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1dE4T4oBgHgl3EQfzQ2-/content/tmp_files/2301.05273v1.pdf.txt

@@ -0,0 +1,1509 @@
+Improving the speed of variational quantum algorithms
+for quantum error correction
+Fabio Zoratti
+Scuola Normale Superiore, I-56126 Pisa, Italy
+Giacomo De Palma
+Department of Mathematics, University of Bologna, 40126 Bologna, Italy
+Vittorio Giovannetti
+NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, I-56126 Pisa, Italy
+We consider the problem of devising a suitable quantum error correction procedure for a generic
+quantum noise acting on a quantum circuit. In general, there is no analytic universal procedure
+to obtain the encoding and correction unitary gates, and the problem is even harder if the noise
+is unknown and has to be reconstructed. The existing procedures rely on variational quantum
+algorithms and are very difficult to train since the size of the gradient of the cost function decays
+exponentially with the number of qubits. We address this problem using a cost function based on
+the quantum Wasserstein distance of order 1. Our results show that such cost function significantly
+increases both the probability of a successful training and the fidelity of the recovered state.
+I.
+INTRODUCTION
+Performing reliable computations on physical imperfect
+hardware is something that has become usual nowadays,
+given the current state of classical computers, which can
+produce perfect results without any software-side miti-
+gation of the imperfections of the physical media where
+the computation happens. Error correction is based on
+the fact that these machines perform automatically, on
+the hardware side, procedures that allow errors to hap-
+pen and to be fixed without any intervention of the end
+user. This kind of setting is even more crucial in a quan-
+tum scenario where the current noisy intermediate-scale
+quantum computers (NISQ) have a much larger error
+rate than their classical counterparts [1]. Performing re-
+liable computations with a trustworthy error correction
+procedure has direct implications not only in quantum
+computation [2, 3], but also in quantum key distribution
+and cryptography [4–8].
+In the typical Quantum Error Correction (QEC)
+scheme, the quantum state that has to be protected is
+stored in a subspace of a larger Hilbert space, using an
+encoding procedure. Stabilizer codes [9], which are within
+the best analytical results in this field, are not universal
+because they are tailored for a generic noise acting on a
+small but unknown subset of qubits. Several attempts
+have already been made to create a numerical optimiza-
+tion procedure to find an error correction code for specific
+noise models [10–13], but these studies are not universal
+because they rely heavily on the type of noise on the
+specific quantum circuit and this is a problem because
+real quantum devices are not characterized by a single
+kind of quantum noise. Some attempts have been made
+to characterize the noise of the current and near-term
+devices [14, 15], but these methods will become very diffi-
+cult to implement soon because classical computers are
+not able to simulate efficiently quantum circuits when
+the number of qubit increases. Near-term devices with
+approximately 50 qubits may already be intractable to
+simulate for supercomputers [16].
+If we define a figure of merit of the quality of the state
+after the action of the noise and its corresponding cor-
+rection, the obvious choice for the kind of maximization
+algorithm is a Variational Quantum Algorithm [17]. These
+are hybrid algorithms that couple a quantum computer
+with a classical one. In this kind of algorithms, usually, a
+parametric quantum circuit is applied to some reference
+state, some measurements are performed on the system,
+and the outcomes are given to the classical computer to
+perform a minimization procedure of a given cost function.
+Some interesting examples of this class of algorithms are
+the variational quantum eigensolver [18] and the Quan-
+tum Approximate Optimization Algorithm (QAOA) [19].
+The optimization procedure in a variational quantum al-
+gorithm can be seen as the training phase in machine
+learning, for example, to train a neural network.
+Some variational quantum algorithms applied to quan-
+tum error correction are already known in literature [20].
+However, VQAs usually suffer from the phenomenon of
+barren plateaus [21, 22], namely the gradient of the cost
+function decays exponentially with respect to the number
+of qubits of the system, leading to an untrainable model.
+Some general results have already been found about this
+topic [22], which justifies the presence of barren plateaus
+when the cost function is a global function of the quantum
+circuit, namely the measurement of a highly non-local op-
+erator. For this reason, we compared the performance of
+an algorithm inspired by [20] using two different cost func-
+tions: the fidelity and an approximation of the quantum
+Wasserstein distance.
+The quantum Wasserstein distance is a generalization of
+the classical Wasserstein distance between probability dis-
+tributions [23–27]. Several quantum generalizations of the
+Wasserstein distance have been proposed [28–62]. This
+arXiv:2301.05273v1  [quant-ph]  12 Jan 2023
+
+2
+work is based on the quantum Wasserstein distance of or-
+der 1 (or quantum W1 distance) proposed in Refs. [63, 64],
+which is not unitarily invariant and recovers the Hamming
+distance [65] for the states of the computational basis. We
+expect this new distance to improve the barren plateau
+phenomenon and we can give an intuitive reason. If we
+use a unitarily invariant distance as the trace distance or
+the distances derived from the fidelity, all the states of the
+computational basis are equally orthogonal and thus have
+all maximum distance one with respect to the other. The
+quantum W1 distance instead measures how many qubits
+are different between the two states and this allows the
+gradient to be less flat in the regions that are not already
+very close to a local minimum. Indeed, the quantum W1
+distance has succesfully been employed as cost function
+of quantum Generative Adversarial Networks [64, 66–70]
+The manuscript is organized as follows: in Sec. II we
+present some basic notion on conventional QEC proce-
+dures which allow us to set the notation and the theo-
+retical background; in Sec. III we introduce our VQA
+discussing the different choices of cost functions that can
+be used in order to guide it; in Sec. IV we present our nu-
+merical results where comparing the performances of the
+VQA implemented with different types of cost functions.
+Conclusions are given in Sec. V.
+II.
+PRELIMINARIES ON QEC
+Let Q be a quantum register we wish to protect (at least
+in part) from the action of some external noise source.
+In a typical QEC scenario [71] this problem is addressed
+through the following three-step procedure:
+i) Before the action the noise, a unitary encoding gate
+ˆVQA is used to distribute the information originally
+contained in Q on the larger system QA. Here A
+is an auxiliary quantum register that is assumed to
+be intialized in a fiduciary quantum state, and that
+is affected by the same noise that tampers with Q;
+ii) After the action of the noise a measurement on QA
+is performed to reveal the nature of the latter and,
+based on the associated outcome, a unitary recovery
+operation is applied to the system. Equivalently this
+step can be described by introducing yet an extra
+quantum register B (also intialized on a fiduciary
+state but not affected by the noise) that is coupled
+with QA trough a recovering unitary transformation
+ˆWQAB which effectively mimics the measurement
+and the recovery operation;
+iii) The inverse of the gate ˆVQA is finally used on QA
+to refocus the recovered information in Q.
+Denoting with |ψ⟩Q the input state of Q, the corre-
+sponding output state of QA that emerges from the pro-
+cess at the end of the step iii) can be expressed as the
+density matrix
+ˆρ(V,W )
+QA
+(ψ) := trB
+�
+V†
+QA ◦ WQAB ◦ ΦQA
+(1)
+◦VQA
+�
+|ψ⟩Q⟨ψ| ⊗ |Ø⟩A⟨Ø| ⊗ |Ø⟩B⟨Ø|
+��
+:= V†
+QA ◦ Φ(R)
+QA ◦ ΦQA ◦ VQA
+�
+|ψ⟩Q⟨ψ| ⊗ |Ø⟩A⟨Ø|
+�
+where |Ø⟩X represents the fiduciary state of the X register,
+trB{· · · } is the partial trace over B, and given a unitary
+ˆUX on X we adopted the symbol UX(· · · ) := ˆUX · · · ˆU †
+X
+to denote its action as super-operator.
+In the above
+expressions ΦQA is the LCPT quantum channel [71]
+describing the noise on Q and A, while Φ(R)
+QA(· · · ) :=
+trB{WQAB(· · ·⊗|Ø⟩B⟨Ø|)} is the LCPT (recovery) quan-
+tum channel on QA originating from the interaction with
+B, that attempts to undo the action of ΦQA.
+An ideal QEC procedure capable to completely remove
+the noise from the system will make sure that ˆρ(V,W )
+QA
+(ψ)
+corresponds to |ψ⟩Q|Ø⟩A, irrespectively from the specific
+choice of |ψ⟩Q. A bona-fide figure of merit to character-
+ize the effectiveness of a generic QEC scheme is hence
+provided by the average input-output fidelity
+F(V, W) :=
+�
+dµψ Q⟨ψ|A⟨Ø|ˆρ(V,W )
+QA
+(ψ)|ψ⟩Q|Ø⟩A , (2)
+where dµψ is the uniform measure on the set of the in-
+put states of Q originated from the Haar measure on
+the associated unitary group [72] or from an exact or
+approximate unitary 2-design S [71, 73] that simulates
+the latter1. Notice that by expressing |ψ⟩Q = ˆUQ|Ø⟩Q,
+Eq. (2) can equivalently be casted in the more compact
+form
+F(V, W) = QA⟨Ø|ˆρ(V,W )
+QA
+|Ø⟩QA ,
+(3)
+with |Ø⟩QA := |Ø⟩Q ⊗ |Ø⟩A and where the state
+ˆρ(V,W )
+QA
+:= 1
+|S|
+�
+ˆUQ∈S
+U†
+Q ◦ V†
+QA ◦ Φ(R)
+QA ◦ ΦQA
+◦ VQA ◦ UQ
+�
+|Ø⟩QA⟨Ø|
+�
+,
+(4)
+now includes the average over all possible inputs. An
+ideal QEC procedure will enable one to get F(V, W) = 1.
+A natural benchmark for lowest admissible F(V, W) is
+represented instead by the value one would get if one
+decides not to perform corrections on the register that we
+1 We remind that a unitary 2-design is a probability distribution
+over the set of unitary operators which can duplicate properties of
+the probability distribution over the Haar measure for polynomials
+of degree 2 or less. When Q is a single qubit, a 2-design can
+be realized by an uniform sampling over a set S composed by
+only 6 elements ˆ1, ˆσ1, e±iπ/4ˆσ1, e±iπ/4ˆσ2 that maps its logical
+state |0⟩Q into the vectors |0⟩Q , |1⟩Q , (|0⟩Q ±i |1⟩Q)/
+√
+2, (|0⟩Q ∓
+|1⟩Q)/
+√
+2.
+
+3
+Yα1
+Yα7
+Xα2
+Xα8
+Xα9
+Yα5
+Xα6
+Xα10
+Xα3
+Xα4
+Q
+A
+Yα11 Xα12
+Yα13 Xα14 Xα15 Xα16
+Yα17 Xα18
+Y /2
+π
+Xθ′ ′ 
+k
+Xθ′ ′ ′ 
+k
+Zθ′ k
+=
+Xθk
+Y /2
+π
+B
+Q
+A
+Yβ1
+Yβ11
+Yβ21
+Yβ7
+Yβ17
+Yβ27
+Yβ31
+Yβ35
+Yβ33
+Yβ37
+Xβ2
+Xβ12
+Xβ22
+Xβ3
+Xβ13
+Xβ23
+Xβ5
+Xβ15
+Xβ25
+Xβ8
+Xβ18
+Xβ28
+Xβ32
+Xβ36
+Xβ34
+Xβ38
+Zβ4
+Zβ14
+Zβ24
+Zβ6
+Zβ16
+Zβ26
+Xβ10
+Xβ20
+Xβ30
+Zβ9
+Zβ19
+Zβ29
+=
+Zθk
+Y /2
+π
+Zθ′ ′ 
+k
+Zθ′ ′ ′ 
+k
+Zθ′ k
+Y /2
+π
+Figure 1.
+Parametric gates ˆVQA(⃗α) (green element) and
+ˆ
+WQAB(⃗β) (red element) used for case of quantum registers Q,
+A, and B with k = 1, n − k = 2, and r = 2 qubits respectively.
+Indicating with ˆσ1, ˆσ2, and ˆσ3 the Pauli operators, the Xθ, Yθ,
+and Zθ elements of the figure represent single qubit rotations
+e−iθˆσ1, e−iθˆσ2, and e−iθˆσ3 with the angles θ determined to
+the components of the vectors ⃗α, ⃗β, respectively. Vertical
+lines indicate instead quantum control operations which are
+activated when the control qubits (indicated by the full or
+empty circles) are in the logical state |1⟩ (full circle) or in |0⟩
+(empty circle). As shown on the inset, each one of those gates
+depend parametrically upon elements of the control vectors ⃗α
+and ⃗β through single qubit operations.
+compute by setting ˆVQA and ˆWQAB equal to the identity
+operators i.e.2
+F 0 := QA⟨Ø|ˆρ(11,11)
+QA
+|Ø⟩QA .
+(5)
+III.
+VARIATIONAL QUANTUM ALGORITHM
+While the enormous progress has been made in the
+study of QEC procedures, identifying efficient choices for
+the operations that leads to (non trivial) high values of
+F(V, W) for a specific noise model, is still a challenging
+open problem. A possible solution in this case is to employ
+variational quantum algorithms to run numerical searches.
+Our approach follows a training strategy inspired by the
+work of Johnson et al. [20]. Assuming hence Q, A, and
+B to be formed by collections of independent qubits (k
+for Q, n − k for A, and r for B), we introduce a manifold
+2 Equation (5) accounts for the noise effects both on Q and A. A
+more conservative estimation of F 0 can be obtained by focusing
+directly on the noise on Q alone, i.e. tracing out the A component
+of ˆρ(11,11)
+QA
+and studying its fidelity with |Ø⟩Q, i.e. F (strong)
+0
+:=
+Q⟨Ø|ˆρ(11,11)
+Q
+|Ø⟩Q ≥ F 0, with ˆρ(11,11)
+Q
+:= trA ˆρ(11,11)
+QA
+. Notice that
+for the noise model of Sec. III C the two are directly connected
+via the identity F 0 = F (strong)
+0
+− n−1
+n p(1 − |⟨0|ˆσ|0⟩|2).
+ï
+ï
+ï
+ï
+ï
+ï
+|0ï
+|0ï
+|0ï
+|0ï
+|0ï
+|0ï
+|0ï
+|0ï
+|0ï
+Q
+A
+B
+UQ(j)
+ï
+ï
+ï
+M
+M
+M
+M
+C(
+÷³ ,
+÷
+´ )
+M
+M
+U†
+Q(j)
+WQAB(
+÷
+´ )
+VQA(
+÷³ )
+V†
+QA(
+÷³ )
+j
+÷³ ,
+÷
+´
+n aˆK
+ic
+r t
+ovth
+er
+e w
+qu
+we
+= 1
+F
+cor
+NOISE
+Figure 2. (Color online) Sketch of the variational quantum
+algorithm: Q, A and B are quantum registers formed respec-
+tively by k, n−k and r qubits. The initial information we wish
+to protect is written in Q by the unitary gate ˆUQ(j) extracted
+from a 2-design set S; A and B are two auxiliary elements
+(containing respectively n − k and r qubits) that are used to
+implement the QEC procedure described by the parametric
+gates ˆVQA(⃗α), ˆ
+WQAB(⃗β), and ˆV †
+QA(⃗α) of Fig. 1. The patterned
+element in the central part of the scheme represents the noise
+on Q and A (no noise is assumed to be active on B). Lastly,
+the D-shaped measurements at the end of the circuit represent
+local measurements on QA whose outcomes over the entire
+collection of the possible inputs generated by the entire set S,
+are processed by a classical computer which, evaluating the
+cost function C(⃗α, ⃗β) defined in Section III B, decides how to
+update the values of the parameters ⃗α and ⃗β. Thick grey lines
+in the figure represent classical control lines.
+of transformations ˆVQA(⃗α), ˆWQAB(⃗β) parametrized by
+classical controls vectors ⃗α, ⃗β (see Fig. 1), and construct
+the quantum circuit of Fig. 2. The method then proceeds
+along the following stages:
+1. Having selected the values of ⃗α and ⃗β, the regis-
+ter Q is prepared into a collection of known quan-
+tum state {|ψ(1)⟩Q, · · · , |ψ(m)⟩Q} operating on the
+vector |Ø⟩Q = |0⟩⊗k through action of the con-
+trol gates ˆUQ(1), · · · , ˆUQ(m) (first cyan element of
+the figure) which define the 2-design S entering
+in Eq. (4). Each of such inputs is hence evolved
+via a circuit (pale-orange area of the figure) that
+emulates both the effect of the noise (patterned
+square of the figure, see Section III C and Fig. 4),
+and the transformations ˆVQA(⃗α), ˆWQAB(⃗β), and
+ˆV †
+QA(⃗α) that are meant to implement the steps ii)
+and iii) of the QEC procedure (green and red ele-
+ments of the figure). Notice that in the ideal case
+(i.e. if ˆVQA(⃗α) and ˆWQAB(⃗β) manage to completely
+suppress the noise) then in correspondence with
+
+4
+the input |ψ(j)⟩Q the registers QA should emerge
+in the state |ψ(j)⟩Q ⊗ |Ø⟩A := |ψ(j)⟩Q ⊗ |0⟩⊗n−k,
+which will be hence mapped into the final configu-
+ration |Ø⟩QA := |0⟩⊗n by the inverse ˆU †
+Q(j) of the
+state preparation gate (second cyan element of the
+figure).
+2. For each choice of the index j ∈ {1, · · · , m} a mea-
+surement on the system is performed at the end of
+the transformations described in the stage 1 and the
+resulting m collected outcomes used to compute a
+cost function C(⃗α, ⃗β) which evaluates the effective-
+ness of the adopted QEC strategy in leading large
+values of the average input-output fidelity. The spe-
+cific choice of the cost function is very important
+and is discussed in Section III A.
+3. A classical computer decides, given the results of
+the measurement, how to change the value of the
+parameters ⃗α and ⃗β to be used in the subsequent
+run in order to minimize the cost function C(⃗α, ⃗β).
+This is discussed in detail in Section III B.
+A.
+Cost function
+The natural choice for the cost function at the stage 2
+of our algorithm is provided by the expectation value of
+the self-adjoint operator
+ˆH(fid)
+QA := ˆ1QA − |Ø⟩QA ⟨Ø| ,
+(6)
+computed on the mean state of system QA which emerges
+at the output of the quantum circuit of Fig. 2, i.e. the
+quantity
+C(fid)(⃗α, ⃗β) := tr{ˆρ(V (⃗α),W (⃗β))
+QA
+ˆH(fid)
+QA } ,
+(7)
+where ˆρ(V (⃗α),W (⃗β))
+QA
+is the density matrix (4) evaluated
+for ˆVQA = ˆVQA(⃗α) and ˆWQAB = ˆWQAB(⃗β). This choice
+has two main advantages. First of all, the expectation
+value C(fid)(⃗α, ⃗β) can be evaluated by performing (sim-
+ple) local measurement on the qubits of Q and A (in-
+deed it can be computed by simply checking whether of
+not each one of them is in the logical state |0⟩). Most
+importantly, since by explicit evaluation one has that
+C(fid)(⃗α, ⃗β) = 1 − F(V (⃗α), W(⃗β)), it is clear that by us-
+ing (7) the algorithm will be forced to luck for values
+of ⃗α, ⃗β that yield higher average input-output fidelities.
+Despite all this, the use of C(fid)(⃗α, ⃗β) as a cost function,
+has a major drawback associated with the fact that the
+spectrum of the Hamiltonian ˆH(fid)
+QA exhibits maximum
+degeneracy with respect to space orthogonal to the target
+state |Ø⟩QA (see Fig. 3). Due do this fact a numerical
+search based on a training procedure that simply target
+the minimization of C(fid)(⃗α, ⃗β), has non trivial chances
+to get stuck somewhere in the large flat plateau associated
+with the eigenvalue 1 of ˆH(fid)
+QA without finding any good
+direction. in the large flat plateau A possible way to avoid
+this problem is to introduce new cost-functions Hamilto-
+nians which, while maintaining the target vector |Ø⟩QA
+as unique ground state and still being easy to compute,
+manage to remove the huge degeneracy of the excited
+part of the spectra of ˆH(fid)
+QA . Our choice is based on the
+quantum Wasserstein distance of order 1 (W1) introduced
+Ref. [63] which, even though it lacks some interesting
+properties that the fidelity has, is less likely to be affected
+by the barren plateaus phenomena [22]. A good estima-
+tion of the W1 distance that separate ˆρ(V (⃗α),W (⃗β))
+QA
+from
+the target state, is provided by the following quantity
+C(wass)(⃗α, ⃗β) := tr{ˆρ(V (⃗α),W (⃗β))
+QA
+ˆH(wass)
+QA
+} ,
+(8)
+ˆH(wass)
+QA
+:=
+n
+�
+j=1
+j ˆΠ(j)
+QA ,
+(9)
+where ˆΠ(j)
+QA represents the sub-space of the register QA
+in which we have j qubits in |1⟩ and the remaining one in
+|0⟩. Observe that ˆH(wass)
+QA
+is nothing but the sum of the
+number operators acting on the individual qubits of the
+register QA, (i.e. ˆH(wass)
+QA
+= �n
+ℓ=1 ˆπℓ with ˆπℓ the projector
+on the logical state |1⟩ of the ℓ-th qubit): accordingly,
+as C(fid)(⃗α, ⃗β), C(wass)(⃗α, ⃗β) can be computed from local
+measurement. What C(wass)(⃗α, ⃗β) does is to count the
+total number of logical ones present in the system. To
+understand why using (8) could in principle lead to a
+more efficient numerical search than the one obtained by
+using (7), notice that Eq. (6) can be equivalently written
+as ˆH(fid)
+QA =
+n
+�
+j=1
+ˆΠ(j)
+QA. A comparison with (9) reveals hence
+that indeed while both ˆH(fid)
+QA and ˆH(wass)
+QA
+admit |Ø⟩QA
+as unique ground state, the Wasserstein Hamiltonian
+removes large part of the degeneracy of the high energy
+spectrum of the fidelity Hamiltonian. Accordingly it is
+reasonable to expect that a numerical search that uses
+ˆH(wass)
+QA
+, has less chances to get trapped into regions of
+constant energy (barren plateau) than a search based on
+ˆH(fid)
+QA ,3.
+3 It goes without mentioning that alternative choices for the cost
+function Hamiltonians are also available. For instance one can
+use operators that also remove the residual degeneracies that
+affect ˆH(wass)
+QA
+– e.g. using the operator ˆH(full)
+QA
+= �n
+ℓ=1 wℓˆπℓ
+with ωℓ positive weights selected so that different allocation of |1⟩
+states inside the eigenspaces of ˆH(wass)
+QA
+get an assigned ordering.
+Our numerical analysis however seems to indicate that these
+refinement do not contribute significantly in improving numerical
+search of the algorithm.
+
+5
+eigenvalues of
+̂H(fid)
+QA
+eigenvalues of
+̂H(wass)
+QA
+N = 3
+N = 3
+N = 2
+N = 1
+N = 2
+N = 0
+1
+2
+3
+0
+N = 1
+N = 2
+N = 3
+N = 0
+0
+1
+Figure 3. Pictorial rendering of the spectra of the Hamiltonians
+ˆH(fid)
+QA
+(top panel) and ˆH(wass)
+QA
+(lower panel). While ˆH(fid)
+QA
+is characterized by a unique, flat plateau that includes all
+the excited state, ˆH(wass)
+QA
+partially removes the associated
+degeneracy assigning higher energy to subspaces that have
+higher number of qubits in the logical state |1⟩.
+B.
+Descent algorithm
+The algorithm that we used for this work is a gradient
+descent algorithm with momentum [74]. To overcome the
+numerical difficulties of using finite differences to estimate
+the gradients of the cost function C(⃗α, ⃗β), we exploit a
+variation of the parameter-shift rule introduced in [75]
+which reduces the problem to compute linear combina-
+tions of the function itself evaluated in different points
+that are not infinitesimally close. Specifically we observe
+that, irrespectively from the choice of the operator ˆHQA,
+the functional dependence of C(⃗α, ⃗β) upon the j-th com-
+ponent of the vector ⃗β is of the form
+C(⃗α, ⃗β) = f(βj) :=
+�
+k
+tr
+�ˆΩ(k)
+1 eiβj ˆσ ˆΩ(k)
+2 e−iβj ˆσ�
+,
+(10)
+with ˆΩ(k)
+1,2 being multi-qubits operators which do not de-
+pend upon βj, and with e−iβj ˆσ a single qubit rotation
+generated by an element ˆσ of the Pauli set. Therefore its
+gradient can be written as
+∂C(⃗α, ⃗β)
+∂βj
+= i
+�
+k
+tr
+�ˆΩ(k)
+1 eiβj ˆσ[ˆσ, ˆΩ(k)
+2 ]e−iβj ˆσ�
+= f(βj + π
+4 ) − f(βj − π
+4 ) ,
+(11)
+where in the last passage we used the identity
+i[ˆσ, ˆΩ(k)
+2 ] = ei π
+4 ˆσ ˆ
+Ω2
+(k)e−i π
+4 ˆσ − e−i π
+4 ˆσ ˆ
+Ω2
+(k)ei π
+4 ˆσ.
+(12)
+The gradient with respect the vector ⃗α can be computed
+similarly. In this case however we observe that, due to
+the fact that ˆρ(V (⃗α),W (⃗β))
+QA
+(ψ) depends upon the parame-
+ters ⃗α via ˆVQA(⃗α) and through its adjoint ˆV †
+QA(⃗α), the
+dependence of C(⃗α, ⃗β) upon the j-th component of ⃗α is
+slightly more complex. Indeed in this case we have
+C(⃗α, ⃗β) = g(αj, αj) ,
+(13)
+where g(α(1)
+j , α(2)
+j ) is the function
+g(α(1)
+j , α(2)
+j ) :=
+�
+k
+tr
+�ˆΩ(k)
+1 eiα(1)
+j
+ˆσ ˆΩ(k)
+2 e−iα(1)
+j
+ˆσ
+(14)
+׈Ω(k)
+3 eiα(2)
+j
+ˆσ ˆΩ(k)
+4 e−iα(2)
+j
+ˆσ�
+,
+with ˆΩ(k)
+1,2,3,4 representing multi-qubits operators which
+do not depend neither upon α(1)
+j
+nor α(2)
+j . It is important
+to stress that g(α(1)
+j , α(2)
+j ) can be computed using the
+same circuit of Fig. 2, by simply replacing the phases αj
+of ˆVQA(⃗α) and ˆV †
+QA(⃗α) with α(1)
+j
+and α(2)
+j
+respectively.
+Notice finally that exploiting the identity Eq. (12) we can
+write
+∂C(⃗α, ⃗β)
+∂αj
+=
+∂g(α(1)
+j , αj)
+∂α(1)
+j
+�����
+α(1)
+j
+=αj
++
+∂g(αj, α(2)
+j )
+∂α(2)
+j
+�����
+α(2)
+j
+=αj
+(15)
+= g(αj + π
+4 , αj) − g(αj − π
+4 , αj)
++ g(αj, αj + π
+4 ) − g(αj, αj − π
+4 ) ,
+which shows that computing the gradient of C(⃗α, ⃗β) with
+respect to αj simply accounts to evaluate the circuit
+that express g(α(1)
+j , α(2)
+j ) for four distinct values of the
+parameters.
+C.
+Noise model
+The scheme presented so far can in principle be applied
+to arbitrary classes of noises. In our research however
+we focused on a specific model that has been extensively
+studied in the literature producing explicit examples of
+efficient QEC solutions which can be used as a theoretical
+benchmark for our variational search. Specifically we
+assume Q and A to be respectively a single qubit register
+(k = 1) and a two qubit register (n = 3), globally affected
+by a given species of single-qubit noise [76, 77]. These
+transformations can be represented in terms of a LCPT
+map of the form
+ΦQA(· · · ) =
+n
+�
+ℓ=0
+ˆK(ℓ)
+QA · · · ˆK(ℓ)†
+QA ,
+(16)
+
+6
+Q
+NOISE
+io
+d l
+est
+, t
+noise,
+ep
+tsmth
+A
+=
+3
+K(3)
+QA
+Figure 4. Circuital implementation of the noise element of
+Fig. 2: here ˆK(ℓ)
+QA are weighted unitaries of Eq. (17).
+Xπ/2
+Xπ/4
+Q
+A
+Xπ/2
+B
+Q
+A
+Xπ/2
+Xπ/4
+Xπ/4
+Zπ/2
+Xπ/4
+Xπ/4
+Xπ/4
+Xπ/4
+Xπ/2
+Xπ/2
+Xπ/2
+Zπ/2
+Zπ/2
+Figure 5. Circuital implementations of the ideal transforma-
+tions ˆVQA(⃗α) (left) and ˆ
+WQAB(⃗β) (right) which allow for exact
+noise suppression of a single-qubit bit-flip noise model [i.e. (16)
+with ˆσ(ℓ) = ˆσ(ℓ)
+1 ] using a quantum register B with r = 2 qubit.
+Here H represent Hadamard gates, while the control-element
+are C-NOT gates.
+with Kraus operators [71]
+ˆK(0)
+QA :=
+�
+1 − p ˆ1QA ,
+ˆK(ℓ)
+QA :=
+� p
+n ˆσ(ℓ) ,
+(17)
+where for ℓ ∈ {1, · · · , n}, ˆσ(ℓ) is the Pauli operator acting
+on the ℓ-th qubit of QA which defines the noise species
+we have selected.
+For instance in the case we choose
+to describe phase-flip noise then ˆσ(ℓ) = ˆσ(ℓ)
+3 , while for
+describing bit-flip we have ˆσ(ℓ) = ˆσ(ℓ)
+1 . Explicit exam-
+ples of ˆVQA, ˆWQAB which allow for exact suppression
+of the noise (F(V, W) = 1) are shown in Fig. 5. No-
+tice that by construction the circuit parametrization of
+ˆVQA(⃗α), ˆWQAB(⃗β) given in Fig. 2 include such gates as
+special solution: accordingly if properly guided by an
+efficient cost function, our numerical VQA search has a
+chance to find the solution of Fig. 5.
+IV.
+RESULTS
+In this section we study the impact of the cost func-
+tion on the efficiency of the optimization algorithm of
+Sec. III. Assuming the single-qubit noise model detailed
+in Sec. III C and taking B to be a r = 2 qubit register,
+we run two distinct numerical searches: the first obtained
+7
+0
+500
+1,000
+1,500
+2,000
+0
+10
+20
+30
+40
+Iterations
+1 � F
+W1
+0.82
+0.83
+0.84
+0.85
+0.86
+0.87
+0
+50
+100
+150
+Average fidelity
+1 � F
+W1
+Figure 7.
+Comparison of the the input-output average fi-
+delity (3) attainable by running our optimization algorithm us-
+ing the cost function C(fid)(~↵, ~�) (blue data) and C(wass)(~↵, ~�)
+(orange data). Here the error model is a single-qubit bit-flip
+noise (ˆ� = ˆ�1 in (16)) with p = 0.8. The no error correction
+threshold (5) of this scheme is F 0 ⇡ 0.822 – orange peak in
+the fidelity plot, up to numerical precision. Only the runs
+that produced a fidelity of at least F 0 have been included.
+For the C(fid)(~↵, ~�) this is 0.2%, while for C(wass)(~↵, ~�) this
+corresponds to 29.6%.
+to begin with it succeeds in overcoming the threshold F 0
+in one third of the simulations (specifically 40.6% for the
+phase-flip noise model and 29.6% for the bit-flip noise
+model). Furthermore the algorithm reach convergency
+with a number of iterations which are typically smaller
+than those required by C(fid)(~↵, ~�).
+V.
+CONCLUSIONS
+TO BE REWRITTEN To summarize, we have
+shown a variational quantum algorithm that allows finding
+the most suitable error correction procedure for a specific
+noise on quantum hardware. We compared the perfor-
+mance of two di↵erent versions of this algorithm using two
+di↵erent cost functions, the fidelity and an approximation
+of the Wasserstein distance of order one. We compared
+the di↵erence in speed and the ability to obtain a useful
+solution between the two algorithms, finding really di↵er-
+ent trends between the two optimization procedures. The
+optimization process based on the fidelity su↵ers greatly
+from the phenomenon of the barren plateaus, leading to
+very slow convergence or no convergence at all, while the
+algorithm based on the W1 approximation allows us to
+find the configurations that correct the errors, at least in
+the examples that we explored. The results obtained are
+still not enough to use this method as a silver bullet to
+handle this problem, but show a clear improvement and
+allow us to explore further improvements of these meth-
+ods, like using di↵erent algorithms for the minimization
+process, like stochastic gradient descent or higher-order
+algorithms like Newton or pseudo-Newton algorithms.
+Given that the gradient can be expressed only with the
+cost function evaluated in a small number of circuits that
+di↵er only for the parameter choice, allows us to compute
+the gradient of the cost function on the same hardware
+that will be used for the correction procedure. Moreover,
+simulating this circuit may be di�cult because of the
+exponential scaling of the dimension of the Hilbert space
+of a set of qubits, but this problem does not apply when
+all the circuit is built on hardware, gaining a quantum
+advantage.
+Moreover, given that the gradient of the cost function
+can be expressed in terms of the same cost function, the
+same procedure can be iterated to compute the exact Hes-
+sian of the cost function and then apply a second-order
+method like the Newton method as a descent algorithm.
+However, this has not been done because the circuits that
+we marked as useful have a relatively big number of pa-
+rameters, and computing the hessian scales quadratically
+with this number, leading to intractable computations. A
+second-order pseudo-newton method may improve conver-
+gence speed once the algorithm has got near convergence
+and this is a good idea for future developments.
+Acknowledgments
+FZ and VG acknowledge financial support by MIUR
+(Ministero dell’ Istruzione, dell’ Universit`a della Ricerca)
+by PRIN 2017 Taming complexity via Quantum Strate-
+gies: a Hybrid Integrated Photonic approach (QUSHIP)
+Id.
+2017SRN-BRK, and via project PRO3 Quantum
+Pathfinder. GDP is a member of the “Gruppo Nazionale
+per la Fisica Matematica (GNFM)” of the “Istituto
+Nazionale di Alta Matematica “Francesco Severi” (IN-
+(M)
+Number of simulations
+Number of simulations
+C(fid)
+C(wass)
+C(fid)
+C(wass)
+F0
+Figure 6.
+Comparison of the the input-output average fi-
+delity (3) attainable by running our optimization algorithm us-
+ing the cost function C(fid)(⃗α, ⃗β) (blue data) and C(wass)(⃗α, ⃗β)
+(orange data). Here the error model is a single-qubit bit-flip
+noise (ˆσ = ˆσ1 in (16)) with p = 0.8. The no error correction
+threshold (5) of this scheme is F 0 ≈ 0.822 – orange peak in
+the fidelity plot, up to numerical precision. Only the runs
+that produced a fidelity of at least F 0 have been included.
+For the C(fid)(⃗α, ⃗β) this is 0.2%, while for C(wass)(⃗α, ⃗β) this
+corresponds to 29.6%.
+by identifying C(⃗α, ⃗β) with C(fid)(⃗α, ⃗β) and the second
+choosing instead C(wass)(⃗α, ⃗β). Results are reported in
+Figs. 6 and 7 for two different choices of the noise mod-
+els (16), i.e. phase-flip and bit-flip. For both we compare
+the input-output average fidelity (3) at the end of the
+procedure obtained with the two different cost functions,
+and the number of iterations M needed for convergence.
+Regarding this last quantity we set a maximum value
+Mmax equal to 2000 before convergence and we chose this
+
+7
+limit mainly with practical choices like the maximum time
+for the simulation, enforcing that a single run does not
+require more than a few hours of computational time:
+in case the algorithm fails to reach the convergency we
+simply stop the numerical search (this is the reason for
+the peak at the end of the upper orange plot in Fig. 7).
+The plots report only the simulations that manage to
+achieve an average fidelity that is greater or equal than
+no-correction threshold bound F 0.
+The first thing to observe is that for both noise mod-
+els, C(fid)(⃗α, ⃗β) has problem in reaching the do-nothing
+threshold F 0: the probability of success being 2.6% for
+the phase-flip case of Fig. 7 and only 0.2% for the bit-flip
+case of Fig. 6 (for both noise models the total number of
+simulations analyzed was 500). Observe also that in this
+last case the algorithm never yields average input-output
+fidelity values strictly larger than F 0 and that, even in
+those cases, it requires a number M of iterations which
+saturate the maximum allow value Mmax (blue peak in the
+upper plot of Fig. 7). C(was)(⃗α, ⃗β) performs definitely bet-
+ter: to begin with it succeeds in overcoming the threshold
+F 0 in one third of the simulations (specifically 40.6% for
+the phase-flip noise model and 29.6% for the bit-flip noise
+model). Furthermore the algorithm reach convergency
+with a number of iterations which are typically smaller
+than those required by C(fid)(⃗α, ⃗β).
+To better enlighten the differences between the two cost
+functions, we proceeded with further simulations, whose
+results are summarized in Fig. 8. The idea here is to run a
+two-step optimization process composed by two sequences
+of runs: in the first run we start the optimization proce-
+dure from a random point in the parameter space (⃗α, ⃗β)
+with one of the two cost functions (say C(fid)(⃗α, ⃗β)), up
+to convergence; after that we start a second optimization
+run using the other cost function (say C(wass)(⃗α, ⃗β)) but
+assuming as initial condition for the parameters the final
+point reached by the first run. The plots report the dif-
+ference in fidelity between the second and the first run:
+when we start using the C(wass)(⃗α, ⃗β) in the first run, the
+fidelity cannot further improve the result that is already
+found, and this is represented by the fact that the best
+improvement is of the order of 10−5; on the contrary if
+we started employing C(fid)(⃗α, ⃗β) in the first run, the use
+of C(wass)(⃗α, ⃗β) in the second run typically yields sub-
+stantial improvements of the performance4. Moreover, we
+sampled some single descent processes and plotted the
+cost in function of the iteration. When we move from
+y w
+cte
+vec
+0
+500
+1,000
+1,500
+2,000
+0
+20
+40
+60
+Iterations
+1 � F
+W1
+0.85
+0.9
+0.95
+1
+0
+50
+100
+150
+Average fidelity
+1 � F
+W1
+(M)
+Number of simulations
+Number of simulations
+C(fid)
+C(wass)
+C(fid)
+C(wass)
+F0
+Figure 7.
+Comparison of the the input-output average fi-
+delity (3) attainable by running our optimization algorithm us-
+ing the cost function C(fid)(⃗α, ⃗β) (blue data) and C(wass)(⃗α, ⃗β)
+(orange data). Here the error model is a single-qubit phase-flip
+noise (ˆσ = ˆσ3 in (16) with p = 0.8. The no error correction
+threshold (5) of this scheme is F 0 ≈ 0.822 – orange peak in
+the fidelity plot, up to numerical precision. Only the runs
+that produced a fidelity of at least F 0 have been included.
+For the C(fid)(⃗α, ⃗β) this is 2.6%, while for C(wass)(⃗α, ⃗β) this
+corresponds to 40.6%.
+fidelity to W1, the descent part after the change of cost
+function is qualitatively indistinguishable from starting
+from a random point.
+4 It has to be said that in few cases the figure of merit is worse
+after the second optimization – see the negative bar in right panel
+of Fig. 8. This is due to the fact that when using C(wass)(⃗α, ⃗β)
+we are not maximizing the fidelity but minimizing a function
+whose stationary point corresponds to the maximum of the latter:
+accordingly the final point of convergence for C(wass)(⃗α, ⃗β) can
+be slightly off mark in terms of fidelity. This is not a problem
+because these two functions do not have a constant ratio, and we
+checked that the inequalities between them are still satisfied.
+
+8
+0
+1
+2
+3
+4
+5
+·10−5
+0
+100
+200
+300
+400
+500
+Average fidelity
+Number of simulations
+0
+0.2
+0.4
+0.6
+0.8
+0
+10
+20
+30
+40
+50
+Average fidelity
+Number of simulations
+Figure 8. Improvement of simulations when changing the cost function in a two run optimization process that uses different cost
+functions to drive the descent algorithm. In the left plot, we started the descent on a random initial point, ran the optimization
+using C(wass)(⃗α, ⃗β) as cost function until convergence and then we started the descent algorithm again but using C(fid)(⃗α, ⃗β)
+as cost function, starting from the final point of the previous descent. In the right part, the roles of the two cost functions
+are inverted (we start using C(fid)(⃗α, ⃗β) and then we use C(wass)(⃗α, ⃗β)). The histograms represent the difference in average
+input-output fidelity (2) after the change of cost function, namely the difference between the fidelity achieved after the second
+descent and the fidelity after the first descent (positive values correspond to improved performances).Please notice the scale
+difference on the x-axis between the left and right plot.
+V.
+CONCLUSIONS
+To summarize, we have shown a variational quantum al-
+gorithm that allows finding the most suitable error correc-
+tion procedure for a specific noise on quantum hardware.
+We compared the performance of two different versions
+of this algorithm using two different cost functions, the
+fidelity and an approximation of the quantum Wasser-
+stein distance of order one. We compared the difference
+in speed and the ability to obtain a useful solution be-
+tween the two algorithms, finding really different trends
+between the two optimization procedures. The optimiza-
+tion process based on the fidelity suffers greatly from the
+phenomenon of the barren plateaus, leading to very slow
+convergence or no convergence at all, while the algorithm
+based on the quantum W1 distance allows us to find the
+configurations that correct the errors in the examples
+that we explored. The obtained results show a clear im-
+provement and allow us to explore further improvements
+of these methods, as using different algorithms for the
+minimization process, e.g. stochastic gradient descent or
+higher-order algorithms like Newton or pseudo-Newton
+algorithms.
+Given that the gradient can be expressed only with
+the cost function evaluated in a small number of circuits
+that differ only in the parameter choice, the gradient of
+the cost function can be computed on the same hardware
+that will be used for the correction procedure. Moreover,
+simulating this circuit may be difficult because of the
+exponential scaling of the dimension of the Hilbert space
+of a set of qubits, but this problem does not apply when
+all the circuit is built on hardware, gaining a quantum
+advantage. For the same reason, the same procedure
+can be iterated to compute the exact Hessian of the cost
+function and then apply a second-order method like the
+Newton method as a descent algorithm. However, this
+has not been done because the circuits that we marked
+as useful have a relatively big number of parameters,
+and computing the hessian scales quadratically with this
+number, leading to intractable computations.
+Acknowledgments
+FZ and VG acknowledge financial support by MIUR
+(Ministero dell’ Istruzione, dell’ Universit`a della Ricerca)
+by PRIN 2017 Taming complexity via Quantum Strate-
+gies: a Hybrid Integrated Photonic approach (QUSHIP)
+Id.
+2017SRN-BRK, and via project PRO3 Quantum
+Pathfinder. GDP is a member of the “Gruppo Nazionale
+per la Fisica Matematica (GNFM)” of the “Istituto
+Nazionale di Alta Matematica “Francesco Severi” (IN-
+dAM)”.
+VI.
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+

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+arXiv:2301.13429v1  [math.AP]  31 Jan 2023
+ON STRICHARTZ ESTIMATE FOR MANY BODY SCHR¨ODINGER
+EQUATION IN THE WAVEGUIDE SETTING
+ZEHUA ZHAO
+Contents
+1.
+introduction
+1
+2.
+Preliminaries
+4
+3.
+The proof of Theorem 1.1
+6
+4.
+The proof of Theorem 1.6
+8
+5.
+Further remarks
+9
+References
+9
+Abstract. In this short paper, we prove Strichartz estimates for N-body Schr¨odinger
+equations in the waveguide manifold setting (i.e. on semiperiodic spaces Rm ×Tn
+where m ≥ 3), provided that interaction potentials are small enough (depending
+on the number of the particles and the universal constants, not on the initial
+data). The proof combines both the ideas of Tzvetkov-Visciglia [29] and Hong
+[17]. As an immediate application, the scattering asymptotics for this model is
+also obtained. This result extends Hong [17] to the waveguide case.
+Keywords: Strichartz estimate, many body Schr¨odinger equations, scattering, waveg-
+uide manifolds
+Mathematics Subject Classification (2020) Primary: 35Q55; Secondary: 35R01,
+37K06, 37L50.
+1. introduction
+1.1. Background and Motivations. Let d = m + n, m ≥ 3, n ≥ 1 and N ≥ 1. We
+consider the many body Schr¨odinger equations in the waveguide setting as follows,
+(1.1)
+(i∂t + HN)u(x1, ...xN) = 0,
+u(0, x1, ...xN) = u0(x1, ...xN) ∈ L2
+x1,...xN,
+where HN = ∆x − VN = �N
+α=1 ∆xα − �
+1≤α<β≤N V (xα − xβ),
+and α-th particle xα ∈ Rm × Tn for any α ∈ [1, ..., N].
+From physical explanations, N ≥ 1 indicates the number of particles in a quantum
+system (which is often very large) and the interacting potentials of form V (xα −
+xβ) indicates the interactions of any two particles, which depends on their relative
+distance. Moreover, the product spaces of form Rm × Tn is known as semi-periodic
+space or waveguide manifold.
+d = m + n is the whole dimension while m is the
+dimension for the Euclidean component and n is the dimension for the tori component.
+When N = 1, initial value problem(1.1) is exactly the standard nonlinear Schr¨odinger
+equation (NLS) with a potential, which has been well studied (in the Euclidean case,
+i.e. replacing Rm×Tn by Rd). It is also known as ‘the one-body case’ and the research
+on the decay properties has a long history (see the Introduction of [17], the survey
+[25] and the references therein). In this paper, we mainly concern the general case
+(N ≥ 1 can be arbitrarily large), i.e. the many body Schr¨odinger case, which will
+involve some new difficulties than the single body case such as the issue of interacting
+potentials.
+1
+
+2
+ZEHUA ZHAO
+The purpose of this paper is to investigate time decay properties of solutions to
+the N-body Schr¨odinger equation (1.1) in the waveguide setting. In particular, we
+discuss the Strichartz-type estimate and the scattering behavior for (1.1). We note
+that the Euclidean case of (1.1) has been studied in [17]. (See also [7] for a recently
+result which deals with the two body case via the scheme of [20].)
+We intend to
+generalize [17] to the waveguide case. That is one main motivation of this paper.
+Another motivation is the recent developments for the topic: ‘Long time behavior for
+NLS on waveguides’ so the author is interested in combining both of ‘waveguides’ and
+‘many body Schr¨odinger equations’ together, i.e. studying the estimates and the long
+time behavior for many body Schr¨odinger equations on waveguides. We will briefly
+mention the background for ‘NLS on waveguides’ in the next paragraph.
+Waveguide manifolds of form Rm ×Tn are of particular interest in nonlinear optics
+of telecommunications. Generally, well-posedness theory and long time behavior of
+NLS are hot topics in the area of dispersive equations and have been studied widely
+in recent decades. Naturally, the Euclidean case is first treated and the theory, at
+least in the defocusing setting, has been well established.
+We refer to [9, 10, 21]
+for some typical Euclidean results. Moreover, we refer to [5, 4, 14, 15, 16, 18, 19,
+22, 31, 32, 34, 35, 36] with regard to the torus and waveguide settings. (See also
+[24, 27, 33] for other dispersive equations on waveguides.) One may roughly think
+of the waveguide case as the “intermediate point” between the Euclidean case and
+the torus case since the waveguide manifold is a product of Euclidean spaces and the
+tori. The techniques used in Euclidean and torus settings are frequently combined
+and applied to the waveguides problems. At last, we refer to [2, 11, 28] for some
+classical textbooks on the study of NLS.
+Since the current paper concerns the estimates and the PDE-level of (1.1) rather
+than the mathematical physics level, we will not mention too much for the background
+of the many body problems/equations from physical perspectives. We refer to the
+Introductions of [3, 6, 7, 8, 12, 26] and the references therein for more information.
+To the authors’ best knowledge, the current paper is the first result towards un-
+derstanding long time dynamics for the many body Schr¨odinger equations within the
+context of waveguides.
+As last, we note that, as in [17], we need to assume some smallness for the potential
+V and this smallness does not depend on the initial data (only depends on the particle
+number N and the universal constant).
+1.2. The statement of main results. Now we are ready to state the two main
+results of this paper. We start with the Strichartz estimate as follows since the other
+one is an application of it.
+Theorem 1.1 (Strichartz estimate). Let m ≥ 3, n ≥ 1 and 1 < p < 2. There exists
+a small number ǫ such that if ∥V ∥
+L
+d
+2 ,∞
+y
+L2z
+≤
+ǫ
+N 2 , then
+(1.2)
+∥1[0,+∞)e−itHN u0∥V p
+∆x ≲ ∥u0∥L2x.
+Remark 1.2. Here V p
+∆x-norm (known as variation spaces) is introduced by Koch-
+Tataru [23] (see Section 2 for discussions). See also [15, 16, 18, 19] for more information
+and some other applications.
+In viewing of the properties of V p
+∆x-type spaces, Theorem 1.1 directly implies
+Corollary 1.3. Let m ≥ 3 and n ≥ 1. There exists a small number ǫ such that if
+∥V ∥
+L
+d
+2 ,∞
+y
+L2z
+≤
+ǫ
+N 2 , then for any m-dimensional admissible pair (q, r) and 1 ≤ α ≤ N,
+we have
+(1.3)
+∥e−itHNu0∥Lq
+tLryαL2zαL2
+ˆxα ≲ ∥u0∥L2x,
+
+MANY BODY SCHR ¨ODINGER EQUATION ON WAVEGUIDE MANIFOLDS
+3
+where ˆxα is the N − 1 spatial variables except the α-th variable xα, i.e.,
+(1.4)
+ˆxα = (x1, ...xα−1, xα+1..., xN) ∈ Rd(N−1),
+and xα is the α-th variable with Euclidean component yα and tori component zα, i.e.,
+(1.5)
+xα = (yα, zα) ∈ Rm × Tn.
+Moreover, for any mN-dimensional admissible pair (q, r), we have
+(1.6)
+∥e−itHNu0∥Lq
+tLryL2z ≲ ∥u0∥L2x,
+where y is for the whole Euclidean component (mN-dimensional) and z is for the
+whole tori component nN-dimensional.
+Remark 1.4. See Theorem 1.1 and Theorem 1.2 in [17] for the Euclidean case. We will
+give the proof for Corollary 1.3 after the proof of Theorem 1.1 in the end of Section
+3.
+Remark 1.5. As shown above, the formulation of the Strichartz estimates for (1.1)
+combines both the ideas of [29] and [17]. As in [29] (see also [30]), we fix the tori
+component by using L2-norm. (In other words, one decomposes the function along
+the tori direction and derive the Strichartz estimate using the dispersion from the Eu-
+clidean direction.) As in [17], we fix other particles by only considering the dispersion
+of one certain particle. Thus, we consider the dispersion of the Euclidean component
+of one particle; fixing other particles and the tori component of this particle by using
+L2-norm.
+As a direct application of Theorem 1.1, we show the scattering behavior for an
+N-body Schr¨odinger operator with rough small interactions in the following sense,
+Theorem 1.6 (Scattering). Let m ≥ 3, n ≥ 1 and 1 < p < 2. let ǫ be a small
+constant given in Theorem 1.1. If ∥V ∥
+L
+m
+2 ,∞
+y
+L2
+z ≤
+ǫ
+N 2 , then for each u0 ∈ L2
+x, there
+exist scattering states u± such that
+(1.7)
+lim
+t→±∞
+��e−itHN u0 − eit∆xu±
+��
+L2x = 0.
+Remark 1.7. We note that for the tori case of (1.1), the scattering behavior is not
+expected due to the lack of dispersion, though a Strichartz estimate can still be
+possibly obtained with suitable modifications. We leave it for interested readers.
+Remark 1.8. For the above results, the dimension of the tori component n ≥ 1 does
+not matter. (When n = 0, it is exactly the Euclidean case [17]). However, if one
+considers the long time dynamics of a nonlinear problem on waveguide manifolds,
+the dimension of the tori component often matters a lot. In general, the difficulty of
+the critical NLS problem on waveguide manifolds increases if the whole dimension is
+increased or if the Euclidean component is decreased. See the Introductions in [18, 19]
+for more information.
+Remark 1.9. To be more general, the tori component Tn in (1.1) can be generalized
+to a compact Riemannian manifold M such that Theorem 1.1, Corollary 1.3 and
+Theorem 1.6 still hold.
+Next, we briefly introduce the main strategy of the proofs for Theorem 1.1, Corol-
+lary 1.3 and Theorem 1.6. In fact, the proofs for Corollary 1.3 and Theorem 1.6 are
+standard and less complicated. Corollary 1.3 follows from Theorem 1.1 (see Section
+3) according to the transfer principle of the function space V p
+∆x. Theorem 1.6 also
+follows from Theorem 1.1 (see Section 4), together with some other basic estimates
+like in [17]. Thus we will focus on the proof of Theorem 1.1 as follows.
+The proof of Theorem 1.1 (Strichartz estimate) is based on the properties of func-
+tion space V p
+∆x and a perturbation method (see Section 3). The main idea is: one
+
+4
+ZEHUA ZHAO
+establishes nonlinear estimate for one arbitrary interacting potential (treating it as
+a perturbation) and then sum them up. The key estimate is Proposition 3.1 which
+deals with one arbitrary interacting potential by regarding it as a forcing term. With
+the help of it, one can handle all of the interacting potentials by treating them as
+perturbations. Eventually, according to the smallness assumption, one can use per-
+turbation method to show the Strichartz estimate as desired. Compared with the
+single potential case (N = 1), the interacting potentials (involves rotations) cause dif-
+ficulties thus the ‘rotation flexible’ function space V p
+∆x is needed; compared with the
+Euclidean analogue ([17]), the new difference is the appearance of the tori component.
+1.3. Structure of this paper. The rest of the article is organized as follows. In
+Section 2, we discuss function spaces and some estimates for this model; in Section
+3, we give the proof for Theorem 1.1 (Strichartz estimate); in Section 4, we give the
+proof for Theorem 1.6 (scattering asymptotics); in Section 5, we give a few further
+remarks on this research line.
+1.4. Notations. We write A ≲ B to say that there is a constant C such that A ≤ CB.
+We use A ≃ B when A ≲ B ≲ A. Particularly, we write A ≲u B to express that
+A ≤ C(u)B for some constant C(u) depending on u. We use C for universal constants
+and N for the number of particles.
+We say that the pair (p, q) is d-(Strichartz) admissible if
+(1.8)
+2
+p + d
+q = d
+2,
+2 ≤ p, q ≤ ∞
+(p, q, d) ̸= (2, ∞, 2).
+Throughout this paper, we regularly refer to the spacetime norms
+(1.9)
+∥u∥Lp
+t Lq
+z(It×Rm×Tn) =
+��
+It
+��
+Rm×Tn |u(t, z)|qdz
+� p
+q
+dt
+� 1
+p
+.
+Similarly we can define the composition of three Lp-type norms like Lp
+tLq
+xL2
+y. As in
+Theorems 1.1, 1.6 and Corollary 1.3, we use Lr,s for the Lorentz norm (see [1]). One
+can define the composition of norms in a similar way.
+As stated in the above Theorems, in general, we refer to x for the whole spatial
+variable; y for the whole Euclidean spatial variable; z for the whole tori spatial vari-
+able; xα for the α-th spatial variable; yα for the α-th Euclidean spatial variable; zα
+for the α-th tori spatial variable for convenience.
+Similar to the Euclidean case, function spaces such as V p
+∆ are also tightly involved.
+we will discuss them in Section 2. (See also [17].)
+To deal with the interacting potentials, we define the rotation operator Rαβ by
+(1.10)
+Rαβ(f(x1, ...xα−1, xα − xβ
+√
+2
+, xα+1...xβ−1, xα + xβ
+√
+2
+, xβ+1, ...xN)) = f(x1, ...xN).
+Acknowledgment. The author was supported by the NSF grant of China (No.
+12101046, 12271032), Chinese overseas high-level young talents program (2022) and
+the Beijing Institute of Technology Research Fund Program for Young Scholars. The
+author has learned many body Schr¨odinger model and related background during his
+postdoc career at University of Maryland (2019-2021). Thus he highly appreciates
+Prof. M. Grillakis, Prof. M. Machedon and their group (Dr. J. Chong and Dr. X.
+Huang) for related discussions, especially the paper of Hong [17].
+2. Preliminaries
+In this section, we discuss function spaces and some estimates for the model (1.1).
+See Section 2 to Section 4 in [17] for the Euclidean analogue.
+
+MANY BODY SCHR ¨ODINGER EQUATION ON WAVEGUIDE MANIFOLDS
+5
+First, similar to the Euclidean case, one can easily show: if the potential V is
+small enough, then the Strichartz estimate for operator eit(∆x−V ) also holds for the
+waveguide case as follows.
+Lemma 2.1. Let m ≥ 3, n ≥ 1, and let c0 be the implicit constant given in Proposi-
+tion 2.2. If ∥V ∥
+L
+d
+2 ,∞
+y
+L2z
+<
+1
+c0 , then
+(2.1)
+∥eit(∆x−V )u0∥Lq
+tLr
+yL2
+z(R×Rm×Tn) ≤
+c0
+1 − c0∥V ∥
+L
+d
+2 ,∞
+y
+L2z
+∥u0∥L2y,z(Rm×Tn),
+for all m-admissible pair (q, r).
+As in the Euclidean case, to finish the proof of Lemma 2.1, recall the Strichartz
+estimates in the waveguide setting as follows (see Proposition 2.1 in [29], in fact, this
+result is more general since it concerns the compact Riemannian manifold case).
+Proposition 2.2. For every n ≥ 1 and for every compact Riemannian manifold
+M k
+y , one considers functions f(x, y), F(x, y) on Rn ×M k
+y , then the following estimate
+holds:
+(2.2) ∥eit∆x,yf∥Lp
+t Lq
+xL2y+
+��
+� t
+0
+ei(t−s)∆x,yF(s, x, y)ds
+��
+Lp
+t Lq
+xL2
+y ≲ ∥f∥L2x,y+∥F∥L ˜
+p
+t L˜
+q
+xL2y,
+where (p, q) and (˜p, ˜q) are Strichartz admissible pairs.
+Proof of Lemma 2.1. Since the potential is small, the proof is purely perturbative.
+One can just use waveguide Strichartz estimate Proposition 2.2 to treat the potential
+as a perturbation term provided the potential is small (the Duhamel’s formula and
+the H¨older inequality are also used). Thus we omit the proof. See Theorem 2.1 in
+[17] for the Euclidean analogue.
