucky to be assigned an excellent tutor for this course, Christoph Chiaffrino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' After providing him with the topic and few references, Christoph has prepared all the tutorials and the corrections autonomously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' His help brought a lot to the course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' I am particularly obliged to all the students who have taken this course at Lmu for many interesting discussions and comments: Enrico Andriolo, Hrólfur Ásmundsson, Daniel Bockisch, Fabrizio Cordonnier, Julian Freigang, Wilfried Kaase, Andriana Makridou, Pouria Mazloumi, Daniel Panea, Martin Rojo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' I am also grateful to all the string theory community for many exchanges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' For discussions related to the topics of this book, I would like to thank more particularly: Costas Bachas, Adel Bilal, Subhroneel Chakrabarti, Atish Dabholkar, Benoit Douçot, Ted Erler, Dileep Jatkar, Carlo Maccaferri, Juan Maldacena, Yuji Okawa, Sylvain Ribault, Raoul Santachiara, Martin Schnabl, Dimitri Skliros, Jakub Vošmera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' I have received a lot of feedback during the different stages of writing this book, and I am obliged to all the colleagues who sent me feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' I am thankful to my colleagues at Lmu for providing a warm and stimulating envir- onment, with special thanks to Livia Ferro for many discussions around coffee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Moreover, the encouragements and advice from Oleg Andreev and Erik Plauschinn have been strong incentives for publishing this book.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The editorial process at Springer has been very smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' I would also like to thank Christian Caron and Lisa Scalone for their help and efficiency during the publishing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' I am also indebted to Stefan Theisen for having supported the publication at Springer and for numerous comments and corrections on the draft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Most of this book was written at the Ludwig–Maximilians–Universität (Lmu, Munich, Germany) where I was supported by a Carl Friedrich von Siemens Research Fellowship of the Alexander von Humboldt Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The final stage has been completed at the University 3 of Turin (Italy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' My research is currently funded by the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No 891169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Finally, writing this book would have been more difficult without the continuous and loving support from Corinne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' November 2020 Harold Erbin 4 Contents Preface 2 1 Introduction 11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1 Strings, a distinguished theory .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2 String theory .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 48 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='5 Gauge-fixed path integral .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 139 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2 Relative cohomology .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 141 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3 Absolute cohomology, states and no-ghost theorem .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 146 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='4 Cohomology for holomorphic and anti-holomorphic sectors .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 147 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='4 Summary .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 167 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='6 Summary .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 169 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='7 Suggested readings .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 190 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3 Decomposition of moduli spaces and degeneration limit .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 191 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='4 Stubs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 194 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='4 Summary .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 195 12.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 215 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3 Fundamental vertices .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 222 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='5 1PI vertices .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1 String product .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 240 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='4 Expansion of the action .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 242 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='5 Relating the equations of motion .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3 Quantum BV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 293 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='4 Suggested readings .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 294 Bibliography 295 Index 310 10 Chapter 1 Introduction Abstract In this chapter, we introduce the main motivations for studying string theory, and why it is important to design a string field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' After describing the central features of string theory, we describe the most important concepts of the worldsheet formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Then, we explain the reasons leading to string field theory (SFT) and outline the ideas which will be discussed in the rest of the book.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1 Strings, a distinguished theory The first and simplest reason for considering theories of fundamental p-branes (fundamental objects extended in p spatial dimensions) can be summarized by the following question: “Why would Nature just make use of point-particles?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' There is no a priori reason forbidding the existence of fundamental extended objects and, according to Gell-Mann’s totalitarian principle, “Everything not forbidden is compulsory.” If a consistent theory cannot be built (after a reasonable amount of effort) or if it contradicts current theories (in their domains of validity) and experiments, then one can support the claim that only point-particles exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' On the other side, if such a theory can be built, it is of primary interest to understand it deeper and to see if it can solve the current problems in high-energy theoretical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The simplest case after the point particle is the string, so it makes sense to start with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' It happens that a consistent theory of strings can be constructed, and that the latter (in its supersymmetric version) contains all the necessary ingredients for a fully consistent high-energy model:1 quantum gravity (quantization of general relativity plus higher-derivative corrections);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' grand unification (of matter, interactions and gravity);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' no divergences, UV finiteness (finite and renormalizable theory);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' fixed number of dimensions (26 = 25 + 1 for the bosonic string, 10 = 9 + 1 for the supersymmetric version);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' existence of all possible branes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' no dimensionless parameters and one dimensionful parameter (the string length ℓs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' It can be expected that a theory of fundamental strings (1-branes) occupies a distinguished place among fundamental p-branes for the following reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 1There are also indications that a theory of membranes (2-branes) in 10+1 dimensions, called M-theory, should exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' No direct and satisfactory description of the latter has been found and we will thus focus on string theory in this book.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 11 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1: Locality of a particle interaction: two different observers always agree on the interaction point and which parts of the worldline are 1- and 2-particle states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Interaction non-locality In a QFT of point particles, UV divergences arise because interactions (defined as the place where the number and/or nature of the objects change) are arbitrarily localized at a spacetime point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In Feynman graphs, such divergences can be seen when the momentum of a loop becomes infinite (two vertices collide): this happens when trying to concentrate an infinite amount of energy at a single point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' However, these divergences are expected to be reduced or absent in a field theory of extended objects: whereas the interaction between particles is perfectly local in spacetime and agreed upon by all observers (Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1), the spatial extension of branes makes the interactions non-local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This means that two different observers will neither agree on the place of the interactions (Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2), nor on the part of the diagram which describes one or two branes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The string lies at the boundary between too much local and too much non-local: in any given frame, the interaction is local in space, but not in spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The reason is that a string is one-dimensional and splits or joins along a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' For p > 1, the brane needs to break/join along an extended spatial section, which looks non-local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Another consequence of the non-locality is a drastic reduction of the possible interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' If an interaction is Lorentz invariant, Lorentz covariant objects can be attached at the vertex (such as momentum or gamma matrices): this gives Lorentz invariants after contracting with indices carried by the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' But, this is impossible if the interaction itself is non-local (and thus not invariant): inserting a covariant object would break Lorentz invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Brane degrees of freedom The higher the number of spatial dimensions of a p-brane, the more possibilities it has to fluctuate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' As a consequence, it is expected that new divergences appear as p increases due to the proliferations of the brane degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' From the worldvolume perspective, this is understood from the fact that the worldvolume theory describes a field theory in (p + 1) dimensions, and UV divergences become worse as the number of dimensions increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The limiting case happens for the string (p = 1) since two-dimensional field theories are well-behaved in this respect (for example, any monomial interaction for a scalar field is power-counting renormalizable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This can be explained by the low-dimensionality of the momentum integration and by the enhancement of symmetries in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Hence, strings should display nice properties and are thus of special interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Worldvolume theory The point-particle (0-brane) and the string (1-brane) are also re- markable in another aspect: it is possible to construct a simple worldvolume field theory (and the associated functional integral) in terms of a worldvolume metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' All components of the latter are fixed by gauge symmetries (diffeomorphisms for the particle, diffeomorph- isms and Weyl invariance for the string).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This ensures the reparametrization invariance of 12 (a) Observers at rest and boosted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (b) Observers close to the speed of light moving in opposite dir- ections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The interactions are widely separated in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2: Non-locality of string interaction: two different observers see the interaction happening at different places (denoted by the filled and empty circles) and they don’t agree on which parts of the worldsheet are 1- and 2-string states (the litigation is denoted by the grey zone).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' the worldvolume without having to use a complicated action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Oppositely, the worldvolume metric cannot be completely gauge fixed for p > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Summary As a conclusion, strings achieve an optimal balance between spacetime and worldsheet divergences, as well as having a simple description with reparametrization in- variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Since the construction of a field theory is difficult, it is natural to start with a worldsheet theory and to study it in the first-quantization formalism, which will provide a guideline for writing the field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In particular, this allows to access the physical states in a simple way and to find other general properties of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' When it comes to the interactions and scattering amplitudes, this approach may be hopeless in general since the topology of the worldvolume needs to be specified by hand (describing the interaction process).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In this respect, the case of the string is again exceptional: because Riemann surfaces have been classified and are well-understood, the arbitrariness is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Combined with the tools of conformal field theory, many computations can be performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Moreover, since the modes of vibrations of the strings provide all the necessary ingredients to describe the Standard model, it is sufficient to consider only one string field (for one type of strings), instead of the plethora found in point-particle field theory (one field for each particle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Similarly, non-perturbative information (such as branes and dualities) could be found only due to the specific properties of strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Coming back to the question which opened this section, higher-dimensional branes of all the allowed dimensions naturally appear in string theory as bound states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Hence, even if the worldvolume formulation of branes with p > 1 looks pathological2, string theory hints towards another definition of these objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 2Entering in the details would take us too far away from the main topic of this book.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Some of the problems found when dealing with (p > 2)-branes are: how to define a Wick rotation for 3-manifolds, the presence of Lorentz anomalies in target spacetime, problems with the spectrum, lack of renormalizability, impossibility to gauge-fix the worldvolume metric [5–11, 41, 44, 45, 62, 127, 152, 156–158, 181, 193].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2 String theory 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1 Properties The goal of this section is to give a general idea of string theory by introducing some concepts and terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The reader not familiar with the points described in this section is advised to follow in parallel some standard worldsheet string theory textbooks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Worldsheet CFT A string is characterized by its worldsheet field theory (Chapter 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3 The worldsheet is parametrized by coordinates σa = (τ, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The simplest description is obtained by endowing the worldsheet with a metric gab(σa) (a = 0, 1) and by adding a set of D scalar fields Xµ(σa) living on the worldsheet (µ = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' , D − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The latter represents the position of the string in the D-dimensional spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' From the classical equations of motion, the metric gab is proportional to the metric induced on the worldsheet from its embedding in spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' More generally, one ensures that the worldsheet metric is non-dynamical by imposing that the action is invariant under (worldsheet) diffeomorphisms and under Weyl transformations (local rescalings of the metric).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The consistency of these conditions at the quantum level imposes that D = 26, and this number is called the critical dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Gauge fixing the symmetries, and thus the metric, leads to the conformal invariance of the resulting worldsheet field theory: a conformal field theory (CFT) is a field theory (possibly on a curved background) in which only angles and not distances can be measured (Chapters 5 to 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This simplifies greatly the analysis since the two-dimensional conformal algebra (called the Virasoro algebra) is infinite-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' CFTs more general than D free scalar fields can be considered: fields taking non-compact values are interpreted as non-compact dimensions while compact or Grassmann-odd fields are interpreted as compact dimensions or internal structure, like the spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' While the light-cone quantization allows to find quickly the states of the theory, the simplest covariant method is the BRST quantization (Chapter 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' It introduces ghosts (and superghosts) associated to the gauge fixing of diffeomorphisms (and local supersymmetry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' These (super)ghosts form a CFT which is universal (independent of the matter CFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The trajectory of the string is denoted by xc(τ, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' It begins and ends respectively at the geometric shapes parametrized by xc(τi, σ) = xi(σ) and by xc(τf, σ) = xf(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Note that the coordinate system on the worldsheet itself is arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The spatial section of a string can be topologically closed (circle) or open (line) (Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3), leading to cylindrical or rectangular worldsheets as illustrated in Figures 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='4 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' To each topology is associated different boundary conditions and types of strings: closed: periodic and anti-periodic boundary conditions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' open: Dirichlet and Neumann boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' While a closed string theory is consistent by itself, an open string theory is not and requires closed strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Spectrum In order to gain some intuition for the states described by a closed string, one can write the Fourier expansion of the fields Xµ (in the gauge gab = ηab and after imposing the equations 3We focus mainly on the bosonic string theory, leaving aside the superstring, except when differences are important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 14 (a) Open string (b) Closed string Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3: Open and closed strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' −−−−−−−−→ Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='4: Trajectory xµ c (τ, σ) of a closed string in spacetime (worldsheet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' It begins and ends at the circles parametrized by xi(σ) and xf(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The worldsheet is topologically a cylinder and is parametrized by (τ, σ) ∈ [τi, τf] × [0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' −−−−−−−−→ Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='5: Trajectory xµ c (τ, σ) of an open string in spacetime (worldsheet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' It begins and ends at the lines parametrized by xi(σ) and xf(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The worldsheet is topologically a rectangle and is parametrized by (τ, σ) ∈ [τi, τf] × [0, ℓ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 15 of motion) Xµ(τ, σ) ∼ xµ + pµτ + i √ 2 � n∈Z∗ 1 n � αµ ne−in(τ−σ) + ¯αµ ne−in(τ+σ)� , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1) where xµ is the centre-of-mass position of the string and pµ its momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='4 Canonical quantization leads to the usual commutator: [xµ, pν] = iηµν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2) With respect to a point-particle for which only the first two terms are present, there are an infinite number of oscillators αµ n and ¯αµ n which satisfy canonical commutation relations for creation n < 0 and annihilation operators n > 0 [αµ m, αν n] = m ηµνδm+n,0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3) The non-zero modes are the Fourier modes of the excitations of the embedded string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The case of the open string is simply obtained by setting ¯αn = αn and p → 2p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The Hamiltonian for the closed and open strings read respectively Hclosed = −m2 2 + N + ¯N − 2 , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='4a) Hopen = −m2 + N − 1 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='4b) where m2 = −pµpµ is the mass of the state (in Planck units), N and ¯N (level operators) count the numbers Nn and ¯Nn of oscillators αn and ¯αn weighted by their mode index n: N = � n∈N nNn , Nn = 1 n α−n · αn , ¯N = � n∈N n ¯Nn , ¯Nn = 1 n ¯α−n · ¯αn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='5) With these elements, the Hilbert space of the string theory can be constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Invariance under reparametrization leads to the on-shell condition, which says that the Hamiltonian vanishes: H |ψ⟩ = 0 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='6) for any physical state |ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Another constraint for the closed string is the level-matching condition (N − ¯N) |ψ⟩ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='7) It can be understood as fixing an origin on the string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The ground state |k⟩ with momentum k is defined to be the eigenstate of the momentum operator which does not contain any oscillator excitation: pµ |k⟩ = kµ |k⟩ , ∀n > 0 : αµ n |k⟩ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='8) A general state can be built by applying successively creation operators |ψ⟩ = � n>0 D−1 � µ=0 (αµ −n)Nn,µ |k⟩ , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='9) 4In the introduction, we set α′ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 16 where Nn,µ ∈ N counts excitation level of the oscillator αµ −n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In the rest of this section, we describe the first two levels of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The ground state is a tachyon (faster-than-light particle) because the Hamiltonian con- straint shows that it has a negative mass (in the units where α′ = 1): closed : m2 = −4 , open : m2 = −1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='10) The first excited state of the open string is found by applying α−1 on the vacuum |k⟩: αµ −1 |k⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='11) This state is massless: m2 = 0 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='12) and since it transforms as a Lorentz vector (spin 1), it is identified with a U(1) gauge boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Writing a superposition of such states |A⟩ = � dDk Aµ(k) αµ −1 |k⟩ , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='13) the coefficient Aµ(k) of the Fourier expansion is interpreted as the spacetime field for the gauge boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Reparametrization invariance is equivalent to the equation of motion k2Aµ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='14) One can prove that the field obeys the Lorentz gauge condition kµAµ = 0 , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='15) which results from gauge fixing the U(1) gauge invariance Aµ −→ Aµ + kµλ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='16) It can also be checked that the low-energy action reproduces the Maxwell action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The first level of the closed string is obtained by applying both α−1 and ¯α−1 (this is the only way to match N = ¯N at this level) αµ −1¯αν −1 |k⟩ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='17) and the corresponding states are massless m2 = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='18) These states can be decomposed into irreducible representations of the Lorentz group � αµ −1¯αν −1 + αν −1¯αµ −1 − 1 D ηµνα−1 · ¯α−1 � |p⟩ , � αµ −1¯αν −1 − αν −1¯αµ −1 � |p⟩ , 1 D ηµναµ −1¯αν −1 |p⟩ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='19) which are respectively associated to the spacetime fields Gµν (metric, spin 2), Bµν (Kalb– Ramond 2-form) and Φ (dilaton, spin 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The appearance of a massless spin 2 particle (with low-energy action being the Einstein–Hilbert action) is a key result and originally raised interest for string theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 17 Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1 (Reparametrization constraints) Reparametrization invariance leads to other constraints than H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' They imply in particular that the massless fields have the correct gauge invariance and hence the correct degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Note that, after taking into account these constraints, the remaining modes correspond to excitations of the string in the directions transverse to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Hence, each vibrational mode of the string corresponds to a spacetime field for a point- particle (and linear superpositions of modes can describe several fields).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This is how string theory achieves unification since a single type of string (of each topology) is sufficient for describing all the possible types of fields encountered in the standard model and in gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' They correspond to the lowest excitation modes, the higher massive modes being too heavy to be observed at low energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Bosonic string theory includes tachyons and is thus unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' While the instability of the open string tachyon is well understood and indicates that open strings are unstable and condense to closed strings, the status of the closed string tachyon is more worrisome (literally interpreted, it indicates a decay of spacetime itself).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In order to solve this problem, one can introduce supersymmetry: in this case, the spectrum does not include the tachyon because it cannot be paired with a supersymmetric partner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Moreover, as its name indicates, the bosonic string possesses only bosons in its spectrum (perturbatively), which is an important obstacle to reproduce the standard model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' By introducing spacetime fermions, supersymmetry also solves this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The last direct advantage of the superstring is that it reduces the number of dimensions from 26 to 10, which makes the compactification easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2 Classification of superstring theories In this section, we describe the different superstring theories (Chapter 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In order to proceed, we need to introduce some new elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The worldsheet field theory of the closed string is made of two sectors, called the left- and right-moving sectors (the αn and ¯αn modes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' While they are treated symmetrically in the simplest models, they are in fact independent (up to the zero-mode) and the corresponding CFT can be chosen to be distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The second ingredient already evoked earlier is supersymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This symmetry associ- ates a fermion to each boson (and conversely) through the action of a supercharge Q |boson⟩ = Q |fermion⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='20) More generally, one can consider N supercharges which build up a family of several bosonic and fermionic partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Since each supercharge increases the spin by 1/2 (in D = 4), there is an upper limit for the number of supersymmetries – for interacting theories with a finite number of fields5 – in order to keep the spin of a family in the range where consistent actions exist: Nmax = 4 without gravity (−1 ≤ spin ≤ 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Nmax = 8 with gravity (−2 ≤ spin ≤ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This counting serves as a basis to determine the maximal number of supersymmetries in other dimensions (by relating them through dimensional reductions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Let’s turn our attention to the case of the two-dimensional worldsheet theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The number of supersymmetries of the closed left- and right-moving sectors can be chosen in- dependently, and the number of charges is written as (NL, NR) (the index is omitted when 5These conditions exclude the cases of free theories and higher-spin theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 18 statements are made at the level of the CFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The critical dimension (absence of quantum anomaly for the Weyl invariance) depends on the number of supersymmetry D(N = 0) = 26, D(N = 1) = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='21) Type II superstrings have (NL, NR) = (1, 1) and come in two flavours called IIA and IIB according to the chiraly of the spacetime gravitini chiralities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' A theory is called heterotic if NL > NR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' we will mostly be interested in the case NL = 1 and NR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='6 In such theories, there cannot be open strings since both sectors must be equal in the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Since the critical dimensions of the two sectors do not match, one needs to get rid of the additional dimensions of the right-moving sector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' this leads to the next topic – gauge groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Gauge groups associated with spacetime gauge bosons appear in two different places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In heterotic models, the compactification of the unbalanced dimensions of the left sector leads to the appearance of a gauge symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The possibilities are scarce due to consistency conditions which ensure a correct gluing with the right-sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Another possibility is to add degrees of freedom – known as Chan–Paton indices – at the ends of open strings: one end transforms in the fundamental representation of a group G, while the other end transforms in the anti-fundamental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The modes of the open string then reside in the adjoint representation, and the massless spin-1 particles become the gauge bosons of the non-Abelian gauge symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Finally, one can consider oriented or unoriented strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' An oriented string possesses an internal direction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' there is a distinction between going from the left to the right (for an open string) or circling in clockwise or anti-clockwise direction (for a closed string).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Such an orientation can be attributed globally to the spacetime history of all strings (interacting or not).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The unoriented string is obtained by quotienting the theory by the Z2 worldsheet parity symmetry which exchanges the left- and right-moving sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Applying this to the type IIB gives the type I theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The tachyon-free superstring theories together with the bosonic string are summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' worldsheet susy D spacetime susy gauge group open string oriented tachyon bosonic (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 0) 26 0 any* yes yes / no yes type I (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 1) 10 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 0) SO(32) yes no no type IIA (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 1) 10 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 1) U(1) (yes)† yes no type IIB (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 1) 10 (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 0) none (yes)† yes no heterotic SO(32) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 0) 10 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 0) SO(32) no yes no heterotic E8 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 0) 10 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 0) E8 × E8 no yes no heterotic SO(16) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 0) 10 (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 0) SO(16) × SO(16) no yes no UV divergences beyond the tachyon (interpreted as closed string dilaton tadpoles) cancel only for the unoriented open plus closed strings with gauge group SO(213) = SO(8192).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' † The parenthesis indicates that type II theories don’t have open strings in the vacuum: they require a D-brane background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This is expected since there is no gauge multiplet in d = 10 (1, 1) or (2, 0) supergravities (the D-brane breaks half of the supersymmetry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1: List of the consistent tachyon-free (super)string theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The bosonic theory is added for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' There are additional heterotic theories without spacetime supersym- metry, but they contain a tachyon and are thus omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 6The case NL < NR is identical up to exchange of the left- and right-moving sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 19 (a) Closed strings (b) Open strings Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='6: Graphs corresponding to 1-loop 4-point scattering after a conformal mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3 Interactions Worldsheet and Riemann surfaces After having described the spectrum and the general characteristics of string theory comes the question of interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The worldsheets obtained in this way are Riemann surfaces, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 1-dimensional complex manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' They are classified by the numbers of handles (or holes) g (called the genus) and external tubes n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In the presence of open strings, surfaces have boundaries: in addition to the handles and tubes, they are classified by the numbers of disks b and of strips m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='7 A particularly important number associated to each surface is the Euler characteristics χ = 2 − 2g − b , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='22) which is a topological invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' It is remarkable that there is a single topology at every loop order when one considers only closed strings, and just a few more in the presence of open strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The analysis is greatly simplified in contrast to QFT, for which the number of Feynman graphs increases very rapidly with the number of loops and external particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Due to the topological equivalence between surfaces, a conformal map can be used in order to work with simpler surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In particular, the external tubes and strips are collapsed to points called punctures (or marked points) on the corresponding surfaces or boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' A general amplitude then looks like a sphere from which holes and disks have been removed and to which marked points have been pierced (Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Amplitudes In order to compute an amplitude for the scattering of n strings (Chapters 3 and 4), one must sum over all the inequivalent worldsheets through a path integral weighted by the CFT action chosen to describe the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='8 At fixed n, the sum runs over the genus g, such that each term is described by a Riemann surface Σg,n of genus g with n punctures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The interactions between strings follow from the graph topologies: since the latter are not encoded into the action, the dependence in the coupling constant must be added by hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' For closed strings, there is a unique cubic vertex with coupling gs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' A direct inspection shows that the correct factor is gn−2+2g s : 7We ignore unoriented strings in this discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' They would lead to an additional object called a cross-cap, which is a place where the surface looses its orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 8For simplicity we focus on closed string amplitudes in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 20 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='7: General Riemann surfaces with boundaries and punctures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' for n = 3 there is one factor gs, and every additional external string leads to the addition of one vertex with factor gs, since this process can be obtained from the n−1 process by splitting one of the external string in two by inserting a vertex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' each loop comes with two vertices, so g-loops provide a factor g2g s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2 (Status of gs as a parameter) It was stated earlier that string theory has no dimensionless parameter, but gs looks to be one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In reality it is determined by the expect- ation value of the dilaton gs = e⟨Φ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Hence the coupling constant is not a parameter defining the theory but is rather determined by the dynamics of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Finally, the external states must be specified: this amounts to prescribe boundary condi- tions for the path integral or to insert the corresponding wave functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Under the conformal mapping which brings the external legs to punctures located at zi, the states are mapped to local operators Vi(ki, zi) inserted at the points zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The latter are built from the CFT fields and are called vertex operators: they are characterized by a momentum kµ which comes from the Fourier transformation of the Xµ fields representing the non-compact dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' These operators are inserted inside the path integral with integrals over the positions zi in order to describe all possible conformal mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Ultimately, the amplitude (amputated Green function) is computed as An(k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' , kn) = � g≥0 gn−2+2g s Ag,n (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='23) where Ag,n = � n � i=1 d2zi � dgabdΨ e−Scft[gab,Ψ] n � i=1 Vi(ki, zi) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='24) is the g-loop n-point amplitude (for simplicity we omit the dependence on the states beyond the momentum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Ψ denotes collectively the CFT fields and gab is the metric on the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The integration over the metrics and over the puncture locations contain a huge redund- ancy due to the invariance under reparametrizations, which means that one integrates over many equivalent surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' To avoid this, Faddeev–Popov ghosts must be introduced and the integral is restricted to only finitely many (real) parameters tλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' They form the moduli space Mg,n of the Riemann surfaces Σg,n whose dimension is dimR Mg,n = 6g − 6 + 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='25) 21 The computation of the amplitude Ag,n can be summarized as: Ag,n = � Mg,n 6g−6+2n � λ=1 dtλ F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='26) The function F(t) is a correlation function in the worldsheet CFT defined on the Riemann surface Σg,n F(t) = � n � i=1 Vi × ghosts × super-ghosts � Σg,n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='27) Note that the (super)ghost part is independent of the choice of the matter CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Divergences and Feynman graphs Formally the moduli parameters are equivalent to Schwinger (proper-time) parameters si in usual QFT: these are introduced in order to rewrite propagators as 1 k2 + m2 = � ∞ 0 ds e−s(k2+m2), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='28) such that the integration over the momentum k becomes a Gaussian times a polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This form of the propagator is useful to display the three types of divergences which can be encountered: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' IR: regions si → ∞ (for k2 + m2 ≤ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' These divergences are artificial for k2 + m2 < 0 and means that the parametrization is not appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Divergences for k2 + m2 = 0 are genuine and translates the fact that quantum effects shift the vacuum and the masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Taking these effects into account necessitates a field theory framework in which renormalization can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' UV: regions si → 0 (after integrating over k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Such divergences are absent in string theories because these regions are excluded from the moduli space Mg,n (see Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='8 for the example of the torus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Spurious: regions with finite si where the amplitude diverges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This happens typically only in the presence of super-ghosts and it translates a breakdown of the gauge fixing condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='10 Since these spurious singularities of the amplitudes are not physical, one needs to ensure that they can be removed, which is indeed possible to achieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Hence, only IR divergences present a real challenge to string theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Dealing with these divergences requires renormalizing the amplitudes, but this is not possible in the standard formulation of worldsheet string theory since the states are on-shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='11 9There is a caveat to this statement: UV divergences reappear in string field theory in Lorentzian signature due to the way the theory is formulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The solution requires a generalization of the Wick rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Moreover, this does not hold for open strings whose moduli spaces contains those regions: in this case, the divergences are reinterpreted in terms of closed strings propagating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 10Such spurious singularities are also found in supergravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 11The on-shell condition is a consequence of the BRST and conformal invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' While the first will be given up, the second will be maintained to facilitate the computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 22 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='8: Moduli space of the torus: Re τ ∈ [−1/2, 1/2], Im τ > 0 and |τ| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3 String field theory 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1 From the worldsheet to field theory The first step is to solve the IR divergences problem is to go off-shell (Chapters 11 and 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This is made possible by introducing local coordinates around the punctures of the Riemann surface (Chapter 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The IR divergences originate from Riemann surfaces close to degeneration, that is, sur- faces with long tubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The latter can be of separating and non-separating types, depending on whether the Riemann surface splits in two pieces if the tube is cut (Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' By exploring the form of the amplitudes in this limit (Chapter 14), the expression naturally separates into several pieces, to be interpreted as two amplitudes (of lower n and g) con- nected by a propagator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The latter can be reinterpreted as a standard (k2 + m2)−1 term, hence solving the divergence problem for k2 + m2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Taking this decomposition seriously leads to identify each contribution with a Feynman graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Decomposing the amplitude recursively, the next step consists in finding the elementary graphs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' the interaction vertices from which all other graphs (and amplitudes) can be built.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' These graphs are the building blocks of the field theory (Chapter 15), with the kinetic term given by the inverse of the propagator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Having Feynman diagrams and a field theory allows to use all the standard tools from QFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' However, this field theory is gauge fixed because on-shell amplitudes are gauge invariant and include only physical states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' For this reason, one needs to find how to re-establish the gauge invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Due to the complicated structure of string theory, the full-fledged Batalin–Vilkovisky (BV) formalism must be used (Chapter 15): it basically amounts to introduce ghosts before the gauge fixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The final stage is to obtain the 1PI effective action from which the physics is more easily extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' But, it is useful to study first the free theory (Chapters 9 and 10) to gain some insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The book ends with a discussion of the momentum-space representation and of background independence (Chapters 16 and 18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The procedure we will follow is a kind of reverse-engineering: we know what is the final 23 (a) Separating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (b) Non-separating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='9: Degeneration of Riemann surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' result and we want to study backwards how it is obtained: on-shell amplitude → off-shell amplitude → Feynman graphs → gauge fixed field theory → BV field theory In standard QFT, one follows the opposite process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3 There are some prescriptions (using for example analytic continuation, the optical theorem, some tricks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' ) to address the problems mentioned above, but there is no general and universally valid procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' A field theory is much more satisfactory because it provides a unique and complete framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' We can now summarise the disadvantages of the worldsheet approach over the spacetime field one: no natural description of (relativistic) multi-particle states;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' on-shell states: – lack of renormalization, – presence of infrared divergences, – scattering amplitudes only for protected states;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' interactions added by hand;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' hard to check consistency (unitarity, causality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' absence of non-perturbative processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Some of these problems can be addressed with various prescriptions, but it is desirable to dispose of a unified and systematic procedure, which is to be found in the field theory description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 24 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2 String field action A string field theory (SFT) for open and closed strings is based on two fields Φ[X(σ)] (open string field) and Ψ[X(σ)] (closed string field) governed by some action S[Φ, Ψ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This action is built from a diagonal kinetic term S0 = 1 2 KΨ(Ψ, Ψ) + 1 2 KΦ(Φ, Φ) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='29) and from an interaction polynomial in the fields Sint = � m,n Vm,n(Φm, Ψn) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='30) where Vm,n is an appropriate product mapping m closed and n open string states to a number (the power is with respect to the tensor product).