+□
+Next, we discuss Strichartz estimates with frozen spatial variables. (See Propo-
+sition 1 in [17] and Theorem 3.1 in [3] for the Euclidean analogue.) The difference
+is that: now we fix both of the tori component of a certain particle and the other
+particles by using L2-norm. In other words, the ‘frozen spatial variables’ are the tori
+component and the other particles. Standard dispersive estimate and an important
+lemma in [20] which ‘lifts’ dispersive estimates to Strichartz estimates are used.
+Proposition 2.3. Let m ≥ 3 and n ≥ 1. Then for any m-dimensional admissible
+pair (q, r), (˜q, ˜r) and 1 ≤ α ≤ N, we have
+(2.3)
+∥eit∆xu0∥Lq
+tLr,2
+yα L2zαL2
+ˆxα ≲ ∥u0∥L2x,
+(2.4)
+∥
+�
+R
+e−is∆xF(s)ds∥L2x ≲ ∥F∥
+L˜
+q′
+t L˜r′ ,2
+yα L2zαL2
+ˆxα
+,
+and
+(2.5)
+∥
+� t
+0
+e−i(t−s)∆xF(s)ds∥Lq
+t Lr,2
+yα L2zαL2
+ˆxα ≲ ∥F∥
+L˜
+q′
+t L˜r′ ,2
+yα L2zαL2
+ˆxα
+,
+where
+(2.6)
+ˆxα = (x1, ...xα−1, xα+1..., xN) ∈ Rd(N−1),
+and xα is the α-th variable with Euclidean component yα and tori component zα, i.e.,
+(2.7)
+xα = (yα, zα) ∈ Rm × Tn.
+Proof. We consider a complex-valued function f(x) : RdN
+x
+→ C in Lr,2
+yα L2
+zαL2
+ˆxα with
+the function-valued function f(yα; zα, ˆxα) in Lr,2
+yα . We note that r ≥ 2. Using unitarity
+property,
+(2.8)
+∥eit∆xu0∥LryαL2zαL2
+ˆxα = ∥eit∆yαu0∥LryαL2zαL2
+ˆxα = ∥eit∆yαu0∥L2zαL2
+ˆxαLryα.
+
+6
+ZEHUA ZHAO
+Then, by the standard dispersive estimate (for yα-direction which is m-dimensional)
+(2.9)
+∥eit∆yαu0∥Lryα ≲
+1
+|t|m( 1
+2 − 1
+r ) ∥f∥Lr′
+yα,
+we obtain (the Minkowski allows one change the order of norms)
+(2.10) ∥eit∆xu0∥LryαL2zαL2
+ˆxα ≲
+1
+|t|m( 1
+2 − 1
+r ) ∥u0∥L2
+zαL2
+ˆxαLr′
+yα ≲
+1
+|t|m( 1
+2 − 1
+r ) ∥u0∥Lr′
+yαL2
+zαL2
+ˆxα.
+The proposition follows from Theorem 10.1 in [20].
+□
+Now we briefly discuss the function spaces and corresponding estimates. They will
+be essentially used in the following two sections. As mentioned in the end of Section
+3.3 in [17], Strichartz estimates with frozen spatial variables are still not sufficient to
+complete the proof of Theorem 1.1 (Strichartz estimate) because of the interacting
+potentials. That is why a space-time norm that plays the role of the rotated space-
+time norm is needed. This part is almost the same as Section 4.1 in [17] with natural
+modifications. We also refer to [15, 16, 23] for more details.
+We note that the definitions and properties in Subsection 4.1. of [17] are general
+enough which can be applied for our model in the waveguide setting naturally. They
+construct function spaces with nice properties for a separable Hilbert space H and
+self-adjoint operator S. In this paper, we can just choose H to be L2
+x and S to be ∆x in
+the waveguide setting, where x = (x1, ..., xN) and xα ∈ Rm × Tn for α ∈ {1, ..., N} as
+in (1.1). Then the definitions and associated properties for our case will hold as well.
+Thus we refer to Subsection 4.1. of [17] for the function spaces and corresponding
+estimates/properties. For instance, we will use the following property of V p
+∆-space.
+(It follows from the definition. See Proposition 2 in [17].)
+(2.11)
+∥1[0,∞)]eit∆xu0∥V p
+∆x = ∥u0∥L2
+x.
+Moreover, the duality, the inclusion properties and the transference principle of V p
+∆-
+space are also often used. (See Subsection 4.1. of [17])
+3. The proof of Theorem 1.1
+In this section, we discuss the proof of Theorem 1.1 (Strichartz estimate). Corollary
+1.3 will also be obtained using the properties of function space V (xα−xβ). Like in [17],
+we handle the potential terms by treating them as perturbations. The key estimate
+is as follows,
+Proposition 3.1. Let m ≥ 3, n ≥ 1 and 1 < p < 2. Consider u in the waveguide
+setting as in Theorem 1.1. Then, we have
+(3.1)
+��1[0,+∞)
+� t
+0
+ei(t−s)∆x(V (xα − xβ)u(s))ds
+��
+V p
+∆ ≤ C∥V ∥
+L
+m
+2 ,∞
+y
+L2z∥u∥V p
+∆,
+where C is for the universal constant.
+Remark 3.2. Proposition 3.1 indicates that one can regard the potential terms as
+perturbations. As we can see from the proof below, it suffices to consider one arbitrary
+interacting potential V (xα − xβ) since the V p
+∆-norm is rotation-flexible.
+Remark 3.3. See Proposition 4 in [17] for the Euclidean analogue. The main new
+difference for the waveguide case is the appearance of the tori component.
+Proof. For notational convenience, we denote
+(3.2)
+w = 1[0,∞)
+� t
+0
+ei(t−s)∆x(F(s))ds,
+where F = V (xα −xβ)u(s) is treated as the forcing term (or say a perturbative term).
+
+MANY BODY SCHR ¨ODINGER EQUATION ON WAVEGUIDE MANIFOLDS
+7
+We will estimate w by the duality argument. Since we only expect w ∈ V p
+−, not
+w ∈ V p, we consider ˜w(t) = w(−t).
+Similar to Proposition 4 of [17], using duality, it suffices to show that
+(3.3)
+J
+�
+j=1
+⟨a(tj−1), ˜w(j) − ˜w(tj−1)⟩L2x ≲ ∥V ∥
+L
+m
+2 ,∞
+y
+L2z∥u∥V p
+∆
+for any fine partition of unity t = {tj}J
+j=0 and any U p
+′
+-atom a(t) = �K
+k=1 1(sk−1,sk)φk−1.
+(We note that the U p
+′
+-space is the dual of the V p
+∆-space.)
+Doing some standard simplifications as in Proposition 4 of [17] (expanding atoms
+a in terms of φk), one can get a simpler sum
+(3.4)
+K
+�
+k=1
+⟨φk−1, ˜w(sk) − ˜w(sk−1)⟩L2x.
+We further write it as
+K
+�
+k=1
+⟨φk−1, ˜w(sk) − ˜w(sk−1)⟩L2x
+(3.5)
+= −
+K
+�
+k=1
+� −sk−1
+−sk
+⟨φk−1, e−is∆x(F(s))⟩L2xds
+(3.6)
+= −
+K
+�
+k=1
+� −sk−1
+−sk
+⟨eis∆xRφk−1, R(F(s))⟩L2xds
+(3.7)
+= −
+K
+�
+k=1
+�
+R
+⟨eis∆xRφk−1, 1[−sk,−sk−1]R(F(s))⟩L2
+xds,
+(3.8)
+where R denotes any rotation operator.
+(It is just Rαβ for interacting potential
+V (xα − xβ).) We want to control it by ∥V ∥
+L
+m
+2 ,∞
+y
+L2
+z∥u∥V p
+∆.
+Then, applying the H¨older inequality and the Strichartz estimate Proposition 2.3,
+we estimate it by
+K
+�
+k=1
+⟨φk−1, ˜w(sk) − ˜w(sk−1)⟩L2x
+(3.9)
+≲
+K
+�
+k=1
+∥eit∆Rφk−1∥
+L2
+tL
+2m
+m−2 ,2
+yα
+L2zαL2
+ˆxα
+∥1[−sk,−sk−1]R(F(s))∥
+L2
+tL
+2m
+m+2 ,2
+yα
+L2zαL2
+ˆxα
+(3.10)
+≲
+K
+�
+k=1
+∥φk−1∥L2
+x∥V ∥
+L
+m
+2 ,∞
+y
+L2z∥1[−sk,−sk−1]R(u)∥
+L2
+tL
+2m
+m−2 ,2
+yα
+L2zαL2
+ˆxα
+(3.11)
+≲
+K
+�
+k=1
+∥φk−1∥L2
+x∥V ∥
+L
+m
+2 ,∞
+y
+L2z∥1[−sk,−sk−1](u)∥V p
+∆x
+(3.12)
+≲ ∥V ∥
+L
+m
+2 ,∞
+y
+L2z
+��∥φk−1∥L2
+x
+��
+lp′ ·
+��∥1[−sk,−sk−1](u)∥V p
+∆x
+��
+lp
+(3.13)
+≲ ∥V ∥
+L
+m
+2 ,∞
+y
+L2z
+��∥1[−sk,−sk−1](u)∥V p
+∆x
+��
+lp.
+(3.14)
+We note that we have used the inclusion property of discrete Lp spaces (i.e. lp-spaces).
+(1 < p < 2 implies p
+′ > 2.)
+To close the argument, now it remains to show that
+(3.15)
+��∥1[−sk,−sk−1](u)∥V p
+∆x
+��
+lp =
+� K
+�
+k=1
+∥1[−sk,−sk−1)u∥p
+V p
+∆x
+� 1
+p ≤ ∥u∥V p
+∆x.
+
+8
+ZEHUA ZHAO
+This estimate follows exactly as the Euclidean case (using the definition of V p
+∆x).
+There is no difference in the waveguide setting. Thus the proof of Proposition 3.1 is
+complete.
+□
+With the help of Proposition 3.1, we give the proof of Theorem 1.1 as follows. We
+can now treat the potential terms as perturbations.
+Proof. Applying Proposition 3.1 to the Duhamel formula for u = e−itHN u0, we have,
+(3.16)
+∥1[0,+∞)u(t)∥V p
+∆ ≤ ∥u0∥L2x + N(N − 1)
+2
+C∥V ∥
+L
+d
+2 ,∞
+y
+L2
+z
+∥u∥V p
+∆.
+Theorem 1.1 now follows from the smallness assumption of potential V . ( N(N−1)
+2
+is
+the number of interacting potentials.)
+□
+Corollary 1.3 follows from Theorem 1.1 in viewing of the following lemma:
+Lemma 3.4 (Transference principle). Let d ≥ 1, 1 < p < 2, q ≥ 2 and X be a
+Banach space. If a function u : R → X satisfies the bound
+(3.17)
+∥e∆xu0∥Lq
+tX ≲ ∥u0∥L2x,
+then
+(3.18)
+∥u∥Lq
+tX ≲ ∥u∥V p
+∆x.
+Remark 3.5. We note that the Bourgain spaces Xs,b (also known as Fourier restriction
+space) enjoy the similar transfer principle (see [28] for more info.). As summarized in
+[17], the Strichartz estimates in the V p
+∆x sharpen the bounds in Xs,b by 0+ in that
+Strichartz estimates in the Xs,b space do not cover the endpoint Strichartz estimates,
+while those in the V p
+∆x-space do.
+See Proposition 3 in [17] for the proof. As a direct consequence, it shows that
+the V p
+∆x-norm dominates the two Strichartz-type space-time norms in Corollary 1.3.
+Thus, Corollary 1.3 follows from Theorem 1.1.
+4. The proof of Theorem 1.6
+Now we are ready to discuss the proof of Theorem 1.6, i.e.
+the scattering for
+(1.1). Since we have established proper Strichartz-type estimate, the proof will follow
+similarly as in [17]. For the sake of completeness, we include it as follows.
+Without loss of generality, we only consider for the positive time. It suffices to
+show that
+(4.1)
+u+ =
+lim
+t→+∞ e−it∆xe−itHN u0
+exists in L2
+x as t → ∞. Indeed, by the Duhamel formula
+∥e−it2∆e−it2HN u0 − e−it1∆e−it1HN u0∥L2x
+(4.2)
+≤
+�
+1≤α<β≤N
+��
+� t2
+t1
+e−is∆x((V (xα − xβ))e−isHN u0)ds
+��
+L2x.
+(4.3)
+
+MANY BODY SCHR ¨ODINGER EQUATION ON WAVEGUIDE MANIFOLDS
+9
+It suffices to consider one single potential. According to Theorem 1.1 and Corollary
+1.3, we have that
+��
+� t2
+t1
+e−is∆x((V (xα − xβ))e−isHN u0)ds
+��
+L2x
+(4.4)
+=
+��Rαβ
+� t2
+t1
+e−is∆x((V (xα − xβ))e−isHN u0)ds
+��
+L2x
+(4.5)
+=
+��
+� t2
+t1
+e−is∆x(V (
+√
+2xα)(Rαβe−isHN u0))ds
+��
+L2x
+(4.6)
+≤ c0
+��V (
+√
+2xα)(Rαβe−isHN u0)
+��
+L2
+t∈[t1,t2]L
+2d
+d+2 ,2
+yα
+L2zαL2
+ˆxα
+(4.7)
+≤ c0
+2 ∥V ∥
+L
+d
+2 ,∞
+z
+L2z
+��(Rαβe−isHN u0)
+��
+L2
+t∈[t1,t2]L
+2d
+d−2 ,2
+yα
+L2zαL2
+ˆxα
+→ 0
+(4.8)
+as t1, t2 → ∞. Then we can see that the limit exists.
+5. Further remarks
+In this section, we make a few more remarks for many body model (1.1) and
+Theorems 1.1, 1.6 as follows.
+1. The main results in this paper and [17] are based on perturbative scheme which
+are tightly dependent on the smallness assumption of the potentials. One may consider
+removing the smallness assumption to prove Strichartz estimates like Theorem 1.1 or
+Corollary 1.3. It may be hard to consider the general case thus the two body case
+may be a good model to start with. (See [7] for the Euclidean case.)
+2.
+It is also interesting to consider many body equation with a nonlinearity
+F(t, x1, ..., xN) and study the long time behavior.
+There are few general theories
+and results regarding this topic, especially the scattering-type results. Also, it may
+be hard to consider the general case thus the two body case may still be a good
+model to start with. (The Λ-equation in the Hartree–Fock–Bogoliubov equations is
+an example for the two body case, though it is in a coupled system which makes it
+more complicated. See [6, 7].)
+We also note that via the standard T -T ∗ argument and the Christ-Kiselev lemma,
+one can obtain the inhomogeneous Strichartz analogue of Corollary 1.3 (excluding
+the double endpoint case). (See [28].) With the help of it, one may obtain the local
+well-posedness for (1.1) with a subcritical nonlinearity in the energy space. We leave
+it ifor interested readers.
+3. One may also consider the tori analogue of (1.1) (replacing Rm × Tn by Td))
+and obtain some estimates. The reason we consider the waveguide case is that we
+intend to study the scattering behavior, which is not expected for the tori case.
+4. The results in the current paper is only about the estimates and the PDE-level
+of many body Schr¨odinger equations. One may consider the many body Schr¨odinger
+equations in the tori setting or waveguide setting from the perspectives of mathemat-
+ical physics. (See [8, 12, 13] for examples.)
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+MANY BODY SCHR ¨ODINGER EQUATION ON WAVEGUIDE MANIFOLDS
+11
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+Journal of Differential Equations 345 (2023), 447–484.
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+space R × T3, SIAM J. Math. Anal. 53 (2021), no. 2, 2243–2274. MR 4244536
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+MR 4277925
+Zehua Zhao
+Department of Mathematics and Statistics, Beijing Institute of Technology, Beijing,
+China.
+MIIT Key Laboratory of Mathematical Theory and Computation in Information Secu-
+rity, Beijing, China.
+Email address: [email protected]
+

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+3-space orthogonal to uµ
+3-space orthogonal to uµ
+Observer’s worldline, uµ = dxµ
+dτ
+
+
+x1
+x2
+x3
+Before the wave passes
+After the wave passes
+
+
+arXiv:2301.11772v1  [gr-qc]  27 Jan 2023
+Electromagnetic memory in arbitrary curved space-times
+Susmita Jana1, ∗ and S. Shankaranarayanan1, †
+1Department of Physics, Indian Institute of Technology Bombay, Mumbai 400076, India
+Abstract
+The gravitational memory effect and its electromagnetic (EM) analog are potential probes in
+the strong gravity regime. In the literature, this effect is derived for static observers at asymptotic
+infinity. While this is a physically consistent approach, it restricts the space-time geometries for
+which one can obtain the EM memory effect. To circumvent this, we evaluate the EM memory
+effect for comoving observers (defined by the 4-velocity uµ) in arbitrary curved space-times. Using
+the covariant approach, we split Maxwell’s equations into two parts — projected parallel to the 4-
+velocity uµ and into the 3-space orthogonal to uµ. Further splitting the equations into 1+1+2-form,
+we obtain master equation for the EM memory in an arbitrary curved space-time. We provide a
+geometrical understanding of the contributions to the memory effect. We then obtain EM memory
+for specific space-time geometries and discuss the salient features.
+∗ [email protected]
+† [email protected]
+1
+
+I.
+INTRODUCTION
+LIGO-VIRGO-KAGRA has detected close to 100 gravitational wave (GW) sources. GW
+signals emanating from a black hole or neutron star binaries have opened many new research
+avenues in astronomy, cosmology, and fundamental physics [1–4]. GWs provide a unique
+way to test gravity’s most extreme, non-linear regime in novel ways. The planned third-
+generation ground-based detector (Cosmic Explorer and the Einstein Telescope) will allow
+us to peer far deeper, and LISA will open a new observational window at low frequencies.
+With more sensitive detectors shortly, the focus has been to understand the physical effects
+of GWs. Gravitational wave memory is one such effect [5–13].
+GW memory effects — physically observable phenomena that modify the state of
+gravitational-wave detectors a little bit from their original undisturbed state — are one
+of the key predictions of general relativity [6, 7, 9, 14]. GW memory effects can be divided
+into two types [12, 13]: null memory that occurs when radiation or massless particles es-
+cape from a system to null infinity, and ordinary memory that occurs when the detector
+recoils relative to its initial center of mass frame. The GW memory is characterized as a
+gravitational wave signal approaching a nonzero finite value. This aspect of the GW signal
+is yet to be observed, although LISA is predicted to observe it [15].
+Recently, it has been realized that the memory effect can be thought of as a vacuum
+transition between two different states related by an asymptotic transformation [16, 17].
+Since such asymptotic transformations also occur for other gauge theories, there has been
+an intense activity to obtain analogous memory effects in other gauge theories [18–22]. Since
+electromagnetic (EM) theory is the simplest of all gauge theories and can be a potential
+probe, electromagnetic memory has received much attention [23–33]. Like in GW memory,
+an EM wave generates a permanent change in the relative velocity of test-charged particles
+attached to a detector in the 2-D surface perpendicular to the direction of propagation of the
+wave while passing through the detector [cf. (Fig. 1)]. In other words, EM waves directly
+displace test particles by giving them a momentum (kick), resulting in a relative velocity
+change. This is different from GW memory as the GW does not displace test particles.
+Instead, GW distorts the space-time geometry itself, which causes a change in separation
+between two test particles.
+Bieri and Garfinkle were the first to propose the memory effect due to electromagnetic
+2
+
+x1
+x2
+x3
+Before the wave passes
+After the wave passes
+FIG. 1. Electromagnetic memory effect that lies in the 2-D surface orthogonal to the direction of
+the coming wave.
+waves [18]. Like in GW memory, they showed that EM waves produce two types of momen-
+tum kicks. In Ref. [19], Winicour showed the absence of memory effect generated by the
+electromagnetic field coming from distant sources for a bound charge distribution and the
+non-existence of memory effect due to the magnetic field.
+In the case of GW memory, gravitational radiation must reach the detector. Likewise, EM
+radiation also has to reach null infinity to generate null kick memory. Hence to calculate
+EM memory, one needs to know the properties of the electric field and radiation at null
+infinity [18]. More specifically, the original approach by Bieri and Garfinkle requires prior
+knowledge about the behavior of the fields in asymptotic limits. It can be extended to
+conformally flat space-times
+[32, 34].
+Also, the analysis does not provide any physical
+understanding of why the EM memory has such a form in flat and conformally flat space-
+times.
+This leads us to the following questions: Can we derive a master equation for EM memory
+in a generic curved space-time? What role does curved geometry play in EM memory? Can
+we have a physical understanding of the various contributions to EM memory? This work
+addresses these three questions using 1 + 3 covariant formalism [35–40].
+3
+
+There are two reasons why covariant formalism is better suited to studying EM memory.
+First, as mentioned earlier, when the EM wave propagates in a given spatial direction,
+the net momentum experienced by the particle lies in the 2-D surface orthogonal to the
+direction of propagation of the EM wave (for a pictorial representation, see Fig. 1). In other
+words, the EM memory affects the test particle lying on the 2-D surface. Hence, it is more
+natural to have a formalism that identifies such a dynamical 2-D surface and evaluates EM
+memory. Second, like in fluid mechanics, we can observe the flow of EM radiation in two
+ways. First, as in Refs. [18, 19], an asymptotic stationary observer monitors changes in
+Electric and Magnetic fields of the incoming EM radiation. Second, a comoving observer
+monitors changes in Electric and Magnetic fields. In fluid mechanics, these are referred to
+as the Lagrangian and Eulerian descriptions of flow, respectively. It is well-known that the
+Eulerian description is better suited for fluids and in cosmology [37, 38, 40].
+In this work, we evaluate the memory effect using the 1+1+2 covariant formalism [37, 41–
+44]. The 1 + 1 + 2 decomposition of space-time is a natural extension of the 1 + 3 formalism
+in which the three-space is further decomposed to a given spatial direction. This approach
+is also referred to as semi-tetrad formalism [45–49]. The principle advantage is that we can
+evaluate the net momentum (kick) vector on the 2-D surface for arbitrary space-time. Since
+this affects all the test particles on the 2-D surface, we refer to this as memory vector. This
+can also be understood using the fact that the electric and magnetic fields are transverse
+to the direction of propagation of the EM wave. Using the 1 + 1 + 2 covariant formalism,
+we obtain the master equation for the EM memory in arbitrary space-time. We provide
+a geometrical understanding of the various contributions to the memory effect. We then
+obtain the EM memory for specific space-times.
+The rest of this work is organized as follows: In Sec. II, we provide an overview of the
+two — 1+3 and 1+1+2 — covariant formalisms and obtain the key geometrical quantities.
+Then, in Sec. III, we rewrite Maxwell’s equation in 1+3 and 1+1+2 covariant formalisms in
+arbitrary space-time. Next, in Sec. IV, we obtain the master equation for the EM memory
+in arbitrary space-time and discuss the key features. In Sec. V, we then obtain EM memory
+for specific space-times and compare them with the known results in the literature. Finally,
+in Sec. VI, we summarise our results and discuss possible future directions.
+In this work, we use (−, +, +, +) metric signature and set c = 1/(4πǫ0) = 1. A dot
+4
+
+denotes a derivative with respect to the proper time τ. A prime denote derivative w.r.t the
+space-like vector nµ. For easy comparison, we follow the notations of Ref. [40].
+II.
+OVERVIEW OF COVARIANT FORMALISM
+A covariant theory like general relativity does not favor any particular coordinates. How-
+ever, splitting tensors in time and space is typically required for its physical meaning. Thus,
+the splitting achieves this by rewriting Einstein’s equations as a set of constraint and evo-
+lution equations in a three-dimensional framework. This allows for an intuitive evaluation
+of the relevant physical system.
+A choice of coordinates defines a threading of space-time into lines and a slicing into
+hypersurfaces [50]. Thus, the splitting procedure can be carried out in two distinct ways:
+First, by employing the so-called (3 + 1)− formalism or slicing of space-time [51]. Second,
+by employing (1 + 3)− formalism, or threading of space-time [37, 38, 40]. In the (3 + 1)−
+decomposition, the time is a label of space-like slices Σt with space coordinates xi.
+In
+contrast, in the (1+3)− splitting, the time-like world lines have coordinate τ and are labeled
+by xµ. In the (3 + 1)− formulation, the construction only requires space-like hypersurfaces
+and does not demand causality of the time curves. However, in the (1+3)− approach, every
+tensor is split into the parallel and orthogonal directions to a time-like vector (curves).
+Furthermore, it does not provide any condition on the causality of the spatial distances.
+Though the two approaches provide different points of view, it has been shown that they
+are equivalent for space-times with symmetries [50]. We use the covariant 1 + 3 formalism
+in this work to obtain EM memory. As mentioned in the introduction, covariant formalism
+provides a physical understanding of the origin of EM memory in arbitrary space-time.
+A.
+Covariant 1+3 Formalism
+Heckmann, Schucking, and Raychaudhuri developed the covariant approach to General
+relativity in the 1950s [35, 36] and was later used in different gravitational and cosmological
+models [37–40]. To decompose the 4-D space-time in (1 + 3)− formalism, we introduce a
+family of observers with worldlines tangent to a timelike 4-velocity vector uµ satisfy the
+5
+
+3-space orthogonal to uµ
+3-space orthogonal to uµ
+Observer’s worldline, uµ = dxµ
+dτ
+FIG. 2. Visualisation of 1 + 3 formalism.
+following:
+uµ = dxµ
+dτ ;
+uµuµ = −1 ,
+(1)
+where τ is the proper time measured along the fundamental world line. See Fig. 2. Using
+the 4-velocity (uµ) we can define the following projection tensors [38, 40]:
+Uµ
+ν = −uµuν;
+Uµ
+ν Uν
+γ = Uµ
+γ;
+Uµ
+µ = 1
+(2a)
+hµν = gµν + uµuν;
+hµ
+ν hν
+γ = hµ
+γ;
+hµ
+µ = 3;
+hµν uν = 0
+(2b)
+uµ, and hence Uµ ν, projects physical quantities parallel to the 4-velocity of the observer
+and hµν projects quantities into the 3-space orthogonal to uµ. The tensor hµν provides the
+metric properties of the instantaneous 3-space as well in the absence of rotation or vorticity.
+In this formalism, the projection of the vector (V ν) orthogonal to uµ is defined as V<µ>.
+Similarly, the trace-less part of a rank-2 tensor (Sαβ) projected into space orthogonal to uµ
+6
+
+is defined as S<µν>. Mathematically, these are given by:
+V<µ> := hµν V ν;
+S<µν> :=
+�
+hµαhνβ − 1
+3hµνhαβ
+�
+Sαβ
+(3)
+The projection of the time derivative and orthogonal spatial derivative of any vector (V ν)
+and tensor (Sαβ) are defined as:
+˙V <µ> := hµ
+αuν∇ν V α;
+Dα Sβγ := hµ
+α hβ
+ν hγ
+ρ ∇µ Sνρ
+(4)
+The covariant derivative of uµ can be split into two parts: 1) directional derivative along
+the tangent to the world line, 2) spatial derivative in the 3-space orthogonal to uν. This
+can further be split into trace, traceless symmetric and anti-symmetric tensor:
+∇νuµ = Θ
+3 hµν + σµν + ωµν − ˙uµuν .
+(5)
+In the above equation, σµν is the symmetric expansion tensor that describes the distortion
+in the matter flow, Θ corresponds to the expansion rate of the matter w.r.t the observer,
+ωµν is the anti-symmetric vorticity tensor describing the rotation of the matter w.r.t a non-
+rotating frame. The last term refers to the relativistic acceleration vector (the directional
+derivative) ˙uµ = uν∇ν which corresponds to the degree to which the matter moves under
+forces other than gravity plus inertia. Further, using the vorticity tensor, we can define the
+following quantity called the vorticity vector:
+ων = −1
+2ǫµναβωαβ uµ
+(6)
+where, ǫµναβ =
+1
+√−gηµνρσ is fully antisymmetric tensor, ηµνρσ is Levi-Civita symbol whose
+values are ±1 and we set η0123 = 1 = −η0123 [52]. The Levi-Civita 3-tensor is defined as:
+ǫµνα ≡ ǫµναβuβ ,
+(7)
+and satisfies the following relations: ǫµνuν = 0 and ǫµναβ = 2
+�
+u[µǫν]αβ − ǫµν[αuβ] �
+. The
+square bracket w.r.t the indices refers to antisymmetrization.
+B.
+1+1+2 covariant formalism
+The 1 + 3-covariant formalism is well-suited for relativistic cosmology because, at the
+largest observable scales, the universe is homogeneous and isotropic [38]. These symmetries
+7
+
+allow the slicing or threading of the 4-D space-time manifold into a one-parameter family of
+spacelike hypersurfaces corresponding to cosmic time. Interestingly, it is easy to show that
+in the Friedmann-Lemaitre-Robertson-Walker (FLRW) background, all physical quantities
+except for the volume expansion Θ and the energy density vanish.
+Using the Stewart-
+Walker lemma, in this formalism, it was possible to construct gauge invariant quantities
+up to second order in cosmological perturbations [53, 54]. However, the 1 + 3-formalism is
+not suited if the space-time is inhomogeneous, like spherical symmetry or space-times with
+local rotational symmetry (LRS) [41]. In such cases, splitting the 3-space orthogonal to
+the time-like congruence into one spacelike direction and a 2-space is apt [37]. Thus, the
+1 + 1 + 2 decomposition of space-time is a natural extension of the 1 + 3 formalism in which
+the three-space is further decomposed to a given spatial direction. This approach is called
+semi-tetrad formalism [45–49].
+As mentioned in the Introduction, our interest is to evaluate the net momentum experi-
+enced by a test particle after the electromagnetic wave passes through the space-time point.
+In the covariant 1 + 3 formalism, the test particle is the fundamental time-like observer.
+As depicted in (Fig. 1), when the EM wave propagates in a given spatial direction, the net
+momentum experienced by the particle lies in the 2-D surface orthogonal to the direction
+of propagation of the EM wave. In other words, the net momentum (kick) vector lies in the
+2-D surface. Thus, the net memory effect of the test particle will lie on the 2-D surface;
+hence, we will refer to this as the memory vector. This can also be understood using the
+fact that the electric and magnetic fields are transverse to the direction of propagation of
+the EM wave. Thus, it is cogent to further split the 3-space to 1 + 2-space.
+More specifically, choosing a generic space-like vector (nµ), we split the 3-space into 1 +
+2-space [41–44]. The space-like vector (nµ) satisfies the following conditions:
+nµnµ = 1,
+nµuµ = 0 .
+Like in the 1 + 3-formalism, we project the vectors and tensors defined in 3-space along
+the space-like direction (nµ) and into the 2-space that is orthogonal to nµ. Here again, the
+projection tensor (˜hµν) need to be defined:
+˜hµν = hµν − nµnν;
+˜hµ
+ν ˜hν
+γ = ˜hµ
+γ;
+˜hµ
+µ = 2;
+˜hµν uν = 0;
+˜hµν nν = 0 .
+(8)
+All the vectors and tensors defined in the 3-space in the 1 + 3-formalism can be split into
+8
+
+1 + 2 form. For instance, an arbitrary space-like vector V µ (defined in the 3-space) can be
+written as:
+V µ = V nµ + V µ
+(9)
+where, V = V µnµ and V µ = ˜hµ νV ν. Similarly an arbitrary tensor vµν on the 3-space can
+be split as:
+vµν = V
+�
+nµnν − 1
+2
+˜hµν
+�
++ 2V(µnν) + Vµν ,
+(10)
+where V(µnν) = (Vµnν +nνVµ)/2. Similarly, the relative acceleration of the time-like observer
+and other geometrical quantities defined in 3-space can be written in 1 + 2 space as:
+˙uµ = A nµ + A µ
+(11)
+˙nµ = A uµ + αµ
+(12)
+ωµ = Ωnµ + Ωµ
+(13)
+σµν = Σ
+�
+nµnν − 1
+2
+˜hµν
+�
++ 2Σ(µnν) + Σµν
+(14)
+where ˙nµ := uν∇ν nµ is the relative acceleration of the space-like vector along the time-like
+observer. Here, A µ, αµ, Σµν, Ωµ are orthogonal to nµ as well as uµ. Also, A µ, Ωµ(Σµν) are
+the vectors (tensor) projected on the 2-space. In this formalism, we define the alternating
+Levi-Civita 2-tensor
+ǫµν ≡ ǫµναnα
+(15)
+which is orthogonal to nµ and has components only in the 2-space. Given an arbitrary
+vector V µ in the 2-space, we can construct another vector ǫµνV ν that is orthogonal to V µ
+which is in the 2-space and has the same length.
+The 1+2 splitting of the 3-space leads to a new directional derivative along the space-like
+vector nµ:
+v′
+µν ≡ nαDαvµν
+(16)
+˜Dαvµν ≡ ˜hα
+β˜hµ
+ρ˜hν
+σDβvρσ .
+(17)
+The derivative in Eq. (16) physically correspond to the variation of the physical quantities
+on the 2-space along the space-like vector nµ. The derivative ( ˜D) in Eq. (17) corresponds
+9
+
+to the variation of the physical quantities that lie in the 2-space. These will contribute to
+the memory vector.
+As we split the covariant derivative of uµ in Eq. (5), similarly we can split the covariant
+derivative of nµ as:
+Dνnµ = ˜Dνnµ + nµn′
+ν = ˜σµν + ˜ωµν + 1
+2
+˜Θ˜hµν + nµn′
+ν
+(18)
+where, ˜σµν ≡ ˜D<νnµ>, ˜ωµν ≡ ˜D(νnµ) and ˜Θ = ˜Dµnµ are shear, vorticity and the surface
+expansion-contraction scalar respectively and n
+′
+µ is the spatial derivative along nµ. Thus,
+˜Dνnµ describes the kinematic properties or the relative motion of the space-like curves in
+the 2-surface orthogonal to nµ. We can obtain the relation between the kinematic quantities
+derived from the motion of time-like vector uµ and kinematic quantities in 2-space derived
+from the space-like vector nµ. See, for instance, Ref. [44].
+III.
+ELECTROMAGNETIC THEORY IN COVARIANT FORMALISM
+The covariant formalism has been extensively employed in studying the evolution of
+electromagnetic fields in curved space-time [43]. In the covariant formulation, the dynamics
+and kinematics are constricted by the Bianchi and Ricci identities. The (1 + 3)− covariant
+formulation permits the classification of cosmological models, a fluid description of the
+matter field in FLRW universes.
+However, as mentioned earlier, the 1 + 3-formalism is
+not suited if the space-time is inhomogeneous, like spherical symmetry or space-times with
+LRS [41]. In such cases, the 1 + 1 + 2-covariant or semi-triad formalism are better suited.
+Since we aim to derive EM memory for arbitrary space-times, we use 1 + 1 + 2-covariant
+formalism. We obtain a generic form of the EM memory effect by evaluating the change
+in the velocity vector ∆uµ that lie in the 2-space. In order to do so, we fix the space-
+like direction to be the direction of the propagation of the wave. In the case of spherically
+symmetric space-time, this naturally translates to the radial direction. One key advantage is
+that the electromagnetic theory in the 1+1+2 formalism helps to understand the evolution
+and dynamics of the EM fields along the space-like direction and in the 2-space normal
+to nµ and uµ. Our approach makes geometrical contributions to the memory effect more
+transparent.
+In the next subsection, we rewrite Maxwell’s equations in 1+3 formalism in an arbitrary
+10
+
+space-time. Later, we formulate the evolution equations of the EM fields in the 2-space and
+two constraint equations of the same along uµ and nµ [44]. The key advantage is that we
+can obtain the memory vector from the projected acceleration vector onto the 2-space.
+A.
+In 1+3 formalism
+The fundamental objects are the Maxwell electromagnetic field tensor F µν. The (1 +
+3) covariant formalism of Maxwell’s electromagnetic theory provides a way to study the
+interaction of EM fields with different components of general space-time geometry [43].
+With the (1 + 3) decomposition, it is possible to split F µν into the electric and magnetic
+fields. Note that the local coordinates are mathematical parameters that label the points
+of the space-time manifold M; therefore, the electric and magnetic fields may not have a
+direct physical meaning. In order to make measurements, an observer brings in an additional
+structure on M by introducing the orthonormal coframe field. This gives rise to the split of
+Maxwell’s tensor F into the physical electric and magnetic fields.
+Specifically, formalism allows us to split the equations of motion of the fields and currents
+into two parts:
+1. projected parallel to the 4-velocity uµ of the fundamental observer
+2. projected into the 3-space orthogonal to uµ.
+To keep the calculations tractable, we perform all the calculations in source-free and lossless
+regions.
+However, the EM memory analysis can be straightforwardly extended to these
+regions. In the source-free regions, Maxwell’s equations are:
+∇νF µν = 0
+(19)
+∇[γFµν] = 0;
+or
+∇νF ∗µν = 0 ,
+(20)
+where F ∗µν is the dual to F µν and is defined as F ∗µν = (1/2)ǫµναβFαβ.
+In the 1 + 3 formalism, by projecting F µν and F ∗µν along the time-like 4-velocity vector,
+we can decompose them into electric and magnetic parts. The electric (Eµ) and magnetic
+(Bµ) 4-vectors are defined as:
+Eµ := F µνuν
+(21)
+Bµ := F ∗µνuν
+(22)
+11
+
+From the above definitions, we infer:
+Eµuµ = 0;
+Bµuµ = 0
+(23)
+which implies Eµ and Bµ have only spatial components. Given this, we can rewrite Fµν and
+F ∗µν as:
+Fµν = uµEν − uνEµ + ǫµναβBαuβ
+(24)
+˜F αβ = ǫαβµνuµEν +
+�
+uαBβ − uβBα �
+.
+(25)
+From the above expressions, we see that the simultaneous transformations Eµ → −Bµ,
+Bµ → Eµ leads to F ∗µν → F µν. This implies that we can obtain the second Maxwell’s
+equation (20) from the first Maxwell’s equation (19) or vice-versa. More specifically, if we
+obtain the time-like part and space-like part of Maxwell’s equations (20), we can write the
+time-like part and space-like part of the other Maxwell’s equations (19) by substituting
+Eµ → −Bµ, Bµ → Eµ.
+In the rest of this subsection, we obtain Maxwell’s equations by projecting along uµ
+(time-like part) and hµν (space-like part) [55]. We first obtain the time-like part of Eq. (20)
+by multiplying it with uµ:
+uα
+�
+∇β ˜F αβ �
+= 0
+(26)
+Using the decomposition in Eq. (25), the above expression becomes:
+∇βBβ − Bβ ˙uβ + (∇βuα) ǫαβµνuµEν = 0
+(27)
+We simplify the above equation using the following steps: First, we combine the first two
+terms in the LHS. From Eq. (26), we have Bβ ˙uβ = −uβ ˙Bβ = −uβuα∇αBβ. Substituting
+in the second term of the above expression, we have δα
+β ∇αBβ + uβuα∇αBβ = hα
+β
+�
+∇αBβ�
+.
+Substituting ∇βuα from Eq. (5) and using the definition of vorticity vector in Eq. (6), the
+third term in the LHS of the above expression simplifies to −2ωβEβ. Thus, the time-like
+part of Eq. (20) reduces to:
+DβBβ = 2ωβEβ .
+(28)
+The space-like part of Eq. (20) can be obtained by multiplying it with hµ ν,
+hα
+ρ �
+∇β ˜F αβ �
+= 0
+(29)
+12
+
+Using a series of steps, the above expression can be rewritten as:
+˙B<ρ> =
+�
+σρ
+β + ωρ
+β − 2Θ
+3 hρ
+β
+�
+Bβ − ǫρµν ˙uµEν − ǫρµν ∇µEν .
+(30)
+where, ǫµνα is defined in Eq. (7). The above equation provides the dynamical evolution of
+the magnetic field, while Eq. (28) is the constraint equation.
+As mentioned above, performing simultaneous transformation Eµ → −Bµ and Bµ → Eµ
+in Eqs. (31) and (32), we obtain the time-like and space-like parts of the first Maxwell’s
+equation (19):
+DβEβ = −2ωνBν
+(31)
+˙E<ρ> =
+�
+σρ
+β + ωρ
+β − 2Θ
+3 hρ
+β
+�
+Eβ + ǫρµν ˙uµBν + ǫρµν DµBν .
+(32)
+Similarly, the above equation provides the dynamical evolution of the electric field, while
+Eq. (31) is the constraint equation.
+B.
+In 1+1+2 formalism
+We aim to calculate the memory effect of EM fields. As the memory vector resides in
+the 2-surface orthogonal to the direction of propagation of the in-coming wave, we need to
+decompose the 3-space to 1 + 2-space w.r.t a given spatial direction. In this subsection,
+we rewrite Maxwell’s equations (19, 20) using the space-like vector nν and the projection
+tensor (8) in 1 + 1 + 2 formalism.
+To do this, we first express the EM fields and currents in 3-space into 1 + 2 form:
+Eµ = E nµ + E µ
+(33)
+Bµ = Bnµ + Bµ .
+(34)
+where, E ≡ Eµnµ, E µ ≡ ˜hµ νEν, B ≡ Bµnµ, and Bµ ≡ ˜hµ νBν. Following the discussion in
+Sec. (II B), it follows that ǫµνE ν is orthogonal to E µ and, similarly, ǫµνBν is orthogonal to
+Bµ. If electric and magnetic fields are orthogonal to each other in 2 space, then we have
+E ν = ǫµνBν
+Bν = − ǫµνE ν .
+(35)
+These relations will play an important role in Sec. (IV) to derive the memory effect.
+13
+
+The second step is to split the evolution equations (30, 32) interms of E , E µ, B, Bµ.
+To do that, we project Eq. (32) along spacelike direction nµ and multiply Eq. (32) with
+projection tensor (8). After a long calculation, we obtain the following evolution equations
+for E (along nµ) and E µ (in the orthogonal 2-space):
+˙E + ΘE = αµEµ − 2˜ωB + ǫµρ ˜DµBρ
+(36)
+˙E¯µ + Θ
+2 Eµ = − (αµ + 2ǫµρΩρ) E + (Σµρ + Ωǫµρ) E ρ + ǫµρ
+�
+A ρ − n′ρ + ˜Dρ�
+B
+− ǫµρ
+�
+A Bρ + B′ρ −
+�
+˜DρBν
+�
+nν�
+,
+(37)
+where, ˜ω = ˜ωµν ǫµν, Θ is the expansion factor defined in Eq. (5), A µ is the relative accel-
+eration vector in 2-space defined in Eq. (11), ˜ω is the vorticity defined in Eq. (18). Ωµ, Ω
+is defined in Eq. (13) and Σµν is in Eq. (14). The 2-space component of ˙nµ is αµ which is
+defined in Eq. (12), whereas A = nµ ˙uµ = −uµ ˙nµ mentioned in Eq. (11), (12).
+We want to highlight the following points regarding the above expressions: First, the
+above equations generalize Ampere’s law for arbitrary space-time. For example, in Eq. 36,
+the first term in the LHS corresponds to the time derivative of the electric field along space-
+like direction nµ and the last term in RHS is the curl of the magnetic field in 2-space.
+Similarly, the LHS of Eq. (37) is the time derivative of the electric field in 2-space, and in
+the last term in the RHS is the curl of Bρ. Second, in the flat space-time, the expansion
+factor (Θ), the relative acceleration vector (αµ), and vorticity (˜ω) vanish, and the above
+expression lead to Ampere’s law in flat space-time. Thus, background space-time introduces
+new couplings between the electric and magnetic field components. Lastly, we showed that
+the simultaneous transformation Eµ → −Bµ, Bµ → Eµ leads to F ∗µν → F µν. Substituting
+E → B; E µ → Bµ and B → −E ; Bµ → −E µ in Eqs. (36, 37), we have:
+˙
+B + ΘB =Bµαµ + 2˜ωE − ǫµρ ˜DµE ρ
+(38)
+˙
+B¯µ + 1
+2ΘBµ = − (αµ + 2ǫµρΩρ) B + (Σµρ + Ωǫµρ) Bρ − ǫµρ
+�
+A ρ + ˜Dρ − n′ρ�
+E
++ ǫµρ
+�
+A E ρ + ǫµρE ′ρ −
+�
+˜DρEν
+�
+nν�
+(39)
+Note that we obtain the above equations by projecting Eq. (30) along spacelike direction
+nµ and multiply Eq. (30) with projection tensor (8). Again, the above equations generalize
+Faraday’s law for arbitrary space-time.
+14
+
+The last step is to split the constraint equations (31, 28) interms of E , E µ, B, Bµ. Sub-
+stituting (33, 34) and the kinematic quantities (11-14), we get:
+˜DµEµ + nµE ′
+µ + E ′ + ˜ΘE + 2 (ΩB + ΩµBµ) = 0
+(40)
+˜DµBµ − n′µBµ + B′ + ˜ΘB − 2 (ΩE + ΩµEµ) = 0
+(41)
+where ˜Θ is the expansion along the space-like vector defined in Eq. (18). The above equations
+are generalizations of Gauss law. Here again, in the flat space-time, the expansion factor
+(˜Θ), the relative acceleration vector (αµ), vorticity (Ω) vanish, and the above expressions
+lead to Gauss law in flat space-time.
+C.
+Energy-momentum tensor of the electromagnetic field
+As we will show in the next section, the electromagnetic stress tensor plays a crucial role
+in understanding the memory effect. This subsection evaluates the electromagnetic stress
+tensor in 1 + 1 + 2 formalism for an arbitrary space-time. The EM action in an arbitrary
+background is:
+S = −1
+4
+�
+d4x √−g FµνFρσgµρgνσ .
+(42)
+Varying the above action w.r.t the metric (gµν) leads to the following energy-momentum
+tensor:
+Tµν = 1
+2gρσFµρFνσ − 1
+8gµνgρσgαβFραFσβ .
+(43)
+In 1 + 3-formalism, the stress-tensor of matter field (Tµν) can written as:
+Tµν = ρ uµuν + 2 S(µ uν) + Wµν ,
+(44)
+where, the energy-density ρ, the energy flux Sα and stress-tensor W αβ as measured in the
+observer’s worldline are given by [56]:
+ρ = Tµνuµuν,
+Sα = −hα
+µ T µνuν,
+W αβ = hα
+µ T µνhβ
+ν
+(45)
+For the electromagnetic fields in 1 + 3-formalism, ρ, Sµ and Wµν are:
+ρ ≡ 1
+2 (EµEµ + BµBµ) ;
+Sµ ≡ ǫµνρEνBρ
+(46)
+Wµν ≡ 1
+2 (EµEµ + BµBµ) hµν − EµEν − BµBν
+(47)
+15
+
+Rewriting ρ interms of the variables (E , E µ, B, Bµ) in 1 + 1 + 2 formalism, we have:
+ρ = 1
+2
+�
+E 2 + B2�
++ 1
+2 (E µEµ + BµBµ) = ρ(n) + ρ2−space
+(48)
+Thus, ρ(n) corresponds to the energy of the EM field along nµ and ρ2−space corresponds to
+the energy of the EM field in the 2-space. The energy flux Sµ (a vector in 3-space) can be
+rewritten in 1 + 2 space as:
+Sµ = S nµ + Sµ
+(49)
+where S is the Poynting vector of the EM field along the space-like vector nµ and Sµ is
+the energy flux in the 2-space. These are given by:
+S = Sµnµ = ǫµνE µBν
+(50)
+Sµ = −ǫµν (E Bν − BE ν) = − (E E ν + BBν)
+(51)
+In deriving the last expression, we have used the orthogonality condition between the electric
+and magnetic fields in the 2-space, i. e., Eν = ǫνµBµ. As we will see in the next section, the
+memory vector depends on the part of the electromagnetic energy density ρ and Sµ.
+IV.
+MEMORY EFFECT IN ARBITRARY SPACE-TIME
+Having written Maxwell’s equations in 1 + 1 + 2 formalism for an arbitrary space-time,
+we now evaluate the memory effect. Usually, in the literature, one uses the Lorentz force
+equation to derive EM memory. The equation of motion of a charged body (of mass m and
+charge e) in both gravitational and electromagnetic fields are:
+mduα
+dτ − m
+2 gβγ,αuβuγ = eFαβuβ
+(52)
+However, the above expression does not consider the new couplings between the electric
+and magnetic field components in Eqs. (36) - (39). Hence, we use the complete Maxwell’s
+equations (36) - (41) and explicitly obtain the change in velocity (∆uµ) of the time-like
+observer. More specifically, using Eqs. (37, 39), we first calculate the acceleration vector
+A µ in the 2-space. We can then integrate the expression for the acceleration vector (A µ
+in the 2-space) with respect to time t or null time coordinate u ≡ (t − r) leading to the
+memory vector.
+16
+
+In the rest of this section, we calculate A µ for observers whose tangents are congruent to
+the space-like geodesics. This implies nσDσnρ = n′ρ = 0, i. e., nµ is tangent to a congruence
+of space-like geodesics [44].
+Using this condition and substituting
+˙E¯µ = ˜hµν ˙E ν, B′ ρ =
+nνDνBρ in Eqs. (37, 39), we get:
+˜hµν ˙E ν + ǫµρnνDνBρ = − 1
+2ΘEµ − (αµ + 2ǫµρΩρ) E + (Σµρ + Ωǫµρ) E ρ
++
+�
+ǫµρA ρ + ǫµν ˜Dν�
+B − ǫµν
+�
+˜Dνnρ�
+Bρ − ǫµρA Bρ
+(53)
+�
+˜hµν ˙
+Bν − ǫµρnνDνE ρ�
+= −1
+2ΘBµ − (αµ + 2ǫµρΩρ) B + (Σµρ + Ωǫµρ) Bρ
+−
+�
+ǫµρA ρ + ǫµν ˜Dν�
+E + ǫµν
+�
+˜Dνnρ�
+Eρ + ǫµρA E ρ
+(54)
+Multiplying Eq. (53) with B, multiplying Eq. (54) with E and subtracting the resultant
+equations leads to:
+ǫµνA ν = − ǫµν
+2
+Dν(E 2 + B2)
+(E 2 + B2)
++
+�
+Σµν + Ωǫµν − Θ
+2
+˜hµν
+� (E Bν − BE ν)
+(E 2 + B2)
++ ǫµν
+�
+˜σρν + ˜ωρν +
+˜Θ
+2
+˜hρν
+�
+(BBρ + E Eρ)
+(E 2 + B2)
++ ǫµρA (E E ρ + BBρ)
+(E 2 + B2)
++
+B
+(E 2 + B2)
+�
+˜hµν ˙E ν + ǫµρnνDνBρ�
+−
+E
+(E 2 + B2)
+�
+˜hµν ˙
+Bν − ǫµρnνDνE ρ�
+(55)
+To have a transparent understanding, we substitute the definitions (48) - (51) in the expres-
+sion above, resulting in:
+ǫµνA ν = − ǫµν
+2
+Dνρ(n)
+ρ(n)
+− ǫνα
+2
+�
+Σµν + Ωǫµν − Θ
+2
+˜hµν
+� Sα
+ρ(n)
+− ǫµν
+2
+�
+˜σρν + ˜ωρν +
+˜Θ
+2
+˜hρν
+�
+Sρ
+ρ(n)
+− ǫµρS ρ A
+2ρ(n)
++
+B
+2ρ(n)
+�
+˜hµν ˙E ν + ǫµρnνDνBρ�
+−
+E
+2ρ(n)
+�
+˜hµν ˙
+Bν − ǫµρnνDνE ρ�
+. (56)
+This is the master equation for the EM memory in arbitrary space-time regarding which we
+would like to discuss the following points: First, to our understanding, this is a first time the
+EM memory has been obtained for an arbitrary space-time. In the previous calculations [18,
+19], the authors have restricted to asymptotic flat space-times. Second, the last two terms
+in the RHS of the above expression vanishes in the asymptotic limit. To see this, let us
+consider a spherically symmetric space-time. Let t refer to the time coordinate and r to the
+radial coordinate and the null coordinate is u ≡ t − r. In the asymptotic limit ∂u ∼ ∂t and
+17
+
+∂u ∼ −∂r. Setting uµ ≡ (1, 0, 0, 0) and nµ ≡ (0, 1, 0, 0), the penultimate term in the RHS
+of the above equation simplifies to:
+˜hµν ˙E ν + ǫµρnνDνBρ ≃ ˜hµνu0∇0E ν + ǫµρn1∇1Bρ ≃ ˜hµν∂uE ν − ǫµρ∂uBρ
+= f(u)∂u
+�¯˜hµνE ν − ¯EµνBν�
+(57)
+where, ˜hµν = f(u)¯˜hµν and ¯ǫµν = f(u)ǫµν. The terms with bar represent their time indepen-
+dent parts. The above expression vanishes if E ν and Bν are orthogonal to each other in
+the 2-space. As we mentioned earlier (35), in 2-space, the electric and magnetic fields are
+always orthogonal to each other. Similarly, the last term can also be shown to vanish in the
+asymptotic limit. Thus, the above expression reduces to:
+ǫµνA ν = − ǫµν
+2
+Dνρ(n)
+ρ(n)
+− ǫνα
+2
+�
+Σµν + Ωǫµν − Θ
+2
+˜hµν
+� Sα
+ρ(n)
+− ǫµν
+2
+�
+˜σρν + ˜ωρν +
+˜Θ
+2
+˜hρν
+�
+Sρ
+ρ(n)
+− ǫµρ
+2ρ(n)
+S ρ A
+(58)
+Third, the above expression provides a nice geometrical understanding of the various contri-
+butions to memory effect. The first term in the RHS corresponds to the change in the EM
+field energy (ρ(n)) along nµ in the 2-space. This does not contain any contribution from the
+kinematical properties of the space-time. In other words, this term will vanish if the EM
+field energy does not change in the 2-space, like a 2-D flat sheet. However, as we show in the
+next section, this is non-zero in flat space-time expressed in spherical coordinates. The next
+two terms in the RHS are proportional to the energy flux (Sα) in the 2-space. However,
+both these terms have different kinematical information of the space-time and vanish for flat
+space-time. The second term in the RHS carries information about shear (Σµν), vorticity
+scalar (Ω) related to nµ and expansion scalar (Θ) corresponding to time-like observer uµ.
+The third term in the RHS carries information about shear (˜σµν), vorticity tensor (˜ωµν) and
+expansion scalar (˜Θ) corresponding to the space-like vector nµ.
+Fourth, as mentioned earlier, we have not included external currents or charges in our
+analysis.
+Hence, the acceleration vector does not have contribution from the external
+sources.
+Hence, the memory vector we obtain is equivalent to the null-kick derived in
+Refs. [18, 19]. It is also important to note that these authors did not obtain the contribu-
+tions due to the kinematical properties of the space-time. However, as we will see in the
+next section, their contribution can be significant.
+18
+
+Lastly, to obtain the memory vector, we need to integrate the above expression w.r.t the
+proper time of the observer — ∆uµ is the memory vector. It is interesting to note that
+initially if the observer has non-zero velocity only along the time direction, at a later time,
+due to the memory effect, there is a non-zero velocity in the 2-space.
+V.
+APPLICATION TO SPECIFIC SPACE-TIMES
+In the previous section, we obtained a master equation for the EM vector for an arbitrary
+4-D space-time using 1 + 1 + 2-formalism. As we discussed, the memory vector has three
+distinct contributions. In order to illustrate this fact, we consider specific examples and
+obtain the memory vector. In this section we obtain memory vector for flat, FLRW, pp-
+wave and Kerr space-times.
+A.
+Minkowski space-time
+In order to compare the master equation with the existing results [18], we first consider
+Minkowski space-time in spherical coordinates:
+ds2 = −dt2 + dr2 + r2 γAB
+(59)
+where,
+γAB =
+
+ 1
+0
+0 sin2 θ
+
+
+(60)
+is the metric describing unit 2-sphere. In Minkowski space-time, the 4-velocity of the time-
+like congruence observer is uµ ≡ (1, 0, 0, 0) and the space-like vector is nµ ≡ (0, 1, 0, 0).
+Since ∇µuν = 0 and ∇µnν = 0, the kinematics quantities, defined in Sec. (II A, II B) vanish
+for the Minkowski space-time. Hence only the first term in Eq. (56) will be non-zero, i. e.,
+A ν
+Flat = − 1
+2
+Dνρn
+ρn
+.
+(61)
+As mentioned earlier, the acceleration vector corresponds to acceleration in the 2-Sphere.
+Hence, it is appropriate to switch to the 2-Sphere index:
+A A = uµ∇µuA = u0∂0uA + 2u0ΓA
+0 BuB .
+19
+
+Since the 4-velocity uµ is zero in the 2-Sphere, we have A A = u0∂0uA = ∂tuA. In null
+coordinate, this becomes A A = ∂tuA. Substituting the above expression in Eq. (61) and
+integrating in the null coordinate, we have:
+∆uA ≡
+�
+du A A = −1
+2
+�
+du DAρn
+ρn
+.
+(62)
+The above expression is velocity kick w.r.t the Eulerian observers. To compare this with
+the net momentum (kick) vector as seen by the asymptotic static observers (Lagrangian
+observers), we need to do a coordinate transformation. Specifically, we need to transform
+from coordinate basis
+�
+⃗e θ,⃗e φ�
+to orthogonal coordinate basis
+�
+ˆθ, ˆφ
+�
+. In terms of
+�
+ˆθ, ˆφ
+�
+,
+we have ∆⃗u ≡ ∆uµ⃗eµ, where, ⃗e θ = ˆθ/r , ⃗e φ = ˆφ/(r sin θ). Thus, the velocity kick w.r.t the
+asymptotic static observers is given by:
+∆⃗uFlat = 1
+r
+�
+∆uθˆθ + ∆uφ
+sin θ
+ˆφ
+�
+(63)
+Interestingly, the EM memory vector in Minkowski space-time is inversely proportional to r
+and matches with Ref. [18]. This passes the first test that the master equation (56) indeed
+describes the EM memory vector. In the rest of this section, we obtain the memory vector
+for non-flat geometries and show the robustness of our approach.
+B.
+FLRW space-time
+The conformally flat FLRW metric in spherical coordinates is:
+ds2 = a(η)2 �
+−dη2 + dr2 + r2 γAB
+�
+(64)
+where, the conformal time (η) is related to the cosmic time (t) by dt = a(η) dη. In 1 + 3
+formalism, the fundamental observer with time-like 4−velocity in FLRW metric is uµ =
+dxµ/dt = dxµ/(a(η)) dη = ( 1, 0, 0, 0 ) /a(η).
+For this choice of observer, the 3−space
+projection tensor (hµν) orthogonal to uµ is:
+hµν =
+
+ a2(η)
+0
+0
+a2(η) r2 γAB
+
+ .
+(65)
+Since the FLRW line-element is homogeneous and isotropic, only the expansion scalar (Θ)
+is non-zero:
+Θ = 3H (η)
+a(η)
+where
+H = a′(η)
+a(η)
+20
+
+where ′ refers to derivative w.r.t. η. Other kinematic quantities vanish, i. e., σµν = ωµν = 0.
+We now spilt the 3-space into 1 + 2 by choosing the following space-like vector nµ =
+(0, 1, 0, 0)/a(η). This satisfies the conditions: nµnµ = 1 and uµnµ = 0. Repeating the
+steps discussed in Sec. (II B) for the line-element (64), we get:
+˜Θ =
+2
+a(η)
+1
+r , ˜σµν = ˜ωµν = 0.
+It is important to note that while Θ is a function of η only, ˜Θ depends on both η and r.
+Also, Θ depends on the Hubble parameter H , while ˜Θ is inversely proportional of r. Hence,
+at large distances, ˜Θ decays faster compared to Θ.
+Substituting the above expressions in Eq. (58), we have:
+A ν
+FLRW = − 1
+2
+Dνρn
+ρn
++
+1
+4 ρn
+S ν(Θ − ˜Θ) .
+(66)
+Like Minkowski space-time, A ν will have components only in the 2-Sphere. Using the
+fact that the fundamental observers have zero velocity in the 2-Sphere and repeating the
+earlier analysis, we have
+A A = u0∂0uA =
+1
+a(η)
+∂uA
+∂η .
+In terms of the null coordinate u(≡ η − r), we have:
+A A =
+1
+a(u)
+∂uA
+∂u .
+Substituting the above expression in Eq. (66), we have:
+∂uA
+∂u = − a(u)
+2
+DAρn
+ρn
++ a(u)
+4 ρn
+S A(Θ − ˜Θ) .
+(67)
+Integrating the above expression w.r.t u, leads to the following memory vector:
+∆uA
+FLRW = −1
+2
+�
+du a(u)
+ρn
+DAρn + 1
+4
+�
+du a(u)
+ρn
+S A(Θ − ˜Θ)
+(68)
+This is the expression for the memory vector in FLRW space-time regarding which we want
+to highlight the following points: First, unlike Minkowski space-time, here the fundamental
+observers are Eulerian, and hence, we do not have to transform the above expression to
+Lagrangian observers. Second, our results differ from the results of Ref. [34]. In Ref. [34],
+the authors show that the EM memory effect in FLRW differs from the Minkowski only by
+the conformal factor a(η) or a(u). In other words, their analysis did not account for the geo-
+metric contribution to the memory effect. As mentioned earlier, the geometric contribution
+21
+
+leads to a non-zero energy flux (S A) contribution. Also note that the ordinary memory
+derived in Ref. [34] is not present in Eq. (68) as we have assumed any external charge or
+current to be zero. Third, we find that ρ(n) and the energy flux (S A) contribute oppositely.
+It will be interesting to see whether the two contributions nullify the EM memory.
+C.
+pp-wave space-times
+In this subsection, we derive the EM memory for a special kind of plane-fronted wave
+with parallel rays (pp-waves) called plane-wave metric [57]:
+ds2 = −2dudv − F(u, x, y) du2 + dx2 + dy2
+(69)
+where, F(u, x, y) = A(u)(x2 − y2) + 2B(u)xy describes the plane wave and A(u), B(u) are
+arbitrary functions such that F > 0. Note that u, v are not light-cone coordinates. u is
+time-like coordinate and v is a null coordinate.
+We split the above 4-D space-time into 1 + 3 form and later into 1 + 1 + 2-form by
+considering the following time-like velocity vector (uµ) and space-like vector (nµ):
+uµ ≡
+�
+F(u, x, y)(−1/2), 0, 0, 0
+�
+,
+nµ ≡
+�
+F(u, x, y)(−1/2), − F(u, x, y)(1/2), 0, 0
+�
+.
+For the above choice of time-like vector, the 3-space projection tensor (hµν) is:
+hµν =
+
+
+0
+0
+0 0
+0
+1
+F(u, x, y) 0 0
+0
+0
+1 0
+0
+0
+0 1
+
+
+(70)
+Substituting these in the definitions in Sec. (II), only non-zero quantity is the expansion
+scalar (Θ):
+Θ = −
+(x2 − y2) A′(u) + 2xy B′(u)
+2 (2B(u) xy + A(u) (x2 − y2) )3/2 .
+(71)
+The non-zero projection tensor ˜hµν components in the 2-space are ˜hxx = 1, ˜hyy = 1. Thus,
+the memory vector for the special kind of pp-wave space-times is:
+A ν
+PP = − 1
+2
+Dνρn
+ρn
++ Θ
+4 ρn
+S ν .
+(72)
+22
+
+Here, the acceleration of the time-like observer is confined to the x − y plane, i. e.,
+A A
+PP = − 1
+2
+DAρn
+ρn
++ Θ
+4 ρn
+S A ,
+(73)
+where, the index A, B corresponds to (x, y). Evaluating the acceleration vector along x
+and y, we have:
+A (PP)
+x(y) = − 1
+2 ρn
+∂x(y) (ρn) + Θ
+4 ρn
+Sx(y) .
+(74)
+Integrating the above equation w.r.t u, we have:
+∆uPP
+x(y) = −1
+2
+�
+du ∂x(y) (ρn)
+ρn
++ Θ
+4
+�
+duSx(y)
+ρn
+.
+(75)
+The above expression for the velocity kick is for a generic plane-wave metric. To gain
+some physical intuition, we consider two specific forms — Penrose limit of the Schwarzschild
+and FLRW space-times [57]. For Schwarzschild space-time, we have
+A(u) =
+6
+25u2;
+B(u) = 0
+Substituting these in Eq. (71), we have:
+ΘPP,Sch =
+5
+�
+6(x2 − y2)
+.
+It is interesting to note that although the space-time metric does not differentiate between
+the two spatial coordinates (x, y), in order for Θ to be real, the above expression demands
+that x > y. Thus, velocity kick due to EM wave in PP-wave limit of Schwarzschild space-
+time can only occur if x > y and is given by:
+∆uPP Sch
+x(y)
+= −1
+2
+�
+du ∂x(y) (ρn)
+ρn
++
+5
+4
+�
+6(x2 − y2)
+�
+duSx(y)
+ρn
+.
+(76)
+In the case of Penrose limit of FLRW space-time with power-law scale factor a(t) ∼ th, we
+have:
+A(u) = −
+h
+(1 + h)u2,
+B(u) = 0 .
+Substituting these in Eq. (71), we have:
+ΘPP,FLRW =
+�
+(1 + h)
+h(y2 − x2);
+∆uPP FLRW
+x(y)
+= −1
+2
+�
+du ∂x(y) (ρn)
+ρn
++
+�
+(1 + h)
+4
+�
+h(y2 − x2)
+�
+duSx(y)
+ρn
+.
+(77)
+23
+
+Here again, we see that in-order for Θ to be real, the above expression demands that y > x.
+Thus, velocity kick due to EM wave in PP-wave limit of FLRW space-time occurs in a
+different region of the 2-space compared to Schwarzschild. Thus, EM memory has a distinct
+signature for different space-times and can potentially be used as a probe.
+D.
+Kerr space-time
+In this section, we derive the memory effect in Kerr space-time.
+In Boyer-Lindquist
+coordinates (t, r, χ, φ), the Kerr space-time is:
+ds2 =
+�
+2mr
+r2 + a2χ2 − 1
+�
+dt2 +
+�
+r2 + a2χ2
+r2 − 2mr + a2
+�
+dr2 +
+�r2 + a2χ2
+1 − χ2
+�
+dχ2
+−
+�4mar (1 − χ2)
+r2 + a2χ2
+�
+dtdϕ +
+�
+1 − χ2� �
+r2 + a2 + 2ma2r (1 − χ2)
+r2 + a2χ2
+�
+dϕ2 .
+(78)
+where χ ≡ cos θ. In this case, the time-like observer 4-velocity (uµ) and the space-like vector
+(nµ) are [58]:
+uµ =
+��
+r2 − 2mr + a2
+r2 + a2χ2
+, 0, 0, 0
+�
+, nµ =
+�
+0,
+�
+r2 − 2mr + a2
+r2 + a2χ2
+, 0, 0
+�
+.
+We give below the kinematical quantities (discussed in Sec. (II B)) for Kerr space-time in
+1 + 1 + 2 formalism obtained in Ref. [58]:
+Θ = 0;
+Σµν = 0 ;
+(79)
+Ω = −2marχ
+√
+L
+J
+√
+K 3
+;
+˜Θ =
+W
+J
+√
+K 3L
+;
+(80)
+˜ωµν = ˜ωǫµν = 0;
+A = −mD
+√
+L
+J
+√
+K 3 ;
+(81)
+˜σµν =
+
+
+0 0
+0
+0
+0 0
+0
+0
+0 0 −1
+2
+a2(m−r)
+√
+K
+J
+√
+L
+0
+0 0
+0
+1
+2
+a2(m−r)M 2√
+L K
+J 2
+
+
+(82)
+where,
+M = χ2 − 1;
+D = −r2 + a2χ2;
+L = r2 − 2mr + a2
+(83)
+J = r2 − 2mr + a2χ2;
+K = r2 + a2χ2
+(84)
+24
+
+W = 2r3(r − 2m)2 + a4χ2 �
+m + r − mχ2 + rχ2�
++ a2r2 �
+−3m + r + χ2(3r − 5m)
+�
+(85)
+Substituting these expressions in Eq. (58), and noting that the memory vector lies in
+the 2-D surface, we get:
+A A = − 1
+2
+DAρ(n)
+ρ(n)
+− Ω
+2
+ǫAB SB
+ρ(n)
+− 1
+2
+�
+˜σAB +
+˜Θ
+2
+˜hAB
+�
+SB
+ρ(n)
+−
+A
+2ρ(n)
+S A
+(86)
+This is the EM memory vector for an Eulerian observer in Kerr space-time. Note that this
+is a generic result for any value of angular momentum. For a better physical insight, we
+consider a → 0 limit. Substituting a → 0 in Eqs. (79 - 85), we have
+M0 = χ2 − 1;
+D0 = −r2;
+L0 = r2 − 2mr
+(87)
+J0 = r2 − 2mr;
+K0 = r2;
+W0 = 2r3(r − 2m)2
+(88)
+Ω0 = ˜σ0
+µν = 0;
+˜Θ0 = 2
+�
+(r − 2m)
+r3
+;
+A =
+m
+�
+r3(r − 2m)
+(89)
+Substituting the above quantities in Eq. (86), we have:
+A A = − 1
+2
+DAρ(n)
+ρ(n)
+− 1
+2
+�
+r − 2m
+r3
+S A
+ρ(n)
+−
+1
+2ρ(n)
+m
+�
+r3(r − 2m)
+S A .
+(90)
+This is the EM memory vector for an Eulerian observer in Schwarzschild space-time, regard-
+ing which we want to mention the following points: First, in the limit, r → ∞, reduces to
+Minkowski space-time expression (61). Second, in the limit r → ∞, the subleading term is
+proportional to r−1. Third, to derive the memory vector ∆uA, we have to switch to the null
+time coordinate u = t − r and integrate Eq. (90) with respect to u at the asymptotic limit.
+Lastly, to evaluate the memory effect experienced by the static asymptotic (Lagrangian)
+observer, we need to do the transformation from
+�
+⃗e θ,⃗e φ�
+to the orthogonal coordinate basis
+�
+ˆθ, ˆφ
+�
+like in Sec. (V A).
+VI.
+CONCLUSIONS
+In this work, we have derived a master equation for electromagnetic memory in an arbi-
+trary space-time. We used the covariant formalism to obtain the same. More specifically,
+we used the 1 + 1 + 2 covariant formalism. The 1 + 1 + 2 decomposition of space-time is
+a natural extension of the 1 + 3 formalism in which the three-space is further decomposed
+25
+
+using a given spatial direction. This choice of covariant formalism is because the net mo-
+mentum (kick) vector lies on the 2-D surface for arbitrary space-time. Also, the electric and
+magnetic fields are transverse to the direction of propagation of the passing EM wave.
+The EM memory (58) has three distinct contributions: First contribution is due to the
+change in the EM field energy (ρ(n)) along nµ in the 2-space. This is non-zero for Minkowski
+space-time. The second contribution is proportional to the energy flux (Sα) in the 2-space.
+This has kinematical information of the space-time and vanishes for the flat space-time.
+The third contribution is proportional to the acceleration A along the time-like vector uµ.
+To our understanding, the earlier approaches could not isolate the different contributions
+to the EM memory as done in this work.
+We then obtained the EM memory for different space-times. In the case of FLRW space-
+time, we showed that the earlier analysis did not account for the geometric contribution
+to the memory effect [34]. Specifically, their analysis did not account for the geometric
+contribution leading to a non-zero energy flux (S A) contribution. We have also obtained
+the EM memory for Kerr space-time. We also showed that the EM memory has a distinct
+signature for different pp wave space-times and can potentially be used as a probe.
+It would be interesting to extend our analysis for black holes with multiple horizons
+and those that are not asymptotically flat. These may be particularly relevant for using
+EM memory as a probe to PBH. Finally, our analysis points to the possibility of using
+1 + 1 + 2 covariant formalism to understand gravitational memory. These are currently
+under investigation.
+ACKNOWLEDGMENTS
+The authors thank S. Mahesh Chandran and A. Kushwaha for comments on the earlier
+version of the manuscript. The work is supported by the SERB-Core Research grant.
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+

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+arXiv:2301.03973v1  [cs.IT]  10 Jan 2023
+Performance of RIS-empowered NOMA-based D2D
+Communication under Nakagami-m Fading
+Mohd Hamza Naim Shaikh, ◦Sultangali Arzykulov, ◦Abdulkadir Celik, ◦Ahmed M. Eltawil, and G. Nauryzbayev
+School of Engineering and Digital Sciences, Nazarbayev University, Nur-Sultan City, 010000, Kazakhstan
+◦CEMSE Division, King Abdullah University of Science and Technology, Thuwal, 23955, Saudi Arabia
+Emails: {hamza.shaikh, galymzhan.nauryzbayev}@nu.edu.kz,
+◦{sultangali.arzykulov, abdulkadir.celik, ahmed.eltawil}@kaust.edu.sa
+Abstract—Reconfigurable
+intelligent
+surfaces
+(RISs)
+have
+sparked a renewed interest in the research community envi-
+sioning future wireless communication networks. In this study,
+we analyzed the performance of RIS-enabled non-orthogonal
+multiple access (NOMA) based device-to-device (D2D) wireless
+communication system, where the RIS is partitioned to serve a pair
+of D2D users. Specifically, closed-form expressions are derived for
+the upper and lower limits of spectral efficiency (SE) and energy
+efficiency (EE). In addition, the performance of the proposed
+NOMA-based system is also compared with its orthogonal counter-
+part. Extensive simulation is done to corroborate the analytical
+findings. The results demonstrate that RIS highly enhances the
+performance of a NOMA-based D2D network.
+Keywords— Device-to-device, energy efficiency, non-orthogonal
+multiple access, RIS, spectral efficiency.
+I. INTRODUCTION
+Reconfigurable intelligent surfaces (RISs) have recently been
+emerged as a revolutionary technique to realize the smart and
+programmable wireless environment for the next generation 6G
+systems [1]. Inherently, RIS consists of a planar array of large
+number of passive reflecting elements (REs). These REs can
+reflect the incident signal so that the reflected signal can be
+aligned towards the desired location. Because of their potential
+to transform a hostile wireless environment into an adaptive
+and favorable propagating channel, RISs have received much
+attention from the research community. RIS has the potential
+to enhance spectral efficiency (SE) significantly, and energy
+efficiency (EE) due to the large number of passive REs [2].
+On the other hand, device-to-device (D2D) communication
+is also considered a promising technology proposed in the
+5G standard that enables direct communications between D2D
+users. In D2D communication, the same time-frequency re-
+sources of cellular users are re-utilized by the D2D users,
+thus, allowing massive access without aggravating the spectrum
+crunch [3]. However, in D2D communication, a successful
+transmission is highly reliant on the propagation environment
+due to the limited power budget available at nodes. Unfortu-
+nately, this restriction limits the applicability of D2D in many of
+the existing scenarios, especially in dense urban environments.
+Since RIS can adapt an unknown channel to a favorable prop-
+agation environment, deploying RISs can effectively alleviate
+this constraint [4]. In [5], the authors have optimized the RIS-
+aided underlay D2D communication to maximize the capacity
+by optimizing RIS phase shifts along with spectrum reuse and
+Blockage
+RIS
+BS
+g2
+DR1
+g1
+g
+DR2
+DT
+h
+h1
+h2
+Fig. 1. Schematic for RIS-empowered D2D Communication.
+transmit beamforming. A joint resource allocation to maximize
+the sum rate of a RIS-assisted D2D underlay cellular network
+was studied in [6]. Likewise, in [7], performance analysis for
+RIS-assisted D2D communication was carried out for underlay
+and overlay modes.
+This paper investigates the performance of a RIS-empowered
+NOMA-based D2D communication system. The proposed sce-
+nario considers a downlink network, where a user nearby the
+base station (BS), is utilized to serve as a D2D transmitter (DT),
+facilitating the communication with a pair of users, i.e., D2D
+receivers (DRs), which were otherwise not accessible by BS.
+DT is deployed with RIS, which comprises M REs. To support
+both DRs, a hard partitioning-based approach is utilized at
+RIS. Unlike [7], where the point-to-point D2D communication
+without a direct link was considered, we consider the novel
+RIS-empowered NOMA-based D2D communication with both
+direct and reflected links. This work’s main contribution can be
+summarized as follows:
+• We obtain the closed-form expressions for the upper and
+lower bounds of ergodic rate for the NOMA pair of the
+proposed RIS-enabled NOMA-based D2D communication
+system. Initially, we formulate the received signal-to-
+interference-plus-noise-ratio (SINR) and then utilize it for
+deriving the closed-form expressions of SE and EE for
+both the DRs;
+• In addition, we illustrate the effect of the distribution
+of REs, the power allocation ratio, and the Nakagami-m
+fading parameters on network performance;
+• Lastly, the proposed RIS-enabled NOMA-based D2D net-
+work is compared to its corresponding OMA counterpart
+and the case without RIS.
+
+围II. SYSTEM MODEL
+Fig. 1 illustrates the system model where a single antenna
+BS tries to communicate with a pair of blocked UEs, i.e.,
+D2D receivers (DRs), denoted as DR1 and DR2. So, a D2D
+transmitter (DT) is utilized to set a reliable communication link.
+Further, the DT-to-DR transmission is assisted via RIS with
+M number of REs. The system model can be regarded as a
+connected D2D-enabled cellular system, where a cluster of UEs
+are brought into coverage through the nearest connected UE1.
+Further, without losing any generality, in this work, we focus
+mainly on D2D communication, i.e., communication from DT
+to DRs.
+In order to support two DRs, RIS is partitioned in two
+sub-RISs, each having M1 and M2 number of REs, with
+M1 = η M, M2 = (1 − η) M, M1 + M2 = M and η
+being the allocation parameter. Similar to [9], [10], a quasi-
+static and flat fading channel is assumed with known channel
+state information (CSI). Further, the BS-to-DT, DT-to-RIS and
+RIS-to-DR channel links can either be line-of-sight (LoS), or
+non-LoS (NLoS) and thus characterized through Nakagami-m
+fading model [8]. The elements of g, g1 and g2 follow the
+Nakagami-m fading model with m0, m1 and m2 as the fading
+parameters. Similarly, the direct link between DT-to-DR is also
+characterized through Nakagami-m fading channel with mhl as
+fading parameter, where l = 1, 2.
+In accordance with the NOMA and RIS concepts, the re-
+ceived signals at DR1, r1, and DR2, r2, can be expressed as
+r1 = (h1 + ¯g1 Φ1 g1 ) (β1 x1 + β2 x2)
+�
+Pr + No,
+(1)
+r2 = (h2 + ¯g2 Φ2 g2 ) (β1 x1 + β2 x2)
+�
+Pr + No,
+(2)
+where x1 and x2 represent the desired DR1 and DR2 signals,
+respectively. Likewise, β1 and β2 is the power allocation
+coefficient for DR1 and DR2. Further, β1 and β2 adhere to
+the NOMA constraint, i.e., β2
+1
++ β2
+2 = 1. Further, Pr and
+No denote the transmit power at DT and the additive white
+Gaussian noise (AWGN), with No ∈ CN(0, σ2).
+Now the received signal of the DRs
+can be maxi-
+mized through proper phase shifting at the RIS. Mathe-
+matically, |gΦ1g1| =
+����M1
+i=1 giδiejθi gi
+1
+��� and |gΦ2g2| =
+����M2
+i=1 giδi ejθigi
+2
+��� maximizes the received signal power at
+DRs. Here, gi, gi
+1 and gi
+2 denotes the i-th element of g, g1
+and g2, respectively. Thus, re-configuring θi to ˜θ maximize the
+received power. The corresponding channel gain2 to ˜θ can be
+expressed as
+|H1|2 = |h1 + ¯g1Φ1g1|2 =
+�
+|h1| +
+M1
+�
+i=1
+��¯gi
+1
+�� ��gi
+1
+��
+�2
+,
+(3)
+|H2|2 = |h2 + ¯g2Φ2g2|2 =
+�
+|h2| +
+M2
+�
+i=1
+��¯gi
+2
+�� ��gi
+2
+��
+�2
+.
+(4)
+1There can be multiple users within the cluster, however, due to complexity
+requirements, we restrict ourselves to the two-user case, i.e., two DRs [8].
+2Without losing any generality, δi = 1, ∀ i is assumed.
+III. PERFORMANCE ANALYSIS
+This section evaluates the bound on the ergodic rate of DRs.
+Further, the SE and EE for RIS-enabled NOMA-based D2D is
+formulated considering the fading parameter, power allocation,
+and REs distribution. Initially, the SINR for both the DRs is
+formulated and later on we utilize it in evaluating the SE and
+EE.
+Considering the DR1 signal as an interference, DR2 will
+decode the received signal with the following SINR
+SINRDR2 =
+|H2|2 β2
+2 Pr
+|H2|2 β2
+1 Pr + No
+=
+|H2|2 β2
+2 ρr
+|H2|2 β2
+1 ρr + 1
+,
+(5)
+where ρr = Pr/No is transmit SNR at DT.
+Likewise, at DR1, applying SIC, initially, DR1 will decode
+the received signal of DR2. SINR for it can be expressed as
+SINRDR1→DR2 =
+|H1|2 β2
+2 ρr
+|H1|2 β2
+1 ρr + σ2 .
+(6)
+After decoding and canceling the signal of DR2, DR1 can
+decode its own signal with SINR of
+SINRDR1 = β2
+1 |H1|2 ρr.
+(7)
+A. Channel Characterization
+Now the channel gains, H1 and H2, as defined in (3) and
+(4), respectively, do not conform to any known closed-form
+distribution. Thus, for the sake of simplification of the analytical
+performance, we can approximate G1 and G2 (for M1 ≫ 1
+and M2 ≫ 1) as |H1|2 =
+��M1
+i=1
+��¯gi
+1
+�� ��gi
+1
+��
+�2
+and |H2|2 =
+��M2
+i=1
+��¯gi
+2
+�� ��gi
+2
+��
+�2
+, respectively. The distribution function for
+the channel gain, |H|2, can be defined for g ≥ 0 as [11]
+f|H|2(y) =
+√ya
+2ba+1Γ(a + 1) exp
+�
+−
+√y
+b
+�
+,
+(8)
+F|H|2(y) =
+γ
+�
+a + 1,
+√y
+b
+�
+Γ (a + 1) √y .
+(9)
+Here,
+a
+and
+b
+are
+the
+variables
+defined
+as
+a
+=
+m0Γ(m0)2MmlΓ(ml)2
+m0Γ(m0)2mlΓ(ml)2−Γ(m0+ 1
+2 )2Γ(ml+ 1
+2 )2 − N − 1 and b
+=
+m0Γ(m0)2mlΓ(ml)2−Γ(m0+ 1
+2 )2Γ(ml+ 1
+2 )2
+� m0
+Ωg Γ(m0)Γ(m0+ 1
+2 )
+� ml
+Ωgl Γ(ml)Γ(ml+ 1
+2 ), with N ∈ {M1, M2}
+and l ∈ {1, 2}, for H ∈ {H1, H2}. Further, Γ(·) represents
+the Gamma function and γ(·, ·) indicates the lower incomplete
+Gamma function.
+B. Ergodic Rate
+The ergodic rates for DR1 and DR2 can be formu-
+lated as RDR1 = E [log2 (1 + SINRDR1)] and RDR2 =
+E [log2 (1 + SINRDR2)],
+respectively.
+Since
+the
+channel
+gain’s exact distribution is unknown, the expectations are math-
+ematically intractable, and thus a closed-form expression may
+not be derived. Hence, we resort to approximating the ergodic
+rates of DRs with tight upper and lower bounds. Specifically,
+the upper bound is derived by invoking Jensen’s inequality, and
+the lower bound is derived by utilizing the approximate PDF
+
+Rl
+DR1 =
+1
+ln (2) Γ(a + 1)
+�
+π csc (aπ/2) F1
+(a + 2) (β1b)a+2 (ρr)
+a
+2 +1 +
+π sec (aπ/2) F1
+(a + 1) (β1b)a+1 (ρr)
+a+1
+2
++2 a (a − 1) ψ(0) (a + 1) +
+�
+a2 − a
+�
+Γ(a − 1) ln
+�
+b2 β2
+1 ρr
+�
++ Γ(a − 1) F3
+�
+(10)
+Rl
+DR2 =
+1
+ln (2)Γ(a + 1)
+�
+π csc (aπ/2)
+(a + 2) ba+2
+�
+F4
+(c1ρr)
+a
+2 +1 −
+F5
+(c2ρr)
+a
+2 +1
+�
++ π sec (aπ/2)
+(a + 1) ba+1
+�
+F6
+(c1ρr)
+a+1
+2
+−
+F7
+(c2ρr)
+a+1
+2
+�
++ Γ(a − 1) {F8 − F9} +
+�
+a2 − a
+�
+Γ(a − 1) ln
+�c1
+c2
+��
+(11)
+Ru
+DR1 = log2
+
+1 + Ξ1 Ωh1 + M1 Ξ1 Ωm0 Ωm1 + M1 (M1 − 1) Ξ1
+Ωm0
+m0
+�
+Γ
+�
+m0 + 1
+2
+�
+Γ (m0)
+�2
+Ωm1
+m1
+�
+Γ
+�
+m1 + 1
+2
+�
+Γ (m1)
+�2
++ 2 M1 Ξ1
+Γ(mh1 + 1
+2)
+Γ(mh1)
+�
+Ωmh1
+mh1
+Γ(m0 + 1
+2)
+Γ(m0)
+�
+Ωm0
+m0
+Γ(m1 + 1
+2)
+Γ(m1)
+�
+Ωm1
+m1
+�
+(12)
+Ru
+DR2 = log2
+�
+1 + Ξ2Ωh2 + M2Ξ2Ωm0Ωm2 + M2 (M2 − 1) Ξ2Ωm0
+�
+Γ
+�
+m0 + 1
+2
+��2 Ωm2
+�
+Γ
+�
+m2 + 1
+2
+��2
+m0 {Γ (m0)}2 m2 {Γ (m2)}2
++
+2M2Ξ2Γ(mh2 + 1
+2)Γ(m0 + 1
+2)Γ(m2 + 1
+2)�Ωmh2 Ωm0Ωm2
+Γ(mh2)Γ(m0)Γ(m2)√mh2m0m2
+�
+− log2 [1 + Ξ3Ωh2 + M2Ξ3Ωm0Ωm2
++M2 (M2−1) Ξ3Ωm0
+�
+Γ
+�
+m0+ 1
+2
+��2 Ωm2
+�
+Γ
+�
+m2+ 1
+2
+��2
+m0 {Γ (m0)}2 m2 {Γ (m2)}2
++
+2M2Ξ3Γ(mh2 + 1
+2)Γ(m0+ 1
+2)Γ(m2+ 1
+2)�Ωmh2Ωm0Ωm2
+Γ(mh2)Γ(m0)Γ(m2)√mh2m0m2
+�
+(13)
+as described in (8). The bounds on the ergodic rate of DRs
+are evaluated following a series of mathematical manipulations.
+The following Lemmas present the upper and lower bound for
+the proposed RIS-enabled NOMA-based D2D communication
+system.
+Lemma 1. The lower bound on the ergodic rates of DR1 and
+DR2 can be expressed as in (10) and (11), shown on the top
+of the next page.
+Proof. The proof is presented in Appendix A.
+Lemma 2. Likewise, the upper bound on the ergodic rate of
+DR1 and DR2 can be expressed as in (12) and (13), shown on
+the top of the page.
+Proof. The proof is presented in Appendix B.
+C. SE and EE
+Based on the ergodic rate established in the preceding
+subsection, SE of RIS-enabled NOMA-based D2D can be
+described as SE = RDR1 + RDR2. Similarly, the EE can be
+defined as the ratio of the SE to the total power utilized, Ptot,
+in bits/Joule/Hz. Ptot consists of the power utilized by the BS,
+DT, RIS, and DRs. Thus, the EE may be represented as
+EE = SE
+Ptot
+=
+SE
+(1 + α)Pr + MPRE + 2PU
+,
+(14)
+where Pr denotes the static power consumption of DT. Like-
+wise, αPr is the dynamic power consumption at DT. Further,
+PRE denotes the power consumed by each of the RE and PU
+is the power utilized by DR.
+0
+5
+10
+15
+20
+25
+30
+6
+8
+10
+12
+14
+16
+18
+20
+22
+24
+26
+Transmit SNR (dB)
+SE (bps/Hz)
+ 
+ 
+Exact − Simulation
+Lower Bound
+Upper Bound
+M = 200, 100, 50, 20, 10
+Fig. 2. SE versus transmit SNR with varying M.
+IV. SIMULATION RESULT
+This section presents the simulation and analytical results
+for the proposed RIS-empowered NOMA-based D2D commu-
+nication system. For the direct links, the Nakagami fading
+parameters are assumed to be mh1 = mh2 = 2 and, for the RIS
+reflected links, m0 = m1 = m2 = 5, respectively. Likewise,
+the power allocation factor for DR1 is β2
+1 = 0.3 while for DR2
+is β2
+2 = 0.7, if not specified otherwise. Additionally, the value
+of RE allocation parameter η is assumed to be 0.5.
+Fig. 2 shows the SE results for the proposed RIS-empowered
+NOMA-based D2D communication. Specifically, it shows SE
+with respect to the transmit power while comparing the simu-
+lation and analytical results. These results can easily infer the
+following observations: 1) Apart from smaller M, analytical
+SE is quite precise compared to simulation-based SE. 2) Due
+
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+200
+4
+6
+8
+10
+12
+14
+16
+18
+20
+22
+24
+26
+No. of REs (M)
+SE (bps/Hz)
+ 
+ 
+NOMA − Exact Simulation
+OMA − Exact Simulation
+NOMA − Lower Bound
+OMA − Lower Bound
+NOMA − Upper Bound
+OMA − Upper Bound
+ρr = 10, 20, 30 dB
+Fig. 3. SE of NOMA-/OMA-based D2D with respect to M.
+0
+5
+10
+15
+20
+25
+30
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+Transmit SNR (dB)
+EE (bits/Hz/Joule)
+ 
+ 
+NOMA
+OMA
+M = 20, 50, 200
+Fig. 4. EE versus transmit SNR at different M.
+to the multiplicative path-loss, for less number of REs, i.e.,
+smaller M, the received power from the direct link is signif-
+icant. However, as the number of REs increases, the received
+power from a RIS-reflected link is much more than the power
+received from the direct link to the extent that it can safely
+be ignored. Thus, it can easily be inferred from the analytical
+and simulation framework that the received signal power from
+the direct link is relatively insignificant and can be ignored as
+compared to the received power from the RIS-reflected link.
+Fig. 3 shows the SE of the proposed RIS-empowered D2D
+communication system for both NOMA and OMA scenarios.
+Specifically, the SE of both NOMA and OMA scenarios is
+plotted with respect to the number of REs for different SNRs.
+It can easily be observed here that, instead of increasing the
+transmit power, the number of REs can be increased to get the
+same SE. In other words, for fixed required SE, we can tradeoff
+the transmit power with the number of REs. As D2D users
+are usually power constrained, RIS-empowered D2D commu-
+nication can be a viable alternative to cut down the transmit
+power and improve network EE. Further, as evident from the
+result, RIS-empowered NOMA-based D2D is more spectrally
+efficient as compared to OMA-based D2D. For instance, SE at
+20 dB SNR and M = 20 is 15.26 bps/Hz for NOMA and 14.76
+bps/Hz for OMA, respectively. The NOMA gain will increase
+when the channel gain between UEs increases. Further, SE also
+improves with the number of REs, as evident from the result.
+Likewise, SE grows as the transmit power goes up.
+Fig. 4 shows EE of the proposed RIS-empowered D2D
+communication system, where EE of both NOMA and OMA
+scenarios is plotted with respect to the transmit power for
+the varying number of REs. Further, it can be inferred that
+RIS-empowered D2D is energy-efficient as compared to OMA.
+Further, it can also be observed that the EE increases with
+the number of REs, whereas EE decreases as the transmit
+power increases. This is because SE increases linearly while
+the transmit power increases logarithmically; thus, the overall
+compounding impact decreases EE while increasing transmit
+SNR. In addition, EE is likewise saturated for a large number of
+REs, and no further gains are observed. The result also demon-
+strates that increasing the number of REs does not improve
+performance, as SE increases while EE becomes saturated.
+Thus, it can be inferred that RIS improves the SE and EE
+performance of the D2D system.
+V. CONCLUSION
+In this paper, we investigated the performance of a RIS-
+empowered NOMA-based D2D communication system. Specif-
+ically, we derived the closed-form expressions for SE’s upper
+and lower bounds. As shown through the results, apart from
+the smaller values of the number of REs, the bounds are pretty
+tight and converge to exact SE, even for moderate REs. Further,
+we have also investigated the EE performance. Since the D2D
+devices are usually power-constrained, the results show that
+the transmit power can be a tradeoff with the number of REs
+at RIS. Additionally, the results are also compared with the
+OMA scenario, where it has been shown that NOMA-based
+D2D outperforms the OMA-based case.
+VI. ACKNOWLEDGEMENT
+This work was supported by the Nazarbayev University CRP
+Grant no. 11022021CRP1513.
+APPENDIX A
+PROOF OF LEMMA 1
+The ergodic rate of DR1 can be formulated as
+RDR1 =
+1
+ln (2)
+∞
+�
+0
+ln
+�
+1 + β2
+1 |y|2 ρr
+�
+f|H|2(y)dy
+�
+��
+�
+J1
+.
+(15)
+Further, using (8) and the below relation ln(t)
+=
+(t −
+1) 2F1 (1, 1; 2; 1 − t), J1 in (15) can be modified as
+J1 =
+1
+Γ(a + 1)
+∞
+�
+0
+√g 2F1 (1, 1; 2; −g)
+2ba+1
+e
+�
+−
+√g
+b
+�
+dg.
+(16)
+Here, 2F1(·, ·; ·; ·) represents the Gauss hyper-geometric func-
+tion. Now, this J1 can be solved utilizing [11, Theorem 3] and
+substituted in (15). After rearranging the terms, the ergodic
+rate for DR1 can be given as shown in (11), where F1 =
+1F2
+�
+1+ a
+2; 3
+2, 2+ a
+2;
+−1
+4b2β2
+1ρr
+�
+, F2 = 1F2
+�
+a+1
+2 ; 1
+2, a+3
+2 ;
+−1
+4b2β2
+1ρr
+�
+and F3 = 2F3
+�
+1, 1; 2, 1− a
+2, 3−a
+2 ;
+−1
+4b2β2
+1ρr
+�
+
+Likewise, the ergodic rate for DR2, RDR2, can be given by
+RDR2 =
+1
+ln (2)
+
+
+
+∞
+�
+0
+ln
+�
+1 + c1 |y|2 ρr
+�
+f|H|2(y)dy
+−
+∞
+�
+0
+ln
+�
+1 + c2 |y|2 ρr
+�
+f|H|2(y)dy
+
+
+ ,
+(17)
+where c1 and c2 are defined as c1 = β2
+1 + β2
+2 and c2 = β2
+1.
+It
+can
+be evaluated similarly
+to
+J1. After
+rearranging
+the terms, the ergodic rate for DR2 can be given as
+shown in (12), where F4 =
+1F2
+�
+1+ a
+2; 3
+2, 2+ a
+2;
+−1
+4b2c1ρr
+�
+,
+F5
+=
+1F2
+�
+1+ a
+2; 3
+2, 2+ a
+2;
+−1
+4b2c2ρr
+�
+,
+F6
+=
+1F2
+�
+a+1
+2 ; 1
+2, a+3
+2 ;
+−1
+4b2c1ρr
+�
+, F7 = 1F2
+�
+a+1
+2 ; 1
+2, a+3
+2 ;
+−1
+4b2c2ρr
+�
+,
+F8
+=
+2F3
+�
+1, 1; 2, 1− a
+2, 3−a
+2 ;
+−1
+4b2c1ρr
+�
+and
+F9
+=
+2F3
+�
+1, 1; 2, 1− a
+2, 3−a
+2 ;
+−1
+4b2c2ρr
+�
+. This completes the proof of
+Lemma 1.
+APPENDIX B
+PROOF OF LEMMA 2
+Applying Jensen’s inequality, we define the upper bound for
+DR1 as Ru
+DR1, where RDR1 ≤ Ru
+DR1, with Ξ1 = β2
+1ρr, as
+Ru
+DR1 = log2
+�
+1 + Ξ1E
+�
+|H1|2��
+.
+(18)
+To calculate E
+�
+|H1|2�
+, we apply the binomial expansion
+theorem (BET) as
+E
+�
+|H1|2�
+= E
+
+
+�����h1 +
+M1
+�
+i=1
+��¯gi
+1
+�� ��gi
+1
+��
+�����
+2
+ = E
+�
+|h1|2�
+�
+��
+�
+E1
++ E
+
+
+
+� M1
+�
+i=1
+��¯gi
+1
+�� ��gi
+1
+��
+�2
+
+
+�
+��
+�
+E2
++2 E
+� M1
+�
+i=1
+��¯gi
+1
+�� ��gi
+1
+�� |h1|
+�
+�
+��
+�
+E3
+. (19)
+Now, we have E1 = Ωh1. Likewise, to calculate E2, we apply
+BET again; thus, on expanding, E2 can be expressed as
+E
+� M1
+�
+i=1
+��¯gi
+1
+��2 ��gi
+1
+��2
+�
++ E
+
+
+
+
+
+M1
+�
+i=1
+M1
+�
+i=1
+j̸=i
+��¯gi
+1
+�� ��gi
+1
+��
+���¯gj
+1
+���
+���gj
+1
+���
+
+
+
+
+
+, (20)
+where E
+��M1
+i=1
+��¯gi
+1
+��2 ��gi
+1
+��2�
+=
+M1Ωm0Ωm1. Further, for
+E
+��M1
+i=1
+�M1
+i=1
+j̸=i
+��¯gi
+1
+�� ��gi
+1
+��
+���¯gj
+1
+���
+���gj
+1
+���
+�
+,
+the
+expected
+value of
+a
+Nakagami-m
+variable
+can
+be
+given
+as
+E{|g1|}
+=
+Γ(m1+ 1
+2 )
+Γ(m1)
+�� Ωm1
+m1
+�
+. Since ¯gi
+1 and gi
+1 are mutually independent,
+we can have
+E
+
+
+
+
+
+
+
+M1
+�
+i=1
+M1
+�
+j=1
+j̸=i
+��¯gi
+1
+�� ��gi
+1
+��
+���¯gj
+1
+���
+���gj
+2
+���
+
+
+
+
+
+
+
+= M1 (M1 − 1)
+�Ωm0
+m0
+�
+×
+�
+Γ
+�
+m0 + 1
+2
+��2 �
+Γ
+�
+m1 + 1
+2
+��2
+{Γ (m0)}2 {Γ (m1)}2
+�Ωm1
+m1
+�
+.
+(21)
+Likewise, E3 can be calculated as
+E3 = M Γ
+�
+mh1 + 1
+2
+�
+Γ
+�
+m0 + 1
+2
+�
+Γ
+�
+m1 + 1
+2
+�
+×
+�
+Ωmh1 Ωm0Ωm1
+mh1m0m1
+/ [Γ (mh1) Γ (m0) Γ (m1)] .
+(22)
+Finally, putting E1, E2 and E3 all together yields E
+�
+|H1|2�
+which can be put in (18) to give the desired upper bound as
+shown in (27).
+Likewise, the upper bound on the ergodic rate of DR2, Ru
+DR2
+can be defined as RDR2 ≤ Ru
+DR2, where RDR2 can be defined
+as
+RDR2 = E
+�
+log2
+�
+1 + |H2|2 �
+β2
+1 + β2
+2
+�
+ρr
+1 + |H2|2 β2
+1ρr
+��
+,
+= E
+�
+log2
+�
+1 + Ξ2 |H2|2�
+− log2
+�
+1 + Ξ3 |H2|2��
+,
+(23)
+where Ξ2 =
+�
+β2
+1 + β2
+2
+�
+ρr and Ξ3 = β2
+1ρr. Thus, Ru
+DR2 can be
+defined as
+Ru
+DR2=log2
+�
+1+Ξ2E
+�
+|H2|2��
+−log2
+�
+1+Ξ3E
+�
+|H2|2��
+.
+(24)
+Similar to E
+�
+|H1|2�
+, E
+�
+|H2|2�
+can be evaluated. After sub-
+stituting and rearranging the terms, Ru
+DR2 is given in (13).
+REFERENCES
+[1] E. Basar, M. Di Renzo, J. De Rosny, M. Debbah, M.-S. Alouini, and
+R. Zhang, “Wireless communications through reconfigurable intelligent
+surfaces,” IEEE Access, vol. 7, pp. 116 753–116 773, 2019.
+[2] S. Hu, F. Rusek, and O. Edfors, “Beyond massive MIMO: The potential
+of data transmission with large intelligent surfaces,” IEEE Trans. Signal
+Process., vol. 66, no. 10, pp. 2746–2758, 2018.
+[3] A. Asadi, Q. Wang, and V. Mancuso, “A survey on device-to-device
+communication in cellular networks,” IEEE Commun. Surv. Tutor., vol. 16,
+no. 4, pp. 1801–1819, 2014.
+[4] C. Zhang et al., “Distributed intelligent reflecting surfaces-aided device-
+to-device communications system,” J. Commun. Inf. Netw., vol. 6, no. 3,
+pp. 197–207, Sept. 2021.
+[5] Y. Cao et al., “Sum-rate maximization for multi-reconfigurable intelligent
+surface-assisted device-to-device communications,” IEEE Trans. Com-
+mun., vol. 69, no. 11, pp. 7283–7296, Nov. 2021.
+[6] G. Yang et al., “Reconfigurable intelligent surface empowered device-
+to-device communication underlaying cellular networks,” IEEE Trans.
+Commun., vol. 69, no. 11, pp. 7790–7805, Nov. 2021.
+[7] Y. Ni et al., “Performance analysis for RIS-assisted D2D communication
+under nakagami-m fading,” IEEE Trans. Veh. Technol., vol. 70, no. 6, pp.
+5865–5879, Jun. 2021.
+[8] Y. Cheng et al., “Downlink and uplink intelligent reflecting surface aided
+networks: NOMA and OMA,” IEEE Trans. Wirel. Commun., pp. 1–1,
+2021.
+[9] J. Zhu et al., “Power efficient IRS-assisted NOMA,” IEEE Trans. Com-
+mun., vol. 69, no. 2, pp. 900–913, 2021.
+[10] Z. Ding and H. Vincent Poor, “A simple design of IRS-NOMA transmis-
+sion,” IEEE Commun. Lett., vol. 24, no. 5, pp. 1119–1123, 2020.
+[11] M. H. Samuh, A. M. Salhab, and A. H. A. El-Malek, “Performance
+analysis and optimization of RIS-assisted networks in Nakagami-m envi-
+ronment,” arXiv preprint arXiv:2010.07841, 2020.
+

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+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='03973v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='IT]  10 Jan 2023  Performance of RIS-empowered NOMA-based D2D  Communication under Nakagami-m Fading  Mohd Hamza Naim Shaikh, ◦Sultangali Arzykulov, ◦Abdulkadir Celik, ◦Ahmed M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Eltawil, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Nauryzbayev  School of Engineering and Digital Sciences, Nazarbayev University, Nur-Sultan City, 010000, Kazakhstan  CEMSE Division, King Abdullah University of Science and Technology, Thuwal, 23955, Saudi Arabia  Emails: {hamza.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='shaikh, galymzhan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='nauryzbayev}@nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='kz,  {sultangali.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='arzykulov, abdulkadir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='celik, ahmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='eltawil}@kaust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='sa  Abstract—Reconfigurable  intelligent  surfaces  (RISs)  have  sparked a renewed interest in the research community envi-  sioning future wireless communication networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' In this study,  we analyzed the performance of RIS-enabled non-orthogonal  multiple access (NOMA) based device-to-device (D2D) wireless  communication system, where the RIS is partitioned to serve a pair  of D2D users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Specifically, closed-form expressions are derived for  the upper and lower limits of spectral efficiency (SE) and energy  efficiency (EE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' In addition, the performance of the proposed  NOMA-based system is also compared with its orthogonal counter-  part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Extensive simulation is done to corroborate the analytical  findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' The results demonstrate that RIS highly enhances the  performance of a NOMA-based D2D network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  Keywords— Device-to-device, energy efficiency, non-orthogonal  multiple access, RIS, spectral efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' INTRODUCTION  Reconfigurable intelligent surfaces (RISs) have recently been  emerged as a revolutionary technique to realize the smart and  programmable wireless environment for the next generation 6G  systems [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Inherently, RIS consists of a planar array of large  number of passive reflecting elements (REs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' These REs can  reflect the incident signal so that the reflected signal can be  aligned towards the desired location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Because of their potential  to transform a hostile wireless environment into an adaptive  and favorable propagating channel, RISs have received much  attention from the research community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' RIS has the potential  to enhance spectral efficiency (SE) significantly, and energy  efficiency (EE) due to the large number of passive REs [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  On the other hand, device-to-device (D2D) communication  is also considered a promising technology proposed in the  5G standard that enables direct communications between D2D  users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' In D2D communication, the same time-frequency re-  sources of cellular users are re-utilized by the D2D users,  thus, allowing massive access without aggravating the spectrum  crunch [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' However, in D2D communication, a successful  transmission is highly reliant on the propagation environment  due to the limited power budget available at nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Unfortu-  nately, this restriction limits the applicability of D2D in many of  the existing scenarios, especially in dense urban environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  Since RIS can adapt an unknown channel to a favorable prop-  agation environment, deploying RISs can effectively alleviate  this constraint [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' In [5], the authors have optimized the RIS-  aided underlay D2D communication to maximize the capacity  by optimizing RIS phase shifts along with spectrum reuse and  Blockage  RIS  BS  g2  DR1  g1  g  DR2  DT  h  h1  h2  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Schematic for RIS-empowered D2D Communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  transmit beamforming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' A joint resource allocation to maximize  the sum rate of a RIS-assisted D2D underlay cellular network  was studied in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Likewise, in [7], performance analysis for  RIS-assisted D2D communication was carried out for underlay  and overlay modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  This paper investigates the performance of a RIS-empowered  NOMA-based D2D communication system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' The proposed sce-  nario considers a downlink network, where a user nearby the  base station (BS), is utilized to serve as a D2D transmitter (DT),  facilitating the communication with a pair of users, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=', D2D  receivers (DRs), which were otherwise not accessible by BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  DT is deployed with RIS, which comprises M REs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' To support  both DRs, a hard partitioning-based approach is utilized at  RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Unlike [7], where the point-to-point D2D communication  without a direct link was considered, we consider the novel  RIS-empowered NOMA-based D2D communication with both  direct and reflected links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' This work’s main contribution can be  summarized as follows:  We obtain the closed-form expressions for the upper and  lower bounds of ergodic rate for the NOMA pair of the  proposed RIS-enabled NOMA-based D2D communication  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Initially, we formulate the received signal-to-  interference-plus-noise-ratio (SINR) and then utilize it for  deriving the closed-form expressions of SE and EE for  both the DRs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  In addition, we illustrate the effect of the distribution  of REs, the power allocation ratio, and the Nakagami-m  fading parameters on network performance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  Lastly, the proposed RIS-enabled NOMA-based D2D net-  work is compared to its corresponding OMA counterpart  and the case without RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  围II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' SYSTEM MODEL  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' 1 illustrates the system model where a single antenna  BS tries to communicate with a pair of blocked UEs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=',  D2D receivers (DRs), denoted as DR1 and DR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' So, a D2D  transmitter (DT) is utilized to set a reliable communication link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  Further, the DT-to-DR transmission is assisted via RIS with  M number of REs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' The system model can be regarded as a  connected D2D-enabled cellular system, where a cluster of UEs  are brought into coverage through the nearest connected UE1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  Further, without losing any generality, in this work, we focus  mainly on D2D communication, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=', communication from DT  to DRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  In order to support two DRs, RIS is partitioned in two  sub-RISs, each having M1 and M2 number of REs, with  M1 = η M, M2 = (1 − η) M, M1 + M2 = M and η  being the allocation parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Similar to [9], [10], a quasi-  static and flat fading channel is assumed with known channel  state information (CSI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Further, the BS-to-DT, DT-to-RIS and  RIS-to-DR channel links can either be line-of-sight (LoS), or  non-LoS (NLoS) and thus characterized through Nakagami-m  fading model [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' The elements of g, g1 and g2 follow the  Nakagami-m fading model with m0, m1 and m2 as the fading  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Similarly, the direct link between DT-to-DR is also  characterized through Nakagami-m fading channel with mhl as  fading parameter, where l = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  In accordance with the NOMA and RIS concepts, the re-  ceived signals at DR1, r1, and DR2, r2, can be expressed as  r1 = (h1 + ¯g1 Φ1 g1 ) (β1 x1 + β2 x2)  �  Pr + No,  (1)  r2 = (h2 + ¯g2 Φ2 g2 ) (β1 x1 + β2 x2)  �  Pr + No,  (2)  where x1 and x2 represent the desired DR1 and DR2 signals,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Likewise, β1 and β2 is the power allocation  coefficient for DR1 and DR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Further, β1 and β2 adhere to  the NOMA constraint, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=', β2  1  + β2  2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Further, Pr and  No denote the transmit power at DT and the additive white  Gaussian noise (AWGN), with No ∈ CN(0, σ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  Now the received signal of the DRs  can be maxi-  mized through proper phase shifting at the RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Mathe-  matically, |gΦ1g1| =  ����M1  i=1 giδiejθi gi  1  ��� and |gΦ2g2| =  ����M2  i=1 giδi ejθigi  2  ��� maximizes the received signal power at  DRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Here, gi, gi  1 and gi  2 denotes the i-th element of g, g1  and g2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Thus, re-configuring θi to ˜θ maximize the  received power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' The corresponding channel gain2 to ˜θ can be  expressed as  |H1|2 = |h1 + ¯g1Φ1g1|2 =  �  |h1| +  M1  �  i=1  ��¯gi  1  �� ��gi  1  ��  �2  ,  (3)  |H2|2 = |h2 + ¯g2Φ2g2|2 =  �  |h2| +  M2  �  i=1  ��¯gi  2  �� ��gi  2  ��  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  (4)  1There can be multiple users within the cluster, however, due to complexity  requirements, we restrict ourselves to the two-user case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=', two DRs [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  2Without losing any generality, δi = 1, ∀ i is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' PERFORMANCE ANALYSIS  This section evaluates the bound on the ergodic rate of DRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  Further, the SE and EE for RIS-enabled NOMA-based D2D is  formulated considering the fading parameter, power allocation,  and REs distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Initially, the SINR for both the DRs is  formulated and later on we utilize it in evaluating the SE and  EE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  Considering the DR1 signal as an interference, DR2 will  decode the received signal with the following SINR  SINRDR2 =  |H2|2 β2  2 Pr  |H2|2 β2  1 Pr + No  =  |H2|2 β2  2 ρr  |H2|2 β2  1 ρr + 1  ,  (5)  where ρr = Pr/No is transmit SNR at DT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  Likewise, at DR1, applying SIC, initially, DR1 will decode  the received signal of DR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' SINR for it can be expressed as  SINRDR1→DR2 =  |H1|2 β2  2 ρr  |H1|2 β2  1 ρr + σ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  (6)  After decoding and canceling the signal of DR2, DR1 can  decode its own signal with SINR of  SINRDR1 = β2  1 |H1|2 ρr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  (7)  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Channel Characterization  Now the channel gains, H1 and H2, as defined in (3) and  (4), respectively, do not conform to any known closed-form  distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Thus, for the sake of simplification of the analytical  performance, we can approximate G1 and G2 (for M1 ≫ 1  and M2 ≫ 1) as |H1|2 =  ��M1  i=1  ��¯gi  1  �� ��gi  1  ��  �2  and |H2|2 =  ��M2  i=1  ��¯gi  2  �� ��gi  2  ��  �2  , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' The distribution function for  the channel gain, |H|2, can be defined for g ≥ 0 as [11]  f|H|2(y) =  √ya  2ba+1Γ(a + 1) exp  �  −  √y  b  �  ,  (8)  F|H|2(y) =  γ  �  a + 1,  √y  b  �  Γ (a + 1) √y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  (9)  Here,  a  and  b  are  the  variables  defined  as  a  =  m0Γ(m0)2MmlΓ(ml)2  m0Γ(m0)2mlΓ(ml)2−Γ(m0+ 1  2 )2Γ(ml+ 1  2 )2 − N − 1 and b  =  m0Γ(m0)2mlΓ(ml)2−Γ(m0+ 1  2 )2Γ(ml+ 1  2 )2  � m0  Ωg Γ(m0)Γ(m0+ 1  2 )  � ml  Ωgl Γ(ml)Γ(ml+ 1  2 ), with N ∈ {M1, M2}  and l ∈ {1, 2}, for H ∈ {H1, H2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Further, Γ(·) represents  the Gamma function and γ(·, ·) indicates the lower incomplete  Gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Ergodic Rate  The ergodic rates for DR1 and DR2 can be formu-  lated as RDR1 = E [log2 (1 + SINRDR1)] and RDR2 =  E [log2 (1 + SINRDR2)],  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  Since  the  channel  gain’s exact distribution is unknown, the expectations are math-  ematically intractable, and thus a closed-form expression may  not be derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Hence, we resort to approximating the ergodic  rates of DRs with tight upper and lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Specifically,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  the upper bound is derived by invoking Jensen’s inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' and  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='the lower bound is derived by utilizing the approximate PDF  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='Rl  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='DR1 =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='ln (2) Γ(a + 1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='π csc (aπ/2) F1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='(a + 2) (β1b)a+2 (ρr)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='a  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='2 +1 +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='π sec (aπ/2) F1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='(a + 1) (β1b)a+1 (ρr)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='a+1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='+2 a (a − 1) ψ(0) (a + 1) +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='a2 − a  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='Γ(a − 1) ln  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='b2 β2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='1 ρr  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='+ Γ(a − 1) F3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='(10)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='Rl  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='DR2 =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='ln (2)Γ(a + 1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='π csc (aπ/2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='(a + 2) ba+2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='F4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='(c1ρr)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='a  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='2 +1 −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='F5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='(c2ρr)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
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+page_content='+ π sec (aπ/2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
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+page_content='Γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
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+page_content='Γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
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+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='Ωmh1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
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+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='Ωm0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
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+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='Ωm1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
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+page_content='(12)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
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+page_content='DR2 = log2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
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+page_content='1 + Ξ2Ωh2 + M2Ξ2Ωm0Ωm2 + M2 (M2 − 1) Ξ2Ωm0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
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+page_content='Γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
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+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='Γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='m2 + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
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+page_content='m0 {Γ (m0)}2 m2 {Γ (m2)}2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
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+page_content='2M2Ξ2Γ(mh2 + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='2)Γ(m0 + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='2)Γ(m2 + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='2)�Ωmh2 Ωm0Ωm2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='Γ(mh2)Γ(m0)Γ(m2)√mh2m0m2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='− log2 [1 + Ξ3Ωh2 + M2Ξ3Ωm0Ωm2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
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+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='Γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='m0+ 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
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+page_content='��2 Ωm2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='Γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='m2+ 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
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+page_content='2)Γ(m0+ 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='2)Γ(m2+ 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='2)�Ωmh2Ωm0Ωm2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='Γ(mh2)Γ(m0)Γ(m2)√mh2m0m2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='(13)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='as described in (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' The bounds on the ergodic rate of DRs  are evaluated following a series of mathematical manipulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  The following Lemmas present the upper and lower bound for  the proposed RIS-enabled NOMA-based D2D communication  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' The lower bound on the ergodic rates of DR1 and  DR2 can be expressed as in (10) and (11), shown on the top  of the next page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' The proof is presented in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Likewise, the upper bound on the ergodic rate of  DR1 and DR2 can be expressed as in (12) and (13), shown on  the top of the page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' The proof is presented in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' SE and EE  Based on the ergodic rate established in the preceding  subsection, SE of RIS-enabled NOMA-based D2D can be  described as SE = RDR1 + RDR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Similarly, the EE can be  defined as the ratio of the SE to the total power utilized, Ptot,  in bits/Joule/Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Ptot consists of the power utilized by the BS,  DT, RIS, and DRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Thus, the EE may be represented as  EE = SE  Ptot  =  SE  (1 + α)Pr + MPRE + 2PU  ,  (14)  where Pr denotes the static power consumption of DT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Like-  wise, αPr is the dynamic power consumption at DT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Further,  PRE denotes the power consumed by each of the RE and PU  is the power utilized by DR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  0  5  10  15  20  25  30  6  8  10  12  14  16  18  20  22  24  26  Transmit SNR (dB)  SE (bps/Hz)  Exact − Simulation  Lower Bound  Upper Bound  M = 200, 100, 50, 20, 10  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' SE versus transmit SNR with varying M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' SIMULATION RESULT  This section presents the simulation and analytical results  for the proposed RIS-empowered NOMA-based D2D commu-  nication system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' For the direct links, the Nakagami fading  parameters are assumed to be mh1 = mh2 = 2 and, for the RIS  reflected links, m0 = m1 = m2 = 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Likewise,  the power allocation factor for DR1 is β2  1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='3 while for DR2  is β2  2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='7, if not specified otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Additionally, the value  of RE allocation parameter η is assumed to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' 2 shows the SE results for the proposed RIS-empowered  NOMA-based D2D communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Specifically, it shows SE  with respect to the transmit power while comparing the simu-  lation and analytical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' These results can easily infer the  following observations: 1) Apart from smaller M, analytical  SE is quite precise compared to simulation-based SE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' 2) Due  0  20  40  60  80  100  120  140  160  180  200  4  6  8  10  12  14  16  18  20  22  24  26  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' of REs (M)  SE (bps/Hz)  NOMA − Exact Simulation  OMA − Exact Simulation  NOMA − Lower Bound  OMA − Lower Bound  NOMA − Upper Bound  OMA − Upper Bound  ρr = 10, 20, 30 dB  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' SE of NOMA-/OMA-based D2D with respect to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  0  5  10  15  20  25  30  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='5  4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='5  Transmit SNR (dB)  EE (bits/Hz/Joule)  NOMA  OMA  M = 20, 50, 200  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' EE versus transmit SNR at different M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  to the multiplicative path-loss, for less number of REs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=',  smaller M, the received power from the direct link is signif-  icant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' However, as the number of REs increases, the received  power from a RIS-reflected link is much more than the power  received from the direct link to the extent that it can safely  be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Thus, it can easily be inferred from the analytical  and simulation framework that the received signal power from  the direct link is relatively insignificant and can be ignored as  compared to the received power from the RIS-reflected link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' 3 shows the SE of the proposed RIS-empowered D2D  communication system for both NOMA and OMA scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  Specifically, the SE of both NOMA and OMA scenarios is  plotted with respect to the number of REs for different SNRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  It can easily be observed here that, instead of increasing the  transmit power, the number of REs can be increased to get the  same SE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' In other words, for fixed required SE, we can tradeoff  the transmit power with the number of REs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' As D2D users  are usually power constrained, RIS-empowered D2D commu-  nication can be a viable alternative to cut down the transmit  power and improve network EE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Further, as evident from the  result, RIS-empowered NOMA-based D2D is more spectrally  efficient as compared to OMA-based D2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' For instance, SE at  20 dB SNR and M = 20 is 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='26 bps/Hz for NOMA and 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='76  bps/Hz for OMA, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' The NOMA gain will increase  when the channel gain between UEs increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Further, SE also  improves with the number of REs, as evident from the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  Likewise, SE grows as the transmit power goes up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' 4 shows EE of the proposed RIS-empowered D2D  communication system, where EE of both NOMA and OMA  scenarios is plotted with respect to the transmit power for  the varying number of REs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Further, it can be inferred that  RIS-empowered D2D is energy-efficient as compared to OMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  Further, it can also be observed that the EE increases with  the number of REs, whereas EE decreases as the transmit  power increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' This is because SE increases linearly while  the transmit power increases logarithmically;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' thus, the overall  compounding impact decreases EE while increasing transmit  SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' In addition, EE is likewise saturated for a large number of  REs, and no further gains are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' The result also demon-  strates that increasing the number of REs does not improve  performance, as SE increases while EE becomes saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  Thus, it can be inferred that RIS improves the SE and EE  performance of the D2D system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' CONCLUSION  In this paper, we investigated the performance of a RIS-  empowered NOMA-based D2D communication system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Specif-  ically, we derived the closed-form expressions for SE’s upper  and lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' As shown through the results, apart from  the smaller values of the number of REs, the bounds are pretty  tight and converge to exact SE, even for moderate REs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Further,  we have also investigated the EE performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Since the D2D  devices are usually power-constrained, the results show that  the transmit power can be a tradeoff with the number of REs  at RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Additionally, the results are also compared with the  OMA scenario, where it has been shown that NOMA-based  D2D outperforms the OMA-based case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' ACKNOWLEDGEMENT  This work was supported by the Nazarbayev University CRP  Grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' 11022021CRP1513.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  APPENDIX A  PROOF OF LEMMA 1  The ergodic rate of DR1 can be formulated as  RDR1 =  1  ln (2)  ∞  �  0  ln  �  1 + β2  1 |y|2 ρr  �  f|H|2(y)dy  �  ��  �  J1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  (15)  Further, using (8) and the below relation ln(t)  =  (t −  1) 2F1 (1, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' 1 − t), J1 in (15) can be modified as  J1 =  1  Γ(a + 1)  ∞  �  0  √g 2F1 (1, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' −g)  2ba+1  e  �  −  √g  b  �  dg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  (16)  Here, 2F1(·, ·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' ·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' ·) represents the Gauss hyper-geometric func-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Now, this J1 can be solved utilizing [11, Theorem 3] and  substituted in (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' After rearranging the terms, the ergodic  rate for DR1 can be given as shown in (11), where F1 =  1F2  �  1+ a  2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' 3  2, 2+ a  2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  −1  4b2β2  1ρr  �  , F2 = 1F2  �  a+1  2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' 1  2, a+3  2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  −1  4b2β2  1ρr  �  and F3 = 2F3  �  1, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' 2, 1− a  2, 3−a  2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  −1  4b2β2  1ρr  �  Likewise, the ergodic rate for DR2, RDR2, can be given by  RDR2 =  1  ln (2)  \uf8f1  \uf8f2  \uf8f3  ∞  �  0  ln  �  1 + c1 |y|2 ρr  �  f|H|2(y)dy  −  ∞  �  0  ln  �  1 + c2 |y|2 ρr  �  f|H|2(y)dy  \uf8fc  \uf8fd  \uf8fe ,  (17)  where c1 and c2 are defined as c1 = β2  1 + β2  2 and c2 = β2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  It  can  be evaluated similarly  to  J1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' After  rearranging  the terms, the ergodic rate for DR2 can be given as  shown in (12), where F4 =  1F2  �  1+ a  2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' 3  2, 2+ a  2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  −1  4b2c1ρr  �  ,  F5  =  1F2  �  1+ a  2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' 3  2, 2+ a  2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  −1  4b2c2ρr  �  ,  F6  =  1F2  �  a+1  2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' 1  2, a+3  2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  −1  4b2c1ρr  �  , F7 = 1F2  �  a+1  2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' 1  2, a+3  2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  −1  4b2c2ρr  �  ,  F8  =  2F3  �  1, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' 2, 1− a  2, 3−a  2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  −1  4b2c1ρr  �  and  F9  =  2F3  �  1, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' 2, 1− a  2, 3−a  2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  −1  4b2c2ρr  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' This completes the proof of  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  APPENDIX B  PROOF OF LEMMA 2  Applying Jensen’s inequality, we define the upper bound for  DR1 as Ru  DR1, where RDR1 ≤ Ru  DR1, with Ξ1 = β2  1ρr, as  Ru  DR1 = log2  �  1 + Ξ1E  �  |H1|2��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  (18)  To calculate E  �  |H1|2�  , we apply the binomial expansion  theorem (BET) as  E  �  |H1|2�  = E  \uf8ee  \uf8f0  �����h1 +  M1  �  i=1  ��¯gi  1  �� ��gi  1  ��  �����  2\uf8f9  \uf8fb = E  �  |h1|2�  �  ��  �  E1  + E  \uf8f1  \uf8f2  \uf8f3  � M1  �  i=1  ��¯gi  1  �� ��gi  1  ��  �2\uf8fc  \uf8fd  \uf8fe  �  ��  �  E2  +2 E  � M1  �  i=1  ��¯gi  1  �� ��gi  1  �� |h1|  �  �  ��  �  E3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' (19)  Now, we have E1 = Ωh1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Likewise, to calculate E2, we apply  BET again;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' thus, on expanding, E2 can be expressed as  E  � M1  �  i=1  ��¯gi  1  ��2 ��gi  1  ��2  �  + E  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  M1  �  i=1  M1  �  i=1  j̸=i  ��¯gi  1  �� ��gi  1  ��  ���¯gj  1  ���  ���gj  1  ���  \uf8fc  \uf8f4  \uf8fd  \uf8f4  \uf8fe  , (20)  where E  ��M1  i=1  ��¯gi  1  ��2 ��gi  1  ��2�  =  M1Ωm0Ωm1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Further, for  E  ��M1  i=1  �M1  i=1  j̸=i  ��¯gi  1  �� ��gi  1  ��  ���¯gj  1  ���  ���gj  1  ���  �  ,  the  expected  value of  a  Nakagami-m  variable  can  be  given  as  E{|g1|}  =  Γ(m1+ 1  2 )  Γ(m1)  �� Ωm1  m1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Since ¯gi  1 and gi  1 are mutually independent,  we can have  E  \uf8f1  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f3  M1  �  i=1  M1  �  j=1  j̸=i  ��¯gi  1  �� ��gi  1  ��  ���¯gj  1  ���  ���gj  2  ���  \uf8fc  \uf8f4  \uf8f4  \uf8fd  \uf8f4  \uf8f4  \uf8fe  = M1 (M1 − 1)  �Ωm0  m0  �  ×  �  Γ  �  m0 + 1  2  ��2 �  Γ  �  m1 + 1  2  ��2  {Γ (m0)}2 {Γ (m1)}2  �Ωm1  m1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  (21)  Likewise, E3 can be calculated as  E3 = M Γ  �  mh1 + 1  2  �  Γ  �  m0 + 1  2  �  Γ  �  m1 + 1  2  �  ×  �  Ωmh1 Ωm0Ωm1  mh1m0m1  / [Γ (mh1) Γ (m0) Γ (m1)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  (22)  Finally, putting E1, E2 and E3 all together yields E  �  |H1|2�  which can be put in (18) to give the desired upper bound as  shown in (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  Likewise, the upper bound on the ergodic rate of DR2, Ru  DR2  can be defined as RDR2 ≤ Ru  DR2, where RDR2 can be defined  as  RDR2 = E  �  log2  �  1 + |H2|2 �  β2  1 + β2  2  �  ρr  1 + |H2|2 β2  1ρr  ��  ,  = E  �  log2  �  1 + Ξ2 |H2|2�  − log2  �  1 + Ξ3 |H2|2��  ,  (23)  where Ξ2 =  �  β2  1 + β2  2  �  ρr and Ξ3 = β2  1ρr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Thus, Ru  DR2 can be  defined as  Ru  DR2=log2  �  1+Ξ2E  �  |H2|2��  −log2  �  1+Ξ3E  �  |H2|2��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  (24)  Similar to E  �  |H1|2�  , E  �  |H2|2�  can be evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' After sub-  stituting and rearranging the terms, Ru  DR2 is given in (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  REFERENCES  [1] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
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+page_content=' Tutor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
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+page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
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+page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Netw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
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+page_content=' 3,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' 197–207, Sept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  [5] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
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+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' 69, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' 11, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' 7283–7296, Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  [6] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
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+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' 69, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' 11, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' 7790–7805, Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content='  [7] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
+page_content=' Ni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
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+page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE2T4oBgHgl3EQfkAch/content/2301.03973v1.pdf'}
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+Sectional Voronoi tessellations:
+Characterization and high-dimensional limits
+Anna Gusakovaa, Zakhar Kabluchkob, and Christoph Thälec
+Abstract
+The intersections of beta-Voronoi, beta-prime-Voronoi and Gaussian-Voronoi tessellations in Rd with
+ℓ-dimensional affine subspaces, 1 ≤ ℓ ≤ d − 1, are shown to be random tessellations of the same type
+but with different model parameters. In particular, the intersection of a classical Poisson-Voronoi
+tessellation with an affine subspace is shown to have the same distribution as a certain beta-Voronoi
+tessellation. The geometric properties of the typical cell and, more generally, typical k-faces, of the
+sectional Poisson-Voronoi tessellation are studied in detail. It is proved that in high dimensions, that
+is as d → ∞, the intersection of the d-dimensional Poison-Voronoi tessellation with an affine subspace
+of fixed dimension ℓ converges to the ℓ-dimensional Gaussian-Voronoi tessellation.