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In particular, it contains the coupling constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Contrary to the worldsheet approach where the cubic interaction looks sufficient, higher-order elementary interactions with m, n ∈ N are typically needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' A second specific feature is that the products also admit a loop (or genus g) expansion: a fundamental n- point interaction is introduced at every loop order g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' These terms are interpreted as (finite) counter-terms needed to restore the gauge invariance of the measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' These two facts come from the decomposition of the moduli spaces in pieces (Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Writing an action for a field Ψ[X(σ)] for which reparametrization invariance holds is highly complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The most powerful method is to introduce a functional dependence in ghost fields Ψ[X(σ), c(σ)] and to extend the BRST formalism to the string field, leading ultimately to the BV formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' While the latter formalism is the most complete and ensures that the theory is consistent at the quantum level, it is difficult to characterize the interactions explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Several constructions which exploit different properties of the theory have been proposed: direct computation by reverse engineering of worldsheet amplitudes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' specific parametrization of the Riemann surfaces (hyperbolic, minimal area);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' analogy with Chern–Simons and Wess–Zumino–Witten (WZW) theories;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' exploitation of the L∞ and A∞ algebra structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' It can be shown that these constructions are all equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' For the superstring, the simplest strategy is to dress the bosonic interactions with data from the super-ghost sector, which motivates the study of the bosonic SFT by itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The main difficulty in working with SFT is that only the first few interactions have been constructed explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Finally, the advantage of the first formulation is that it provides a general formulation of SFT at the quantum level, from which the general structure can be studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3 Expression with spacetime fields To obtain a more intuitive picture and to make contact with the spacetime fields, the field is expanded in terms of 1-particle states in the momentum representation |Ψ⟩ = � n � dDk (2π)D ψα(k) |k, α⟩ , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='31) where α denotes collectively the discrete labels of the CFT eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The coefficients ψα(k) of the CFT states |k, α⟩ are spacetime fields, the first ones being the same as those found in the first-quantized picture (Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1) ψα = {T, Gµν, Bµν, Φ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='32) 25 Then, inserting this expansion in the action gives an expression like S[T, Gµν, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The exact expression of this action is out of reach and only the lowest terms are explicitly computable for a given CFT background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Nonetheless, examining the string field action indicates what is the generic form of the action in terms of the spacetime fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' One can then study the properties of such a general QFT: since it is more general than the SFT (expanded) action, any result derived for it will also be valid for SFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This approach is very fruitful for studying properties related to consistency of QFT (unitarity, soft theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' ) and this can provide helpful phenomenological models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In conclusion, SFT can be seen as a regular QFT with the following properties: infinite number of fields;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' non-local interaction (proportional to e−k2#);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' the amplitudes agree with the worldsheet amplitudes (when the latter can be defined);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' genuine (IR) divergences agree but can be handled with the usual QFT tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='4 Applications The first aspect is the possibility to use standard QFT techniques (such as renormalization) to study – and to make sense of – string amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In this sense, SFT can be viewed as providing recipes for computing quantities in the worldsheet theory which are otherwise not defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This program has been pushed quite far in the last years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Another reason to use SFT is gauge invariance: it is always easier to describe a sys- tem when its gauge invariance is manifest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' We have explained that string theory contains Yang–Mills and graviton fields with the corresponding (spacetime) gauge invariances (non- Abelian gauge symmetry and diffeomorphisms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In fact, these symmetries are enhanced to an enormous gauge invariance when taking into account the higher-spin fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This invari- ance is hidden in the standard formulation and cannot be exploited fully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' On the other hand, the full gauge symmetry is manifest in string field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Finally, the worldvolume description of p-brane is difficult because there is no analogue of the Polyakov action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' If one could find a first-principle description of SFT which does not rely on CFT and first-quantization, then one may hope to generalize it to build a brane field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' We can summarize the general motivations for studying SFT: field theory (second-quantization);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' more rigorous and constructive formulation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' make gauge invariance explicit (L∞ algebras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' use standard QFT techniques (renormalization, analyticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' ) → remove IR divergences, prove consistency (Cutkosky rules, unitarity, soft theorems, background independence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' worldvolume theory ill-defined for (p > 1)-branes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Beyond these general ideas, SFT has been developed in order to address different questions: worldsheet scattering amplitudes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' effective actions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' map of the consistent backgrounds (classical solutions, marginal deformations, RR fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 26 collective, non-perturbative, thermal, dynamical effects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' symmetry breaking effects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' dynamics of compactification;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' proof of dualities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' proof of the AdS/CFT correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The last series of points is still out of reach within the current formulation of SFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' However, the last two decades have seen many important develoments developments: construction of the open, closed and open-closed superstring field theories: – 1PI and BV actions and general properties [73, 165, 166, 213, 214, 216, 218, 220, 222, 225, 226, 230], – dressing of bosonic products using the WZW construction and homotopy al- gebra [18, 19, 67–70, 74–76, 79, 80, 94, 111, 131, 134, 140–146, 180], – light-cone super-SFT [114–117], – supermoduli space [175, 241];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' hyperbolic and minimal area constructions [38, 102, 103, 162–164, 183];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' open string analytic solutions [77, 78];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' level-truncation solutions [135–137];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' field theory properties [34, 43, 150, 187, 221, 223, 224];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' spacetime effective actions [65, 153, 154, 248];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' defining worldsheet scattering amplitudes [184–186, 215, 216, 219, 227–229];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' marginal and RR deformations [35, 229, 248].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Recent reviews are [42, 71, 72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='4 Suggested readings For references about different aspects in this chapter: Differences between the worldvolume and spacetime formalisms – and of the associated first- and second-quantization – for the particle and string [124, chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 1, 265, chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' General properties of relativistic strings [92, 265].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Divergences in string theory [42, 217, 256, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Motivations for building a string field theory [192, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 27 Part I Worldsheet theory 28 Chapter 2 Worldsheet path integral: vacuum amplitudes Abstract In this chapter, we develop the path integral quantization for a generic closed string theory in worldsheet Euclidean signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' We focus on the vacuum amplitudes, leaving scattering amplitudes for the next chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This allows to focus on the definition and gauge fixing of the path integral measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The exposition differs from most traditional textbooks in three ways: 1) we consider a general matter CFT, 2) we consider the most general treatment (for any genus) and 3) we don’t use complex coordinates but always a covariant parametrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The derivation is technical and the reader is encouraged to not stop at this chapter in case of difficulties and to proceed forward: most concepts will be reintroduced from a different point of view later in other chapters of the book.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1 Worldsheet action and symmetries The string worldsheet is a Riemann surface W = Σg of genus g: the genus counts the number of holes or handles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Coordinates on the worldsheet are denoted by σa = (τ, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' When there is no risk of confusion, σ denotes collectively both coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Since closed strings are considered, the Riemann surface has locally the topology of a cylinder, with the spatial section being circles S1 with radius taken to be 1, such that σ ∈ [0, 2π), σ ∼ σ + 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1) The string is embedded in the D-dimensional spacetime M with metric Gµν through maps Xµ(σa) : W → M with µ = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' , D − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The Nambu–Goto action is the starting point of the worldsheet description: SNG[Xµ] = 1 2πα′ � d2σ � det Gµν(X)∂Xµ ∂σa ∂Xν ∂σb , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2) where α′ is the Regge slope (related to the string tension and string length).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' However, quantizing this action is difficult because it is highly non-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' To solve this problem, a Lagrange multiplier is introduced to remove the squareroot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This auxiliary field corresponds to an intrinsic worldsheet metric gab(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The worldsheet dynamics is described by the Polyakov action: SP[g, Xµ] = 1 4πα′ � d2σ√g gabGµν(X)∂Xµ ∂σa ∂Xν ∂σb , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3) 29 which is classically equivalent to the Nambu–Goto action (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In this form, it is clear that the scalar fields Xµ(σ) (µ = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' D − 1) characterize the string theory under consideration in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' First, by specifying some properties of the spacetime in which the string propagates (for example, the number of dimensions is determined by the number of fields Xµ), second, by describing the internal degrees of freedom (vibration modes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1 But, nothing prevents to consider a more general matter content in order to describe a different spacetime or different degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In Polyakov’s formalism, the worldsheet geometry is endowed with a metric gab(σ) together with a set of matter fields living on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The scalar fields Xµ can be described by a general sigma model which encodes the embedding of the string in the D non-compact spacetime dimensions, and other fields can be added, for example to describe compactified dimensions or (spacetime) spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Different sets of fields (and actions) correspond to different string theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' However, to describe precisely the different possibilities, we first have to understand the constraints on the worldsheet theories and to introduce conformal field theories (Part I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In this chapter (and in most of the book), the precise matter content is not important and we will denote the fields collectively as Ψ(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Before discussing the symmetries, let’s introduce a topological invariant which will be needed throughout the text: the Euler characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' It is computed by integrating the Riemann curvature R of the metric gab over the surface Σg: χg := χ(Σg) := 2 − 2g = 1 4π � Σg d2σ√g R, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='4) where g is the genus of the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Oriented Riemann surfaces without boundaries are completely classified (topologically or as complex manifolds) by their Euler characteristics χg, or equivalently by their genus g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In order to describe a proper string theory, the worldsheet metric gab(σ) should not be dynamical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This means that the worldsheet has no intrinsic dynamics and that no supplementary degrees of freedom are introduced when parametrizing the worldsheet with a metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' A solution to remove these degrees of freedom is to introduce gauge symmetries with as many gauge parameters as there are of degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The simplest symmetry is invariance under diffeomorphisms: indeed, the worldsheet theory is effectively a QFT coupled to gravity and it makes sense to require this invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Physically, this corresponds to the fact that the worldsheet spatial coordinate σ used along the string and worldsheet time are arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' However, diffeomorphisms alone are not sufficient to completely fix the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Another natural candidate is Weyl invariance (local rescalings of the metric).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' A diffeomorphism f ∈ Diff(Σg) acts on the fields as σ′a = f a(σb), g′(σ′) = f ∗g(σ), Ψ′(σ′) = f ∗Ψ(σ), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='5) where the star denotes the pullback by f: this corresponds simply to the standard coordinate transformation where each tensor index of the field receives a factor ∂σa/∂σ′b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In particular, the metric and scalar fields transform explicitly as g′ ab(σ′) = ∂σc ∂σ′a ∂σd ∂σ′b gcd(σ), X′µ(σ′) = Xµ(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='6) The index µ is inert since it is a target spacetime index: from the worldsheet point of view, it just labels a collection of worldsheet scalar fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Infinitesimal variations are generated by vector fields on Σg: δξσa = ξa, δξΨ = LξΨ, δξgab = Lξgab, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='7) 1Obviously, the vibrational modes are also constrained by the spacetime geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 30 where Lξ is the Lie derivative2 with respect to the vector field ξ ∈ diff(Σg) ≃ TΣg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The Lie derivative of the metric is Lξgab = ξc∂cgab + gac∂bξc + gbc∂aξc = ∇aξb + ∇bξa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='8) The Lie algebra generates only transformations in the connected component Diff0(Σg) of the diffeomorphism group which contains the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Transformations not contained in Diff0(Σg) are called large diffeomorphisms: this in- cludes reflections, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The quotient of the two groups is called the modular group Γg (also mapping class group or MCG): Γg := π0 � Diff(Σg) � = Diff(Σg) Diff0(Σg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='9) It depends only on the genus g of the Riemann surface, but not on the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' It is an infinite discrete group for genus g ≥ 1 surfaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' in particular, Γ1 = SL(2, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' A Weyl transformation e2ω ∈ Weyl(Σg) corresponds to a local rescaling of the metric and leaves the other fields unaffected3 g′ ab(σ) = e2ω(σ)gab(σ), Ψ′(σ) = Ψ(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='10) The exponential parametrization is generally more useful, but one should remember that it is e2ω and not ω which is an element of the group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The infinitesimal variation reads δωgab = 2ω gab, δωΨ = 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='11) where ω ∈ weyl(Σ) ≃ F(Σg) is a function on the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Two metrics related in this way are said to be conformally equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The conformal structure of the Riemann surface is defined by Conf(Σg) := Met(Σg) Weyl(Σg), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='12) where Met(Σg) denotes the space of all metrics on Σg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Each element is a class of conformally equivalent metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Diffeomorphisms have two parameters ξa (vector field) and Weyl invariance has one, ω (function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Hence, this is sufficient to locally fix the three components of the metric (symmetric matrix) and the total gauge group of the theory is the semi-direct product G := Diff(Σg) ⋉ Weyl(Σg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='13) Similarly, the component connected of the identity is written as G0 := Diff0(Σg) ⋉ Weyl(Σg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='14) The semi-direct product arises because the Weyl parameter is not inert under diffeo- morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Indeed, the combination of two transformations is g′ = f ∗� e2ωg � = e2f ∗ωf ∗g, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='15) such that the diffeomorphism acts also on the conformal factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 2For our purpose here, it is sufficient to accept the definition of the Lie derivative as corresponding to the infinitesimal variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 3For simplicity, we consider only fields which do not transform under Weyl transformations, which excludes fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 31 The combination of transformations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='15) can be chosen to fix the metric in a convenient gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' For example, the conformal gauge reads gab(σ) = e2φ(σ)ˆgab(σ), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='16) where ˆgab is some (fixed) background metric and φ(σ) is the conformal factor, also called the Liouville field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Fixing only diffeomorphisms amount to keep φ arbitrary: the latter can then be fixed with a Weyl transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' For instance, one can adopt the conformally flat gauge ˆgab = δab, φ arbitrary (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='17) with a diffeomorphism, and then reach the flat gauge ˆgab = δab, φ = 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='18) with a Weyl transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Another common choice is the uniformization gauge where ˆg is taken to be the metric of constant curvature on the sphere (g = 0), on the plane (g = 1) or on the hyperbolic space (g > 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' All these gauges are covariant (both in spacetime and worldsheet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1 (Active and passive transformations) Usually, symmetries are described by active transformations, which means that the field is seen to be changed by the transform- ation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' On the other hand, gauge fixing is seen as a passive transformation, where the field is expressed in terms of other fields (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' a different parametrization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' These are mathem- atically equivalent since both cases correspond to inverse elements, and one can choose the most convenient representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' We will use indifferently the same name for the parameters to avoid introducing minus signs and inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2 (Topology and gauge choices) While it is always possible to adopt locally the flat gauge (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='18), it may not be possible to extend it globally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The can be seen intuitively from the fact that the sign of the curvature is given by the one of 1 − g, but the curvature of the flat metric is zero: curvature must then be localized somewhere and this prevents from using a single coordinate patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The final step is to write an action Sm[g, Ψ] for the matter fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' According to the previous discussion, it must have the following properties: local in the fields;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' renormalizable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' non-linear sigma models for the scalar fields;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' periodicity conditions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' invariant under diffeomorphisms (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' invariant under Weyl transformations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The latter two conditions are summarized by Sm[f ∗g, f ∗Ψ] = Sm[g, Ψ], Sm[e2ωg, Ψ] = Sm[g, Ψ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='19) The invariance under diffeomorphisms is straightforward to enforce by using only covariant objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Since the scalar fields represent embedding of the string in spacetime, the non- linear sigma model condition means that spacetime is identified with the target space of the sigma model, of which D dimensions are non-compact, and the spacetime metric appears 32 in the matter action as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The isometries of the target manifold metric become global symmetries of Sm: while they are not needed in this chapter, they will have their importances in other chapters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Finally, to make the action consistent with the topology of the worldsheet, the fields must satisfy appropriate boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' For example, the scalar fields Xµ must be periodic for the closed string: Xµ(τ, σ) ∼ Xµ(τ, σ + 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='20) Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3 (2d gravity) The setup in two-dimensional gravity is exactly similar, except that the system is, in general, not invariant under Weyl transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' As a consequence, one component of the metric (usually taken to be the Liouville mode) remains unconstrained: in the conformal gauge, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='16) only ˆg is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The symmetries (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='19) of the action have an important consequence: they imply that the matter action is conformally invariant on flat space gab = δab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' A two-dimensional conformal field theory (CFT) is characterized by a central charge cm: roughly, it is a measure of the quantum degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The central charge is additive for decoupled sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In partic- ular, the scalar fields Xµ contribute as D, and it is useful to define the perpendicular CFT with central charge c⊥ as the matter which does not describe the non-compact dimensions: cm = D + c⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='21) This will be discussed in length in Part I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' For this chapter and most of the book, it is sufficient to know that the matter is a CFT of central charge cm and includes D scalar fields Xµ: matter CFT parameters: D, cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='22) The energy–momentum is defined by Tm,ab := − 4π √g δSm δgab .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='23) The variation of the action under the transformations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='7) vanishes on-shell if the energy– momentum tensor is conserved ∇aTm,ab = 0 (on-shell).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='24) On the other hand, the variation under (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='11) vanishes off-shell (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' without using the equations of motion) if the energy–momentum tensor is traceless: gabTm,ab = 0 (off-shell).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='25) The conserved charges associated to the energy–momentum tensor generate worldsheet translations P a := � dσ T 0a m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='26) The first component is identified with the worldsheet Hamiltonian P 0 = H and generates time translations, the second component generates spatial translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='4 (Tracelessness of the energy–momentum tensor) In fact, the trace can also be proportional to the curvature gabTm,ab ∝ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='27) Then, the equations of motion are invariant since the integral of R is topological.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The theory is invariant even if the action is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Importantly, this happens for fields at the quantum level (Weyl anomaly), for the Weyl ghost field (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='4) and for the Liouville theory (two-dimensional gravity coupled to conformal matter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 33 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2 Path integral The quantization of the system is achieved by considering the path integral, which yields the genus-g vacuum amplitude (or partition function): Zg := � dggab Ωgauge[g] Zm[g], Zm[g] := � dgΨ e−Sm[g,Ψ] (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='28) at fixed genus g (not to be confused with the metric).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The integration over gab is performed over all metrics of the genus-g Riemann surface Σg: gab ∈ Met(Σg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The factor Ωgauge[g] is a normalization inserted in order to make the integral finite: it depends on the metric (but only through the moduli parameters, as we will show later) [53, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 931], which explains why it is included after the integral sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Its value will be determined in the next section by requiring the cancellation of the infinities due to the integration over the gauge parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This partition function corresponds to the g-loop vacuum amplitude: interactions and their associated scattering amplitudes are discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In order to perform the gauge fixing and to manipulate the path integral (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='28), it is necessary to define the integration measure over the fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Because the space is infinite- dimensional, this is a difficult task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' One possibility is to define the measure implicitly through Gaussian integration over the field tangent space (see also Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' A Gaus- sian integral involves a quadratic form, that is, an inner-product (or equivalently a metric) on the field space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The explanation is that a metric also defines a volume form, and thus a measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' To reduce the freedom in the definition of the inner-product, it is useful to introduce three natural assumptions: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' ultralocality: the measure is invariant under reparametrizations and defined point-wise, which implies that it can depend on the fields but not on their derivatives;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' invariant measure: the measure for the matter transforms trivially under any sym- metry of the matter theory by contracting indices with appropriate tensors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' free-field measure: for fields other than the worldsheet metric and matter (like ghosts, Killing vectors, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' ), the measure is the one of a free field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This means that the inner-product is obtained by contracting the worldsheet indices of the fields with a tensor built only from the worldsheet metric, by contracting other indices (like spacetime) with some invariant tensor (like the spacetime metric), and finally by integrating over the worldsheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' We need to distinguish the matter fields from those appearing in the gauge fixing proced- ure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The matter fields live in the representation of some group under which the inner-product is invariant: this means that it is not possible to define each field measure independently if the exponential of inner-products does not factorize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' As an example, on a curved back- ground: dX ̸= � µ dXµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' However, we will not need to write explicitly the partition function for performing the gauge fixing: it is sufficient to know that the matter is a CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In the gauge fixing procedure, different types of fields (including the metric) appear which don’t carry indices (beyond the worldsheet indices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Below, we focus on defining a measure for each of those single fields (and use free-field measures according to the third condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Considering the finite elements δΦ1 and δΦ2 of tangent space at the point Φ of the state of fields, the inner-product (·, ·)g and its associated norm | · |g read (δΦ1, δΦ2)g := � d2σ√g γg(δΦ1, δΦ2), |δΦ|2 g := (δΦ, δΦ)g, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='29) where γg is a metric on the δΦ space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' It is taken to be flat for all fields except the metric itself, that is, independent of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The dependence in the metric ensures that the inner-product is 34 diffeomorphism invariant, which in turns will lead to a metric-dependent but diffeomorphism invariant measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The functional measure is then normalized by a Gaussian integral: � dgδΦ e− 1 2 (δΦ,δΦ)g = 1 � det γg .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='30) This, in turn, induces a measure on the field space itself: � dΦ � det γg (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='31) The determinant can be absorbed in the measure, such that � dgδΦ e− 1 2 (δΦ,δΦ)g = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='32) In fact, this normalization and the definition of the inner-product is ambiguous, but the ultralocality condition allows to fix uniquely the final result (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Moreover, such a free-field measure is invariant under field translations Φ(σ) −→ Φ′(σ) = Φ(σ) + ε(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='33) The most natural inner-products for single scalar, vector and symmetric tensor fields are (δf, δf)g := � d2σ√g δf 2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='34a) (δV a, δV a)g := � d2σ√g gabδV aδV b, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='34b) (δTab, δTab)g := � d2σ√g GabcdδTabδTcd, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='34c) where the (DeWitt) metric for the symmetric tensor is Gabcd := Gabcd ⊥ + u gabgcd, Gabcd ⊥ := gacgbd + gadgbc − gabgcd, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='35) with u a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The first term G⊥ is the projector on the traceless component of the tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Indeed, consider a traceless tensor gabTab = 0 and a pure trace tensor Λgab, then we have: GabcdTcd = Gabcd ⊥ Tcd = 2Tab, Gabcd(Λgcd) = 2u (Λgab).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='36) While all measures are invariant under diffeomorphisms, only the vector measure is invariant under Weyl transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This implies the existence of a quantum anomaly (the Weyl or conformal anomaly): the classical symmetry is broken by quantum effects because the path integral measure cannot respect all the classical symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Hence, one can expect difficulties for imposing it at the quantum level and ensuring that the Liouville mode in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='16) remains without dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The metric variation (symmetric tensor) is decomposed in its trace and traceless parts δgab = gab δΛ + δg⊥ ab, δΛ = 1 2 gabδgab, gabδg⊥ ab = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='37) In this decomposition, both terms are decoupled in the inner-product |δgab|2 g = 4u|δΛ|2 g + |δg⊥ µν| 2 g, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='38) 35 where the norm of δΛ is the one of a scalar field (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='34a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The norm for δg⊥ ab is equivalent to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='34c) with u = 0 (since it is traceless).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Requiring positivity of the inner-product for a non-traceless tensor imposes the following constraint on u: u > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='39) One can absorb the coefficient with u in δΛ, which will just contribute as an overall factor: its precise value has no physical meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The simple choice u = 1/4 sets the coefficient of |δΛ|2 g to 1 in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='38) (another common choice is u = 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Ultimately, this implies that the measure factorizes as dggab = dgΛ dgg⊥ ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='40) Computation – Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='38) Gabcd δgabδgcd = � Gabcd ⊥ + u gabgcd�� gab δΛ + δg⊥ ab �� gcd δΛ + δg⊥ cd � = � 2u gcd δΛ + Gabcd ⊥ δg⊥ ab �� gcd δΛ + δg⊥ cd � = 4u (δΛ)2 + Gabcd ⊥ δg⊥ abδg⊥ cd = 4u δΛ2 + 2gacgbdδg⊥ abδg⊥ cd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='5 Another common parametrization is Gabcd = gacgbd + c gabgcd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='41) It corresponds to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='35) up to a factor 1/2 and setting u = 1 + 2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='6 (Matter and curved background measures) As explained previously, mat- ter fields carry a representation and the inner-product must yield an invariant combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In particular, spacetime indices must be contracted with the spacetime metric Gµν(X) (which is the non-linear sigma model metric appearing in front of the kinetic term) for a general curved background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' For example, the inner-product for the scalar fields Xµ is (δXµ, δXµ)g = � d2σ√g Gµν(X)δXµδXν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='42) It is not possible to normalize anymore the measure to set det G(X) = 1 like in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='32) since it depends on the fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' On the other hand, this factor is not important for the manipu- lations performed in this chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Any ambiguity in the measure will again corresponds to a renormalization of the cosmological constant [53, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 923].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Moreover, as explained above, it is not necessary to write explicitly the matter partition function as long as it describes a CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3 Faddeev–Popov gauge fixing The naive integration over the space Met(Σg) of all metrics of Σg (note that the genus is fixed) leads to a divergence of the functional integral since equivalent configurations (f ∗g, f ∗Ψ) ∼ (g, Ψ), (e2ωg, Ψ) ∼ (g, Ψ) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='43) gives the same contribution to the integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This infinite redundancy causes the integral to diverge, and since the multiple counting is generated by the gauge group, the infinite contribution corresponds to the volume of the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The Faddeev–Popov procedure is a 36 means to extract this volume by separating the integration over the gauge and physical degrees of freedom d(fields) = Jacobian × d(gauge) × d(physical).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='44) The space of fields (g, Ψ) is divided into equivalence classes and one integrates over only one representative of each class (gauge slice), see Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This change of variables introduces a Jacobian which can be represented by a partition function with ghost fields (fields with a wrong statistics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This program encounters some complications since G is a semi-direct product and is non-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1 – Gauge redundancy A finite-dimensional integral which mimics the problem is Z = � R2 dx dy e−(x−y)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='45) One can perform the change of variables r = x − y, y = a (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='46) such that Z = � R da � ∞ 0 e−r2 = √π 2 Vol(R), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='47) and Vol(R) is to be interpreted as the volume of the gauge group (translation by a real number a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='7 Mathematically, the Faddeev–Popov procedure consists in identifying the or- bits (class of equivalent metrics) under the gauge group G and to write the integral in terms of G-invariant objects (orbits instead of individual metrics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This can be done by decompos- ing the tangent space into variations generated by G and its complement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Then, one can define a foliation of the field space which equips it with a fibre bundle structure: the base is the push-forward of the complement and the fibre corresponds to the gauge orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The integral is then defined by selecting a section of this bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1 Metrics on Riemann surfaces According to the above procedure, each metric gab ∈ Met(Σg) has to be expressed in terms of gauge parameters (ξ and ω) and of a metric ˆgab which contains the remaining gauge- independent degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' As there are as many gauge parameters as metric com- ponents (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1), one could expect that there are no remaining physical parameters and then that ˆg is totally fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' But, this is not the case and the metric ˆg depends on a finite number of parameters ti (moduli).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The reason for this is topological: while locally it is always possible to completely fix the metric, topological obstructions may prevent doing it globally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This means that not all conformal classes in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='12) can be (globally) related by a diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The quotient of the space of metrics by gauge transformations is called the moduli space Mg := Met(Σg) G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='48) Accordingly, its coordinates ti with i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' , dimR Mg are called moduli parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The Teichmüller space Tg is obtained by taking the quotient of Met(Σg) with the component 37 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1: The space of metrics decomposed in gauge orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Two metrics related by a gauge transformation lie on the same orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Choosing a gauge slice amounts to pick one metric in each orbit, and the projection gives the space of metric classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' connected to the identity Tg := Met(Σg) G0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='49) The space Tg is the covering space of Mg: Mg = Tg Γg , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='50) where Γg is the modular group defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Both spaces can be endowed with a complex structure and are finite-dimensional [172]: Mg := dimR Mg = dimR Tg = � � � � � 0 g = 0, 2 g = 1, 6g − 6 g ≥ 2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='51) In particular, their volumes are related by � Mg dMgt = 1 ΩΓg � Tg dMgt (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='52) where ΩΓg is the volume of Γg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' We will need to extract volumes of different groups, so it is useful to explain how they are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' A natural measure on a connected group G is the Haar measure dg, which is the unique left-invariant measure on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Integrating the measure gives the volume of the group ΩG := � G dg = � G d(hg), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='53) for any h ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Given the Lie algebra g of the group, a general element of the algebra is a linear combinations of the generators Ti with coefficients αi α = αiTi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='54) 38 Group elements can be parametrized in terms of α through the exponential map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Moreover, since a Lie group is a manifold, it is locally isomorphic to Rn: this motivates the use of a flat metric for the Lie algebra, such that ΩG = � dα := � � i dαi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='55) Finally, it is possible to perform a change of coordinates from the Lie parameters to co- ordinates x on the group: the resulting Jacobian is the Haar measure for the coordinates x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='8 While Tg is a manifold, this is not the case of Mg for g ≥ 2, which is an orbifold: the quotient by the modular group introduces singularities [173].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='9 (Moduli space and fundamental domain) Given a group acting on a space, a fundamental domain for a group is a subspace such that the full space is generated by act- ing with the group on the fundamental domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Hence, one can view the moduli space Mg as a fundamental domain (sometimes denoted by Fg) for the group Γg and the space Tg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In the conformal gauge (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='16), the metric gab can be parametrized by gab = ˆg(f,φ) ab (t) := e2f ∗φf ∗ˆgab(t) = f ∗� e2φˆgab(t) � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='56) where φ := ω and t denotes the dependence in the moduli parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' To avoid surcharging the notations, we will continue to write g when there is no ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In coordinates, this is equivalent to: gab(σ) = ˆg(f,φ) ab (σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' t) := e2φ(σ)ˆg′ ab(σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' t), ˆg′ ab(σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' t) = ∂σ′c ∂σa ∂σ′d ∂σb ˆgcd(σ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='57) Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='10 Strictly speaking, the matter fields also transform and one should write Ψ = Ψ(f) := f ∗ ˆΨ and include them in the change of integration measures of the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' But, this does not bring any particular benefits since these changes are trivial because the matter is decoupled from the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='11 Although the metric cannot be completely gauge fixed, having just a finite- dimensional integral is much simpler than a functional integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In higher dimensions, the gauge fixing does not reduce that much the degrees of freedom and a functional integral over ˆg remains (in similarity with Yang–Mills theories).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The corresponding infinitesimal transformations are parametrized by (φ, ξ, δti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The variation of the metric (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='56) can be expressed as δgab = 2φ gab + ∇aξb + ∇bξa + δti∂igab, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='58) which is decomposed in a reparametrization (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='7), a Weyl rescaling (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='11), and a contri- bution from the variations of the moduli parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The latter are called Teichmüller deformations and describe changes in the metric which cannot be written as a combination of diffeomorphism and Weyl transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Only the last term is written with a delta because the parameters ξ and φ are already infinitesimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' There is an implicit sum over i and we have defined ∂i := ∂ ∂ti .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='59) 39 According to the formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='55), the volumes ΩDiff0[g] and ΩWeyl[g] of the diffeomorph- isms connected to the identity and Weyl group are ΩDiff0[g] := � dgξ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='60a) ΩWeyl[g] := � dgφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='60b) The full diffeomorphism group has one connected component for each element of the modular group Γg, according to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='9): the volume ΩDiff[g] of the full group is the volume of the component connected to the identity times the volume ΩΓg ΩDiff[g] = ΩDiff0[g] ΩΓg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='60c) We have written that the volume depends on g: but, the metric itself is parametrized in terms of the integration variables, and thus the LHS of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='60) cannot depend on the variable which is integrated over: ΩDiff0 can depend only on φ and ΩWeyl only on ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' But, all measures (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='34b) are invariant under diffeomorphisms, and thus the result cannot depend on ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Moreover, the measure for vector is invariant under Weyl transformation, which means that ΩDiff0 does not depend on φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This implies that the volumes depend only on the moduli parameters ΩDiff0[g] := ΩDiff0[e2φˆg] = ΩDiff0[ˆg], ΩWeyl[g] := ΩWeyl[Lξˆg] = ΩWeyl[ˆg].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='61a) For this reason, it is also sufficient to take the normalization factor Ωgauge to have the same dependence: Ωgauge[g] := Ωgauge[ˆg].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='61b) These volumes are also discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Computation – Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='61) ΩDiff0[e2φˆg] = � de2φLξˆgξ = � de2φˆgξ = � dˆgξ = ΩDiff0[ˆg], ΩWeyl[Lξˆg] = � de2φLξˆgφ = � de2φˆgφ = ΩWeyl[ˆg].