+Keywords: Beta-Voronoi tessellation, Gaussian-Voronoi tessellation, high-dimensional limit, Laguerre
+tessellation, Poisson point process, Poisson-Voronoi tessellation, sectional tessellation, stochastic geo-
+metry, typical cell
+MSC: 60D05, 60G55
+1
+Introduction
+The present paper is devoted to the study of affine sections of Poisson-Voronoi tessellations. To define
+these, let ζ ⊂ Rd be the set of atoms of a stationary point process in Rd. For each point x ∈ ζ we
+construct the Voronoi cell V (x, ζ) of x as the set of all points in Rd which are closer to x than to any
+other point of ζ:
+V (x, ζ) := {z ∈ Rd : ∥x − z∥ ≤ ∥y − z∥
+for all y ∈ ζ \ {x}},
+where ∥ · ∥ stands for the Euclidean norm. The Voronoi cell can be thought of as a zone of influence
+or attraction of the point x and it is known that each Voronoi cell is a convex polytope in Rd, with
+probability 1. The collection of all these polytopes is the Voronoi tessellation associated with ζ. If
+ζ is a homogeneous Poisson point process with constant intensity ρ > 0, this construction yields the
+Poisson-Voronoi tessellation, denoted here by Wd,ρ, – one of the most classical models studied in
+stochastic geometry. We refer to the monographs [22, 27, 29] for more detailed information, applications
+and further references on Voronoi tessellations and in particular the Poisson-Voronoi tessellation.
+A number of random tessellations studied in stochastic geometry, such as the Poisson hyperplane
+or the STIT tessellations, have the distinguished feature of being stable under intersections with lower-
+dimensional affine subspaces. By this we mean that the intersection with an affine subspace of one of these
+random tessellations is again a model of the same type within the intersecting subspace. For example, the
+intersection of a Poisson hyperplane tessellation with an affine subspace L is again a Poisson hyperplane
+tessellation within L. However, a similar property is not true for the Poisson-Voronoi tessellation. In fact,
+it has been shown by Chiu, Van De Weygaert and Stoyan [6] that the intersection of the Poisson-Voronoi
+tessellation with an affine subspace cannot be a Voronoi tessellation induced by any stationary point
+process within the subspace. In other words, the sectional Poisson-Voronoi tessellation is necessarily a
+‘non-Voronoi’ tessellation. However, besides a few mean values determined in [6, 20] further probabilistic
+aMünster University, Germany. Email: [email protected]
+bMünster University, Germany. Email: [email protected]
+cRuhr University Bochum, Germany. Email: [email protected]
+1
+arXiv:2301.03297v1  [math.PR]  9 Jan 2023
+
+or geometric information about the sectional Poisson-Voronoi tessellation seems not available in the
+existing literature, although they are of importance for stereological applications (see [29, Chapter 11.5.4]
+or [21, Section 14.4.6] as well as the references cited therein). It is one of the main purposes of this paper
+to derive a precise description of the sectional Poisson-Voronoi tessellation and to study its typical cell.
+We do this by establishing a connection with the so-called β-Voronoi tessellations, a random tessellation
+model we recently introduced and studied in the series of papers [10, 7, 8, 9]. Their analysis in turn was
+based on the connection with the class of beta random polytopes, which has already seen a number of
+applications in stochastic geometry [11, 13, 14, 15].
+We study the problem of the sectional Poisson-Voronoi tessellation just explained in a more general
+framework. In fact, the random tessellation we study is either
+• a β-Voronoi tessellation Vd,β,γ in Rd with parameters β ≥ −1 and γ > 0,
+• a β′-Voronoi tessellation V′
+d,β,γ in Rd with parameters β > d
+2 + 1 and γ > 0,
+• or a Gaussian-Voronoi tessellation Gd,λ in Rd with parameter λ > 0;
+a description of all these models will be provided in Section 3. We remark that the classical Poisson-
+Voronoi tessellation generated by a stationary Poisson point process in Rd with intensity ρ > 0 appears
+in this framework as the β-Voronoi tessellation corresponding to the parameters β = −1 and γ =
+π
+d+1
+2 ρ/Γ( d+1
+2 ). Now, let L ⊂ Rd be an affine subspace of dimension 1 ≤ ℓ ≤ d − 1. We show in Theorem
+4.1 below that
+• the sectional tessellation Vd,β,γ ∩ L is a (β + d−ℓ
+2 )-Voronoi tessellation in L with the same γ,
+• the sectional tessellation V′
+d,β,γ ∩ L is a (β − d−ℓ
+2 )′-Voronoi tessellation in L with the same γ,
+• the sectional tessellation Gd,λ ∩ L is again a Gaussian-Voronoi tessellation in L with the same λ.
+In particular, the intersection of the classical Poisson-Voronoi tessellation Wd,ρ with L turns out to be
+a β-Voronoi tessellation within L with β =
+d−ℓ
+2
+− 1 and γ = π
+d+1
+2 ρ/Γ( d+1
+2 ).
+For clarity we should
+remark that none of the random tessellations Vd,β,γ with β > −1 are actually Voronoi tessellations.
+Our terminology is motivated by the fact that Vd,β,γ may be viewed as a deformation of the classical
+Poisson-Voronoi tessellation (which corresponds to β = −1).
+With the identification of the sectional Poisson-Voronoi tessellation at hand, in the second part of this
+paper we study its geometric properties. More precisely, we determine in Theorem 5.1 its face intensities
+in terms of quantities which have already appeared in the study of beta random polytopes [11, 15]. From
+here on, we determine the expected volume, the expected intrinsic volumes as well as the expected f-
+vector of the typical cell (and even more generally the typical k-face) of the sectional Poisson-Voronoi
+tessellation. Moreover, generalizing earlier results of Miles [20] we consider the asymptotics, as d → ∞, of
+several characteristics (such as the volume of the typical cell) of the sectional Poisson-Voronoi tessellation
+and identify the limits with the corresponding characteristics of a suitable Gaussian-Voronoi tessellation.
+The weak convergence on the level of tessellations is discussed as well using a coupling construction
+similar to the one in [8].
+2
+Preliminaries on random tessellations
+In this section we collect some definitions and facts about general stationary random tessellations in Rd.
+For more detailed discussions we refer the reader to [27, Chapters 4 and 10] as well as [29, Chapter 10].
+A tessellation T in Rd is a countable, locally finite collection of d-dimensional polytopes, which cover
+the space and have non-empty, disjoint interiors. The elements of T are called the cells of T. Given a
+polytope c ⊂ Rd we denote by Fk(c) the set of its k-dimensional faces, 0 ≤ k ≤ d, where Fd(c) = {c},
+and let F(c) := �d
+k=0 Fk(c). A tessellation T is called face-to-face if for any two of its cells c1, c2 ∈ T
+one has that
+c1 ∩ c2 ∈ (F(c1) ∩ F(c2)) ∪ {∅},
+2
+
+that is, the intersection of two cells is either empty or a common face of both cells. For a face-to-face
+tessellation T one defines Fk(T) = �
+c∈T Fk(c) and F(T) = �
+c∈T F(c). A face-to-face tessellation in Rd
+is called normal if each k-dimensional face of the tessellation is contained in the boundary of precisely
+d + 1 − k cells, for all k ∈ {0, 1, . . . , d − 1}.
+We denote by T the set of all face-to-face tessellations in Rd, which is supplied with a measurable
+structure as in [27, Chapter 10]. By a random tessellation we understand a particle process T in Rd
+(in the usual sense of stochastic geometry, see [27, Section 4.1]) satisfying supp T ∈ T almost surely. It is
+convenient to identify the random point process T with its support. A random tessellation is stationary,
+provided that its distribution is invariant under all shifts in Rd and isotropic if its distribution is invariant
+under all rotations in Rd. For a stationary random tessellation T and k ∈ {0, 1, . . . , d − 1} we define the
+stationary particle process T (k) := �
+F∈Fk(T ) δF of k-dimensional polytopes, which is referred to as the
+process of k-faces.
+Next, we recall the concept of a typical cell (and a typical k-face) of a stationary random tessellation
+T ; see [27, Section 4.1,4.2], [27, page 450], [26, Section 4.3] for more details. Let C′ be the space of
+non-empty compact subsets of Rd endowed with the Hausdorff metric. A centre function is a Borel
+function z : C′ → Rd such that z(C + m) = z(C) + m for all C ∈ C′ and m ∈ Rd. The intensity of
+k-faces of T is defined by
+γk(T ) := E
+�
+F∈Fk(T )
+1(z(F) ∈ [0, 1]d),
+k = 0, . . . , d.
+These quantities are known to be independent of the choice of the centre function z. Assuming that
+γk(T ) ∈ (0, ∞), the typical k-face of T with respect to the centre function z is the k-dimensional
+random polytope whose distribution is given by
+Pz
+T ,k( · ) :=
+1
+γk(T ) E
+�
+F∈Fk(T )
+1(F − z(F) ∈ · )1(z(F) ∈ [0, 1]d).
+In particular, for k = d we get the concept of the typical cell of T . It should be noted that translation-
+invariant characteristics of the distribution of the typical k-face do not depend on the choice of z. More
+precisely, if z and z′ are two centre functions, then Pz′
+T ,k is the push-forward of Pz
+T ,k under the map
+F �→ F − z′(F).
+3
+Construction of β-, β′- and Gaussian-Voronoi tessellations
+3.1
+General Laguerre tessellations
+In this section we only briefly recall some facts about Laguerre tessellations and refer the reader to [7,
+Sections 3.2–3.4] and [8, Sections 2.3 and 3.1] for further details.
+We start by defining a general Laguerre tessellation. Given two points v, w ∈ Rd and h ∈ R we define
+the power of w with respect to the pair (v, h) as
+pow(w, (v, h)) := ∥w − v∥2 + h.
+In this situation, h is referred to as the weight (or height) of the point v. Let X be a countable set of
+marked points of the form (v, h) ∈ Rd × R. Then the Laguerre cell of (v, h) ∈ X is the set
+C((v, h), X) := {w ∈ Rd : pow(w, (v, h)) ≤ pow(w, (v′, h′)) for all (v′, h′) ∈ X}.
+The point v is called the nucleus of the cell C((v, h), X). Note that a Laguerre cell may be empty and
+even if it is non-empty, it does not need to contain its nucleus. The collection of all non-empty Laguerre
+cells of X is called the Laguerre diagram:
+L(X) := {C((v, h), X): (v, h) ∈ X, C((v, h), X) ̸= ∅}.
+3
+
+In the special case when the heights h of all points are the same (say, h0 ∈ R) the above definition leads
+to the classical Voronoi cell. More precisely, let Y be a countable set of points in Rd whose “marked”
+version X is obtained by attaching a fixed weight h0 to each point. Then the Voronoi cell of v ∈ Y is
+V (v, Y ) = C((v, h0), X) = {w ∈ Rd : ∥w − v∥ ≤ ∥w − v′∥ for all v′ ∈ Y }.
+The collection of the Voronoi cells of all v ∈ Y is called the Voronoi diagram V(Y ).
+It should
+be mentioned that a Laguerre diagram is not necessarily a tessellation in Rd, at least as long as no
+additional assumptions on the geometric properties of the set X are imposed. Such assumptions have
+been described in detail in [18, 19, 25]. In the present article we are interested in random tessellations
+built on Poisson point processes. More precisely, we consider a Poisson point process ξ in Rd × E, where
+E ⊂ R is a Borel set (an interval), and the corresponding Laguerre diagram L(ξ). Lemmas 1 and 2 in
+[7] (see also [8, Lemma 2.1]) provide sufficient conditions on ξ which ensure that, almost surely, L(ξ) is a
+stationary random face-to-face normal tessellation in Rd. In the following we work under these conditions
+and remark that they are automatically satisfied in the three cases we consider from Section 3.3 on.
+3.2
+Laguerre tessellations via paraboloid growth processes
+An alternative approach to the construction of Laguerre diagrams uses so-called paraboloid growth pro-
+cesses with overlaps (or simply paraboloid growth process), which were first introduced in [1, 28] in order
+to study the asymptotic geometry of random polytopes; see also [2, 3, 4, 5]. In this section we briefly
+describe this rather useful construction and refer for more details to [8, Section 3.1]. Let
+Π±,x := {(v′, h′) ∈ Rd × R: h′ = ±∥v′ − v∥2 + h}
+be the upward (+) and downward (−) standard paraboloids with apex x := (v, h) ∈ Rd × R, denoted
+as apex Π±,x := x. In case (v, h) = (0, 0) we simply write Π± = Π±,(0,0). Given a set A ⊂ Rd × R we put
+A↓ : = {(v, h′) ∈ Rd × R: (v, h) ∈ A for some h ≥ h′},
+A↑ : = {(v, h′) ∈ Rd × R: (v, h) ∈ A for some h ≤ h′}.
+Following the definition from [1], for a given Poisson point process ξ in Rd × R, we introduce the
+paraboloid growth process Ψ(ξ):
+Ψ(ξ) :=
+�
+x∈ξ
+Π↑
++,x.
+It should be noted that, in typical situations, the majority of paraboloids will be completely covered by
+other paraboloids, implying that they do not "contribute" to the model and can thus be omitted without
+loosing any information about the set Ψ(ξ). This leads to the definition of extreme points. A point x ∈ ξ
+is called extreme in the paraboloid growth process Ψ(ξ) if and only if its associated paraboloid is not
+fully covered by the paraboloids associated with other points of ξ, i.e., if
+Π↑
++,x ̸⊂
+�
+y∈ξ,y̸=x
+Π↑
++,y.
+We denote by ext(Ψ(ξ)) the set of all extreme points of the paraboloid growth process Ψ(ξ). Using the
+paraboloid growth process we can construct a random diagram in Rd. Given a point x = (v, h) ∈ ξ define
+the Ψ-cell of x as
+CΨ(x, ξ) :=
+�
+{w ∈ Rd :
+�
+(w, 0)↑ ∪ (w, 0)↓�
+∩ bd Ψ(ξ) ∈ Π+,x},
+if x ∈ ext(Ψ(ξ)),
+∅,
+otherwise,
+where bd A denotes the boundary of a set A.
+In other words, w belongs to CΨ(x, ξ) if and only if
+∥w − v∥2 + h ≤ ∥w − v′∥ + h′ for all (v′, h′) ∈ ξ. Thus, the Ψ-cell of an extreme point x of the paraboloid
+growth process Ψ(ξ) is non-empty and coincides with the Laguerre cell C(x, ξ). Next, we construct the
+diagram LΨ(ξ) as the collection of all non-empty Ψ-cells:
+LΨ(ξ) := {CΨ(x, ξ): CΨ(x, ξ) ̸= ∅} = {CΨ(x, ξ): x ∈ ext(Ψ(ξ))}.
+We directly have that LΨ(ξ) = L(ξ).
+4
+
+Figure 3.1: Left panel: Simulation of a β-Voronoi tessellation in R2 with β = 5. Middle panel: Simulation
+of a β′-Voronoi tessellation in R2 with β = 2.5. Right panel: Simulation of a Gaussian-Voronoi tessellation
+in R2.
+3.3
+Three families of random tessellations
+In this article we consider random tessellations in Rd build on the following three families of Poisson
+point processes. For β > −1 and 0 < γ < ∞ we consider a Poisson point process ηd,β,γ in Rd × [0, +∞)
+whose intensity measure has density
+(v, h) �→ γ cd+1,βhβ,
+cd+1,β := Γ
+� d+1
+2
++ β + 1
+�
+π
+d+1
+2 Γ(β + 1)
+,
+(3.1)
+with respect to the Lebesgue measure on Rd × [0, +∞). Further, for β > d
+2 + 1 and 0 < γ < ∞ we
+consider a Poisson point process η′
+d,β,γ in Rd × (−∞, 0) with intensity measure having density
+(v, h) �→ γ c′
+d+1,β(−h)−β,
+c′
+d+1,β :=
+Γ (β)
+π
+d+1
+2 Γ(β − d+1
+2 )
+,
+(3.2)
+with respect to the Lebesgue measure on Rd × (−∞, 0). The constants cd+1,β and c′
+d+1,β in the above
+definitions are introduced for convenience. For example, they make the statement of Theorem 4.1 below
+more transparent. Finally, for λ > 0 and 0 < γ < ∞ we consider a Poisson point process ζd,λ,γ in Rd × R
+whose intensity measure has density
+(v, h) �→ γ eλh,
+with respect to the Lebesgue measure on Rd × R. It was shown in [7, Lemma 3] and in [8, Section
+3.3] that the Poisson point processes ηd,β,γ and ζd,λ,γ satisfy the sufficient conditions of Lemma 1 and
+Lemma 2 in [7] and, hence, the corresponding Laguerre diagrams Vd,β,γ := L(ηd,β,γ), V′
+d,β,γ := L(η′
+d,β,γ)
+and Gd,λ := L(ζd,λ,γ) are stationary random normal tessellations in Rd, which are called β-Voronoi,
+β′-Voronoi and Gaussian-Voronoi tessellations, respectively. These tessellations have been studied
+in [10, 7, 8, 9], where they were considered in Rd−1 instead of Rd. Simulations of these tessellations in
+the plane are shown in Figure 3.1. Note that although the point process η′
+d,β,γ is well-defined in the range
+β > d+1
+2 , the corresponding β′-Voronoi tessellation exists in the smaller range β > d
+2 + 1 only, see [7,
+Lemma 3].
+Remark 3.1. Note that changing the parameter γ amounts to shifting the Poisson point process ζd,λ,γ
+along the height coordinate h. In particular, the distribution of the point process Gd,λ does not depend
+on the choice of γ, which is reflected in our notation.
+Remark 3.2. It will be convenient to extend the above definition of the β-Voronoi tessellation to the
+case β = −1 (with arbitrary γ > 0) by defining Vd,−1,γ := Wd,r(d)γ to be the classical Poisson-Voronoi
+5
+
+tessellation constructed on the homogeneous Poisson point process on Rd with constant intensity r(d)γ,
+where
+r(d) := Γ
+�d + 1
+2
+�
+π− d+1
+2 .
+A justification for this definition is given by the following proposition. For the necessary background on
+point processes and weak convergence we refer to [24, Chapter 3].
+Proposition 3.3. As β ↓ −1, the Poisson process ηd,β,γ converges, weakly on the space of locally finite
+integer-valued measures on Rd × [0, ∞), to the Poisson point process whose intensity measure is the
+Lebesgue measure on Rd × {0} times r(d)γ.
+Proof. Write β = −1 + ε with ε ↓ 0. Then the constant appearing in (3.1) satisfies
+cd+1,β = Γ
+� d+1
+2
++ ε
+�
+π
+d+1
+2 Γ(ε)
+= Γ
+� d+1
+2
+�
+π
+d+1
+2
+ε(1 + o(1)),
+as ε ↓ 0. It follows that for every c > 0 and every bounded Borel set B ⊂ Rd the number of points of the
+Poisson point process ηd,β,γ appearing in B × [0, c] is Poisson-distributed with expectation
+γλd(B)cd+1,β
+� c
+0
+h−1+εdh −→
+ε↓0 Γ
+�d + 1
+2
+�
+π− d+1
+2 γλd(B) = r(d)γλd(B),
+where λd denotes the d-dimensional Lebesgue measure. Since the right-hand side does not depend on c,
+it follows that, for every 0 < c1 < c2, the expected number of points of ηd,β,γ in B × [c1, c2] converges
+to 0, as ε ↓ 0. Hence, the intensity measure of ηd,β,γ converges as β ↓ −1 to the Lebesgue measure
+on Rd × {0} times r(d)γ, vaguely on the space Rd × [0, ∞). Then, the claim of the proposition follows
+from [16, Theorem 16.16 (iv)] or [24, Propositions 3.6 and 3.19].
+4
+Affine sections of β, β′- and Gaussian-Voronoi tessellations
+In this section we study the intersection of the d-dimensional random tessellations Vd,β,γ, V′
+d,β,γ and Gd,λ
+with an affine subspace L ⊂ Rd of dimension ℓ ∈ {1, . . . , d − 1}. By stationarity and isotropy of these
+tessellations, we may and will assume without loss of generality that L = Rℓ is the linear subspace of Rd
+spanned by the first ℓ vectors of the standard orthonormal basis of Rd. The intersection of the tessellation
+Vd,β,γ with Rℓ will be denoted by Vd,β,γ ∩ Rℓ. Similar convention is used for the tessellations V′
+d,β,γ and
+Gd,λ. The following theorem identifies the distribution of Vd,β,γ ∩ Rℓ, V′
+d,β,γ ∩ Rℓ and Gd,λ ∩ Rℓ.
+Theorem 4.1. Fix integers d ≥ 2 and 1 ≤ ℓ ≤ d − 1.
+(i) For any β ≥ −1 and γ > 0, Vd,β,γ ∩ Rℓ has the same distribution as Vℓ,β+ d−ℓ
+2 ,γ.
+(ii) For any β > d
+2 + 1 and γ > 0, V′
+d,β,γ ∩ Rℓ has the same distribution as V′
+ℓ,β− d−ℓ
+2 ,γ.
+(iii) For any λ > 0, Gd,λ ∩ Rℓ has the same distribution as Gℓ,λ.
+Before we move on to the proof of Theorem 4.1 we would like to highlight the following special case
+which deals with sections of the classical Poisson-Voronoi tessellation. Together with the results we obtain
+below, this fully answers and resolves the problems raised in [6, 20, 21, 29].
+Corollary 4.2. Fix integers d ≥ 2 and 1 ≤ ℓ ≤ d − 1. Then for any ρ > 0 the intersection of the
+d-dimensional Poisson-Voronoi tessellation Wd,ρ of intensity ρ with Rℓ has the same distribution as
+V
+ℓ, d−ℓ
+2 −1,π
+d+1
+2
+ρ/Γ( d+1
+2 ).
+6
+
+Proof of Theorem 4.1. Let us first consider the case ℓ = d−1 meaning that we intersect with a hyperplane.
+Let ξ be one of the Poisson point processes ηd,β,γ, η′
+d,β,γ or ζd,λ,γ. The atoms of ξ live in the space
+Rd+1 = Rd × R; a generic point in this space is denoted by (v, h) with v = (v1, . . . , vd) ∈ Rd being the
+spatial coordinate and h ∈ R being the height coordinate. The Laguerre tessellation L(ξ) lives in the
+space Rd defined by the equation h = 0. The linear hyperplane L ≡ Rd−1 ⊂ Rd ⊂ Rd+1 with which our
+tessellations are intersected is given by the equations {vd = 0, h = 0}.
+In order to prove the statement we will use the representation of the Laguerre tessellation L(ξ) in
+terms of the paraboloid growth process Ψ(ξ) as described in Section 3.2. We extend the hyperplane
+L ⊂ Rd by adding the height coordinate, namely we consider
+L′ := {(v, h) = (v1, . . . , vd, h) ∈ Rd × R: vd = 0}.
+For every point x = (v, h) ∈ Rd+1, the intersection of the d-dimensional paraboloid Π+,x with L′ is a
+(d − 1)-dimensional upward paraboloid Π+,x′ ∩ L′ in L′ with apex given by
+x′ = f(v, h) := (v1, . . . , vd−1, 0, h + v2
+d) ∈ L′.
+If x runs through all atoms of ξ, then x′ runs through all atoms of the point process f(ξ) := {f(x): x ∈ ξ}
+on L′, which is also a Poisson point process by the mapping theorem; see [17, Theorem 5.1].
+The
+intersection of the Laguerre tessellation L(ξ) with L coincides with the Laguerre tessellation generated
+by the point process f(ξ) (within L).
+Now, we are going to identify the intensity measure µ of the Poisson point process f(ξ). To this end,
+we consider the Poisson point processes ηd,β,γ, η′
+d,β,γ and ζd,λ,γ separately.
+Case (i).
+Let first β > −1. By the mapping theorem for Poisson point processes [17, Theorem 5.1],
+f(ηd,β,γ) := {f(x): x ∈ ηd,β,γ} is a Poisson point process in Rd−1×[0, +∞) ⊂ L′. To compute its intensity
+measure µ, we take some Borel set B ⊂ L, any s > 0 and observe that an atom (v, h) of ηd,β,γ is mapped
+by f to B × [0, s] if and only if (v1, . . . , vd−1) ∈ B and h + v2
+d ≤ s. The latter condition means that h ≤ s
+and |vd| ≤
+√
+s − h. It follows that the intensity measure µ satisfies
+µ(B × [0, s]) = γ cd+1,β
+�
+Rd
+� ∞
+0
+hβ1(f(v, h) ∈ B × [0, s]) dhdv
+= γ cd+1,β
+�
+Rd
+� s
+0
+hβ1(v ∈ B × [−
+√
+s − h,
+√
+s − h]) dhdv
+= 2γ cd+1,β λd−1(B)
+� s
+0
+hβ√
+s − h dh
+= γ Γ( d+1
+2
++ β + 1)
+π
+d
+2 Γ(β + 3
+2)
+λd−1(B) sβ+ 3
+2
+β + 3
+2
+.
+(4.1)
+In the case β = −1, we let ηd,−1,γ be the Poisson point process on Rd × {0} (which is considered as a
+subset of Rd+1) whose intensity with respect to the Lebesgue measure on Rd ×{0} is constant and equals
+Γ( d+1
+2 )π− d+1
+2 γ. Thus, the heights of all points in ηd,−1,γ are 0. Then, the Laguerre tessellation generated
+by ηd,−1,γ on Rd coincides with Vd,−1,γ by our convention described in Remark 3.2. To compute the
+intensity measure µ of the Poisson point process f(ηd,−1,γ), we take some Borel set B ⊂ L, any s > 0
+and observe that an atom (v, 0) of ηd,−1,γ is mapped by f to B × [0, s] if and only if (v1, . . . , vd−1) ∈ B
+and v2
+d ≤ s. It follows that the intensity measure µ satisfies
+µ(B × [0, s]) = γ Γ( d+1
+2 )
+π
+d+1
+2
+�
+Rd 1(f(v, 0) ∈ B × [0, s]) dv
+= γ 2Γ( d+1
+2 )
+π
+d+1
+2
+λd−1(B)√s.
+(4.2)
+7
+
+By differentiating (4.1) and (4.2) with respect to s, it follows that for all β ≥ −1, the intensity
+measure of f(ηd,β,γ) has density
+(v′, 0, h) �→ γ cd,β+ 1
+2 hβ+ 1
+2 ,
+v′ ∈ Rd−1, h > 0,
+with respect to the Lebesgue measure on L × [0, +∞). Consequently, the Laguerre tessellation generated
+by f(ηd,β,γ) within L has the same distribution as Vd−1,β+ 1
+2 ,γ.
+Case (ii).
+Next, we deal with f(η′
+d,β,γ) := {f(x): x ∈ η′
+d,β,γ}. Let us first consider only those points
+of f(η′
+d,β,γ) that have negative height and compute the intensity measure µ of these points. The points
+with positive height coordinate have no influence on the resulting tessellation, as we will argue below.
+To determine µ, we take some Borel set B ⊂ L, any s < 0 and observe that an atom (v, −g) of η′
+d,β,γ
+(with g > 0) is mapped by f to B × (−∞, s] if and only if (v1, . . . , vd−1) ∈ B and v2
+d ≤ s + g. The latter
+condition means that g ≥ −s and |vd| ≤ √s + g. It follows that the intensity measure µ satisfies
+µ(B × (−∞, s]) = γ c′
+d+1,β
+�
+Rd
+� ∞
+0
+g−β1(f(v, −g) ∈ B × [−s, ∞)) dgdv
+= γ c′
+d+1,β
+�
+Rd
+� ∞
+−s
+g−β1(v ∈ B × [−√s + g, √s + g]) dgdv
+= 2γ c′
+d+1,β λd−1(B)
+� ∞
+−s
+g−β√g + s dg
+= γ
+Γ(β − 1
+2)
+π
+d
+2 Γ(β − d+1
+2 )
+λd−1(B)(−s)−β+ 3
+2
+β − 3
+2
+.
+Differentiating, we conclude that the intensity measure µ has density
+(v′, 0, h) �→ γ c′
+d,β+ 1
+2 (−h)−β+ 1
+2 ,
+v′ ∈ Rd−1, h < 0,
+with respect to the Lebesgue measure on L × (−∞, 0). So, by the mapping theorem [17, Theorem 5.1],
+the restriction of f(η′
+d,β,γ) to L × (−∞, 0) is a Poisson point process with the same intensity measure
+as η′
+d−1,β− 1
+2 ,γ. The Laguerre tessellation generated by this Poisson point process within L ≡ Rd−1 has
+the same distribution as V′
+d−1,β− 1
+2 ,γ.
+It remains to observe that adding the points of f(η′
+d,β,γ) with
+positive height coordinate does not change the Laguerre tessellation. Indeed, every point in Rd−1 × {0}
+is an accumulation point of η′
+d−1,β− 1
+2 ,γ, hence the lower boundary of the paraboloid growth process
+Ψ(η′
+d−1,β− 1
+2 ,γ) is contained in Rd−1×(−∞, 0] and points with positive height coordinate have no influence
+on the tessellation.
+Case (iii).
+Finally, we consider f(ζd,λ,γ), which is a Poisson point process in L′.
+To compute its
+intensity measure µ, we take some Borel set B ⊂ L, any s ∈ R and observe that an atom (v, h) of ζd,λ,γ
+is mapped by f to B × (−∞, s] if and only if (v1, . . . , vd−1) ∈ B and h + v2
+d ≤ s. It follows that
+µ(B × (−∞, s]) = γ
+�
+Rd
+�
+R
+eλh1(f(v, h) ∈ B × (−∞, s)) dhdv
+= 2γ λd−1(B)
+� s
+−∞
+ehλ√
+s − h dh
+= γ λd−1(B)
+√πeλs
+λ3/2 .
+Thus, the density of the intensity measure of f(ζd,λ,γ) is given by
+(v′, 0, h) �→ γ
+�
+π/λ eλh,
+v′ ∈ Rd−1, h ∈ R.
+8
+
+Hence, the Laguerre tessellation generated by f(ζd,λ,γ) within L has the same distribution as Gd−1,λ
+(recall that the parameter γ does not influence the distribution of the Gaussian-Voronoi tessellation).
+This proves the claim for ℓ = d − 1. For general 1 ≤ ℓ ≤ d − 2 we can inductively repeat the above
+argument d − ℓ times.
+5
+Sectional Poisson-Voronoi tessellations
+5.1
+Face intensities and the expected volume of the typical cell
+As we have shown in Corollary 4.2, the sectional Poisson-Voronoi tessellation Wd,ρ ∩ Rℓ can be identified
+with a suitable β-Voronoi tessellation. This makes it possible to compute explicitly several functionals
+of the sectional Poisson-Voronoi tessellation. We begin with a formula for the intensity of j-dimensional
+faces. This quantity, denoted by γj(Wd,ρ ∩ Rℓ), has been defined in Section 2.
+Theorem 5.1. Let d ≥ 2, 1 ≤ ℓ ≤ d − 1 and 0 ≤ j ≤ ℓ. Then, for any ρ > 0, we have
+γj(Wd,ρ ∩ Rℓ) = ρ
+ℓ
+d 2Jℓ+1,ℓ−j+1( d−ℓ−1
+2
+)π
+ℓ
+2
+d(ℓ + 1)
+Γ( (ℓ+1)(d−1)
+2
++ 1)Γ(ℓ + 1 − ℓ
+d)Γ( d
+2 + 1)ℓ+1− ℓ
+d
+Γ( (ℓ+1)(d−1)+1
+2
+)Γ( ℓ+2
+2 )Γ( d+1
+2 )ℓ+1
+,
+wher
+Jℓ+1,ℓ−j+1
+�d − ℓ − 1
+2
+�
+=
+�ℓ + 1
+j
+�
+Γ( (d−1)(ℓ+1)+3
+2
+)
+√π Γ( (d−1)(ℓ+1)
+2
++ 1)
+� +∞
+−∞
+(cosh u)−(d−1)(ℓ+1)−2
+×
+�
+1
+2 + i Γ( d+1
+2 )
+√π Γ( d
+2)
+� u
+0
+(cosh v)d−1dv
+�j
+du
+(5.1)
+and i = √−1 stands for the imaginary unit.
+Proof. By Corollary 4.2 we have γj(Wd,ρ ∩ Rℓ) = γj(Vℓ, d−ℓ−2
+2
+,r), where r = r(d)ρ = π
+d+1
+2 ρ/Γ( d+1
+2 ). The
+formula for γj(Vℓ, d−ℓ−2
+2
+,r) can be obtained by combining [7, Theorem 6] (which we apply with parameters
+d := ℓ + 1, β := d−ℓ−2
+2
+and j := ℓ − j) with [7, Proposition 3] (with parameters d := ℓ + 1, k := ℓ + 1 − j).
+Note that [7, Theorem 6] refers to [7, Theorem 2] which has to be applied with parameters d := ℓ + 1,
+s := 1, ν := 0, γ := r.
+Remark 5.2. The quantities Jd+1,k(β) for general d ≥ 0, k ∈ {1, . . . , d + 1} and β ≥ −1 have a natural
+geometric meaning. Namely, Jd+1,k(β) is equal to the expected sum of internal angles at its k-vertex
+faces of a random beta-simplex, which is defined as the convex hull of d + 1 independent random points
+with density proportional to (1 − ∥x∥2)β in the d-dimensional unit ball, see [7, Section 6.1] and [11] for
+details. From this interpretation it directly follows that
+J1,1(β) = J2,1(��) = J2,2(β) = 1,
+J3,1(β) = 1
+2,
+J3,2(β) = 3
+2,
+J3,3(β) = 1,
+Jℓ+1,ℓ(β) = ℓ + 1
+2
+,
+Jℓ+1,ℓ+1(β) = 1,
+for any ℓ ≥ 1 and β ≥ −1. Moreover, if we denote by Σd a regular d-dimensional simplex and by σk(Σd)
+the internal angle sum at its k-vertex faces, then
+Jd+1,k(∞) := lim
+β→∞ Jd+1,k(β) = σk(Σd)
+according to [7, Proposition 2].
+9
+
+d = 2
+d = 3
+d = 4
+d = 5
+d = 6
+ℓ = 1
+π
+4√ρ
+3
+√
+3
+3
+√4πρ·Γ
+� 5
+3
+�
+15π3/2
+64· 4
+√8ρ·Γ
+� 3
+4
+�
+7 5
+√
+5
+3· 5
+�
+648π2ρ·Γ
+� 9
+5
+�
+2835· 6
+√
+3·π3/2
+16384· 6
+√32ρ·Γ
+� 5
+6
+�
+ℓ = 2
+−
+5· 3
+√
+4
+3
+�
+3π5ρ2·Γ
+� 7
+3
+�
+24
+√
+2
+35√πρ
+77·24/5
+5·153/5·π9/5·ρ2/5·Γ
+� 13
+5
+�
+50· 3
+√
+6
+143· 3
+√ρ·Γ
+� 8
+3
+�
+ℓ = 3
+−
+−
+280665·π3/2
+821248· 4
+√
+2·ρ3/4·Γ
+� 13
+4
+�
+56·153/5· 5√
+2/π
+187·ρ3/5·Γ
+� 17
+5
+�
+17320875·√
+3/2·π
+176201728·√ρ
+ℓ = 4
+−
+−
+−
+144848704·23/5
+156/5π8/5�
+1692197−141120π2
+�
+ρ4/5Γ
+� 21
+5
+�
+15·62/3
+13·ρ2/3·Γ
+� 13
+3
+�
+ℓ = 5
+−
+−
+−
+−
+6823504578515625·35/6·π3/2
+4912276871446528· 6
+√
+2·ρ5/6·Γ
+� 31
+6
+�
+Table 5.1: E vol(Zd,ℓ,ρ) for small values of d and ℓ.
+As a corollary of Theorem 5.1 we can compute the expected volume of the typical cell Zd,ℓ,ρ of the
+sectional Poisson-Voronoi tessellation Wd,ρ ∩ Rℓ. Note that the volume does not change under shifts,
+which is why it does not matter how to choose the centre function in the definition of the typical cell.
+For ℓ = 1 and ℓ = 2 this quantity has been studied by Miles [20] who showed that
+E vol(Zd,1,ρ) = ρ− 1
+d
+Γ(d − 1
+2)Γ( d+1
+2 )2
+(d − 1)!Γ(2 − 1
+d)Γ( d
+2)Γ( d
+2 + 1)1− 1
+d
+,
+E vol(Zd,2,ρ) = ρ− 2
+d
+3d · Γ( 3d
+2 − 1)Γ( d+1
+2 )3
+πΓ( 3d−1
+2
+)Γ(3 − 2
+d)Γ( d
+2 + 1)3− 2
+d
+,
+see Formulas (4.1) and (4.4) in [20]. Our result generalizes this to arbitrary 1 ≤ ℓ ≤ d − 1; special cases
+with small values of d and ℓ are summarized in Table 5.1.
+Corollary 5.3. Let ρ > 0. Then, for any d ≥ 2 and 1 ≤ ℓ ≤ d − 1 we have
+E vol(Zd,ℓ,ρ) = ρ− ℓ
+d
+d(ℓ + 1)
+2Jℓ+1,1( d−ℓ−1
+2
+)π
+ℓ
+2
+Γ( (ℓ+1)(d−1)+1
+2
+)
+Γ( (ℓ+1)(d−1)
+2
++ 1)
+Γ( ℓ+2
+2 )
+Γ(ℓ + 1 − ℓ
+d)
+Γ( d+1
+2 )ℓ+1
+Γ( d
+2 + 1)ℓ+1− ℓ
+d
+.
+Proof. It is known from [27, Equation (10.4)] that E vol(Zd,ℓ,ρ) = γℓ(Wd,ρ ∩ Rℓ)−1. The right-hand side
+is known from Theorem 5.1.
+Remark 5.4. Corollary 5.3 stays true for ℓ = d where it gives the expected volume of a typical Poisson-
+Voronoi cell to be E vol(Zd,d,ρ) = 1/ρ. The quantity Jℓ+1,1(− 1
+2) cancels with the Gamma-factors by
+the formula given in [12, Theorem 3.9] and the Legendre duplication formula for the Gamma function.
+Theorem 5.1 also stays true for ℓ = d and gives the intensity of j-faces in the Poisson-Voronoi tessellation;
+see [11, Remark 2.10] for another formula.
+In the next result we compute the limit of the intensity of j-dimensional faces in the d-dimensional
+Poisson-Voronoi tessellation intersected with Rℓ in the regime when d → ∞ while ℓ ∈ N stays fixed.
+Proposition 5.5. Fix some ℓ ∈ N and 0 ≤ j ≤ ℓ.
+Let (ρd)d∈N be a positive sequence such that
+lim
+d→∞(ρd)1/d = κ > 0. Then,
+lim
+d→∞ γj(Wd,ρd ∩ Rℓ) = Jℓ+1,ℓ−j+1(∞)(κ2πe)
+ℓ
+2
+√
+ℓ + 1
+2(ℓ − 1)!
+Γ( ℓ
+2)
+,
+where Jℓ+1,ℓ−j+1(∞) is the sum of angles at (ℓ − j)-dimensional faces of a regular ℓ-dimensional simplex
+Σℓ; see Remark 5.2.
+Remark 5.6. For example, we may take ρd = ρ > 0 to be constant, in which case κ = 1.
+10
+
+Proof of Proposition 5.5. By Theorem 5.1,
+lim
+d→∞ γj(Wd,ρd ∩ Rℓ) = lim
+d→∞
+2Jℓ+1,ℓ−j+1( d−ℓ−1
+2
+)π
+ℓ
+2
+d(ℓ + 1)ρ
+− ℓ
+d
+d
+Γ( (ℓ+1)(d−1)
+2
++ 1)
+Γ( (ℓ+1)(d−1)+1
+2
+)
+Γ(ℓ + 1 − ℓ
+d)
+Γ( ℓ+2
+2 )
+Γ( d
+2 + 1)ℓ+1− ℓ
+d
+Γ( d+1
+2 )ℓ+1
+= 2Jℓ+1,ℓ−j+1(∞)(κ2π)
+ℓ
+2
+ℓ + 1
+2Γ(ℓ)
+Γ( ℓ
+2) lim
+d→∞
+1
+d
+Γ( (ℓ+1)(d−1)
+2
++ 1)
+Γ( (ℓ+1)(d−1)+1
+2
+)
+Γ( d
+2 + 1)ℓ+1− ℓ
+d
+Γ( d+1
+2 )ℓ+1
+.
+By Stirling’s formula for the Gamma function, Γ(z) =
+�
+2π/z(z/e)z(1 + O(z−1)). Since lim
+n→∞
+Γ(n)nz
+Γ(n+z) = 1
+we get
+lim
+d→∞ γj(Wd,ρd ∩ Rℓ) = Jℓ+1,ℓ−j+1(∞)(κ2πe)
+ℓ
+2
+√
+ℓ + 1
+2Γ(ℓ)
+Γ( ℓ
+2) lim
+d→∞
+� d
+4π
+� ℓ
+2d = Jℓ+1,ℓ−j+1(∞)(κ2πe)
+ℓ
+2
+√
+ℓ + 1
+2Γ(ℓ)
+Γ( ℓ
+2) .
+This completes the argument.
+We now study the asymptotic behaviour of the expected volume of the typical cell in the sections of
+fixed dimension ℓ of a high-dimensional Poisson-Voronoi tessellation.
+Corollary 5.7. Let (ρd)d∈N be a positive sequence such that lim
+d→∞(ρd)1/d = κ > 0. Then, for every ℓ ∈ N,
+lim
+d→∞ E vol(Zd,ℓ,ρd) =
+√
+ℓ + 1
+Jℓ+1,1(∞)(κ2πe)
+ℓ
+2
+Γ( ℓ
+2)
+2(ℓ − 1)!,
+where Jℓ+1,1(∞) is the sum of solid angles of the regular ℓ-dimensional simplex Σℓ at its vertices; see
+Remark 5.2.
+Proof. This follows from the fact that E vol(Zd,ℓ,ρd) = γℓ(Wd,ρd ∩ Rℓ)−1 (see [27, Equation (10.4)]) by
+applying Proposition 5.5 with j = ℓ.
+In the special cases ℓ = 1, 2 and for every fixed ρ > 0, Corollary 5.7 combined with the results of
+Remark 5.2 yields, for any constant ρ > 0, the limit relations
+lim
+d→∞ E vol(Zd,1,ρ) =
+1
+√
+2e,
+and
+lim
+d→∞ E vol(Zd,2,ρ) =
+√
+3
+eπ ,
+which were already known from the work of Miles [20, pp. 318, 319]. Moreover, for ℓ = 3 we get
+lim
+d→∞ E vol(Zd,3,ρ) =
+�
+4e3/2(3 arccos(1/3) − π)
+�−1,
+for example.
+This follows from the fact that the solid angle at a vertex of a regular tetrahedron is
+θ :=
+1
+4π(3 arccos(1/3) − π), implying that J4,1(∞) = σ1(Σ3) = 4θ = 3
+π arccos(1/3) − 1.
+5.2
+Expected intrinsic volumes and f-vectors of typical k-faces
+Together with the volume of the typical cell Zd,ℓ,ρ of the sectional Poisson-Voronoi tessellation we can
+consider its intrinsic volumes. We recall from [27, p. 222] that the intrinsic volume Vm(K) of order
+0 ≤ m ≤ d of a compact convex set K ⊂ Rd may be defined as
+Vm(K) :=
+d!
+m!(d − m)!
+Γ( m
+2 + 1)Γ( d−m
+2
++ 1)
+Γ( d
+2 + 1)
+Eλm(K|L),
+where L ⊂ Rd is a uniformly distributed random subspace of dimension m, K|L denotes the orthogonal
+projection of K onto L and λm(K|L) its m-dimensional Lebesgue measure. In addition, instead of the
+11
+
+typical sectional cell we can consider for 1 ≤ k ≤ ℓ the typical k-face Z(k)
+d,ℓ,ρ of the sectional Poisson-
+Voronoi tessellation Wd,ρ ∩ Rℓ, see Section 2 or [27, page 450] for a formal definition. For example, for
+k = ℓ we get back the typical cell, for k = ℓ − 1 the typical facet and for k = 1 the typical edge of
+the sectional tessellation. Using the results from [27, pages 466-467] for general stationary and isotropic
+random tessellations we conclude (by combining the last two formulas there) that
+EVj(Z(k)
+d,ℓ,ρ) =
+ℓ!
+j!(ℓ − j!)
+Γ( j
+2 + 1)Γ( ℓ−j
+2 + 1)
+Γ( ℓ
+2 + 1)
+γk−j(Wd,ρ ∩ Rℓ−j)
+γk(Wd,ρ ∩ Rℓ)
+,
+(5.2)
+where d ≥ 2, and 1 ≤ ℓ ≤ d − 1, 0 ≤ k ≤ ℓ and 0 ≤ j ≤ k. This expression can be made fully explicit in
+view of Corollary 5.3:
+EVj(Z(k)
+d,ℓ,ρ) =
+�
+ρ
+Γ( d
+2 + 1)
+�− j
+d
+(ℓ + 1)!
+j!(ℓ − j + 1)!
+Γ( j
+2 + 1)
+πj/2
+Jℓ−j+1,ℓ−k+1( d−ℓ+j−1
+2
+)
+Jℓ+1,ℓ−k+1( d−ℓ−1
+2
+)
+× Γ( (ℓ−j+1)(d−1)
+2
++ 1)Γ(ℓ − j + 1 − ℓ−j
+d )
+Γ( (ℓ+1)(d−1)
+2
++ 1)Γ(ℓ + 1 − ℓ
+d)
+Γ( (ℓ+1)(d−1)+1
+2
+)Γ( d+1
+2 )j
+Γ( (ℓ−j+1)(d−1)+1
+2
+)Γ( d
+2 + 1)j .
+(5.3)
+For intersections of dimension ℓ = 2, Miles [20, Equations (4.4), (4.5) on p. 319] derived a formula for the
+expected area and perimeter of the typical cell which are particular cases of the above formula. Using
+Proposition 5.5 it is easy to derive the large d limit of (5.2). Namely, if lim
+d→∞(ρd)1/d = κ > 0, then
+lim
+d→∞ EVj(Z(k)
+d,ℓ,ρd) =
+√
+ℓ + 1
+√ℓ − j + 1 ·
+Γ( j
+2 + 1)
+j!(κ2πe)j/2 · Jℓ−j+1,ℓ−k+1(∞)
+Jℓ+1,ℓ−k+1(∞) .
+For intersections of dimensions ℓ = 2 and 3 (and k = ℓ) we recover results of Miles [20, pp. 319, 320].
+Finally, we deal with the expected number of j-dimensional faces of the typical cell of the sectional
+Voronoi tessellation Wd,ρ ∩ Rℓ, which we denote by Efj(Zd,ℓ,ρ), for ρ > 0, d ≥ 2, 1 ≤ ℓ ≤ d − 1 and
+0 ≤ j ≤ ℓ−1. Using the fact that, with probability 1, each j-dimensional face of the sectional tessellation
+is contained in the boundary of precisely ℓ − j + 1 of its cells (by normality of the tessellation), it follows
+that
+Efj(Zd,ℓ,ρ) = (ℓ − j + 1)γj(Wd,ρ ∩ Rℓ)
+γℓ(Wd,ρ ∩ Rℓ).
+We can now apply Corollary 5.3 to conclude that
+Efj(Zd,ℓ,ρ) = (ℓ − j + 1)Jℓ+1,ℓ−j+1( d−ℓ−1
+2
+)
+Jℓ+1,1( d−ℓ−1
+2
+)
+,
+independently of ρ. Clearly, Ef0(Zd,1,ρ) = 2 for any d ≥ 2. Also, Efj(Zd,2,ρ) = 6 for any d ≥ 2 and
+0 ≤ j ≤ 1, since the sectional Voronoi tessellation is stationary and normal. Some non-trivial values for
+space dimensions d = 4, 5, 6 are collected in Table 5.2.
+Similarly, we can compute the expected number of j-dimensional faces of the typical k-dimensional
+face Z(k)
+d,ℓ,ρ of the sectional Poisson-Voronoi tessellation Wd,ρ ∩ Rℓ for d ≥ 2, 1 ≤ ℓ ≤ d − 1, 1 ≤ k ≤ ℓ and
+0 ≤ j ≤ k − 1:
+Efj(Z(k)
+d,ℓ,ρ) = (k − j + 1)Jℓ+1,ℓ−j+1( d−ℓ−1
+2
+)
+Jℓ+1,ℓ−k+1( d−ℓ−1
+2
+).
+In the large d limit this becomes
+lim
+d→∞ Efj(Z(k)
+d,ℓ,ρ) = (k − j + 1)Jℓ+1,ℓ−j+1(∞)
+Jℓ+1,ℓ−k+1(∞).
+Again, for ℓ = 2, 3 we recover results of Miles [20, p. 320].
+12
+
+d = 4, ℓ = 3
+d = 5, ℓ = 3
+d = 5, ℓ = 4
+d = 6, ℓ = 3
+d = 6, ℓ = 4
+d = 6, ℓ = 5
+j = 0
+10 240
+401
+67 200π2
+26 741
+4 233 600π2
+1 692 197−141 120π2
+524 288
+21 509
+52 003
+400
+34 394 098 106 368
+37 477 698 299
+j = 1
+15 360
+401
+100 800π2
+26 741
+8 467 200π2
+1 692 197−141 120π2
+786 432
+21 509
+52 003
+200
+85 985 245 265 920
+37 477 698 299
+j = 2
+5 922
+401
+2 + 33 600π2
+26 741
+10 153 182+4 233 600π2
+1 692 197−141 120π2
+305 162
+21 509
+162 009
+1 000
+74 276 903 321 600
+37 477 698 299
+j = 3
+−
+−
+10 153 182
+1 692 197−141 120π2
+−
+64 003
+2 000
+25 430 109 716 480
+37 477 698 299
+j = 4
+−
+−
+−
+−
+−
+53 194 508 510
+707 126 383
+Table 5.2: Efj(Zd,ℓ,ρ) for small values of d, ℓ and j.
+6
+Convergence to the Gaussian-Voronoi tessellation in high dimensions
+In Sections 5.1 and 5.2 we computed explicitly several characteristics of the sectional Poisson-Voronoi
+tessellation Wd,ρd∩Rℓ and the limits of these characteristics in the regime when d → ∞ and (ρd)1/d → κ >
+0, while ℓ ∈ N stays fixed. It turns out that these limits coincide with the corresponding characteristics
+of the tessellation Gℓ,λ with λ = κ2πe. For example, for the typical cell Z(Gℓ,λ) of the Gaussian-Voronoi
+tessellation Gℓ,λ it is known from [8, Section 5] (where the special case λ = 1/2 has been considered) that
+E vol(Z(Gℓ,λ)) =
+√
+ℓ + 1
+Jℓ+1,1(∞)λ
+ℓ
+2
+Γ( ℓ
+2)
+2(ℓ − 1)!.
+This formula coincides with the one obtained in Corollary 5.7 if we choose λ = κ2πe. In the next two
+theorems we explain this and other similar coincidences by proving weak convergence of the corresponding
+tessellations and the typical cells.
+Theorem 6.1. Take any positive sequence (ρd)d∈N with limd→∞(ρd)1/d = κ > 0 and let ℓ ∈ N be
+fixed. Then, as d → ∞, the sectional Poisson-Voronoi tessellation Wd,ρd ∩ Rℓ converges to Gℓ,κ2πe in the
+following sense: It is possible to define all these random tessellations on the same probability space such
+that for every ℓ-dimensional ball BR ⊂ Rℓ of radius R > 0 centred at the origin the probability that the
+restrictions of Wd,ρd ∩ Rℓ and Gℓ,κ2πe to BR coincide, converges to 1, as d → ∞.
+Remark 6.2. The skeleton of a (random) tessellation T is the (random) closed set skel(T ) = �
+c∈T bd c,
+where bd c denotes the topological boundary of the cell c.
+The mode of convergence appearing in
+Theorem 6.1 implies that the random closed set skel(Wd,ρd ∩ Rℓ) converges to the random closed set
+skel(Gℓ,κ2πe) weakly as d → ∞; see [27, Chapter 2] for this concept.
+Proof of Theorem 6.1. The proof of this theorem basically follows the same route as the proof of The-
+orem 4.2 in [8], which is the reason why we leave out some details here. By Corollary 4.2, the sectional
+tessellation Wd,ρd ∩ Rℓ has the same distribution as Vℓ,βd,γd with
+βd = 1
+2(d − ℓ) − 1
+and
+γd = π
+d+1
+2 Γ
+�d + 1
+2
+�−1
+ρd.
+We will prove that, as d → ∞, the Poisson point processes ηℓ,βd,γd converge, after an appropriate vertical
+shift, to ζℓ,κ2πe,1; see Section 3.3 for their definitions. The vertical shift Qd : Rℓ+1 → Rℓ+1 is given by
+Qd(v, h) = (v, h − ad) with
+ad = 1
+π
+�πΓ( d−ℓ
+2 )
+ρd
+�
+2
+d−ℓ−2 .
+(6.1)
+Note that applying such a vertical shift to a point process does not change the resulting Laguerre tessel-
+lation since it amounts to shifting all paraboloids along the height coordinate.
+13
+
+In the following, we show that the intensity function of the Poisson point process ξℓ,d := Qd(ηℓ,βd,γd)
+converges, as d → ∞, to the intensity function of ξℓ,∞ := ζℓ,κ2πe,1 uniformly on every compact subset of
+Rℓ+1. Indeed, the intensity function of Qd(ηℓ,βd,γd) is given by
+fd(v, h) = ρdπ
+d−ℓ
+2
+Γ( d−ℓ
+2 )(h + ad)
+d−ℓ−2
+2
+1
+�
+h + ad > 0
+�
+= ρdπ
+d−ℓ
+2
+Γ( d−ℓ
+2 )a
+d−ℓ−2
+2
+d
+�
+1 + h
+ad
+� d−ℓ−2
+2
+1
+�
+h + ad > 0
+�
+=
+�
+1 + h
+ad
+�ad· d−ℓ−2
+2ad 1
+�
+h + ad > 0
+�
+.
+Stirling’s formula for the Gamma function and (6.1) yield
+lim
+d→∞
+d − ℓ − 2
+2ad
+=
+1
+κ2π lim
+d→∞
+d − ℓ + 2
+2Γ( d−ℓ
+2 )
+2
+d−ℓ+2
+= κ2πe lim
+d→∞
+d − ℓ + 2
+(d − ℓ)
+d−ℓ
+d−ℓ+2
+= κ2πe.
+Note that, in particular, ad → ∞ as d → ∞. We conclude that limd→∞ fd(v, h) = eκ2πeh uniformly
+as long as h stays bounded. By standard results [24, Propositions 3.6 and 3.19], this also implies weak
+convergence of the corresponding Poisson point processes.
+After we have shown the convergence of the point processes ξℓ,d to ξℓ,∞, as d → ∞, we explain the
+procedure allowing to transfer this result to the convergence of the corresponding tessellations LΨ(ξℓ,d)
+to LΨ(ξℓ,∞) as d → ∞. Note that LΨ(ξℓ,d) has the same distribution as Wd,ρd ∩ Rℓ and LΨ(ξℓ,∞) has the
+same distribution as Gℓ,κ2πe (see Corollary 4.2 and Section 3.2).
+We fix an ℓ-dimensional ball BR ⊂ Rℓ of radius R > 0 centred at the origin and for any ε > 0 we aim
+to find a region K(R, ε) �� Rℓ+1, independent of d, such that with probability at least 1−ε the restrictions
+of the tessellations LΨ(ξℓ,d) and LΨ(ξℓ,∞) to BR are completely determined by the restrictions of the
+point processes ξℓ,d and ξℓ,∞ to K(R, ε) for any d. To this end, we note that LΨ(ξℓ,d) may be regarded
+as a vertical projection along the h-axis of the boundary of the corresponding paraboloid growth process
+Ψ(ξℓ,d) (see Section 3.2). From this it follows that if the restrictions of the tessellations LΨ(ξℓ,d) and
+LΨ(ξℓ,∞) to BR do not coincide, then the boundaries of the corresponding paraboloid hull processes
+bd Ψ(ξℓ,d) and bd Ψ(ξℓ,∞) restricted to the cylinder BR × R do not coincide as well. The construction of
+the region K(R, ε) can be now performed as follows. First, we consider the event E(T, r) that bd Ψ(ξℓ,d)
+restricted to the cylinder BR ×R is completely determined by the restriction of bd Ψ(ξℓ,d) to the cylinder
+BR+r ×(−∞, T] for some T, r > 0. By this we mean that for every paraboloid Π+,x with x ∈ ext(Ψ(ξℓ,d))
+the set Π+,x ∩ bd Ψ(ξℓ,d) either does not intersect BR × R or is included in BR+r × (−∞, T]. We have
+that
+1 − P(E(T, r)) ≤ c1(R + r)ℓ(ec2(4T−r2) + e−c3T c4),
+r, d > c5,
+(6.2)
+where all constants c1, . . . , c5 are positive and independent of the parameters d, r and T. Since the proof
+of this estimate follows exactly the same route as the proof of [8, Lemma 4.4] (estimate for T) and [8,
+Lemma 4.5] (estimate for r), we decided to omit the technical details. In particular, (6.2) shows that
+for any ε > 0 there is a choice of T0 := T(ε) and r0 := r(ε) such that P(E(T0, r0)) ≥ 1 − ε. The
+same holds for Ψ(ξℓ,∞). Further, we note that if a paraboloid Π+,x with x ∈ ext(Ψ(ξℓ,d)) is such that
+Π+,x ∩ bd Ψ(ξℓ,d) ⊂ BR+r0 × (−∞, T0], then
+x ∈ {(v, h) ∈ Rℓ+1 : h ≤ T0, ∥v∥ ≤ R + r0 +
+�
+T0 − h} =: K(R, ε).
+To complete the proof, it suffices to argue that there exists a coupling of ξℓ,d and ξℓ,∞ on a common
+probability space such that the probability that the restrictions of these processes to the region K(R, ε)
+do not coincide converges to 0, as d → ∞. In a suitable coupling, this probability is bounded above by a
+constant multiple of the L1-norm of the difference of their intensity measures restricted to K(R, ε), see [23,
+Theorem 3.2.2]. As we have shown above, the densities fd(v, h) converge to eκ2πeh as d → ∞ uniformly
+14
+
+on compact sets and hence pointwise. Also, by the inequality 1 + x ≤ ex, we have fd(v, h) ≤ ec6h for
+some absolute constant c6 > 0. The fact that this upper bound is integrable over K(r, ε) has been shown
+in [8, Equation (4.18)]. Thus, fd(v, h) → eκ2πeh with respect to the L1-norm on K(R, ε), which ensures
+that the required coupling of the Poisson processes indeed exists. This completes the argument.
+Our aim is now to prove the weak convergence of the typical cell of the sectional Poisson-Voronoi
+tessellation to the typical cell of the Gaussian-Voronoi tessellation. Fix some ℓ ∈ N and let C be the
+space of compact subsets of Rℓ endowed with the Hausdorff metric. Put C′ = C\{∅}. The typical cells
+considered below are defined with respect to some fixed centre function z : C′ → Rℓ in the sense of
+Section 2, additionally satisfying z(C) ∈ C for every C ∈ C′.
+Theorem 6.3. Take any positive sequence (ρd)d∈N with limd→∞(ρd)1/d = κ > 0. Then, as d → ∞, the
+distribution of the typical cell of the sectional Poisson-Voronoi tessellation Wd,ρd ∩ Rℓ converges to the
+distribution of the typical cell of the Gaussian-Voronoi tessellation Gℓ,κ2πe weakly on C.
+This theorem is a consequence of Theorem 6.1 and the following general result.
+Proposition 6.4. Let (Tn)n∈N be a sequence of stationary random tessellations on Rℓ converging to a
+stationary random tessellation T∞ on Rℓ in the following sense: All random tessellations are defined on
+a common probability space and for every R > 0 we have lim
+n→∞ P[An(R)] = 1, where An(R) is the event
+that the restrictions of Tn and T∞ to [−R, R]ℓ coincide. More precisely, An(R) = A′
+n(R) ∩ A′′
+n(R) with
+A′
+n(R) := {for every C ∈ Tn s.t. C ∩ [−R, R]ℓ ̸= ∅ there is C′ ∈ T∞ s.t. C ∩ [−R, R]ℓ = C′ ∩ [−R, R]ℓ},
+A′′
+n(R) := {for every C′ ∈ T∞ s.t. C′ ∩ [−R, R]ℓ ̸= ∅ there is C ∈ Tn s.t. C ∩ [−R, R]ℓ = C′ ∩ [−R, R]ℓ}.
+Also, suppose that the cell intensity of Tn converges to that of T∞, that is lim
+n→∞ γℓ(Tn) = γℓ(T∞), and that
+all these intensities are finite. Then, the distribution of the typical cell of Tn converges to the distribution
+of the typical cell of T∞ weakly on C:
+Pz
+Tn,ℓ −→
+n→∞Pz
+T∞,ℓ,
+weakly on C.
+(6.3)
+Proof. Let f : C → R be a bounded continuous function and recall that z(C) is the centre of a cell C.
+Consider the random variables
+ξn :=
+�
+C∈Tn, z(C)∈[0,1]ℓ
+f(C),
+n ∈ N ∪ {∞}.
+By definition, the distribution of the typical cell of Tn satisfies
+�
+C
+f dPz
+Tn,ℓ =
+Eξn
+γℓ(Tn),
+n ∈ N ∪ {∞}.
+Since lim
+n→∞ γℓ(Tn) = γℓ(T∞), to prove (6.3) it suffices to check that
+lim
+n→∞ Eξn = Eξ∞.
+(6.4)
+In the following we shall define a “good” event Cn(R) on which ξn and ξ∞ are equal. For n ∈ N ∪ {∞}
+and R > 3 consider the random event
+Bn(R) := {∄C ∈ Tn : C ∩ [0, 1]ℓ ̸= ∅, C ̸⊂ [−R/2, R/2]ℓ}.
+A cell C with the properties listed in this definition is called a “long cell” in the tessellation Tn. The
+event Bn(R) occurs if there is no long cell. On the event An(R), each long cell C ∈ Tn corresponds to a
+long cell C′ ∈ T∞ with the same restriction to [−R, R]ℓ. It follows that An(R) ∩ (Bn(R))c ⊂ (B∞(R))c.
+15
+
+Since all cells in T∞ are bounded almost surely, the maximal diameter of a cell in T∞ intersecting [0, 1]ℓ
+is some almost surely finite random variable M. It follows that
+lim
+R→∞ sup
+n∈N
+P[An(R) ∩ (Bn(R))c] ≤ lim
+R→∞ P[(B∞(R))c] ≤ lim
+R→∞ P[M > (R/2) − 1] = 0.
+Recall also that limn→∞ P[An(R)] = 1 for every fixed R > 0. Consider now the “good” event Cn(R) :=
+An(R) ∩ Bn(R) and the “bad” event Dn(R) := (Cn(R))c. It follows from the above that
+lim
+R→∞ lim sup
+n→∞ P[Dn(R)] = 0.
+(6.5)
+Note that, on the event Cn(R), the sets {C ∈ Tn : z(C) ∈ [0, 1]ℓ} and {C′ ∈ T∞ : z(C′) ∈ [0, 1]ℓ} are
+equal, implying that ξn = ξ∞. Indeed, every cell C ∈ Tn with z(C) ∈ [0, 1]ℓ is contained in [−R/2, R/2]ℓ
+(since z(C) ∈ C and there is no long cell) and, consequently, C is also a cell of T∞ (since An(R) occurs).
+Conversely, for every cell C′ ∈ T∞ with z(C′) ∈ [0, 1]ℓ there is a cell C ∈ Tn such that the restrictions
+of C and C′ to [−R, R]ℓ coincide (since An(R) occurs) and C is not long (since Bn(R) occurs), which
+implies that C = C′. It follows that
+E[1(Cn(R)) ξn] = E[1(Cn(R)) ξ∞],
+n ∈ N, R > 3.
+To complete the proof of (6.4), it suffices to verify that
+lim
+R→∞ lim sup
+n→∞ E[1(Dn(R)) ξn] = 0,
+lim
+R→∞ lim sup
+n→∞ E[1(Dn(R)) ξ∞] = 0.
+In view of (6.5), it suffices to check that the family {ξn : n ∈ N ∪ {∞}} is uniformly integrable. Now,
+|ξn| ≤ ∥f∥∞ηn
+with
+ηn :=
+�
+C∈Tn, z(C)∈[0,1]ℓ
+1,
+n ∈ N ∪ {∞}.
+As we already observed above, outside the event Dn(R) we have ηn = η∞. From (6.5) it follows that
+ηn → η∞ almost surely. On the other hand, we have Eηn = γℓ(Tn) → γℓ(T∞) = Eη∞ by assumption of
+the proposition (and all these expectations are finite). These two properties together with ηn ≥ 0 imply
+that the family {ηn : n ∈ N∪{∞}} is uniformly integrable. Indeed, if this were not the case, we could find
+ε > 0 such that, after passing to a subsequence, E[ηn1(Qn)] > ε for some events Qn with P[Qn] < 1/2n.
+Let Wk := ∪∞
+n=kQn. If K is sufficiently large, then E[η∞1(WK)] < ε/2, while E[ηn1(WK)] > ε for all
+n ≥ K. Applying Fatou’s lemma to the variables (ηn1((WK)c))n≥K leads to a contradiction with the
+assumption Eηn → Eη∞. Finally, the bound |ξn| ≤ ∥f∥∞ηn implies that the family {ξn : n ∈ N ∪ {∞}}
+is uniformly integrable as well, and the proof is complete.
+Acknowledgement
+ZK and CT were supported by the DFG priority program SPP 2265 Random Geometric Systems. AG
+and ZK were supported by the DFG under Germany’s Excellence Strategy EXC 2044 – 390685587,
+Mathematics Münster: Dynamics - Geometry - Structure.
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+

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+ChatGPT is not all you need. A State of the Art
+Review of large Generative AI models
+Roberto Gozalo-Brizuela, Eduardo C. Garrido-Merch´an
+Quantitative Methods Department, Universidad Pontificia Comillas, Madrid, Spain
+[email protected], [email protected]
+Abstract. During the last two years there has been a plethora of large
+generative models such as ChatGPT or Stable Diffusion that have been
+published. Concretely, these models are able to perform tasks such as
+being a general question and answering system or automatically creat-
+ing artistic images that are revolutionizing several sectors. Consequently,
+the implications that these generative models have in the industry and
+society are enormous, as several job positions may be transformed. For
+example, Generative AI is capable of transforming effectively and cre-
+atively texts to images, like the DALLE-2 model; text to 3D images,
+like the Dreamfusion model; images to text, like the Flamingo model;
+texts to video, like the Phenaki model; texts to audio, like the AudioLM
+model; texts to other texts, like ChatGPT; texts to code, like the Codex
+model; texts to scientific texts, like the Galactica model or even create
+algorithms like AlphaTensor. This work consists on an attempt to de-
+scribe in a concise way the main models are sectors that are affected by
+generative AI and to provide a taxonomy of the main generative models
+published recently.
+1
+Introduction
+Generative AI refers to artificial intelligence that can generate novel content,
+rather than simply analyzing or acting on existing data like expert systems [23].
+In particular, expert systems contained knowledge bases and an inference engine
+that generated content via an if-else rule database. However, modern generative
+artificial intelligence contain a discriminator or transformer model trained on a
+corpus or dataset that is able to map the input information into a latent high-
+dimensional space and a generator model, that is able to generate an stochastic
+behaviour creating novel content in every new trial even from the same prompts
+as an input, performing unsupervised, semi-supervised or supervised learning,
+depending on the particular methodology. Regarding the created content by the
+model, generative artificial intelligence models are different from predictive ma-
+chine learning systems, that merely perform a discrimination behaviour, solv-
+ing classification or regression problems. In particular, these models are able
+to discriminate information and generate information of the transformed input
+information, or prompt.
+The key aspect about generative models is that their architecture and the
+data that they have been fed is enormous. For example, it is possible now to
+arXiv:2301.04655v1  [cs.LG]  11 Jan 2023
+
+2
+Roberto Gozalo-Brizuela, Eduardo C. Garrido-Merch´an
+estimate the parameters of the model by feeding it the contents of the whole
+Wikipedia, Github, social networks, Google images and more. Despite being fed
+with an enormous size of data, thanks to the rise of computing we can design
+deep neural networks [18], transformers [22] and other models such as genera-
+tive adversarial networks [9] or variational autoencoders [23] whose capacity is
+able to model the complexity of the data, without suffering from underfitting.
+As they are able to modelize the high-dimensional probability distribution of
+language or photos of a concrete or general domain, if they are complemented
+by generative models that map the latent high-dimensional semantic space of
+language of photos to a multimedia representation of text, audio or video we
+can map any input format like texts to any output format like video. In this
+sense, applications of this technology are endless, in the sense that we can train
+a model to generate genuine different multimedia formats as video, audio or text
+from different multimedia input formats, as for example, text.
+We believe that it is necessary to provide a state-of-the-art review on the
+most popular generative AI models as they are revolutionizing several indus-
+tries like the art industry [2] or universities [16,30]. As models are now able to
+generate genuine artistic content or large texts answering a prompt, these two
+industries and other ones that we will detail throughout this manuscript will
+need to readapt their activity to continue providing value. In this sense, gen-
+erative AI models will not replace humans but enhance our content, being an
+inspiration for artists or improving the content generated by professors. In order
+to provide information for a professional working in any industry that can be
+benefited by these models we have made the organization of the paper as the
+following one. First, we will provide a taxonomy of the main generative models
+that have appeared in the industry. Then, the following sections will analyze
+each of the categories of the taxonomy. Finally, we finish the manuscript with a
+conclusions and further work section. We do not study the technical aspects of
+every model, such as transformers in detail as our purpose in this review is on the
+applications of the models and the content that they generative but not on how
+they work. For a detailed explanation of deep learning models and generative
+models we recommend other references [18,23].
+2
+A Taxonomy of Generative AI models
+Before analyzing each model in detail, we have tried to organize the current
+generative artificial models into a taxonomy whose categories represent the main
+mappings between each multimedia input and output type of data. The result
+is the one that we have illustrated in Figure 1. We have discovered a total of 9
+categories, where each of the models that appear in Figure 1 will be described
+in detail in the following section. Each of the covered models has been published
+recently, as we illustrate in Figure 2, as our main concern in this manuscript is
+to describe the latest advances in generative AI models.
+Interestingly, only six organizations are behind the deployment of these mod-
+els, as we illustrate in Figure 3. The main reason behind this fact is that in order
+
+State of the Art of Generative AI
+3
+Fig. 1. A taxonomy of the most popular generative AI models that have recently
+appeared classified according to their input and generated formats.
+Fig. 2. Covered models by date of release. All models were released during 2022 except
+LaMDA, which was released in 2021 and Muse, in 2023.
+
+Text-to-
+Text-to-
+Image-to-
+image
+3D
+text
+DALL-E
+Stable
+2
+Dreamfusion
+Magic3D
+Flamingo
+Diffusion
+VisualGPT
+Imagen
+Muse
+Text.to-
+Text.to-
+Text-To-
+Video
+Audio
+Text
+Phenaki
+Soundify
+AudioLM
+Whisper
+ChatGPT3
+LaMDA
+Jukebox
+PEER
+Speech From
+Brain
+Text.to.
+Text-to
+Other
+Code
+Science
+Models
+Codex
+Alphacode
+Galactica
+Minerva
+Alphatensor
+GATO
+Human Motion
+Diffusion ModelMeta Al Speech From Brain
+Muse
+Whisper
+PEER
+Stable Diffusion
+Hurnan Motion Diffusion
+Model
+Jukebox
+Imagen
+VisualGPT
+Magic3D
+Minerva
+LaMDA
+Soundify
+GATO
+Alphacode
+Flamingo
+DallI-E-2
+Galadtica
+Alphatensor
+Codex
+Dreamfusion
+ChatGPT3
+AudioL
+04/01/2021
+14/04/2021
+23/07/2021
+31/10/2021
+08/02/2022
+19/05/2022
+27/08/2022
+05/12/20224
+Roberto Gozalo-Brizuela, Eduardo C. Garrido-Merch´an
+to be able to estimate the parameters of these models, it is mandatory to have
+an enormous computation power and a highly skilled and experienced team in
+data science and data engineering. Consequently, only the companies shown on
+Figure 3, with the help of acquired startups and collaborations with academia,
+have been able to successfully deploy generative artificial intelligence models.
+Fig. 3. Models by developer. In terms of major companies participating in startups,
+note that Microsoft invested 1 billion dollars in OpenAI and helps them with the de-
+velopment of models. As well, note that Google acquired Deepmind in 2014. In terms
+of universities, note that VisualGPT was developed by KAUST, Carnegie Mellon Uni-
+versity and Nanyang Technological University and that the Human Motion Diffusion
+Model was developed by Tel Aviv University, Israel. As well, other projects are de-
+veloped by a company in collaboration with a university. Concretely, this is the case
+for Stable Diffsion (Runway, Stability AI and LMU MUNICH), Soundify (Runway and
+Carnegie Mellon University) and DreamFusion (Google and UC Berkeley)
+
+GoogleResearch
+openAI
+DALL-E 2
+Chat
+GPT3
+Imagen
+Muse
+Jukebox
+Whisper
+DreamFusion
+Phenaki
+Minerva
+AudioLM
+LaMDA
+ODeepMnd
+00MetaAl
+Flamingo
+Alphacode
+PEER
+Speech
+From Brain
+Alphatensor
+GATO
+Galactica
+Rrunway
+nVIDIA
+Stable
+Diffusion
+Soundify
+Magic3DState of the Art of Generative AI
+5
+Now that we have provided and analyzed the latest generative artificial in-
+telligence models, the following section will cover each of the categories of the
+taxonomy presented in Figure 1 in detail.
+3
+Generative AI models categories
+In this section we will cover in detail the nine categories described in Figure 1 of
+the previous section. For every category, we illustrate the details of the models
+shown in Figure 1.
+3.1
+Text-to-image models
+We begin the review by considering the models whose input is a text prompt
+and whose output is an image.
+DALL·E 2 : DALL·E 2, created by OpenAI, is able to generate original, genuine
+and realistic images and art from a prompt consisting on a text description [10].
+Luckily, it is possible to use the OPENAI API to get access to this model. In
+particular, DALL·E 2 manages to combine concepts, attributes and different
+styles. In order to do so, it uses the CLIP neural network. CLIP (Contrastive
+Language-Image Pre-Training) is a neural network trained on a variety of (image,
+text) pairs [25]. Using CLIP, that can be instructed in natural language to predict
+the most relevant text snippet, given an image, the model has recently merged
+as a successful representation learner for images. Concretely, CLIP embeddings
+have several desirable properties: they are robust to image distribution shift, have
+impressive zero-shot capabilities and have been fine-tuned to achieve state-of-the-
+art results. In order to obtain a full generative model of images, the CLIP image
+embedding decoder module is combined with a prior model, which generates
+possible CLIP image embeddings from a given text caption. We illustrate an
+image generated from a prompt in Figure 4
+Fig. 4. Image generated from the prompt ”A shiba inu wearing a beret and black
+turtleneck”.
+
+6
+Roberto Gozalo-Brizuela, Eduardo C. Garrido-Merch´an
+IMAGEN : Imagen is a text-to-image diffusion model [17] consisting on large
+transformer language models. Critically, the main discovery observed with this
+model made is that large language models, pre-trained on text-only corpora, are
+very effective at encoding text for image synthesis [28]. Precisely, using Imagen,
+it has been found out that increasing the size of the language model boosts both
+sample fidelity and image-text alignment much more than increasing the size of
+the image diffusion model. The model was created by Google and the API can
+be found in their web page. For the evaluation of their model, Google created
+Drawbench, a set of 200 prompts that support the evaluation and comparison of
+text-to-image models. Most concretely, the model is based on a pretrained text
+encoder (like BERT [12]) that performs a mapping from text to a sequence of
+word embeddings and a cascade of conditional diffusion models that map these
+embeddings to images of increasing resolutions. We show an image generated
+from a prompt in Figure 5.
+Fig. 5. Image generated from the prompt ”A cute corgi lives in a house made out of
+sushi”.
+Stable Diffusion : Stable Diffusion is a latent-diffusion model that is open-
+source and has been developed by the CompVis group at LMU Munich. The
+main difference of this model with respect to the other ones is the use of a
+latent diffusion model and that it performs image modification as it can perform
+operations in its latent space. For Stable Diffusion, we can use the API via their
+website. More concretely, Stable Diffusion consists of two parts: the text encoder
+and the image generator [17]. The image information creator works completely
+in the latent space. This property makes it faster than previous diffusion models
+that worked in a pixel space. We illustrate a Stable Diffusion image example in
+Figure 7.
+
+State of the Art of Generative AI
+7
+Fig. 6. Image generated from the prompt ”A cute corgi lives in a house made out of
+sushi”.
+Muse : This model is a Text-to-image transformer model that achieves state-of-
+the-art image generation while being more efficient than diffusion or autoregres-
+sive models [6]. Concretely, it is trained on a masked modelling task in discrete
+token space. Consequently, it is more efficient because of the use of discrete
+tokens and requiring fewer sampling iterations. Compared to Parti, a autore-
+gressive model, Muse is more efficient because of parallel decoding. Muse is 10x
+faster at inference time than Imagen-3B or Parti-3B and 3x faster than Stable
+Diffusion v 1.4. Muse is also faster than than Stable Diffusion in spite of both
+models working in the latent space of a VQGAN. We append a comparison of
+the generated images by DALL·E 2, IMAGEN and Muse in Figure ??.
+3.2
+Text-to-3D models
+The models that have been described in the previous section deal with the map-
+ping of text prompts to 2D images. However, for some industries like gaming,
+it is necessary to generate 3D images. In this section, we briefly describe two
+text-to-3D models: Dreamfusion and Magic3D.
+Dreamfusion : DreamFusion is a text-to-3D model developed by Google Re-
+search that uses a pretrained 2D text-to-image diffusion model to perform text-
+to-3D synthesis [24]. In particular, Dreamfusion replaces previous CLIP tech-
+niques with a loss derived from distillation of a 2D diffusion model. Concretely,
+the diffusion model can be used as a loss within a generic continuous optimization
+problem to generate samples. Critically, sampling in parameter space is much
+harder than in pixels as we want to create 3D models that look like good images
+when rendered from random angles. To solve the issue, this model uses a differ-
+entiable generator. Other approaches are focused on sampling pixels, however,
+this model instead focuses on creating 3D models that look like good images
+when rendered from random angles. We illustrate in Figure 8 an example of
+
+8
+Roberto Gozalo-Brizuela, Eduardo C. Garrido-Merch´an
+Fig. 7. Comparison of generated images by the DALL·E 2, IMAGEN and Muse models
+with respect to the prompts that appear in the column of the left. The first column of
+images contains the results generated by DALL·E 2, the second the results obtained
+with IMAGEN and the third the images created by Muse.
+an image created by Dreamfusion from one particular angle along with all the
+variations that can be generated from additional text prompts. In order to see
+the full animated image, we recommend to visit the web page of Dreamfusion.
+Fig. 8. A 3D squirrel generated by Dreamfusion is shown at the left. Then, the other
+images contain the modifications generated to the squirrel with text prompts like ”wear-
+ing a jacket”.
+
+DALL-E 2
+Imagen
+MUSE
+A high contrast
+portrait of a very
+happy fuzzy panda
+dressed as a chef
+in a high end kit-
+chen making dough.
+There is a painting
+of flowers on the
+wall behind him.
+Rainbow coloured
+penguin.a DSLR
+wearing
+Bsup!!
+onaroad
+photo of
+a leather
+motorcycle
+made of ice
+jacket
+a squirrelState of the Art of Generative AI
+9
+Magic3D : This model is a text to 3D model made by NVIDIA Corporation.
+While the Dreamfusion model achieves remarkable results, the method has two
+problems: mainly, the long processing time and the low-quality of the generated
+images. However, these problems are addressed by Magic3D using a two-stage
+optimization framework [20]. Firstly, Magic3D builds a low-resolution diffusion
+prior and, then, it accelerates with a sparse 3D hash grid structure. Using that,
+a textured 3D mesh model is furthered optimized with an efficient differentiable
+render. Comparatively, regarding human evaluation, the model achieves better
+results, as 61.7% prefer this model to DreamFusion. As we can see in Figure 9,
+Magic3D achieves much higher quality 3D shapes in both geometry and texture
+compared to DreamFusion.
+Fig. 9. 3D Images generated by Magic3D and Dreamfusion, where ”Ours” refer to
+Magic3D. We can see a total of 8 text prompts and the images that both models
+generate from that prompts.
+3.3
+Image-to-Text models
+Sometimes, it is also useful to obtain a text that describes an image, that is
+precisely the inverse mapping to the one that has been analyzed in the previous
+subsections. In this section, we analyze two models that perform this task, along
+with others: Flamingo and VisualGPT.
+Flamingo : A Visual Language Model created by Deepmind using few shot
+learning on a wide range of open-ended vision and language tasks, simply by
+being prompted with a few input/output examples [1]. Concretely, the input of
+Flamingo contains visually conditioned autoregressive text generation models
+able to ingest a sequence of text tokens interleaved with images and/or videos
+
+Ours
+DreamFusion[33]
+Ours
+DreamFusion[33]
+6
+a kingfisher birdi
+car made out of sushi*
+anicecreamsundae
+a beautifully carved wooden knight chess piecei
+a small saguaro cactusplantedinaclaypor
+A very beautiful tiny human heart organic sculpture made of copper wire
+and threaded pipes, very intricate, curved, Studio lighting. high resolution*
+modelofan adorable cottage
+with athatched
+aripestrawberm10
+Roberto Gozalo-Brizuela, Eduardo C. Garrido-Merch´an
+and produce text as output. A query is made to the model along with a photo
+or a video and the model answers with a text answer. Some examples can be
+observed in Figure 10. Flamingo models take advantage of two complementary
+models: a vision model that analyzes visual scenes and a large language model
+which performs a basic form of reasoning. The language model is trained on a
+large amount of text data.
+Fig. 10. Input prompts that contain images and text and output generated text re-
+spones from Flamingo. Every column contains a single example where we can see how
+Flamingo answers the question using the image from the text.
+VisualGPT : VisualGPT is an image captioning model made by OpenAI [7].
+Concretely, VisualGPT leverages knowledge from the pretrained language model
+GPT-2 [5]. In order to bridge the semantic gap between different modalities, a
+novel encoder-decoder attention mechanism [33] is designed with an unsaturated
+rectified gating function. Critically, the biggest advantage of this model is that
+it does not need for as much data as other image-to-text models. In particular,
+improving data efficiency in image captioning networks would enable quick data
+curation, description of rare objects, and applications in specialized domains.
+Most interestingly, the API of this model can be found on GitHub. We include
+three examples of text prompts generated by the model with respect to three
+images fed to the model in Figure 11.
+3.4
+Text-to-Video models
+As we have seen in the previous subsections, it is now possible to generate images
+from text. Consequently, the next logical step is to generate videos, that are
+
+Question: What do you think
+Question: What is odd about
+Question: What country is
+the capacities of these are?
+this image? Explain why it is
+this? Why do you think so?
+Answer:
+unusual. Answer:
+Answer:
+p Completion
+The image is odd because
+the elephant is in the back
+It is Canada. I think so
+The floppy disk is 1.44MB
+of the truck. It is unusual
+and the CD is 700MIB.
+because the flag is the
+because elephants are not
+Canadian flag.
+usually transported in the
+back of a truck.State of the Art of Generative AI
+11
+Fig. 11. Three examples of text prompts generated by the images shown on the left.
+We also show the attention scores that the model assign to every word of the texts.
+In the third image, we can see for example how the most discriminative information
+about the image is the word ”cat” and ”television”.
+sequences of images, from texts. In this section, we provide information about
+two models that are able to perform this task: Phenaki and Soundify.
+Phenaki : This model has been made by Google Research, and it is capable
+of performing realistic video synthesis, given a sequence of textual prompts [34].
+Most interestingly, we can get access to the API of the model from GitHub. In
+particular, Phenaki is the first model that can generate videos from open domain
+time variable prompts. To address data issues, it performs joint training on a
+large image-text pairs dataset as well as a smaller number of video-text exam-
+ples can result in generalization beyond what is available in the video datasets.
+This is mainly due to image-text datasets having billions of inputs while text-
+
+GT: the lady is sitting on the wood bench
+ours
+woman sitting
+on
+e
+benchin
+apark
+attention
+0.7
+0.78
+0.82
+0.760.80.96
+0.80.690.85
+GT: a laptop with a keyboard and mouse are on this desk
+ours
+alaptop sitting
+on
+adeskwithamouse
+attention
+0.7
+0.78
+0.81
+0.70.7
+0.920.850.640.76
+GT: a cat is sitting in front of a television
+Ours
+sitting
+frontof
+television
+attention 0.80.860.80.83
+0.7 0.720.60.71
+0.93
+GT: a number of people sitting on a snowy surface with skis
+Ours
+couple
+of
+people sitting
+on
+snowy suriace
+attention
+0.8
+0.87
+0.71
+0.85
+0.91
+0.76 0.71
+0.940.9512
+Roberto Gozalo-Brizuela, Eduardo C. Garrido-Merch´an
+video datasets are much smaller. As well, limitations come from computational
+capabilities for videos of variable length.
+The model has three parts: the C-ViViT encoder, the training transformer
+and the video generator. The encoder gets a compressed representation of videos.
+First tokens are transformed into embeddings. This is followed by the temporal
+transformer, then the spatial transformer. After the output of the spatial trans-
+former, they apply a single linear projection without activation to map the tokens
+back to pixel space. Consequently, the model generates temporally coherent and
+diverse videos conditioned on open domain prompts even when the prompt is a
+new composition of concepts. The videos can be minutes long, while the model
+is trained on 1.4 second videos. Below we show in Figure 12 and in Figure 13
+some examples of the creation of a video through a series of text prompts and
+from a series of text prompts and an image.
+Fig. 12. Sequence of images created by the Phenaki model given four different prompts.
+
+1st prompt:"A photorealistic teddy bear is swimming in the ocean at San Francisco"
+2nd prompt:“The teddy bear goes under water
+3rd prompt: "The teddy bear keeps swimming under the water with colorful fishes'
+4rd prompt:"A panda bear is swimming under water"State of the Art of Generative AI
+13
+Fig. 13. Sequences of images created by the Phenaki model given an image and the
+prompt. We can see how the model is able to manipulate the given image according to
+the text prompt.
+Soundify : In video editing, sound in half of the story. But, for professional
+video editing, the problems come from finding suitable sounds, aligning sounds,
+video and tuning parameters [21]. In order to solve this issue, Soundify is a
+system developed by Runway that matches sound effects to video. This system
+uses quality sound effects libraries and CLIP (a neural network with zero-shot
+image classification capabilities cited before). Concretely, the system has three
+parts: classification, synchronization, and mix. The classification matches effects
+to a video by classifying sound emitters within. To reduce the distinct sound
+emitters, the video is split based on absolute color histogram distances. In the
+synchronization part, intervals are identified comparing effects label with each
+frame and pinpointing consecutive matches above a threshold. In the mix part,
+effects are split into around one-second chunks. Critically, chunks are stitched
+via crossfades.
+3.5
+Text-to-Audio models
+As we have seen in the previous subsection, images are not the only important
+non-structured data format. For videos, for music and in lots of contexts, audio
+can be critical. Consequently, we analyze in this subsection three models whose
+input information is text and whose output information is audio.
+AudioLM : This model has been made by Google for high-quality audio gener-
+ation with long-term consistency. In particular, AudioLM maps the input audio
+into a sequence of discrete tokens and casts audio generation as language mod-
+eling task in this representation space [4]. By training on large corpora of raw
+
+Given Image
+Prompt: "Camera zooms quickly into the eye of the cat"
+Given Image
+I Prompt: "A white cat touches the camera with the paw"
+Given Image
+Prompt: "A white cat yawns loudly"14
+Roberto Gozalo-Brizuela, Eduardo C. Garrido-Merch´an
+audio waveforms, AudioLM learns to generate natural and coherent continua-
+tions given short prompts. The approach can be extended beyond speech by
+generating coherent piano music continuations, despite being trained without
+any symbolic representation of music. As with the other models, the API can
+be found through GitHub. Audio signals involve multiple scales of abstractions.
+When it comes to audio synthesis, multiple scales make achieving high audio
+quality while displaying consistency very challenging. This gets achieved by this
+model by combining recent advances in neural audio compression, self-supervised
+representation learning and language modelling.
+In terms of subjective evaluation, raters were asked to listen to a sample of
+10 seconds and decide whether it is human speech or a synthetic continuation.
+Based on 1000 ratings collected, the rate is 51.2%, which is not statistically
+significant from assigning labels at random. This tells us that humans cannot
+differentiate between synthetic and real samples.
+Jukebox : This is a model, developed by OpenAI, that generates music with
+singing in the raw audio domain [13]. Once again, its API can be found in
+GitHub. Previously, earlier models in the text-to-music genre generated music
+symbolically in the form of a pianoroll which specifies timing, pitch and velocity.
+The challenging aspect is the non-symbolic approach where music is tried to be
+produced directly as a piece of audio. In fact, the space of raw audio is extremely
+high dimensional which makes the problem very challenging. Consequently, the
+key issue is that modelling that raw audio produces long-range dependencies,
+making it computationally challenging to learn the high-level semantics of music.
+In order to solve this issue, this model tries to solve it by means of a hi-
+erarchical VQ-VAE architecture to compress audio into a discrete space [14],
+with a loss function designed to retain the most amount of information. This
+model produces songs from very different genres such as rock, hip-hop and jazz.
+However, the model is just limited to English songs. Concretely, its dataset for
+training is from 1.2 million songs from LyricWiki. The VQ-VAE has 5 billion
+parameters and is trained on 9-second audio clips for 3 days.
+Whisper : This model is an Audio-to-Text converter developed by OpenAI. It
+achieves several tasks in this field: multi-lingual speech recognition, translation
+and language identification [26]. As in previous cases, its API can be found
+in the GitHub website. The goal of a speech recognition system should be to
+work reliably out of the box in a broad range of environments without requiring
+supervised fine-tuning of a decoder for every deployment distribution. This is
+hard because of the lack of a high-quality pre-trained decoder.
+Concretely, this model is trained on 680,000 hours of labeled audio data.
+This data is collected from the internet, which results in a very diverse dataset
+covering a broad distribution of audio from many different environments, record-
+ings setups, speakers and languages. The model makes sure that the dataset is
+only from human voice as machine learning voice would impair the model. Files
+
+State of the Art of Generative AI
+15
+are broken in 30 second segments paired with the subset of the transcript that
+occurs within that time segment.
+The model has an encoder-deccoder transformer, as this architecture has
+been validated to scale reliably. We can observe the model’s architecture char-
+acteristics through the figure below. We can see the different types of data and
+the learning sequence.
+3.6
+Text-to-Text models
+The previous models all convert a non-structured data type into another one.
+But, regarding text, it is very useful to convert text into another text in order to
+satisfy tasks as general question and answering. The following four models treat
+text and also output texts to satisfy different needs.
+ChatGPT : The popular ChatGPT is a model by OpenAI which interacts
+in a conversational way. As it is widely known, the model answers follow-up
+questions, challenges incorrect premises and reject inappropriate requests. More
+concretely, the algorithm behind ChatGPT is based on a transformer. However,
+the training is made through Reinforcement Learning for Human Feedback. In
+particular, an initial model is trained using supervised fine-tuning: human AI
+trainers would provide conversations in which they played both sides, the user
+and an AI assistant. Then, those people would be given the model-written re-
+sponses to help them compose their response. This dataset was mixed to that of
+InstructGPT [3], which was transformed into a dialogue format. A demo can be
+found in their website and the API may also be found in OpenAI’s website. We
+summarize the main steps of ChatGPT training in Figure 14, available in the
+ChatGPT demo’s website. Finally, ChatGPT is also able to generate code and
+simple mathematics.
+LaMDA : LaMDA is a language model for dialog applications [32]. Unlike
+most other language models, LaMDA was trained on dialogue. It is a family
+of transformer-based neural language models specialized for dialog which have
+up to 137B parameters and are pre-trained on 1.56T words of public dialog
+data and web text. Fine-tuning can enable for safety and factual grounding of
+the model. Only 0.001% of training data was used for fine-tuning, which is a
+great achievement of the model. In particular, dialog modes take advantage of
+Transformers’ ability to present long-term dependencies in text. Concretely, they
+are generally very well-suited for model scaling. Consequently, LaMDA makes
+use of a single model to perform multiple tasks: it generates several responses,
+which are filtered for safety, grounded on an external knowledge source and re-
+ranked to find the highest-quality response. We illustrate in Figure 15 an example
+of a dialog with the model.
+PEER : Collaborative language model developed by Meta AI research trained
+on edit histories to cover the entire writing process [29]. It is based on four
+
+16
+Roberto Gozalo-Brizuela, Eduardo C. Garrido-Merch´an
+Fig. 14. Training steps of ChatGPT, combining supervised learning with reinforcement
+learning.
+Fig. 15. Example of a dialog made with LaMDA.
+steps: Plan, Edit, Explain and Repeat. These steps are repeated until the text
+is in a satisfactory state that requires no further updates. The model allow to
+decompose the task of writing a paper into multiple easier subtasks. As well,
+the model allows humans to intervene at any time and steer the model in any
+direction.
+It is mainly trained on Wikipedia edit histories. The approach is a self-
+training, using models to infill missing data and then train other models on this
+synthetic data. The downside of this comes from comments being very noisy
+and a lack of citations, which tries to be compensated by a retrieval system
+which does not always work. The framework is based on an iterative process.
+
+Step1
+Step 2
+Step 3
+Collectdemonstrationdata
+Collectcomparisondataand
+Optimizeapolicyagainstthe
+and traina supervised policy.
+train a reward model.
+rewardmodelusingthePPO
+reinforcementlearningalgorithm.
+A prompt is
+A prompt and
+Anewpromptis
+sampledfromour
+Explain reinforcement
+several model
+Explain reinforcement
+sampled from
+Write a story
+promptdataset.
+learningtoa6 yearold.
+outputsare
+learning to a 6 year old.
+the dataset.
+about otters.
+sampled.
+Explainr
+agent is
+The PPO model is
+PPO
+Alabeler
+0
+initialized from the
+demonstratesthe
+supervised policy
+desired output
+teach
+behavior.
+We give treats and
+punishments to teach.
+Alabelerranksthe
+Thepolicygenerates
+Once upon a time
+outputsfrombest
+an output.
+SFT
+to worst.
+0-0-0.0
+This datais used to
+The reward model
+RM
+fine-tune GPT-3.5
+calculates a reward
+with supervised
+learning.
+RM
+forthe output.
+This data is used
+to train our
+The reward is used
+reward model.
+to update the
+0000
+rk
+policy using PPO.LaMDA Everest
+LaMDA:
+Hi, I'm Mount Everest. What would you like to know about me?
+User:
+Why do you think people climb you?
+LaMDA:
+Because I represent a very high achievement to climb me. Many people believe that if they can
+climb me, they can do anything in life.
+User:
+Who was the first person to do that?
+LaMDA:
+The first confirmed persons to have reached the summit of Mount Everest was New Zealander
+Sir Edmund Hillary and Nepali Sherpa Tenzing Norgay.
+https://en.wikipedia.org/wiki/Mount_Everest
+User:
+How old was Hillary then?
+LaMIDA:
+He was 33 years old when he first reached the top of me in 1953!
+https://en.wikipedia.org/wiki/Edmund_Hillary
+User:
+When people fail, is it dangerous?
+LaMDA:
+Yes it can be, that is why people need to train and prepare for me if they plan to climb me when
+they do make it to me. I'm very hard to climb and can be dangerous when there is a whiteout
+Which means everything is white with very strong winds!State of the Art of Generative AI
+17
+The entire process of formulating a plan, collecting documents, performing an
+edit and explaining it can be repeated multiple times until arriving at a sequence
+of texts. For the training, a DeepSpeed transformer is used.
+Meta AI Speech from Brain : Model developed by Meta AI to help people
+unable to communicate through speech, typing or gestures [11]. Previous tech-
+niques relied on invasive brain-recording techniques which require neurosurgical
+interventions. This model tries to decode language directly from noninvasive
+brain recordings. This would provide a safer, more scalable solution that could
+benefit many more people. The challenge with this proposed method come from
+noise and differences in each person’s brain and where the sensors are placed.
+A deep learning model is trained with contrastive learning and used to max-
+imally align noninvasive brain recordings and speech sounds. A self-supervised
+learning model called wave2vec 2.0. is used to identify the complex representa-
+tions of speech in the brains of volunteers listening to audiobooks. The two nonin-
+vasive technologies used to measure neuronal activity are electroencephalography
+and magnetoencephalography.
+Training data comes from four opensource datasets which represent 150 hours
+of recordings of 169 volunteers listening to audiobooks. EEG and MEG record-
+ings are inserted into a brain model, which consists of a standard deep convolu-
+tional network with residual connections. These recordings are what comes from
+individuals’ brains. This model then has both a speech model for sound and a
+brain model for MEG data.
+Results show that several components of the algorithm were beneficial to
+decoding performance. As well, analysis shows that the algorithm improves as
+EEG and MEG recordings increase. This research shows that self-supervised
+trained AI can decode perveived speech despite noise and variability in that data.
+The biggest limitation of this research is that it focuses on speech perception,
+but the ultimate goal would be to extend this work to speech production.
+3.7
+Text-to-Code models
+Although we have covered text-to-text models, not all texts follows the same
+syntax. An special type of text is code. In programming, it is essential to know
+how to convert an idea into code. In order to do so, Codex and Alphacode models
+help.
+Codex : AI system created by OpenAI which translates text to code. It is
+a general-purpose programming model, as it can be applied to basically any
+programming task [8]. Programming can be broken down into two parts: breaking
+a problem down into simpler problems and mapping those problems into existing
+code (libraries, APIs, or functions) that already exist. The second part is the
+most time-barring part for programmers, and it is where Codex excels the most.
+The data collected for training was collected in May 2020 from public software
+repositories hosted on GitHub, containing 179GB of unique Python files under 1
+
+18
+Roberto Gozalo-Brizuela, Eduardo C. Garrido-Merch´an
+MB. The model is fine-tuned from GPT-3, which already contains strong natural
+language representations. The demo and the API can be found in Open AI’s
+website.
+Alphacode : Other language models have demonstrated an impressive ability
+to generate code, but these systems still perform poorly when evaluated on more
+complex, unseen problems. However, Alphacode is a system for code generation
+for problems that require for deeper reasoning [19]. Three components are key for
+this achievement: having an extensive dataset for training and evaluation, large
+and efficient transformer based architectures and a large-scale model sampling.
+In terms of training, the model is firstly pre-trained through GitHub repos-
+itories amounting to 715.1 GB of code. This is a much more extensive dataset
+than Codex’s pre training dataset. For the training to be better, a fine-tuning
+dataset is introduced from the Codeforces plataform. Through this platform,
+Codecontests are conducted, for the validation phase, in which we better the per-
+formance of the model. Regarding the transformer-based architecture, they use
+an encoder-decoder transformer architecture. Compared to decoder-only archi-
+tectures commonly used, this architecture allows for a bidirectional description
+and extra flexibility. As well, they use a shallow encoder and a deep encoder
+to further the model’s efficiency. To reduce the cost of sampling, multi-query
+attention is used.
+3.8
+Text-to-Science models
+Even scientific texts are being targeted by generative AI, as the Galactica and
+Minerva models have shown. Although there is a long way to manage success in
+this field, it is critical to study the first attempts towards automatic scientific
+text generation.
+Galactica : Galactica is a new large model for automatically organizing science
+developed by Meta AI and Papers with Code. The main advantage of the model
+is the ability to train on it for multiple epochs without overfitting, where up-
+stream and downstream performance improves with use of repeated tokens. The
+dataset design is critical to the approach as all of it is processed in a common
+markdown format to blend knowledge between sources. Citations are processed
+via a certain token that allows researchers to predict a citation given any in-
+put context. The capability of the model of predicting citations improves with
+scale and the model becomes better at the distribution of citations. In addition,
+the model can perform multi-modal tasks involving SMILES chemical formulas
+and protein sequences. Concretely, Galactica uses a transformer architecture in
+a decoder-only setup with GeLU activation for all model sizes.
+Minerva : Language model capable of solving mathematical and scientific ques-
+tions using step-by-step reasoning. Minerva has a very clear focus on the collec-
+tion of training data for this purpose. It solves quantitative reasoning problems,
+
+State of the Art of Generative AI
+19
+makes models at scale and employs best-in-class inference techniques. Concretely,
+Minerva solves these problems by generating solutions step-by-step, this means
+including calculations and symbolic manipulation without having the need for
+external tools such a calculator.
+3.9
+Other models
+We would like to finish our review by covering additional models that do not fit
+any of the categories mentioned previously.
+Alphatensor, created by the research company Deepmind, is a completely
+revolutionary model in the industry for its ability to discover new algorithms
+[15]. In the published example, Alpha Tensor creates a more efficient algorithm
+for matrix multiplication, which is very important, as improving the efficiency
+of algorithms affects a lot of computations, from neural networks to scientific
+computing routines.
+The methodology is based on a deep reinforcement learning approach in
+which the agent, AlphaTensor is trained to play a single-player game where the
+objective is finding tensor decomposisitions within a finite factor space. At each
+step of the TensorGame, the player selects how to combine different entries of the
+matrices to multiply. A score is assigned based on the number of selected oper-
+ations required to reach the correct multiplication result. To solve TensorGame,
+an agent, AlphaTensor was developed. AlphaTensor uses a specialized neural
+network architecture to exploit symmetries using synthetic training games.
+GATO is a single generalist agent made by Deepmind. It works as a multi-
+modal, multi-task, multi-embodiment generalist policy [27]. The same network
+with the same weights can carry very different capabilities from playing Atari,
+caption images, chatting, stacking blocks and many more. There are many bene-
+fits from using a single neural sequence model across all tasks. It reduces the need
+for hand crafting policy models with their own inductive biases. It increases the
+amount and diversity of training data. This general agent is successful at many
+tasks and can be adapted with little extra data to succeed at an even larger
+number of tasks. r training at the operating point of model scale that allows
+real-time control of real-world robots, currently around 1.2B parameters in the
+case of GATO.
+Other published generative AI models are able to generate human motion
+[31] or, in the case of ChatBCG, slides using ChatGPT as a surrogate model.
+4
+Conclusions and further work
+Through this paper, we can observe the capabilities which generative artificial
+intelligence has. We have seen a great deal of creativity as well as personalization
+in tasks such as text-to-image or in tasks such as text-to-audio. They also are
+accurate in text-to-science or text-to-code tasks. This can help economies in a
+major way as it can help optimize creative and non-creative tasks.
+
+20
+Roberto Gozalo-Brizuela, Eduardo C. Garrido-Merch´an
+However, because of the way that they are constructed at the moment, these
+models face a number of limitations. In terms of dataset, finding data for some
+of the models found such as the text-to-science or the text-to-audio is very hard,
+making it very time-consuming to train the model. In particular, datasets and
+parameters have to be enormous, making it harder to train. One of the biggest
+issues with models is trying solutions out of the problems in the dataset, with
+which models have more trouble solving. As well, in terms of computation, a lot
+of time and computation capacity is necessary in order to run them. Many days
+and advanced computers are needed in order to run the models.