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='12 (Free-field measure for the Liouville mode) The explicit measure (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='60b) of the Liouville mode is complicated since the inner-product contains an exponential of the field: |δφ|2 = � d2σ√g δφ2 = � d2σ � ˆg e2φδφ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='62) It has been proposed by David–Distler–Kawai [40, 55], and later checked explicitly [50, 51, 160], how to rewrite the measure in terms of a free measure weighted by an effective action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The latter is identified with the Liouville action (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In principle, we could follow the standard Faddeev–Popov procedure by inserting a delta function for the gauge fixing condition Fab := gab − ˆg(f,φ) ab (t), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='63) with ˆg(f,φ) ab (t) defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' However, we will take a detour to take the opportunity to study in details manipulations of path integrals and to understand several aspects of the 40 geometry of Riemann surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In any case, several points are necessary even when going the short way, but less apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In order to make use of the factorization (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='40) of the integration measure, the variation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='58) is decomposed into its trace (first term) and traceless parts (last two terms) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='37) δgab = 2˜Λ gab + (P1ξ)ab + δti µiab, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='64) where4 (P1ξ)ab = ∇aξb + ∇bξa − gab∇cξc, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='65a) µiab = ∂igab − 1 2 gab gcd∂igcd, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='65b) ˜Λ = Λ + 1 2 δti gab∂igab, Λ = φ + 1 2 ∇cξc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='65c) The objects µi are called Beltrami differentials and correspond to traceless Teichmüller deformations (the factor of 1/2 comes from the symmetrization of the metric indices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The decomposition emphasizes which variations are independent from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In particular, changes to the trace of the metric due to a diffeomorphism generated by ξ or a modification of the moduli parameters can be compensated by a Weyl rescaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' One can use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='40) to replace the integration over gab by one over the gauge parameters ξ and φ and over the moduli ti since they contain all the information about the metric: Zg = � dMgt dg ˜Λ dg(P1ξ) Ωgauge[g]−1 Zm[g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='66) It is tempting to perform the change of variables (P1ξ, ˜Λ) −→ (ξ, φ) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='67) such that dg(P1ξ) dg ˜Λ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='= dgξ dgφ ∆FP[g] (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='68) where ∆FP[g] is the Jacobian of the transformation ∆FP[g] = det ∂(P1ξ, ˜Λ) ∂(ξ, φ) = det �P1 0 ⋆ 1 � = det P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='69) But, one needs to be more careful: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The variations involving P1ξ and δti are not orthogonal and, as a consequence, the measure does not factorize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' P1 has zero-modes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' vectors such that P1ξ = 0, which causes the determinant to vanish, det P1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' A rigorous analysis will be performed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2 and will lead to additional factors in the path integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Next, if the actions and measures were invariant under diffeomorphisms and Weyl trans- formations (which amounts to replace g by ˆg everywhere), it would be possible to factor out the integrations over the gauge parameters and to cancel the corresponding infinite factors thanks to the normalization Ωgauge[g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' A new problem arises because the measures are not Weyl invariant as explained above and one should be careful when replacing the metric (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 4For comparison, Polchinski [193] defines P1 with an overall factor 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 41 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2 Reparametrizations and analysis of P1 The properties of the operator P1 are responsible for both problems preventing a direct factorization of the measure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' for this reason, it is useful to study it in more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The operator P1 is an object which takes a vector v to a symmetric traceless 2-tensor T, see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='65a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Conversely, its adjoint P † 1 can be defined from the scalar product (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='34c) (T, P1v)g = (P † 1 T, v)g, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='70) and takes symmetric traceless tensors to vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In components, one finds (P † 1 T)a = −2∇bTab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='71) The Riemann–Roch theorem relates the dimension of the kernels of both operators [172]: dim ker P † 1 − dim ker P1 = −3χg = 6g − 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='72) Teichmüller deformations We first need to characterize Teichmüller deformations, the variations of moduli parameters which lead to transformations of the metric independent from diffeomorphisms and Weyl rescalings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This means that the different variations must be orthogonal for the inner-product (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' First, the deformations must be traceless, otherwise they can be compensated by a Weyl transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The traceless metric variations δg which cannot be generated by a vector field ξ are perpendicular to P1ξ (otherwise, the former would a linear combination of the latter): (δg, P1ξ)g = 0 =⇒ (P † 1 δg, ξ)g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='73) Since ξ is arbitrary, this means that the first argument vanishes P † 1 δg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='74) Metric variations induced by a change in the moduli ti are in the kernel of P † 1 δg ∈ ker P † 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='75) Elements of ker P † 1 are called quadratic differentials and a basis (not necessarily orthonor- mal) of ker P † 1 is denoted as: ker P † 1 = Span{φi}, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' , dim ker P † 1 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='76) (these should not be confused with the Liouville field).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The dimension of ker P † 1 is in fact equal to the dimension of the moduli space (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='51): dimR ker P † 1 = Mg = � � � � � 0 g = 0, 2 g = 1, 6g − 6 g > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='77) The last two terms in the variation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='64) of δgab are not orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Let’s introduce the projector on the complement space of ker P † 1 Π := P1 1 P † 1 P1 P † 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='78) 42 The moduli variations can then be rewritten as δti µi = δti (1 − Π)µi + δti Πµi = δti (1 − Π)µi + δti P1ζi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='79) The ζi exist because Πµi ∈ Im P1, and they read ζi := 1 P † 1 P1 P † 1 µi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='80) The first term can be decomposed on the quadratic differential basis (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='76) (1 − Π)µi = φj(M −1)jk(φk, µi)g (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='81) where Mij := (φi, φj)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='82) Ultimately, the variation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='64) becomes δgab = (P1 ˜ξ)ab + 2˜Λ gab + Qiab δti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='83) where ˜ξ = ξ + ζiδti, Qiab = φjab (M −1)jk(φk, µi)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='84) Correspondingly, the norm of the variation splits in three terms since each variation is orthogonal to the others: |δg|2 g = |δ˜Λ| 2 g + |P1 ˜ξ| 2 g + |Qiδti|2 g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='85) Since the norm is decomposed as a sum, the measure factorizes: dggab = dg ˜Λ dg(P1 ˜ξ) dg(Qiδti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='86) One can then perform a change of coordinates (˜ξ, ˜Λ, Qiδti) −→ (ξ, Λ, δti), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='87) where Λ was defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='65c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The goal of this transformation is to remove the dependence in the moduli from the measures on the Weyl factor and vector fields, and to recover a finite- dimensional integral over the moduli: dg ˜Λ dg(P1 ˜ξ) dg(Qiδti) = dMgt dgΛ dg(P1ξ) det(φi, µj)g � det(φi, φj)g , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='88) where the determinants correspond to the Jacobian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The role of the determinant in the denominator is to ensure a correct normalization when the basis is not orthonormal (in particular, it ensures that the Jacobian is independent of the basis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Plugging this result in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='28) gives the partition function as Zg = � Tg dMgt 1 Ωgauge[ˆg] � dgΛ dg(P1ξ) det(φi, µj)g � det(φi, φj)g Zm[g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='89) The ti are integrated over the Teichmüller space Tg defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='49) because the vectors ξ generate only reparametrizations connected to the identity, and thus the remaining freedom lies in Met(Σg)/G0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Next, we study how to perform the changes of variables to remove P1 from the measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 43 Conformal Killing vectors In this section, we focus on the dgΛ dg(P1ξ) part of the measure and we make contact with the rest at the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Infinitesimal reparametrizations generated by a vector field ξa produce only transform- ations close to the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' For this reason, integrating over all possible vector fields yields the volume (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='60a) of the component of the diffeomorphism group connected to the identity: � dgξ = ΩDiff0[ˆg].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='90) Remember that the volume depends only on the moduli, but obviously not on ξ (integrated over) nor φ (the inner-product (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='34b) is invariant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' But, due to the existence of zero-modes, one gets an integration over a subset of all vector fields, and this complicates the program, as we discuss now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Zero-modes ξ(0) of P1 are called conformal Killing vectors (CKV) ξ(0) ∈ Kg := ker P1 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='91) and satisfy the conformal Killing equation (see also Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1): (P1ξ(0))ab = ∇aξ(0) b + ∇bξ(0) a − gab∇cξ(0)c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='92) CKVs correspond to reparametrizations which can be absorbed by a change of the con- formal factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' They should be removed from the ξ integration in order to not double-count the corresponding metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The dimension of the zero-modes CKV space depends on the genus [172]: Kg := dimR Kg = dimR ker P1 = � � � � � 6 g = 0, 2 g = 1, 0 g > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='93) The associated transformations will be interpreted later (Chapter 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The groups generated by the CKVs are g = 0 : K0 = SL(2, C), g = 1 : K1 = U(1) × U(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='94) Note that the first group is non-compact while the second is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' A general vector ξ can be separated into a zero-mode part and its orthogonal complement ξ′: ξ = ξ(0) + ξ′, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='95) such that (ξ(0), ξ′)g = 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='96) for the inner-product (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='34b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Because zero-modes are annihilated by P1, the correct change of variables in the partition function (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='66) maps to ξ′ only: (P1ξ, Λ) −→ (ξ′, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='97) Integrating over ξ at this stage would double count the CKV (since they are already described by the φ integration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The appropriate Jacobian reads dgΛ dg(P1ξ) = dgφ dgξ′ ∆FP[g], (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='98) where the Faddeev–Popov determinant is ∆FP[g] = det′ ∂(P1ξ, Λ) ∂(ξ′, φ) = det′ P1 = � det′ P1P † 1 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='99) 44 the prime on the determinant indicating that the zero-modes are excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This brings the partition function (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='89) to the form Zg = � Tg dMgt Ωgauge[ˆg]−1 � dgφ dgξ′ det(φi, µj)g � det(φi, φj)g ∆FP[g]Zm[g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='100) Computation – Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='98) The Jacobian can be evaluated directly: ∆FP[g] = det′ ∂(P1ξ, Λ) ∂(ξ′, φ) = det′ � P1 0 1 2∇ 1 � = det′ P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='101) As a consequence of det′ P † 1 = det′ P1, the Jacobian can be rewritten as: � det′ P † 1 P1 = det′ P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='102) It is instructive to derive this result also by manipulating the path integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Con- sidering small variations of the fields, one has: 1 = � dgδΛ dg(P1δξ) e−|δΛ|2 g−|P1δξ′|2 g = ∆FP[g] � dgδφ dgδξ′ e−|δφ+ 1 2 ∇cδξc|2 g−|P1δξ′|2 g = ∆FP[g] � dgδφ dgδξ′ e−|δφ|2 g−(δξ′,P † 1 P1δξ′)g = ∆FP[g] � det′ P † 1 P1 �−1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' That the expression is equal to 1 follows from the normalization of symmetric tensors and scalars (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='34) (the measures appearing in the path integral (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='89) arises without any factor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The third equality holds because the measure is invariant under translations of the fields, and we used the definition of the adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The volume of the group generated by the vectors orthogonal to the CKV is denoted as Ω′ Diff0[g] := Ω′ Diff0[ˆg] = � dgξ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='103) As explained in the beginning of this section, one should extract the volume of the full Diff0 group, not only the volume Ω′ Diff0[g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Since the two sets of vectors are orthogonal, we can expect the measures, and thus the volumes, to factorize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' However, a Jacobian can and does arise: its role it to take into account the normalization of the zero-modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Denoting by ψi a basis (not necessarily orthonormal) for the zero-modes ker P1 = Span{ψi}, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' , Kg, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='104) the change of variables ξ′ −→ ξ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='105) reads dgξ′ = 1 � det(ψi, ψj)g dgξ Ωckv[g], (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='106) 45 where Ωckv[g] is the volume of the CKV group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The determinant is necessary when the basis is not orthonormal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The relation between the gauge volumes is then ΩDiff0[g] = � det(ψi, ψj)g Ωckv[g] Ω′ Diff0[g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='107) Note that the CKV volume is given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='111) and depends only on the topology but not on the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' By using arguments similar to the ones which lead to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='61), one can expect that each term is independently invariant under Weyl rescaling: this is indeed true (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Computation – Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='106) Let’s expand ξ(0) on the zero-mode basis ξ(0) = αiψi, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='108) where the αi are real numbers, such that one can write the changes of variables ξ −→ (ξ′, αi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='109) The Jacobian is computed from 1 = � dξ e−|ξ|2 g = J � dξ(0) dξ′ e−|ξ′|2 g−|ξ(0)| 2 g = J � � i dαi e−αiαj(ψi,ψj)g � dξ′ e−|ξ′|2 g = J (det(ψi, ψj)g)−1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Note that the integration over the αi is a standard finite-dimensional integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This gives dξ = � det(ψi, ψj)g dξ′ � i dαi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='110) Since nothing depends on the αi, they can be integrated over as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='53), giving the volume of the CKV group Ωckv[g] = � � i dαi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='111) Replacing the integration over ξ′ thanks to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='106), the path integral becomes Zg = � Tg dMgt Ωgauge[ˆg]−1 � dgφ dgξ det(φi, µj)g � det(φi, φj)g Ωckv[g]−1 � det(ψi, ψj)g ∆FP[g] Zm[g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='112) Since the matter action and measure, and the Liouville measure are invariant under reparametrizations, one can perform a change of variables (f ∗ˆg, f ∗φ, f ∗Ψ) −→ (ˆg, φ, Ψ) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='113) such that everything becomes independent of f (or equivalently ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Since the measure for ξ is Weyl invariant, it is possible to separate it from the rest of the expression, which yields an overall factor of ΩDiff0[g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This brings the partition function to the form Zg = � Tg dMgt ΩDiff0[ˆg] Ωgauge[ˆg] � dgφ det(φi, µj)g � det(φi, φj)g Ωckv[g]−1 � det(ψi, ψj)g ∆FP[g] Zm[g] (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='114) 46 where the same symbol is used for the metric gab := g(φ) ab = e2φˆgab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='115) Since the expression is invariant under the full diffeomorphism group Diff(Σg) and not just under its component Diff0(Σg), one needs to extract the volume of the full diffeomorph- ism group before cancelling it with the normalization factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Otherwise, there is still an over-counting the configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Using the relation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='60c) leads to: Zg = 1 ΩΓg � Tg dMgt ΩDiff[ˆg] Ωgauge[ˆg] � dgφ det(φi, µj)g � det(φi, φj)g Ωckv[g]−1 � det(ψi, ψj)g ∆FP[g] Zm[g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='116) The volume ΩΓg can be factorized outside the integral because it depends only on the genus and not on the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Finally, using the relation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='52), one can replace the integration over the Teichmüller space by an integration over the moduli space Zg = � Mg dMgt ΩDiff[ˆg] Ωgauge[ˆg] � dgφ det(φi, µj)g � det(φi, φj)g Ωckv[g]−1 � det(ψi, ψj)g ∆FP[g] Zm[g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='117) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3 Weyl transformations and quantum anomalies The next question is whether the integrand depends on the Liouville mode φ such that the Weyl volume can be factorized out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' While the matter action has been chosen to be Weyl invariant – see the condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='19) – the measures cannot be defined to be Weyl invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This means that there is a Weyl (or conformal) anomaly, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' a violation of the Weyl invariance due to quantum effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Since the techniques needed to derive the results of this section are outside the scope of this book, we simply state the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' It is possible to show that the Weyl anomaly reads [53, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 929]5 ∆FP[e2φˆg] � det(φi, φj)e2φˆg = e cgh 6 SL[ˆg,φ] ∆FP[ˆg] � det(ˆφi, ˆφj)ˆg (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='118a) Zm[e2φˆg] = e cm 6 SL[ˆg,φ]Zm[ˆg], (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='118b) where SL is the Liouville action SL[ˆg, φ] := 1 4π � d2σ � ˆg � ˆgab∂aφ∂bφ + ˆRφ � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='119) where ˆR is the Ricci scalar of the metric ˆgab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' These relations require to introduce counter- terms, discussed further in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The coefficients cm and cgh are the central charges respectively of the matter and ghost systems, with: cgh = −26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='120) This value will be derived in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The inner-products between φi and µj, and between the ψi, and the CKV volume are independent of φ [172, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2, 53, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 931] det(φi, µj)e2φˆg = det(ˆφi, ˆµj)ˆg, det(ψi, ψj)e2φˆg = det(ψi, ψj)ˆg, Ωckv[e2φˆg] = Ωckv[ˆg].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='121) 5The relation is written for Zm since the action is invariant and is not affected by the anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 47 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='13 (Weyl and gravitational anomalies) The Weyl anomaly translates into a non-zero trace of the quantum energy–momentum tensor ⟨gµνTµν⟩ = c 12 R, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='122) where c is the central charge of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The Weyl anomaly can be traded for a gravita- tional anomaly, which means that diffeomorphisms are broken at the quantum level [122].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Inserting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='118) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='117) yields Zg = � Mg dMgt ΩDiff[ˆg] Ωgauge[ˆg] det(φi, ˆµj)ˆg � det(φi, φj)ˆg Ωckv[ˆg]−1 � det(ψi, ψj)ˆg ∆FP[ˆg] Zm[ˆg] � dgφ e− cL 6 SL[ˆg,φ], (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='123) with the Liouville central charge cL := 26 − cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='124) The critical “dimension” is defined to be the value of the matter central charge cm such that the Liouville central charge cancels cL = 0 =⇒ cm = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='125) If the number of non-compact dimensions is D, it means that the central charge (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='21) of the transverse CFT satisfies c⊥ = 26 − D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='126) In this case, the integrand does not depend on the Liouville mode (because ΩDiff is invariant under Weyl transformations) and the integration over φ can be factored out and yields the volume of the Weyl group (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='60b) � dgφ = ΩWeyl[ˆg].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='127) Then, taking Ωgauge[ˆg] = ΩDiff[ˆg] × ΩWeyl[ˆg] (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='128) removes the infinite gauge contributions and gives the partition function Zg = � Mg dMgt det(φi, ˆµj)ˆg � det(φi, φj)ˆg Ωckv[ˆg]−1 � det(ψi, ψj)ˆg ∆FP[ˆg] Zm[ˆg].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='129) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='4 Ambiguities, ultralocality and cosmological constant Different ambiguities remain in the previous computations, starting with the definitions of the measures (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='32) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='34), then in obtaining the volume of the diffeomorphism (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='60a) and Weyl (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='60b) groups, and finally in deriving the conformal anomaly (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='118).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' These different ambiguities can be removed by renormalizing the worldsheet cosmological constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This implies that the action Sµ[g] = � d2σ√g (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='130) must be added to the classical Lagrangian, where µ0 is the bare cosmological constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This means that Weyl invariance is explicitly broken at the classical level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' After performing all the manipulations, µ0 is determined by removing all ambiguities and enforcing invariance under the Weyl symmetry at the quantum level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This amounts to set the renormalized cosmological constant to zero (since it breaks the Weyl symmetry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The possibility to 48 introduce a counter-term violating a classical symmetry arises because the symmetry itself is broken by a quantum anomaly, so there is no reason to enforce it in the classical action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' We now review each issue separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' First, consider the inner-product of a single tensor (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='32): the determinant det γg depends on the metric and one should be more careful when fixing the gauge or integrating over all metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' However, ultralocality implies that the determinant can only be of the form [53, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 923] � det γg = e−µγ Sµ[g], (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='131) for some µγ ∈ R, since Sµ is the only renormalizable covariant functional depending on the metric but not on its derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The effect is just to redefine the cosmological constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Second, the volume of the field space can be defined as the limit λ → 0 of a Gaussian integral [53, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 931]: ΩΦ = lim λ→0 � dgΦ e−λ (Φ,Φ)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='132) Due to ultralocality, the Gaussian integral should again be of the form � dgΦ e−λ (Φ,Φ)g = e−µ(λ) Sµ[g], (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='133) for some constant µ(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Hence, the limit λ → 0 gives ΩΦ = � dgΦ = e−µ(0) Sµ[g], (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='134) which can be absorbed in the cosmological constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' However, the situation is more com- plicated if Φ = ξ, φ since the integration variables also appear in the measure, as it was also discussed before (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' But, in that case, it cannot appear in the expression of the volume in the LHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Moreover, invariances under diffeomorphisms for both measures, and under Weyl rescalings for the vector measure, imply that the LHS can only depend on the moduli through the background metric ˆg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The diffeomorphism and Weyl volumes can be written in terms of e−ˆµ Sµ[ˆg]: since there is no counter-term left (the cosmological constant counter-term is already fixed to cancel the coefficient of Sµ[g]), it is necessary to divide by Ωgauge to cancel the volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Finally, the computation of the Weyl anomaly (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='118) yields divergent terms of the form lim ϵ→0 1 ϵ � d2σ√g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='135) These divergences are canceled by the cosmological constant counter-term, see [54, app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='A] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='5 Gauge-fixed path integral As a conclusion of this section, we found that the partition function (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='28) can be written as Zg = � Mg dMgt det(φi, ˆµj)ˆg � det(φi, φj)ˆg Ωckv[ˆg]−1 � det(ψi, ψj)ˆg ∆FP[ˆg] Zm[ˆg], (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='136a) = � Mg dMgt � det(φi, ˆµj)2 ˆg det(φi, φj)ˆg det′ ˆP † 1 ˆP1 det(ψi, ψj)ˆg Zm[ˆg] Ωckv[ˆg].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='136b) after gauge fixing of the worldsheet diffeomorphisms and Weyl rescalings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' It is implicit that the factors for the CKV and moduli are respectively absent for g > 1 and g < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' For g = 0 the CKV group is non-compact and its volume is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' It looks like the partition vanishes, but there are subtleties which will be discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 49 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='14 (Weil–Petersson metric) When the metric is chosen to be of constant curvature ˆR = −1, the moduli measure together with the determinants form the Weil– Petersson measure d(WP) = � Mg dMgt det(φi, ˆµj)ˆg � det(φi, φj)ˆg .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='137) In (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='136), the background metric ˆgab is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' However, the derivation holds for any choice of ˆgab: as a consequence, it makes sense to relax the gauge fixing and allow it to vary while adding gauge symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The first symmetry is background diffeomorphisms: σ′a = ˆf a(σb), ˆg′(σ′) = f ∗ˆg(σ), φ′(σ′) = f ∗φ(σ), Ψ′(σ′) = f ∗Ψ(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='138) This symmetry is automatic for Sm[ˆg, Ψ] since Sm[g, Ψ] was invariant under (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Similarly, the integration measures are also invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' A second symmetry is found by inspecting the decomposition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='56) gab = f ∗� e2φˆgab(t) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='139) which is left invariant under a background Weyl symmetry (also called emergent): g′ ab(σ) = e2ω(σ)gab(σ), φ′(σ) = φ(σ) − ω(σ), Ψ′(σ) = Ψ(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='140) Let us stress that it is not related to the Weyl rescaling (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='10) of the metric gab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The background Weyl rescaling (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='140) is a symmetry even when the physical Weyl rescaling (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='10) is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Together, the background diffeomorphisms and Weyl symmetry have three gauge parameters, which is sufficient to completely fix the background metric ˆg up to moduli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In fact, the combination of both symmetries is equivalent to invariance under the physical diffeomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' To prove this statement, consider two metrics g and g′ related by a diffeomorphism F and both gauge fixed to pairs (f, φ, ˆg) and (f ′, φ′, ˆg′): g′ ab = F ∗gab, g′ ab = f ′∗� e2φ′ˆg′ ab � , gab = f ∗� e2φˆgab � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='141) Then, the gauge fixing parametrizations are related by background symmetries ( ˆF, ω) as ˆF = f ′−1 ◦ F ◦ f, φ′ = ˆF ∗(φ − ω), ˆg′ ab = ˆF ∗(e2ωˆgab).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='142) Moreover, this also implies that there is a diffeomorphism ˜f = F ◦ f such that g′ is gauge fixed in terms of (φ, ˆg): g′ ab = ˜f ∗� e2φˆgab � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='143) Computation – Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='142) The functions F, f, f ′, φ, φ′ and the metrics gab, g′ ab, ˆgab and ˆg′ ab are all fixed and one must find ˆF and ω such that the relations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='141) are compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' First, one rewrites g′ ab in terms of ˆgab and compare with the expression with ˆg′ ab: g′ ab = F ∗gab = F ∗� f ∗� e2φˆgab �� = F ∗� f ∗� e2(φ−ω)e2ωˆgab �� = f ′∗� e2φ′ˆg′ ab � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In the third equality, we have introduced ω because ˆg′ ab = ˆF ∗ˆgab is not true in general since there are 3 independent components but ˆF has only 2 parameters, so we cannot just define f ′ = F ◦ f and φ′ = φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This explains the importance of the emergent Weyl symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 50 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='15 (Gauge fixing and field redefinition) Although it looks like we are un- doing the gauge fixing, this is not exactly the case since the original metric is not used any- more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' One can understand the procedure of this section as a field redefinition: the degrees of freedom in gab are repackaged into two fields (φ, ˆgab) adapted to make some properties of the system more salient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' A new gauge symmetry is introduced to maintain the number of degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The latter helps to understand the structure of the action on the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Finally, in this context, the Liouville action is understood as a Wess–Zumino action, which is defined as the difference between the effective actions evaluated in each metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Another typical use of this point of view is to rewrite a massive vector field as a massless gauge field together with an axion [195].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='16 (Two-dimensional gravity) In 2d gravity, one does not work in the crit- ical dimension (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='125) and cL ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Thus, the Liouville mode does not decouple: the con- formal anomaly breaks the Weyl symmetry at the quantum level which gives dynamics to gravity, even if it has no degree of freedom classically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' As a consequence, one chooses Ωgauge = ΩDiff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Since the role of the classical Weyl symmetry is not as important as for string theory, it is even not necessary to impose it classically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This leads to consider non-conformal matter [21, 22, 31, 82, 83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Following the arguments from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1, the existence of the emergent Weyl symmetry (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='140) implies that the total action Sgrav[ˆg, φ] + Sm[ˆg, Ψ] must be a CFT for a flat background ˆg = δ, even if the two actions are not independently CFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='4 Ghost action 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1 Actions and equations of motion It is well-known that a determinant can be represented with two anticommuting fields, called ghosts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The fields carry indices dictated by the map induced by the operator of the Faddeev– Popov determinant: one needs a symmetric and traceless anti-ghost bab and a vector ghost ca fields: ∆FP[g] = � d′ gb d′ gc e−Sgh[g,b,c], (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='144) where the prime indicates that the ghost zero-modes are omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The ghost action is Sgh[g, b, c] := 1 4π � d2σ√g gabgcdbac(P1c)bd (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='145a) = 1 4π � d2σ√g gab� bac∇bcc + bbc∇acc − bab∇ccc� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='145b) The ghosts ca and anti-ghosts bab are associated respectively to the variations due to the diffeomorphisms ξa and to the variations perpendicular to the gauge slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The normalization of 1/4π is conventional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In Minkowski signature, the action is multiplied by a factor i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Since bab is traceless, the last term of the action vanishes and could be removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' However, this implies to consider traceless variations of the bab when varying the action (to compute the equations of motion, the energy–momentum tensor, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' On the other hand, one can keep the term and consider unconstrained variation of bab (since the structure of the action will force the variation to have the correct symmetry), which is simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' A last possibility is to introduce a Lagrange multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' These aspects are related to the question of introducing a ghost for the Weyl symmetry, which is described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The equations of motion are (P1c)ab = ∇acb + ∇bca − gab∇ccc = 0, (P † 1 b)a = −2∇bbab = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='146) 51 Hence, the classical solutions of b and c are respectively mapped to the zero-modes of the operators P † 1 and P1, and they are thus associated to the CKV and Teichmüller parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The energy–momentum tensor is T gh ab = −bac∇bcc − bbc∇acc + cc∇cbab + gabbcd∇ccd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='147) Its trace vanishes off-shell (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' without using the b and c equations of motion) gabT gh ab = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='148) which shows that the action (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='145) is invariant under Weyl transformations Sgh[e2ωg, b, c] = Sgh[g, b, c].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='149) The action (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='145) also has a U(1) global symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The associated conserved charge is called the ghost number and counts the number of c ghosts minus the number of b ghosts, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Ngh(b) = −1, Ngh(c) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='150a) The matter fields are inert under this symmetry: Ngh(Ψ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='150b) In terms of actions, the path integral (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='136) can be rewritten as Zg = � Mg dMgt det(φi, ˆµj)ˆg � det(φi, φj)ˆg Ωckv[ˆg]−1 � det(ψi, ψj)ˆg � dˆgΨ d′ ˆgb d′ ˆgc e−Sm[ˆg,Ψ]−Sgh[ˆg,b,c].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='151) One can use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='136) or (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='151) indifferently: the first is more appropriate when using spectral analysis to compute the determinant explicitly, while the second is more natural in the context of CFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2 Weyl ghost Ghosts have been introduced for the reparametrizations (generated by ξa) and the traceless part of the metric (the gauge field associated to the transformation): one may wonder why there is not a ghost cw associated to the Weyl symmetry along with an antighost for the trace of the metric (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' the conformal factor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This can be understood from several viewpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' First, the relation between a metric and its transformation – and the corresponding gauge fixing condition – does not involve any derivative: as such, the Jacobian is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Second, one could choose F ⊥ ab = √ggab − � ˆgˆgab = 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='152) as a gauge fixing condition instead of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='63), and the trace component does not appear anywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Finally, a local Weyl symmetry is not independent from the diffeomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='17 (Local Weyl symmetry) The topic of obtaining a local Weyl symmetry by gauging a global Weyl symmetry (dilatation) is very interesting [86, chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 15, 113].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Under general conditions, one can express the new action in terms of the Ricci tensor (or of the curvature): this means that the Weyl gauge field and its curvature are composite fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Moreover, one finds that local Weyl invariance leads to an off-shell condition while diffeo- morphisms give on-shell conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This explains why one imposes only Virasoro constraints (associated to reparametrizations) and no constraints for the Weyl symmetry in the covariant quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 52 However, it can be useful to introduce a ghost field cw for the Weyl symmetry nonetheless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In view of the previous discussion, this field should appear as a Lagrange multiplier which ensures that bab is traceless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Starting from the action (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='145), one finds S′ gh[g, b, c, cw] = 1 4π � d2σ√g gab� bac∇bcc + bbc∇acc + 2babcw � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='153) where bab is not traceless anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The ghost cw is not dynamical since the action does not contain derivatives of it, and it can be integrated out of the path integral to recover (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='145).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The equations of motion for this modified action are ∇acb + ∇bca + 2gabcw = 0, ∇abab = 0, gabbab = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='154) Contracting the first equation with the metric gives cw = −1 2 ∇aca, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='155) and thus cw is nothing else than the divergence of the ca field: the Weyl ghost is a composite field (this makes connection with Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='17) – see also (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='65c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The energy–momentum tensor of the ghosts with action (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='153) is T ′gh ab = − � bac∇bcc + bbc∇acc + 2babcw � − ∇c(babcc) + 1 2 gabgcd� bce∇dce + bde∇cce + 2bcdcw � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='156) The trace of this tensor gabT ′gh ab = −gab∇c(babcc) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='157) does not vanish off-shell, but it does on-shell since gabbab = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This implies that the theory is Weyl invariant even if the action is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' It is interesting to contrast this with the trace (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='148) when the Weyl ghost has been integrated out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The equations of motion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='146) and energy–momentum tensor (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='147) for the action (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='145) can be easily derived by replacing cw by its solution in the previous formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Computation – Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='156) The first parenthesis comes from varying gab, the second from the covariant derivatives, the last from the √g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The second term comes from gab� bacδ∇bcc + bbcδ∇acc� = 2gabbacδ∇bcc = 2gabbacδΓc bdcd = gabbacgce� ∇bδgde + ∇dδgbe − ∇eδgbd � cd = bab� ∇aδgbc + ∇cδgab − ∇bδgac � cc = bab∇cδgabcc, where two terms have cancelled due to the symmetry of bab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Integrating by part gives the term in the previous equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Note that the integration on the Weyl ghost yields a delta function � dgcw e−(cw,gabbab)g = δ � gabbab � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='158) 53 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3 Zero-modes The path integral (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='151) excludes the zero-modes of the ghosts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' One can expect them to be related to the determinants of elements of ker P1 and ker P † 1 with Grassmann coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' They can be included after few simple manipulations (see also Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' It is simpler to first focus on the b ghost (to avoid the problems related to the CKV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The path integral (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='151) can be rewritten as Zg = � Mg dMgt Ωckv[ˆg]−1 � det(ψi, ψj)ˆg � dˆgΨ dˆgb d′ ˆgc Mg � i=1 (b, ˆµi)ˆg e−Sm[ˆg,Ψ]−Sgh[ˆg,b,c].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='159) In this expression, c zero-modes are not integrated over, only the b zero-modes are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This is the standard starting point on Riemann surfaces with genus g ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The inner-product reads explicitly (b, ˆµi)ˆg = � d2σ � ˆg Gabcd ⊥ babˆµi,cd = � d2σ � ˆg gacgbdbabˆµi,cd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='160) Computation – Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='159) Since the zero-modes of b are in the kernel of P † 1 , it means that the quadratic differentials (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='76) also provide a suitable basis: b = b0 + b′, b0 = b0iφi, where the b0i are Grassmann-odd coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The first step is to find the Jacobian for the changes of variables b → (b′, b0i): 1 = � dˆgb e−|b|2 ˆg = J � dˆgb′ � i db0i e−|b′|2 ˆg−|b0iφi|2 = J � det(φi, φj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Next, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='151) has no zero-modes, so one must insert Mg of them at arbitrary positions σ0 j to get a non-vanishing result when integrating over dMgb0i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The result of the integral is: � dMgb0i � j b0(σ0 j ) = � dMgb0i � j � b0iφi(σ0 j ) � = det φi(σ0 j ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The only combination of the φi which does not vanish is the determinant due to the anti-symmetry of the Grassmann numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Combining both results leads to: dˆgb′ � det(φi, φj)ˆg = dˆgb det φi(σ0 j ) Mg � j=1 b(σ0 j ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='161) The locations positions σ0 j are arbitrary (in particular, the RHS does not depend on them since the LHS does not either).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Note that more details are provided in Ap- pendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' An even simpler result can be obtained by combining the previous formula with the factor det(φi, ˆµj)ˆg: dˆgb′ det(φi, ˆµj)ˆg � det(φi, φj)ˆg = dˆgb Mg � j=1 (b, ˆµj)ˆg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='162) 54 This follows from Mg � j=1 b(σ0 j ) = Mg � j=1 � b0iφi(σ0 j ) � = det φi(σ0 j ) Mg � j=1 b0i, det(φi, ˆµj)ˆg Mg � j=1 b0i = Mg � j=1 � b0i(φi, ˆµj)ˆg � = Mg � j=1 (b0iφi, ˆµj)ˆg = Mg � j=1 (b, ˆµj)ˆg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Note that the previous manipulations are slightly formal: the symmetric traceless fields bab and φi,ab carry indices and there should be a product over the (two) independent components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This is a trivial extension and would just make the notations heavier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Similar manipulations lead to a new expression which includes also the c zero-mode (but which is not very illuminating): Zg = � Mg dMgt Ωckv[ˆg]−1 det ψi(σ0 j ) � dˆgΨ dˆgb dˆgc Kc g � j=1 ϵab 2 ca(σ0 j )cb(σ0 j ) × Mg � i=1 (ˆµi, b)ˆg e−Sm[ˆg,Ψ]−Sgh[ˆg,b,c].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='163) The σ0a j are Kc g = Kg/2 fixed positions and the integral does not depend on their values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Note that only Kc g positions are needed because the coordinate is 2-dimensional: fixing 3 points with 2 components correctly gives 6 constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Then, ψi(σ0a j ) is a 6-dimensional matrix, with the rows indexed by i and the columns by the pair (a, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The expression cannot be simplified further because the CKV factor is infinite for g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This is connected to a fact mentioned previously: there is a remaining gauge symmetry which is not taken into account c −→ c + c0, P1c0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='164) A proper account requires to gauge fix this symmetry: the simplest possibility is to insert three or more vertex operators – this topic is discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Finally, note that the same question arises for the b-ghost since one has the symmetry b −→ b + b0, P † 1 b0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='165) That there is no problem in this case is related to the presence of the moduli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='5 Normalization In the previous sections, the closed string coupling constant gs did not appear in the ex- pressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Loops in vacuum amplitudes are generated by splitting of closed strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' By inspecting the amplitudes, it seems that there are 2g such splittings (Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2), which would lead to a factor g2g s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' However, this is not quite correct: this result holds for a 2-point function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Gluing the two external legs to get a partition function (that is, taking the trace) leads to an additional factor g−2 s (to be determined later), such that the overall factor is g2g−2 s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The fact that it is the appropriate power of the coupling constant can be more easily understood by considering n-point amplitudes (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The normalization of the path integral can be completely fixed by unitarity [193].