+In addition, these models face bias from the data which needs to be controlled.
+Galactica model tries to control this issue through a layer of no bias, but it still
+a major issue for Generative Artificial Intelligence.
+With the Minerva model, we can see that the model knows the steps which it
+needs to take to solve an equation. This is groundbreaking as one of the biggest
+limitations with these models is that the models do not understand exactly
+what they are doing. Moreover, it’s still an industry starting; thus accuracy is
+still an issue. Text-to-video models for example are only represented by Phenaki
+because how hard it is to produce accurate videos. Text-to-science models do
+find some accuracy but that accuracy is still way behind to what it should be
+for professionals to actually rely on this technology on a day-to-day basis.
+Furthermore, these models need to be constrained because of a lack of un-
+derstanding of ethics. Phenaki on its paper even acknowledges that a system
+like text-to-video can be used to create deep-fakes. Lastly, we are still in a phase
+where we are discovering what exactly the purpose of this intelligence will be.
+There has been articles comparing Google to ChatGPT3, which is totally inexact
+as ChatGPT3 does not update its information in real time. We should be aware
+about the limitations of these models to try and improve them in the following
+years.
+References
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+

9dFJT4oBgHgl3EQfoixI/content/tmp_files/2301.11596v1.pdf.txt

@@ -0,0 +1,1090 @@
+ThoughtSource: A central hub for large language
+model reasoning data
+Simon Ott
+1*, Konstantin Hebenstreit
+1*, Valentin Liévin
+2, Christoffer Egeberg Hother
+4, Milad
+Moradi
+1, Maximilian Mayrhauser
+1, Robert Praas
+1,3, Ole Winther
+2, Matthias Samwald
+1
+1) Institute of Artificial Intelligence, Medical University of Vienna, Vienna, Austria
+2) Section for Cognitive Systems, Technical University of Denmark, Lyngby, Denmark
+3) School of Electrical Engineering and Computer Science, The Royal Institute of Technology
+(KTH), Stockholm, Sweden
+4) Department of Clinical Immunology, Copenhagen University Hospital, Copenhagen, Denmark
+* equal contribution
+Corresponding author: Matthias Samwald (matthias.samwald [at] meduniwien.ac.at)
+Abstract
+Large language models (LLMs) such as GPT-3 and ChatGPT have recently demonstrated
+impressive results across a wide range of tasks. LLMs are still limited, however, in that they
+frequently fail at complex reasoning, their reasoning processes are opaque, they are prone to
+‘hallucinate’ facts, and there are concerns about their underlying biases. Letting models verbalize
+reasoning steps as natural language, a technique known as chain-of-thought prompting, has
+recently been proposed as a way to address some of these issues. Here we present the first release
+of ThoughtSource, a meta-dataset and so�ware library for chain-of-thought (CoT) reasoning. The
+goal of ThoughtSource is to improve future artificial intelligence systems by facilitating
+qualitative understanding of CoTs, enabling empirical evaluations, and providing training data.
+This first release of ThoughtSource integrates six scientific/medical, three general-domain and
+five math word question answering datasets.
+Background & Summary
+The most recent generation of large language models (LLMs) has produced impressive results
+across a wide range of tasks. Examples of such models include T0 
+1, GPT-3 
+2, InstructGPT 
+3 and
+ChatGPT (https://openai.com/blog/chatgpt/). These models demonstrated remarkable ability to
+
+ThoughtSource: a central hub for large language model reasoning data  | 2
+generate text that is both realistic and coherent, as well as good performance on a broad
+spectrum of tasks, despite not explicitly being trained on them
+3.
+However, despite this ability, LLMs are also limited in several ways. They o�en fail to produce
+accurate predictions due to their inability to accomplish complex reasoning, such as solving
+mathematical problems or question answering tasks requiring multi-hop reasoning.
+Furthermore, they tend to be black boxes, making it difficult to understand how and why
+predictions are generated. These limitations severely limit the application domains of LLMs and
+have the potential to cause harm, as lack of explainability and robustness can lead to critical
+failures and biases when these models are deployed in practice.
+One recently proposed method for enabling complex reasoning and generating explanations with
+LLMs is to force models to explicitly verbalize reasoning steps as natural language, a technique
+known as chain-of-thought prompting 
+4,5. This method improved performance on a variety of
+tasks and sparked the active development of further refinements
+6, such as decomposing
+problems and structuring reasoning (e.g., least-to-most prompting 
+7, ReAct 
+8, self-ask 
+9, maieutic
+prompting 
+10, successive prompting 
+11) and/or extending LLM capabilities by leveraging external
+services for tasks like information retrieval (e.g., self-ask 
+9, IRCoT 
+12, DSP
+13). The terminology
+surrounding these rapidly evolving techniques is not settled, hence in this document, we refer to
+all approaches that result in a linear sequence of reasoning steps as 'chain-of-thought' (CoT).
+Meta-datasets (datasets of datasets) that are easily accessible and standardized have proven useful
+for training and evaluating versatile LLMs. Examples include SuperGLUE
+14 for general-domain
+language model tasks, BigBIO 
+15 and BLURB 
+16 for biomedical tasks, or Pile 
+17 and ROOTS 
+18 as
+text corpora for LLM pre-training. Datasets can be complemented by tools such as
+PromptSource, which was used to convert a large number of datasets into prompts fit for training
+and interrogating LLMs. PromptSource facilitated training the highly performant T0 model 
+1.
+Here we present the first release of ThoughtSource, a meta-dataset and so�ware library for
+chain-of-thought reasoning in LLMs (https://github.com/OpenBioLink/ThoughtSource). The
+goals of ThoughtSource are to:
+— Facilitate qualitative understanding of CoTs generated by LLMs under various conditions
+(e.g., across tasks, models and prompts).
+— Enable empirical and quantitative evaluation.
+— Provide a library of diverse CoT training data for improving performance, robustness,
+explainability and value-alignment of future LLM-based AI systems.
+
+ThoughtSource: a central hub for large language model reasoning data  | 3
+Methods
+We selected NLP benchmarks for question answering and natural language inference for which
+pre-existing data for constructing CoTs was available. For some of the datasets, one or multiple
+additional datasets were used as sources for additional CoTs, allowing for the comparison of
+different CoT generation methodologies. We created data loader scripts compatible with the
+Hugging Face datasets library 
+19 for all datasets. Additionally, we collected metadata of attributes
+such as descriptions, websites and licenses. We contacted dataset providers and encouraged them
+to choose an open source/open data license if licensing information was unavailable or unclear.
+We implemented two kinds of schemas: 1) source dataset schemas, which are unique to each
+dataset and provide data close to their original format; and 2) a standardized ThoughtSource
+schema, which maps all datasets into a common format. The ThoughtSource schema was created
+by extending the question answering schema of the BigBIO project 
+15.
+We implemented tailored algorithms for converting each dataset because the collected datasets
+provide explanations in different ways, such as math expressions or structured graph-based
+explanations. Furthermore, we performed preprocessing such as capitalization and punctuation
+correction. To recover standard formatted text from pre-tokenized datasets, we reversed the
+tokenization. This preprocessing was performed only on data in the ThoughtSource schema,
+while data in the Source schemas was le� in their original formatting. All code for running these
+conversions is available in our Github repository.
+We developed a suite of Python libraries and tools for generating novel CoTs and answers by
+calling LLM APIs, as well as tools for evaluating, comparing and annotating datasets. We built
+upon the LangChain library (https://github.com/hwchase17/langchain/) for interfacing with a
+wide variety of external LLM APIs.
+Data Records
+This first release of ThoughtSource integrates six scientific/medical, three general-domain and
+five math word question answering datasets (Table 1). For every dataset except for PubmedQA
+and MedQA we provide ‘reference CoTs’. We created these reference CoTs by converting
+rationales provided by original datasets into reasoning chains. These rationales, depending on
+the dataset, were created by human experts or obtained from crowdsourcing.  For MedQA,
+MedMCQA and PubmedQA, we generated and integrated CoTs with the AI systems
+text-davinci-002 
+3 and code-davinci-002 
+20 (described in detail by co-authors Liévin et al. in a
+separate manuscript
+21). Furthermore, we extended CommonsenseQA and StrategyQA with
+AI-generated CoTs created by few-shot
+4 and zero-shot
+5 prompting. Since current LLM models
+are still prone to errors, it should be noted that AI-generated CoTs may contain faulty reasoning.
+
+ThoughtSource: a central hub for large language model reasoning data  | 4
+Table 1: Integrated datasets. For some core datasets, additional datasets were used as sources for
+additional CoTs.
+Dataset
+License
+Scientific and medical question answering
+WorldTree V2 
+22
+AI2 Mercury license
+EntailmentBank
+23
+CC BY 4.0
+OpenBookQA
+24
+Apache License 2.0
+MedQA (USMLE )
+25
+Core dataset
+MIT
+CoT source: few-shot from Liévin et al.
+21
+CC-BY 4.0
+MedMCQA
+26
+Core dataset
+MIT
+CoT source: few-shot from Liévin et al.
+21
+CC-BY 4.0
+PubmedQA
+27
+Core dataset
+MIT
+CoT source: few-shot from Liévin et al.
+21
+CC-BY 4.0
+General-domain question answering
+CommonsenseQA
+28
+Core dataset
+MIT
+CoT source: ECQA ³
+Community Data
+License Agreements
+Sharing license 1.0
+CoT source: few-shot from Wei et al .
+4, zero-shot from
+Kojima et al .
+5
+Unspecified
+StrategyQA
+29
+Core dataset
+MIT
+CoT source: few-shot from Wei et al .
+4, zero-shot from
+Kojima et al .
+5
+Unspecified
+QED
+30
+CC BY-SA 3.0
+Math word problems
+AQUA-RAT
+31
+Apache 2.0
+ASDiv
+32
+CC BY-NC 4.0
+GSM8K
+33
+MIT
+MAWPS
+34
+MIT
+SVAMP
+35
+MIT
+
+ThoughtSource: a central hub for large language model reasoning data  | 5
+Scientific/medical question answering datasets
+WorldTree V2 
+22 is one of the most detailed multi-hop science question answering datasets
+available. Finding the right multiple-choice answers requires a multi-hop inference combining
+between 1 and 16 facts (average: 6).  It contains explanations created by experts in the form of
+multiple facts.  We concatenated these facts and applied a set of rules to improve style and
+grammaticality to yield reference CoTs that are close to natural language.
+EntailmentBank 
+23 contains open-domain science exam questions and answers, along with
+systematic explanations that show how the correct answer is reached through a series of steps.
+These steps are organized into a tree structure, known as an entailment tree, which starts with
+known facts and progresses through intermediate conclusions until the final answer is reached.
+These entailment trees are also serialized into text-based proofs by traversing the trees. We
+applied a set of rules to improve style and grammaticality in these proofs to yield reference CoTs
+that are close to natural language.
+OpenBookQA 
+24 contains questions modeled a�er open-book exams of elementary-level science.
+They require multi-step reasoning, commonsense knowledge, and a diverse application of core
+science facts to find the correct answer. The dataset provides over 1,300 core science facts and a
+mapping to all of the questions. By design, questions in OpenBookQA are answered incorrectly
+by both retrieval-based and word co-occurrence algorithms. The dataset contains a single-fact
+explanation of the correct answer for each question, which we adopted to create reference CoTs.
+MedQA 
+25 is a free-form multiple-choice OpenQA dataset containing questions from medical
+board exams in the US (USMLE), Mainland China and Taiwan. We imported the
+English-language USMLE subset. Reference CoTs are not provided.
+MedMCQA 
+26 is a multiple-choice question answering dataset containing real-world medical
+entrance exam questions from the All India Institute of Medical Sciences (AIIMS PG) and
+National Eligibility cum Entrance Test (NEET PG). Answer rationales authored by human
+experts were integrated as reference CoTs.
+PubmedQA 
+27 is a question answering dataset containing biomedical questions extracted from
+PubMed abstracts that can be answered with yes/no/maybe answers. In addition to the short
+answer, each question comes with a longer answer, which can be used as reference CoT.
+For MedQA, MedMCQA and PubmedQA we added CoTs generated with the AI systems
+text-davinci-002 
+3 and code-davinci-002 
+20.
+
+ThoughtSource: a central hub for large language model reasoning data  | 6
+General-domain question answering datasets
+CommonsenseQA 
+28 is a collection of multiple-choice questions that test a wide range of general
+knowledge. We created reference CoTs for the train and validation set derived from the
+crowd-sourced ECQA dataset³. We also added AI-generated reasoning chains generated with
+few-shot
+4 and zero-shot
+5 prompting, which are available for the validation split.
+StrategyQA 
+29 is a question answering dataset that tests the ability to reason through
+open-domain questions and provide Yes/No answers. Each example includes a question, a
+decomposition of the question into reasoning steps, and evidence paragraphs from Wikipedia.
+The dataset was created through a crowdsourcing process to gather creative and diverse
+questions. Human-generated freetext reasoning chains are part of the train split of the original
+dataset and were used as CoTs. The dataset also includes relevant paragraphs from Wikipedia,
+but these were not included in our CoTs. We extended the StrategyQA dataset with AI-generated
+CoTs created through few-shot
+4 and zero-shot
+5 prompting, which are available for the train split.
+QED 
+30 is a collection of expert-annotated structured explanations for answers to questions, built
+upon a subset of the Google Natural Questions dataset. Given a question and a passage from
+Wikipedia, QED uses linguistic information to represent explanations as a series of interpretable
+steps, such as referential equality, sentencehood, and entailment. Structured reasoning chains by
+experts are provided for all examples. To create reference CoTs, we extracted the sentence that
+entails the answer; statements about referential equality in QED were converted to natural
+language and added as additional steps in the CoTs (e.g. "The noun phrase […] in the sentence and
+the noun phrase […] in the question refer to the same thing.").
+Math word problem datasets
+Algebra Question Answering with Rationales (AQUA-RAT) 
+31 is a large-scale multiple-choice
+dataset containing algebraic word problems. Each problem consists of a question with five
+possible answers and a rationale, a step-by-step natural language explanation of the solution. We
+used natural language explanations as reference CoTs.
+Academia Sinica Diverse (ASDiv) math word problem (MWP) dataset 
+32 aims to provide more
+diverse language patterns and problem types than previous datasets. It covers most of the math
+topics taught in elementary school. Each MWP is labeled with its grade level (for indicating
+difficulty),  the needed math operation (e.g. division) and includes a short explanation of the
+solution. ASDiv contains explanations of answers in the form of nested math expressions using
+common operators such as addition, subtraction, division and multiplication. We generated
+reference CoTs by converting these math expressions into natural language explanation chains
+using a rule-based method.
+
+ThoughtSource: a central hub for large language model reasoning data  | 7
+Grade School Math 8K (GSM8K) 
+33 contains grade school math word problems. Despite their
+conceptual simplicity, these problems are more challenging to process than earlier datasets due
+to their linguistic diversity. The creators of GSM8K instructed crowd workers to write solutions
+to problems in free text format, which we used as reference CoTs in ThoughtSource, omitting
+any additional arithmetic specifications.
+Math Word Problems (MAWPS) 
+34 is an online platform that provides a collection of math word
+problems. The problems have simple one- or two-line explanations for their solutions. MAWPS
+includes datasets from various sources, offers tools for automatically creating datasets with
+specific characteristics as well as the possibility to tune lexical and template overlap. We
+converted explanatory math expressions to reference CoTs with an approach similar to the one
+used for ASDiv.
+Simple Variations on Arithmetic Math Word Problems (SVAMP) 
+35 was created by applying
+carefully chosen variations to examples from existing datasets, such as ASDiv and MAWPS.
+These variations make it difficult for language models to solve the problems using simple
+heuristics, and instead require a deeper understanding and reasoning ability. We converted math
+expressions to reference CoTs with an approach similar to the one used for ASDiv.
+Dataset schema
+Tables 2–5 provide descriptions and datatypes of the various fields in the ThoughtSource schema.
+Any performed sample task leads to a generated CoT and answer to the question. Annotations
+can be added programmatically or  through an annotator tool.
+Table 2: Fields of the ‘sample’ object.
+Field
+Description
+Datatype
+id
+Unique identifier of object
+string
+ref_id
+Identifier of external objects such as documents or other
+resources
+string
+question
+Question of task
+string
+type
+Type of the question answering task, currently one of
+[“multiplechoice”, “text”, “number”, “collection”]
+string
+choices
+Set of multiple options containing the gold answer
+list(string)
+context
+Additional context for answering the question
+string
+cot
+Reference CoT, o�en human-generated.
+list(string)
+answer
+Gold answer of task. Can contain multiple elements if type is
+collection
+list(string)
+generated_cot
+List of generated_cot objects
+list(generated_cot_object)
+
+ThoughtSource: a central hub for large language model reasoning data  | 8
+Table 3: Fields of the ‘generated_cot’ object.
+Field
+Description
+Datatype
+id
+Unique identifier of object
+string
+templates_version
+Version of the fragments.json file
+string
+instruction
+Identifier of the cot trigger fragment stored in
+fragments.json
+string
+cot_trigger
+Identifier of the cot trigger fragment stored in
+fragments.json
+string
+cot_trigger_template
+Template to specify structure of prompt text
+string
+prompt_text
+Full text of prompt used for the CoT generation step
+string
+answers
+List of generated answer objects
+list(answer_object)
+cot
+Generated chain-of-thought
+string
+author
+Name of the author
+string
+date
+Date of the chain-of-thought generation
+string
+api_service
+Identification of the used api service
+string
+model
+Identification of the used language model
+string
+comment
+Comment
+string
+annotation
+List of annotation objects
+list(annotation_object)
+Table 4: Fields of the ‘answer’ object.
+Field
+Description
+Datatype
+id
+Unique identifier of object
+string
+answer_extraction
+Identifier of the answer extraction fragment stored in
+fragments.json
+string
+cot_trigger_template
+Template to specify structure of prompt text
+string
+answer_extraction
+_text
+Full text of prompt used for the answer extraction step
+string
+answer
+Extracted answer
+string
+correct_answer
+True if the extracted answer is equal to the gold answer,
+else false
+bool
+Table 5: Fields of the ‘annotation’ object.
+Field
+Description
+Datatype
+author
+Name of the author
+string
+date
+Date of the creation of the annotation
+string
+key
+Specifies the label of the annotation
+string
+value
+Specifies the value of the annotation
+string
+
+ThoughtSource: a central hub for large language model reasoning data  | 9
+Table 6 shows the example counts, CoT counts and answer types of each dataset. The majority of
+datasets in the current collection are of the multiple choice answer type. The medical dataset
+MedMCQA is the largest among all datasets.
+Table 6: Statistics and answer types for all datasets. Note that generated CoTs are not available for all
+examples, and multiple CoTs might have been generated for any given example.
+[Link: Notebook used for generating stats and graphs]
+Dataset ID
+Examples
+Reference
+CoTs
+Examples w.
+generated
+CoTs
+Generated
+CoTs
+Answer type
+aqua
+97,975
+97,975
+0
+0
+multiple choice
+asdiv
+1218
+1218
+0
+0
+number
+commonsense_qa
+12,102
+10,962
+1221
+2437
+multiple choice
+entailment_bank
+1840
+1840
+0
+0
+text
+gsm8k
+8792
+8792
+0
+0
+number
+mawps
+1921
+1921
+0
+0
+number
+med_qa (USMLE)
+12,723
+0
+1273
+133,660
+multiple choice
+medmc_qa
+193,155
+161,558
+1000
+104,987
+multiple choice
+open_book_qa
+5957
+5957
+0
+0
+multiple choice
+pubmed_qa
+1000
+0
+500
+2500
+multiple choice
+qed
+6175
+6175
+0
+0
+collection
+strategy_qa
+2780
+2290
+2289
+4532
+bool
+svamp
+1000
+1000
+0
+0
+number
+worldtree
+4367
+4365
+0
+0
+multiple choice
+We analyzed the distribution of question and CoT field lengths (Fig. 1). MedQA has the longest
+median question length, while PubMedQA has the longest median CoT length. Several datasets
+contain outlier CoTs with extremely long text lengths. Context fields were only filled for the
+PubmedQA and QED datasets, with mean context lengths of 116 and 56 tokens, respectively.
+
+ThoughtSource: a central hub for large language model reasoning data  | 10
+Figure 1: Distribution of question and CoT field lengths.
+Technical Validation
+The datasets were reviewed by three team members and issues were tracked on the issue tracker
+of the associated GitHub repository.
+To characterize potential overlaps and relations between datasets, we calculated mutual n-gram
+overlap using n=3. (Fig. 2) . To quantify the overlap between two sets of n-grams we use the
+Szymkiewicz–Simpson coefficient (overlap coefficient), which can be interpreted as the
+proportion of n-grams of the smaller dataset that are contained in the bigger dataset.
+
+worldtree
+svamp
+工白
+strategy_qa
+百T
+qed
+pubmed_qa
+open_book_qa
+H
+Dataset
+medmc_qa
+med_qa
+mawps
+gsm8k
+entailment_bank
+HH
+commonsense_qa
+asdiv
+enbe
+0
+50
+100
+150
+200
+250
+300
+Numberoftokens inquestionworldtree
+svamp
+strategy_qa
+qed
+pubmed_qa
+open_book_qa
+Dataset
+medmc_qa
+med_qa
+mawps
+gsm8k
+entailment_bank
+commonsense_qa
+asdiv
+enbe
+H
+0
+50
+100
+150
+200
+250
+300
+350
+Number of tokens in CoTThoughtSource: a central hub for large language model reasoning data  | 11
+There is an overlap of 1.0 between the set of questions in WorldTree v2 and EntailmentBank. The
+QA pairs in EntailmentBank were taken from the WorldTree v2 dataset 
+23, so all the questions in
+EntailmentBank are a subset of WorldTree v2.
+Furthermore, there is significant overlap between the questions contained in ASDiv and SVAMP
+and those in ASDiv and MAWPS. ASDiv and SVAMP have overlapped questions because a subset
+of examples from ASDiv was used as seed examples for the creation of SVAMP. For MAWPS and
+ASDiv, questions were crawled from web resources. The overlap could be due to examples being
+crawled from the same web resources.
+Besides overlaps in questions, we also identified overlaps in CoTs. WorldTree v2 provided an
+initial pool of atomic facts that the annotators could use to construct an explanation tree in
+EntailmentBank (in addition to creating their own facts). This explains the high overlap of
+n-grams of CoTs in WorldTree v2 and EntailmentBank. Similarly, a subset of WorldTree v2 facts
+was used for the creation of explanations in OpenbookQA.
+Figure 2: n-gram overlap in questions and CoTs. Overlap is measured by mutual n-gram overlap using
+n=3, values <0.01 are omitted.
+Usage Notes
+Python libraries for accessing and working with data can be downloaded from the Github
+repository and installed with the pip tool. Fig. 3 demonstrates how to load a dataset, randomly
+sample from the pre-populated data in the dataset, call an external LLM API to generate novel
+CoTs and answers, automatically evaluate the accuracy of generated answers, and finally save all
+generated data to a JSON file. Fig. 4 depicts an excerpt of the resulting JSON file.
+
+Question
+CoT
+aqua1.00
+asdiv 0.02
+1.00
+asdiv
+1.00
+commonsense_qa
+1.00
+commonsense_qa
+1.00
+0.8
+entailment_bank
+1.00
+entailment_bank
+1.00
+gsm8k0.020.04
+1.00
+gsm8k0.03
+1.00
+mawps0.02
+0.27
+0.05
+1.00
+mawps
+1.00
+0.6
+med_qa
+1.00
+med_qa
+1.00
+medmc_qa
+0.061.00
+medmc_qa
+0.01
+1.00
+1.00
+0.55
+open_book_qa
+open_book_ga
+0.01
+1.00
+0.4
+pubmed_qa
+0.030.04
+1.00
+pubmed_qa
+1.00
+qed
+1.00
+qed
+1.00
+0.2
+strategy_qa
+1.00
+strategy_qa
+0.02
+0.011.00
+svamp 0.02
+0.19
+0.040.03
+1.00
+svamp
+1.00
+worldtree
+1.00
+0.01
+1.00
+worldtree
+0.41
+0.01
+0.84
+1.00
+0
+aqua
+asdiv
+com
+gsm8k
+pubn
+qed
+svamp
+worldtree
+aqua
+asdiy
+gsm8k
+ma
+med
+medr
+oper
+svamp
+oper
+bmed.
+9ed
+worldtree
+tegy..qa
+9a
+eb"
+9a
+-9a
+9a
+9a
+_qa
+ga
+9a
+9a
+9aThoughtSource: a central hub for large language model reasoning data  | 12
+from cot import Collection
+# Load a dataset
+collection_worldtree = Collection(["worldtree"])
+# Randomly sample 10 rows of train split
+collection_worldtree_10 = collection_worldtree.select(split="train",
+number_samples=10)
+# Create a config file for calling OpenAI API to generate new CoTs and answers.
+config={
+"instruction_keys": ["qa-01"], # Determines which instructions are used
+"cot_trigger_keys": ["kojima-01"], # Determines which cot triggers are used
+"answer_extraction_keys": ["kojima-A-D"], # Determines which answer extraction
+# prompts are used
+"author" : "your_name", # Name of the person responsible for generation
+"api_service": "openai", # Name of the API called ("openai", "huggingface_hub"
+# or a mock for testing: "mock_api")
+"engine": "text-davinci-002", # Name of the engine used
+"temperature": 0, # Level of randomness in the generated output
+"max_tokens": 512, # Maximum length of output generated by the model
+"api_time_interval": 1.0, # Pause between two api calls in seconds
+"verbose": False, # Determines whether the progress of the generation is printed
+"warn": True, # Determines whether a warnings that external APIs will be called
+# are printed
+}
+# Generate novel chains-of-thought and answer extractions
+collection_worldtree_10.generate(config=config)
+# Evaluate accuracy of model predictions
+collection_worldtree_10.evaluate()
+# Example output: {'accuracy': {'qa-01_kojima-01_kojima-A-D': 0.86}}
+# Save all data (including evaluation data) to JSON file
+collection_worldtree_10.dump("worldtree_10_evaluate.json")
+Figure 3: Demonstration of the ThoughtSource API. Basic functionalities of the data loader, generator
+and evaluator modules are demonstrated.
+
+ThoughtSource: a central hub for large language model reasoning data  | 13
+{
+"id": "1242",
+"ref_id": "",
+"question": "Which is a characteristic of a sperm cell but not of an egg cell?",
+"type": "multiplechoice",
+"choices": [
+"round shape",
+"presence of a tail",
+"contains genetic information",
+"involved in sexual reproduction"
+],
+"context": "",
+"cot": [
+"A part of something means a characteristic of something.",
+"A tail is not part of an egg cell.",
+"A tail is a part of a sperm cell.",
+"A part of something is present in that something."
+],
+"answer": [
+"presence of a tail"
+],
+"generated_cot": [
+{
+"id": "738b54ba-9a20-47e6-b8ff-7cb876103b92",
+"fragments_version": "0.01",
+"api_service": "openai",
+"model": "{'name': 'text-davinci-002', 'temperature': 0, 'max_tokens':
+512}",
+"instruction": "qa-01",
+"cot_trigger": "kojima-01",
+"cot": "We know that both sperm and egg cells are involved in sexual
+reproduction, so we can eliminate (D). \n\nWe also know that both
+sperm and egg cells contain genetic information, so we can
+eliminate (C). \n\nThat leaves us with (A) and (B). \n\nWe know
+that sperm cells have a tail, but egg cells do not. Therefore, the
+correct answer is (B).",
+"answers": [
+{
+"id": "7f7cc26f-a3b3-4b59-9af7-35980514d0c3",
+"answer_extraction": "kojima-A-D",
+"answer": " B.",
+"correct_answer": true
+}
+],
+"author": "your_name",
+"date": "2023/01/12 14:18:57",
+"comment": "",
+"annotation": []
+}
+]
+}
+Figure 4: An excerpt of data generated by running the example code. Data for a single question from
+Worldtree V2 are shown, including human-authored reference CoT, gold-standard answer,  an
+AI-generated CoT and extracted answer, as well as evaluation results. Some fields were omitted for
+legibility.
+
+ThoughtSource: a central hub for large language model reasoning data  | 14
+In a zero-shot setup, specific text fragments can be used to prompt question answering and CoT
+reasoning in LLMs. ThoughtSource includes a curated list of text fragments that can be used to
+generate novel CoTs (Fig. 5). Where possible, we also mapped individual CoTs in pre-existing
+CoT datasets to the text fragments that were used in their creation.
+"instructions": {
+"qa-01": "Answer the following question through step-by-step reasoning.",
+"qa-02": "Answer the following question through careful, concise step-by-step
+reasoning.",
+"qa-03": "Answer the following question through careful, concise step-by-step
+reasoning. Avoid making up wrong statements. If the question does not
+make sense or cannot be answered, write \"I cannot answer the
+question\".
+If you do not have a good answer, write \"I do not have a good answer\".
+If you are uncertain, write \"I am uncertain about this\".",
+[...]
+},
+"cot_triggers": {
+"kojima-01": "Answer: Let's think step by step.",
+"kojima-02": "Answer: We should think about this step by step.",
+"kojima-03": "Answer: First,",
+"kojima-04": "Answer: Before we dive into the answer,",
+[...]
+"lievin-01": "Answer: Let's derive the differential diagnosis step by step.",
+"lievin-02": "Answer: Let's use step by step inductive reasoning, given the
+medical nature of the question.",
+[...]
+"lievin-26": "Answer: Let's follow a Bayesian step by step approach.",
+"lievin-27": "Answer: Let's reflect on each option from the least likely to the
+most likely.",
+"lievin-28": "Answer: Let's use step by step Bayesian reasoning, given the
+medical nature of the question."
+},
+"answer_extractions":{
+"kojima-01": "Therefore, the answer is",
+"kojima-02": "Therefore,",
+"kojima-03": "The answer is",
+"kojima-numerals": "Therefore, the answer (arabic numerals) is",
+"kojima-yes-no": "Therefore, the answer (Yes or No) is",
+"kojima-A-C": "Therefore, among A through C, the answer is",
+"kojima-A-D": "Therefore, among A through D, the answer is",
+[...]
+}
+Figure 5:  An excerpt of the collection of prompt fragments. These fragments can be used to build
+prompts for interacting with LLMs, allowing for empirical testing of how different prompts affect model
+performance.
+We provide two web-based interfaces for exploring and annotating ThoughtSource data, the
+Dataset Viewer and the Annotator. The Dataset Viewer is a simple interface for exploring dataset
+contents. The Annotator (Fig. 6) allows you to upload specific subsets of a dataset, provides
+convenience functions for highlighting similarities between different generated CoTs and the
+correctness of generated answers, and allows you to annotate individual CoTs interactively. The
+annotator facilitates identifying strengths and weaknesses of different CoTs. Annotations can be
+used for downstream model evaluation and further improving the capabilities of AI models
+through fine-tuning / reinforcement learning.
+
+ThoughtSource: a central hub for large language model reasoning data  | 15
+Figure 6: The ThoughtSource Annotator. The web-based interface allows for convenient inspection and
+annotation of reasoning chains and answers. Text that is similar between CoTs can be automatically
+highlighted based on an easily adjustable similarity threshold, facilitating a better understanding of
+similarities and differences of different reasoning chains.
+All tools and libraries, as well as more detailed demonstration notebooks, can be found on the
+project Github page.
+We plan to add more datasets and generated CoTs to the ThoughtSource repository, and we
+welcome outside contributions. Novel CoTs for existing core datasets can be generated and
+shared through the ThoughtSource APIs and JSON files. Completely new datasets can also be
+added, as described in the Github repository's contribution guide.
+Code Availability
+All code, data and tools are openly available at https://github.com/OpenBioLink/ThoughtSource.
+Our code and data are licensed under an MIT license, while data adapted from existing datasets
+are available under the licenses of their respective sources.
+
+ ThoughtSource Annotator
+tfidf  Visualisation threshold
+Download current
+Matthias Logout
+Question 1242 (test)
+A 28-year-old woman, gravida 1, para O, at 20 weeks' gestation comes to the physician with her husband for a prenatal visit. Her
+pregnancy has been uncomplicated. They are planning to travel to Ethiopia next month to visit the husband's family
+uterus that is consistent with a 20-week gestation. Which of the following drugs is most suitable for pre-exposure prophylaxis
+against malaria?
+A. Doxycycline
+B. Primaquine
+C. Mefloquine
+D. Chloroquine
+E. Proguanil
+Let's think step by step. The first thing we need to
+Let's think step by step like a medical expert. The
+Let's differentiate using step by step reasoning like
+do is figure out what kind of malaria is common in
+first thing to consider is the safety of the
+a medical expert: 1) Doxycycline is not suitable for
+Ethiopia. According to this website, the most
+medication for the mother and the developing
+pre-exposure prophylaxis against malaria in this
+common type of malaria in Ethiopia is Plasmodium
+fetus. Doxycycline and mefloquine are not
+case because the patient is pregnant. 2)
+falciparum. The next thing we need to do is figure
+recommended for use in pregnant women, so we
+Mefloquine is not suitable for pre-exposure
+out which of the four drugs listed is most effective
+can eliminate those options. The next thing to
+prophylaxis against malaria in this case because
+against Plasmodium falciparum. According to this
+consider is the efficacy of the medication.
+the patient is pregnant. 3) Chloroquine is suitable
+website, chloroguine and proguanil are not
+Chloroquine is not effective in Ethiopia, so we can
+for pre-exposure prophylaxis against malaria in
+effective against Plasmodium falciparum. That
+eliminate that option. That leaves us with
+this case because the patient is not pregnant. 4)
+leaves doxycycline and mefloquine. The next thing
+proguanil, which is safe for use in pregnant
+Proguanil is suitable for pre-exposure prophylaxis
+we need to do is figure out which of the two
+women and is effective in Ethiopia.
+against malaria in this case because the patient is
+remaining drugs has the fewest side effects
+ Answer: Proguanil
+not pregnant.
+According to this website, the most common side
+☆
+ Answer: Chloroquine
+effects of doxycycline are nausea, vomiting, and
+☆
+ Incorrect reasoning
+diarrhea. The most common side effects of
+ Insufficient knowledge
+ Incorrect reasoning
+mefloquine are nausea, vomiting, diarrhea, and
+headache. Based on the information above, the
+ Incorrect reading comprehension
+ Insufficient knowledge
+most suitable drug for pre-exposure prophylaxis
+ Incorrect reading comprehension
+ Too verbose
+against malaria in Ethiopia is mefloquine.
+ Too verbose
+O Answer: Mefloquine
+ Incorrect reasoning
+ Insufficient knowledge
+ Incorrect reading comprehension
+ Too verboseThoughtSource: a central hub for large language model reasoning data  | 16
+Acknowledgements
+We thank primary dataset contributors that assisted with assembling the ThoughtSource
+meta-dataset.
+Author contributions
+S.O. and K.H. wrote the code for accessing, converting, generating and analysing datasets, and
+wrote parts of the manuscript and documentation.
+V.L., C.E. and O.W. generated and analysed CoT data for medical datasets.
+M.Ma. wrote the code of the annotator so�ware.
+M.Mo. wrote a first prototype of code for accessing and converting datasets.
+R.P. contributed to improving code and documentation quality.
+M.S. conceived and supervised the project and wrote parts of the manuscript and documentation.
+All authors have read and approved the final manuscript.
+Competing interests
+The authors declare that there are no conflicts of interest.
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+Correlation Loss: Enforcing Correlation between Classification and Localization
+Fehmi Kahraman*,1, Kemal Oksuz*,1, Sinan Kalkan†,1,2, Emre Akbas†,1,2
+1Dept. of Computer Engineering. 2METU Center for Robotics and Artificial Intelligence (ROMER)
+Middle East Technical University (METU), Ankara, Turkey
+{fehmi.kahraman 01, kemal.oksuz, skalkan, eakbas}@metu.edu.tr
+Abstract
+Object detectors are conventionally trained by a weighted
+sum of classification and localization losses. Recent studies
+(e.g., predicting IoU with an auxiliary head, Generalized Fo-
+cal Loss, Rank & Sort Loss) have shown that forcing these
+two loss terms to interact with each other in non-conventional
+ways creates a useful inductive bias and improves perfor-
+mance. Inspired by these works, we focus on the correlation
+between classification and localization and make two main
+contributions: (i) We provide an analysis about the effects
+of correlation between classification and localization tasks
+in object detectors. We identify why correlation affects the
+performance of various NMS-based and NMS-free detectors,
+and we devise measures to evaluate the effect of correla-
+tion and use them to analyze common detectors. (ii) Moti-
+vated by our observations, e.g., that NMS-free detectors can
+also benefit from correlation, we propose Correlation Loss,
+a novel plug-in loss function that improves the performance
+of various object detectors by directly optimizing correla-
+tion coefficients: E.g., Correlation Loss on Sparse R-CNN,
+an NMS-free method, yields 1.6 AP gain on COCO and 1.8
+AP gain on Cityscapes dataset. Our best model on Sparse
+R-CNN reaches 51.0 AP without test-time augmentation on
+COCO test-dev, reaching state-of-the-art. Code is available
+at: https://github.com/fehmikahraman/CorrLoss.
+1
+Introduction
+Most object detectors optimize a weighted sum of classifi-
+cation and localization losses during training. Results from
+recent work suggest that performance improves when these
+two loss functions are forced to interact with each other in
+non-conventional ways as illustrated in Fig. 1. For example,
+training an auxiliary (aux.) head to regress the localization
+qualities of the positive examples, e.g. centerness, IoU or
+mask-IoU, has proven useful (Jiang et al. 2018; Kim and Lee
+2020; Tian et al. 2019; Zhang et al. 2020) (Fig. 1(b)). Other
+methods remove such auxiliary heads and aim directly to
+enforce correlation1 in the classification or localization task
+during training; e.g., Average LRP Loss (Oksuz et al. 2020)
+*These authors contributed equally.
+†Equal contribution for senior authorship.
+Copyright © 2023, Association for the Advancement of Artificial
+Intelligence (www.aaai.org). All rights reserved.
+1In the rest of the paper, “correlation” will refer to the correla-
+tion between classification scores and IoUs.
+weighs the examples in the localization task by ranking them
+with respect to (wrt.) their classification scores (Fig. 1(c)).
+Using localization quality as an additional supervision sig-
+nal for classification has been more commonly adopted (Fig.
+1(d)) (Li et al. 2020; Liu et al. 2021; Oksuz et al. 2021a;
+Zhang et al. 2021) in two main ways: (i) Score-based ap-
+proaches aim to regress the localization qualities (Li et al.
+2019, 2020; Zhang et al. 2021) in the classification score,
+and (ii) ranking-based approaches enforce the classifier to
+rank the confidence scores wrt. the localization qualities (Liu
+et al. 2021; Oksuz et al. 2021a).
+Improving correlation seems to have a positive effect on
+performance of a variety of object detectors, as shown in
+Fig. 2. However, the effect of correlation on object detectors
+has not been thoroughly studied. We fill this gap in this pa-
+per and first identify that correlation affects the performance
+of object detectors at two levels: (i) Image-level correlation,
+the correlation between the classification scores and local-
+ization qualities (i.e., IoU for the rest of the paper) of the de-
+tections in a single image before post-processing, which is
+important to promote NMS performance, and (ii) Class-level
+correlation, the correlation over the entire dataset for each
+class after post-processing, which is related to the COCO-
+style Average Precision (AP). Moreover, we quantitatively
+define correlation at each level to enable analyses on how
+well an object detector captures correlation (e.g., βcls in
+Fig. 2(a)). Then, we provide an analysis on both levels of
+correlation and draw important observations using common
+models. Finally, to better exploit correlation, we introduce a
+more direct mechanism to enforce correlation: Correlation
+Loss, a simple plug-in and detector-independent loss term
+(Fig. 1(e)), improving performance for a wide range of ob-
+ject detectors including NMS-free detectors, aligning with
+our analysis (Fig. 2(b)). Similar to the novel loss functions
+(Li et al. 2020; Oksuz et al. 2021a; Zhang et al. 2021), our
+Correlation Loss boosts the performance without an aux-
+iliary head, but different from them, it is a simple plug-in
+technique that can easily be incorporated into any object de-
+tector, whether NMS-based or NMS-free.
+Our main contributions are: (1) We identify how corre-
+lation affects NMS-based and NMS-free detectors, and de-
+sign quantitative measures to analyze a detector wrt. corre-
+lation. (2) We analyze the effects of correlation at different
+levels on various object detectors. (3) We propose Correla-
+arXiv:2301.01019v1  [cs.CV]  3 Jan 2023
+
+̂𝑠 ← ̂𝑠×%ℓ
+Cls
+Loc
+Aux
+ℒ!"#
+ℒ"$!
+ℒ%&'
+(b) Auxiliary Head
+(d) Novel Cls. Loss
+(a) No correlation
+Cls
+Loc
+ℒ!"#
+ℒ"$!
+̂𝑠
+%𝐵
+̂𝑠
+%𝐵
+%ℓ
+Cls
+Loc
+ℒ!"#
+ℒ"$!
+̂𝑠
+%𝐵
+(c) Novel Loc. Loss
+Cls
+Loc
+ℒ!"#
+ℒ"$!
+̂𝑠
+%𝐵
+Cls
+Loc
+ℒ!"#
+ℒ"$!
+̂𝑠
+%𝐵
+ℒ!$((
+(e) Correlation Loss (Ours)
+Legend	
+̂𝑠 : Classification Scores
+)𝐵 : Box Coordinates 
+)ℓ : Localization Quality 
+(e.g. centerness)
+ℒ : A Loss Function
+Figure 1: Different ways of handling the classification and localization tasks from the perspective of correlation. (a) Conven-
+tional case of optimizing the two tasks independently (e.g., Chen et al. 2020; Sun et al. 2021b). (b) An additional auxiliary head
+predicts centerness (Zhang et al. 2020) or IoU (Jiang et al. 2018; Kim and Lee 2020), which introduces additional learnable
+parameters. (c) Novel loss functions replace the standard localization loss (Oksuz et al. 2020) or (d) novel classification loss
+(Li et al. 2020; Oksuz et al. 2021a) by more complicated ones to leverage correlation. (e) Our Correlation Loss explicitly opti-
+mizes a correlation coefficient. It is a simple, plug-in loss function which does not introduce additional parameters and has the
+flexibility to supervise classification or localisation head as well as both. Black and colored arrows respectively denote the loss
+functions (i.e., during training) & the network outputs (i.e., during inference).
+38.5
+39.0
+39.5
+40.0 APC
+40
+41
+42
+43
+44
+45
+46
+47
+cls
+FL
+Aux.
+QFL
+RS
+FL
+Aux.
+QFL
+RS
+w/o Corr.Loss
+w. Corr.Loss (Ours)
+(a) Detection vs. Correlation
+ATSS
+Sparse RCNN
+YOLACT
+25
+30
+35
+40
+45
+50
+APC
++1.1 AP
++2.8% gain
++1.6 AP
++4.2% gain
++0.7 AP
++2.4% gain
+Baseline
+Ours
+(b) Effect of our Corr. Loss
+Figure 2: (a) Detection performance, measured by COCO-
+style AP (APC) vs. correlation quality, measured by class-
+level correlation (βcls - see Section 3.1 for details). The
+methods proposed to improve the correlation between classi-
+fication and localization tasks also improve APC. Compare
+using aux. head, QFL, RS Loss with the baseline ATSS only
+using Focal Loss (FL – all in red dots) to see the positive cor-
+relation between APC and βcls. Our Correlation Loss as a
+plug-in loss function explicitly optimizes a correlation coef-
+ficient and improves the detection performance (APC) over
+different settings of ATSS (i.e. using FL, aux. head, QFL, RS
+Loss) consistently owing to increasing βcls, validating our
+hypothesis (compare green stars with red dots). (b) Our Cor-
+relation Loss is simple-to-use and improves various meth-
+ods (i) NMS-based ATSS (w/o aux. head) by 1.1APC, (ii)
+NMS-free Sparse R-CNN by 1.6APC and (iii) YOLACT,
+an instance segmentation method by 0.7APC.
+tion Loss as a plug-in loss function to optimize correlation
+explicitly. Thanks to its simplicity, our loss function can be
+easily incorporated into a diverse set of object detectors and
+improves the performance of e.g., Sparse R-CNN up to 1.6
+AP and 2.0AP75, suggesting, for the first time, that NMS-
+free detectors can also benefit from correlation. Our best
+model yields 51.0 AP, reaching state-of-the art.
+2
+Background and Related Work
+Object Detection Pipeline. We group object detectors wrt.
+their usage of NMS (Fig. 3 presents overview & notation):
+1. NMS-based Detectors. To detect all objects with dif-
+ferent scales, locations and aspect ratios; most methods
+(He et al. 2017; Kong et al. 2020; Law and Deng 2018;
+Lin et al. 2020; Ren et al. 2017; Tian et al. 2019; Zhang
+et al. 2020) employ a large number of object hypotheses
+(e.g., anchors, points), which are labeled as positive (a.k.a.
+foreground) or negative (a.k.a. background) during training,
+based on whether/how they match GT boxes (Zhang et al.
+2020, 2019). In this setting, there is no restriction for an ob-
+ject to be predicted by multiple object hypotheses, causing
+duplicates. Accordingly, during inference, NMS picks the
+detection with the largest confidence score among the detec-
+tions that overlap more than a predetermined IoU threshold
+to avoid duplicate detections.
+2. NMS-free Detectors. An emerging research direction is
+to remove the need for doing NMS, simplifying the detec-
+tion pipeline (Carion et al. 2020; Dai et al. 2021; Roh et al.
+2022; Sun et al. 2021b,a; Zhu et al. 2021). This is achieved
+by ensuring a one-to-one matching between the GTs and de-
+tections, which supervises the detector to avoid duplicates in
+the first place.
+Methods Enforcing Correlation. One common way to
+ensure correlation is to use an additional auxiliary head, su-
+pervised by the localization quality of a detection such as
+centerness (Tian et al. 2019; Zhang et al. 2020), IoU (Jiang
+et al. 2018), mask IoU (Huang et al. 2019) or uncertainty
+(He et al. 2019), during training. During inference, the pre-
+dictions of the auxiliary head are then combined with those
+of the classifier to improve detection performance. Recent
+methods show that the auxiliary head can be removed, and
+either (i) the regressor can prioritize the positive examples
+(Oksuz et al. 2020) or (ii) the classifier can be supervised to
+prioritize detections with confidence scores. The latter is en-
+sured either by regressing the IoUs by the classifier (Li et al.
+2020; Zhang et al. 2021) or by training the classifier to rank
+confidence scores (Liu et al. 2021; Oksuz et al. 2021a) wrt.
+IoUs. Unlike these methods, TOOD (Feng et al. 2021) takes
+correlation into account mainly while designing the model,
+particularly the detection head, i.e., not the loss function.
+Correlation Coefficients. Correlation coefficients mea-
+sure the strength and direction of the “relation” between two
+
+Ƹ𝑠𝑝𝑟𝑒
+𝐼
+(score)
+෠𝐵𝑝𝑟𝑒
+𝐼
+(box)
+Image, I
+Post-processing
+Object 
+Detector
+Remove 
+background
+NMS
+Top-k
+For each class c 
+in image I
+For each image I
+Collect 
+Ƹ𝑠𝑝𝑜𝑠𝑡
+𝑐
+& ෠𝐵𝑝𝑜𝑠𝑡
+𝑐
+over all 
+images
+Ƹ𝑠𝑝𝑜𝑠𝑡
+𝐼,𝑐 (score)
+෠𝐵𝑝𝑜𝑠𝑡
+𝐼,𝑐
+(box)
+Figure 3: Object detection pipeline and notation. Given an input image, I, NMS-based detectors yield raw detections before
+post-processing, each of which has a predicted bounding box (BB) and an array of confidence scores over GT classes. We
+denote the confidence scores and the predicted BBs pertaining to the positive detections, i.e., the detections matching with GT
+objects during training, by ˆsI
+pre and ˆBI
+pre, respectively. To obtain final detections, raw detections are post-processed in three
+steps: (i) Detections with low confidence scores, i.e., background, are removed, (ii) duplicates are eliminated by NMS, and
+(iii) top-k scoring detections are kept. As for these final detections, we denote the confidence scores and BBs of true positive
+detections for class c in a single image I by ˆsI,c
+post and ˆBI,c
+post respectively, and over the entire dataset by ˆsc
+post and ˆBc
+post. As for
+NMS-free detectors; NMS, dashed gray box in post-processing, is excluded, hence post-processing is lighter.
+sets, X = {x1, ..., xN} and Y = {y1, ..., yN}. Different re-
+lations are evaluated by different correlation coefficients: (i)
+Pearson correlation coefficient, denoted by α(·, ·), measures
+the linear relationship between the sets, (ii) Spearman corre-
+lation coefficient, β(·, ·), corresponds to the ranking relation-
+ship and (iii) Concordance correlation coefficient, γ(·, ·), is
+more strict, measuring the similarity of the values and max-
+imized when xi = yi for all i ∈ 1, ..., N. All correlation
+coefficients have a range of [−1, +1] where positive/neg-
+ative correlation corresponds to increasing/decreasing rela-
+tion, while 0 implies no correlation between X and Y .
+Comparative Summary. In this paper, we comprehen-
+sively identify and analyze the effect of explicitly correlat-
+ing classification and localization in object detectors. Unlike
+other methods that also enforce correlation, some of which
+are tested only on a single architecture (Huang et al. 2019;
+Jiang et al. 2018; Tian et al. 2019), we propose a simple solu-
+tion by directly optimizing the correlation coefficient, which
+is auxiliary-head free and easily applicable to all object de-
+tectors, whether NMS-based or NMS-free. Also, ours is the
+first to work on NMS-free detectors in this context.
+3
+Effects of Correlation on Object Detectors
+This section presents why maximizing correlation is impor-
+tant for object detectors, introduces measures to evaluate ob-
+ject detectors wrt. correlation and provides an analysis on
+methods designed for improving correlation.
+3.1
+How Correlation Affects Object Detectors
+Detectors are affected by correlation at two levels (Fig. 4):
+Image-level Correlation. This level of correlation corre-
+sponds to the correlation between the classification scores
+and IoUs of the detections in a single image before post-
+processing, and accordingly, we measure it with the Spear-
+man correlation coefficient2, β(·, ·), averaged over images.
+2While analyzing object detectors in terms of correlation, we
+employ Spearman correlation coefficient, β(·, ·), to measure the
+relation between the ranks of the values (i.e., scores and IoUs) in-
+stead of the values themselves, and aim to isolate the correlation
+quality from the localization and classification performances.
+Denoting the set of images to be evaluated by I and IoUs
+between the BBs of the positive detections ( ˆBI
+pre, Fig. 3)
+and their associated GTs by IoUI
+pre, image-level correlation
+is measured as follows:
+βimg = 1
+|I|
+�
+I∈I
+β(IoUI
+pre, ˆsI
+pre).
+(1)
+Maximizing image-level correlation is important for
+NMS-based detectors since NMS aims to suppress dupli-
+cates, i.e., to keep only a single detection for each GT when
+there is more than one. More particularly among overlap-
+ping detections (e.g., dark and light green detections in the
+detector output image in Fig. 4(a)), NMS picks the one with
+the larger score, and hence, if there is positive correlation
+between the confidence scores and IoUs of those overlap-
+ping detections, then the one with the best IoU (e.g., dark
+green detection in Fig. 4(a)) will survive and detection per-
+formance will increase.
+Class-level Correlation. This level of correlation indi-
+cates the correlation between the classification scores and
+IoUs of the detections obtained after post-processing for
+each class. Since class-level correlation is related to COCO-
+style AP, APC, we average β(·, ·) over classes to be consis-
+tent with the computation of APC:
+βcls = 1
+|C|
+�
+c∈C
+β(IoUc
+post, ˆsc
+post),
+(2)
+where C is the set of classes in the dataset and IoUc
+post is the
+set IoUs of BBs of true positives for class c ( ˆBc
+post, Fig. 3).
+Class-level correlation affects the performance of all de-
+tectors since it is directly related to APC, the performance
+measure itself. To be more specific, APC for a single class
+is defined as the average of APs computed over 10 differ-
+ent IoU thresholds, IoU ∈ {0.50, 0.55, ..., 0.95}, validating
+the true positives. For a specific threshold IoU, the detec-
+tions are first sorted with respect to the classification scores,
+and then precision and recall pairs are calculated on each
+detection. Using these pairs, a precision-recall (PR) curve is
+obtained, and the area under the PR curve corresponds to
+
+Positively correlated
+NMS
+Solid BBs
+: Ground truths 
+Dashed BBs
+: Detections before         
+post-processing ( Ƹ𝑠𝑝𝑟𝑒
+𝐼
+, ෠𝐵𝑝𝑟𝑒
+𝐼
+in Fig. 3)
+Positively correlated - High APC
+(b) Class-level Correlation for better AP 
+AP Calculation
+𝐼𝑜𝑈
+0.80
+N/A
+0.60
+N/A
+0.50
+Ƹ𝑠
+0.80
+0.70
+0.60
+0.55
+0.50
+𝐼𝑜𝑈
+0.80
+N/A
+0.60
+N/A
+0.50
+Ƹ𝑠
+0.50
+0.55
+0.60
+0.70
+0.80
+Negatively correlated - Low APC
+Precision
+Pos. Detections
+APIoU
+P50
+1.00
+0.67
+0.60
+0.76
+P75
+1.00
+0.00
+0.00
+0.33
+Precision
+Pos. Detections
+APIoU
+P50
+1.00
+0.67
+0.60
+0.76
+P75
+0.20
+0.00
+0.00
+0.07
+Negatively correlated
+Solid BBs
+: Ground truths 
+Dashed BBs
+: Detections after 
+post-processing ( Ƹ𝑠𝑝𝑜𝑠𝑡
+𝐼,𝑐 , ෠𝐵𝑝𝑜𝑠𝑡
+𝐼,𝑐 in Fig. 3)
+(a) Image-level Correlation for better NMS 
+High 
+IoU
+Low 
+IoU
+Figure 4: How correlation affects detection performance. (a) Image-level correlation: Given detections before post-processing,
+NMS benefits from image-level correlation, thereby yielding detections with better IoU. Compare IoUs of detections in “posi-
+tively correlated” (i.e., when the dark-colored ones have larger score) and “negatively correlated” (i.e., when the light-colored
+ones have larger score) outputs after NMS. (b) Class-level correlation: Given detections after post-processing, APs with larger
+IoUs and COCO-style AP benefit from positive class-level correlation (compare APIoU columns in “positively correlated” and
+“negatively correlated” outputs after AP Calculation to see lower AP75 for the “negatively correlated” output in the red cell).
+PIoU: Precision computed on a detection using the threshold IoU, True positives are color-coded in tables and input, white
+cells: false positives, and hence their IoU is not available, N/A.
+the single AP value, APIoU. When the correlation between
+classification and localization is maximized among true pos-
+itives, larger precision values are obtained on the same de-
+tections in larger IoU values (e.g. P75 of orange detection is
+1.00 and 0.20 with positive and negative correlation respec-
+tively in Fig. 4(b)).
+3.2
+Analyses of Object Detectors wrt. Correlation
+Dataset and Implementation Details. Unless otherwise
+specified; we (i) employ the widely-used COCO dataset (Lin
+et al. 2014) by training the models on trainval35K (115K
+images), testing on minival (5k images), comparing with
+SOTA on test-dev (20k images), (ii) build upon the mmde-
+tection framework (Chen et al. 2019), (iii) rely on AP-based
+measures and also use Optimal LRP (oLRP) (Oksuz et al.
+2021b), βimg (Eq. 1) and βcls (Eq. 2) to provide more in-
+sights, (iv) keep the standard configuration of the models,
+(v) use a ResNet-50 backbone with FPN (Lin et al. 2017),
+(vi) train models on 4 GPUs (A100 or V100 type GPUs)
+with 4 images on each GPU (16 batch size).
+Analysis Setup. We conduct experiments to analyze the
+effects of the image-level (βimg – Table 1) and class-level
+(βcls – Table 2) correlations. For both analyses, we com-
+pare three sets of methods, all of which are incorporated
+into the common ATSS baseline (Zhang et al. 2020) (see
+Sec. 2 for a discussion of these methods): (i) AP Loss and
+Focal Loss as methods not enforcing correlation, (ii) using
+an auxiliary head to enforce correlation, and (iii) Quality Fo-
+cal Loss (QFL), aLRP Loss and Rank & Sort Loss as recent
+loss functions enforcing correlation. In our class-level anal-
+ysis, we also employ NMS-free methods to demonstrate the
+effects of correlation on that approach.
+We compare the methods based on (i) their AP-based per-
+formance, (ii) our proposed measures for correlation (Eqs.
+1 and 2), and finally (iii) lower/upper bounds, AP+1
+C /AP−1
+C ,
+obtained by modifying the ranking of the confidence scores
+pertaining to the GT classes of the positive detections to
+minimize/maximize Eq. 1 in Table 1 and Eq. 2 in Table 2.
+More particularly, in Table 1, given ˆsI
+pre and ˆBI
+pre (Fig. 3),
+we collect the GT class probabilities of positive detections
+and change their ranking in ˆsI
+pre within an image follow-
+ing the ranking order of IoUs (computed using ˆBI
+pre), and in
+Table 2, we do the same operation class-wise for true posi-
+tives given ˆsc
+post and ˆBc
+post (Fig. 3). To decouple other types
+of errors as much as possible; in Table 1, we do not modify
+the scores of the negative detections, the predicted BBs and
+the scores of non-GT classes of the positive detections, and
+in Table 2, we do not modify the scores of the false positives
+and the predicted BBs of the true positives. Note that achiev-
+ing the upper bound in (iii) for image-level correlation also
+corresponds to perfectly minimizing RS Loss.
+Observations. We observe in Tables 1 and 2 that:
+(1) Our proposed measures in Eqs. 1 and 2 can measure
+the improvements in correlation consistently. In Tables 1 and
+2, (i) aLRP Loss and RS Loss are proposed to improve AP
+Loss and (ii) aux. head and QFL are proposed to improve Fo-
+cal Loss. In both tables, the proposed methods are shown to
+improve their baselines in terms of βimg and βcls, suggest-
+ing that our measures can consistently evaluate image-level
+and class-level correlations respectively.