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The above factor has a nice geometrical interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Defining Φ0 = ln gs (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='166) 55 and remembering the expression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='4) of the Euler characteristics χg = 2 − 2g, the coupling factor can be rewritten as g2g−2 s = e−Φ0χg = exp � −Φ0 4π � d2σ√gR � = e−Φ0SEH[g], (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='167) where SEH is the Einstein–Hilbert action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This action is topological in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Hence, the coupling constant can be inserted in the path integral simply by shifting the action by the above term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This shows that string theory on a flat target spacetime is completely equivalent to matter minimally coupled to Einstein–Hilbert gravity with a cos- mological constant (tuned to impose Weyl invariance at the quantum level).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The advantage of describing the coupling power in this fashion is that it directly generalizes to scattering amplitudes and to open strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The parameter Φ0 is interpreted as the expectation value of the dilaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Replacing it by a general field Φ(Xµ) is a generalization of the matter non-linear sigma model, but this topic is beyond the scope of this book.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2: g-loop partition function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='6 Summary In this chapter, we started with a fairly general matter CFT – containing at least D scalar fields Xµ – and explained under which condition it describes a string theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The most important consequence is that the matter 2d QFT must in fact be a 2d CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' We then continued by describing how to gauge fix the integration over the surfaces and we identified the remaining degrees of freedom – the moduli space Mg – up to some residual redundancy – the conformal Killing vector (CKV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Then, we showed how to rewrite the result in terms of ghosts and proved that they are also a CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This means that a string theory can be com- pletely described by two decoupled CFTs: a universal ghost CFT and a theory-dependent matter CFT describing the string spacetime embedding and the internal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The advantage is that one can forget the path integral formalism altogether and employ only CFT techniques to perform the computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This point of view will be developed for off- shell amplitudes (Chapter 11) in order to provide an alternative description of how to build amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' It is particularly fruitful because one can also consider matter CFTs which do not have a Lagrangian description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In the next chapter, we describe scattering amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='7 Suggested readings Numerous books have been published on the worldsheet string theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Useful (but not required) complements to this chapter and subsequent ones are [151, 265] for introductory texts and [24, 47, 48, 128, 193] for more advanced aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The definition of a field measure from a Gaussian integral and manipulations thereof can be found in [100, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1, 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1, 172, chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 14, 191, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 56 The most complete explanations of the gauge fixing procedure are [100, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1, 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1, 24, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='4, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2, 193, chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 5, 48, 124, chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The original derivation can be found in [52, 161].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' For the geometry of the moduli space, see [172, 173].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Ultralocality and its consequences are described in [53, 191] (see also [98, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The use of a Weyl ghost is shown in [240, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 8, 258, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 57 Chapter 3 Worldsheet path integral: scattering amplitudes Abstract In this chapter, we generalize the worldsheet path integral to compute scattering amplitudes, which corresponds to insert vertex operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The gauge fixing from the previous chapter is generalized to this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In particular, we discuss the 2-point amplitude on the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Finally, we introduce the BRST symmetry and motivate some properties of the BRST quantization, which will be performed in details later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The formulas in this chapter are all covariant: they will be rewritten in complex coordinates in the next chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1 Scattering amplitudes on moduli space In this section, we describe the scattering of n strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The momentum representation is more natural for describing interactions, especially in string theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Therefore, each string is characterized by a state Vαi(ki) with momentum ki and some additional quantum numbers αi (i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' , n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' We start from the worldsheet path integral (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='28) before gauge fixing: Zg = � dggab Ωgauge[g] Zm[g], Zm[g] = � dgΨ e−Sm[g,Ψ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1 Vertex operators and path integral The external states are represented by infinite semi-tubes attached to the surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Under a conformal mapping, the tubes can be mapped to points called punctures on the worldsheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' At g loops, the resulting space is a Riemann surface Σg,n of genus g with n punctures (or marked points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The external states are represented by integrated vertex operators Vα(ki) := � d2σ � g(σ) Vα(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2) The vertex operators Vα(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' σ) are built from the matter CFT operators and from the world- sheet metric gab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The functional dependence is omitted to not overload the notation, but one should read Vα(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' σ) := Vα[g, Ψ](k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The integration over the state positions is neces- sary because the mapping of the tube to a point is arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Another viewpoint is that it is needed to obtain an expression invariant under worldsheet diffeomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The vertex operators described general states which not necessarily on-shell: this restriction will be found later when discussing the BRST invariance of scattering amplitudes (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 58 Following Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='5, the Einstein–Hilbert action with boundary term SEH[g] := 1 4π � d2σ√g R + 1 2π � ds k = χg,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3) is inserted in the path integral equals the Euler characteristics χg,n (the g in χg,n denotes the genus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' On a surface with punctures, the latter is shifted by the number of punctures (which are equivalent to boundaries or disks) with respect to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='4): χg,n := χ(Σg,n) = 2 − 2g − n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='4) This gives the normalization factor: g−χg,n s = e−Φ0SEH[g], Φ0 := ln gs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='5) The correctness factor can be verified by inspection of the Riemann surface for the scat- tering of n string at g loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In particular, the string coupling constant is by definition the interaction strength for the scattering of 3 strings at tree-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Moreover, the tree-level 2-point amplitude contains no interaction and should have no power of gs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This factor can also be obtained by unitarity [193].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' By inserting these factors in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='28), the g-loop n-point scattering amplitude is described by: Ag,n({ki}){αi} := � dggab Ωgauge[g] dgΨ e−Sm[g,Ψ]−Φ0SEH[g] n � i=1 �� d2σi � g(σi) Vαi(ki;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' σi) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='6) The σi dependence of each √g will be omitted from now on since no confusion is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The following equivalent notations will be used: Ag,n({ki}){αi} := Ag,n(k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' , kn)α1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=',αn := Ag,n � Vα1(k1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' , Vαn(kn) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='7) The complete (perturbative) amplitude is found by summing over all genus: An(k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' , kn)α1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=',αn = ∞ � g=0 Ag,n(k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' , kn)α1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=',αn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='8) We omit a genus-dependent normalization which can be determined from unitarity [193].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Sometimes, it is convenient to extract the factor e−Φ0χg,n of the amplitude Ag,n to display explicitly the genus expansion, but we will not follow this convention here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Since each term of the sum scales as Ag,n ∝ g2g+n−2 s , this expression clearly shows that worldsheet amplitudes are perturbative by definition: this motivates the construction of a string field theory from which the full non-perturbative S-matrix can theoretically be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Finally, the amplitude (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='6) can be rewritten in terms of correlation functions of the matter QFT integrated over worldsheet metrics: Ag,n({ki}){αi} = � dggab Ωgauge[g] e−Φ0SEH[g] � n � i=1 d2σi √g � n � i=1 Vαi(ki;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' σi) � m,g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='9) The correlation function plays the same role as the partition function in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This shows that string expressions are integrals of CFT expressions over the space of worldsheet metrics (to be reduced to the moduli space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' We address a last question before performing the gauge fixing: what does (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='6) computes exactly: on-shell or off-shell?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Green functions or amplitudes?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' if amplitudes, the S-matrix 59 or just the interacting part T (amputated Green functions)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The first point is that a path integral over connected worldsheets will compute connected processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' We will prove later, when discussing the BRST quantization, that string states must be on-shell (Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2) and that it corresponds to setting the Hamiltonian (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='26) to zero: H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='10) From this fact, it follows that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='28) must compute amplitudes since non-amputated Green functions diverge on-shell (due to external propagators).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Finally, the question of whether it computes the S-matrix S = 1 + iT, or just the interacting part T is subtler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' At tree-level, they agree for n ≥ 3, while T = 0 for n = 2 and S reduces to the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This difficulty (discussed further in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2) is thus related to the question of gauge-fixing tree-level 2-point amplitude (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' It has long been believed that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='28) computes only the interacting part (amputated Green functions), but it has been understood recently that this is not correct and that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='28) computes the S-matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1 (Scattering amplitudes in QFT) Remember that the S-matrix is separ- ated as: S = 1 + iT, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='11) where 1 denotes the contribution where all particles propagate without interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The connected components of S and T are denoted by Sc and T c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The n-point (connected) scattering amplitudes An for n ≥ 3 can be computed from the Green functions Gn through the LSZ prescription (amputation of the external propagators): An(k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' , kn) = Gn(k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' , kn) n � i=1 (k2 i + m2 i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='12) The path integral computes the Green functions Gn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' perturbatively, they are obtained from the Feynman rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' They include a D-dimensional delta function Gn(k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' , kn) ∝ δ(D)(k1 + · · · + kn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='13) The 2-point amputated Green function T2 computed from the LSZ prescription vanishes on-shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' For example, considering a scalar field at tree-level, one finds: T2 = G2(k, k′) (k2 + m2)2 ∼ (k2 + m2) δ(D)(k + k′) −−−−−−→ k2→−m2 0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='14) since G2(k, k′) = δ(D)(k + k′) k2 + m2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='15) Hence, T2 = 0 and the S-matrix (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='11) reduces to the identity component Sc 2 = 12 (which is a connected process).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' There are several way to understand this result: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The recursive definition of the connected S-matrix Sc from the cluster decomposition principle requires a non-vanishing 2-point amplitude [121, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='5, 251, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3, 63, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The 2-point amplitude corresponds to the normalization of the 1-particle states (overlap of a particle state with itself, which is non-trivial) [250, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='4, 239, chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' A single particle in the far past propagating to the far future without interacting is a connected and physical process [63, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 133].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' It is required by the unitarity of the 2-point amplitude [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 60 These points indicate that the 2-point amplitude is proportional to the identity in the mo- mentum representation [121, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 212, 250, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='5] A2(k, k′) = 2k0 (2π)D−1δ(D−1)(k − k′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='16) The absence of interactions implies that the spatial momentum does not change (the on- shell condition implies that the energy is also conserved).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This relation is consistent with the commutation relation of the operators with the Lorentz invariant measure1 [a(k), a†(k′)] = 2k0 (2π)D−1δ(D−1)(k − k′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='17) That this holds for all particles at all loops can be proven using the Källen–Lehman repres- entation [121, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 212].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' On the other hand, the identity part in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='11) is absent for n ≥ 3 for connected amp- litudes: Sc n = T c n for n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This shows that the Feynman rules and the LSZ prescription compute only the interacting part T of the on-shell scattering amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The reason is that the derivation of the LSZ formula assumes that the incoming and outgoing states have no overlap, which is not the case for the 2-point function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' A complete derivation of the S-matrix from the path integral is more involved [121, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='5, 260, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='7, 81] (see also [37]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The main idea is to consider a superposition of momentum states (here, in the holomorphic representation [260, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='4]) φ(α) = � dD−1k α(k)∗a†(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='18) They contribute a quadratic piece to the connected S-matrix and, setting them to delta func- tions, one recovers the above result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2 Gauge fixing: general case The Faddeev–Popov gauge fixing of the worldsheet diffeomorphisms and Weyl rescaling (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='15) goes through also in this case if the integrated vertex operators are diffeomorphism and Weyl invariant: δξVαi(ki) = δξ � d2σ√g Vαi(ki;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' σ) = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='19a) δωVαi(ki) = δω � d2σ√g Vαi(ki;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' σ) = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='19b) with the variations defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='7) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Diffeomorphism invariance is straightforward if the states are integrated worldsheet scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' However, if the states are classically Weyl invariant, they are not necessary so at the quantum level: vertex operators are composite operators, which need to be renormalized to be well-defined at the quantum level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Renor- malization introduces a scale which breaks Weyl invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Enforcing it to be a symmetry of the vertex operators leads to constraints on the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' We will not enter in the details since it depends on the matter CFT and we will assume that the operators Vαi(ki) are indeed Weyl invariant (see [193, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='6] for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In the rest of this book, we will use CFT techniques developed in Chapter 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The Einstein–Hilbert action is clearly invariant under both symmetries since it is a topological quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 1If the modes are defined as ˜a(k) = a(k)/ √ 2k0 such that [˜a(k), ˜a†(k′)] = (2π)D−1δ(D−1)(k − k′), then one finds ˜ A2(k, k′) = (2π)D−1δ(D−1)(k − k′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 61 Following the computations from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3 leads to a generalization of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='136) with the vertex operators inserted for the amplitude (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='6): Ag,n({ki}){αi} = g−χg,n s � Mg dMgt det(φi, ˆµj)ˆg � det(φi, φj)ˆg Ωckv[ˆg]−1 � det(ψi, ψj)ˆg × � n � i=1 d2σi � ˆg � n � i=1 ˆVαi(ki;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' σi) � m,ˆg .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='20) The hat on the vertex operators indicates that they are evaluated in the background metric ˆg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The next step is to introduce the ghosts: following Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='4, the generalization of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='159) is Ag,n({ki}){αi} = g−χg,n s � Mg dMgt Ωckv[ˆg]−1 � det(ψi, ψj)ˆg � dˆgb d′ ˆgc Mg � i=1 (b, ˆµi)ˆg e−Sgh[ˆg,b,c] × � n � i=1 d2σi � ˆg � n � i=1 ˆVαi(ki;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' σi) � m,ˆg .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='21) For the moment, only the b ghosts come with zero-modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Then, c zero-modes can be introduced in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='21) Ag,n = g−χg,n s � Mg dMgt Ωckv[ˆg]−1 det ψi(σ0 j ) � dˆgb dˆgc Kc g � j=1 ϵab 2 ca(σ0 j )cb(σ0 j ) Mg � i=1 (ˆµi, b)ˆg e−Sgh[ˆg,b,c] × � n � i=1 d2σi � ˆg � n � i=1 ˆVαi(ki;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' σi) � m,ˆg , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='22) by following the same derivation as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='163).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The formulas (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='21) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='22) are the correct starting point for all g and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In particular, the c ghosts are not paired with any vertex (a condition often assumed or presented as mandatory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This fact will help resolve some difficulties for the 2-point function on the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Remember that there is no CKV and no c zero-mode for g ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' For the sphere g = 0 and the torus g = 1, there are CKVs, indicating that there is a residual symmetry in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='21) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='22), which is the global conformal group of the worldsheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' It can be gauge fixed by imposing conditions on the vertex operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2 The simplest gauge fixing condition amounts to fix the positions of Kc g vertex operators through the Faddeev–Popov trick: 1 = ∆(σ0 j ) � dξ Kc g � j=1 δ(2)(σj − σ0(ξ) j ), σ0(ξ) j = σ0 j + δξσ0 j , δξσ0 j = ξ(σ0 j ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='23) where ξ is a conformal Killing vector, and the variation of σ was given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' We find that ∆(σ0 j ) = det ψi(σ0 j ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='24) A priori, the positions σ0 j are not the same as the one appearing in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='163) (since both sets are arbitrary): however, considering the same positions allows to cancel the factor (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='24) with the same one in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='163).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 2In fact, it is only important to gauge fix for the sphere because the volume of the group is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' On the other hand, the volume of the CKV group for the torus is finite-dimensional such that dividing by Ωckv is not ambiguous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 62 Computation – Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='24) The first step is to compute ∆ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' For this, we decompose the CKV ξ on the basis (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='104) ξ(σ0 j ) = αiψi(σ0 j ) and write the Gaussian integral: 1 = � Kc g � j=1 d2δσj e −� j(δσj,δσj) = ∆ � Kg � j=1 dαi e −� j,i,i′(αiψi(σj),αi′ψi′(σj)) = ∆ � det ψi(σj) �−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Again, we have reduced rigour in order to simplify the manipulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' After inserting the identity (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='23) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='22), one can integrate over Kc g vertex operator positions to remove the delta functions – at the condition that there are at least Kc g operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' As a consequence, we learn that the proposed gauge fixing works only for n ≥ 1 if g = 1 or n ≥ 3 if g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This condition is equivalent to χg,n = 2 − 2g − n < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='25) In this case, the factors det ψi(σ0 j ) cancel and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='21) becomes Ag,n({ki}){αi} = g−χg,n s � Mg dMgt � dˆgb dˆgc Kc g � j=1 ϵab 2 ca(σ0 j )cb(σ0 j ) Mg � i=1 (ˆµi, b)ˆg e−Sgh[ˆg,b,c] × � n � i=Kc g+1 d2σi � ˆg � Kc g � j=1 ˆVαj(kj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' σ0 j ) n � i=Kcg+1 ˆVαi(ki;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' σi) � m,ˆg .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='26) The result may be divided by a symmetry factor if the delta functions have solutions for several points [193, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Performing the gauge fixing for the other cases (in particular, g = 0, n = 2 and g = 1, n = 0) is more subtle (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3 and [193]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The amplitude can be rewritten in two different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' First, the ghost insertions can be rewritten in terms of a ghost correlation functions Ag,n({ki}){αi} = g−χg,n s � Mg dMgt � n � i=Kc g+1 d2σi � ˆg � Kc g � j=1 ϵab 2 ca(σ0 j )cb(σ0 j ) Mg � i=1 (ˆµi, b)ˆg � gh,ˆg × � Kc g � j=1 ˆVαj(kj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' σ0 j ) n � i=Kc g+1 ˆVαi(ki;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' σi) � m,ˆg .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='27) This form is particularly interesting because it shows that, before integration over the mod- uli, the amplitudes factorize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This is one of the main advantage of the conformal gauge, since the original complicated amplitude (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='6) for a QFT on a dynamical spacetime reduces to the product of two correlation functions of QFTs on a fixed curved background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In fact, the situation is even simpler when taking a flat background ˆg = δ since both the ghost and mat- ter sectors are CFTs and one can employ all the tools from two-dimensional CFT (Part I) to perform the computations and mostly forget about the path integral origin of these formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This approach is particularly fruitful for off-shell (Chapter 11) and superstring amplitudes (Chapter 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 63 Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2 (Amplitudes in 2d gravity) The derivation of amplitudes for 2d gravity follows the same procedure, up to two differences: 1) there is an additional decoupled (before moduli and position integrations) gravitational sector described by the Liouville field, 2) the matter and gravitational action are not CFTs if the original matter was not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' A second formula can be obtained by bringing the c-ghost on top of the matter vertex operators which are at the same positions Ag,n({ki}){αi} = g−χg,n s � Mg dMgt � n � i=Kc g+1 d2σi � ˆg � Mg � i=1 ˆBi Kc g � j=1 ˆ Vαj(kj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' σ0 j ) n � i=Kc g+1 ˆVαi(ki;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' σi) � ˆg , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='28) and where ˆ Vαj(kj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' σ0 j ) := ϵab 2 ca(σ0 j )cb(σ0 j ) ˆVαj(kj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' σ0 j ), ˆBi := (ˆµi, b)ˆg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='29) The operators Vαi(ki;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' σ0 j ) (a priori off-shell) are called unintegrated operators, by opposition to the integrated operators Vαi(ki).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' We will see that both are natural elements of the BRST cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' To stress that the ˆBi insertions are really an element of the measure, it is finally possible to rewrite the previous expression as Ag,n({ki}){αi} = g−χg,n s � Mg×Cn−Kcg � Mg � i=1 ˆBi dti Kc g � j=1 ˆ Vαi(ki;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' σ0 j ) n � i=Kc g+1 ˆVαi(ki;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' σi) d2σi � ˆg � ˆg .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='30) The result (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='28) suggests a last possibility for improving the expression of the amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Indeed, the different vertex operators don’t appear symmetrically: some are integrated over and other come with c ghosts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Similarly, the two types of integrals have different roles: the moduli are related to geometry while the positions look like external data (vertex operators).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' However, punctures can obviously be interpreted as part of the geometry, and one may wonder if it is possible to unify the moduli and positions integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' It is, in fact, possible to put all vertex operators and integrals on the same footing by considering the amplitude to be defined on the moduli space Mg,n of genus-g Riemann surfaces with n punctures instead of just Mg [193] (see also Section 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3 Gauge fixing: 2-point amplitude As discussed at the end of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1, it has long been believed that the tree-level 2-point amplitude vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' There were two main arguments: there are not sufficiently many vertex operators 1) to fix completely the SL(2, C) invariance or 2) to saturate the number of c- ghost zero-modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Let’s review both points and then explain why they are incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' We will provide the simplest arguments, referring the reader to the literature [66, 208] for more general approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' For simplicity, we consider the flat metric ˆg = δ and an orthonormal basis of CKV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The two weight-(1, 1) matter vertex operators are denoted as Vk(z, ¯z) and Vk′(z′, ¯z′) such that the 2-point correlation function on the sphere reads (see Chapters 6 and 7 for more details): ⟨Vk(z, ¯z)Vk′(z′, ¯z′)⟩S2 = i (2π)Dδ(D)(k + k′) |z − z′|4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='31) The numerator comes from the zero-modes ei(k+k′)·x for a target spacetime with a Lorentzian signature [48, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 866, 193] (required to make use of the on-shell condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 64 Review of the problem The tree-level amplitude (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='20) for n = 2 reads: A0,2(k, k′) = CS2 Vol K0,0 � d2zd2z′ ⟨Vk(z, ¯z)Vk′(z′, ¯z′)⟩S2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='32) where K0,n is the CKV group of Σ0,n, the sphere with n punctures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In particular, the group of the sphere without puncture is K0,0 = PSL(2, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The normalization of the amplitude is CS2 = 8πα′−1 for gs = 1 [193, 249].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Since there are two insertions, the symmetry can be partially gauge fixed by fixing the positions of the two punctures to z = 0 and z′ = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In this case, the amplitude (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='32) becomes: A0,2(k, k′) = CS2 Vol K0,2 ⟨Vk(∞, ∞)Vk′(0, 0)⟩S2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='33) where K0,2 = R∗ +×U(1) is the CKV group of the 2-punctured sphere – containing dilatations and rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3 Since the volume of this group is infinite Vol K0,2 = ∞, it looks like A0,2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' However, this forgets that the 2-point correlation function (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='31) contains a D-dimensional delta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The on-shell condition implies that the conservation of the momentum k + k′ = 0 is automatic for one component, such that the numerator in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='33) contains a divergent factor δ(0): A0,2(k, k′) = (2π)D−1δ(D−1)(k + k′) CS2 2πi δ(0) Vol K0,2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='34) Hence, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='33) is of the form A0,2 = ∞/∞ and one should be careful when evaluating it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The second argument relies on a loophole in the understanding of the gauge fixed amp- litude (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The result (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='28) is often summarized by saying that one can go from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='20) to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='28) by replacing Kc g integrated vertices � V by unintegrated vertices c¯cV in order to saturate the ghost zero-modes and to obtain a non-zero result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' For g = 0, this requires 3 unintegrated vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' But, since there are only two operators in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='32), this is impossible and the result must be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' However, this is also incorrect because it is always possible to insert 6 c zero-modes, as show the formulas (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='163) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Indeed, they are part of how the path integral measure is defined and do not care of the matter operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The question is whether they can be attached to vertex operators (for aesthetic reasons or more pragmatically to get natural states of the BRST cohomology).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' To find the correct result with ghosts requires to start with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='27) and to see how this can be simplified when there are only two operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Computation of the amplitude In this section, we compute the 2-point amplitude from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='33): A0,2(k, k′) = CS2 Vol K0,2 ⟨Vk(∞, ∞)Vk′(0, 0)⟩S2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='35) The volume of K0,2 reads (by writing a measure invariant under rotations and dilatations, but not translations nor special conformal transformations) [53, 60]: Vol K0,2 = � d2z |z|2 = 2 � 2π 0 dσ � ∞ 0 dr r , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='36) 3The subgroup and the associated measure depend on the locations of the two punctures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 65 by doing the change of variables z = reiσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Since the volume is infinite, it must be regu- larized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' A first possibility is to cut-off a small circle of radius ϵ around r = 0 and r = ∞ (corresponding to removing the two punctures at z = 0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' A second possibility consists in performing the change of variables r = eτ and to add an imaginary exponential: Vol K0,2 = 4π � ∞ 0 dr r = 4π � ∞ −∞ dτ = 4π lim ε→0 � ∞ −∞ dτ eiετ = 4π × 2π lim ε→0 δ(ε), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='37) such that the regularized volume reads Volε K0,2 = 8π2 δ(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='38) In fact, τ can be interpreted as the Euclidean worldsheet time on the cylinder since r corresponds to the radial direction of the complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Since the worldsheet is an embedding into the target spacetime, both must have the same signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' As a consequence, for the worldsheet to be also Lorentzian, the formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='37) must be analytically continued as ε = −iE and τ = it such that VolM,E K0,2 = 8π2i δ(E), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='39) where the subscript M reminds that one considers the Lorentzian signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Inserting this expression in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='34) and taking the limit E → 0, it looks like the two δ(0) will cancel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' However, we need to be careful about the dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Indeed, the worldsheet time τ and energy E are dimensionless, while the spacetime time and energy are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Thus, it is not quite correct to cancel directly both δ(0) since they don’t have the same dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In order to find the correct relation between the integrals in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='37) and of the zero-mode in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='31), we can look at the mode expansion for the scalar field (removing the useless oscillators): X0(z, ¯z) = x0 + i 2 α′k0 ln |z|2 = x0 + iα′k0τ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='40) where the second equality follows by setting z = eτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' After analytic continuation k0 = −ik0 M, X0 = iX0 M, x0 = ix0 M and τ = it, we find [265, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 186]: X0 M = x0 M + α′k0 Mt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='41) This indicates that the measure of the worldsheet time in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='39) must be rescaled by 1/α′k0 M such that: VolM K0,2 −→ 8π2i δ(0) α′k0 M = CS2 2πi δ(0) 2k0 M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='42) This is equivalent to rescale E by α′k0 and to use δ(ax) = a−1δ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Ultimately, the 2-point amplitude becomes (removing the subscript on k0): A0,2(k, k′) = 2k0(2π)D−1δ(D−1)(k + k′) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='43) and matches the QFT formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' We see that taking into account the scale of the coordinates is important to reproduce this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The computation displayed here presents some ambiguities because of the regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' However, this ambiguity can be fixed from unitarity of the scattering amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' A more general version of the Faddeev–Popov gauge fixing has been introduced in [66] to avoid dealing altogether with infinities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' It is an interesting question whether these techniques can be extended to the compute the tree-level 1- and 0-point amplitudes on the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In most cases, the 1-point amplitude is expected to vanish since 1-point correlation functions 66 of primary operators other than the identity vanish in unitary CFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='4 The 0-point function corresponds to the sphere partition function: the saddle point approximation to leading order allows to relate it to the spacetime action evaluated on the classical solution φ0, Z0 ∼ e−S[φ0]/ℏ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Since the normalization is not known and because S[φ0] is expected to be infinite, only comparison between two spacetimes should be meaningful (à la Gibbons– Hawking–York [190, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In particular, for Minkowski spacetime we find naively Z0 ∼ δ(D)(0) Vol K0 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='44) which is not well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This question has no yet been investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Expression with ghosts There are different ways to rewrite the 2-point amplitude in terms of ghosts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In all cases, one correctly finds the 6 insertions necessary to get a non-vanishing result since, by definition, it is always possible to rewrite the Faddeev–Popov determinant in terms of ghosts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' A first approach is to insert 1 = � d2z δ(2)(z) inside (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='32) to mimic the presence of a third operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This is equivalent to use the identity ⟨0| c−1¯c−1c0¯c0c1¯c1 |0⟩ = 1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='45) inside (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='33), leading to: A0,2(k, k′) = CS2 Vol K0,2 ⟨Vk(∞, ∞)c0¯c0 Vk′(0, 0)⟩S2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='46) where Vk(z, ¯z) = c¯cVk(z, ¯z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This shows that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='16) can also be recovered using the correct insertions of ghosts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The presence of c0¯c0 can be expected from string field theory since they appear in the kinetic term (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='115).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The disadvantage of this formula is to still contain the infinite volume of the dilatation group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' It is also possible to introduce ghosts for the more general gauge fixing presented in [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' An alternative approach has been proposed in [208].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2 BRST quantization The symmetries of a Lagrangian dictate the possible terms which can be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This continues to hold at the quantum level and the counter-terms introduced by renormalization are constrained by the symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' However, if the path integral is gauge fixed, the original symmetry is no more available for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Fortunately, one can show that there is a global symmetry (with anticommuting parameters) remnant of the local symmetry: the BRST symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' It ensures consistency of the quantum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' It also provides a direct access to the physical spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The goal of this section is to provide a general idea of the BRST quantization for the worldsheet path integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' A more detailed CFT analysis and the consequence for string theory are given in Chapter 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The reader is assumed to have some familiarity with the BRST quantization in field theory – a summary is given in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 4The integral over the zero-mode gives a factor δ(D)(k) which implies k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' At zero momentum, the time scalar X0 is effectively described by unitary CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' However, there can be some subtleties when considering marginal operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 67 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1 BRST symmetry The partition function (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='159) is not the most suitable to display the BRST symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The first step is to restore the dependence in the original metric gab by introducing a delta function Zg = � Mg dMgt Ωckv[g] � dggab dgΨ dgb d′ gc δ �√ggab − � ˆgˆgab � Mg � i=1 (φi, b)g e−Sm[g,Ψ]−Sgh[g,b,c].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='47) Note that it is necessary to use the traceless gauge fixing condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='152) as it will become clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The delta function is Fourier transformed in an exponential thanks to an auxiliary bosonic field: Zg = � Mg dMgt Ωckv[g] � dggab dgBab dgΨ dgb d′ gc Mg � i=1 (φi, b)g e−Sm[g,Ψ]−Sgf[g,ˆg,B]−Sgh[g,b,c] (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='48) where the gauge-fixing action reads: Sgf[g, ˆg, B] = − i 4π � d2σ Bab�√ggab − � ˆgˆgab � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='49) Varying the action with respect to the auxiliary field Bab, called the Nakanish–Lautrup field, produces the gauge-fixing condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The BRST transformations are δϵgab = iϵ Lcgab, δϵΨ = iϵ LcΨ, δϵca = iϵ Lcca, δϵbab = ϵ Bab, δϵBab = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='50) where ϵ is a Grassmann parameter (anticommuting number) independent of the position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' If the traceless gauge fixing (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='152) is not used, then Bab is not traceless: in that case, the variation δϵbab will generate a trace, which is not consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Since the transformations act on the matter action Sm as a diffeomorphism with vector ϵca, it is obvious that it is invariant by itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' It is easy to show that the transformations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='50) leave the total action invariant in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The invariance of the measure is given in [193].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3 (BRST transformations with Weyl ghost) One can also consider the ac- tion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='153) with the Weyl ghost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In this case, the transformation law of the metric is modified and the Weyl ghost transforms as a scalar: δϵgab = iϵ Lcgab + iϵ gabcw, δϵcw = iϵ Lccw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='51) The second term in δϵgab is a Weyl transformation with parameter ϵcw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Moreover, bab and Bab are not symmetric traceless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The equation of motion for the auxiliary field is Bab = i Tab := i � T m ab + T gh ab � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='52) where the RHS is the total energy–momentum tensor (matter plus ghosts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Integrating it out imposes the gauge condition gab = ˆgab and yields the modified BRST transformations δϵΨ = iϵ LcΨ, δϵca = iϵ Lcca, δϵbab = iϵ Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='53) Without starting with the path integral (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='48) with auxiliary field, it would have been difficult to guess the transformation of the b ghost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Since ca is a vector, one can also write δϵca = ϵ cb∂bca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='54) 68 Associated to this symmetry is the BRST current ja B and the associated conserved BRST charge QB QB = � dσ j0 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='55) The charge is nilpotent Q2 B = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='56) and, through the presence of the c-ghost in the BRST transformation, the BRST charge has ghost number one Ngh(QB) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='57) Variations of the matter fields can be written as δϵΨ = i [ϵQB, Ψ]±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='58) Note that the energy–momentum tensor is BRST exact Tab = [QB, bab].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='59) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2 BRST cohomology and physical states Physical state |ψ⟩ are elements of the absolute cohomology of the BRST operator: |ψ⟩ ∈ H(QB) := ker QB Im QB , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='60) or, more explicitly, closed but non-exact states: QB |ψ⟩ = 0, ∄ |χ⟩ : |ψ⟩ = QB |χ⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='61) The adjective “absolute” is used to distinguish it from two other cohomologies (relative and semi-relative) defined below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Two states of the cohomology differing by an exact state represent identical physical states: |ψ⟩ ∼ |ψ⟩ + QB |Λ⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='62) This equivalence relation, translated in terms of spacetime fields, correspond to spacetime gauge transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In particular, it contains the (linearized) reparametrization invari- ance of the spacetime metric in the closed string sector, and, for the open string sector, it contains Yang–Mills symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' We will find that it corresponds to the gauge invariance of free string field theory (Chapter 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' However, physical states satisfy two additional constraints (remember that bab is traceless symmetric): � dσ bab |ψ⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='63) These conditions are central to string (field) theory, so they will appear regularly in this book.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' For this reason, it is useful to provide first some general motivations, and to refine the analysis later since the CFT language will be more appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Moreover, these two conditions will naturally emerge in string field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In order to introduce some additional terminology, let’s define the following quantities:5 b+ := � dσ b00, b− := � dσ b01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='64) 5The objects b± are zero-modes of the b ghost fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' They correspond (up to a possible irrelevant factor) to the modes b± 0 in the CFT formulation of the ghost system (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='132).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 69 The semi-relative and relative cohomologies H−(QB) and H0(QB) are defined as6 H−(QB) = H(QB) ∩ ker b−, H0(QB) = H−(QB) ∩ ker b+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='65) The first constraint arises as a consequence of the topology of the closed string worldsheet: the spatial direction is a circle, which implies that the theory must be invariant under translations along the σ direction (the circle is invariant under rotation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' However, choosing a parametrization implies to fix an origin for the spatial direction: this is equivalent to a gauge fixing condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' As usual, this implies that the corresponding generator Pσ of worldsheet spatial translations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='26) must annihilate the states: Pσ |ψ⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='66) This is called the level-matching condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='59), this can be rewritten as Pσ |ψ⟩ = � dσ T01 |ψ⟩ = � dσ {QB, b01} |ψ⟩ = QB � dσ b01 |ψ⟩ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='67) since QB |ψ⟩ = 0 for a state |ψ⟩ in the cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The simplest way to enforce this condition is to set the state on which QB acts to zero:7 b− |ψ⟩ = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='68) which is equivalent to one of the conditions in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='63).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The second condition does not follow as simply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The Hilbert space can be decomposed according to b+ as H− := H↓ ⊕ H↑, H↓ := H0 := H− ∩ ker b+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='69) Indeed, b+ is a Grassmann variable and generates a 2-state system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In the ghost sector, the two Hilbert spaces are generated from the ghost vacua | ↓⟩ and | ↑⟩ obeying b+ | ↓⟩ = 0, b+ | ↑⟩ = | ↓⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='70) The action of the BRST charge on states |ψ↓⟩ ∈ H↓ and |ψ↑⟩ ∈ H↑ follow from these relations and from the commutation relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='59): QB |ψ↓⟩ = H |ψ↑⟩ , QB |ψ↑⟩ = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='71) where H is the worldsheet Hamiltonian defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' To prove this relation, start first with H |ψ↑⟩, then use (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='59)) to get the LHS of the first condition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' then apply QB to get the second condition (using that QB commutes with H, and b+ with any other operators building the states).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' For H ̸= 0, the state |ψ↓⟩ is not in the cohomology and |ψ↑⟩ is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Thus, the exact and closed states are Im QB = � |ψ↑⟩ ∈ H↑ | H |ψ↑⟩ ̸= 0 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='72a) ker QB = � |ψ↑⟩ ∈ H↑ � ∪ � |ψ↓⟩ ∈ H↓ | H |ψ↓⟩ = 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='72b) This implies that eigenstates of H in the cohomology satisfy the on-shell condition: H |ψ⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='73) 6The BRST cohomologies described in this section are slightly different from the ones used in the rest of this book.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' To distinguish them, indices are written as superscripts in this section, and as subscripts otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 7The reverse is not true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' We will see in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2 the relation between the two conditions in more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 70 This is consistent with the fact that scattering amplitudes involve on-shell states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In this case, |ψ↑⟩ is not exact and is thus a member of the cohomology H(QB), as well as |ψ↓⟩ since it becomes close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' But, the Hilbert space H↑ must be rejected for two reasons: there would be an apparent doubling of states and scattering amplitudes would behave badly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The first problem arises because one can show that the cohomological subspaces of each space are isomorphic: H↓(QB) ≃ H↑(QB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Hence, keeping both subspaces would lead to a doubling of the physical states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' For the second problem, consider an amplitude where one of the external state is built from |ψ↑⟩: the amplitude vanishes if the states are off-shell since the state |ψ↑⟩ is exact, but it does not vanish on-shell [193, ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This means that it must be proportional to δ(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' But, general properties in QFT forbid such dependence in the amplitude (only poles and cuts are allowed, except if D = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Projecting out the states in H↑ is equivalent to require b+ |ψ⟩ = 0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='74) for physical states, which is the second condition in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='63).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In fact, this condition can be obtained very similarly as the b− = 0 condition: using the expression of H (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='26) and the commutation relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='59), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='73) is equivalent to QB � dσ b00 |ψ⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='75) Hence, imposing (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='74) allows to automatically ensure that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='73) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Since the on-shell character (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='73) of the BRST states and of the BRST symmetry are intimately related to the construction of the worldsheet integral, one can expect difficulty for going off-shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3 Summary In this chapter, we derived general formulas for string scattering amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The general BRST formalism has been summarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Moreover, we gave general motivations for restrict- ing the absolute cohomology to the smaller relative cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In Chapter 8, a more precise derivation of the BRST cohomology is worked out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' It includes also a proof of the no-ghost theorem: the ghosts and the negative norm states (in Minkowski signature) are unphysical particles and should not be part of the physical states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This theorem asserts that it is indeed the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' It will also be the occasion to recover the details of the spectrum in various cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='4 Suggested readings The delta function approach to the gauge fixing is described in [193, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3, 151, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2], with a more direct computation is in [128].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The most complete references for scattering amplitudes in the path integral formalism are [53, 193].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Computation of the tree-level 2-point amplitude [66, 208] (for discussions of 2-point function, see [53, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 936–7, 207, 60, 61, 48, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 863–4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The BRST quantization of string theory is discussed in [155, 39, 193, chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' For a general discussion see [105, 247, 251].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The use of an auxiliary field is considered in [252, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 71 Chapter 4 Worldsheet path integral: complex coordinates Abstract In the two previous chapters, the amplitudes computed from the worldsheet path integrals have been written covariantly for a generic curved background metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In this chapter, we start to use complex coordinates and finally take the background metric to be flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This is the usual starting point for computing amplitudes since it allows to make contact with CFTs and to employ tools from complex analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' We first recall few facts on 2d complex manifolds before briefly describing how to rewrite the scattering amplitudes in complex coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1 Geometry of complex manifolds Choosing a flat background metric simplifies the computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' However, we have seen in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3 that there is a topological obstruction to get a globally flat metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The solution is to work with coordinate patches (σ0, σ1) = (τ, σ) such that the background metric ˆgab is flat in each patch (conformally flat gauge): ds2 = gabdσadσb = e2φ(τ,σ)� dτ 2 + dσ2� , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1) or gab = e2φδab, ˆgab = δab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2) To simplify the notations, we remove the dependence in the flat metric and the hat for quantities (like the vertex operators) expressed in the background metric when no confusion is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Introducing complex coordinates z = τ + iσ, ¯z = τ − iσ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3a) τ = z + ¯z 2 , σ = z − ¯z 2i , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3b) the metric reads1 ds2 = 2gz¯zdzd¯z = e2φ(z,¯z)|dz|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='4) 1In Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1, we provide more details on the relation between the worldsheet (viewed as a cylinder or a sphere) and the complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 72 The metric and its inverse can also be written in components: gz¯z = e2φ 2 , gzz = g¯z¯z = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='5a) gz¯z = 2e−2φ, gzz = g¯z¯z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='5b) Equivalently, the non-zero components of the background metric are ˆgz¯z = 1 2, ˆgz¯z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='6) An oriented two-dimensional manifold is a complex manifold: this means that there exists a complex structure, such that the transition functions and changes of coordinates between different patches are holomorphic at the intersection of the two patches: w = w(z), ¯w = ¯w(¯z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='7) For such a transformation, the Liouville mode transforms as e2φ(z,¯z) = ���� ∂w ∂z ���� 2 e2φ(w, ¯ w) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='8) such that ds2 = e2φ(w, ¯ w)|dw|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='9) This shows also that a conformal structure (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='12) induces a complex structure since the transformation law of φ is equivalent to a Weyl rescaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The integration measures are related as d2σ := dτdσ = 1 2 d2z, d2z := dzd¯z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='10) Due to the factor of 2 in the expression, the delta function δ(2)(z) also gets a factor of 2 with respect to δ(2)(σ) δ(2)(z) = 1 2 δ(2)(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='11) Then, one can check that � d2z δ(2)(z) = � d2σ δ(2)(σ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='12) The basis vectors (derivatives) and one-forms can be found using the chain rule: ∂z = 1 2 (∂τ − i∂σ), ∂¯z = 1 2 (∂τ + i∂σ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='13a) dz = dτ + idσ, d¯z = dτ − idσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='13b) The Levi–Civita (completely antisymmetric) tensor is normalized by ϵ01 = ϵ01 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='14a) ϵz¯z = i 2, ϵz¯z = −2i, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='14b) remembering that it transforms as a density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Integer indices run over local frame coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The different tensors can be found from the tensor transformation law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' For example, the components of a vector V a in both systems are related by V z = V 0 + iV 1, V ¯z = V 0 − iV 1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='15) 73 such that V = V 0∂0 + V 1∂1 = V z∂z + V ¯z∂¯z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='16) For holomorphic coordinate transformations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='7), the components of the vector do not mix: V w = ∂w ∂z V z, V ¯ w = ∂ ¯w ∂¯z V ¯z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='17) This implies that the tangent space of the Riemann surface is decomposed into holomorphic and anti-holomorphic vectors:2 TΣg ≃ TΣ+ g ⊕ TΣ− g , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='18a) V z∂z ∈ TΣ+ g , V ¯z∂¯z ∈ TΣ− g , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='18b) as a consequence of the existence of a complex structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Similarly, the components of a 1-form ω – which is the only non-trivial form on Σg – can be written in terms of the real coordinates as: ωz = 1 2 (ω0 − iω1), ω¯z = 1 2 (ω0 + iω1) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='19) such that ω = ω0dσ0 + ω1dσ1 = ωzdz + ω¯zd¯z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='20) Hence, a 1-form is decomposed into complex (1, 0)- and (0, 1)-forms: T ∗Σg ≃ Ω1,0(Σg) ⊕ Ω0,1(Σg), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='21a) ωzdz ∈ Ω1,0(Σg), ω¯zd¯z ∈ Ω0,1(Σg), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='21b) since both components will not mixed under holomorphic changes of coordinates (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Fi- nally, the metric provides an isomorphism between TΣ+ g and Ω0,1(Σg), and between TΣ− g and Ω1,0(Σg), since it can be used to lower/raise an index while converting it from holo- morphic to anti-holomorphic, or conversely: Vz = gz¯zV ¯z, V¯z = gz¯zV z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='22) This can be generalized further by considering components with more indices: all anti- holomorphic indices can be converted to holomorphic indices thanks to the metric: T q++p− ���� z···z z···z ���� p++q− = (gz¯z)p−(gz¯z)q−T q+ ���� z···z q− ���� ¯z···¯z z···z ���� p+ ¯z···¯z ���� p− .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='23) Hence, it is sufficient to study (p, q)-tensors with p upper and q lower holomorphic indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In this case, the transformation rule under (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='7) reads T q ���� w···w w···w ���� p = �∂w ∂z �n T q ���� z···z z···z ���� p , n := q − p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='24) The number n ∈ Z is called the helicity or rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3 The set of helicity-n tensors is denoted by T n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The first example is vectors (or equivalently 1-forms): V z ∈ T 1, Vz ∈ T −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The second most useful case is traceless symmetric tensors, which are elements of T ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Consider a 2However, at this stage, each component can still depend on both z and ¯z: V z = V z(z, ¯z) and V ¯z = V ¯z(z, ¯z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 3In fact, it is even possible to consider n ∈ Z + 1/2 to describe spinors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 74 traceless symmetric tensor T ab = T ba and gabT ab = 0: this implies T 01 = T 10 and T 00 = −T 11 in real coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The components in complex coordinates are: T zz = 2(T 00 + iT 01) ∈ T 2, T ¯z¯z = 2(T 00 − iT 01) ∈ T −2, T z z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='25) Note that Tzz = gz¯zgz¯zT ¯z¯z = 1 2(T 00 − iT 01), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='26) and T z z = gz¯zT z¯z ∈ T 0 corresponds to the trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Computation – Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='25) T zz = �∂z ∂τ �2 T 00 + � ∂z ∂σ �2 T 11 + 2 ∂z ∂τ ∂z ∂σ T 01 = T 00 − T 11 + 2i T 01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Stokes’ theorem in complex coordinates follows directly from (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='10): � d2z (∂zvz + ∂¯zv¯z) = −i � � dz v¯z − d¯zvz� = −2i � ∂R (vzdz − v¯zd¯z), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='27) where the integration contour is anti-clockwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' To obtain this formula, note that d2x = 1 2d2z and ϵz¯z = i/2, such that the factor 1/2 cancels between both sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2 Complex representation of path integral In the previous section, we have found that tensors of a given rank are naturally decomposed into different subspaces thanks to the complex structure of the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Accordingly, complex coordinates are natural and one can expect most objects in string theory to split similarly into holomorphic and anti-holomorphic sectors (or left- and right-moving).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This will be particularly clear using the CFT language (Chapter 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The main difficulty for this program is due to the matter zero-modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In this section, we focus on the path integral measure and expression of the ghosts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' There is, however, a subtlety in displaying explicitly the factorization: the notion of “holomorphicity” depends on the metric (because the complex structure must be compatible with the metric for an Hermitian manifold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Since the metric depends on the moduli which are integrated over in the path integral, it is not clear that there is a consistent holomorphic factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' We will not push the question of achieving a global factorization further (but see Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1) to focus instead on the integrand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The latter is local (in moduli space) and there is no ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The results of the previous section indicate that the basis of Killing vectors (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='104) and quadratic differentials (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='76) split into holomorphic and anti-holomorphic components: ψi(z, ¯z) = ψz i ∂z + ψ¯z i ∂¯z, φi(z, ¯z) = φi,zz(dz)2 + φi,¯z¯z(d¯z)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='28) Similarly, the operators P1 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='65a) and P † 1 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='71) also split: (P1ξ)zz = 2∇zξz = ∂zξ¯z, (P1ξ)¯z¯z = 2∇¯zξ¯z = ∂¯zξz, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='29a) (P † 1 T)z = −2∇zTzz = −4 ∂¯zTzz, (P † 1 T)¯z = −2∇¯zT¯z¯z = −4 ∂zT¯z¯z (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='29b) for arbitrary vector ξ and traceless symmetric tensor T (in the background metric).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' As a consequence, the components of Killing vectors and quadratic differentials are holomorphic or anti-holomorphic as a function of z: ψz = ψz(z), ψ¯z = ψ¯z(¯z), φzz = φzz(z), φ¯z¯z = φ¯z¯z(¯z), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='30) 75 such that it makes sense to consider a complex basis instead of the previous real basis: ker P1 = Span{ψK(z)} ⊕ Span{ ¯ψK(¯z)}, K = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' , Kc g, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='31a) ker P † 1 = Span{φI(z)} ⊕ Span{¯φI(¯z)}, I = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' , Mc g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='31b) The last equation can inspire to search for a similar rewriting of the moduli parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In fact, the moduli space itself is a complex manifold and can be endowed with complex coordinates [173, 193]: mI = t2I−1 + it2I, ¯mI = t2I−1 − it2I, I = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' , Mc g (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='32) with the integration measure dMgt = d2Mc gm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='33) The last ingredient to rewrite the vacuum amplitudes (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='136) is to obtain the determin- ants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The inner-products of vector and traceless symmetric fields also factorize: (T1, T2) = 2 � d2σ � ˆg ˆgacgbdT1,abT2,cd = 4 � d2z � T1,zzT2,¯z¯z + T1,¯z¯zT2,zz � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='34a) (ξ1, ξ2) = � d2σ � ˆg ˆgabξaξb = 1 4 � d2z � ξz 1ξ¯z 2 + ξ¯z 1ξz 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='34b) All inner-products are evaluated in the flat background metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' For (anti-)holomorphic fields, only one term survives in each integral: since each field appears twice in the determinants (φi, φj) and (φi, φj), the final expression is a square, which cancels against the squareroot in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='136).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The remaining determinant involves the Beltrami differential (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='65b): µizz = ∂i¯gzz, µi¯z¯z = ∂i¯g¯z¯z (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='35) (¯gzz = 0 in our coordinates system, but its variation under a shift of moduli is not zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The basis can be changed to a complex basis such that the determinant of inner-products between Beltrami and quadratic differentials is a modulus squared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' All together, the different formulas lead to the following rewriting of the vacuum amplitude : Zg = � Mg d2Mc gm | det(φI, µJ)|2 | det(φI, ¯φJ)| det′ P † 1 P1 | det(ψI, ¯ψJ)| Zm[δ] Ωckv[δ], (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='36) where the absolute values are to be understood with respect to the basis of P1 and P † 1 , for example |f(mI)|2 := f(mI)f( ¯mI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The same reasoning can be applied to the ghosts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The c and b ghosts are respectively a vector and a symmetric traceless tensor, both with two independent components: it is customary to define c := cz, ¯c := c¯z, b := bzz, ¯b := b¯z¯z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='37) In that case, the action (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='145) reads Sgh[g, b, c] = 1 2π � d2z � b∂¯zc + ¯b∂z¯c � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='38) The action is the sum of two holomorphic and anti-holomorphic contributions and it is independent of φ(z, ¯z) as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In fact, the equations of motion are ∂zc = 0, ∂zb = 0, ∂¯z¯c = 0, ∂¯z¯b = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='39) 76 such that b and c (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' ¯b and ¯c) are holomorphic (anti-holomorphic) functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Then, the integration measure is simply Mg � i=1 Bi dti = Mc g � I=1 BI ¯BI dmI ∧ ¯mI, BI := (µI, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='40) Note that BI does not contain ¯b(¯z), it is built only from b(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Finally, the vacuum amplitude (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='163) reads Zg = � Mg d2Mc gm Ωckv[δ]−1 | det ψI(z0 j )|2 � d(b,¯b) d(c, ¯c) Kc g � j=1 c(z0 j )¯c(¯z0 j ) Mc g � I=1 |(µI, b)|2 e−Sgh[b,c] Zm[δ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='41) The c insertions are separated in holomorphic and anti-holomorphic components because, at the end, only the zero-modes contribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The measures are written as d(b,¯b) and d(c, ¯c) because proving that they factorize is difficult (Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1 (Holomorphic factorization) It was proven in [15, 27, 33] (see [173, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 9, 53, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' VII, 237, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 3] for reviews) that the ghost and matter path integrals can be globally factorized, up to a factor due to zero-modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Such a result is suggested by the factorization of the inner-products, which imply a factorization of the measures: the caveat is due to the zero-mode determinants and matter measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Interestingly, the factorization is possible only in the critical dimension (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='125).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3 Summary In this chapter, we have introduced complex notations for the fields, path integral and moduli space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='4 Suggested readings Good references for this chapter are [24, 53, 172, 173, 193].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Geometry of complex manifolds is discussed in [24, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2, 172, chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 14, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 77 Chapter 5 Conformal symmetry in D dimensions Abstract Starting with this chapter, we discuss general properties of conformal field the- ories (CFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The goal is not to be exhaustive, but to provide a short introduction and to gather the concepts and formulas that are needed for string theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' However, the subject is presented as a standalone topic such that it can be of interest for a more general public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The conformal group in any dimension is introduced in this chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The specific case D = 2, which is the most relevant for the current book, is developed in the following chapters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1 CFT on a general manifold In this chapter and in the next one, we discuss CFTs as QFTs living on a spacetime M, independently from string theory (there is no reference to a target spacetime).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' As such, we will use spacetime notations together with some simplifications: coordinates are written as xµ with µ = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' , D − 1 and time is written as x0 = t (x0 = τ) in Lorentzian (Euclidean) signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Given a metric gµν on a D-dimensional manifold M, the conformal group CISO(M) is the set of coordinate transformations (called conformal symmetries or isometries) xµ −→ x′µ = x′µ(x) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1) which leaves the metric invariant up to an overall scaling factor: gµν(x) −→ g′ µν(x′) = ∂xρ ∂x′µ ∂xσ ∂x′ν gρσ(x) = Ω(x′)2gµν(x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2) This means that angles between two vectors u and v are left invariant under the transform- ation: u · v |u| |v| = u′ · v′ |u′| |v′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3) It is often convenient to parametrize the scale factor by an exponential Ω := eω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='4) Considering an infinitesimal transformation δxµ = ξµ, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='5) 78 the condition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2) becomes the conformal Killing equation δgµν = Lξgµν = ∇µξν + ∇νξµ = 2 d gµν∇ρξρ, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='6) such that the scale factor is Ω2 = 1 + 2 d ∇ρξρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='7) The vector fields ξ satisfying this equation are called conformal Killing vectors (CKV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Con- formal transformations form a global subgroup of the diffeomorphism group: the generators of the transformations do depend on the coordinates, but the parameters do not (for an internal global symmetry, both the generators and the parameters don’t depend on the coordinates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The conformal group contains the isometry group ISO(M) of M as a subgroup, corres- ponding to the case Ω = 1: ISO(M) ⊂ CISO(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='8) These transformations also preserve distances between points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The corresponding generators of infinitesimal transformations are called Killing vectors and satisfies the Killing equation δgµν = Lξgµν = ∇µξν + ∇νξµ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='9) They form a subalgebra of the CKV algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' An important point is to be made for the relation between infinitesimal and finite trans- formations: with spacetime symmetries it often happens that the first cannot be exponenti- ated into the second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The reason is that the (conformal) Killing vectors may be defined only locally, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' they are well-defined in a given domain but have singularities outside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' When this happens, they do not lead to an invertible transformation, which cannot be an element of the group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' These notions are sometimes confused in physics and the term of “group” is used instead of “algebra”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' We shall be careful in distinguishing both concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1 (Isometries of M ⊂ Rp,q) In order to find the conformal isometries of a manifold M which is a subset of Rp,q defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='10), it is sufficient to restrict the trans- formations of Rp,q to the subset M [206].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In the process, not all global transformations generically survive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' On the other hand, the algebra of local (infinitesimal) transformations for M and Rp,q are identical since M is locally like Rp,q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2 CFT on Minkowski space In this section, we consider the case where M = Rp,q (D = p + q) and where g = η is the flat metric with signature (p, q): η = diag(−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' , −1 � �� � q , 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' , 1 � �� � p ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='10) The conformal Killing equation becomes � ηµν∆ + (D − 2)∂µ∂ν � ∂ · ϵ = 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='11) where ∆ is the D-dimensional Beltrami–Laplace operator for the metric ηµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The case D = 2 is relegated to the next chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' For D > 2, one finds the following transformations: translation: ξµ = aµ, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='12a) rotation & boost: ξµ = ωµ νxν, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='12b) dilatation: ξµ = λ xµ, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='12c) SCT: ξµ = bµx2 − 2b · x xµ, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='12d) 79 where ωµν is antisymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The rotations include Lorentz transformations and SCT means “special conformal transformation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' All parameters {aµ, ωµν, λ, bµ} are constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The generators are respectively denoted by {Pµ, Jµν, D, Kµ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The finite translations and rotations form the Poincaré group SO(p, q), while the conformal group can be shown to be SO(p + 1, q + 1): ISO(Rp,q) = SO(p, q), CISO(Rp,q) = SO(p + 1, q + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='13) The dimension of this group is dim SO(p + 1, q + 1) = 1 2 (p + q + 2)(p + q + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='14) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3 Suggested readings References on higher-dimensional CFTs are [54, 196, 203, 206, 236].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 80 Chapter 6 Conformal field theory on the plane Abstract Starting with this chapter, we focus on two-dimensional Euclidean CFTs on the complex plane (or equivalently the sphere).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' We start by describing the geometry of the sphere and the relation to the complex plane and to the cylinder, in order to make contact with the string worldsheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Then, we discuss classical CFTs and the Witt algebra obtained by classifying the conformal isometries of the complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Then, we describe quantum CFTs and introduce the operator formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This last section is the most important for this book as it includes information on the operator product expansion, Hilbert space, Hermitian and BPZ conjugations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' As described at the beginning of Chapter 5, we use spacetime notations for the coordin- ates, but follow otherwise the normalization for the worldsheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In particular, integrals are normalized by 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' However, the spatial coordinate on the cylinder is still written as σ to avoid confusions: xµ = (τ, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1 The Riemann sphere 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1 Map to the complex plane The Riemann sphere Σ0, which is diffeomorphic to the unit sphere S2, has genus g = 0 and is thus the simplest Riemann surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Its most straightforward description is obtained by mapping it to the extended1 complex plane ¯C (also denoted ˆC), which is the complex plane z ∈ C to which the point at infinity z = ∞ is added: ¯C = C ∪ {∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1) One speaks about “the point at infinity” because all the points at infinity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' the points z such that |z| → ∞) lim r→∞ r eiθ := ∞ (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2) are identified (the limit is independent of θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The identification can be understood by mapping (say) the south pole to the origin of the plane and the north pole to infinity2 (Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1) through the stereographic projection z = eiφ cot θ 2, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3) 1This qualification will often be omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 2Note that the points are distinguished in order to write the map, but they have nothing special by themselves (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' they are not punctures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 81 −−−−−−−−→ Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1: Map from the Riemann sphere to the complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The south and north poles are denoted by the letter S and N, and the equatorial circle by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' where (θ, φ) are angles on the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Any circle on the sphere is mapped to a circle in the complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Conversely, the Riemann sphere can be viewed as a compactification of the complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Introducing Cartesian coordinates (x, y) related to the complex coordinates by3 z = x + iy, ¯z = x − iy, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='4a) x = z + ¯z 2 , y = z − ¯z 2i , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='4b) the metric reads ds2 = dx2 + dy2 = dzd¯z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='5) The relations between the derivatives in the two coordinate systems are easily found: ∂ := ∂z = 1 2 (∂x − i∂y), ¯∂ := ∂¯z = 1 2 (∂x + i∂y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='6) The indexed form will be used when there is a risk of confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' If the index is omitted then the derivative acts directly to the field next to it, for example ∂φ(z1)∂φ(z2) := ∂z1∂z2φ(z1)φ(z2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='7) Generically, the meromorphic and anti-meromorphic parts of a object will be denoted without and with a bar, see (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='55) for an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The extended complex plane ¯C can be covered by two coordinate patches z ∈ C and w ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In the first, the point at infinity (north pole) is removed, in the second, the origin (south pole) is removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' On the overlap, the transition function is w = 1 z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='8) This description avoids to work with the infinity: studying the behaviour of f(z) at z = ∞ is equivalent to study f(1/w) at w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Since any two-dimensional metric is locally conformally equivalent to the flat metric, it is sufficient to work with this metric in each patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This is particularly convenient for the Riemann sphere since one patch covers it completely except for one point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 3General formulas can be found in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1 by replacing (τ, σ) with (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In most cases, the conformal factor is set to zero (φ = 0) in this chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 82 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2 Relation to the cylinder – string theory The worldsheet of a closed string propagating in spacetime is locally topologically a cylinder R × S1 of circumference L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In this section, we show that the cylinder can also be mapped to the complex plane – and thus to the Riemann sphere – after removing two points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Since the cylinder has a clear physical interpretation in string theory, it is useful to know how to translate the results from the plane to the cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' It makes also sense to define two-dimensional models on the cylinder independently of a string theory interpretation since the compactification of the spatial direction from R to S1 regulates the infrared divergences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Moreover, it leads to a natural definition of a “time” and of an Hamiltonian on the Euclidean plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Denoting the worldsheet coordinates in Lorentzian signature by (t, σ) with4 t ∈ R, σ ∈ [0, L), σ ∼ σ + L, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='9) the metric reads ds2 = −dt2 + dσ2 = −dσ+dσ−, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='10) where the light-cone coordinates dσ± = dt ± dσ (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='11) have been introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' It is natural to perform a Wick rotation from the Lorentzian time t to the Euclidean time τ = it, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='12) and the metric becomes ds2 = dτ 2 + dσ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='13) It is convenient to introduce the complex coordinates w = τ + iσ, ¯w = τ − iσ (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='14) for which the metric is ds2 = dwd ¯w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='15) Note that the relation to Lorentzian light-cone coordinates are w = i(t + σ) = iσ+, ¯w = i(t − σ) = iσ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='16) Hence, an (anti-)holomorphic function of w ( ¯w) depends only on σ+ (σ−) before the Wick rotation: this leads to the identification of the left- and right-moving sectors with the holo- morphic and anti-holomorphic sectors of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The cylinder can be mapped to the complex plane through z = e2πw/L, ¯z = e2π ¯ w/L, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='17) and the corresponding metric is ds2 = � L 2π �2 dzd¯z |z|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='18) A conformal transformation brings this metric to the flat metric (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The conventions for the various coordinates and maps vary in the different textbooks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' We have gathered in Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1 the three main conventions and which references use which.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 4Consistently with the comments at the beginning of Chapter 5, the Lorentzian worldsheet time is denoted by t instead of τM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 83 −−−−−→ −−−−−→ Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2: Map from the cylinder to the sphere with two tubes, to the 2-punctured sphere Σ0,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The map from the cylinder to the plane is found by sending the bottom end (corres- ponding to the infinite past t → −∞) to the origin of the plane, and the top end (infinite future t → ∞) to the infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Since the cylinder has two boundaries (its two ends) the map excludes the point z = 0 and z = ∞ and one really obtains the space ¯C − {0, ∞} = C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This space can, in turn, be mapped to the 2-punctured Riemann sphere Σ0,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The physical interpretation for the difference between Σ0 and Σ0,2 is simple: since one considers the propagation of a string, it means that the worldsheet corresponds to an amp- litude with two external states, which are the mapped to the sphere as punctures (Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Removing the external states (yielding the tree-level vacuum amplitude) cor- responds to gluing half-sphere (caps) at each end of the cylinder (Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Then, it can be mapped to the Riemann sphere without punctures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' As a consequence, the properties of tree-level string theory are found by studying the matter and ghost CFTs on the Riemann sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Scattering amplitudes are computed through correlation functions of appropriate op- erators on the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This picture generalizes to higher-genus Riemann surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Moreover, since local properties of the CFT (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' the spectrum of operators) are determined by the conformal algebra, they will be common to all surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Mathematically, a difference between Σ0 and Σ0,2 had to be expected since the sphere has a positive curvature (and χ = −2) but the cylinder is flat (with χ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Punctures contribute negatively to the curvature (and thus positively to the Euler characteristics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1 The coordinate z is always used as a coordinate on the complex plane, but the corresponding metric may be different – compare (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='5) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' As explained previously, this does not matter since the theory is insensitive to the conformal factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2 Classical CFTs In this section, we consider an action S[Ψ] which is conformally invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' We first identify and discuss the properties of the conformal algebra and group, before explaining how a CFT is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 84 −−−−−−−−→ Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3: Map from the cylinder with two caps (half-spheres) to the Riemann sphere Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1 Witt conformal algebra Since the Riemann sphere is identified with the complex plane, they share the same conformal group and algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Consider the metric (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='5) ds2 = dzd¯z, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='19) then, any meromorphic change of coordinates z −→ z′ = f(z), ¯z −→ ¯z′ = ¯f(¯z) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='20) is a conformal transformation since the metric becomes ds2 = dz′d¯z′ = ���� df dz ���� 2 dzd¯z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='21) However, only holomorphic functions which are globally defined on ¯C are elements of the group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' At the algebra level, any holomorphic function f(z) regular in a domain D gives a well-defined transformation in this domain D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Hence, the algebra is infinite-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' On the other hand, f(z) is only meromorphic on C generically: it cannot be exponentiated to a group element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' We first characterize the algebra and then obtain the conditions to promote the local transformations to global ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Since the transformations are defined only locally, it is sufficient to consider an infinites- imal transformation δz = v(z), δ¯z = ¯v(¯z), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='22) where v(z) is a meromorphic vector field on the Riemann sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Indeed, the conformal Killing equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='6) in D = 2 is equivalent to the Cauchy–Riemann equations: ¯∂v = 0, ∂¯v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='23) The vector field admits a Laurent series v(z) = � n∈Z vnzn+1, ¯v(¯z) = � n∈Z ¯vn¯zn+1, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='24) and the vn and ¯vn are to be interpreted as the parameters of the transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' A basis of vectors (generators) is: ℓn = −zn+1∂z, ¯ℓn = −¯zn+1∂¯z, n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='25) One can check that each set of generators satisfies the Witt algebra [ℓm, ℓn] = (m − n)ℓm+n, [¯ℓm, ¯ℓn] = (m − n)¯ℓm+n, [ℓm, ¯ℓn] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='26) 85 Since there are two commuting copies of the Witt algebra, it is natural to extend the ranges of the coordinates from C to C2 and to consider z and ¯z as independent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In particular, this gives a natural action of the product algebra over C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This procedure will be further motivated when studying CFTs since the holomorphic and anti-holomorphic parts will generally split, and it makes sense to study them separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Ultimately, phys- ical quantities can be extracted by imposing the condition ¯z = z∗ at the end (the star is always reserved for the complex conjugation, the bar will generically denote an independent variable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In that case, the two algebras are also related by complex conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Note that the variation of the metric (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='6) under a meromorphic change of coordinates (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='22) becomes δgz¯z = ∂v + ¯∂¯v, δgzz = δg¯z¯z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='27) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2 PSL(2, C) conformal group The next step is to determine the globally defined vectors and to study the associated group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' First, the conditions for a vector v(z) to be well-defined at z = 0 are lim |z|→0 v(z) < ∞ =⇒ ∀n < −1 : vn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='28) The behaviour at z = ∞ can be investigated thanks to the map z = 1/w v(1/w) = dz dw � n vnw−n−1, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='29) where the additional derivative arises because v is a vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Then, the regularity conditions at z = ∞ are lim |z|→∞ v(z) = lim |w|→0 dz dw v(1/w) = − lim |w|→0 v(1/w) w2 < ∞ =⇒ ∀n > 1 : vn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='30) As a result, the globally defined generators are {ℓ−1, ℓ0, ℓ1} ∪ {¯ℓ−1, ¯ℓ0, ¯ℓ1} (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='31) where ℓ−1 = −∂z, ℓ0 = −z∂z, ℓ1 = −z2∂z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='32) It is straightforward to check that they form two copies of the sl(2, C) algebra [ℓ0, ℓ±1] = ∓ℓ±1, [ℓ1, ℓ−1] = 2ℓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='33) The global conformal group is sometimes called Möbius group: PSL(2, C) := SL(2, C)/Z2 ∼ SO(3, 1), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='34) where the additional division by Z2 is clearer when studying an explicit representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' It corresponds with ker P1 defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='91): K0 = PSL(2, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='35) A matrix representation of SL(2, C) is g = �a b c d � , a, b, c, d ∈ C, det g = ad − bc = 1, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='36) 86 which shows that this group has six real parameters K0 := dim SL(2, C) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='37) The associated transformation on the complex plane reads fg(z) = az + b cz + d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='38) The quotient by Z2 is required since changing the sign of all parameters does not change the transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' These transformations have received different names: Möbius, projective, homographic, linear fractional transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Holomorphic vector fields are then of the form v(z) = β + 2αz + γz2, ¯v(¯z) = ¯β + 2¯α¯z + ¯γ¯z2, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='39) where a = 1 + α, b = β, c = −γ, d = 1 − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='40) The finite transformations associated to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='12) are: translation: fg(z) = z + a, a ∈ C, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='41a) rotation: fg(z) = ζ z, |ζ| = 1, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='41b) dilatation: fg(z) = λ z, λ ∈ R, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='41c) SCT: fg(z) = z cz + 1, c ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='41d) Investigation leads to the following association between the generators and transformations: translation: ℓ−1 and ¯ℓ−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' dilatation (or radial translation): (ℓ0 + ¯ℓ0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' rotation (or angular translation): i(ℓ0 − ¯ℓ0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' special conformal transformation: ℓ1 and ¯ℓ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The inversion defined by inversion: I+(z) := I(z) := 1 z (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='42a) is not an element of SL(2, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' However, the inversion with a minus sign I−(z) := −I(z) = I(−z) = −1 z (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='42b) is a SL(2, C) transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' A useful transformation is the circular permutation of (0, 1, ∞): g∞,0,1(z) = 1 1 − z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='43) 87 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3 Definition of a CFT A CFT is characterized by its set of (composite) fields (also called operators) O(z, ¯z) which correspond to any local expression constructed from the fields Ψ appearing in the Lagrangian and of their derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='5 For example, in a scalar field theory, the simplest operators are of the form ∂mφn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Among the operators, two particular categories are distinguished according to their trans- formation laws: primary operator: ∀f meromorphic : O(z, ¯z) = �df dz �h �d ¯f d¯z �¯h O′� f(z), ¯f(¯z) � , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='44) quasi-primary (or SL(2, C) primary) operator: ∀f ∈ PSL(2, C) : O(z, ¯z) = �df dz �h �d ¯f d¯z �¯h O′� f(z), ¯f(¯z) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='45) The parameters (h, ¯h) are the conformal weights of the operator O (both are independent from each other), and combinations of them give the conformal dimension ∆ and spin s: ∆ := h + ¯h, s := h − ¯h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='46) The conformal weights correspond to the charges of the operator under ℓ0 and ¯ℓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' We will use “(h, ¯h) (quasi-)primary” as a synonym of “(quasi-)primary field with conformal weight (h, ¯h)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2 (Complex conformal weights) While we consider h, ¯h ∈ R, and more spe- cifically h, ¯h ≥ 0 for a unitary theory (which is the case of string theory except for the re- parametrization ghosts), theories with h, ¯h ∈ C make perfectly sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' One example is the Liouville theory with complex central charge c ∈ C [200, 202] (central charges are defined below, see (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='58)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Primaries and quasi-primaries are hence operators which have nice transformations re- spectively under the algebra and group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Obviously, a primary is also a quasi-primary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' These transformations are similar to those of a tensor with h holomorphic and ¯h anti-holomorphic indices (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Another point of view is that the object O(z, ¯z) dzhd¯z ¯h (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='47) is invariant under local / global conformal transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The notation f ◦ O indicates the complete change of coordinates, including the tensor transformation law and the possible corrections if the operator is not primary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='6 For a primary field, we have: f ◦ O(z, ¯z) := f ′(z)h ¯f ′(¯z) ¯h O′� f(z), ¯f(¯z) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='48) We stress that it does not correspond to function composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Under an infinitesimal transformations δz = v(z), δ¯z = ¯v(¯z), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='49) 5Not all CFTs admit a Lagrangian description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' But, since we are mostly interested in string theories defined from Polyakov’s path integral, it is sufficient to study CFTs with a Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 6In fact, one has f ◦ O := f∗O in the notations of Chapter 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 88 a primary operator changes as δO(z, ¯z) = (h ∂v + v ∂)O(z, ¯z) + (¯h ¯∂¯v + ¯v ¯∂)O(z, ¯z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='50) The transformation of a non-primary field contains additional terms, see for example (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='89).