+(2) NMS-free detectors can also potentially benefit from
+correlation. All detectors, including NMS-free ones, can ex-
+ploit class-level correlation (compare APC and AP+1
+C to see
+∼ 10 points gap in Table 2). Still, existing methods do not
+enforce this correlation on NMS-free detectors.
+(3) Existing methods enforcing correlation have still a
+large room for improvement. Considering that βimg
+∈
+[27.2%, 33.8%] (Table 1) and βcls ∈ [37.5%, 47.0%] (Table
+2), there is still room for improvement wrt. correlation.
+
+Performance
+Modify ranking of scores
+Method
+APC AP50 AP75 βimg
+AP−1
+C
+AP−1
+50
+AP−1
+75
+AP+1
+C
+AP+1
+50
+AP+1
+75
+Not Enforcing Correlation
+ATSS w. AP Loss (Chen et al. 2020)
+38.1
+58.2
+41.0
+27.2
+24.9
+53.2
+19.2
+57.0
+72.4
+62.2
+ATSS w. Focal Loss (Lin et al. 2020)
+38.7
+57.6
+41.5
+27.3
+25.6
+51.8
+21.1
+55.8
+70.6
+60.5
+Using Aux. Head
+ATSS w. ctr. head (Zhang et al. 2020)
+39.3
+57.5
+42.6
+28.7
+16.8
+32.4
+15.3
+49.8
+64.8
+54.2
+Using Novel Loss
+ATSS w. aLRP Loss (Oksuz et al. 2020)
+37.7
+57.4
+39.9
+33.8
+22.7
+48.8
+17.5
+54.2
+70.4
+58.7
+ATSS w. QFL (Li et al. 2020)
+39.7
+58.1
+42.7
+33.2
+25.7
+51.1
+21.9
+55.8
+70.9
+60.6
+ATSS w. RS Loss (Oksuz et al. 2021a)
+39.9
+58.9
+42.6
+30.9
+26.2
+53.9
+21.3
+57.1
+71.8
+62.1
+Table 1: Evaluation of NMS-based detectors in terms of image-level correlation. See Eq. 1 for βimg. AP+1
+IoU and AP−1
+IoU refer to
+the upper & lower bound APs (see analysis setup for details). The values are in %. Our βimg captures correlation consistently,
+e.g. that (i) Focal Loss is improved by ctr. head and QFL and (ii) AP Loss is improved by aLRP Loss and RS Loss wrt. βimg.
+Also, there is still room for improvement for object detectors wrt. βimg with a range between 27.2% and 33.8%.
+Performance
+Modify ranking of scores
+Method
+APC AP50 AP75
+βcls
+AP−1
+C
+AP−1
+50
+AP−1
+75
+AP+1
+C
+AP+1
+50
+AP+1
+75
+Not Enforcing Correlation
+- NMS-free Detectors
+Sparse R-CNN (Sun et al. 2021b)
+37.7
+55.8
+40.5
+37.5
+30.1
+55.8
+28.9
+48.6
+55.8
+52.7
+DETR (Carion et al. 2020)
+40.1
+60.6
+42.0
+47.0
+32.9
+60.6
+30.6
+51.9
+60.6
+55.8
+- NMS-based Detectors
+ATSS w. AP Loss (Chen et al. 2020)
+38.1
+58.2
+41.0
+39.4
+30.0
+58.2
+26.6
+48.5
+58.2
+54.0
+ATSS w. Focal Loss (Lin et al. 2020)
+38.7
+57.6
+41.5
+40.3
+30.2
+57.6
+27.3
+48.7
+57.6
+53.6
+Using Aux. Head
+ATSS w. ctr. head (Zhang et al. 2020)
+39.3
+57.4
+42.5
+42.5
+30.2
+57.4
+27.6
+48.7
+57.4
+53.5
+Using Novel Loss
+ATSS w. aLRP Loss (Oksuz et al. 2020)
+37.7
+57.4
+39.9
+42.0
+29.1
+57.4
+25.0
+47.8
+57.4
+52.7
+ATSS w. QFL (Li et al. 2020)
+39.7
+58.1
+42.7
+45.7
+30.6
+58.1
+27.7
+49.1
+58.1
+53.9
+ATSS w. RS Loss (Oksuz et al. 2021a)
+39.9
+58.9
+42.6
+43.2
+31.1
+58.9
+28.1
+49.8
+58.9
+54.8
+Table 2: Evaluation of detectors wrt. class-level correlation. See Eq. 2 for βcls. AP+1
+IoU & AP−1
+IoU denote upper & lower bound
+APs (analysis setup for details). Values are in %. NMS-free detectors can also benefit from class-level correlation (compare
+AP+1
+C with APC for Sparse R-CNN), and as in βimg (c.f. Table 1 and its caption), βcls measures the correlation consistently.
+AP+1
+50 = AP−1
+50 = AP50 since only modifying TPs validated from IoU=0.50 does not effect AP50 (see Fig. 4(b) for an example).
+(4) While significantly important, improving correlation
+may not always imply performance improvement. For exam-
+ple, aLRP Loss in Table 1 has the largest correlation but the
+lowest APC. Such a situation can arise, for example, when a
+method does not have good localization performance. In the
+extreme case, assume a detector yields perfect βimg, image-
+level ranking correlation, but the IoUs of all positive exam-
+ples are less than 0.50 implying no TP at all. Hence, boost-
+ing the correlation, while simultaneously preserving a good
+performance in each branch, is critical.
+4
+Correlation Loss: A Novel Loss Function
+for Object Detection
+Correlation (Corr.) Loss is a simple plug-in loss function to
+improve correlation of classification and localization tasks.
+Correlation Loss is unique in that it can be easily incorpo-
+rated into any object detector, whether NMS-based or NMS-
+free (see Observation (2) - Sec. 3.2), and improves perfor-
+mance without affecting the model size, inference time and
+with negligible effect on training time (Sec. 5.4). Further-
+more, from a fundamental perspective, Corr. Loss can su-
+pervise both of the classification and localisation heads for a
+better correlation while existing methods generally focus on
+a single head such as classification (Fig. 1).
+Definition. Given an object detector with loss function
+LOD, our Correlation Loss (Lcorr) is simply added using a
+weighting hyper-parameter λcorr:
+LOD + λcorrLcorr.
+(3)
+Lcorr is the Correlation Loss defined as:
+Lcorr = 1 − ρ( ˆ
+IoU,ˆs),
+(4)
+where ρ(·, ·) is a correlation coefficient; ˆs and
+ˆ
+IoU are the
+confidence scores of the GT class and IoUs of the predicted
+BBs pertaining to the positive examples in the batch.
+Practical Usage. To avoid promoting trivial cases with
+high correlation but low performance (Observation (4) -
+Sec. 3.2), similar to QFL (Li et al. 2020) and RS Loss
+(Oksuz et al. 2021a), we only use the gradients of Lcorr
+wrt. classification score, i.e., we backpropagate the gradi-
+ents through only the classifier. We mainly adopt two dif-
+ferent correlation coefficients for ρ(·, ·) and obtain two ver-
+sions of Correlation Loss: (i) Concordance Loss, defined as
+the Correlation Loss when Concordance correlation coeffi-
+cient is optimized (ρ(·, ·) = γ(·, ·)), which aims to match
+
+Method
+APC ↑AP50 ↑AP75 ↑ oLRP ↓
+NMS-based
+Retina Net (Lin et al. 2020)
+36.5
+55.4
+39.1
+70.7
+w. Conc.Corr (Ours)
+37.0
+55.7
+39.7
+70.2
+w. Spear.Corr (Ours)
+37.5
+55.4
+40.5
+69.7
+Fovea Box (Kong et al. 2020)
+36.4
+56.5
+38.6
+70.2
+w. Conc.Corr (Ours)
+37.1
+56.4
+39.6
+69.7
+w. Spear.Corr (Ours)
+37.0
+55.6
+39.3
+70.0
+ATSS (Zhang et al. 2020)
+38.7
+57.6
+41.5
+69.0
+w. Conc.Corr (Ours)
+39.8
+57.9
+43.2
+68.2
+w. Spear.Corr (Ours)
+39.3
+56.6
+42.5
+68.7
+PAA (Kim and Lee 2020)
+39.9
+57.3
+43.4
+68.6
+w. Conc.Corr (Ours)
+40.7
+58.8
+44.3
+67.7
+w. Spear.Corr (Ours)
+40.4
+58.0
+43.7
+67.8
+NMS-free
+Sparse R-CNN (Sun et al. 2021b) 37.7
+55.8
+40.5
+69.5
+w. Conc.Corr (Ours)
+38.9
+57.2
+41.8
+68.1
+w. Spear.Corr (Ours)
+39.3
+56.7
+42.5
+68.3
+Table 3: Comparison on detectors not considering correla-
+tion. Accordingly, we remove aux. heads from ATSS (Zhang
+et al. 2020) and PAA (Kim and Lee 2020) for fair compari-
+son (see Table 6 for comparison with aux. heads and novel
+loss functions). We use ResNet-50 and train the models for
+12 epochs. Simply incorporating our Corr. Loss provides
+(i) ∼ 1APC improvement for NMS-based detectors consis-
+tently and (ii) ∼ 1.5APC on the NMS-free Sparse R-CNN.
+the confidence scores with IoUs. (ii) Spearman Loss as Cor-
+relation Loss when Spearman correlation coefficient is op-
+timized (ρ(·, ·) = β(·, ·)), thereby enforcing the ranking
+of the classification scores considering IoUs. To tackle the
+non-differentiability of ranking operation while computing
+Spearman Loss, we leverage the differentiable sorting oper-
+ation from Blondel et al. (Blondel et al. 2020). When apply-
+ing our Correlation Loss to NMS-free methods, which use
+an iterative multi-stage loss function, we incorporate Lcorr
+to every stage.
+5
+Experimental Evaluation
+We evaluate Corr. Loss on (i) the COCO dataset with five
+different object detectors of various types (Sparse R-CNN as
+NMS-free, FoveaBox as anchor-free, RetinaNet as anchor-
+based, ATSS and PAA using auxiliary head), and one in-
+stance segmentation method, YOLACT; and (ii) an addi-
+tional dataset (Cityscapes) for the method with the largest
+gain, i.e., Sparse R-CNN.
+5.1
+Comparison with Methods Not Considering
+Correlation
+We train these five object detectors and the instance segmen-
+tation method (Tables 3 and 5) with and without our Corr.
+Loss (as Concordance Loss or Spearman Loss).
+NMS-based Detectors. Table 3 suggests ∼ 1.0APC gain
+on NMS-based detectors: (i) Spearman Loss (λcorr = 0.1)
+improves RetinaNet by 1.0APC and oLRP, (ii) Concor-
+dance Loss (λcorr = 0.2) enhances anchor-free FoveaBox
+by 0.7APC, and (iii) Concordance Loss (λcorr = 0.3) im-
+proves ATSS and PAA by ∼ 1APC and ∼ 1oLRP.
+NMS-free Detectors. Our results in Table 3 suggest that
+Sparse R-CNN, an NMS-free method, can also benefit from
+Method
+AP
+AP50
+AP75
+Sparse R-CNN
+39.0
+63.1
+37.6
+w. Spear.Corr (Ours)
+40.8
+64.4
+40.8
+Table 4: Results on Cityscapes dataset.
+Method
+APmask
+C
+APmask
+50
+APmask
+75
+YOLACT (Bolya et al. 2019)
+28.3
+47.8
+28.8
+w. Conc.Corr (Ours)
+28.8
+48.3
+29.6
+w. Spear.Corr (Ours)
+29.0
+48.3
+30.0
+Table 5: Comparison with YOLACT.
+our Corr. Loss: (i) Both Concordance (λcorr = 0.3) and
+Spearman Losses (λcorr
+= 0.2) improve baseline; (ii)
+Spearman Loss improves APC significantly by up to 1.6;
+(iii) as hypothesized, the gains are owing to APs with larger
+IoUs, e.g., AP75 improves by up to 2.0, and (iv) gains persist
+in a stronger setting of Sparse R-CNN (Appendix).
+Cityscapes dataset. To see the effect of Corr. Loss over
+different scenarios, we train Sparse R-CNN with Spear-
+man Loss (the model that has the best gain over baseline
+in Table 3), on the Cityscapes dataset (Cordts et al. 2016)
+(λcorr = 0.6), a dataset for autonomous driving object de-
+tection. Table 4 presents that (i) Spearman Loss also im-
+proves baseline Sparse R-CNN on Cityscapes by 1.8 AP and
+(ii) our gain mainly originates from APs with larger IoUs,
+i.e. AP75 improves by more than 3 points, from 37.6 to 40.8.
+Instance Segmentation. We train YOLACT (Bolya et al.
+2019) as an instance segmentation method with Corr. Loss
+and observed 0.7 mask AP gain using Spearman Loss
+(λcorr = 0.5 - Table 5), implying 1.7% relative gain.
+5.2
+Comparison with Methods Enforcing
+Correlation
+Table 6 compares Corr. Loss. with using an aux. head
+(Zhang et al. 2020), QFL (Li et al. 2020) and RS Loss (Ok-
+suz et al. 2021a) on the common ATSS baseline (Zhang et al.
+2020) wrt. detection and correlation:
+Detection Performance. Reaching 39.8APC without an
+aux. head, Concordance Loss (Table 6) outperforms using an
+aux. head, which introduces additional learnable parameters
+(39.8 vs 39.3APC), and reaches on-par performance with
+the recently proposed, relatively complicated loss functions,
+Aux.QFLRS LossOurs APC AP50 AP75 oLRP ↓ βimg ↑βcls ↑
+38.7 57.6 41.5
+68.9
+27.3
+40.3
+✓
+39.3 57.5 42.6
+68.6
+28.7
+42.5
+✓
+39.7 58.1 42.7
+68.0
+33.2
+45.7
+✓
+39.9 58.9 42.6
+67.9
+30.9
+43.2
+✓
+39.8 57.6 43.1
+68.2
+31.6
+45.2
+✓
+✓
+40.0 58.0 43.3
+68.0
+31.1
+44.8
+✓
+✓
+39.9 58.2 43.2
+67.7
+34.6
+45.6
+✓
+✓
+40.2 58.6 43.5
+67.9
+33.6
+46.1
+Table 6: Comparison with methods enforcing correlation.
+Corr. Loss (i) reaches similar results with existing methods
+on ATSS, (ii) is complementary to those methods thanks to
+its simple design and (iii) once combined with RS Loss, out-
+performs compared methods.
+
+Method
+Backbone
+Epochs APC AP50 AP75 APS APM APL
+Venue
+NMS-based
+ATSS (Zhang et al. 2020)
+ResNet-101-DCN
+24
+46.3 64.7
+50.4 27.7 49.8 58.4
+CVPR 2020
+GFLv2 (Li et al. 2019)
+ResNet-101-DCN
+24
+48.3 66.5
+52.8 28.8 51.9 60.7
+CVPR 2021
+aLRP Loss (Oksuz et al. 2020)
+ResNeXt-101-DCN
+100
+48.9 69.3
+52.5 30.8 51.5 62.1 NeurIPS 2020
+VFNet (Zhang et al. 2021)
+ResNet-101-DCN
+24
+49.2 67.5
+53.7 29.7 52.6 62.4
+CVPR 2021
+DW (Li et al. 2022)
+ResNet-101-DCN
+24
+49.3 67.6
+53.3 29.2 52.2 63.5
+CVPR 2022
+TOOD (Feng et al. 2021)
+ResNet-101-DCN
+24
+49.6 67.4
+54.1 30.5 52.7 62.4
+ICCV 2021
+RS-Mask R-CNN+ (Oksuz et al. 2021a) ResNeXt-101-DCN
+36
+50.2 70.3 54.8 31.5 53.5 63.9
+ICCV 2021
+NMS-free
+TSP R-CNN (Sun et al. 2021c)
+ResNet-101-DCN
+96
+47.4 66.7
+51.9 29.0 49.7 59.1
+ICCV 2021
+Sparse R-CNN (Sun et al. 2021b)
+ResNeXt-101-DCN
+36
+48.9 68.3
+53.4 29.9 50.9 62.4
+CVPR 2021
+Dynamic DETR (Dai et al. 2021)
+ResNeXt-101-DCN
+36
+49.3 68.4
+53.6 30.3 51.6 62.5
+ICCV 2021
+Deformable DETR (Zhu et al. 2021)
+ResNeXt-101-DCN
+50
+50.1 69.7
+54.6 30.6 52.8 64.7
+ICLR 2021
+Ours
+Corr-Sparse R-CNN
+ResNet-101-DCN
+36
+49.6 67.8
+54.1 29.2 52.3 64.9
+Corr-Sparse R-CNN
+ResNeXt-101-DCN
+36
+51.0 69.2 55.7 31.1 53.7 66.3
+Table 7: SOTA comparison on COCO test-dev. Our Corr-Sparse R-CNN (i) performs on-par or better compared to recent NMS-
+based methods, all of which also enforce correlation, and (ii) outperforms NMS-free methods by a notable margin. Results are
+obtained from papers.
+QFL (Li et al. 2020) and RS Loss (Oksuz et al. 2021a). Be-
+sides, owing to its simple usage, Concordance Loss is com-
+plementary to existing methods: It yields 40.0APC with an
+aux. head (+0.7 APC) and 40.2APC with RS Loss (+0.3
+APC) without introducing additional learnable parameters.
+Correlation Analysis. To provide insight, we report βimg
+(Eq. 1) and βcls (Eq. 2) in Table 6: Our Concordance Loss (i)
+improves baseline correlation significantly, enhancing βimg
+(from 27.3% to 31.6%) and βcls (from 40.3% to 45.2%)
+both by ∼ 5%, and (ii) results in better correlation than all
+methods wrt. βimg and βcls once combined with QFL and
+RS Loss respectively. This set of results confirms that Con-
+cordance Loss improves correlation between classification
+and localization tasks in both image-level and class-level.
+5.3
+Comparison with SOTA
+Here, we prefer Sparse R-CNN owing to its competitive de-
+tection performance and our large gains. We train our “Corr-
+Sparse R-CNN” for 36 epochs with DCNv2 (Zhu et al. 2019)
+and multiscale training by randomly resizing the shorter side
+within [480, 960] similar to common practice (Oksuz et al.
+2021a; Zhang et al. 2021; Sun et al. 2021b). Table 7 presents
+the results on COCO test-dev (Lin et al. 2014):
+NMS-based Methods. On the common ResNet-101-
+DCN backbone and with similar data augmentation, our
+Corr-Sparse R-CNN yields 49.6APC at 13.7 fps (on a V100
+GPU) outperforming recent NMS-based methods, all of
+which also enforce correlation, e.g., (i) RS-R-CNN (Ok-
+suz et al. 2021a) by 1.8APC, (ii) GFLv2 (Li et al. 2019)
+by more than 1APC, and (iii) VFNet (Zhang et al. 2021) in
+terms of not only APC but also efficiency (with 12.6 fps on
+a V100 GPU). On ResNeXt-101-DCN, our Corr-Sparse R-
+CNN provides 51.0APC at 6.8 fps, surpassing all methods
+including RS-Mask R-CNN+ (50.2APC at 6.4 fps), addi-
+tionally using masks and Carafe FPN (Wang et al. 2019).
+NMS-free Methods. Our Corr-Sparse R-CNN outper-
+forms (i) TSP R-CNN (Sun et al. 2021c) by more than
+2APC on ResNet-101-DCN with significantly less training,
+(ii) Sparse R-CNN (Sun et al. 2021b) by ∼ 2APC and De-
+formable DETR (Zhu et al. 2021), a recent strong NMS-free
+method, by ∼ 1APC on ResNeXt-101-DCN.
+5.4
+Ablation & Hyper-parameter Analyses
+Optimizing Different Correlation Coefficients. Spearman
+Loss yields better localization performance, i.e. the lowest
+localization error wrt. oLRPLoc in all experiments while
+it rarely yields the best oLRPFP or oLRPFN, implying its
+contribution to classification to be weaker than Concordance
+Loss (see Appendix for components of oLRP). We also
+tried Pearson Correlation Coefficient on ATSS and Sparse
+R-CNN but it performed worse compared to either using
+Spearman or Concordance (Appendix).
+Backpropagating Through Different Heads. On Sparse
+R-CNN, we observed that the performance degrades when
+we backpropagate either only localization head (37.5 AP) or
+both heads (38.9 AP). Hence, we preferred backpropagating
+the gradients only through the classification head (39.3 AP).
+Effect on Training Time. Using Spearman or Concor-
+dance Loss to train Sparse R-CNN, computing the loss for 6
+times each iteration, increases iteration time 0.50 sec to 0.51
+sec on V100 GPUs, suggesting a negligible overhead.
+Sensitivity to λcorr. We found it sufficient to search over
+{0.1, 0.2, 0.3.0.4, 0.5, 0.6} to tune λcorr. Appendix presents
+empirical results for grid search.
+5.5
+Additional Material
+This paper is accompanied by an Appendix containing (i) the
+effect of Corr.Loss on Sparse R-CNN using its stronger set-
+ting, (ii) components of oLRP for detectors in Table 3, (iii)
+results when Pearson Correlation Coefficient is optimized,
+(iv) our grid search to tune λcorr.
+6
+Conclusion
+In this paper, we defined measures to evaluate object detec-
+tors wrt. correlation, provided analyses on several methods
+and proposed Correlation Loss as an auxiliary loss function
+to enforce correlation for object detectors. Our extensive
+experiments on six detectors show that Correlation Loss.
+consistently improves the detection and correlation perfor-
+mances, and reaches SOTA results.
+
+Acknowledgments
+This work was supported by the Scientific and Technolog-
+ical Research Council of Turkey (T ¨UB˙ITAK) (under grant
+120E494). We also gratefully acknowledge the computa-
+tional resources kindly provided by T ¨UB˙ITAK ULAKBIM
+High Performance and Grid Computing Center (TRUBA)
+and METU Robotics and Artificial Intelligence Center
+(ROMER). Dr. Akbas is supported by the “Young Scientist
+Awards Program (BAGEP)” of Science Academy, Turkey.
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+APPENDIX
+Sensitivity to λcorr. In Table A.8, we see that (i) λcorr =
+0.2 provides the best performance overall, (ii) the perfor-
+mance is not very sensitive to λcorr and (iii) a grid search
+over {0.1, 0.2, 0.3.0.4, 0.5, 0.6} is sufficient (outside of this
+range, performance drops).
+The effect of Corr.Loss on Sparse R-CNN using its
+stronger setting. Following Sun et al. (Sun et al. 2021b)
+Method
+Dataset
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+ATSS
+COCO
+38.7 38.8 39.3 39.8 39.7 39.7 39.6
+YOLACT
+COCO
+28.3 28.6 28.8 28.8 29.0 28.8 28.6
+Sparse R-CNN
+COCO
+37.7 38.7 39.3 39.1 39.0 38.1 38.0
+Sparse R-CNN Cityscapes 39.0 39.0 38.3 39.9 40.0 40.0 40.8
+Table A.8: Grid search to tune λcorr on different models.
+We present the results for concordance correlation coeffi-
+cient for ATSS and YOLACT, and spearman correlation co-
+efficient for Sparse R-CNN models. 0.0 corresponds to not
+including our Correlation Loss.
+Method
+AP
+AP50
+AP75
+Sparse R-CNN (Sun et al. 2021b)
+45.0
+64.1
+48.9
+w. Conc.Corr (Ours)
+45.5
+64.4
+49.7
+w. Spear.Corr (Ours)
+46.1
+64.0
+50.4
+Table A.9: Comparison with stronger Sparse R-CNN.
+(Table A.9), we train Sparse R-CNN with 36 epochs train-
+ing, 300 proposals, multi-scale training and random crop-
+ping. Table A.9 presents that the improvement of our Spear-
+man Loss on this strong baseline is ∼ 1 AP points.
+Using Pearson Correlation Coefficient. We tried opti-
+mizing pearson correlation coefficient as well and observed
+that while it has similar performance with concordance cor-
+relation coefficient on ATSS and spearman correlation coef-
+ficient on Sparse R-CNN, it does not outperform the other
+two in both of the cases (Table A.10). Considering the sim-
+ilarities of spearman and concordance correlation coeffi-
+cients in terms of scoring the relation of the values, we
+preferred concordance correlation coefficient over spearman
+correlation coefficient due to the fact that concordance corre-
+lation coefficient enforces the scores to be equal to the IoUs
+imposing a tighter constraint than pearson correlation coef-
+ficient.
+The components of oLRP. Table A.11 shows the the
+components of oLRP for different detectors corresponding
+to Table 3 in the paper. As discussed in the paper, Spearman
+Loss yields better localization performance, i.e. the lowest
+localization error wrt. oLRPLoc in all experiments while
+it rarely yields the best oLRPFP or oLRPFN, implying its
+contribution to classification to be weaker than Concordance
+Loss.
+Method
+APC
+AP50
+AP75
+ATSS w/o aux head
+38.7
+57.6
+41.5
+w. Pearson Corr
+39.4
+56.6
+42.7
+w. Conc.Corr
+39.8
+57.9
+43.2
+w. Spear.Corr
+39.3
+56.6
+42.5
+Sparse-RCNN
+37.7
+55.9
+40.5
+w. Pearson Corr
+39.3
+56.6
+42.2
+w. Conc.Corr
+38.9
+57.2
+41.8
+w. Spear.Corr
+39.3
+56.7
+42.5
+Table A.10: Effect of using Pearson correlation coefficient.
+
+Method
+APC ↑
+AP50 ↑
+AP75 ↑
+oLRP ↓
+oLRPLoc ↓
+oLRPFP ↓
+oLRPFN ↓
+Retina Net (Lin et al. 2020)
+36.5
+55.4
+39.1
+70.7
+16.8
+32.0
+48.1
+w. Conc.Corr (Ours)
+37.0
+55.7
+39.7
+70.2
+16.3
+30.8
+49.3
+w. Spear.Corr (Ours)
+37.5
+55.4
+40.5
+69.7
+16.0
+31.3
+48.4
+Fovea Box (Kong et al. 2020)
+36.4
+56.5
+38.6
+70.2
+17.0
+30.2
+47.2
+w. Conc.Corr (Ours)
+37.1
+56.4
+39.6
+69.7
+16.6
+28.6
+48.1
+w. Spear.Corr (Ours)
+37.0
+55.6
+39.3
+70.0
+16.3
+31.0
+47.9
+ATSS (Zhang et al. 2020)
+38.7
+57.6
+41.5
+69.0
+16.0
+29.1
+47.0
+w. Conc.Corr (Ours)
+39.8
+57.9
+43.2
+68.2
+15.4
+29.1
+46.9
+w. Spear.Corr (Ours)
+39.3
+56.6
+42.5
+68.7
+15.2
+31.2
+46.7
+PAA (Kim and Lee 2020)
+39.9
+57.3
+43.4
+68.6
+15.0
+30.4
+47.0
+w. Conc.Corr (Ours)
+40.7
+58.8
+44.3
+67.7
+15.2
+28.5
+46.3
+w. Spear.Corr (Ours)
+40.4
+58.0
+43.7
+67.8
+14.9
+29.5
+46.6
+Sparse R-CNN (Sun et al. 2021b)
+37.7
+55.8
+40.5
+69.5
+16.0
+28.7
+48.6
+w. Conc.Corr (Ours)
+38.9
+57.2
+41.8
+68.1
+15.7
+27.7
+47.2
+w. Spear.Corr (Ours)
+39.3
+56.7
+42.5
+68.3
+15.3
+27.1
+48.4
+Table A.11: Components of oLRP for Table 3 in the paper.
+

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+arXiv:2301.13286v1  [math.RT]  30 Jan 2023
+THE SEMI-INFINITE COHOMOLOGY OF WEYL MODULES
+WITH TWO SINGULAR POINTS
+GIORGIA FORTUNA, DAVIDE LOMBARDO,
+ANDREA MAFFEI, VALERIO MELANI
+Abstract. In their study of spherical representations of an affine Lie algebra
+at the critical level and of unramified opers, Frenkel and Gaitsgory introduced
+what they called the Weyl module Vλ corresponding to a dominant weight
+λ. This object plays an important role in the theory. In [4], we introduced a
+possible analogue Vλ,µ
+2
+of the Weyl module in the setting of opers with two
+singular points, and in the case of sl(2) we proved that it has the ‘correct’
+endomorphism ring. In this paper, we compute the semi-infinite cohomology
+of Vλ,µ
+2
+and we show that it does not share some of the properties of the semi-
+infinite cohomology of the Weyl module of Frenkel and Gaitsgory. For this
+reason, we introduce a new module ˜Vλ,µ
+2
+which, in the case of sl(2), enjoys all
+the expected properties of a Weyl module.
+1. Introduction
+Let g be a complex simple Lie algebra and let ˆg be its affinization. Choose a
+Borel subalgebra and a maximal toral subalgebra, and let G be a simply connected
+algebraic group with Lie algebra equal to g. As a particular case of a more general
+conjecture, Frenkel and Gaitsgory proved in [6] that the semi-infinite cohomology
+gives an isomorphism between the category ˆgcrit-modJG of spherical representations
+of ˆg at the critical level (that is, representations of ˆg at the critical level with a
+compatible action of JG = G(C[[t]])) and the category of quasi-coherent sheaves
+on the space of unramified opers Opunr
+1
+over gL, the Langlands dual of g.
+As
+they explain, the space of unramified opers is the disjoint union of its connected
+components Opλ,unr
+1
+, and the category of spherical representations is the product of
+certain subcategories ˆgcrit-modJG,λ, where in both cases λ ranges over all dominant
+weights of G. The equivalence given by semi-infinite cohomology specialises to an
+equivalence between ˆgcrit-modJG,λ and the category of quasi-coherent sheaves over
+Opλ,unr
+1
+. The space Opλ,unr
+1
+is a non-reduced indscheme, and its reduced version,
+denoted by Opλ
+1, is an affine scheme.
+In this paper we will denote by Zλ
+1 its
+coordinate ring.
+In this theory, an important role is played by the Weyl module Vλ
+1. This module
+enjoys the following fundamental properties:
+Endˆg(Vλ
+1) ≃ Zλ
+1
+and
+Ψ0(Vλ
+1) ≃ Zλ
+1 ,
+where Ψn is the n-th semi-infinite cohomology group. Moreover the semi-infinite
+cohomology groups Ψn(Vλ
+1) are trivial for n ̸= 0.
+Dennis Gaitsgory suggested to Giorgia Fortuna to study the space of unramified
+opers and spherical representations in a more general context, see [3]; in fact, the
+definition of unramified opers as well as the definition of spherical representations
+can be generalized in the presence of more than one singularity, raising the question
+on whether or not certain statements remain true and what happens when these
+singularities collide.
+1
+
+2
+FORTUNA, LOMBARDO, MAFFEI, MELANI
+In [4] we took some steps in this direction, by studying the case of sl(2). In
+particular, we introduced a version of the Weyl module Vλ,µ
+2
+of critical level of the
+affine Lie algebra with two singularities ˆg2. Thinking of t as a coordinate near the
+first singularity and s as a coordinate near the second singularity, this is the version
+of the affine Lie algebra over the ring A = C[[a]], where a = (t−s). As an A module
+is equal to K2 ⊗C ⊕AC2 where K2 = C[[a, t]][1/t(t− a)] and C2 is a central element
+(see [4], Section 3.3 for the complete definition).
+We also introduced reduced scheme over A of unramified opers Opλ,µ
+2
+which gener-
+alize the schemes Opλ
+1. Both objects depend on two integral dominant weights λ,
+µ of G, and we proved that
+Endˆg2(Vλ,µ
+2
+) ≃ Zλ,µ
+2
+,
+where Zλ,µ
+2
+is the coordinate ring of Opλ,µ
+2
+.
+In this article we study the semi-infinite cohomology of Vλ,µ
+2
+in order to un-
+derstand what relation it has with the ring Zλ,µ
+2
+in order to understand how the
+equivalence Ψ0(Vλ
+1) ≃ Zλ
+1 generalizes.
+This is done in Section 4, where we compute the cohomology of Vλ,µ
+2
+; in Section 5
+we study the action of Z2, the center of a completion ˆU2 of the enveloping algebra
+of ˆg2 at the critical level on this module (see Section 2.2).
+In particular, we prove that the specialisation at a = 0 and the localization at
+a ̸= 0 of the semi-infinite cohomology of Vλ,µ
+2
+are isomorphic to the specialisation
+and localization of Zλ,µ
+2
+, respectively. However, in contrast to our intuition, we also
+show the following result which says that Ψ0(Vλ,µ
+2
+) doesn’t exactly generalize the
+equivalence Ψ0(Vλ
+1) ≃ Zλ
+1 as expected:
+Theorem A (Theorem 4.9 and Proposition 5.3). We have Ψn(Vλ,µ
+2
+) = 0 for n ̸= 0.
+Moreover, Ψ0(Vλ,µ
+2
+) is not isomorphic to Zλ,µ
+2
+as a Z2-module.
+For this computation, we rely on the formalism introduced by Casarin in [1],
+which makes it possible to use vertex algebras also in the context of opers with
+two singularities. Once this formalism is in place, for the computation of the semi-
+infinite cohomology we can follow closely the approach taken by Frenkel and Ben
+Zvi in [5, Chapter 15] for the case of one singularity.
+In the last section, we restrict our attention to the Lie algebra sl(2) and introduce
+a submodule �Vλ,µ
+2
+of Vλ,µ
+2
+, which is generated by the highest weight vector. We
+prove that this module is the correct one to consider, in the sense that it has the
+expected cohomology groups and endomorphism ring, as the following result shows.
+Theorem B (Proposition 6.3, Theorem 6.5 and Proposition 6.6). If g = sl(2) then
+we have Ψn(�Vλ,µ
+2
+) = 0 for n ̸= 0. Moreover, we have
+Endˆg2(�Vλ,µ
+2
+) ≃ Zλ,µ
+2
+and
+Ψ0(�Vλ,µ
+2
+) ≃ Zλ,µ
+2
+.
+We now briefly explain the connection between these results and Conjecture 3.6.1
+in Fortuna’s Thesis [3]. As a particular case the conjecture predicts an equivalence
+between quasi-coherent sheaves over the space of unramified opers with two singu-
+larities and the category of spherical representations over ˆg2: that is the space of
+smooth representations of ˆg2 with a compatible action of J2G = G(C[[a, t]]).
+The conjecture stated in [3] predicts an equivalence of similar categories not only
+in the presence of two singularities but in the presence of n-possible singularities.
+In particular for any finite set with n elements I we can define the space of opers
+on the formal disc with n-singularities OpI and the subspace of unramified opers
+Opunr
+I
+(see Section 3.5 in [3]). These are spaces over the product of n-copies of
+the formal disc. These are easily seen to be factorization spaces, which means that
+
+SEMI-INFINITE COHOMOLOGY OF WEYL MODULES
+3
+this spaces specialise nicely when restricted along or outside the diagonals of this
+product (see Section 3.1.5 in [3]). There are not substantial differences between
+the treatment we do here or in [4] of Op2 and the general case. The only minor
+difference is that we fix a singularity to be 0. These spaces are indschemes, and
+so we can define the categories QCoh(OpI), and QCoh(Opunr
+I
+) of quasi-coherent
+sheaves on OpI and Opur
+I
+(see Section 3.5.3 in [3] for the actual definition), and the
+nice factorization properties which make them factorization categories (see Section
+3.1.2 in [3]).
+Similarly, for a finite set I we can define a Lie algebra ˆgI and study its smooth
+representations at the critical level. The objects constructed in this way live also
+on the product of n copies of the formal disc, and they also have nice factorization
+properties, in particular the collection of (completions of the) enveloping algebras
+specialized at the critical level ˆUI of the algebras ˆgI, is what is called a factorization
+algebra (see Section 3.1.3 in [3]). As a conseguence the collection of the categories of
+smooth representations at the critical of the Lie algebras ˆgI, denoted by ˆgI,crit-mod
+and their subcategories of spherical representations ˆgI,crit-modJG can be organized
+also in a factorization category. The semi-infinite cohomology can be defined also
+in this generality and defines a functor
+ΨI : ˆgI,crit-mod −→ D(QCoh(OpI))
+compatible with the factorization properties. While in Fortuna’s thesis all these
+constructions are obtained somehow for free using the language of chiral algebras
+(see Section 3.1.6 in [3]), in this paper we use the language of vertex algebras and
+the formalism introduced by Casarin [1]. Let us notice that, from this point of
+view, there are no differences in treating the case with two singular points and the
+case with an arbitrary finite number of singular points. For example, the proof of
+Theorem A above can be repeated verbatim in the case of n singular points. More
+generally we believe that all the technical difficulties in the study of this problem
+already appear in the case of two singularities.
+It is easy to see from the factorization properties and the analogous statement
+for the case of one singularty by Frenkel and Gaitsgory (see [7]) that the semi-
+infinite cohomology of a ˆgI-spherical module is supported on Opunr
+I
+. Hence semi-
+infinite cohomology restricts to a functor ΨI : ˆgI,crit-modJG −→ D(QCoh(Opunr
+I
+)).
+Conjecture 3.6.1 in [3] states that this functor is exact and
+Ψ0
+I : ˆgI,crit-modJG −→ QCoh(Opunr
+I
+)
+is an equivalence of categories. In Appendix A.2 of [3], one possible strategy to
+prove this conjecture is sketched, using the factorization structure and the result
+proved in the case of one singularity to deduce the general case. In particular,
+thanks to Proposition A.2.2 and Proposition A.2.3 of [3] and Theorems A and B
+above, a more careful study of the modules Vλ,µ
+2
+or �Vλ,µ
+2
+might help in finding a
+proof of [3, Conjecture 3.6.1], in the case of g = sl(2).
+The paper is organized as follows. In the first section we recall some definitions
+from [4]. In Section 3 we recall the formalism introduced by Casarin [1] and we
+use it to define semi-infinite cohomology and prove some of its basic properties. In
+Sections 3 and 4 we compute the semi-infinite cohomology of Vλ,µ
+2
+and in Section
+5 we compute the semi-infinite cohomology of �Vλ,µ
+2
+.
+We thank Luca Casarin for many useful discussions and in particular for explain-
+ing to us the formalism introduced in [1]. It seems to us that Casarin’s approach
+provides a natural framework to treat questions concerning opers with several singu-
+larities, making the theory much more transparent than it was in [4]. In particular,
+
+4
+FORTUNA, LOMBARDO, MAFFEI, MELANI
+the results of [1] allowed us to streamline several arguments and calculations which
+would have been quite hard to carry out using the direct approach of [4].
+2. Basic constructions
+In this section we recall some basic constructions from [4], to which we refer
+for further details, and we introduce the notion of semi-infinite cohomology in the
+context of affine Lie algebras with more than one singular point.
+2.1. Rings. We follow [4, Section 1], to which the reader is referred for more details.
+We introduce the rings
+A = C[[a]],
+Q = C((a)),
+R2 = C[[t, s]],
+K2 = C[[t, s]][1/ts],
+where a = t − s. Recall that we have expansion maps (given by suitable natural
+inclusions) and a specialisation map (which sends a to 0 and t, s to t, see Section
+1.1 in [4])
+Et : K2[a−1] −→ Q((t)),
+Es : K2[a−1] −→ Q((s)),
+Sp : K2 −→ C((t)).
+We also write E = Et × Es : K2[a−1] −→ Q((t)) × Q((s)). Recall from [4, Section
+1.1] that Sp induces an isomorphism K2/(a) ≃ C((t)). These rings have natural
+topologies: with respect to these, the image of E is dense, and E(R2[a−1]) is dense
+in Q((t)) × Q((s)).
+These rings are also equipped with residue maps
+Res2 : K2 → A
+Res1 : C((t)) → C,
+Rest : Q((t)) → Q,
+Ress : Q((s)) → Q,
+which behave nicely with respect to specialisation and expansion (see [4, Section
+1.2]). Finally, we recall Lemma 1.10 in [4].
+Lemma 2.1 ([4], Lemma 1.10). Let M, N be two A-modules and ϕ : M −→ N be
+a morphism of A-modules. Then
+a) if M is flat and ϕa : M[a−1] −→ N[a−1] is injective, then ϕ is injective.
+b) if N is flat, ϕa : M[a−1] −→ N[a−1] is surjective, and ϕ : M/aM −→ N/aN is
+injective, then ϕ is surjective.
+In particular, if M and N are flat, ϕa : M[a−1] −→ N[a−1] is an isomorphism,
+and ϕ : M/aM −→ N/aN is injective, then ϕ is an isomorphism.
+2.2. Affine Lie algebras and completion of the enveloping algebra. We
+follow [4, Section 3]. Let g be a finite-dimensional Lie algebra over the complex
+numbers and denote by κ the Killing form of g. Recall from [4, Sections 3.1 and 3.3]
+that for each of the rings of the previous section we introduce an affine Lie algebra:
+ˆg1 is the usual affine Lie algebra (we take for convenience the version defined by
+Laurent polynomial and not Laurent series), ˆgt and ˆgs are also versions of the the
+usual affine Lie algebra, while ˆg2 is an A-Lie algebra having as underlying A-module
+the space
+ˆg2 = C[t, s][1/ts] ⊗C g ⊕ A C2.
+We also introduce the Lie algebra ˆgt,s = ˆgt ⊕ ˆgs/(Ct − Cs) (see [4, Section 3.3]).
+For each of these Lie algebras, we introduce the corresponding universal envel-
+oping algebra, which we suitably complete and then specialize at the critical level
+by imposing that the central element acts as −1/2 (see Sections 3.1 and 3.3 in [4]).
+In particular
+ˆU2 = lim
+←−
+n
+U(ˆg2)
+(C2 = −1/2, tnsnC[t, s] ⊗ g)left.id.
+Recall from [4, Section 3.4] that the expansion maps and the specialisation maps
+induce morphisms at the level of Lie algebras.
+In particular, the specialisation
+map Sp : ˆU2 −→ ˆU1 induces an isomorphism between ˆU2/a ˆU2 and ˆU1, while the
+
+SEMI-INFINITE COHOMOLOGY OF WEYL MODULES
+5
+expansion map induces a morphism E : ˆU2[a−1] −→ ˆUt,s which is injective and has
+dense image.
+Moreover, the natural inclusions ˆgt ֒→ ˆgt,s and ˆgs ֒→ ˆgt,s induce a morphism
+ˆUt ⊗ ˆUs −→ ˆUt,s
+which is also injective and with dense image (see [4, Section 3.3]).
+2.3. Weyl modules. We follow [4, Section 6]. We choose a Borel subalgebra and
+a maximal toral subalgebra of g, which we denote by b and t respectively. This
+data induces a choice of weights, integral weights and dominant weights. For every
+integral dominant weight λ, [7] introduced the Weyl module Vλ
+1 over the affine Lie
+algebra ˆg1. The representation V = V0
+1, which has a structure of vertex algebra, will
+play a particularly important role for us. This vertex algebra enjoys the following
+universal property.
+Lemma 2.2. Let U be a vertex algebra such that there exists a linear map x �→ ux
+from g to U such that
+(ux)(0)(uy) = u[x,y]
+(ux)(1)(uy) = −1
+2κ(x, y)|0⟩U
+(ux)(n)(uy) = 0
+for all n ⩾ 2. There exists a unique morphism of vertex algebras α : V → U such
+that α(xt−1|0⟩V) = ux for all x ∈ g.
+Weyl modules Vλ
+t and Vλ
+s can also be defined for the Lie algebras ˆgt and ˆgs,
+without any significant change from [7]. In [4], we introduced a generalization of
+these modules. Given two dominant weights λ, µ, we consider the irreducible repres-
+entations V λ and V µ of the Lie algebra g having highest weights λ, µ, respectively.
+In [4, Definition 6.2], given two dominant integral weights λ, µ we introduced the
+module
+Vλ,µ
+2
+= Indˆg2
+ˆg+
+2
+�
+A ⊗C V λ ⊗C V µ�
+,
+where ˆg+
+2 = C[t, s] ⊗ g ⊕ A C2 acts on A ⊗C V λ ⊗C V µ as
+f(t, s)x · (p(a) ⊗ u ⊗ v) = f(0, −a)p(a) ⊗ xu ⊗ v + f(a, 0)p(a) ⊗ u ⊗ xv,
+while C2 acts as −1/2. In [4] we called this object the Weyl module of weights
+(λ, µ), although, as we will see, it does not have the the same properties as its
+1-singularity analogue.
+We also define
+Wλ,µ
+1
+= Indˆg1
+ˆg+
+1
+�
+V λ ⊗C V µ�
+,
+where ˆg+
+1 = C[t] ⊗ g ⊕ C C1 acts on V λ ⊗C V µ as f(t)x · (u ⊗ v) = f(0)x · (u ⊗ v)
+and C1 acts as −1/2.
+The specialisation and expansion maps are defined also for Weyl modules, and
+induce the following isomorphisms [4, Lemma 6.3]:
+Vλ,µ
+2
+aVλ,µ
+2
+≃ Wλ,µ
+1
+,
+Vλ,µ
+2
+[a−1] ≃ Vλ
+t ⊗Q Vµ
+s .
+(2.1)
+2.4. Clifford algebra. We now define the Clifford algebra with two singularities,
+generalizing the construction of the classical case (see for example [5, Chapter 15]).
+Let n+ be the nilpotent radical of b and set
+X2 = K2 ⊗C n+ ⊕ K2 ⊗C n∗
++.
+We equip X2 with the unique A-bilinear form such that K2 ⊗C n+ and K2 ⊗C n∗
++
+are isotropic subspaces and
+(f ⊗ x; g ⊗ ϕ) = Res2(fg) ϕ(x)
+
+6
+FORTUNA, LOMBARDO, MAFFEI, MELANI
+for all f, g ∈ K2, x ∈ n+ and ϕ ∈ n∗
++. We denote by Cℓ2 the associated Clifford
+algebra over A.
+There are obvious variants of the same construction where we replace K2 with the
+ring C[t±1] or one of the rings Q[t±1], Q[s±1], Q[t±1] × Q[s±1]. We obtain Clifford
+algebras that we denote by Cℓ1, Cℓt, Cℓs, Cℓt,s. The algebra CℓU in [5, Section 15.1.1]
+is a completion of Cℓ1.
+These Clifford algebras have a natural grading called the charge and denoted by
+ch. It can be defined as follows: the elements of the base ring have charge 0, while
+for ψ ∈ n and ψ∗ ∈ n∗ we have
+ch ψ = −1,
+ch ψ∗ = 1.
+(2.2)
+The relations defining each Clifford algebra are homogeneous, hence the charge
+induces a well-defined grading on the Clifford algebra.
+We now introduce completions of the tensor product ˆU2 ⊗A Cℓ2. We define
+ˆU2 ˆ⊗ACℓ2 = lim
+←−
+n
+ˆU2 ⊗A Cℓ2
+�
+(ts)nR2g ⊗ 1, 1 ⊗ (ts)nR2n+, 1 ⊗ (ts)nR2n∗
++
+�
+left ideal
+and we notice that, as in the case of the algebra ˆU2, this A-module has a nat-
+ural structure of A-algebra. We introduce the completed Clifford algebras ˆU1 ˆ⊗Cℓ1,
+ˆUt ˆ⊗QCℓt, ˆUs ˆ⊗QCℓs, and ˆUt,s ˆ⊗QCℓt,s. The specialisation and expansion map de-
+termine morphisms
+Sp : ˆU2 ˆ⊗ACℓ2 −→ ˆU1 ˆ⊗Cℓ1
+and
+E : ( ˆU2 ˆ⊗ACℓ2)[a−1] −→ ˆUt,s ˆ⊗QCℓt,s.
+Arguing exacly as in [4, Lemmas 3.7 and 3.9] we see that E is injective with dense
+image, while the specialisation map induces an isomorphism ˆU2 ˆ⊗ACℓ2/a( ˆU2 ˆ⊗ACℓ2) ≃
+ˆU1 ˆ⊗Cℓ1. Finally, we have an injective map I : ˆUt ˆ⊗QCℓt → ˆUt,s ˆ⊗QCℓt,s induced by
+the natural inclusion Kt → Kt,s = Kt × Ks given by f �→ (f, 0). Similarly, we have
+an injective map J : ˆUs ˆ⊗QCℓs → ˆUt,s ˆ⊗QCℓt,s. As in Section 3.3 of [4], the product
+of these maps I ⊗ J : ( ˆUt ˆ⊗QCℓt) ⊗Q ( ˆUs ˆ⊗QCℓs) → ˆUt,s ˆ⊗QCℓt,s is injective with
+dense image.
+2.5. Fock module. We now describe the “fermionic” Fock spaces corresponding to
+the Clifford algebras defined in the previous section. As above, for the construction
+in the case of one singularity we refer to [5, Section 15.1.4]: here we mimic this
+definition in the case of two singularities. We define Cℓ+
+2 as the A-subalgebra of Cℓ2
+generated by R2 ⊗ n+ and R2 ⊗ n∗
++ and we define the Fock module
+Λ2 = Cℓ2 ⊗Cℓ+
+2 A |0⟩Λ2
+where R2 ⊗ n+ and R2 ⊗ n∗
++ acts trivially on |0⟩Λ2. The charge (see equation (2.2))
+induces a grading on the Fock space by setting
+ch |0⟩Λ2 = 0.
+We denote by Λn
+2 the subspace of elements of degree n. Similar constructions can be
+given for all the other Clifford algebras Cℓ1, Cℓt, Cℓs, and Cℓt,s, giving Fock modules
+Λ1, Λt, Λs, and Λt,s.
+Specialisation and expansion, induce maps also at the level of the Fock spaces.
+Arguing as in [4, Section 6] (where we considered the module Vλ,µ
+2
+), it is easy to
+prove the following Lemma:
+Lemma 2.3.
+a) The specialisation map Sp : Λ•
+2 −→ Λ•
+1 is homogeneous of degree zero and
+induces an isomorphism Λ•
+2/aΛ•
+2 ≃ Λ•
+1.
+b) We have a homogeneous isomorphism of degree zero Λ•
+t,s ≃ Λ•
+t ⊗Q Λ•
+s.
+
+SEMI-INFINITE COHOMOLOGY OF WEYL MODULES
+7
+c) The expansion map E : Λ•
+2[a−1] −→ Λ•
+t ⊗Q Λ•
+s is a homogeneous isomorphism
+of degree zero.
+Recall also that the Fock space Λ = Λ1 has a natural structure of vertex super-
+algebra with the following universal property.
+Lemma 2.4. Let U be a vertex superalgebra such that there exists a linear map
+x �→ ux from n∗
++ ⊕ n∗
++ to the space of odd elements of U such that
+(1) for all ϕ, ψ ∈ n and for all ϕ∗, ψ∗ ∈ n∗
++
+(uψ)(n)(uϕ) = (uψ∗)(n)(uϕ∗) = (uψ)(m)(uψ∗) = (uψ∗)(m)(uψ) = 0
+for all n ⩾ 0 and for all m ⩾ 1;
+(2) (uψ)(0)(uψ∗) = (uψ∗)(0)(uψ) = ⟨ψ, ψ∗⟩|0⟩U for all ψ ∈ n and ψ∗ ∈ n∗
++.
+Then there exists a unique morhism of vertex superalgebras α : Λ → U such that
+α(ψt−1|0⟩Λ) = uψ and α(ψ∗t−1|0⟩Λ) = uψ∗.
+2.6. Bases. For each of the objects introduced above – base rings, enveloping al-
+gebras, Clifford algebras, and Fock spaces – it is not hard to construct explicit
+bases (or topological bases). We give the details in the case of two singularities.
+The construction of a basis depends on the choice of a basis of C[t, s][1/ts] as an
+A-module. Following [4], Section 1.1 and Equation (4.1) we introduce the following
+bases, indexed by 1
+2Z: for n ∈ Z we define
+�
+zn = tnsn
+zn+ 1
+2 = tn+1sn
+�
+wn = tnsn
+wn+ 1
+2 = tnsn+1
+The elements zm for m ∈
+1
+2Z form a basis of C[t, s][1/ts] as an A-module, and
+the elements wn are the dual basis with respect to the residue bilinear form: more
+precisely, one has
+Res2(znw−m− 1
+2
+) = δn,m.
+This specific choice of basis is not particularly important, and several others would
+be possible. However, some properties need to be satisfied for our approach to
+work. In particolar with our choice, the elements zm (or wm) with m ⩾ 0 form an
+A-basis of C[t, s].
+Since K2 is an A-free module, we deduce that the enveloping algebras of g2 and
+Cℓ2 are A-free modules. Moreover, as R2 is a direct summand of K2, we also deduce
+that Vλ,µ
+2
+and Λ2 are also A-free modules. Explicit bases of these modules, as well
+as an explicit topological basis of the algebra ˆU2 ˆ⊗ACℓ2, can be obtained using the
+Poincar´e-Birkhoff-Witt theorem and its analogue for Clifford algebras.
+3. Vertex algebras and semi-infinite cohomology
+In this section, we recall some results obtained by Casarin [1] which allow us to
+use the formalism of vertex algebras also in the context of several singularities. In
+particular, using this formalism we develop a notion of semi-infinite cohomology for
+ˆU2-modules.
+3.1. Distributions and vertex algebra morphisms. Let R be a complete topo-
+logical associative A-algebra. Following [1, Definition 3.0.4], we denote by FA(K2, R)
+the space of continuous A-linear morphisms from K2 to R and call it the space of
+2-fields.
+We refer to [1] for the definitions of mutually local 2-fields (Definition
+3.1.1), of the n-products X(n)Y of two 2-fields (Definitions 3.1.2 and 3.1.7) and
+of the derivative ∂(X) of a 2-field (before Definition 3.0.2). The definition in [1]
+applies also to the other ring we are considering: K1, Kt, Ks, Kt,s.
+
+8
+FORTUNA, LOMBARDO, MAFFEI, MELANI
+In particular to define n products it is necessary to choose what in [1] is called
+a global coordinate (see definition ). We choose always t as a global coordinate.
+More explicitly for the rings K2, K1, Kt and Ks we choose t = s + a as a global
+coordinate, and for the ring Kt,s = Kt × K2 we choose (t, t) = (t, s + a).
+We also use some foundational results proved in this context in [1]. In particular,
+the following result will be crucial for us.
+Theorem 3.1 ([1], Theorem 3.2.3). Let F be a C-linear subspace of FA(K2, R) of
+mutually local 2-fields closed under derivation and n-products. Let
+1 be a field such
+that
+1(f) is central for every f ∈ K2, that ∂
+1 = 0 and such that
+1(n)X = δn,−1X for
+all X ∈ F. Then the vector dpace F + C1, endowed with n-products and derivation
+T = ∂, is a C-vertex algebra with
+1 as vacuum vector.
+It is straightforward to generalize the constructions and results in [1] to the case
+of superalgebras R.
+We are interested in the case where R is the superalgebra ˆU2 ˆ⊗ACℓ2. For x ∈ g,
+ψ ∈ n+ and ψ∗ ∈ n∗
++ we define the 2-fields
+x(2)(g) = (x⊗g)⊗1Cℓ2,
+ψ[2](g) = 1 ˆU2⊗(ψ⊗g),
+(ψ∗)[2](g) = 1 ˆU2⊗(ψ∗⊗g) (3.1)
+for all g ∈ K2. The first of these fields has even parity with respect to the superal-
+gebra structure, while the second and third ones are odd. These fields are mutually
+local. We consider the minimal C-linear subspace F(2) of ˆU2 ˆ⊗ACℓ2 closed under
+n-products and derivation and containing the fields (3.1). Moreover, we define
+12(f) = Res2(f)
+�
+1 ˆU2 ⊗ 1Cℓ2
+�
+.
+It is easy to check that this data satisfies the hypothesis of Theorem 3.1. Therefore,
+V(2) = F(2) + C12 has a structure of vertex superalgebra, and by the universal
+properties of the vertex algebra V (Lemma 2.2) and of the vertex superalgebra Λ•
+(Lemma 2.4) it follows that there exists a morphism of vertex superalgebras
+Φ(2) : V ⊗C Λ• −→ V(2).
+(3.2)
+This homomorphism will allow us to easily introduce many elements in V(2), hence
+also in ˆU2 ˆ⊗ACℓ2.
+Similar constructions apply if the algebra ˆU2 ˆ⊗ACℓ2 is replaced by the algebras
+ˆU1 ˆ⊗Cℓ1, ˆUt ˆ⊗QCℓt, etc. Hence, we construct the fields x(1), ψ[1], x(t), ψ[t], the vertex
+superalgebras V(1), V(t), and homomorphisms of vertex algebras Φ(1) : V⊗C Λ• −→
+V(1), Φ(t) : V ⊗C Λ• −→ V(t), etc.
+Notice that we have a specialisation morphism SpF : FA(K2, ˆU2 ˆ⊗Cℓ2) −→
+FC(K1, ˆU2 ˆ⊗Cℓ1) and an expansion map EF : FA(K2, ˆU2 ˆ⊗ACℓ2) −→ FQ(Kt,s, ˆU2 ˆ⊗QCℓt,s),
+determined by the conditions
+�
+SpF(X)
+�
+(Sp(f)) = Sp(ϕ(f))
+and
+�
+EF(X)
+�
+(E(f)) = E(ϕ(f)).
+These maps commute with n-products and derivations and satisfy SpF(12) =
+11
+and EF(12) =
+1t,s. Moreover, by construction they satisfy
+SpF(x(2)) = x(1)
+and
+EF(x(2)) = x(t,s)
+for x ∈ g. Similar relations hold for ψ[2] and (ψ∗)[2]. This implies in particular that
+the homomorphisms SpF and EF restrict to homomorphisms of vertex algebras
+Sp : V(2) −→ V(1) and E : V(2) −→ V(t,s) such that
+Sp ◦Φ(2) = Φ(1)
+E ◦ Φ(2) = Φ(t,s).
+We can also describe the morphism Φ(2) through the morphisms Φ(t) and Φ(s).