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3 (Higher-genus Riemann surfaces) According to Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1, all Riemann surfaces Σg share the same conformal algebra since locally they are all subsets of R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' On the other hand, one finds that no global transformations are defined for g > 1, and only the subgroup U(1) × U(1) survives for the torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The most important operator in a CFT is the energy–momentum tensor Tµν, if it exists as a local operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' According to Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1, this tensor is conserved and traceless ∇νTµν = 0, gµνTµν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='51) The traceless equation in components reads gµνTµν = 4 Tz¯z = Txx + Tyy = 0 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='52) which implies that the off-diagonal component vanishes in complex coordinates Tz¯z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='53) Then, the conservation equation yields ∂zT¯z¯z = 0, ∂¯zTzz = 0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='54) such that the non-vanishing components Tzz and T¯z¯z are respectively holomorphic and anti- holomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This motivates the introduction of the notations: T(z) := Tzz(z), ¯T(¯z) := T¯z¯z(¯z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='55) This is an example of the factorization between the holomorphic and anti-holomorphic sec- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Currents are local objects and thus one expects to be able to write an infinite number of such currents associated to the Witt algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Applying the Noether procedure gives Jv(z) := J ¯z v (z) = −T(z)v(z), ¯Jv(¯z) := Jz v (¯z) = − ¯T(¯z)¯v(¯z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='56) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3 Quantum CFTs The previous section was purely classical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The quantum theory is first defined through the path integral Z = � dΨ e−S[Ψ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='57) We will also develop an operator formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The latter is more general than the path integral and allows to work without reference to path integrals and Lagrangians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This is particularly fruitful as it extends the class of theories and parameter ranges (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2) which can be studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 89 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='1 Virasoro algebra As discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3, field measures in path integrals display a conformal anom- aly, meaning that they cannot be defined without introducing a scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This anomaly can be traded for a gravitational anomaly by introducing counter-terms in the action [87, 95, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2, 96, 101, 122, 129].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' As a consequence, the Witt algebra (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='26) is modified to its central extension, the Virasoro algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='7 The generators in both sectors are denoted by {Ln} and {¯Ln} and are called Virasoro operators (or modes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The algebra is given by: [Lm, Ln] = (m − n)Lm+n + c 12 m(m − 1)(m + 1)δm+n, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='58a) [¯Lm, ¯Ln] = (m − n)¯Lm+n + ¯c 12 m(m − 1)(m + 1)δm+n, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='58b) [Lm, ¯Ln] = 0, [c, Lm] = 0, [¯c, ¯Lm] = 0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='58c) where c, ¯c ∈ C are the holomorphic and anti-holomorphic central charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Consistency of the theory on a curved space implies ¯c = c, but there is otherwise no constraint on the plane [95, 246].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The sl(2, C) subalgebra is not modified by the central extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' This means that states are still classified by eigenvalues of (h, ¯h) of (L0, ¯L0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='4 In most models relevant for string theory, one finds that the central charges are real, c, ¯c ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Moreover, unitarity requires them to be positive c, ¯c > 0, and only reparametrization ghosts do not satisfy this condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' On the other hand, it makes perfect sense to discuss general CFTs for c, ¯c ∈ C (the Liouville theory is such an example [200, 202]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='2 Correlation functions A n-point correlation function is defined by � n � i=1 Oi(zi, ¯zi) � = � dΨ e−S[Ψ] n � i=1 Oi(zi, ¯zi), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='59) choosing a normalization such that ⟨1⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The path integral defines the time-ordered product (on the cylinder) of the corresponding operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Invariance under global transformations leads to strong constraints on the correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' For quasi-primary fields, they transform under SL(2, C) as � n � i=1 Oi(zi, ¯zi) � = n � i=1 �df dz (zi) �hi �df d¯z (¯zi) �¯hi × � n � i=1 Oi � f(zi), ¯f(¯zi) � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='60) Considering an infinitesimal variation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='50) yields a differential equation for the n-point function δ � n � i=1 Oi(zi, ¯zi) � = n � i=1 � hi∂iv(zi) + v(zi)∂i + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' � � n � i=1 Oi(zi, ¯zi) � = 0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='61) where ∂i := ∂zi and v is a vector (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='39) of sl(2, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' These equations are sufficient to determine 7That the central charge in the Virasoro algebra indicates a diffeomorphism anomaly can be understood from the fact that 90 completely the forms of the 1-, 2- and 3-point functions of quasi-primaries: ⟨Oi(zi, ¯zi)⟩ = δhi,0δ¯hi,0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='62a) ⟨Oi(zi, ¯zi)Oj(zj, ¯zj)⟩ = δhi,hjδ¯hi,¯hj gij z2hi ij ¯z2¯hi ij , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='62b) ⟨Oi(zi, ¯zi)Oj(zj, ¯zj)Ok(zk, ¯zk)⟩ = Cijk zhi+hj−hk ij zhj+hk−hi jk zhi+hk−hj ki × 1 ¯z ¯hi+¯hj−¯hk ij ¯z ¯hj+¯hk−¯hi jk ¯z ¯hi+¯hk−¯hj ki , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='62c) where we have defined zij = zi − zj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='63) The coefficients Cijk are called structure constants and the matrix gij defines a metric (Zamolodchikov metric) on the space of fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The metric is often taken to be diagonal gij = δij, which amounts to use an orthonormal eigenbasis of L0 and ¯L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The vanishing of the 1-point function of a non-primary quasi-primary holds only on the plane: for example the value on the cylinder can be non-zero since the map is not globally defined – see in particular (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='167).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='5 (Logarithmic CFTs) Logarithmic CFTs display a set of unusual proper- ties [84, 85, 90, 97, 130].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' In particular, the correlation functions are not of the form displayed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The most striking feature of those theories is that the L0 operator is non- diagonalisable (but it can be set in the Jordan normal form).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content='6 (Fake identity) Usually, the only primary operator with h = ¯h = 0 is the identity 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' While this is always true for unitary theories, there are non-unitary theories (c ≤ 1 Liouville theory, SLE, loop models) where there is another field (called the indicator, marking operator, or also fake identity) with h = ¯h = 0 [13, 46, 99, 112, 182, 200, 202].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' The main difference between both fields is that the identity is a degenerate field (it has a null descendant), whereas the other operator with h = ¯h = 0 is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Such theories will not be considered in this book.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Operators with h = ℏ = 0 can also be built by comining several CFTs, and they play a very important role in string theory since they describe on-shell states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19AzT4oBgHgl3EQfuP0B/content/2301.01686v1.pdf'} +page_content=' Finally, the 4-point function is determined up to a function of a single variable x and its complex conjugate: � 4 � i=1 Oi(zi, ¯zi) � = f(x, ¯x) � i min{ℓ, ℓ′}, +S0 ≃ +min{a0,b0} +� +k=0 +V (a0 + b0 − 2k), +and, for t = 1, . . . , min{ℓ, ℓ′}, we have +(i) If V1 = Z(a0, ℓ) and V2 = Z(b0, ℓ′), then +St ≃ V (a0 + b0 + tm). +(ii) If V1 = Z(a0, ℓ) and V2 = Z(bℓ′, ℓ′)∗, then +St ≃ +� +0, +if tm > b0 − a0; +V (b0 − a0 − tm), +if tm ≤ b0 − a0. +(iii) If V1 = Z(aℓ, ℓ)∗ and V2 = Z(bℓ′, ℓ′)∗, then St = 0. + +UNISERIAL REPRESENTATIONS OF THE LIE ALGEBRA sl(2) ⋉ hn +13 +One of the main steps towards proving the above theorem was to prove +certain instances of the following conjecture (see [7, Conjecture 3.4]). +Conjecture 4.2. Let V1 = E(a, b) and V2 = E(c, d) (two uniserial sl(2) ⋉ +a(m)-modules of length 2) and assume that a < c, or a = c and b ≤ d. Then +S2 = 0 in all cases and S1 = 0 except in the following cases. +• Case 1: [a, b] = [0, m]. Here S1 ≃ V (d). +• Cases 2: Here a > 0. +– Case 2.1: a+b = c+d = m with d−a = b−c ≥ 0. Here S1 ≃ V (d−a). +– Case 2.2: b − a = d − c = m. Here S1 ≃ V (d + a). +– Case 2.3: b−a = c−d = m with d−a = c−b ≥ 0. Here S1 ≃ V (d−a). +• Case 3: [c, d] = [b, a]. Here S1 ≃ V (0). +In order to prove Theorem 4.1, we proved in [7, Theorem 3.3] the cases +2.2 and 2.3 (and certain converse statement). Now, in this paper, we need +to prove part of case 1 and case 2.1 of the conjecture (together with certain +converse statement) in order to prove Theorems 4.6 and 4.7. This is estab- +lished in Theorem 4.3 below. We point out that this theorem leaves out +the uniserial +� +sl(2) ⋉ a(3) +� +-modules E(3, 4) and E(4, 3), and a consequence +of this is that Theorems 4.6 and 4.7 are restricted to the standard faith- +ful modules, leaving the exceptional faithful uniserial +� +sl(2) ⋉ h2 +� +-module +FU(4,3,4) out of our results. +Theorem 4.3. Let V1 = E(a, b) with a + b = m and a, b ̸= 0. Let V2 = +V (c)⊕V (d) be the socle decomposition of a uniserial +� +sl(2)⋉V (m) +� +-module. +Then +(i) If V2 ≃ E(c, d) with c + d = m and 0 < a ≤ c < m, then, as sl(2)- +modules, +S1(V1, V2) ≃ +� +V (d − a), +if d − a = b − c ≥ 0 +0, +otherwise. +(ii) If V2 ≃ Z(c, 1) ≃ E(c, c + m), then, as sl(2)-modules, +S1(V1, V2) ≃ +� +V (b), +if c = 0 +0, +if c ̸= 0 . +(iii) If V2 ≃ Z(d, 1)∗ ≃ E(d + m, d), then S1(V1, V2) = 0. +The proof of this result is very technical and it will be given in §6. +4.2. The faithful case. We will now focus on the tensor product of two +uniserial +� +sl(2)⋉hn +� +-modules where one of the factors is a standard faithful +uniserial. +Let V = V (a0) ⊕ V (a1) ⊕ V (a2) be the socle decomposition of a faithful +uniserial +� +sl(2)⋉hn +� +-module and set W = V (b0)⊕. . .⊕V (bℓ), with ℓ ≥ 1, be +the socle decomposition of a (not necessarily faithful) uniserial +� +sl(2) ⋉ hn +� +- +module. By Theorems 3.2 and 3.3 we have that +hn(m) · V (ai) ⊂ V (ai−1) and hn(m) · V (bj) ⊂ V (bj−1) ⊕ V (bj−2) +z · V (ai) ⊂ V (ai−2) and z · V (bj) ⊂ V (bj−2) + +14 +LEANDRO CAGLIERO AND IV´AN G´OMEZ RIVERA +for all i = 0, 1, 2 and 0 ≤ j ≤ ℓ (for convenience we assume V (ai) = V (bj) = +0 if i, j < 0). +Given v ∈ V (ai) ⊗ V (bj), let +(4.3) +esv = (esv)1 + (esv)2 + (esv)3 and zv = (zv)1 + (zv)2 +where +(esv)1 ∈ V (ai−1) ⊗ V (bj), +(zv)1 ∈ V (ai−2) ⊗ V (bj), +(esv)2 ∈ V (ai) ⊗ V (bj−1), +(zv)2 ∈ V (ai) ⊗ V (bj−2), +(esv)3 ∈ V (ai) ⊗ V (bj−2). +Note that (esv)3 = 0 if W is not isomorphic to E4 and that (zv)2 = 0 if W +is not a faithful uniserial module. +Lemma 4.4. Let V1 = V (a0)⊕V (a1)⊕V (a2) and V2 = V (b0)⊕. . . ⊕V (bℓ), +with ℓ ≥ 1, be the socle decomposition of two uniserial +� +sl(2) ⋉ hn +� +-modules, +where V1 is faithful (not necessarily standard). If v0 ∈ V (ai0) ⊗ V (bj0) is a +highest weight vector then: +(i) (esv0)1 = 0 for all s = 0, . . . , m if and only if i0 = 0. +(ii) (esv0)2 = 0 for all s = 0, . . . , m if and only if j0 = 0. +(iii) (zv0)1 = 0 if and only if i0 ̸= 2. +(iv) If V2 is faithful, then (zv0)2 = 0 if and only if j0 ̸= 2. +Proof. Since the action of hn(m) on any uniserial +� +sl(2) ⋉ hn +� +is the same +as the action of am in the corresponding +� +sl(2) ⋉ am +� +-module, cases (i) and +(ii) are immediate consequences of [7, Lemma 3.1]. +By symmetry, it sufficient to prove (iii) to obtain (iv), so let us prove (iii). +If c is the weight of v0, we can assume that v0 = v +ai0,bj0,c +0 +. It follows from +(2.4) and (4.3) that +(zv0)1 = (zv +ai0,bj0,c +0 +)1 += +� +i+j= ai0+bj0−c +2 +CG(ai0 +2 , ai0 +2 − i; bj0 +2 , bj0 +2 − j | c +2, c +2) zv +ai0 +i +⊗ v +bj0 +j +. +(4.4) +From the definition of the modules FU ± +a +for n = 1; FU(0,3,0),FU(1,4,1), +FU(1,2,1) and FU(4,3,4) for n = 2; and the modules FU(0,m,0), FU(1,m+1,1) +and FU1,m−1,1 for n ≥ 3 (here m = 2n − 1), we know that zv +ai0 +i += 0 if +i0 ̸= 2. Therefore, if i0 ̸= 2 then +(zv0)1 = 0. +On the other hand, if i0 = 2, we know from (3.5) that zva2 +i += λva0 +i , where λ +is a non-zero scalar independent of i, 0 ≤ i ≤ a2. Thus, the equation (4.4) +becomes +(zv0)1 = λ +� +i+j= a2+bj0−c +2 +CG(a2 +2 , a2 +2 − i; +bj0 +2 , +bj0 +2 − j | c +2, c +2) va0 +i +⊗ v +bj0 +j +. + +UNISERIAL REPRESENTATIONS OF THE LIE ALGEBRA sl(2) ⋉ hn +15 +In this sum, the term corresponding to i = 0, has a non-zero Clebsch-Gordan +coefficient, indeed +CG(a2 +2 , a2 +2 ; bj0 +2 , c−a2 +2 +| c +2, c +2) = +� +(c+1)! a2! +�a2+bj0+c +2 ++1 +� +! +� a2+c−bj0 +2 +� +! +̸= 0. +Since all terms are linearly independent, we obtain (zv0)1 ̸= 0. +□ +Proposition 4.5. Let V1 = V (a0) ⊕ V (a1) ⊕ V (a2) and V2 = V (b0) ⊕ . . . ⊕ +V (bℓ), with ℓ ≥ 1, be the socle decomposition of two uniserial +� +sl(2) ⋉ hn +� +- +modules, where V1 is standard faithful. Then, S0 = V (a0) ⊗ V (b0) and +(i) St = 0 for all t > min{2, ℓ}. If St ̸= 0, t = 1, 2, then it is irreducible +as sl(2)-module and if v is a non-zero highest weight vector in St of +weight µ, then v = �t +i=0 vi with vi a non-zero highest weight vector +in V (ai) ⊗ V (bt−i), of weight µ, for all i = 0, . . . , t. +(ii) S2 = 0 if V2 is non-faithful. +(iii) S1(V1, V2) ≃ S1 +� +E(a0, a1), E(b0, b1) +� +. +(iv) If V2 is also standard faithful, then S2 ̸= 0 if and only if V1 ≃ V2 +(that is ai = bi, i = 0, 1, 2) and in this case S2 ≃ V (0). +Proof. The proof of this proposition is very similar to that of Proposition +3.2 in [7]. We fix t > 0 and we assume that there is a non-zero highest +weight vector u of weight µ, +u = +� +i+j=t +ui,j ∈ +� � +i+j=t +V (ai) ⊗ V (bj) +�hn ̸= 0, +ui,j ∈ V (ai) ⊗ V (bj). +Since V (ai) ⊗ V (bj) is an sl(2)-submodule, it follows that ui,j is either zero +or a highest weight vector of weight µ. Let +Iµ +t = {(i, j) : 0 ≤ i ≤ 2, 0 ≤ j ≤ ℓ, i + j = t and ui,j ̸= 0}. +Since u ̸= 0, it follows that Iµ +t ̸= ∅ and +u = +� +(i,j)∈Iµ +t +qi,j vai,bj,µ +0 +for certain non-zero scalars 0 ̸= qi,j ∈ F. Now, it follows from items (i) and +(ii) in Lemma 4.4 (see the details in [7][Proposition 3.2]) that +(4.5) +Iµ +t = {(0, t), (1, t − 1), . . . , (t, 0)}. +Now, again, items (i) and (ii) in Lemma 4.4 imply that such a non-zero u +cannot exist if t > min{2, ℓ} and thus St = 0. This proves (i). Furthermore, +(ii) follows similarly by applying item (iii) in Lemma 4.4. +(iii) is clear from the definition of S1. +Let us prove (iv). Assume that V2 = V (b0) ⊕ V (b1) ⊕ V (b2) is standard +faithful, and suppose that +(4.6) +u = q0,2 va0,b2,µ +0 ++ q1,1 va1,b1,µ +0 ++ q2,0 va2,b0,µ +0 +̸= 0 +is a highest weight vector of weight µ in S2. We already know that q0,2, q1,1, q2,0 ̸= +0 and, moreover, we must have +q0,2 va0,b2,µ +0 ++ q1,1 va1,b1,µ +0 +∈ S1 +� +E(a0, a1), E(b1, b2)) ̸= 0 + +16 +LEANDRO CAGLIERO AND IV´AN G´OMEZ RIVERA +and +q1,1 va1,b1,µ +0 ++ q2,0 va2,b0,µ +0 +∈ S1 +� +E(a1, a2), E(b0, b1)) ̸= 0. +Theorems 4.1 and 4.3 imply that it is impossible to have +S1 +� +E(a0, a1), E(b1, b2) +� +̸= 0 +and +S1 +� +E(a1, a2), E(b0, b1) +� +̸= 0 +unless (a0, a1, a2) = (b0, b1, b2). +This follows by considering all the cases +with (a0, a1, a2) and (b0, b1, b2) running over (see §3.6) +(i) if n = 1 (that is m = 1) +(k0, k0 + 1, k0), +k0 ≥ 0; +(k0, k0 − 1, k0), +k0 ≥ 1; +(ii) if n ≥ 2 (that is m ≥ 3), +(0, m, 0), (1, m + 1, 1), (1, m − 1, 1); +(it saves time noticing that E(a0, a1)∗ ≃ E(a1, a2) and E(b0, b1)∗ ≃ E(b1, b2)). +Finally, let (a0, a1, a2) = (b0, b1, b2). We know, from Theorems 4.1 and +4.3, that +S1 +� +E(a0, a1), E(b1, b2) +� +≃ S1 +� +E(a1, a2), E(b0, b1) +� +≃ V (0). +This implies that there is, up to a scalar, a unique element u as in (4.6) +such that hn(m)u = 0. This implies that zu = 0 and hence u ∈ S2. This +completes the proof. +□ +We are now in a position to prove the main theorems of the paper. +Theorem 4.6. Let V1 = V (a0)⊕V (a1)⊕V (a2) and V2 = V (b0)⊕. . .⊕V (bℓ) +be the socle decomposition of two uniserial +� +sl(2) ⋉ hn +� +-modules, where V1 +is standard faithful and V2 is of type Z. Then +soc(V1 ⊗ V2) = S0 ⊕ S1 +where, as sl(2)-modules, +S0 = soc(V1) ⊗ soc(V2) ≃ +min{a0,b0} +� +k=0 +V (a0 + b0 − 2k) +and the following tables describe S1 as sl(2)-modules (recall that m = 2n−1): +Case n = 1 (m = 1). +V1\V2 +Z(b0, ℓ) +Z(bℓ, ℓ)∗ +FU + +a0 +V (a0 + b0 + m). +V (b1 − a0), +if b1 ≥ a0, ℓ ≥ 1; +0, +otherwise. +FU − +a0 +a0 ≥ 1 +V (a0 − b1), +if a0 ≥ b1, ℓ ≥ 1; +0, +otherwise. +0. +Case n > 1 (m ≥ 3). + +UNISERIAL REPRESENTATIONS OF THE LIE ALGEBRA sl(2) ⋉ hn +17 +V1\V2 +Z(b0, ℓ) +Z(bℓ, ℓ)∗ +FU(a0,a0+m,a0) +a0 = 0, 1 +V (a0 + b0 + m). +V (b1 − a0), +if a0 ≤ b1; +0, +if a0 > b1. +FU(1,m−1,1) +V (m − 1), +if b0 = 0; +0, +otherwise. +0. +Proof. We know from Proposition 4.5 that +soc(V1 ⊗ V2) ≃ soc(V1) ⊗ soc(V2) ⊕ S1 +(in particular we know that St(V1, V2) = 0 for all t ≥ 2). The decomposition +of soc(V1) ⊗ soc(V2) follows from the Clebsch-Gordan formula. +We now +describe S1 in each case. +Let us consider the submodules +U1 = V (a0) ⊕ V (a1) ⊂ V1, and +U2 = V (b0) ⊕ V (b1) ⊂ V2. +We know that U1 and U2 are non-faithful uniserial +� +sl(2) ⋉ hn +� +-modules. +From Proposition 4.5 item (iii) we know that +S1(V1, V2) ≃ S1(U1, U2), +If V1 is FU + +a0 or FU(0,m,0) or FU(1,1+m,1) then U1 ≃ Z(a0, 1) and S1 is +obtained from Theorem 4.1. If V1 = FU − +a0, then U1 = Z(a0 − 1, 1)∗ and S1 +is also obtained from Theorem 4.1. +Finally, if V1 = FU(1,m−1,1), then U1 = E(1, m − 1) and Theorem 4.3 +implies the remaining cases. +□ +Theorem 4.7. Let V1 = V (a0) ⊕ V (a1) ⊕ V (a2) and V2 = V (b0) ⊕ V (b1) ⊕ +V (b2) be the socle decomposition of two standard faithful uniserial +� +sl(2) ⋉ +hn +� +-modules. Then +soc(V1 ⊗ V2) = S0 ⊕ S1 ⊕ S2 +where +S0 = soc(V1) ⊗ soc(V2) ≃ +min{a0,b0} +� +k=0 +V (a0 + b0 − 2k) +and the following tables describe S1 and S2 as sl(2)-modules (m = 2n − 1): +Case n = 1 (m = 1), structure of S1. +V1\V2 +FU(b0,b0+1,b0) +FU(b0,b0−1,b0) +b0 ≥ 1 +FU(a0,a0+1,a0) +V (a0 + b0 + 1). +V (b0 − a1), +if a1 ≤ b0; +0, +otherwise. +FU(a0,a0−1,a0) +a0 ≥ 1 +V (a0 − b1), +if b1 ≤ a0; +0, +otherwise. +0. + +18 +LEANDRO CAGLIERO AND IV´AN G´OMEZ RIVERA +Case n = 1 (m = 1), structure of S2. +V1\V2 +FU(b0,b0+1,b0) +FU(b0,b0−1,b0) +b0 ≥ 1 +FU(a0,a0+1,a0) +V (0), +if a0 = b0; +0, +if a0 ̸= b0. +0. +FU(a0,a0−1,a0) +a0 ≥ 1 +0. +V (0), +if a0 = b0; +0, +if a0 ̸= b0. +Case n > 1 (m ≥ 3), structure of S1. +V1\V2 +FU(b0,b0+m,b0) +b0 = 0, 1 +FU(1,m−1,1) +FU(a0,a0+m,a0) +a0 = 0, 1 +V (a0 + b0 + m). +V (m − 1), +if a0 = 0; +0, +otherwise. +FU(1,m−1,1) +V (m − 1), +if b0 = 0; +0, +otherwise. +V (m − 2). +Case n > 1 (m ≥ 3), structure of S2. +V1\V2 +FU(b0,b0+m,b0) +b0 = 0, 1 +FU(1,m−1,1) +FU(a0,a0+m,a0) +a0 = 0, 1 +V (0), +if a0 = b0; +0, +if a0 ̸= b0. +0. +FU(1,m−1,1) +0. +V (0). +Proof. As in the previous theorem, we know from Proposition 4.5 that +soc(V1 ⊗ V2) ≃ soc(V1) ⊗ soc(V2) ⊕ S1 ⊕ S2 +and the decomposition of soc(V1)⊗soc(V2) follows from the Clebsch-Gordan +formula. We now describe S1 and S2 in each case. +First, we consider S1. We know from Proposition 4.5 that +S1(V1, V2) ≃ S1 +� +E(a0, a1), E(b0, b1) +� +. +If V1 = FU + +a0 and V2 = FU + +b0, we know from Theorem 4.1 that +S1(V1, V2) = S1(Z(a0, 1), Z(b0, 1)) = V (a0 + b0 + m). +If V1 = FU + +a0 and V2 = FU − +b0, we have +S1(V1, V2) = +� +S1(Z(a0, 1), Z(b1, 1)∗), +if m = 1; +S1(Z(a0, 1), E(1, 1 − m)), +if m > 1 and b0 = 1; + +UNISERIAL REPRESENTATIONS OF THE LIE ALGEBRA sl(2) ⋉ hn +19 +and it follows from Theorems 4.1 and 4.3 that +S1(V1, V2) = + + + + + +V (b1 − a0) +if m = 1; +V (m − 1), +if m > 1, a0 = 0 and b0 = 1; +0, +otherwise. +This completes the description of S1. +The result for S2 follows from item (iv) in Proposition 4.5. +□ +5. Intertwining operators +In this section we obtain, from Theorems 4.6 and 4.7, the space of in- +tertwining operators between the uniserial representations of g = sl(2) ⋉ hn +considered in the previous section. +Recall that, for any pairs of g-modules U1 and U2, we know that +Homg(U1, U2) ≃ (U ∗ +1 ⊗ U2)g = +� +(U ∗ +1 ⊗ U2)hn�sl(2). +It follows that Homg(U1, U2) is isomorphic to soc(U ∗ +1 ⊗ U2)sl(2) (see (4.1)). +Thus, we must identify the cases in which St(U ∗ +1 , U2)sl(2) ̸= 0 for t = 0, 1, 2. +So let V = V (a0) ⊕ V (a1) ⊕ V (a2) (a0 = a2) and W = V (b0) ⊕ . . . ⊕ V (bℓ) +be the socle decomposition of two uniserial +� +sl(2) ⋉ hn +� +-modules, where V +is standard faithful and W is either of type Z or standard faithful. Recall +that V ∗ ≃ V and that the socle decomposition of W ∗ is V (bℓ) ⊕ . . . ⊕ V (b0). +• Case Ssl(2) +0 +̸= 0, W of type Z. +It follows from Theorem 4.6 that S0(V ∗, W)sl(2) ̸= 0 if and only if +a0 = b0 (and equal to a2). Also, Theorem 4.6 implies that, in these cases, +we have dim S0(V ∗, W)sl(2) = 1 and St(V ∗, W)sl(2) = 0, t = 1, 2. Hence +Homg(V, W) is 1-dimensional and it is described by the following arrow +V = V (a0) ⊕ V (a1) ⊕ V (a0) +↓ +W = V (b0) ⊕ · · · ⊕ V (bℓ). +Similarly, from Theorem 4.6 we know that S0(W ∗, V )sl(2) ̸= 0 if and +only if a0 = bℓ and in these cases Homg(W, V ) is 1-dimensional and it is +described by the following arrow +W = V (b0) ⊕ · · · ⊕V (bℓ) +↓ +V = V (a0) ⊕ V (a1) ⊕ V (a0). +• Case Ssl(2) +0 +̸= 0 or Ssl(2) +2 +̸= 0, W standard faithful. +It follows from Theorem 4.7 that S0(V, W)sl(2) ̸= 0 if and only if a0 = b0. +Hence in what follows, a0 = a2 = b0 = b2. In this case, S1(V, W)sl(2) = 0 +and dim S0(V, W)sl(2) = 1 (note that if V ≃ W, that is a1 = b1, then +Theorem 4.7 says that dim S2(V, W)sl(2) = 1, see below). The non-zero + +20 +LEANDRO CAGLIERO AND IV´AN G´OMEZ RIVERA +g-morphism corresponding to the 1-dimensional space S0(V, W)sl(2) is de- +scribed by the following arrow +V = V (a0) ⊕ V (a1) ⊕ V (a0) +↓ +W = V (b0) ⊕ V (b1) ⊕ V (b0). +This is clearly not an isomorphism. In the particular case when V ≃ W, +the isomorphism is the g-morphism corresponding to the 1-dimensional +space S2(V, W)sl(2) (see also Proposition 4.5). Thus +dim Homg(V, W) = +� +1, +if V ̸≃ W (that is a1 ̸= b1); +2, +if V ≃ W (that is a1 = b1). +(Since here V and W are of the same type, we do not need to consider +Homg(W, V ).) +• Case Ssl(2) +1 +̸= 0. +It follows from Theorems 4.6 and 4.7 that dim Ssl(2) +1 += 1 and in all these +cases, St(V ∗, W)sl(2) = 0, t = 0, 2. Hence Homg(V, W) (or Homg(W, V )) +is 1-dimensional. We describe all these cases below: +◦ Cases with W of type Z and Homg(V, W) ̸= 0. +(i) n = 1, a0 = b1, V = FU − +a0, W = Z(b0, ℓ), ℓ ≥ 1. +V = V (a0) ⊕ V (a0 − 1) ⊕ V (a0) +↓ +↓ +W = V (b0) ⊕ V (b1) ⊕ · · · ⊕ V (bℓ). +(ii) n = 1, a0 = b1, V = FU + +a0, W = Z(bℓ, ℓ)∗, ℓ ≥ 1. +V = V (a0) ⊕ V (a0 + 1) ⊕ V (a0) +↓ +↓ +W = V (b0) ⊕ V (b1) ⊕ · · · ⊕ V (bℓ). +(iii) n > 1, a0 = 0, 1, V = FU + +(a0,a0+m,a0), W = Z(b1, 1)∗. +V = V (a0) ⊕ V (a0 + m) ⊕ V (a0) +↓ +↓ +W = V (b0) ⊕ V (b1). +◦ Cases with W of type Z and Homg(W, V ) ̸= 0. +(i) n = 1, a0 = bℓ−1, V = FU + +a0, W = Z(b0, ℓ), ℓ ≥ 1. +W = V (b0) ⊕ · · · ⊕ V (bℓ−1) ⊕ V (bℓ) +↓ +↓ +V =V (a0) ⊕ V (a0 + 1) ⊕ V (a0). +(ii) n = 1, a0 = bℓ−1, V = FU − +a0, W = Z(bℓ, ℓ)∗, ℓ ≥ 1. +W = V (b0) ⊕ · · · ⊕ V (bℓ−1) ⊕ V (bℓ) +↓ +↓ +V =V (a0) ⊕ V (a0 − 1) ⊕ V (a0). + +UNISERIAL REPRESENTATIONS OF THE LIE ALGEBRA sl(2) ⋉ hn +21 +(iii) n > 1, V = FU(a0,a0+m,a0), a0 = 0, 1, W = Z(a0, 1). +W = V (a0) ⊕ V (a0 + m) +↓ +↓ +V =V (a0) ⊕ V (a0 + m) ⊕ V (a0). +◦ Cases with W standard faithful and Homg(V, W) ̸= 0. +(i) n = 1, a0 = b1, V = FU − +a0, W = FU + +b0. +V = V (a0) ⊕ V (a0 − 1) ⊕ V (a0) +↓ +↓ +W = V (b0) ⊕ V (b0 + 1) ⊕ V (b0). +(ii) n = 1, a0 = b1, V = FU + +a0, W = FU − +b0. +V = V (a0) ⊕ V (a0 + 1) ⊕ V (a0) +↓ +↓ +W = V (b0) ⊕ V (b0 − 1) ⊕ V (b0). +6. Proof of Theorem 4.3 +Let µ be a possible highest weight in S1. We start with some general +considerations and next we will work out each case. +We know from Proposition 4.5 that µ must be highest weight in both +V (a) ⊗ V (d) and V (b) ⊗ V (c), that is +(6.1) +|a − d|, |b − c| ≤ µ ≤ a + d, b + c +and µ ≡ a+d ≡ b+c mod 2. We also know that µ is indeed highest weight +in S1 if and only if there is a linear combination +u0 = q1va,d,µ +0 ++ q2vb,c,µ +0 +, +with q1, q2 ̸= 0 (see item (i) in Proposition 4.5), that is annihilated by es +for all s = 0, . . . , m. Indeed this implies that u0 is also annihilated by z and +thus in S1 (see (4.1)). +We now describe esva,d,µ +0 +and esvb,c,µ +0 +. +On the one hand we have (see (2.4)) +(6.2) +va,d,µ +0 += +� +i,j +CG(a +2, a +2 − i; d +2, d +2 − j | µ +2, µ +2 ) va +i ⊗ vd +j + +22 +LEANDRO CAGLIERO AND IV´AN G´OMEZ RIVERA +and thus (see (3.2)) +esva,d,µ +0 += +� +i,j +CG(a +2, a +2 − i; d +2, d +2 − j | µ +2 , µ +2) va +i ⊗ esvd +j += +� +i,j,k +(−1)jCG(a +2, a +2 − i; d +2, d +2 − j | µ +2, µ +2 ) +× CG( c +2, c +2 − k; d +2, − d +2 + j | m +2 , m +2 − s) va +i ⊗ vc +k += +� +i,j,k +(−1)kCG(a +2, a +2 − i; d +2, d +2 − k | µ +2 , µ +2) +× CG( c +2, c +2 − j; d +2, − d +2 + k | m +2 , m +2 − s) va +i ⊗ vc +j. +(6.3) +In this sum, if the coefficient of va +i ⊗ vc +j is not zero then we must have +a +2 − i + d +2 − k = µ +2 , +c +2 − j − d +2 + k = m +2 − s. +(6.4) +On the other hand, we have (see (2.4)) +(6.5) +vb,c,µ +0 += +� +i,j +CG( b +2, b +2 − i; c +2, c +2 − j | µ +2 , µ +2 ) vb +i ⊗ vc +j +and thus (see (3.2)) +esvb,c,µ +0 += +� +i,j +CG( b +2, b +2 − i; c +2, c +2 − j | µ +2, µ +2) esvb +i ⊗ vc +j. += +� +i,j,k +(−1)iCG( b +2, b +2 − i; c +2, c +2 − j | µ +2, µ +2 ) +× CG(a +2, a +2 − k; b +2, − b +2 + i | m +2 , m +2 − s) va +k ⊗ vc +j += +� +i,j,k +(−1)kCG( b +2, b +2 − k; c +2, c +2 − j | µ +2, µ +2) +× CG(a +2, a +2 − i; b +2, − b +2 + k | m +2 , m +2 − s) va +i ⊗ vc +j. +(6.6) +In this sum, if the coefficient of va +i ⊗ vc +j is not zero then we must have +b +2 − k + c +2 − j = µ +2 , +a +2 − i − b +2 + k = m +2 − s. +(6.7) +Either (6.4) or (6.7) imply +(6.8) +i + j = a + c − m − µ +2 ++ s, +(recall that 0 ≤ i ≤ a and 0 ≤ j ≤ c). Now we consider all the cases. +(i) The case V2 = E(c, d) with c + d = m and 0 < a ≤ c: Here +µ = d + a − 2p, 0 ≤ p ≤ min{a, d} + +UNISERIAL REPRESENTATIONS OF THE LIE ALGEBRA sl(2) ⋉ hn +23 +and it follows from (6.8) that +(6.9) +0 ≤ i + j = p − d + s. +The sum (6.3) is +esva,d,µ +0 += +� +i,j,k +(−1)kCG(a +2, a +2 − i; d +2, d +2 − k | a+d +2 +− p, a+d +2 +− p) +× CG(m−d +2 , m−d +2 +− j; d +2, − d +2 + k | m +2 , m +2 − s) va +i ⊗ vc +j. +In this sum, it follows from (6.4) that +k = p − i +j = p − d + s − i. +The conditions k ≥ 0 and 0 ≤ j ≤ m − d imply p − m + s ≤ i ≤ min{p, p − +d + s}. Therefore, we obtain (see (2.9) and (2.12)) +esva,d,µ +0 += +min{p,p−d+s} +� +i=max{0,p−m+s} +(−1)p−iCG(a +2, a +2 − i; d +2, d +2 − p + i | a+d +2 +− p, a+d +2 +− p) +× CG(m−d +2 , m−d +2 +− p + d − s + i; d +2, − d +2 + p − i | m +2 , m +2 − s) va +i ⊗ vc +p−d+s−i += +min{p,p−d+s} +� +i=max{0,p−m+s} +(−1)p +� +(a + d − 2p + 1)! p! (a − i)! (d − p + i)! +(a − p)! (d − p)! (a + d − p + 1)! i! (p − i)! +× +� +(m − s)! s! (m − d)! d! +m! (m − p − s + i)! (p − i)! (d − p + i)! (p − d + s − i)! va +i ⊗ vc +p−d+s−i. +Thus, +esva,d,µ +0 += (−1)p +� +(m − d)! (a + d − 2p + 1)! p! d! (m − s)! s! +(a − p)! (d − p)! (a + d − p + 1)! m! +wa,d,µ +s +with +wa,d,µ +s += +min{p,p−d+s} +� +i=max{0,p−m+s} +� +(a − i)! +i! (p − i)!2 (m − p − s + i)! (p − d + s − i)! va +i ⊗vc +p−d+s−i. +On the other hand, the sum (6.6) is +esvb,c,µ +0 += +� +i,j,k +(−1)kCG(m−a +2 , m−a +2 +− k; m−d +2 , m−d +2 +− j | a+d +2 +− p, a+d +2 +− p) +× CG(a +2, a +2 − i; m−a +2 , − m−a +2 ++ k | m +2 , m +2 − s) va +i ⊗ vc +j. +In this sum, it follows from (6.7) that +j = p − d + s − i +k = m − a − s + i + +24 +LEANDRO CAGLIERO AND IV´AN G´OMEZ RIVERA +and the condition k ≥ 0 implies i ≥ a − m + s, and condition j ≥ 0 implies +p − d + s ≥ i. Therefore, we obtain (see (2.9) and (2.12)) +esvb,c,µ +0 += +min{a,p−d+s} +� +i=max{0,a−m+s} +(−1)m−a−s+i +× CG(m−a +2 , − m−a +2 ++ s − i; m−d +2 , m−d +2 +− p + d − s + i | a+d +2 +− p, a+d +2 +− p) +× CG(a +2, a +2 − i; m−a +2 , m−a +2 +− s + i | m +2 , m +2 − s) va +i ⊗ vc +p−d+s−i += +min{a,p−d+s} +� +i=max{0,a−m+s} +� +(a + d − 2p + 1)! (p + m − a − d)! (s − i)! (m − p − s + i)! +(d − p)! (a − p)! (m − a − s + i)! (p − d + s − i)! (m − p + 1)! +× +� +s! (m − s)! a! (m − a)! +m! (a − i)! (m − a − s + i)! i! (s − i)! va +i ⊗ vc +a−p+s−i. +Thus +esvb,c,µ +0 += +� +(m − s)! s! (a + d − 2p + 1)! (p + m − a − d)! a! (m − a)! +(d − p)! (a − p)! (m − p + 1)! m! +wb,c,µ +s +where +wb,c,µ +s += +min{a,p−d+s} +� +i=max{0,a−m+s} +� +(m − p − s + i)! +i! (m − a − s + i)!2 (p − d + s − i)! (a − i)! +va +i ⊗vc +p−d+s−i. +Now, if a ≤ d and p = a, we have, for all 0 ≤ s ≤ m, +wa,d,µ +s += +min{a,a−d+s} +� +i=max{0,a−m+s} +� +1 +i! (a − i)! (m − a − s + i)! (a − d + s − i)! va +i ⊗ vc +p−d+s−i += wb,c,µ +s +. +This shows that +u0 = (−1)a √ +d + 1 va,d,µ +0 +− +√ +b + 1 vb,c,µ +0 +is, indeed, a highest weight vector, of weight µ = d − a = b − c, in S1. +If p < a then, for s = d, the sum defining wa,d,µ +d +has the index i running +up to i = a while the sum defining wb,c,µ +d +has the index i running only up to +i = p. In both cases, all the coefficients are non-zero, and thus {wa,d,µ +1 +, wb,c,µ +1 +} +is linearly independent. This shows that there is no possible µ in S1 and +thus S1 = 0. This completes the proof in this case. +(ii) The case V2 = E(c, d) with d = c+m: Since a < m ≤ d and by equation +(6.1) we have +µ = b + c − 2p, 0 ≤ p ≤ min{c, b}, +µ = a + d − 2p′, 0 ≤ p′ ≤ a. +This implies p′ − p = a and hence p′ = a and p = 0. This yields +µ = b + c = d − a. + +UNISERIAL REPRESENTATIONS OF THE LIE ALGEBRA sl(2) ⋉ hn +25 +First we prove that, if c = 0, then S1(V1, V2) ≃ V (b). In this case, (6.3) +becomes +esva,d,µ +0 += +� +i,k +(−1)kCG(m−b +2 , m−b +2 +− i; m +2 , m +2 − k | b +2, b +2) +× CG(0, 0; m +2 , − m +2 + k | m +2 , m +2 − s) va +i ⊗ v0 +0 +(the index j is 0). It follows from (6.4) that +k = m − s +i = −b + s. +Therefore, esva,d,µ +0 += 0 if s < b and, for s ≥ b we have (see (2.9) and (2.11)) +esva,d,µ +0 += (−1)m−sCG(m−b +2 , − m−b +2 ++ m − s; m +2 , − m +2 + s | b +2, b +2) +× CG(0, 0; m +2 , m +2 − s | m +2 , m +2 − s) va +s−b ⊗ v0 +0 += +� +(b + 1) (m − b)! s! +(m + 1)! (s − b)! +va +s−b ⊗ v0 +0. +On the other hand, since c = 0 and µ = b, (6.6) becomes (see also (3.2) +or (3.3)) +esvb,c,µ +0 += esvb +0 ⊗ v0 +0 += + + + +CG(a +2, a +2 − (s − b); b +2, − b +2 | m +2 , m +2 − s) va +s−b ⊗ v0 +0, +if s ≥ b; +0, +if s < b. += + + + +� +a! s! +(s−b)! m! va +s−b ⊗ v0 +0, +if s ≥ b; +0, +if s < b. +This shows that +u0 = +√ +b + 1 va,d,µ +0 +− +√ +m + 1 vc,d,µ +0 +is, indeed, a highest weight vector of weight µ = b, in S1. +Now, suppose that c ̸= 0 and set s = m. Recall that µ = b + c = d − a. +When we consider the sum (6.3), it follows from (6.4) that +j = k = a − i. + +26 +LEANDRO CAGLIERO AND IV´AN G´OMEZ RIVERA +The condition 0 ≤ j ≤ c implies a − c ≤ i ≤ a and hence (6.3) becomes (see +(2.11)) +emva,d,µ +0 += +a +� +i=max{0,a−c} +(−1)a−i CG(a +2, a +2 − i; d +2, d +2 − (a − i) | d−a +2 , d−a +2 ) +× CG( c +2, c +2 − (a − i); d +2, − d +2 + a − i | m +2 , − m +2 ) va +i ⊗ vc +a−i += +a +� +i=max{0,a−c} +(−1)i+d−a +� +(c + b + 1) (m − b)! (c + b + i)! +(c + m + 1)! i! +× +� +(m + 1) c! (c + b + i)! +(c + m + 1)! (c − m + b + i)! va +i ⊗ vc +a−i. +On the other hand, when we consider the sum (6.6), it follows from (6.7) +that +j = a − i = −k +and the condition k ≥ 0 implies that k = j = 0 and i = a = m − b. Thus, +(6.6) is +emvb,c,µ +0 += CG( b +2, b +2; c +2, c +2 | c+b +2 , c+b +2 ) CG(m−b +2 , − m−b +2 ; b +2, − b +2 | m +2 , − m +2 ) va +a ⊗ vc +0 += va +a ⊗ vc +0. +Since c ̸= 0, the sum in emva,d,µ +0 +has at least two non-zero terms, while +the sum in emvc,d,µ +0 +has a single non-zero term, and thus {emva,d,µ +0 +, emvc,b,µ +0 +} +is linearly independent. This completes the proof in this case. +(iii) The case V2 = E(c, d) with c = d + m: Since b < m ≤ c, it follows from +(6.1) that +µ = b + c − 2p, 0 ≤ p ≤ b +µ = a + d − 2p′, 0 ≤ p′ ≤ min{a, d}. +This implies p − p′ = b and hence the only option is p = b, p′ = 0 and this +yields +µ = a + d = c − b. +We compute now esva,d,µ +0 +. It follows from (6.8) and (6.4) that +k = −i +j = s − i, +and since k ≥ 0, we have k = i = 0 and j = s. Therefore, (6.3) becomes +(see also (2.9) and (2.10)) +esva,d,µ +0 += CG(a +2, a +2; d +2, d +2 | a+d +2 , a+d +2 ) CG(d+m +2 , d+m +2 +− s; d +2, − d +2 | m +2 , m +2 − s) va +0 ⊗ vc +s += +� +(d+m−s)! (m+1)! +(d+m+1)! (m−s)! va +0 ⊗ vc +s. +As always, the reader should check that all the numbers under the factorial +sign are non-negative. + +UNISERIAL REPRESENTATIONS OF THE LIE ALGEBRA sl(2) ⋉ hn +27 +We compute now esvb,c,µ +0 +. It follows from (6.8) and (6.7) that +j = s − i +k = m − a − s + i +and conditions k ≥ 0 and j ≥ 0 imply s ≥ i ≥ s − (m − a). Thus, it follows +from (6.6), (2.11) and (2.9) that +esvb,c,µ +0 += +min{a,s} +� +i=max{0,s−(m−a)} +CG(m−a +2 , − m−a +2 ++ s − i; d+m +2 , d+m +2 +− (s − i) | a+d +2 , a+d +2 ) +× CG(a +2, a +2 − i; m−a +2 , m−a +2 +− s + i | m +2 , m +2 − s) va +i ⊗ vc +s−i += +min{a,s} +� +i=max{0,s−(m−a)} +(−1)s−i� +(d+m−s+i)! (m−a)! (a+d+1) +(d+m+1)! (m−a−s+i)! +× +� +a! (m−a)! (m−s)! s! +i! (s−i)! m! (a−i)! (m−a−s+i)! va +i ⊗ vc +s−i +Again, note that all the numbers under the factorial sign are non-negative. +For s = m − a = b, the sum giving esvb,c,µ +0 +has the index i running from +i = 0 to i = min{a, b} ̸= 0, while the sum giving esva,d,µ +0 +has the index i +running only up to i = 0. In both cases, all the coefficients are non-zero, +and thus {esvb,c,µ +0 +, esva,d,µ +0 +} is linearly independent. This shows that there is +no possible µ in S1 and thus S1 = 0. This completes this case and the proof +of the theorem. +References +[1] I. Assem, D. Simson, and A. Skowro´nski. Elements of the Representation Theory of +Associative Algebras: 1. Techniques of the Representation Theory. Cambridge Uni- +versity Press, New York (USA), UK, 2007. +[2] M. Auslander, I. Reiten, and S. O. Smalø. Representation Theory of Artin Algebras. +Cambridge University Press, New York (USA), Melbourne (Australia), 1995. +[3] K. Bongartz and B. Huisgen-Zimmermann. The geometry of uniserial representations +of algebras II. Alternate viewpoints and uniqueness. J. Pure Appl. Algebra, 157:23–32, +2001. +[4] L. Cagliero, L. Guti´errez Frez, and F. Szechtman. Classification of finite dimen- +sional uniserial representations of conformal Galilei algebras. Journal of Mathematical +Physics, 57(101706), 2016. +[5] L. Cagliero, L. Guti´errez Frez, and F. Szechtman. Free 2-step nilpotent Lie algebras +and indecomposable modules. Comm. Algebra, 46:2990–3005, 2018. +[6] L. Cagliero, F. Levstein, and F. Szechtman. Nilpotency degree of the nilradical of +a solvable Lie algebra on two generators and uniserial modules associated to free +nilpotent Lie algebras. Journal of Algebra, 585:447–483, 2021. +[7] L. Cagliero and I. G´omez Rivera. Tensor products and intertwining operators for unis- +erial representations of the Lie algebras sl(2) ⋉ V (m). submitted (ArXiv:2201.10605). +[8] L. Cagliero and F. Szechtman. The classification of uniserial sl(2)⋉V (m)-modules and +a new interpretation of the Racah-Wigner 6j-symbol. Journal Algebra, 386:142–175, +2013. +[9] L. Cagliero and F. Szechtman. On the theorem of the primitive element with appli- +cations to the representation theory of associative and Lie algebras. Canad. Math. +Bull., 57:735–748, 2014. + +28 +LEANDRO CAGLIERO AND IV´AN G´OMEZ RIVERA +[10] L. Cagliero and F. Szechtman. Classification of linked indecomposable modules of a +family of solvable Lie algebras over an arbitrary field of characteristic 0. J. of Algebra +and Its Applications, 15(1650029), 2016. +[11] L. Cagliero and F. Szechtman. Indecomposable modules of 2-step solvable Lie algebras +in arbitrary characteristic. Comm. Algebra, 44:1–10, 2016. +[12] P. Casati. Irreducible sln+1-representations remain indecomposable restricted to some +Abelian subalgebras. Journal Lie Theory, 20:393–407, 2010. +[13] P. Casati. The classification of the perfect cyclic sln+1 ⋉ Cn+1-modules. Journal of +Algebra, 476:311–343, 2017. +[14] P. Casati, S. Minniti, and V. Salari. Indecomposable representations of the Diamond +Lie algebra. Journal of Mathematical Physics, 51(033515):20pp, 2010. +[15] P. Casati, A. Previtali, and F. Szechtman. Indecomposable modules of a family of +solvable lie algebras. Linear Algebra and its Applications, 531:423–446, 2017. +[16] V. Chari and A. Moura. The restricted Kirillov-Reshetikhin modules for the current +and twisted current algebras. Commun. Math. Phys., 266:431–454, 2006. +[17] A. Douglas and H. de Guise. Some nonunitary, indecomposable representations of the +Euclidean algebra e(3). J. Phys. A: Math. Theor., 43(085204):13pp, 2010. +[18] A. Douglas, D. Kahrobaei, and J. Repka. Classification of embeddings of abelian +extensions of Dn into En+1. J. Pure Appl. Algebra, 217:1942–1954, 2013. +[19] A. Douglas and A. Premat. A class of nonunitary, finite dimensional representations +of the euclidean algebra e(2). Communications in Algebra, 35:1433–1448, 2007. +[20] T. Finis. Appendix to the paper “Some uniserial representations of certain special +linear groups” by P. Sin and J. G. Thompson. J. Algebra, 398:461–468, 2014. +[21] B. Huisgen-Zimmermann. The geometry of uniserial representations of finite dimen- +sional algebras. III: Finite uniserial type. Trans. Amer. Math. Soc., 348:4775–4812, +1996. +[22] H. P. Jakobsen. Indecomposable finite-dimensional representations of a class of Lie +algebras and Lie superalgebras, volume 2027. Lecture Notes in Math., Springer, Hei- +delberg, 2011. +[23] L. Morotti. Irreducible tensor products of representations of covering groups of sym- +metric and alternating groups. Represent. Theory, 25:543–593, 2021. +[24] T. Nakayama. On Frobeniusean algebras II. Ann. of Math., 42:1–21, 1941. +[25] Z. Nazemian, A. Ghorbani, and M. Behboodi. Uniserial dimension of modules. J. +Algebra, 399:894–903, 2014. +[26] A. Piard. Sur des repr´esentations ind´ecomposables de dimension finie de SL(2).R2. +Journal of Geometry and Physics, 3:1–53, 1986. +[27] G. Puninski. Serial rings. Kluwer Academic Publishers, Dordrecht, 2001. +[28] P. Sin and J. G. Thompson. Some uniserial representations of certain special linear +groups. J. Algebra, 398:448–460, 2014. +[29] D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii. Quantum theory of +angular momentum. World Scientific, Singapore, 1989. +FaMAF-CIEM (CONICET), Universidad Nacional de C´ordoba, Medina Al- +lende s/n, Ciudad Universitaria, 5000 C´ordoba, Rep´ublica Argentina. +Email address: cagliero@famaf.unc.edu.ar +FaMAF-CIEM (CONICET), Universidad Nacional de C´ordoba, Medina Al- +lende s/n, Ciudad Universitaria, 5000 C´ordoba, Rep´ublica Argentina. +Email address: ivan.gomez.rivera@mi.unc.edu.ar + diff --git a/1dFIT4oBgHgl3EQf3yva/content/tmp_files/load_file.txt b/1dFIT4oBgHgl3EQf3yva/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..056136aa69d3b648e6c193ffc815680a0abc0544 --- /dev/null +++ b/1dFIT4oBgHgl3EQf3yva/content/tmp_files/load_file.txt @@ -0,0 +1,1212 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf,len=1211 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='11383v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='RT] 26 Jan 2023 TENSOR PRODUCTS AND INTERTWINING OPERATORS BETWEEN TWO UNISERIAL REPRESENTATIONS OF THE GALILEAN LIE ALGEBRA sl(2) ⋉ hn LEANDRO CAGLIERO AND IV´AN G´OMEZ RIVERA Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Let sl(2) ⋉ hn, n ≥ 1, be the Galilean Lie algebra over a field of characteristic zero, where hn is the Heisenberg Lie algebra of dimension 2n+ 1, and sl(2) acts on hn so that hn ≃ V (2n− 1)⊕ V (0) as sl(2)-modules (here V (k) denotes the irreducible sl(2)-module of highest weight k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' The isomorphism classes of uniserial � sl(2)⋉hn � modules are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' In this paper we study the tensor product of two uniserial represen- tations of sl(2) ⋉ hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Among other things, we obtain the sl(2)-module structure of the socle of V ⊗W and we describe the space of intertwining operators Homsl(2)⋉hn(V, W ), where V and W are uniserial representa- tions of sl(2) ⋉ hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' This article extends a previous work in which we obtained analogous results for the Lie algebra sl(2) ⋉ am where am is the abelian Lie algebra and sl(2) acts so that am ≃ V (m − 1) as sl(2)- modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Introduction This article is part of a project whose general goal is to understand to what extent there is a class of finite-dimensional representations of (non- semisimple) Lie algebras that is small enough, so that it members can be described in a reasonably efficient way in terms of uniserial representations, and large enough to include many representations that appear in problems of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' This naturally leads to consider the tensor products of unise- rial representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' A typical example of a representation we would like to describe thoroughly is a cohomology space associated to an algebra (as- sociative algebra or non-semisimple Lie algebra), viewed as a module over its whole Lie algebra of derivations (which is, in general, non-semisimple).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Understanding this action becomes specially important if the cohomology space has a Gerstenhaber or a Poisson structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Why uniserial representations as building blocks?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' We recall that, for associative algebras, the class of uniserial modules is very relevant, a foun- dational result here is due to T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Nakayama [24] (see also [1] or[2]) and it states that every finitely generated module over a serial ring is a direct sum of uniserial modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' For more information in the associative case we refer the reader mainly to [1, 2, 27], and also [3, 21, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' In Lie algebra case, even 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' 17B10, 18M20, 22E27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' non-semisimple Lie algebras, uniserial representations, socle, tensor product, intertwining operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' This research was partially supported by an NSERC grant, CONICET PIP 112-2013- 01-00511, PIP 112-2012-01-00501, MinCyT C´ordoba, FONCYT Pict2013 1391, SeCyT- UNC 33620180100983CB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' 1 2 LEANDRO CAGLIERO AND IV´AN G´OMEZ RIVERA though very little is known about uniserial representations, in the articles [8, 9, 10, 11, 5, 4, 6, 15] we and other authors have classified all finite dimen- sional uniserial representations for some different families of Lie algebras g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' These classifications show that the class of uniserial representations of g is rather small and treatable in the universe of all the indecomposable modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Also, in [20], it is shown how the infinite dimensional uniserial representa- tions of certain special linear groups obtained in [28] appear naturally in cohomology spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' We point out that many other authors work on the idea of describing or classifying a special class of indecomposable representations of (non- semisimple) Lie algebras whose members might be used as building blocks for describing more general representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' For instance, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Piard [26] ana- lyzed thoroughly the indecomposable modules U, of the complex Lie algebra sl(2)⋉C2, such that U/rad(U) is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' More recently, various families of indecomposable modules over various types of non-semisimple Lie alge- bras have been constructed and/or classified, see for instance [12, 13, 14, 16, 19, 17, 18, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' This article and [7] are motivated by the challenge of describing, in a standard way, the tensor product of two uniserial g-modules in terms of uniserial g-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' In contrast to the Nakayama case, these tensor products are not at all a direct sum of uniserials, there are many cases where the tensor product of two uniserial g-modules is an indecomposable g-module but not uniserial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' In this paper, all Lie algebras and representations con- sidered are assumed to be finite dimensional over a field F of characteristic zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' For n ≥ 0, we denote by an the abelian Lie algebra of dimension n, by hn the Heisenberg Lie algebra of dimension 2n + 1, and by V (n) the irreducible sl(2)-module with highest weight n (dim V (n) = n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' In addition, for n ≥ 1, sl(2) ⋉ an denotes the Lie algebra obtained by letting sl(2) act so that an ≃ V (n − 1), and sl(2) ⋉ hn denotes the Lie algebra where hn ≃ V (2n − 1) ⊕ V (0) as sl(2)-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' We notice that sl(2) ⋉ a2n−1 is isomorphic to the quotient sl(2) ⋉ hn mod its 1-dimensional center z � sl(2) ⋉ hn � ≃ V (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' In this work we study the structure of the tensor product of two uniserial representations of sl(2)⋉hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' The classification of all the isomorphism classes of uniserial � sl(2) ⋉ hn � modules was obtained in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' This classification is reviewed with details in §3 and a rough description of it is the following: Non-faithful � sl(2) ⋉ hn � modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Since z � sl(2) ⋉ hn � acts trivially on them, they are in correspondence with the uniserial � sl(2)⋉a2n−1 � modules In turn, these where classified in [8]: – A general family E(a, b), where a, b are non-negative integers with cer- tain restrictions (depending on n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' The composition length of E(a, b) is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' – A general family Z(a, ℓ) and its duals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Here a and ℓ are a non-negative integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' The composition length of Z(a, ℓ) is ℓ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' – Some exceptional modules with composition lengths 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' The modules Z(a, ℓ) and their duals are referred to as modules of type Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' UNISERIAL REPRESENTATIONS OF THE LIE ALGEBRA sl(2) ⋉ hn 3 Faithful � sl(2) ⋉ hn � modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' All of them have composition length 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' – For n = 1: Two families denoted by FU + a and FU − a , with a an integer that satisfies a ≥ 0 and a ≥ 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' – For n = 2: Only four equivalence classes, they are denoted by FU(0,3,0), FU(1,4,1), FU(1,2,1) and FU(4,3,4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' – For any n ≥ 3: Only three equivalence classes, they are denoted by FU(0,m,0), FU(1,m+1,1) and FU(1,m−1,1), here m = 2n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' All faithful uniserial modules (except FU(4,3,4), n = 2) are, in some sense, of a similar type and are referred to as standard faithful modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' The � sl(2) ⋉ h2 � module FU(4,3,4) is quite exceptional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' The modules E(a, b) constitute the building blocks of all the other unise- rial modules: all uniserials can be obtained by combining the modules E(a, b) in a subtle way governed by the zeros of the 6j-symbols (see [5, 8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' As a consequence, the � sl(2) ⋉ hn � module structure of the tensor product of two uniserial representations of sl(2) ⋉ hn depends strongly on the � sl(2) ⋉ hn � module structure of E(a, b) ⊗ E(c, d) which is already quite involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' In [7] we stated a conjecture that provides the description of the socle of E(a, b)⊗E(c, d) for any a, b, c, d (see Conjecture 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='2 below) and we proved the part of it that was necessary to obtain the socle of V ⊗W and the intertwining operators Homsl(2)⋉hn(V, W) where V and W are uniserial representations of sl(2) ⋉ hn of type Z (in fact, we dealt in [7] with uniserial representations of sl(2) ⋉ am, instead of sl(2) ⋉ hn, recall that, for modules of type Z, the action of z � sl(2) ⋉ hn � is trivial).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' In particular we proved that the socle of V ⊗ W is multiplicity free as sl(2)-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' As an application of these results, we proved in [7] that if V and W are � sl(2) ⋉ hn � modules of type Z, then V and W are determined from V ⊗ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Moreover, we provided a procedure to identify the corresponding parameters a and ℓ of V and W from V ⊗ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' This is a rare property, even if the factors are irreducible, it is not frequent that the factors V and W are determined from V ⊗ W (see [23] and references within).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' In this paper we extend the results of [7] obtaining the socle of V ⊗ W and the intertwining operators Homsl(2)⋉hn(V, W) when both V and W are standard faithful uniserial � sl(2) ⋉ hn � modules, or when one of them is standard faithful and the other one is uniserial of type Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' In contrast to the non-faithful case, in the standard faithful case it may happen that the socle of V ⊗W is not multiplicity free as sl(2)-modules (this occurs when V = W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' As a consequence, if V and W are isomorphic standard faithful uniserials, then the space of intertwining operators Homsl(2)⋉hn(V, W) is 2-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' The main step toward these results requires to make a considerable advance in the proof of Conjecture 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' See the comments after it to know what is still open about this conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' In §2 we review some basic facts about uniserial representations of Lie algebras and recall all the necessary definitions and formulas involving the Clebsch-Gordan coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' In §3 we review the classification of all uniserial representations of the Lie algebras sl(2)⋉an (obtained in [8]) and sl(2)⋉hn (obtained in [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' The main section 4 LEANDRO CAGLIERO AND IV´AN G´OMEZ RIVERA of the paper is §4, and we obtain in it the sl(2)-module structure of the socle of the tensor product of two (non-exceptional) uniserial � sl(2) ⋉ hn � modules V and W: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='1 recalls the case when V and W are of type Z (obtained in [7]), Conjecture 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='2 deals with all the possible cases of composition length 2, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='3 confirms part of Conjecture 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='2 needed to prove Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='6 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='7, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='6 gives the socle when V is of type Z and W is standard faithful, and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='7 describes the socle when both V and W are standard faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Finally, in §5 we obtain the space of intertwining operators from the results in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Our proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='3 is technical and long, it requires to consider some linear systems with entries given by the Clebsch-Gordan coefficients, and thus we decided to devote §6 to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' The Clebsch-Gordan coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Recall that F is a field of char- acteristic zero and that all Lie algebras and representations are assumed to be finite dimensional over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Let (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='1) e = � 0 1 0 0 � , h = � 1 0 0 −1 � , f = � 0 0 1 0 � be the standard basis of sl(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Let V (a) be the irreducible sl(2)-module with highest weight a ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' We fix a basis {va 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' , va a} of V (a) relative to which the basis {e, h, f} acts as follows: e va k = � a 2 �a 2 + 1 � − �a 2 − k + 1 � �a 2 − k � va k−1, h va k =(a − 2k)va k, f va k = � a 2 �a 2 + 1 � − �a 2 − k − 1 � �a 2 − k � va k+1, where 0 ≤ k ≤ a and va −1 = va a+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' The basis {va 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' , va a} has been chosen in a convenient way to introduce below the Clebsch-Gordan coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Note that if we denote by (x)a the matrix of x ∈ sl(2) relative to the basis {va 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' , va a}, then {(e)1, (h)1, (f)1} are as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='1), and (e)2 = \uf8eb \uf8ed 0 √ 2 0 0 0 √ 2 0 0 0 \uf8f6 \uf8f8 , (h)2 = \uf8eb \uf8ed 2 0 0 0 0 0 0 0 −2 \uf8f6 \uf8f8 , (f)2 = \uf8eb \uf8ed 0 0 0 √ 2 0 0 0 √ 2 0 \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' This means that we may assume that {v2 0, v2 1, v2 2} = {−e, √ 2 2 h, f}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' We know that V (a) ≃ V (a)∗ as sl(2)-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' More precisely, if {(va 0)∗, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' , (va a)∗} is the dual basis of {va 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' , va a} then the map V (a) → V (a)∗ va k �→ (−1)a−k(va a−k)∗ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='2) gives an explicit sl(2)-isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' It is well known that the decomposition of the tensor product of two irreducible sl(2)-modules V (a) and V (b) is (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='3) V (a) ⊗ V (b) ≃ V (a + b) ⊕ V (a + b − 2) ⊕ · · · ⊕ V (|a − b|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' UNISERIAL REPRESENTATIONS OF THE LIE ALGEBRA sl(2) ⋉ hn 5 This is the well known Clebsch-Gordan formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' The Clebsch-Gordan coefficients CG(j1, m1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' j2, m2 | j3, m3) are defined below and they provide an explicit sl(2)-embedding V (c) → V (a) ⊗ V (b) which is the following V (c) → V (a) ⊗ V (b) vc k �→ va,b,c k where, by definition, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='4) va,b,c k = � i,j CG(a 2, a 2 − i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' b 2, b 2 − j | c 2, c 2 − k) va i ⊗ vb j, where the sum runs over all i, j such that a 2 − i + b 2 − j = c 2 − k (in fact, we could let i, j run freely since the Clebsch-Gordan coefficient involved is zero if a 2 − i + b 2 − j ̸= c 2 − k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Since (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='5) Hom(V (b), V (a)) ≃ V (b)∗ ⊗ V (a) ≃ V (a) ⊗ V (b) it follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='4) that the map V (c) → Hom(V (b), V (a)) given by vc k �→ � i,j CG(a 2, a 2 − i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' b 2, b 2 − j | c 2, c 2 − k) va i ⊗ vb j, �→ � i,j (−1)b−jCG(a 2, a 2 − i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' b 2, b 2 − j | c 2, c 2 − k) va i ⊗ (vb b−j)∗, �→ � i,j (−1)jCG(a 2, a 2 − i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' b 2, − b 2 + j | c 2, c 2 − k) (vb j)∗ ⊗ va i (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='6) is an sl(2)-module homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' We now recall briefly the basic definitions and facts about the Clebsch- Gordan coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' We will mainly follow [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Given three non-negative integers or half-integers j1, j2, j3, we say that they satisfy the triangle condition if j1 + j2 + j3 is an integer and they can be the side lengths of a (possibly degenerate) triangle (that is |j1 − j2| ≤ j3 ≤ j1 + j2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' We now define (see [29, §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='2, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (1)]) ∆(j1, j2, j3) = � (j1 + j2 − j3)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (j1 − j2 + j3)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (−j1 + j2 + j3)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (j1 + j2 + j3 + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' if j1, j2, j3 satisfies the triangle condition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' otherwise, we set ∆(j1, j2, j3) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' If in addition m1, m2 and m3 are three integers or half-integers then the corresponding Clebsch-Gordan coefficient CG(j1, m1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' j2, m2|j3, m3) 6 LEANDRO CAGLIERO AND IV´AN G´OMEZ RIVERA is zero unless m1 + m2 = m3 and |mi| ≤ ji for i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' In this case, the following formula is valid for m3 ≥ 0 and j1 ≥ j2 (see [29, §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='2, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (3)]) CG(j1, m1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' j2, m2 | j3, m3) = ∆(j1, j2, j3) � (2j3 + 1) × � (j1 + m1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (j1 − m1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (j2 + m2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (j2 − m2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (j3 + m3)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (j3 − m3)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' × � r (−1)r r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='(j1+j2−j3−r)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='(j1−m1−r)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='(j2+m2−r)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='(j3−j2+m1+r)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (j3−j1−m2+r)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=', where the sum runs through all integers r for which the argument of every factorial is non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' If either m3 < 0 or j1 < j2 we have CG(j1, m1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' j2, m2 | j3, m3) = (−1)j1+j2−j3 CG(j1, −m1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' j2, −m2 | j3, −m3) = (−1)j1+j2−j3 CG(j2, m2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' j1, m1 | j3, m3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='7) In addition, it also holds (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='8) CG(j1, m1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' j2, m2 | j3, m3) = (−1)j1−m1 � 2j3 + 1 2j2 + 1 CG(j1, m1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' j3, −m3 | j2, −m2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' In the following sections, we will need the following particular values of the Clebsch-Gordan coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Here, a, b are integers and i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' , a, j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' , b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='9) CG(a 2, a 2 −i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' b 2, b 2 −j | a+b 2 , a+b 2 −i−j) = � a!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='b!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (a + b − i − j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (i + j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (a + b)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (a − i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (b − j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='10) CG(a 2, a 2 − i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' b 2, j − b 2 | a−b 2 , a−b 2 − i + j) = (−1)j � (a − i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' b!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (a − b + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (a + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (b − j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (a − b − i + j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (i − j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=', (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='11) CG(a 2, i − a 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' b 2, b 2 − j | b−a 2 , b−a 2 + i − j) = (−1)aCG( b 2, b 2 − j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' a 2, i − a 2 | b−a 2 , b−a 2 + i − j) = (−1)j � (b − j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' a!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (b − a + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (b + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (a − i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (b − a − j + i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (j − i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=', (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='12) CG(a 2, a 2 − i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' b 2, b 2 − j | a+b 2 − i − j, a+b 2 − i − j) = (−1)i � (a + b − 2i − 2j + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (i + j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (a − i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (b − j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (a + b − i − j + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (a − i − j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (b − i − j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='. 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Uniserial representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Given a Lie algebra g, a g-module V is uniserial if it admits a unique composition series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' In other words, V is uniserial if the socle series 0 = soc0(V ) ⊂ soc1(V ) ⊂ · · · ⊂ socn(V ) = V UNISERIAL REPRESENTATIONS OF THE LIE ALGEBRA sl(2) ⋉ hn 7 is a composition series of V , that is, the socle factors soci(V )/soci−1(V ) are irreducible for all 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Recall that soc1(V ) = soc(V ) is the sum of all irreducible g-submodules of V and soci(V )/soci−1(V ) = soc(V/soci−1(V )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Note that for uniserial modules, the composition length of V coincides with its socle length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' If the Levi decomposition of g is g = s⋉r, (with r the solvable radical and s semisimple) we may choose irreducible s-submodules Vi ⊂ V , 1 ≤ i ≤ n, such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='13) V = V1 ⊕ · · · ⊕ Vn with Vi ≃ soci(V )/soci−1(V ) as s-modules and rVi ⊂ V1 ⊕ · · · ⊕ Vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' In fact, if [s, r] = r, then rVi ⊂ V1 ⊕ · · · ⊕ Vi−1, see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' We say that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='13) is the socle decomposition of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' We point out that, in the socle decomposition of a g-module, the order of the summands is relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' The proof of the following lemma can be found in [7, Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Assume that r = [s, r] and let V be a g-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Then (1) soc(V ) = V r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (2) If V = V1⊕· · ·⊕Vn is a vector space decomposition such that soc(V ) = V1 and rVk ⊂ Vk−1 for all k = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' , n, then sock(V ) = V1 ⊕ · · · ⊕ Vk for all k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Uniserial representations of sl(2) ⋉ hn 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' The Lie algebra sl(2) ⋉ hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Let us fix n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' We recall that the Heisenberg Lie algebra hn is the (2n + 1)-dimensional vector space with basis {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' , xn, x′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' , x′ n, z} with non-zero brackets [xi, x′ i] = z for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' It is clear that the center of hn is generated by z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' We know that sl(2) acts by derivations on hn in such a way that hn ≃ V (m) ⊕ V (0), m = 2n − 1, as sl(2)-modules, where V (0) corresponds to the center of hn and we may assume that V (m) corresponds to the subspace generated by {xi, x′ i : i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' From now on, m will always be 2n−1 and thus am = hn/Fz as Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' In addition, we will denote by hn(m) the subspace of hn isomorphic to V (m) as sl(2)-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Hence, as sl(2)-modules, we have am ≃ hn(m) ≃ V (m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' We may assume that z corresponds to the basis element v0 0 of V (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Sim- ilarly, let us denote by {e0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' , em} the basis of hn(m) corresponding to the basis {vm 0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' , vm m} of V (m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' We may assume that z and {e0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' , em} have been chosen such that the bracket in hn is given by the projection V (m) ⊗ V (m) → V (0) (dual to the embedding (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='4)), that is 8 LEANDRO CAGLIERO AND IV´AN G´OMEZ RIVERA [ei, em−i] = CG(m 2 , m 2 − i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' m 2 , − m 2 + i | 0, 0) z = (−1)i � 1 m+1 z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='1) It is clear that Fz is also the center of sl(2) ⋉ hn and � sl(2) ⋉ hn � /Fz ≃ sl(2) ⋉ am as Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' In [4], it is obtained the classification, up to isomorphism, of all uniserial representations of the Lie algebra sl(2) ⋉ hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' It is straightforward to see that a uniserial representation of sl(2) ⋉ hn is faithful if and only if z acts non-trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Therefore, the classification is given in two stages: the non- faithful and the faithful ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' The non-faithful ones are the same as those of the Lie algebra sl(2) ⋉ am ≃ sl(2) ⋉ V (m) which were classified earlier in [8, Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' We now recall this classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' The non-faithful � sl(2) ⋉ hn � modules E(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' If a and b are non- negative integers such that m 2 , a 2, b 2 satisfy the triangle condition, it follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='3) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='5) that, up to scalar, there is a unique sl(2)-module ho- momorphism r = V (m) → Hom(V (b), V (a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' This produces an action of r on V (a)⊕V (b) such that r maps V (a) to 0 and V (b) to V (a) as follows (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='2) es vb j = a � i=0 (−1)j CG(a 2, a 2 − i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' b 2, − b 2 + j | m 2 , m 2 − s) va i , s = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Note that this is, except for a sign, the same as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Note also that the above sum has, in fact, at most one summand, that is (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='3) es vb j = \uf8f1 \uf8f2 \uf8f3 0, if i ̸= j + s + a−b−m 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (−1)jCG(a 2, a 2 − i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' b 2, − b 2 + j | m 2 , m 2 − s) va i , if i = j + s + a−b−m 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' This action, combined with the action of sl(2) defines a uniserial � sl(2) ⋉ am � module structure with composition length 2 on E(a, b) = V (a) ⊕ V (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' It is straightforward to see that E(a, b)∗ ≃ E(b, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' The action given in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='2) is the main building block for all other uniserial � sl(2) ⋉ am � modules as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Non-faithful � sl(2) ⋉ hn � modules of type Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' The above construc- tion can be extended to arbitrary composition length V (a0) ⊕ V (a1) ⊕ · · · ⊕ V (aℓ) only when the sequence {ai} is monotonic (increasing or decreasing) and |ai − ai−1| = m, for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' , ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' More precisely, for the “increasing case” UNISERIAL REPRESENTATIONS OF THE LIE ALGEBRA sl(2) ⋉ hn 9 let α and ℓ be non-negative integers and let Z(α, ℓ) be the � sl(2) ⋉ am � module defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='4) Z(α, ℓ) = V (α) ⊕ V (α + m) ⊕ · · · ⊕ V (α + ℓm) as sl(2)-module with action of r sending 0 ←− V (α) ←− V (α + 2m) ←− · · · ←− V (α + ℓm) as indicated in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='2) (with a = α + (i − 1)m, b = α + im, for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' , ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' We point out that the above sequence serves as an indication of the action of r, there is no chain complex involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' We notice that Z(α, 0) = V (α) (r acts trivially) and Z(α, 1) = E(α, α + m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' The “decreasing case” corresponds to the dual modules Z(α, ℓ)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' The modules Z(α, ℓ) and Z(α, ℓ)∗ are called of type Z and they are the unique isomorphism classes of uniserial � sl(2) ⋉ am � modules of composition length ℓ + 1 for ℓ ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Non-faithful � sl(2) ⋉ hn � modules of exceptional type (compo- sition lengths 2, 3 and 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' The modules E(a, b) with |a − b| ̸= m are not of type Z and we consider them of exceptional type (of composition lengths 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' For composition lengths 3 and 4 there are very few possible ways to “combine” the modules E(a, b) so that we do not fall in type Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' For composition length equal to 3, given 0 ≤ c < 2m and c ≡ 2m mod 4, let E3(c) = V (0) ⊕ V (m) ⊕ V (c) as sl(2)-modules with action of r sending 0 V (0) V (m) V (c) with the maps V (c) → V (m) and V (m) → V (0) given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' For composition length equal to 4, if m ≡ 0 mod 4, there is a family of � sl(2)⋉am � modules, parameterized by a non-zero scalar t ∈ F, with a fixed socle decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' This is defined by E4(t) = V (0) ⊕ V (m) ⊕ V (m) ⊕ V (0) as sl(2)-modules with action of r, sending each irreducible component as shown by the arrows 0 V (0) V (m) V (m) V (0) where the horizontal arrows are given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='2) and the bent arrow is t times (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Classification of all non-faithful � sl(2)⋉hn � modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' As we said at the beginning of the section, a uniserial representation of sl(2) ⋉ hn is faithful if and only if z acts non-trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Thus, the non-faithful uniserial � sl(2) ⋉ hn � modules are in correspondence, via the projection sl(2) ⋉ hn → sl(2) ⋉ am, with the uniserial representations of sl(2) ⋉ am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' These were classified in [8, Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='1], we summarize that result in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' 10 LEANDRO CAGLIERO AND IV´AN G´OMEZ RIVERA Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' The following list describes all the isomorphism classes of non-faithful uniserial representations of sl(2) ⋉ hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Length 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Z(a, 0) = V (a), a ≥ 0 (here r acts trivially).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Length 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' E(a, b), with a + b ≡ m mod 2 and 0 ≤ |a − b| ≤ m ≤ a + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Length 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Z(a, 2), Z(a, 2)∗, a ≥ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' and E3(c) with c ≡ 2m mod 4 and 0 ≤ c < 2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Length 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Z(a, 3), Z(a, 3)∗, a ≥ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' and E4(t), with t ∈ F (this exists only if m ≡ 0 mod 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Length ℓ ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Z(a, ℓ − 1), Z(a, ℓ − 1)∗, a ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Faithful � sl(2) ⋉ hn � modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' The faithful uniserial � sl(2) ⋉ hn � modules were classified, up to isomorphism, in [4, Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='5 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' It turns out that there are no faithful uniserial � sl(2) ⋉ hn � modules of composition length different from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Moreover, if V = V (a0) ⊕ V (a1) ⊕ V (a2) is socle decomposition (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='1) of a faithful uniserial � sl(2)⋉hn � module, then a0 = a2 and an explicit representative of each class can be obtained by conveniently combining the modules E(a, b) for some specific values of a and b as we explain below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Let us start with the sl(2)-module V = V (a0) ⊕ V (a1) ⊕ V (a2) with a2 = a0 such that m 2 , a0 2 , a1 2 satisfy the triangle condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' We now indicate how to obtain an action of hn = hn(m) ⊕ Fz on V so that V becomes a faithful uniserial � sl(2) ⋉ hn � module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Although we know that a2 = a0 we keep the notation a2 because we need to indicate that V (a0) is the socle of V and V (a2) corresponds to the third socle factor of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' First, let hn(m) act on V as follows 0 ←− V (a0) ←− V (a1) ←− V (a2) where the actions V (a1) → V (a0) and V (a2) → V (a1) are given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='2) (with a = a0, b = a1 and a = a1, b = a2 respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' This action of hn(m) on V can be extended to hn only in the following cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' In all of them, a0 = a2 and z acts as an sl(2)-isomorphism V (a2) → V (a0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (i) For n = 1 (that is m = 1), (a0, a1, a2) must be (a0, a0 + 1, a0), a0 ≥ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (a0, a0 − 1, a0), a0 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Let us call, respectively, FU + a0 and FU − a0 the first and second � sl(2)⋉ hn � modules above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (ii) For n = 2 (that is m = 3), (a0, a1, a2) must be (0, 3, 0), (1, 4, 1), (1, 2, 1), (4, 3, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' We call these modules FU(0,3,0), FU(1,4,1), FU(1,2,1) and FU(4,3,4) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' UNISERIAL REPRESENTATIONS OF THE LIE ALGEBRA sl(2) ⋉ hn 11 (iii) If n ≥ 3 (that is m ≥ 5), (a0, a1, a2) must be (0, m, 0), (1, m + 1, 1), (1, m − 1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' We call these modules FU(0,m,0), FU(1,m+1,1) and FU(1,m−1,1) re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' In these modules, the action of the center Fz is given by z va2 j = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 −2√m + 1 a + 1 va0 j , if V = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 FU + a and m = 1, FU(0,3,0), FU(1,4,1) and m = 3, FU(0,m,0), FU(1,m+1,1) and m ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' 2√m + 1 a + 1 va0 j , if V = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 FU − a and m = 1, FU(1,2,1) and m = 3, FU(1,m−1,1) and m ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' −4 5va0 j , if V = FU(4,3,4) and m = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='5) The next theorem summarizes this information and was proved in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' The following list describes all the isomorphism classes of faithful uniserial representations of sl(2) ⋉ hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (i) For n = 1: FU + a , a ≥ 0, and FU − a , a ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (ii) For n = 2: FU(0,3,0), FU(1,4,1), FU(1,2,1) and FU(4,3,4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (iii) For n ≥ 3: FU(0,m,0), FU(1,m+1,1) and FU(1,m−1,1) (m = 2n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Each of these modules is isomorphic to its own dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' We close this section pointing out that all the faithful uniserial � sl(2)⋉hn � modules, except FU(4,3,4), belong, in some sense, to the same kind of modules (we will say something more about this after Conjecture 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Therefore, we introduce the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' All faithful uniserial � sl(2) ⋉ hn � modules that are not isomorphic to FU(4,3,4) will be referred to as standard faithful uniserials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' The tensor product of two uniserial � sl(2) ⋉ hn � modules In [7, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='5] we obtained the sl(2)-module structure of the socle of the tensor product of two uniserial � sl(2) ⋉ a(m) � modules of type Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Therefore (see §3), we already know the sl(2)-module structure of the socle of the tensor product V1 ⊗ V2 of two uniserial � sl(2) ⋉ hn � modules in the cases where V1 and V2 are non-faithful of type Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' We summarize this in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' In this section we obtain the sl(2)-module structure of the socle of the tensor product V1 ⊗ V2 when both V1 and V2 are standard faithful uniserial modules, or when one of them is standard faithful and the other one is uniserial of type Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' From this, we derive a complete description of the space of intertwining operators Homsl(2)⋉hn(V1, V2) in all these cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' 12 LEANDRO CAGLIERO AND IV´AN G´OMEZ RIVERA 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' The non-faithful case and the crucial conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Let us recall some of the results obtained in [7], mainly Conjecture 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='4 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='5 that describes the sl(2)-module structure of the socle of the tensor product of two uniserial � sl(2) ⋉ a � modules of type Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' If U is a � sl(2)⋉hn � module, since hn = [sl(2), hn], it follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='2 that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='1) soc(U) = U hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Therefore, if V = V (a0) ⊕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' ⊕ V (aℓ) and W = V (b0) ⊕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' ⊕ V (bℓ′) are the socle decomposition of two � sl(2) ⋉ hn � modules, then soc(V ⊗ W) = ℓ+ℓ′ � t=0 \uf8eb \uf8edsoc(V ⊗ W) ∩ � i+j=t V (ai) ⊗ V (bj) \uf8f6 \uf8f8 = ℓ+ℓ′ � t=0 \uf8eb \uf8ed � i+j=t V (ai) ⊗ V (bj) \uf8f6 \uf8f8 hn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='2) For t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' , ℓ + ℓ′, we define St = St(V, W) = \uf8eb \uf8ed� i+j V (ai) ⊗ V (bj) \uf8f6 \uf8f8 hn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Hence, as sl(2)-modules, soc(V ⊗ W) = ℓ+ℓ′ � t=0 St and S0 = soc(V ) ⊗ soc(W) = V (a0) ⊗ V (b0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' The following theorem is the same as [7, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='5] but stated in terms of non-faithful uniserial � sl(2) ⋉ hn � modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Let V1 = V (a0) ⊕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' ⊕ V (aℓ) and V2 = V (b0) ⊕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' ⊕ V (bℓ′) be socle decomposition of two non-faithful uniserial � sl(2) ⋉ hn � modules of type Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Then, St = 0 for all t > min{ℓ, ℓ′}, S0 ≃ min{a0,b0} � k=0 V (a0 + b0 − 2k), and, for t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' , min{ℓ, ℓ′}, we have (i) If V1 = Z(a0, ℓ) and V2 = Z(b0, ℓ′), then St ≃ V (a0 + b0 + tm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (ii) If V1 = Z(a0, ℓ) and V2 = Z(bℓ′, ℓ′)∗, then St ≃ � 0, if tm > b0 − a0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' V (b0 − a0 − tm), if tm ≤ b0 − a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (iii) If V1 = Z(aℓ, ℓ)∗ and V2 = Z(bℓ′, ℓ′)∗, then St = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' UNISERIAL REPRESENTATIONS OF THE LIE ALGEBRA sl(2) ⋉ hn 13 One of the main steps towards proving the above theorem was to prove certain instances of the following conjecture (see [7, Conjecture 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Conjecture 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Let V1 = E(a, b) and V2 = E(c, d) (two uniserial sl(2) ⋉ a(m)-modules of length 2) and assume that a < c, or a = c and b ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Then S2 = 0 in all cases and S1 = 0 except in the following cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Case 1: [a, b] = [0, m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Here S1 ≃ V (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Cases 2: Here a > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' – Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='1: a+b = c+d = m with d−a = b−c ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Here S1 ≃ V (d−a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' – Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='2: b − a = d − c = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Here S1 ≃ V (d + a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' – Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='3: b−a = c−d = m with d−a = c−b ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Here S1 ≃ V (d−a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Case 3: [c, d] = [b, a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Here S1 ≃ V (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' In order to prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='1, we proved in [7, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='3] the cases 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='3 (and certain converse statement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Now, in this paper, we need to prove part of case 1 and case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='1 of the conjecture (together with certain converse statement) in order to prove Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='6 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' This is estab- lished in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' We point out that this theorem leaves out the uniserial � sl(2) ⋉ a(3) � modules E(3, 4) and E(4, 3), and a consequence of this is that Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='6 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='7 are restricted to the standard faith- ful modules, leaving the exceptional faithful uniserial � sl(2) ⋉ h2 � module FU(4,3,4) out of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Let V1 = E(a, b) with a + b = m and a, b ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Let V2 = V (c)⊕V (d) be the socle decomposition of a uniserial � sl(2)⋉V (m) � module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Then (i) If V2 ≃ E(c, d) with c + d = m and 0 < a ≤ c < m, then, as sl(2)- modules, S1(V1, V2) ≃ � V (d − a), if d − a = b − c ≥ 0 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (ii) If V2 ≃ Z(c, 1) ≃ E(c, c + m), then, as sl(2)-modules, S1(V1, V2) ≃ � V (b), if c = 0 0, if c ̸= 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (iii) If V2 ≃ Z(d, 1)∗ ≃ E(d + m, d), then S1(V1, V2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' The proof of this result is very technical and it will be given in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' The faithful case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' We will now focus on the tensor product of two uniserial � sl(2)⋉hn � modules where one of the factors is a standard faithful uniserial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Let V = V (a0) ⊕ V (a1) ⊕ V (a2) be the socle decomposition of a faithful uniserial � sl(2)⋉hn � module and set W = V (b0)⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='⊕V (bℓ), with ℓ ≥ 1, be the socle decomposition of a (not necessarily faithful) uniserial � sl(2) ⋉ hn � module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' By Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='3 we have that hn(m) · V (ai) ⊂ V (ai−1) and hn(m) · V (bj) ⊂ V (bj−1) ⊕ V (bj−2) z · V (ai) ⊂ V (ai−2) and z · V (bj) ⊂ V (bj−2) 14 LEANDRO CAGLIERO AND IV´AN G´OMEZ RIVERA for all i = 0, 1, 2 and 0 ≤ j ≤ ℓ (for convenience we assume V (ai) = V (bj) = 0 if i, j < 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Given v ∈ V (ai) ⊗ V (bj), let (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='3) esv = (esv)1 + (esv)2 + (esv)3 and zv = (zv)1 + (zv)2 where (esv)1 ∈ V (ai−1) ⊗ V (bj), (zv)1 ∈ V (ai−2) ⊗ V (bj), (esv)2 ∈ V (ai) ⊗ V (bj−1), (zv)2 ∈ V (ai) ⊗ V (bj−2), (esv)3 ∈ V (ai) ⊗ V (bj−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Note that (esv)3 = 0 if W is not isomorphic to E4 and that (zv)2 = 0 if W is not a faithful uniserial module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Let V1 = V (a0)⊕V (a1)⊕V (a2) and V2 = V (b0)⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' ⊕V (bℓ), with ℓ ≥ 1, be the socle decomposition of two uniserial � sl(2) ⋉ hn � modules, where V1 is faithful (not necessarily standard).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' If v0 ∈ V (ai0) ⊗ V (bj0) is a highest weight vector then: (i) (esv0)1 = 0 for all s = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' , m if and only if i0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (ii) (esv0)2 = 0 for all s = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' , m if and only if j0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (iii) (zv0)1 = 0 if and only if i0 ̸= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (iv) If V2 is faithful, then (zv0)2 = 0 if and only if j0 ̸= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Since the action of hn(m) on any uniserial � sl(2) ⋉ hn � is the same as the action of am in the corresponding � sl(2) ⋉ am � module, cases (i) and (ii) are immediate consequences of [7, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' By symmetry, it sufficient to prove (iii) to obtain (iv), so let us prove (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' If c is the weight of v0, we can assume that v0 = v ai0,bj0,c 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' It follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='4) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='3) that (zv0)1 = (zv ai0,bj0,c 0 )1 = � i+j= ai0+bj0−c 2 CG(ai0 2 , ai0 2 − i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' bj0 2 , bj0 2 − j | c 2, c 2) zv ai0 i ⊗ v bj0 j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='4) From the definition of the modules FU ± a for n = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' FU(0,3,0),FU(1,4,1), FU(1,2,1) and FU(4,3,4) for n = 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' and the modules FU(0,m,0), FU(1,m+1,1) and FU1,m−1,1 for n ≥ 3 (here m = 2n − 1), we know that zv ai0 i = 0 if i0 ̸= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Therefore, if i0 ̸= 2 then (zv0)1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' On the other hand, if i0 = 2, we know from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='5) that zva2 i = λva0 i , where λ is a non-zero scalar independent of i, 0 ≤ i ≤ a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Thus, the equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='4) becomes (zv0)1 = λ � i+j= a2+bj0−c 2 CG(a2 2 , a2 2 − i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' bj0 2 , bj0 2 − j | c 2, c 2) va0 i ⊗ v bj0 j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' UNISERIAL REPRESENTATIONS OF THE LIE ALGEBRA sl(2) ⋉ hn 15 In this sum, the term corresponding to i = 0, has a non-zero Clebsch-Gordan coefficient, indeed CG(a2 2 , a2 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' bj0 2 , c−a2 2 | c 2, c 2) = � (c+1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' a2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' �a2+bj0+c 2 +1 � !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' � a2+c−bj0 2 � !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Since all terms are linearly independent, we obtain (zv0)1 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' □ Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Let V1 = V (a0) ⊕ V (a1) ⊕ V (a2) and V2 = V (b0) ⊕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' ⊕ V (bℓ), with ℓ ≥ 1, be the socle decomposition of two uniserial � sl(2) ⋉ hn � modules, where V1 is standard faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Then, S0 = V (a0) ⊗ V (b0) and (i) St = 0 for all t > min{2, ℓ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' If St ̸= 0, t = 1, 2, then it is irreducible as sl(2)-module and if v is a non-zero highest weight vector in St of weight µ, then v = �t i=0 vi with vi a non-zero highest weight vector in V (ai) ⊗ V (bt−i), of weight µ, for all i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' , t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (ii) S2 = 0 if V2 is non-faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (iii) S1(V1, V2) ≃ S1 � E(a0, a1), E(b0, b1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (iv) If V2 is also standard faithful, then S2 ̸= 0 if and only if V1 ≃ V2 (that is ai = bi, i = 0, 1, 2) and in this case S2 ≃ V (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' The proof of this proposition is very similar to that of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='2 in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' We fix t > 0 and we assume that there is a non-zero highest weight vector u of weight µ, u = � i+j=t ui,j ∈ � � i+j=t V (ai) ⊗ V (bj) �hn ̸= 0, ui,j ∈ V (ai) ⊗ V (bj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Since V (ai) ⊗ V (bj) is an sl(2)-submodule, it follows that ui,j is either zero or a highest weight vector of weight µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Let Iµ t = {(i, j) : 0 ≤ i ≤ 2, 0 ≤ j ≤ ℓ, i + j = t and ui,j ̸= 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Since u ̸= 0, it follows that Iµ t ̸= ∅ and u = � (i,j)∈Iµ t qi,j vai,bj,µ 0 for certain non-zero scalars 0 ̸= qi,j ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Now, it follows from items (i) and (ii) in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='4 (see the details in [7][Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='2]) that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='5) Iµ t = {(0, t), (1, t − 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' , (t, 0)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Now, again, items (i) and (ii) in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='4 imply that such a non-zero u cannot exist if t > min{2, ℓ} and thus St = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' This proves (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Furthermore, (ii) follows similarly by applying item (iii) in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (iii) is clear from the definition of S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Let us prove (iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Assume that V2 = V (b0) ⊕ V (b1) ⊕ V (b2) is standard faithful, and suppose that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='6) u = q0,2 va0,b2,µ 0 + q1,1 va1,b1,µ 0 + q2,0 va2,b0,µ 0 ̸= 0 is a highest weight vector of weight µ in S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' We already know that q0,2, q1,1, q2,0 ̸= 0 and, moreover, we must have q0,2 va0,b2,µ 0 + q1,1 va1,b1,µ 0 ∈ S1 � E(a0, a1), E(b1, b2)) ̸= 0 16 LEANDRO CAGLIERO AND IV´AN G´OMEZ RIVERA and q1,1 va1,b1,µ 0 + q2,0 va2,b0,µ 0 ∈ S1 � E(a1, a2), E(b0, b1)) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='3 imply that it is impossible to have S1 � E(a0, a1), E(b1, b2) � ̸= 0 and S1 � E(a1, a2), E(b0, b1) � ̸= 0 unless (a0, a1, a2) = (b0, b1, b2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' This follows by considering all the cases with (a0, a1, a2) and (b0, b1, b2) running over (see §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='6) (i) if n = 1 (that is m = 1) (k0, k0 + 1, k0), k0 ≥ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (k0, k0 − 1, k0), k0 ≥ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (ii) if n ≥ 2 (that is m ≥ 3), (0, m, 0), (1, m + 1, 1), (1, m − 1, 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (it saves time noticing that E(a0, a1)∗ ≃ E(a1, a2) and E(b0, b1)∗ ≃ E(b1, b2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Finally, let (a0, a1, a2) = (b0, b1, b2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' We know, from Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='3, that S1 � E(a0, a1), E(b1, b2) � ≃ S1 � E(a1, a2), E(b0, b1) � ≃ V (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' This implies that there is, up to a scalar, a unique element u as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='6) such that hn(m)u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' This implies that zu = 0 and hence u ∈ S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' □ We are now in a position to prove the main theorems of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Let V1 = V (a0)⊕V (a1)⊕V (a2) and V2 = V (b0)⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='⊕V (bℓ) be the socle decomposition of two uniserial � sl(2) ⋉ hn � modules, where V1 is standard faithful and V2 is of type Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Then soc(V1 ⊗ V2) = S0 ⊕ S1 where, as sl(2)-modules, S0 = soc(V1) ⊗ soc(V2) ≃ min{a0,b0} � k=0 V (a0 + b0 − 2k) and the following tables describe S1 as sl(2)-modules (recall that m = 2n−1): Case n = 1 (m = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' V1\\V2 Z(b0, ℓ) Z(bℓ, ℓ)∗ FU + a0 V (a0 + b0 + m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' V (b1 − a0), if b1 ≥ a0, ℓ ≥ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' FU − a0 a0 ≥ 1 V (a0 − b1), if a0 ≥ b1, ℓ ≥ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Case n > 1 (m ≥ 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' UNISERIAL REPRESENTATIONS OF THE LIE ALGEBRA sl(2) ⋉ hn 17 V1\\V2 Z(b0, ℓ) Z(bℓ, ℓ)∗ FU(a0,a0+m,a0) a0 = 0, 1 V (a0 + b0 + m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' V (b1 − a0), if a0 ≤ b1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' 0, if a0 > b1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' FU(1,m−1,1) V (m − 1), if b0 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' We know from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='5 that soc(V1 ⊗ V2) ≃ soc(V1) ⊗ soc(V2) ⊕ S1 (in particular we know that St(V1, V2) = 0 for all t ≥ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' The decomposition of soc(V1) ⊗ soc(V2) follows from the Clebsch-Gordan formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' We now describe S1 in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Let us consider the submodules U1 = V (a0) ⊕ V (a1) ⊂ V1, and U2 = V (b0) ⊕ V (b1) ⊂ V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' We know that U1 and U2 are non-faithful uniserial � sl(2) ⋉ hn � modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' From Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='5 item (iii) we know that S1(V1, V2) ≃ S1(U1, U2), If V1 is FU + a0 or FU(0,m,0) or FU(1,1+m,1) then U1 ≃ Z(a0, 1) and S1 is obtained from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' If V1 = FU − a0, then U1 = Z(a0 − 1, 1)∗ and S1 is also obtained from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Finally, if V1 = FU(1,m−1,1), then U1 = E(1, m − 1) and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='3 implies the remaining cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' □ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Let V1 = V (a0) ⊕ V (a1) ⊕ V (a2) and V2 = V (b0) ⊕ V (b1) ⊕ V (b2) be the socle decomposition of two standard faithful uniserial � sl(2) ⋉ hn � modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Then soc(V1 ⊗ V2) = S0 ⊕ S1 ⊕ S2 where S0 = soc(V1) ⊗ soc(V2) ≃ min{a0,b0} � k=0 V (a0 + b0 − 2k) and the following tables describe S1 and S2 as sl(2)-modules (m = 2n − 1): Case n = 1 (m = 1), structure of S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' V1\\V2 FU(b0,b0+1,b0) FU(b0,b0−1,b0) b0 ≥ 1 FU(a0,a0+1,a0) V (a0 + b0 + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' V (b0 − a1), if a1 ≤ b0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' FU(a0,a0−1,a0) a0 ≥ 1 V (a0 − b1), if b1 ≤ a0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' 18 LEANDRO CAGLIERO AND IV´AN G´OMEZ RIVERA Case n = 1 (m = 1), structure of S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' V1\\V2 FU(b0,b0+1,b0) FU(b0,b0−1,b0) b0 ≥ 1 FU(a0,a0+1,a0) V (0), if a0 = b0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' 0, if a0 ̸= b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' FU(a0,a0−1,a0) a0 ≥ 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' V (0), if a0 = b0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' 0, if a0 ̸= b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Case n > 1 (m ≥ 3), structure of S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' V1\\V2 FU(b0,b0+m,b0) b0 = 0, 1 FU(1,m−1,1) FU(a0,a0+m,a0) a0 = 0, 1 V (a0 + b0 + m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' V (m − 1), if a0 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' FU(1,m−1,1) V (m − 1), if b0 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' V (m − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Case n > 1 (m ≥ 3), structure of S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' V1\\V2 FU(b0,b0+m,b0) b0 = 0, 1 FU(1,m−1,1) FU(a0,a0+m,a0) a0 = 0, 1 V (0), if a0 = b0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' 0, if a0 ̸= b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' FU(1,m−1,1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' V (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' As in the previous theorem, we know from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='5 that soc(V1 ⊗ V2) ≃ soc(V1) ⊗ soc(V2) ⊕ S1 ⊕ S2 and the decomposition of soc(V1)⊗soc(V2) follows from the Clebsch-Gordan formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' We now describe S1 and S2 in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' First, we consider S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' We know from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='5 that S1(V1, V2) ≃ S1 � E(a0, a1), E(b0, b1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' If V1 = FU + a0 and V2 = FU + b0, we know from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='1 that S1(V1, V2) = S1(Z(a0, 1), Z(b0, 1)) = V (a0 + b0 + m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' If V1 = FU + a0 and V2 = FU − b0, we have S1(V1, V2) = � S1(Z(a0, 1), Z(b1, 1)∗), if m = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' S1(Z(a0, 1), E(1, 1 − m)), if m > 1 and b0 = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' UNISERIAL REPRESENTATIONS OF THE LIE ALGEBRA sl(2) ⋉ hn 19 and it follows from Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='3 that S1(V1, V2) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 V (b1 − a0) if m = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' V (m − 1), if m > 1, a0 = 0 and b0 = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' This completes the description of S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' The result for S2 follows from item (iv) in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Intertwining operators In this section we obtain, from Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='6 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='7, the space of in- tertwining operators between the uniserial representations of g = sl(2) ⋉ hn considered in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Recall that, for any pairs of g-modules U1 and U2, we know that Homg(U1, U2) ≃ (U ∗ 1 ⊗ U2)g = � (U ∗ 1 ⊗ U2)hn�sl(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' It follows that Homg(U1, U2) is isomorphic to soc(U ∗ 1 ⊗ U2)sl(2) (see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Thus, we must identify the cases in which St(U ∗ 1 , U2)sl(2) ̸= 0 for t = 0, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' So let V = V (a0) ⊕ V (a1) ⊕ V (a2) (a0 = a2) and W = V (b0) ⊕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' ⊕ V (bℓ) be the socle decomposition of two uniserial � sl(2) ⋉ hn � modules, where V is standard faithful and W is either of type Z or standard faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Recall that V ∗ ≃ V and that the socle decomposition of W ∗ is V (bℓ) ⊕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' ⊕ V (b0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Case Ssl(2) 0 ̸= 0, W of type Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' It follows from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='6 that S0(V ∗, W)sl(2) ̸= 0 if and only if a0 = b0 (and equal to a2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Also, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='6 implies that, in these cases, we have dim S0(V ∗, W)sl(2) = 1 and St(V ∗, W)sl(2) = 0, t = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Hence Homg(V, W) is 1-dimensional and it is described by the following arrow V = V (a0) ⊕ V (a1) ⊕ V (a0) ↓ W = V (b0) ⊕ · · · ⊕ V (bℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Similarly, from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='6 we know that S0(W ∗, V )sl(2) ̸= 0 if and only if a0 = bℓ and in these cases Homg(W, V ) is 1-dimensional and it is described by the following arrow W = V (b0) ⊕ · · · ⊕V (bℓ) ↓ V = V (a0) ⊕ V (a1) ⊕ V (a0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Case Ssl(2) 0 ̸= 0 or Ssl(2) 2 ̸= 0, W standard faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' It follows from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='7 that S0(V, W)sl(2) ̸= 0 if and only if a0 = b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Hence in what follows, a0 = a2 = b0 = b2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' In this case, S1(V, W)sl(2) = 0 and dim S0(V, W)sl(2) = 1 (note that if V ≃ W, that is a1 = b1, then Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='7 says that dim S2(V, W)sl(2) = 1, see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' The non-zero 20 LEANDRO CAGLIERO AND IV´AN G´OMEZ RIVERA g-morphism corresponding to the 1-dimensional space S0(V, W)sl(2) is de- scribed by the following arrow V = V (a0) ⊕ V (a1) ⊕ V (a0) ↓ W = V (b0) ⊕ V (b1) ⊕ V (b0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' This is clearly not an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' In the particular case when V ≃ W, the isomorphism is the g-morphism corresponding to the 1-dimensional space S2(V, W)sl(2) (see also Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Thus dim Homg(V, W) = � 1, if V ̸≃ W (that is a1 ̸= b1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' 2, if V ≃ W (that is a1 = b1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (Since here V and W are of the same type, we do not need to consider Homg(W, V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=') Case Ssl(2) 1 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' It follows from Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='6 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='7 that dim Ssl(2) 1 = 1 and in all these cases, St(V ∗, W)sl(2) = 0, t = 0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Hence Homg(V, W) (or Homg(W, V )) is 1-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' We describe all these cases below: Cases with W of type Z and Homg(V, W) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (i) n = 1, a0 = b1, V = FU − a0, W = Z(b0, ℓ), ℓ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' V = V (a0) ⊕ V (a0 − 1) ⊕ V (a0) ↓ ↓ W = V (b0) ⊕ V (b1) ⊕ · · · ⊕ V (bℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (ii) n = 1, a0 = b1, V = FU + a0, W = Z(bℓ, ℓ)∗, ℓ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' V = V (a0) ⊕ V (a0 + 1) ⊕ V (a0) ↓ ↓ W = V (b0) ⊕ V (b1) ⊕ · · · ⊕ V (bℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (iii) n > 1, a0 = 0, 1, V = FU + (a0,a0+m,a0), W = Z(b1, 1)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' V = V (a0) ⊕ V (a0 + m) ⊕ V (a0) ↓ ↓ W = V (b0) ⊕ V (b1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Cases with W of type Z and Homg(W, V ) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (i) n = 1, a0 = bℓ−1, V = FU + a0, W = Z(b0, ℓ), ℓ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' W = V (b0) ⊕ · · · ⊕ V (bℓ−1) ⊕ V (bℓ) ↓ ↓ V =V (a0) ⊕ V (a0 + 1) ⊕ V (a0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (ii) n = 1, a0 = bℓ−1, V = FU − a0, W = Z(bℓ, ℓ)∗, ℓ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' W = V (b0) ⊕ · · · ⊕ V (bℓ−1) ⊕ V (bℓ) ↓ ↓ V =V (a0) ⊕ V (a0 − 1) ⊕ V (a0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' UNISERIAL REPRESENTATIONS OF THE LIE ALGEBRA sl(2) ⋉ hn 21 (iii) n > 1, V = FU(a0,a0+m,a0), a0 = 0, 1, W = Z(a0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' W = V (a0) ⊕ V (a0 + m) ↓ ↓ V =V (a0) ⊕ V (a0 + m) ⊕ V (a0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Cases with W standard faithful and Homg(V, W) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (i) n = 1, a0 = b1, V = FU − a0, W = FU + b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' V = V (a0) ⊕ V (a0 − 1) ⊕ V (a0) ↓ ↓ W = V (b0) ⊕ V (b0 + 1) ⊕ V (b0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (ii) n = 1, a0 = b1, V = FU + a0, W = FU − b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' V = V (a0) ⊕ V (a0 + 1) ⊕ V (a0) ↓ ↓ W = V (b0) ⊕ V (b0 − 1) ⊕ V (b0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='3 Let µ be a possible highest weight in S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' We start with some general considerations and next we will work out each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' We know from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='5 that µ must be highest weight in both V (a) ⊗ V (d) and V (b) ⊗ V (c), that is (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='1) |a − d|, |b − c| ≤ µ ≤ a + d, b + c and µ ≡ a+d ≡ b+c mod 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' We also know that µ is indeed highest weight in S1 if and only if there is a linear combination u0 = q1va,d,µ 0 + q2vb,c,µ 0 , with q1, q2 ̸= 0 (see item (i) in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='5), that is annihilated by es for all s = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Indeed this implies that u0 is also annihilated by z and thus in S1 (see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' We now describe esva,d,µ 0 and esvb,c,µ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' On the one hand we have (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='4)) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='2) va,d,µ 0 = � i,j CG(a 2, a 2 − i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' d 2, d 2 − j | µ 2, µ 2 ) va i ⊗ vd j 22 LEANDRO CAGLIERO AND IV´AN G´OMEZ RIVERA and thus (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='2)) esva,d,µ 0 = � i,j CG(a 2, a 2 − i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' d 2, d 2 − j | µ 2 , µ 2) va i ⊗ esvd j = � i,j,k (−1)jCG(a 2, a 2 − i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' d 2, d 2 − j | µ 2, µ 2 ) × CG( c 2, c 2 − k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' d 2, − d 2 + j | m 2 , m 2 − s) va i ⊗ vc k = � i,j,k (−1)kCG(a 2, a 2 − i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' d 2, d 2 − k | µ 2 , µ 2) × CG( c 2, c 2 − j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' d 2, − d 2 + k | m 2 , m 2 − s) va i ⊗ vc j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='3) In this sum, if the coefficient of va i ⊗ vc j is not zero then we must have a 2 − i + d 2 − k = µ 2 , c 2 − j − d 2 + k = m 2 − s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='4) On the other hand, we have (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='4)) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='5) vb,c,µ 0 = � i,j CG( b 2, b 2 − i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' c 2, c 2 − j | µ 2 , µ 2 ) vb i ⊗ vc j and thus (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='2)) esvb,c,µ 0 = � i,j CG( b 2, b 2 − i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' c 2, c 2 − j | µ 2, µ 2) esvb i ⊗ vc j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' = � i,j,k (−1)iCG( b 2, b 2 − i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' c 2, c 2 − j | µ 2, µ 2 ) × CG(a 2, a 2 − k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' b 2, − b 2 + i | m 2 , m 2 − s) va k ⊗ vc j = � i,j,k (−1)kCG( b 2, b 2 − k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' c 2, c 2 − j | µ 2, µ 2) × CG(a 2, a 2 − i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' b 2, − b 2 + k | m 2 , m 2 − s) va i ⊗ vc j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='6) In this sum, if the coefficient of va i ⊗ vc j is not zero then we must have b 2 − k + c 2 − j = µ 2 , a 2 − i − b 2 + k = m 2 − s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='7) Either (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='4) or (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='7) imply (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='8) i + j = a + c − m − µ 2 + s, (recall that 0 ≤ i ≤ a and 0 ≤ j ≤ c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Now we consider all the cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (i) The case V2 = E(c, d) with c + d = m and 0 < a ≤ c: Here µ = d + a − 2p, 0 ≤ p ≤ min{a, d} UNISERIAL REPRESENTATIONS OF THE LIE ALGEBRA sl(2) ⋉ hn 23 and it follows from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='8) that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='9) 0 ≤ i + j = p − d + s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' The sum (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='3) is esva,d,µ 0 = � i,j,k (−1)kCG(a 2, a 2 − i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' d 2, d 2 − k | a+d 2 − p, a+d 2 − p) × CG(m−d 2 , m−d 2 − j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' d 2, − d 2 + k | m 2 , m 2 − s) va i ⊗ vc j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' In this sum, it follows from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='4) that k = p − i j = p − d + s − i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' The conditions k ≥ 0 and 0 ≤ j ≤ m − d imply p − m + s ≤ i ≤ min{p, p − d + s}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Therefore, we obtain (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='9) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='12)) esva,d,µ 0 = min{p,p−d+s} � i=max{0,p−m+s} (−1)p−iCG(a 2, a 2 − i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' d 2, d 2 − p + i | a+d 2 − p, a+d 2 − p) × CG(m−d 2 , m−d 2 − p + d − s + i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' d 2, − d 2 + p − i | m 2 , m 2 − s) va i ⊗ vc p−d+s−i = min{p,p−d+s} � i=max{0,p−m+s} (−1)p � (a + d − 2p + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (a − i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (d − p + i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (a − p)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (d − p)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (a + d − p + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (p − i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' × � (m − s)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (m − d)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' d!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (m − p − s + i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (p − i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (d − p + i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (p − d + s − i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' va i ⊗ vc p−d+s−i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Thus, esva,d,µ 0 = (−1)p � (m − d)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (a + d − 2p + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' d!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (m − s)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (a − p)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (d − p)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (a + d − p + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' wa,d,µ s with wa,d,µ s = min{p,p−d+s} � i=max{0,p−m+s} � (a − i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (p − i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='2 (m − p − s + i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (p − d + s − i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' va i ⊗vc p−d+s−i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' On the other hand, the sum (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='6) is esvb,c,µ 0 = � i,j,k (−1)kCG(m−a 2 , m−a 2 − k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' m−d 2 , m−d 2 − j | a+d 2 − p, a+d 2 − p) × CG(a 2, a 2 − i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' m−a 2 , − m−a 2 + k | m 2 , m 2 − s) va i ⊗ vc j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' In this sum, it follows from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='7) that j = p − d + s − i k = m − a − s + i 24 LEANDRO CAGLIERO AND IV´AN G´OMEZ RIVERA and the condition k ≥ 0 implies i ≥ a − m + s, and condition j ≥ 0 implies p − d + s ≥ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Therefore, we obtain (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='9) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='12)) esvb,c,µ 0 = min{a,p−d+s} � i=max{0,a−m+s} (−1)m−a−s+i × CG(m−a 2 , − m−a 2 + s − i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' m−d 2 , m−d 2 − p + d − s + i | a+d 2 − p, a+d 2 − p) × CG(a 2, a 2 − i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' m−a 2 , m−a 2 − s + i | m 2 , m 2 − s) va i ⊗ vc p−d+s−i = min{a,p−d+s} � i=max{0,a−m+s} � (a + d − 2p + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (p + m − a − d)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (s − i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (m − p − s + i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (d − p)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (a − p)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (m − a − s + i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (p − d + s − i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (m − p + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' × � s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (m − s)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' a!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (m − a)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (a − i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (m − a − s + i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (s − i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' va i ⊗ vc a−p+s−i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Thus esvb,c,µ 0 = � (m − s)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (a + d − 2p + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (p + m − a − d)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' a!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (m − a)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (d − p)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (a − p)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (m − p + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' wb,c,µ s where wb,c,µ s = min{a,p−d+s} � i=max{0,a−m+s} � (m − p − s + i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (m − a − s + i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='2 (p − d + s − i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (a − i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' va i ⊗vc p−d+s−i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Now, if a ≤ d and p = a, we have, for all 0 ≤ s ≤ m, wa,d,µ s = min{a,a−d+s} � i=max{0,a−m+s} � 1 i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (a − i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (m − a − s + i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (a − d + s − i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' va i ⊗ vc p−d+s−i = wb,c,µ s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' This shows that u0 = (−1)a √ d + 1 va,d,µ 0 − √ b + 1 vb,c,µ 0 is, indeed, a highest weight vector, of weight µ = d − a = b − c, in S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' If p < a then, for s = d, the sum defining wa,d,µ d has the index i running up to i = a while the sum defining wb,c,µ d has the index i running only up to i = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' In both cases, all the coefficients are non-zero, and thus {wa,d,µ 1 , wb,c,µ 1 } is linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' This shows that there is no possible µ in S1 and thus S1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' This completes the proof in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (ii) The case V2 = E(c, d) with d = c+m: Since a < m ≤ d and by equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='1) we have µ = b + c − 2p, 0 ≤ p ≤ min{c, b}, µ = a + d − 2p′, 0 ≤ p′ ≤ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' This implies p′ − p = a and hence p′ = a and p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' This yields µ = b + c = d − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' UNISERIAL REPRESENTATIONS OF THE LIE ALGEBRA sl(2) ⋉ hn 25 First we prove that, if c = 0, then S1(V1, V2) ≃ V (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' In this case, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='3) becomes esva,d,µ 0 = � i,k (−1)kCG(m−b 2 , m−b 2 − i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' m 2 , m 2 − k | b 2, b 2) × CG(0, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' m 2 , − m 2 + k | m 2 , m 2 − s) va i ⊗ v0 0 (the index j is 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' It follows from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='4) that k = m − s i = −b + s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Therefore, esva,d,µ 0 = 0 if s < b and, for s ≥ b we have (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='9) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='11)) esva,d,µ 0 = (−1)m−sCG(m−b 2 , − m−b 2 + m − s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' m 2 , − m 2 + s | b 2, b 2) × CG(0, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' m 2 , m 2 − s | m 2 , m 2 − s) va s−b ⊗ v0 0 = � (b + 1) (m − b)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (m + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (s − b)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' va s−b ⊗ v0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' On the other hand, since c = 0 and µ = b, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='6) becomes (see also (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='2) or (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='3)) esvb,c,µ 0 = esvb 0 ⊗ v0 0 = \uf8f1 \uf8f2 \uf8f3 CG(a 2, a 2 − (s − b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' b 2, − b 2 | m 2 , m 2 − s) va s−b ⊗ v0 0, if s ≥ b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' 0, if s < b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' = \uf8f1 \uf8f2 \uf8f3 � a!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (s−b)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' va s−b ⊗ v0 0, if s ≥ b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' 0, if s < b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' This shows that u0 = √ b + 1 va,d,µ 0 − √ m + 1 vc,d,µ 0 is, indeed, a highest weight vector of weight µ = b, in S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Now, suppose that c ̸= 0 and set s = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Recall that µ = b + c = d − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' When we consider the sum (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='3), it follows from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='4) that j = k = a − i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' 26 LEANDRO CAGLIERO AND IV´AN G´OMEZ RIVERA The condition 0 ≤ j ≤ c implies a − c ≤ i ≤ a and hence (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='3) becomes (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='11)) emva,d,µ 0 = a � i=max{0,a−c} (−1)a−i CG(a 2, a 2 − i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' d 2, d 2 − (a − i) | d−a 2 , d−a 2 ) × CG( c 2, c 2 − (a − i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' d 2, − d 2 + a − i | m 2 , − m 2 ) va i ⊗ vc a−i = a � i=max{0,a−c} (−1)i+d−a � (c + b + 1) (m − b)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (c + b + i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (c + m + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' × � (m + 1) c!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (c + b + i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (c + m + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (c − m + b + i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' va i ⊗ vc a−i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' On the other hand, when we consider the sum (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='6), it follows from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='7) that j = a − i = −k and the condition k ≥ 0 implies that k = j = 0 and i = a = m − b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Thus, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='6) is emvb,c,µ 0 = CG( b 2, b 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' c 2, c 2 | c+b 2 , c+b 2 ) CG(m−b 2 , − m−b 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' b 2, − b 2 | m 2 , − m 2 ) va a ⊗ vc 0 = va a ⊗ vc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Since c ̸= 0, the sum in emva,d,µ 0 has at least two non-zero terms, while the sum in emvc,d,µ 0 has a single non-zero term, and thus {emva,d,µ 0 , emvc,b,µ 0 } is linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' This completes the proof in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (iii) The case V2 = E(c, d) with c = d + m: Since b < m ≤ c, it follows from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='1) that µ = b + c − 2p, 0 ≤ p ≤ b µ = a + d − 2p′, 0 ≤ p′ ≤ min{a, d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' This implies p − p′ = b and hence the only option is p = b, p′ = 0 and this yields µ = a + d = c − b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' We compute now esva,d,µ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' It follows from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='8) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='4) that k = −i j = s − i, and since k ≥ 0, we have k = i = 0 and j = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Therefore, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='3) becomes (see also (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='9) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='10)) esva,d,µ 0 = CG(a 2, a 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' d 2, d 2 | a+d 2 , a+d 2 ) CG(d+m 2 , d+m 2 − s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' d 2, − d 2 | m 2 , m 2 − s) va 0 ⊗ vc s = � (d+m−s)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (m+1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (d+m+1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (m−s)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' va 0 ⊗ vc s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' As always, the reader should check that all the numbers under the factorial sign are non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' UNISERIAL REPRESENTATIONS OF THE LIE ALGEBRA sl(2) ⋉ hn 27 We compute now esvb,c,µ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' It follows from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='8) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='7) that j = s − i k = m − a − s + i and conditions k ≥ 0 and j ≥ 0 imply s ≥ i ≥ s − (m − a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Thus, it follows from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='6), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='11) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='9) that esvb,c,µ 0 = min{a,s} � i=max{0,s−(m−a)} CG(m−a 2 , − m−a 2 + s − i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' d+m 2 , d+m 2 − (s − i) | a+d 2 , a+d 2 ) × CG(a 2, a 2 − i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' m−a 2 , m−a 2 − s + i | m 2 , m 2 − s) va i ⊗ vc s−i = min{a,s} � i=max{0,s−(m−a)} (−1)s−i� (d+m−s+i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (m−a)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (a+d+1) (d+m+1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (m−a−s+i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' × � a!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (m−a)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (m−s)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (s−i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (a−i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' (m−a−s+i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' va i ⊗ vc s−i Again, note that all the numbers under the factorial sign are non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' For s = m − a = b, the sum giving esvb,c,µ 0 has the index i running from i = 0 to i = min{a, b} ̸= 0, while the sum giving esva,d,µ 0 has the index i running only up to i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' In both cases, all the coefficients are non-zero, and thus {esvb,c,µ 0 , esva,d,µ 0 } is linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' This shows that there is no possible µ in S1 and thus S1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' This completes this case and the proof of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' References [1] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Assem, D.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='unc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='ar FaMAF-CIEM (CONICET), Universidad Nacional de C´ordoba, Medina Al- lende s/n, Ciudad Universitaria, 5000 C´ordoba, Rep´ublica Argentina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content=' Email address: ivan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='gomez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFIT4oBgHgl3EQf3yva/content/2301.11383v1.pdf'} +page_content='rivera@mi.' metadata={'source': 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b/29E1T4oBgHgl3EQfSAP8/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..f3a2f36094a4a133153839fc115b36a175fa75e0 --- /dev/null +++ b/29E1T4oBgHgl3EQfSAP8/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:9e6e19c89e51b5dd5711513cd5121be93a5cf304a13641912fd5b6e853eadcce +size 5111853 diff --git a/2NA0T4oBgHgl3EQfM_8t/content/tmp_files/2301.02139v1.pdf.txt b/2NA0T4oBgHgl3EQfM_8t/content/tmp_files/2301.02139v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..cbbd030efab142037758553d5533618221e03ad2 --- /dev/null +++ b/2NA0T4oBgHgl3EQfM_8t/content/tmp_files/2301.02139v1.pdf.txt @@ -0,0 +1,2510 @@ +arXiv:2301.02139v1 [math.QA] 5 Jan 2023 +COIDEAL SUBALGEBRAS OF POINTED AND CONNECTED HOPF ALGEBRAS +G.-S. ZHOU +Abstract. Let H be a pointed Hopf algebra with abelian coradical. Let A ⊇ B be left (or right) coideal +subalgebras of H that contain the coradical of H. We show that A has a PBW basis over B, provided that +H satisfies certain mild conditions. In the case that H is a connected graded Hopf algebra of characteristic +zero and A and B are both homogeneous of finite Gelfand-Kirillov dimension, we show that A is a graded +iterated Ore extension of B. These results turn out to be conceptual consequences of a structure theorem +for each pair S ⊇ T of homogenoeus coideal subalgebras of a connected graded braided bialgebra R +with braiding satisfying certain mild conditions. The structure theorem claims the existence of a well- +behaved PBW basis of S over T. The approach to the structure theorem is constructive by means of a +combinatorial method based on Lyndon words and braided commutators, which is originally developed +by V. K. Kharchenko [19] for primitively generated braided Hopf algebras of diagonal type. Since in our +context we don’t priorilly assume R to be primitively generated, new methods and ideas are introduced to +handle the corresponding difficulties, among others. +Contents +Introduction +2 +Notation and conventions +5 +1. +Preliminaries +5 +1.1. +PBW generators +6 +1.2. +Braided structures +7 +1.3. +Lyndon words +9 +2. +Braided calculus on free algebras +11 +3. +Bounded comultiplications on free algebras +17 +4. +Coideal subalgebras of connected graded braided bialgebras +21 +5. +Coideal subalgebras of pointed Hopf algebras +25 +6. +Coideal subalgebras of connected Hopf algebras +28 +7. +Proof of Proposition 3.4 +31 +8. +Proof of Proposition 4.7 +35 +Acknowledgments +37 +Appendix A. +Lyndon ideals of free algebras +38 +References +43 +2010 Mathematics Subject Classification. 16Txx, 68R15, 16P90, 16E65, 16S15, 16W50. +Key words and phrases. pointed Hopf algebra, connected Hopf algebra, braided bialgebra, coideal subalgebra, Lyndon word. +1 + +2 +G.-S. ZHOU +Introduction +The (one-sided) coideal subalgebras of a Hopf algebra H have been an important research focus +since the classical work on the commutative case in the last century [11]. Recall that a left (resp. right) +coideal subalgebra of H is a subalgebra A of H with ∆H(A) ⊆ H ⊗ A (resp. ∆H(A) ⊆ A ⊗ H). They are +conceptually the right context to define and understand quantum homogeneous spaces. Many studies of +coideal subalgebras have concentrated on this aspect. Among others, freeness and faithfully flatness of +a Hopf algebra over its coideal subalgebras are investegated [43, 32, 34, 41]; fundamental homological +properties and invariants of coideal subalgebras are characterized [23, 29]. Nevertheless, a big reason +that coideal subalgebras became more and more important is that many Hopf algebras, prominantly the +quantized enveloping algebras Uq(g), do not have “enough” Hopf subalgebras. For example, coideal +subalgebras are employed to develop a Galois theory for Hopf algebra actions [13, 47]. Particularly +noteworthy is the theory of quantum symmetric pairs originally developed by G. Letzter as coideal +subalgebras of quantized enveloping algebras [26, 27]. This theory has been evolving rapidly over the +past decade, led to a flurry of work aiming to extend many quantum group related constructions to the +setting of quantum symmetric pairs, see the recent survey by W. Wang [46] and the references there. +In this paper, we consider the coideal subalgebras of pointed Hopf algebras with abelian coradical. +This class of Hopf algebras includes many important examples, and under some finiteness conditions, +the general structure of them have been explored in deep via the lifting method introduced by N. +Andruskiewitsch and H.-J. Schneider in [2]. This paper will focus on the construction of PBW bases +for the coideal subalgebras. A well-behaved PBW basis or even the existence of a PBW basis is +manifestly useful in the understanding of an algebra. There are many publications on the construction +of PBW bases for Hopf algebras, notably for quantized enveloping algebras [31, 40, 25]. By means of a +remarkable combinatorial method based on Lyndon words, V. K. Kharchenko has managed to construct +a PBW basis for each character Hopf algebra [19] and later for each of their coideal subalgebras that +containing the coradical [20]. Note that character Hopf algebras are all pointed with abelian coradical. +Kharchenko’s PBW basis is one of the keystones in the study of Nichols algebras of diagonal type. +There is a complete classification of right coideal subalgebras of the Borel part of Uq(g) containing the +coradical in the case that q is not a root of unity [14]. When g is of type An, Bn and Gn, it has been +already obtained in [22], [21] and [38] respectively by similar methods in a parallel way. Kharchenko’s +PBW basis also plays a significant role in all of these works. Following the idea of Kharchenko, we +prove the following general statement on the PBW bases of the coideal subalgebras. +Theorem A (Theorem 5.1). Let H be a pointed Hopf algebra with G = G(H) abelian. Assume that +one of the following two conditions hold: (1) H is locally finite as a kG-module under the adjoint +action of kG on H, and the base field k is algebraically closed; (2) H is generated over kG by a set of +semi-invariants of H. Let A ⊇ B be left (resp. right) coideal subalgebras of H that contain G. Then +there exists a family {zξ}ξ∈Ξ of elements in A and a map h : Ξ → N∞ := N∪{∞} such that ({zξ}ξ∈Ξ, h, <) +is a system of PBW generators of A over B for each total order < on Ξ. + +COIDEAL SUBALGEBRAS OF POINTED AND CONNECTED HOPF ALGEBRAS +3 +Here, G = G(H) is the group of group-like elements of H, and by a semi-invariant of H we mean +an element x ∈ H such that kx is stable under the adjoint action of kG on H. For the notion of systems +of PBW generators of an algebra over its subalgebras, see Definition 1.1. The setting of condition (1) +covers the case that G is finite and k is algebraically closed; and the setting of condition (2) includes +character Hopf algebras, which in turn covers quantized enveloping algebras, all their generalizations, +bosonizations of Nichols algebras of diagonal type, and so on. Note that the setting of condition (1) +is more general than that of condition (2) when k is algebraically closed. Theorem A generalizes and +strengthens the main results of [20] in several aspects. First of all, we greatly expand the class of Hopf +algebras; secondly, we have considerably more freedom on the pairs A ⊇ B, while in loc. cit. either +A = H or B = kG; finally and most importantly, we get rid of the ad-invariant condition on coideal +subalgebras, which is assumed in loc. cit. and seems to be very restrictive. +A connected Hopf algebra is a Hopf algebra with coradical of dimension one. Clearly, they are +pointed Hopf algebras with abelian coradical. Though the condition of being connected is quite restric- +tive, it has rich examples from diverse fields of mathematics, such as universal enveloping algebras of +Lie algebras, the coordinate rings of unipotent algebraic groups, the Hopf algebras of symmmectric +functions, of quasi-symmetric functions and of permutations, and so on. Recently, connected Hopf al- +gebras of finite Gelfand-Kirillov dimension (GK dimension, for short) has been the subject of a series +of papers. Those of GK dimension up to 4 were completely classified, when the base field is alge- +braically closed of characteristic zero [51, 45]; the subclass of iterated Hopf Ore extensions has been +characterized [7, 9, 28, 50]; some fundamental properties of themself are extended to their coideal sub- +algebras [5]; and an example of connected graded Hopf algebra of GK dimension 5 is founded which is +neither commutative nor cocommutative [6]. Finite-dimensional connected Hopf algebras in positive +characteristic also have been studied [12, 35]. As a motivation of this work, the author and his col- +labrators have constructed a well-behaved PBW basis for any connected graded Hopf algebra over any +of its homogeneous Hopf subalgebras when the base field is of characteristic zero; and consequently +it turns out that those of finite GK dimension are iterated Hopf Ore extensions of their homogeneous +Hopf subalgebras [28, 50]. In this paper, we extend the PBW basis to the setting of coideal subalgebras +(Theorem 6.1), and prove the following result in the viewpoint of Ore extensions. +Theorem B (Theorem 6.4). Assume that k is of characteristic 0. Let H be a connected Γ-graded Hopf +algebra, where Γ is a nontrivial free abelian monoid. Let A ⊇ B be homogeneous left (resp. right) +coideal subalgebras of H, which are of finite GK dimension m and n respectively. Then there is a +sequence B = K0 ⊂ K1 ⊂ · · · ⊂ Km−n = A of homogeneous left (resp. right) coideal subalgebras of H +such that each Ki is a graded Ore extension of Ki−1 of derivation type. Moreover, if A and B are Hopf +subalgebras of H then Ki can be chosen to be Hopf subalgebras of H. +Here, an Ore extension S = R[x; φ, δ] is called of derivation type if φ = idR. In addition to Theorem +B, we also generalize some of the main results of [5, 51], either from connected Hopf algebras to +their coideal subalgebras or from algebraically closed fields of characteristic zero to arbitrary fields of +characteristic zero. For example, coideal subalgebras of connected Hopf algebras of characteristic zero + +4 +G.-S. ZHOU +are shown to be deformations of polynomial algebras (Theorem 6.5). Particularly noteworthy is that +our argument is purely algebraic, while in [5, 51] some geometric facts are employed. +The contexts of the above two theorems seem to be of different nature. However, we manage to +handle them in a same front by taking adavantage of the theory of braided Hopf algebras. Obviously, +connected graded Hopf algebras may be considerd to be braided by the flipping map. For a pointed +Hopf algebra H with coradical K, it is well-known that the associated graded Hopf algebra gr(H) with +respect to the coradical filtration of H is isomorphic to the bosonization (or the Radford biproduct) of +R := gr(H)co K and K [37]. Note that R is a connected graded braided Hopf algebra, and the structure +as well as the coideal subalgebras of H are closely related to that of R. It turns out that Theorem A and +Theorem B are conceputal consequences of the following result. +Theorem C (Theorem 4.3). Let R = (R, τ) be a connected Γ-graded braided bialgebra, where Γ is +a nontrivial free abelian monoid. Assume that τ = τχ for some bicharacter χ of Γ. Let A ⊇ B +be homogeneous left (resp. right) coideal subalgebras of R. Then there exists a family {zξ}ξ∈Ξ of +homogeneous elements of A, a map h : Ξ → N∞ and a total order ⊳ on Ξ such that +(1) ({zξ}ξ∈Ξ, h, <) is a system of PBW generators of A over B for each total order < on Ξ. +(2) For each ξ ∈ Ξ, the subalgebra A⊴ξ of A generated by { zη | η ∈ Ξ, η ⊴ ξ } over B has +{ zr1 +ξ1 · · · zrm +ξm | m ≥ 0, ξ1 ⊳ · · · ⊳ ξm ⊴ ξ, ξi ∈ Ξ, ri < h(ξi), i = 1, . . ., m } +as a basis of left B-module as well as of right B-module. +(3) For each ξ ∈ Ξ, the subalgebra A⊴ξ is a left (resp. right) coideal of R, and it is a subcoalgebra +of R in the case that A, B are both subcoalgebras of R and χ2 = εΓ. +(4) For each ξ ∈ Ξ with h(ξ) < ∞, it follows that zh(ξ) +ξ +∈ � +δ⊳ξ A⊴δ. +(5) For all ξ, η ∈ Ξ with η ⊳ ξ, it follows that [zξ, zη]τ ∈ � +δ⊳ξ A⊴δ (resp. [zξ, zη]τ−1 ∈ � +δ⊳ξ A⊴δ). +(6) For each ξ ∈ Ξ with h(ξ) < ∞, the scalar aξ = χ(deg(zξ), deg(zξ)) is a root of unity. More +precisely, if k is of characteristic 0 then aξ is of order h(ξ), and in particular, aξ � 1; and if k is +of characteristic p > 0 then aξ is of order h(ξ)/ps for some integer s ≥ 0. +We refer to Example 1.10 and the paragraph before it for the notions and notation related to bichar- +acters of abelian monoids. Particularly, εΓ denotes the trivial bicharacter of Γ. +Now let us sketch the proof of Theorem C. The approach is constructive, following the ideas used +in [19, 44, 50, 28]. First we choose a set X of homogeneous generators for R such that Y1 := X ∩ B and +Y2 := X ∩ A generate B and A respectively; and then fix an appropriate well order < on X so that Y1 +and Y2 are both closed (see Definition 3.6), among others. Let k⟨X⟩ be the free algebra on X. Note that +there is a canonical braiding on k⟨X⟩ induced from that of R. It is also denoted by τ and makes k⟨X⟩ a +braided algebra. Denote by I the ideal of defining relations on X for R. Then we may identify R with +k⟨X⟩/I as graded algebras canonically. In addition, let NI be the set of I-irreducible Lyndon words on +X with respect to the graded lex order associated to <. By the standard Gr¨obner basis theory, we may +further define a height map hI : L → N∞ (see Definition 2.6), where L is the set of Lyndon words on +X. By the combinatorial properties of Lyndon words and the braided commutator of polynomials, one +may construct from X a new family of homogeneous generators (zγ)γ∈NI for R (Lemma 2.5). Next we + +COIDEAL SUBALGEBRAS OF POINTED AND CONNECTED HOPF ALGEBRAS +5 +employ the coalgebra structure of R. Since A and B are left (resp. right) coideals of R, it is easy to lift +∆R into a graded algebra homomorphism ∆ : k⟨X⟩ → k⟨X⟩ ⊗τ k⟨X⟩ that is right (resp. left) bounded +(see Definition 3.1). Generally, ∆(x) � 1 ⊗ x + x ⊗ 1 because x is not necessarily primitive in R; and +moreover ∆ is not necessarily coassociative in the sense that (∆ ⊗ id) ◦ ∆ = (id ⊗∆) ◦ ∆. Nevertheless, +as an immediate consequence of a technical result (Proposition 3.4), the one-sided boundedness of ∆ +is sufficient to deduce that this new family of generators satisfy some desirable conditions. Due to the +combinatorial feature of Lyndon words and the closedness of Y1 and Y2, we further show that (zγ)γ∈Ξ1 +and (zγ)γ∈Ξ2, where Ξi := NI ∩ ⟨Yi⟩, are well-behaved families of generators of B and A respectively +(Corollary 3.7). Combine Proposition 3.4 and Corollary 3.7, as well as some other ideas (particularly, +Proposition 2.7), we conclude that the family (zγ)γ∈Ξ, where Ξ := Ξ2\Ξ1, together with the restrictions +of the height map hI and the lexicographic order y, one has x >lex x2 but xy 0 for each x ∈ X. Then for words u, v ∈ ⟨X⟩, if l(u) ≤ l(v) and u lex wv for every factorization u = vw with v, w � 1. +(2) u >lex w for every factorization u = vw with v, w � 1. +(3) v >lex w for every factorization u = vw with v, w � 1. +Proof. (2) ⇒ (3) is clear, and by [19, Lemma 2] one has (1) ⇔ (2). In the following we assume (3) +and to see (2). Let u = vw with v, w � 1. Then there is an integer s ≥ 0 and a word w′ ∈ ⟨X⟩ such that +vsw′ = w and v is not a prefix of w′. It follows that v ≥lex vs+1 >lex w′ and therefore vw′ >lex w′. Now +we can conclude that u = vsvw′ >lex vsw′ = w as desired. +□ +Definition 1.17. A word u ∈ ⟨X⟩ is called Lyndon if u is nonempty and satisfies the equivalent condi- +tions listed in Proposition 1.16. The set of all Lyndon words on X is denoted by L = L(X). +Remark 1.18. In [18, 30], u lex v (which holds in priori when w2 � 1), then w1w2w3 is a Lyndon word. + +COIDEAL SUBALGEBRAS OF POINTED AND CONNECTED HOPF ALGEBRAS +11 +(L2) ([30, Proposition 5.1.3]) Let u be word of length ≥ 2. Then u is a Lyndon word if and only if +uL and uR are both Lyndon words and uL >lex uR. +(L3) ([30, Proposition 5.1.4]) Let u, v be Lyndon words. Then Sh(uv) = (u, v) if and only if either u +is a letter or u is of length ≥ 2 with uR ≤lex v. +(L4) ([30, Theorem 5.1.5]) Every word u can be written uniquely as a nondecreasing product of +Lyndon words (the Lyndon decomposition): +u = u1u2 · · ·ur, +ui ∈ L, u1 ≤lex u2 ≤lex · · · ≤lex ur. +The words ui ∈ L appearing in the decomposition are called the Lyndon atoms of u. +(L5) ([18, Lemma 4.5]) If a Lyndon word v is a factor of a word u then it is a factor of some Lyndon +atom of u. +(L6) ([19, Lemma 4]) Let u1 >lex u2 >lex u′ be nonempty words. If u1u2 and u′ are Lyndon words, +then u1u2u′ >lex u1u′ >lex u′ and u1u2u′ >lex u2u′ >lex u′. +(L7) ([19, Lemma 5]) Let u = u1 · · · um and v = v1 · · · vn be two nonempty words in Lyndon decom- +position. Then u lex w. +Lemma 1.21. Let u, v ∈ ⟨X⟩+ with v Lyndon. The following statements are equivalent: +(1) u ≺ vn for some n ≥ 1. +(2) u lex v. +(3) [[u]ρ, [k⟨X⟩⊴v]ρ]ρ ⊆ [k⟨X⟩⊴uv]ρ for each pair of Lyndon words u >lex v. +(4) [[u]ρ, [k⟨X⟩⊳v]ρ]ρ ⊆ [k⟨X⟩⊳uv]ρ for each pair of Lyndon words u >lex v. +Proof. Part (1) is [19, Lemma 7]; Part (2) is by [19, Lemma 6, Lemma 7]; Next we show Part (3). Let +w1, . . ., wr be a finite sequence of Lyndon words that ≤lex v. One has +[[u]ρ, [w1]ρ · · · [wr]ρ]ρ ∈ +r� +i=1 +k · +� +[w1]ρ · · · [wi−1]ρ · [[u]ρ, [wi]ρ]ρ · [wi+1]ρ · · · [wr]ρ +� +by the braided derivation equation. Note that wi ≤lex v glex · · · >glex ur. By +Lemma 2.3, the leading word of f is u1, which is impossible. +Now we show these residue classes span FγA. It suffices to show they span the residue classes of all +words of degree ≤ γ. We show this by induction on words with respect to the graded lex order. Clearly, +it is true for the empty word, which is the smallest element with respect to 1. Let +∆([uL]ρ) += +1 ⊗ [uL]ρ + [uL]ρ ⊗ 1 + +� +f ′ +i ⊗ f ′′ +i + hL +∆([uR]ρ) += +1 ⊗ [uR]ρ + [uR]ρ ⊗ 1 + +� +g′ +j ⊗ g′′ +j + hR, +where f ′ +i , f ′′ +i , g′ +j, g′′ +j are homogeneous in each variable with f ′ +i ⊗ f ′′ +i +∈ (k⟨X⟩⊳uL ++ +⊗ k⟨X⟩+)deg(uL) and +g′ +j ⊗ g′′ +j ∈ k⟨X⟩⊳uR ++ +⊗ k⟨X⟩+)deg(uR), hL ∈ (k⟨X⟩+ ⊗ k⟨X⟩+) 1 follows readily from the decomposition ∆([un]ρ) = ∆([u]ρ)∆([un−1]ρ) and the fact that +[u]ρ[k⟨X⟩⊳u]ρ ⊆ [k⟨X⟩⊳u]ρ + [k⟨X⟩⊳u]ρ[u]ρ, which is again by Lemma 2.4 (4). +Now assume t > 0. Write w = w1w2 with w1 the first Lyndon atom of w. The desired formula then +follows readily from the decomposition ∆([wun]ρ) = ∆([w1]ρ)∆([w2un]ρ). +□ + +COIDEAL SUBALGEBRAS OF POINTED AND CONNECTED HOPF ALGEBRAS +19 +The next result is the keystone of this paper. For an ideal I of k⟨X⟩, a word w ∈ ⟨X⟩ and an index +γ ∈ Γ, we write ⟨X|I⟩⊳w +γ +:= DI ∩ ⟨X⟩⊳w +γ +and ⟨X|I⟩⊴w +γ +:= DI ∩ ⟨X⟩⊴w +γ . +Proposition 3.4. Let τ be an X-diagonal braiding on kX. Let I be an ideal of k⟨X⟩ and R := k⟨X⟩/I. +Let ∆ be a left (resp. right) bounded τ-comultiplication on k⟨X⟩ such that ∆(I) ⊆ I ⊗ k⟨X⟩ + k⟨X⟩ ⊗ I. +Let ρ = τ−1 (resp. ρ = τ). Then +(1) Every I-restricted word on X is I-irreducible, that is CI = DI. +(2) For each Lyndon word u on X of finite height n ≥ 1, +[u]n +ρ ∈ [k⟨X|I⟩⊳u +deg(un)]ρ + k⟨X⟩ 0 then τu,u is of order hI(u)/ps for some integer s ≥ 0. +Proof. To make the reading more fluent, the proof will be addressed in Section 7. +□ +Remark 3.5. In [50] and [19], there are similar results as above in the special cases that τ is the flipping +map and that k⟨X⟩ is length graded, τ is an X-diagonal braiding and ∆ = ∆s, respectively. The above +proposition generalizes and strengthens these works in several aspects. +Definition 3.6. A subset Y of X is called closed if y < x for each y ∈ Y and each x ∈ X\Y. +Corollary 3.7. Assume the notations and conditions of Proposition 3.4. Assume in addition that I is +homogeneous. Let Y be a closed subset of X, J := I ∩ k⟨Y⟩ and A := k⟨Y⟩/J. Then +(1) hJ(u) = hI(u) for every Lyndon word u on Y. +(2) NJ = NI ∩ ⟨Y⟩, CJ = DJ = DI ∩ ⟨Y⟩ and OJ = OI ∩ ⟨Y⟩. +(3) For each Lyndon word u on Y of finite height n ≥ 1 (with respect to J), +[u]n +ρ ∈ [k⟨Y|J⟩⊳u +deg(un)]ρ + J. +(4) For each pair of J-irreducible Lyndon words u, v ∈ NJ with v 0 then aξ is of order h(ξ)/ps for some integer s ≥ 0. + +22 +G.-S. ZHOU +Proof. One may choose homogeneous subspaces U1 ⊆ B+, U2 ⊆ A+ and U3 ⊆ R+ such that U1 +generates B, U2 ∩ B = 0 and U1 + U2 generates A, U3 ∩ A = 0 and V := U1 + U2 + U3 generates +R. Choose a homogeneous basis Xi of Ui for i = 1, 2, 3. Let X = X1 ∪ X2 ∪ X3. The braiding on +kX = V (denoted by τV) is clearly X-diagonal. Further, equip X with a well order < as follows. First +fix an admissible well order <Γ on Γ and a well order 0 then aξ is of order h(ξ)/ps for some integer s ≥ 0. +Proof. Let V ⊆ R+ be a homogeneous braided subspace of R which is of diagonal type and generates +R. By Lemma 4.4, one may fix a homogeneous basis X of V so that the braiding τV on V is X-diagonal. +Further, equip X with a well order < as follows. First fix an admissible well order <Γ on Γ and a well +order