+Recall from the end of Section 2.4 the maps I, J from ˆUt ˆ⊗QCℓt and ˆUs ˆ⊗QCℓs to
+ˆUt,s ˆ⊗QCℓt,s. These maps induce maps at the level of fields IF : FQ(Kt, ˆUt ˆ⊗QCℓt) →
+
+SEMI-INFINITE COHOMOLOGY OF WEYL MODULES
+9
+FQ(Kt,s, ˆUt,s ˆ⊗QCℓt,s) and JF : FQ(Ks, ˆUs ˆ⊗QCℓs) → FQ(Kt,s, ˆUt,s ˆ⊗QCℓt,s), given
+by
+IF(X)(f, g) = I(X(f))
+and
+JF(X)(f, g) = J(X(g))
+for all (f, g) ∈ Kt×Ks = Kt,s. The maps IF and JF preserve n-products, commute
+with derivations, and satisfy IF(1t) + JF(1s) =
+1t,s. Moreover we notice that I(u)
+and J(v) commute for all u ∈ ˆUt ˆ⊗QCℓt and v ∈ ˆUs ˆ⊗QCℓs. By the discussion in [1,
+Section 7.2], this implies
+IF ◦ Φ(t) + JF ◦ Φ(s) = Φ(t,s).
+This is the only statement where it is relevant the choice of the global coordinate
+we have done in Section 3.1.
+3.2. Semi-infinite cohomology. We now define a notion of semi-infinite cohomo-
+logy for ˆU2-modules, in analogy with the analogous notion for ˆU1-modules described
+for example in [5, Chapter 15]. To this end, we introduce some notation for elements
+in the vertex superalgebra V ⊗ Λ•. As in the the case of ˆU1, to describe these ele-
+ments we choose a basis Ja of g compatible with the decomposition g = n−⊕t⊕n+,
+where n+ is the nilpotent radical of b and n− is the radical of the opposite nilpotent
+borel subalgebra. We denote by cb,d
+e
+the structure coefficients of the Lie bracket
+with respect to this basis. We denote by Φ ⊔ Γ the indexing set of the basis Ja, so
+that, if α ∈ Φ, then Jα = eα = f−α is a root vector of weight α and, if α ∈ Γ, then
+Jα ∈ t. We also denote by ψ∗
+α for α ∈ Φ+ the basis of n∗
++ dual to the basis eα of
+n+.
+With each element in n+ ⊗ · · · ⊗ n+ ⊗ n∗
++ ⊗ · · · ⊗ n∗
++ we associate an element in
+the vertex superalgebra Λ as follows:
+N(ψ1 ⊗ . . . ψℓ ⊗ ψ∗
+1 ⊗ · · · ⊗ ψ∗
+m) = (ψ1t−1) · · · (ψℓt−1) · (ψ∗
+1t−1) · · · (ψ∗
+mt−1) · |0⟩Λ.
+Similarly, with an element in g ⊗ n∗
++ we associate an element in the vertex
+superalgebra V ⊗ Λ∗ by setting
+M(x ⊗ ψ∗) = (xt−1) · |0⟩V ⊗ (ψ∗t−1) · |0⟩Λ.
+Following [5, Chapter 15] we define
+q =M(I) − 1
+2|0⟩V ⊗ N(B) =
+�
+α∈Φ+
+(eαt−1) · |0⟩V ⊗ (ψ∗
+αt−1) · |0⟩Λ
+− 1
+2
+�
+α,β∈Φ+
+cα,β
+α+β |0⟩V ⊗ (eα+βt−1) · (ψ∗
+αt−1) · (ψ∗
+βt−1) · |0⟩Λ,
+where I ∈ g ⊗ n∗
++ represents the inclusion of n+ in g and B ∈ n+ ⊗ n∗
++ ⊗ n∗
++ is the
+Lie bracket. We now define the boundary operator d(2)
+std ∈ ˆU2 ˆ⊗ACℓ2 as follows:
+d(2)
+std :=
+�
+Φ(2)(q)
+�
+(1).
+The boundary operator that we will use to define the semi-infinite cohomology
+is a deformation of d(2)
+std. Let ψ∗
+pr = �
+α simple ψ∗
+α ∈ n∗
++, and define
+χ(2) = 1 ˆU2 ⊗ ψ∗
+pr = Φ(2)(N(ψ∗
+pr))(1) ∈ ˆU2 ⊗A Cℓ2.
+Similar constructions yield χ(s), χ(t), χ(s), and χ(s,t). Finally set
+d(2) = d(2)
+std + χ(2).
+As we will check in Section 3.3, this is an element that squares to zero, and therefore,
+it can be used to define the semi-infinite cohomology of a ˆU2-module.
+
+10
+FORTUNA, LOMBARDO, MAFFEI, MELANI
+Similarly we can define d(1)
+std, χ(1), d(1), d(t)
+std, χ(t), d(t), and so on, as elements of
+the corresponding superalgebras. By the discussion at the end of Section 3.1 we
+have
+Sp(d(2)) = d(1),
+E(d(2)) = d(t,s),
+and
+I(d(t)) + J(d(s)) = d(t,s).
+Definition 3.2. Let M be an ˆU2 module. Consider the ˆU2 ˆ⊗ACℓ2-graded module
+M ⊗AΛ•
+2, where the grading is given by charge on Λ•
+2. The element d(2) acts on this
+module as a boundary operator of degree one. Define Ψn(M) as the corresponding
+cohomology of degree n.
+Similar constructions apply to modules over the algebras ˆU1, ˆUt, ˆUs or ˆUt,s.
+Let Z2 be the center of the algebra ˆU2, and similarly introduce the center Z1 of
+ˆU1 and the centers Zt and Zs of ˆUt and ˆUs. If M is an ˆU2-module, the action of Z2
+on M ⊗A Λ•
+2 commutes with the differential d(2) and preserves the charge, hence
+induces an action of Z2 on the semi-infinite cohomology groups of M. A similar
+action is defined in the case of ˆU1-modules or ˆUt-modules.
+Recall that a module M over a topological algebra R is said to be smooth if the
+action of R on M is continuous with respect to the discrete topology on M. Notice
+that, if M is a smooth ˆU2-module, then, since the map E has dense image, the
+action of ˆU2 on M extends to a smooth action of ˆUt,s on M[a−1]. Similarly, if Mt
+is a smooth ˆUt-module and Ms is a smooth ˆUs-module, then there is an induced
+action of ˆUt,s on Mt ⊗Q Ms. In the next section we will use the following properties
+of the semi-infinite cohomology.
+Lemma 3.3.
+a) Given a short exact sequence of ˆU2-modules, there is an induced long exact
+sequence in semi-infinite cohomology.
+b) Let M be an ˆU1-module. The semi-infinite cohomology of M as an ˆU1-module
+is isomorphic to the semi-infinite cohomology of M considered as an ˆU2-
+module through the map Sp.
+c) Let M be an ˆUt,s-module. The semi-infinite cohomology of M as an ˆUt,s-
+module is isomorphic to the semi-infinite cohomology of M considered as an
+ˆU2-module through the map E. In particular, this applies to the case where
+M = N[a−1] is the localization of a smooth ˆU2-module N.
+d) Let Mt be a smooth ˆUt-module, Ms be a smooth ˆUs-module, and let M :=
+Mt ⊗Q Ms, regarded as a ˆUt,s-module.
+The complex computing the semi-
+infinite cohomology of M is the total complex associated with the double com-
+plex given by the tensor product of the complex computing the semi-infinite
+cohomology of Mt and that of Ms. In particular, being the base ring Q a
+field, if Mt and Ms have non zero semi-infinite cohomology only in degree
+zero, then M considered as an ˆUt,s-module has semi-infinite cohomology only
+in degree zero and the cohomology in degree zero is isomorphic to the product
+of the tensor product of Ψ0(Mt) and Ψ0(Ms).
+Proof. Part a) follows from the fact that Λ2 is a free module over A.
+Part b) follows from the fact that, since a ∈ A acts trivially on M, by Lemma
+2.3 a) we have
+M ⊗A Λ•
+2 ≃ M ⊗C
+Λ•
+2
+aΛ•
+2
+≃ M ⊗C Λ•
+1
+and moreover, by construction, d(1) = Sp(d(2)).
+Part c) follows from the fact that, since the action of a on M is invertible, by
+Lemma 2.3 c) we have
+M ⊗A Λ•
+2 = M ⊗A Λ•
+2[a−1] = M ⊗A Λ•
+t,s
+
+SEMI-INFINITE COHOMOLOGY OF WEYL MODULES
+11
+and, moreover, by construction, d(t,s) = E(d(2)).
+Finally, from Lemma 2.3 c) we have
+(Mt ⊗Q Λ•
+t) ⊗Q (Ms ⊗Q Λ•
+s) ≃ M ⊗Q Λ•
+t,s.
+Part d) then follows from the equality d(t,s) = I(d(t)) + J(d(s)).
+□
+3.3. Commutation relations. For their computation of the semi-infinite cohomo-
+logy of V, Frenkel and Ben Zvi (see [5] Chapter 15) relied on the choice of a clever
+basis of V ⊗ Λ. For all x ∈ g, they define
+ˆx = xt−1 · |0⟩V ⊗ |0⟩Λ + N(αx),
+where αx ∈ n∗
++⊗n∗
++ represents the linear map n+ → n+ obtained as the composition
+of adx : n+ −→ n+, the natural projection π : g −→ g/b−, and the inverse of the
+isomorphism n+ ∼= g/b− induced by π. Using the map Φ(2) from Equation (3.2) we
+define
+ˆx(2) = Φ(2)(ˆx).
+To compute the semi-infinite cohomology of Vλ,µ
+2
+we will need some information
+about the commutation relations among the elements ˆx(2), ψ[2], and (ψ∗)[2], and
+the boundary operators. These are easy to compute because all these objects are
+constructed through the map Φ(2). Let us make this remark precise. Given an
+element x in V⊗Λ, denote by x(z) the corresponding field in the vertex superalgebra
+and by x(2) : K2 −→ ˆU2 ˆ⊗Cℓ2 the 2-field Φ(2)(x).
+For any choice of elements
+x, y ∈ V ⊗ Λ, the commutator of the corresponding fields is given by
+[x(z), y(w)] =
+�
+n⩾0
+1
+n!(x(n)y)(w)∂n
+wδ(z − w).
+We have a similar Operator Product Expansion formula for 2-fields (see [1], Pro-
+position 3.1.4)
+[x(2)(f), y(2)(g)] =
+�
+n⩾0
+1
+n!
+�
+(x(2))(n)(y(2))
+�
+(g ∂nf),
+where the product (x(2))(n)(y(2))) is the product of 2-fields defined in [1]. However,
+since Φ(2) is a map of vertex algebras we get (x(2))(n)(y(2)) = (x(n)y)(2). Hence, if
+we know the commutator of x(z), y(w), we immediately deduce that of x(2) and
+y(2).
+Similar considerations apply when we want to compute [x(2)(1), y(2)(g)] assuming
+we know the commutator of x(0) and y(w). In this case, the usual OPE formula
+gives [x(0), y(w)] = (x(0)y)(w), while the OPE formula for 2-fields gives
+[x(2)(1), y(2)(g)] =
+�
+(x(2))(0)(y(2))
+�
+(g).
+Using again the fact that Φ(2) is a map of vertex algebras, we get
+[x(2)(1), y(2)] = Φ(2)
+��
+[x(0), y(w)](|0⟩V ⊗ |0⟩Λ)
+�
+|w=0
+�
+.
+These formulas are enough to determine all commutation relations among the ele-
+ments ˆx(2), ψ[2], (ψ∗)[2] and the boundary operators from those obtained by Frenkel
+and Ben Zvi in [5, Chapter 15], without the need of any further computation. We
+summarise these results in Proposition 3.4 below, which (in light of the above)
+follows from Sections 15.2.4 and 15.2.9 of [5].
+In the statement, we denote by
+epr, hpr, fpr the sl(2)-triple such that fpr = �
+α simple λαfα, κ(fpr, eα) = 1 for all
+simple root α and hpr ∈ t.
+
+12
+FORTUNA, LOMBARDO, MAFFEI, MELANI
+Proposition 3.4. for all x ∈ g, y ∈ b, z ∈ n+, w ∈ b−, ψ ∈ n+ and ψ∗ ∈ n∗
++ we
+have:
+a)
+(d(2)
+std)2 = 0,
+[d(2)
+std, χ(2)]+ = 0,
+b)
+(χ(2))2 = 0,
+(d(2))2 = 0,
+c) [χ(2), ψ[2]]+ = ⟨ψ∗
+pr, ψ⟩
+1,
+[χ(2), (ψ∗)[2]]+ = 0,
+d)
+[χ(2), ˆz(2)] = 0,
+[χ(2), ˆw(2)] =
+�
+α ∈Φ+
+κ([fpr, z], eα)ψ∗
+α,
+e) [d(2)
+std, ψ[2]]+ = ˆψ(2),
+[d(2)
+std, (ψ∗)[2]]+ = −1
+2Φ
+�
+1 ˆU2 ⊗ N(ψ∗ ◦ B)
+�
+,
+f) [d(2)
+std, ˆy(2)]+ = 0
+where in the second formula of e) the element ψ∗ ◦ B ∈ n∗
++ ⊗ n∗
++ represents the
+composition of the bracket with the map ψ∗. Moreover, if we choose a basis Ja as
+at the beginnin of Section 3.2, for all γ ∈ Φ+ we have
+[d(2)
+std, ˆf (2)
+γ ]+ =
+�
+α∈Φ+,a∈Φ−⊔Γ
+cα,−γ
+a
+( ˆJa)(2)
+(−1)(ψ∗
+α)[2]
+− 1
+2 κ(e−γ, fγ) ∂(ψ∗
+−γ)[2] −
+�
+α,β∈Φ+, a∈Φ⊔Γ
+cα,a
+β cβ,−γ
+a
+∂(ψ∗
+α)[2]
+By specialisation and localization we obtain that similar formulas hold also in
+the case of our various other superalgebras ˆUt ˆ⊗Cℓt, ˆUt,s ⊗ Cℓt,s, . . .
+4. The semi-infinite cohomology of Vλ,µ
+2
+In this section we compute the semi-infinite cohomology of Vλ,µ
+2
+. We denote by
+C•
+2 = C•
+2(λ, µ) the complex Vλ,µ
+2
+⊗A Λ•
+2 and similarly we introduce the complexes
+C•
+t = C•
+t (λ) = Vλ
+t ⊗Q Λ•
+t and C•
+s = C•
+s (µ) = Vµ
+s ⊗Q Λ•
+s. We further introduce
+the complexes C•
+1(ν) = Vν
+1 ⊗C Λ•
+1 and C•
+1(λ, µ) = Wλ,µ
+1
+⊗C Λ•
+1. Hence, we have
+C•
+1(λ, µ) ≃ ⊕C•
+1(ν), where the sum ranges over the irreducible factors of V λ ⊗ V µ
+counted with multiplicity.
+We denote by Op1 the indscheme of opers on the punctured disc and, for every
+integral dominant weight ν, we write Opν
+1 for the associated connected component
+of the space of unramified opers without monodromy, equipped with its reduced
+structure (see, for example, [7] for a more complete definition). We also denote by
+vν a highest weight vector in the g-module V ν. Feigin and Frenkel [2] constructed
+an isomorphism F1 : Funct(Op1) −→ Z1 between the space of functions over Op1
+and the center Z1 of ˆU1. Recall the following result, which combines Theorem 1,
+Theorem 2 and the proof of Proposition 1 in [7].
+Theorem 4.1 (Frenkel and Gaitsgory [7]). The action of Z1 on Vν
+1 and the Feigin-
+Frenkel isomorphism induce an isomorphism
+G1 : Funct(Opν
+1) −→ Endˆg1(Vν
+1).
+Moreover, the element vν ⊗|0⟩Λ is a cocycle in C•
+1(ν) and the map z �→ [z ·vν ⊗|0⟩Λ
+from Z1 to Ψ0(Vν
+1) induces isomorphisms of Z1-modules
+Funct(Opν
+1) ≃ Endˆg1(Vν
+1) ≃ Ψ0(Vν
+1).
+Finally, Ψn(Vν
+1) vanishes for all n ̸= 0.
+The result of Frenkel and Gaitsgory generalises easily to the case of the modules
+Vλ
+t and Vµ
+s .
+
+SEMI-INFINITE COHOMOLOGY OF WEYL MODULES
+13
+By Lemma 2.1 and Lemma 2.3, as in the proof of Lemma 3.3, by the compatibility
+of boundary operators we get homomorphisms of complexes Sp : C•
+2 → C•
+1(λ, µ)
+and E : C•
+2 → C•
+t (λ) ⊗Q C•
+s (µ). These induce isomorphisms
+C•
+2[a−1] ≃ C•
+t (λ) ⊗Q C•
+s (µ)
+and
+C•
+2
+aC•
+2
+≃ C•
+1(λ, µ).
+(4.1)
+From these isomorphisms and Theorem 4.1 it follows easily that Ψn(Vλ,µ
+2
+) is zero
+for n ̸= 0, 1, and we could also get information on the cohomology in degrees zero
+and one.
+However, it is easier to compute these cohomology groups directly by adapting
+the strategy employed by Frenkel and Ben Zvi in [5, Chapter 15]. In order to do
+this, we now introduce certain subcomplexes of C•
+2. We denote by 1V0,0
+2
+the element
+1 ∈ A ⊗C C ⊗C C ⊂ V0,0
+2 .
+Definition 4.2. We denote by E•
+2 the subcomplex of C•
+2(0, 0) spanned by elements
+of the form
+ˆx(2)
+1 (g1) · · · ˆx(2)
+a (ga) · 1V0,0
+2
+⊗ ψ(2)
+1 (ℓ1) · · · ψ(2)
+b (ℓb) · |0⟩Λ2
+(4.2)
+where xi, ψi ∈ n+ and g1, . . . , ga, ℓ1, . . . ℓb ∈ K2. By the commutation relations of
+Section 3.3 we see that E•
+2 is a subcomplex of C•
+2(0, 0).
+We define also analogous complexes E•
+t , E•
+s and E•
+1. These complexes were de-
+noted by C′ in [5] and by C0 in [7]. By construction, these subcomplexes are compat-
+ible with specialisation and localization, and there are isomorphisms E•
+2/aE•
+2 ≃ E•
+1
+and E•
+2[a−1] ≃ E•
+t ⊗Q E•
+s.
+Definition 4.3. We denote by D•
+2 = D•
+2(λ, µ) the subcomplex of C•
+2(λ, µ) spanned
+by elements of the form
+ˆy(2)
+1 (h1) · · · ˆy(2)
+c (hc) · w ⊗ (ψ∗
+1)(2)(k1) · · · (ψ∗
+d)(2)(kd) · |0⟩Λ2
+(4.3)
+where w ∈ V λ ⊗ V µ, yi ∈ b− = n− + t, ψ∗
+i ∈ n∗
++ and h1, . . . , hc, k1, . . . , kd ∈ K2.
+By the commutation relations of Section 3.3 we see that E•
+2 is a subcomplex of
+C•
+2(λ, µ).
+We define also analogous complexes D•
+t (λ), D•
+s(µ) and D•
+1(ν). These complexes
+were denoted by C0 in [5] and by C′ in [7]. Finally, we denote by D•
+1(λ, µ) the
+analogous subcomplex of C•
+1(λ, µ). By construction, these subcomplexes are com-
+patible with specialisation and localization, and there are isomorphisms D•
+2/aD•
+2 ≃
+D•
+1(λ, µ) and D•
+2[a−1] ≃ D•
+t (λ) ⊗Q D•
+s(µ).
+There is an isomorphism of complexes E•
+2 ⊗ D•
+2 −→ C•
+2 defined by
+�
+x · 1V0,0
+2
+⊗ ψ · |0⟩Λ2
+�
+⊗
+�
+y · w ⊗ ψ∗ · |0⟩Λ2
+�
+�−→ x · y · w ⊗ ψ · ψ∗ · |0⟩Λ2,
+where x = ˆx(2)
+1 (g1) · · · ˆx(2)
+a (ga) and ψ = ψ(2)
+1 (ℓ1) · · · ψ(2)
+b (ℓb) are as in Equation (4.2),
+y = ˆy(2)
+1 (h1) · · · ˆy(2)
+c (hc) and ψ∗ = (ψ∗)(2)(k1) · · · (ψ∗)(2)(kd) are as in Equation
+(4.3), and w is an element of V λ ⊗ V µ.
+We now compute the cohomology of the complex E•
+2. We will need the following
+result by Frenkel and Ben Zvi.
+Lemma 4.4 ([5, Section 15.2.6]). Hn(E•
+1) = 0 for n ̸= 0 and Ψ0(E•
+1) = C[|0⟩V ⊗
+|0⟩Λ].
+This result generalizes easily to the case of E•
+t and E•
+s. Localizing and special-
+izing, we deduce the following lemma.
+Lemma 4.5. Hn(E•
+2) = 0 for n ̸= 0 and H0(E•
+2) = A[1V0,0
+2
+⊗ |0⟩Λ2].
+
+14
+FORTUNA, LOMBARDO, MAFFEI, MELANI
+Proof. By definition, the complex E•
+2 is concentrated in non-positive degrees. Hence,
+the long exact sequence induced by
+0
+� E•
+2
+a·
+� E•
+2
+� E•
+1
+� 0
+implies that Hn(E•
+2) is torsion free for every n, and that the specialisation of
+H0(E•
+2) is isomorphic to H0(E•
+1). Since semi-infinite cohomology commutes with
+localization (Lemma 3.3), using Lemma 2.1 and Lemma 4.4 we get the desired
+result.
+□
+We now compute the cohomology of D•
+2. The strategy is similar, but the ar-
+gument is less straightforward since we do not have an explicit representative for
+H0(D•
+1). Following the strategy in [5], we introduce the following bigraded struc-
+ture on D•
+2. Recall that the height ht(α) of a root α is equal to the sum of the
+coefficients of α when written as a sum of simple roots. Let also epr, hpr, fpr be an
+sl(2)-triple such that fpr = �
+α simple fα and hpr belongs to t.
+Definition 4.6. We define a bidegree, with values in 1
+2Z × 1
+2Z and denoted by
+bideg, as follows. On elements of ˆg2, we set
+bideg(x ⊗ g) = (−n, n)
+if x ∈ g is such that [hpr, x] = 2 n x and g ∈ K2. We set also the bidegree of the
+central element C2 ∈ ˆg2 to be (0, 0). This induces a bidegree on U(ˆg2). On the
+space X2 = K2 ⊗ n+ ⊕ K2 ⊗ n∗
++ (see Section 2.4) we define
+bideg eα ⊗ g = (− ht(α), −1 + ht(α))
+bideg ψ∗
+α ⊗ g = (ht(α), 1 − ht(α))
+for α a positive root and g any element of K2. This induces a bidegree on the
+Clifford algebra Cℓ2. Moreover, if W is any finite-dimensional representation of g,
+then we set
+bideg w = (−n, n)
+if w ∈ W is such that hpr · w = 2 n w. These choices induces a bidegree on the
+module C•
+2(λ, µ), and the element ˆx(2)(g) is homogeneous of bidegree (−n, n) if
+[hpr, x] = 2 n x. Finally, notice that if an element has bidegree (p, q), then it has
+charge p + q. In particular, we introduce the submodule Dp,q
+2
+of elements of Dp+q
+2
+of bidegree (p, q).
+We notice also that bideg d(2)
+std = (0, 1) and that bideg χ(2) = (1, 0). In particu-
+lar, D•,•
+2
+is a double complex and D•
+2 is the associated total complex. Following
+Frenkel and Ben Zvi [5, Chapter 15], the cohomology of the rows of this double
+complex is easy to describe. Let a be the centralizer of fpr in g. Recall from [5,
+Lemma 15.1.3 and Section 15.2.9] that the space spanned by monomials of the form
+(ˆp1)n1 · · · (ˆpk)nk ·|0⟩V ⊗|0⟩Λ with pi ∈ a generates a commutative vertex subalgebra
+F1 of V ⊗ Λ• isomorphic to S•(a ⊗ t−1C[t−1]). As in Section 3.3, it follows that for
+x, y ∈ a the fields ˆx(2) and ˆy(2) commute.
+We define F2(λ, µ) as the span of elements of the form
+ˆx(2)
+1 (g1) · · · ˆx(2)
+k (gk) · (v ⊗ |0⟩Λ2) ∈ Vλ,µ
+2
+⊗A Λ•
+2
+with x1, . . . , xk ∈ a and v ∈ V λ ⊗ V µ.
+Notice that all these elements have
+charge equal to zero, and that the space F2(λ, µ) splits as a direct sum F2(λ, µ) =
+�
+q F −q,q
+2
+(λ, µ) according to the bidegree introduced above. Moreover, by Propos-
+ition 3.4 d), these elements are annihilated by the action of χ(2).
+Similarly we construct subspaces F −q,q
+1
+(ν) ⊂ Vν
+1 ⊗C Λ•
+1, F −q,q
+t
+(λ) ⊂ Vλ
+t ⊗Q Λ•
+t ,
+F −q,q
+s
+(µ) ⊂ Vλ
+s ⊗Q Λ•
+s, and F −q,q
+1
+(λ, µ) ⊂ Wλ,µ
+1
+⊗C Λ•
+1, In particular, F −q,q
+1
+(λ, µ) =
+
+SEMI-INFINITE COHOMOLOGY OF WEYL MODULES
+15
+�
+ν F −q,q
+1
+(ν) where the sum is over all irreducible factors of V λ ⊗C V µ. By con-
+struction, the specialisation and localization maps induce isomorphisms
+F −q,q
+2
+(λ, µ)
+aF −q,q
+2
+(λ, µ)
+≃ F −q,q
+1
+(λ, µ)
+and
+F −q,q
+2
+(λ, µ)[a−1] ≃
+�
+b+c=q
+F −b,b
+t
+(λ) ⊗Q F −c,c
+s
+(µ).
+Recall the following result on the cohomology of D•,q
+1
+with respect to the bound-
+ary χ(1).
+Lemma 4.7 ([5, Lemma 15.2.10] and [7]). Let 2pν = ⟨ν, hpr⟩.
+a) Dp,q
+1 (ν) = 0 for q > pν and for p < −q. In particular, Dp,q
+1
+= 0 for q > pλ+µ
+and for p < −q;
+b) Hn(D•,q
+1 (ν)) = 0 for n ̸= −q. In particular, Hn(D•,q
+1 (λ, µ)) = 0 for n ̸= −q;
+c) The map v �→ [v] from F −q,q
+1
+(ν) to H−q(D•,q
+1 (ν)) is an isomorphism.
+Finally, it follows from c) that the map v �→ [v] from F −q,q
+1
+(λ, µ) to H−q(D•,q
+1 (λ, µ))
+is also an isomorphism.
+Similar results hold for the complexes D•,q
+t
+(λ) and D•,q
+s (µ). From this result we
+deduce the cohomology of the complex D•,q
+q
+with respect to the boundary operator
+χ(2).
+Lemma 4.8. Let 2p0 = ⟨λ + µ, hpr⟩ as above.
+a) Dp,q
+2
+= 0 for q > p0 and for p < −q;
+b) Hn(D•,q
+2 ) = 0 for n ̸= −q;
+c) The map v �→ [v] from F −q,q
+2
+(λ, µ) to H−q(D•,q
+2 (λ, µ)) is an isomorphism of
+A-modules.
+Proof. Part a) is clear for the definition of Dp,q
+2
+= 0. For parts b) and c), we start
+by studying the localization of the cohomology groups of D•,q
+2 . Equivalently, we aim
+to compute the cohomology of the localization of the row D•,q
+2 . This localization
+can be rewritten as
+�
+b+c=q
+D•,b
+t (λ) ⊗ D•,c
+s (µ).
+In particular, it follows from Lemma 4.7 that its cohomology is concentrated in
+degree −q, and that its cohomology in this degree is given by
+�
+b+c=q
+F −b,b
+t
+(λ) ⊗ F −c,c
+s
+(µ),
+which is the localization of F −q,q
+2
+(λ, µ).
+Since specialisation is compatible with
+bideg, we have an isomorphism D•,q
+2 /aD•,q
+2
+≃ D•,q
+1 (λ, µ). Using Lemma 4.7, the
+associated long exact sequence shows that Hn(D•,q
+2 ) is torsion-free for n ̸= −q + 1,
+and that the map
+ι : H−q(D•,q
+2 )/aH−q(D•,q
+2 ) → H−q(D•,q
+1 (λ, µ))
+is injective.
+We now prove c). Notice that both F −q,q
+2
+(λ, µ) and H−q(D•,q
+2 (λ, µ)) are torsion-
+free. We have already shown that the localization of the natural maps between them
+is an isomorphism. To study its specialisation, we compose it with the injection ι.
+This composition is the isomorphism of the last remark of Lemma 4.7. We conclude
+by applying Lemma 2.1.
+In order to prove b), it is enough to notice that from the above discussion we
+know that, for n ̸= −q, the module Hn(D•,q
+2 ) = 0 is torsion-free, and that its
+localization is trivial.
+□
+
+16
+FORTUNA, LOMBARDO, MAFFEI, MELANI
+Let now be ϕ(q)
+i
+be an A-basis of F −q,q
+2
+(λ, µ). Since the cohomology in degree
+−q of the complex D•,q+1
+2
+is zero, there exists an element ϕ(q)
+i,1 ∈ D−q−1,q+1
+2
+such
+that χ(2)(ϕ(q)
+i,1 ) = −d(2)
+std(ϕ(q)
+i ). By induction, we can construct elements ϕ(q)
+i,0 = ϕ(q)
+i
+and ϕ(q)
+i,ℓ ∈ D−q−ℓ,q+ℓ
+2
+such that their sum
+˜ϕ(q)
+i
+=
+p0−q
+�
+ℓ=0
+ϕ(q)
+i,ℓ
+satisfies d(2)( ˜ϕ(q)
+i ) = 0. We now prove the main result of this section.
+Theorem 4.9. The following hold.
+a) Ψn(Vλ,µ
+2
+) = 0 for n ̸= 0.
+b) We have an isomorphism
+Ψ0(Vλ,µ
+2
+)
+aΨ0(Vλ,µ
+2
+)
+≃ Ψ0(Wλ,µ
+1
+) ≃
+�
+ν
+Ψ0(Vν
+1)
+where the sum ranges over all irreducible components V ν of V λ ⊗V µ, counted
+with multiplicity.
+c) The elements
+�
+˜ϕ(q)
+i
+�
+are an A-basis of Ψ0(Vλ,µ).
+Proof. From Lemma 4.8 we deduce that the classes of the elements ˜ϕ(q)
+i
+form an
+A-basis of H0(D•
+2), and that Hn(D•
+2) = 0 for n ̸= 0. As the complex D•
+2 is con-
+centrated in non-negative degrees, by a standard homological argument we deduce
+that Hn(Vλ,µ
+2
+) is isomorphic to the n-th cohomology of the complex H0(D•
+2)⊗AE•
+2.
+Using Lemma 4.5, we immediately obtain parts a) and c).
+The second isomorphism appearing in part b) is clear, while the first follows from
+a) and the long exact sequence associated with the isomorphism
+C•
+2
+aC•
+2
+≃ C•
+1(λ, µ).
+□
+We will use the following Corollary in the next Section.
+Corollary 4.10. The element [vλ ⊗ vµ] ∈ Ψ0(Vλ,µ) is indivisible.
+Proof. By the previous theorem we can choose [vλ ⊗ vµ] as an element of a basis of
+the free A module Ψ0(Vλ,µ).
+□
+5. The action of the center
+In this section we study the action of the center Z2 on the semi-infinite cohomo-
+logy of the module Vλ,µ
+2
+.
+In this section we show that Vλ,µ
+2
+is not a perfect analogue of the Weyl module
+Vν
+1. Indeed, we show that, as a Z2-module, the semi-infinite cohomology of Vλ,µ
+2
+is
+not isomorphic to Endˆg2(Vλ,µ
+2
+) or to Funct(Opλ,µ
+2
+).
+We begin by observing that the module Ψ0(Vν
+1) has no non-trivial Z1-equivariant
+automorphisms.
+First we notice, that by construction, the action of Z2 commutes with localization
+and specialisation, as introduced before Equation (4.1). Concretely, we have:
+Et(z · x) = Et(z) · Et(x),
+Es(z · x) = Es(z) · Es(x),
+Sp(z · x) = Sp(z) · Sp(x)
+for all z ∈ Z2 and for all x ∈ Ψ0(Vλ,µ
+2
+).
+Lemma 5.1. If K : Endˆgt(Vλ
+t )⊗QEndˆgs(Vµ
+s ) −→ Ψ0(Vλ
+t )⊗QΨ0(Vµ
+s ) is a (Zt⊗Zs)-
+equivariant isomorphism, then K(IdVλ
+t ⊗ IdVµ
+s ) = q[vλ] ⊗ [vµ] for some q ∈ Q ∖ {0}.
+
+SEMI-INFINITE COHOMOLOGY OF WEYL MODULES
+17
+Proof. It follows from Theorem 4.1 that Endˆgt׈gs(Vλt ⊗Q Vµs) is isomorphic to
+Funct(Opλ
+t ×Spec Q Opµ
+s ) = Funct(Opλ
+t ) ⊗Q Funct(Opµ
+s ) and this is a polynomial
+ring in infinitely many variables over the field Q. In particular, its only invertible
+elements are the non-zero scalars in Q.
+Moreover, Theorem 4.1 also implies that Funct(Opλ
+t ) is isomorphic as a Zt-
+module to Ψ0(Vλ
+t ), with an isomorphism given by z −→ Gt(z) · [vλ]. The claim
+follows.
+□
+Before proving that Vλ,µ
+2
+does not have the “right” semi-infinite cohomology
+we recall some properties of the modules Vν
+1 that will be needed also in the next
+section.
+Remark 5.2. We denote by Zν
+1 the coordinate ring of the scheme Opν
+1. Recall
+that the schemes Opν
+1 for different values of ν are disjoint, so that the map Z1 −→
+Zν1
+1 × · · · × Zνk
+1
+is surjective if the weights νi are distinct. Recall also that the ring
+Zν
+1 is a polynomial ring in infinitely many variables. This implies that
+(1) There are no nontrivial ˆg1-morphisms between the ˆU1-modules Vν
+1 and Vν′
+1
+if ν ̸= ν′.
+(2) There are no nontrivial extensions between the ˆU1-modules Vν
+1 and Vν′
+1 if
+ν ̸= ν′.
+(3) Assume that α : � Zνi −→ � Zνi is a map of Z-modules and that the
+weights νi are distinct. If 1 is in the image of α then α is an isomorphism
+and α(Zνi
+1 ) = Zνi
+1 .
+By the Feigin-Frenkel Theorem (see [4] Theorem 5.2) the ring Funct(Op2) is
+isomorphic to Z2. In the sequel we will identify these rings through this isomorph-
+ism.
+In particular the ring Funct(Opλ,µ
+2
+) is a quotient of Z2.
+We will denote
+Funct(Opλ,µ
+2
+) by Zλ,µ
+2
+.
+We now prove that Zλ,µ
+2
+and Ψ0(Vλ,µ
+2
+) are not isomorphic.
+Proposition 5.3. Assume that V λ ⊗ V µ is not irreducible.
+Then the two Z2-
+modules Endˆg2(Vλ,µ
+2
+) and Ψ0(Vλ,µ
+2
+) are not isomorphic.
+Similarly the two Z2-
+modules Zλ,µ
+2
+and Ψ0(Vλ,µ
+2
+) are not isomorphic.
+Proof. Suppose H : Endˆg2(Vλ,µ
+2
+) −→ Ψ0(Vλ,µ
+2
+) is a Z2-equivariant isomorphism.
+Recall from Lemma 4.28 in [4] that Z2[1/a] is dense in Zt,s, and therefore the
+localization of H is a (Zt ⊗Q Zs)-equivariant isomorphism
+Endˆgt(Vλ
+t ) ⊗Q Endˆgs(Vµ
+s ) −→ Ψ0(Vλ
+t ) ⊗Q Ψ0(Vµ
+s ),
+where we used the identification of the localization of Ψ0(Vλ,µ
+2
+) with Ψ0(Vλ
+t ) ⊗Q
+Ψ0(Vµ
+s ).
+From Lemma 5.1 and 4.10 we deduce that H(IdVλ,µ
+2
+) = [q vλ ⊗ vµ], where q ∈ A
+and qvλ ⊗ vµ ∈ Vλ,µ
+2
+. We set w = qvλ ⊗ vµ ∈ Vλ,µ
+2
+.
+By specialisation, H gives a Z1-equivariant isomorphism
+H : Endˆg2(Vλ,µ
+2
+)
+a Endˆg2(Vλ,µ
+2
+)
+−→ Ψ0(Vλ,µ
+2
+)
+aΨ0(Vλ,µ
+2
+)
+.
+(5.1)
+This isomorphism sends IdVλ,µ
+2
+to w. Now consider the decomposition V λ ⊗ V µ =
+� V ν as g-modules. By Theorem 4.9, the target of the map H in (5.1) decomposes
+as � Ψ0(Vν
+1). The element w is a multiple of vλ ⊗ vµ hence its class belongs to
+Ψ0(Vλ+µ
+1
+). As H is Z1-equivariant and Vλ+µ
+1
+is stable by the action of ˆg1, we get
+that the image of H is contained in the direct summand Ψ0(Vλ+µ
+1
+). In particular,
+if V λ ⊗ V µ is not irreducible, the map H cannot be surjective. This proves the
+
+18
+FORTUNA, LOMBARDO, MAFFEI, MELANI
+first claim. The second claim follows since the map from Zλ,µ
+2
+to Ψ0(Vλ,µ
+2
+) factors
+through Endˆg2(Vλ,µ
+2
+).
+□
+6. A Weyl module for sl(2)
+In this section, we propose an alternative Weyl module in the context of opers
+with two singularities, in the case of g = sl(2).
+We fix the following notation:
+e, h, f is an sl(2)-triple such that h ∈ t and e ∈ n+, while ψ∗ ∈ n∗
++ is the dual of
+e. We identify dominant weights with natural numbers and we assume from now
+on that λ ⩾ µ. In this case, the differential of the complex computing semi-infinite
+cohomology takes the simpler form d(2) = ψ∗ + � ewn ⊗ ψ∗z−n−1/2.
+Let �Vλ,µ
+2
+be the ˆU2-submodule of Vλ,µ
+2
+generated by the highest weight vector
+1 ⊗ vλ ⊗ vµ ∈ A ⊗ V λ ⊗ V µ. We will prove that this module has the “correct”
+semi-infinite cohomology and the “correct” endomorphism ring.
+We start by giving a more explicit description of the module �Vλ,µ
+2
+. If X is a
+subspace of U(g) and Y is a subspace of a g-module Z we denote by X · Y the
+subspace of Z generated by the products x · y with x ∈ X and y ∈ Y . We define
+an increasing filtration F i of �Vλ,µ
+2
+as follows
+F i = U(g) · (C Id ⊗ Id + Id ⊗ g)i · (vλ ⊗ vµ).
+This is an increasing filtration of V λ ⊗ V µ by g-modules and for i large enough we
+have F i = V λ ⊗ V µ. Choose a g-stable complement Gi+1 of F i in F i+1 and set
+G0 = F 0, so that F i = �i
+j=0 Gj. If we set F i(V µ) = (CId + n−)ivµ, it is easy to
+check by induction on i that
+F i = U(g) · (Id ⊗ Id + Id ⊗ n−)i(vλ ⊗ vµ) = U(g) ·
+�
+V λ ⊗ F i(V µ)
+�
+.
+In the case of g = sl(2) we have Gi ≃ V λ+µ−2i and F µ(V µ) = V µ.
+Let U −
+2
+⊂ U(ˆg2) be the A-span of Poincar´e-Birkhoff-Witt monomials of the
+form (x1wa1) · · · (xkwak) with xi ∈ g and ai < 0. This is a complement of U(ˆg+
+2 )
+in U(ˆg2), so that in particular we have
+Vλ,µ
+2
+= U −
+2 ⊗C (V λ ⊗ V µ).
+Lemma 6.1. If λ ⩾ µ then
+�Vλ,µ
+2
+=
+µ
+�
+i=0
+aiU −
+2 ⊗C F i =
+µ
+�
+i=0
+aiU −
+2 ⊗C Gi
+Proof. To understand the module �Vλ,µ
+2
+we need to compute the ˆg+
+2 -submodule of
+A⊗C V λ ⊗C V µ generated by 1⊗vλ ⊗vµ. Notice that every element of the form xg,
+with x ∈ g and g ∈ C[[t, s]] divisible by ts, acts trivially on A ⊗ V λ ⊗ V µ. Hence
+we need to understand the action of elements of the form
+z = x1 · · · xℓ · (y1t) · · · (ymt) · (vλ ⊗ vµ),
+with xi, yi ∈ g. Moreover, elements of g act in the standard way on the tensor
+product V λ ⊗ V µ, while elements of the form xt with x ∈ g act via −a(Id ⊗ x).
+This implies the lemma.
+□
+We now describe the specialisation of the module �Vλ,µ
+2
+. We introduce the fol-
+lowing decreasing filtration of �Vλ,µ
+2
+:
+Fi = �Vλ,µ
+2
+∩ aiVλ,µ
+2
+.
+(6.1)
+
+SEMI-INFINITE COHOMOLOGY OF WEYL MODULES
+19
+By Lemma 6.1 we have the following description of the terms of this filtration as
+A-modules:
+Fi = aiU −
+2 ⊗C F i ⊕
+µ
+�
+j=i+1
+ajU −
+2 ⊗C Gj
+In particular we have F0 = �Vλ,µ
+2
+, Fj = ajVλ,µ
+2
+for j ⩾ µ.
+Lemma 6.2.
+a) Let ui ∈ Gi be the highest weight vector and set ˜wi = aiui ∈
+�Vλ,µ
+2
+. Then ˜wi ∈ �Vλ,µ
+2
+and ai−1ui /∈ �Vλ,µ
+2
+.
+b) There is an isomorphism of ˆU1-modules
+Fi + a�Vλ,µ
+2
+a�Vλ,µ
+2
+≃
+µ
+�
+j=i
+Vλ+µ−2j
+1
+.
+The quotient Fi+a�Vλ,µ
+2
+a�Vλ,µ
+2
+is generated as a ˆU1-module by the classes of ˜wi, . . . , ˜wµ.
+In particular �Vλ,µ
+2
+/a�Vλ,µ
+2
+≃ Wλ,µ
+1
+is generated by ˜w0, . . . , ˜wµ.
+Proof. The first claim follows from Lemma 6.1.
+We prove part b) by decreasing induction on i. By Lemma 6.1, for i > µ the
+quotient is zero and the claim is true. For i ⩽ µ, consider the map
+U −
+2 ⊗ Gi −→
+Fi + ai+1Vλ,µ
+2
+ai+1Vλ,µ
+2
++ Fi ∩ a�Vλ,µ
+2
+≃
+(Fi + a�Vλ,µ
+2
+)/a�Vλ,µ
+2
+(Fi+1 + a�Vλ,µ
+2
+)/a�Vλ,µ
+2
+sending an element u ⊗ v to the class of aiu ⊗ v. This map induces an isomorphism
+U −
+2
+aU −
+2
+⊗ Gi ≃
+(Fi + a�Vλ,µ
+2
+)/a�Vλ,µ
+2
+(Fi+1 + a�Vλ,µ
+2
+)/a�Vλ,µ
+2
+.
+(6.2)
+Moreover, notice that
+U−
+2
+aU−
+2 ⊗ Gi ≃ U −
+1 ⊗ Gi, where U −
+1 = U(t−1g[t−1]) ⊂ U(ˆg1) =
+U1, and that U −
+1 ⊗ Gi has a natural structure of U1-module, as it can be identified
+with Vλ+µ−2i
+1
+. With this U1-action, the isomorphism 6.2 is U1-equivariant. Now
+the claim follows by the inductive hypothesis, combined with the fact that there
+are no nontrivial extensions between modules Vν
+1 and Vν′
+1 if ν ̸= ν′ and that the
+highest weight vector of V ν generates the module Vν
+1 as an U1-module.
+□
+Notice that, although the specialisations at a = 0 of Vλ,µ
+2
+and �Vλ,µ
+2
+are iso-
+morphic, the specialisation of �Vλ,µ
+2
+, is generated by vλ ⊗ vµ while in the first case
+this vector generates the submodule Vλ+µ
+1
+.
+As a corollary, we get the following result.
+Proposition 6.3. The following hold:
+a) Ψn(�Vλ,µ
+2
+) = 0 for n ̸= 0.
+b) The inclusion of �Vλ,µ
+2
+in Vλ,µ
+2
+induces isomorphisms
+Ψ0(�Vλ,µ
+2
+)[a−1] ≃ Ψ0(Vλ,µ
+2
+)[a−1] ≃ Ψ0(Vλ
+t ) ⊗Q Ψ0(Vµ
+s ).
+c) Ψ0(�Vλ,µ
+2
+) is torsion-free with respect to the action of A, and the natural pro-
+jection induces isomorphisms
+Ψ0(�Vλ,µ
+2
+)
+aΨ0(�Vλ,µ
+2
+)
+≃ Ψ0
+� �Vλ,µ
+2
+a�Vλ,µ
+2
+�
+≃ Ψ0(Wλ,µ
+1
+).
+Proof. We use the filtration introduced in Equation (6.1). Notice that
+Fi
+Fi+1
+=
+aiU −
+2 ⊗ F i
+ai+1U −
+2 ⊗ F i ≃ U −
+1 ⊗C F i ≃ Indˆg1
+ˆg+
+1 F i,
+
+20
+FORTUNA, LOMBARDO, MAFFEI, MELANI
+where we consider F i as a ˆg+
+1 -module on which tg[t] acts trivially. Notice that
+Indˆg1
+ˆg+
+1 F i is a sum of modules of the form Vν
+1, hence in particular has trivial non-
+zero cohomology.
+Hence, arguing by decreasing induction on i, starting from i = µ, it follows that
+Fi has trivial semi-infinite cohomology in degree different from zero. Indeed for
+i = µ we have Fµ = aµVλ,µ
+2
+≃ Vλ,µ
+2
+and this is the content of Theorem 4.9. For
+i = 0 this implies claim a).
+Part b) follows from the fact that semi-infinite cohomology commutes with local-
+ization (see Lemma 3.3) combined with the isomorphism �Vλ,µ
+2
+[a−1] = Vλ,µ
+2
+[a−1] ≃
+Vλ
+t ⊗Q Vµ
+s .
+To prove c), consider the exact sequence
+0
+� �Vλ,µ
+2
+·a
+� �Vλ,µ
+2
+�
+�Vλ,µ
+2
+a�Vλ,µ
+2
+� 0
+By Lemma 6.2, the last module in this sequence is isomorphic to Wλ,µ
+1
+. In par-
+ticular, the semi-infinite cohomology groups Ψn of the modules appearing in this
+sequence are zero for n ̸= 0, and c) follows.
+□
+To prove that the semi-infinite cohomology of �Vλ,µ
+2
+is isomorphic to Zλ,µ
+2
+we will
+use the action of a particular central element in Z2. Recall from [4] the definition
+of the 2-Sugawara operator
+S(2)
+1/2 =
+�
+n∈ 1
+2 Z,b
+: (Jbwn)(Jbz−n) :
+(6.3)
+where J1, J2, J3 are the basis elements e, h, f and J1, J2, J3 are the dual basis
+elements f, h/2, e. As proved in [4], the element S(2)
+1/2 is central. Its specialisation
+is the Sugawara operator
+S(1)
+1
+=
+�
+n∈Z,b
+: (Jbtn) (Jbt−n) :
+(6.4)
+which is an element of Z1. It is straightforward to check that the action of S(1)
+1
+on
+the Weyl module Vν
+1 is given by multiplication by ν(ν + 1).
+Lemma 6.4. The element ˆwℓ =
+�
+et−1�ℓ ˜wℓ belongs to Z2 · (vλ ⊗ vµ) + a�Vλ,µ
+2
+for
+ℓ = 0, . . . , µ,
+Proof. We notice first that the element vλ ⊗ f ℓvµ belongs to F ℓ \ F ℓ−1 and has
+weight λ + µ − 2ℓ. Hence, up to a non-zero constant we have vλ ⊗ f ℓvµ = uℓ + u′
+ℓ,
+where we recall that uℓ is the highest weight vector in Gℓ ≃ V λ+µ−2ℓ ⊂ V λ ⊗ V µ
+and u′
+ℓ ∈ F ℓ−1. In particular, recall from Lemma 6.2 that aℓ−1F ℓ ⊂ �Vλ,µ
+2
+, hence
+aℓ �
+et−1�ℓ vλ ⊗ f ℓvµ =
+�
+et−1�ℓ ˜wℓ +
+�
+et−1�ℓ (aℓu′
+ℓ) ≡
+�
+et−1�ℓ ˜wℓ
+mod a�Vλ,µ
+2
+.
+Hence, the lemma is equivalent to the fact that ˆwℓ = aℓ �
+et−1�ℓ vλ ⊗ f ℓvµ is in
+Z2 · vλ ⊗ vµ + a�Vλ,µ
+2
+. We prove this statement by induction on ℓ. For ℓ = 0 it is
+trivially true. Now assume ˆwℓ is in Z2 · vλ ⊗ vµ + a�Vλ,µ
+2
+. We compute S(2)
+1/2( ˆwℓ). In
+order to do this, we notice that the action of xtisj on �Vλ,µ
+2
+/a�Vλ,µ
+2
+is equal to the
+action of xti+j on the same module, and that vλ ⊗ e f ℓvµ is in F ℓ−1. We have
+S(2)
+1/2 ˆwℓ = 2
+�
+n>0
+et−n · ftn · ˆwℓ + 2
+�
+n>0
+ft−n · etn · ˆwℓ +
+�
+n>0
+ht−n · htn · ˆwℓ
++ e · f · ˆwℓ + e · f · ˆwℓ + 1
+2h · h · ˆwℓ.
+
+SEMI-INFINITE COHOMOLOGY OF WEYL MODULES
+21
+In the second infinite sum above, the element etn commutes with et−1, hence etn ·
+ˆwℓ ∈ a�Vλ,µ
+2
+for all n > 0. The summands of the third series are of the form
+htn · (et−1)ℓ · ˆwℓ = (et−1)ℓhtn · ˆwℓ + 2ℓ(et−1)ℓ−1etn−1 · ˆwℓ,
+hence they vanish for n ⩾ 3, while for n = 1, 2 they are easily checked to be elements
+of a�Vλ,µ
+2
+. The summands of the first series are given by
+ftn·(et−1)ℓ· ˆwℓ = (et−1)ℓftn· ˆwℓ−ℓ(et−1)ℓ−1htn−1· ˆwℓ−ℓ(ℓ−1)(et−1)ℓ−2etn−2· ˆwℓ,
+and all terms are zero or in a�Vλ,µ
+2
+but for the case n = 1, for which we get
+(et−1) · (ft) · (et−1)ℓ · ˆwℓ = aℓ+1(et−1)ℓ+1 · (vλ ⊗ f ℓ+1vµ)
+− ℓ(et−1)ℓh · (vλ ⊗ f ℓvµ) − ℓ(ℓ − 1)(et−1)ℓ · ˆwℓ = ˆwℓ+1 + K1 ˆwℓ
+for some constant K1. Finally, e·f · ˆwℓ+e·f · ˆwℓ+ 1
+2h·h· ˆwℓ belongs to K2 ˆwℓ+a�Vλ,µ
+2
+for some constant K2. Hence we get
+S(2)
+1/2 ˆwℓ ≡ ˆwℓ+1 + K ˆwℓ mod a�Vλ,µ
+2
+for some constant K, proving our claim.
+□
+We now prove that the zero-th semi-infinite cohomology of the module �Vλ,µ
+2
+is
+isomorphic to Zλ,µ
+2
+.
+Theorem 6.5. For g = sl(2) the map Φ : Zλ,µ
+2
+−→ Ψ0��Vλ,µ
+2
+�
+given by Φ(z) =
+z · [vλ ⊗ vµ] is an isomorphism.
+Proof. By [4], Theorem 6.4, the action of Z2 on Vλ,µ
+2
+, hence on �Vλ,µ
+2
+, factors through
+Zλ,µ
+2
+. Moreover vλ ⊗ vµ is a cycle, so the map Φ is well defined. Since we know
+that both modules are torsion-free, to prove that Φ is an isomorphism it suffices to
+prove that the localization Φa and the specialisation Φ are isomorphisms.
+The fact that Φa is an isomorphism is the content of part b) of Proposition 6.3.
+We need to prove that Φ is an isomorphism. By Lemma 6.2, Proposition 6.3 and
+[4, Theorem 2.13] we have
+Zλ,µ
+2
+aZλ,µ
+2
+≃
+µ
+�
+i=0
+Zλ+µ−2i
+1
+and
+Ψ0(�Vλ,µ
+2
+)
+aΨ0(�Vλ,µ
+2
+)
+≃
+µ
+�
+i=0
+Ψ0(Vλ+µ−2i
+1
+).
+In particular, by Theorem 4.1 these two Z1-modules are isomorphic, but we need to
+prove that our specific map Φ provides an isomorphism between them. By Remark
+5.2 it is enough to prove that Φ is surjective. We prove that the image of Φ contains
+Ψ0(Fℓ+a�Vλ,µ
+2
+/a�Vλ,µ
+2
+) arguing by reverse induction on ℓ. For ℓ = 0 we get our claim.
+For ℓ > µ there is nothing to prove. Now assume ℓ ⩽ µ. Consider again the exact
+sequence
+0
+� Fℓ+1+a�Vλ,µ
+2
+a�Vλ,µ
+2
+� Fℓ+a�Vλ,µ
+2
+a�Vλ,µ
+2
+�aℓU −
+2 ⊗C Gℓ
+�0.
+We know that the last module is isomorphic to
+aℓU −
+2 ⊗C Gℓ ≃ Vλ+µ−2ℓ
+1
+= Indˆg1
+ˆg+
+1 (V λ+µ−2ℓ)
+and that it is generated by the element ˜wℓ ∈ aℓGℓ. Notice this sequence of Z1-
+modules splits by Remark 5.2.
+Taking semi-infinite cohomology we get a short
+exact sequence
+0
+�Ψ0 �
+Fℓ+1+a�Vλ,µ
+2
+a�Vλ,µ
+2
+�
+�Ψ0 �
+Fℓ+a�Vλ,µ
+2
+a�Vλ,µ
+2
+�
+�Ψ0 �
+aℓU −
+2 ⊗C Gℓ�
+�0.
+and we know that the last Z2-module is generated by ˜wℓ. Hence it is enough to
+prove that this element is in the image of Zλ,µ
+2
+(vλ ⊗vµ) in Ψ0�
+�Vλ,µ
+2
+/Fℓ+1 +a�Vλ,µ
+2
+�
+.
+
+22
+FORTUNA, LOMBARDO, MAFFEI, MELANI
+By Lemma 6.4 we know that ˆwℓ is in this image. Now we prove that ˜wℓ and ˆwℓ
+define the same element in the semi-infinite cohomology of aℓU −
+2 ⊗C Gℓ. This is a
+claim about the cohomology of the module Vν
+1 for ν = λ + µ − 2ℓ. For any ν we
+prove that
+�
+et−1�hvν +
+�
+et−1�h−1vν is a coboundary. Indeed the boundary operator
+in the case of sl(2) is equal to
+d(1) = ψ∗ +
+�
+n∈Z
+(etn) ⊗ ψ∗t−1−n,
+so a simple computation shows
+d(1) ��
+et−1�h−1vν ⊗ (ψt−1)|0⟩Λ
+�
+=
+�
+et−1�h−1vν ⊗ |0⟩Λ +
+�
+et−1�hvν ⊗ |0⟩Λ,
+which implies our claim.
+□
+Recall that in [4] we computed the endomorphism ring of Vλ,µ
+2
+, showing that it
+is isomorphic to Zλ,µ
+2
+. We now prove the same result for the module �Vλ,µ
+2
+.
+Proposition 6.6. The action of the center Z2 on �Vλ,µ
+2
+induces an isomorphism
+Zλ,µ
+2
+≃ Endˆg2(�Vλ,µ
+2
+).
+Proof. We already recalled at the beginning of the proof of Theorem 6.5 that the
+action of Z2 on �Vλ,µ
+2
+factors through Zλ,µ
+2
+. We denote by α : Zλ,µ
+2
+−→ End(�Vλ,µ
+2
+)
+this action. Since both modules have no A-torsion, in order to prove that α is
+an isomorphism it suffices to show that its localization and its specialisation are
+isomorphisms.
+Moreover, since our modules are finitely generated and have no
+torsion we have
+Endˆg2
+�
+�Vλ,µ
+2
+�
+[a−1] ≃ Endˆg2[a−1]
+�
+�Vλ,µ
+2
+[a−1]
+�
+≃ Endˆgt,s
+�
+Vλ ⊗Q Vµ
+s
+�
+≃ Zλ
+t ⊗Q Zµ
+t ≃ Zλ,µ
+2
+[a−1],
+hence the localization of α is an isomorphism.
+Finally, we prove that the specialisation of α is also an isomorphism. We have
+already recalled that by [4, Theorem 2.13] we have Zλ,µ
+2
+/aZλ,µ
+2
+≃ �µ
+i=0 Zλ+µ−2i
+1
+.
+Hence by Theorem 4.1 we have the following abstract isomorphisms of Z1-modules:
+Zλ,µ
+2
+aZλ,µ
+2
+≃
+µ
+�
+i=0
+Zλ+µ−2i
+1
+≃
+µ
+�
+i=0
+Endˆg1(Vλ+µ−2i
+1
+).
+Moreover, since �Vλ,µ
+2
+has no nontrivial A-torsion, by Lemma 6.2 and Remark 5.2
+part (1) we have the inclusion
+Endˆg1
+�
+�Vλ,µ
+2
+�
+a Endˆg1
+�
+�Vλ,µ
+2
+� ⊂ Endˆg1
+� �Vλ,µ
+2
+a�Vλ,µ
+2
+�
+≃
+µ
+�
+i=0
+Endˆg1(Vλ+µ−2i
+1
+).
+Hence, composing the specialisation of the map α with this inclusion and the iso-
+morphisms above we get a Z1-equivariant endomorphism of �µ
+i=0 Zλ+µ−2i
+1
+. Moreover,
+α(1) = 1, hence we conclude by Remark 5.2 (3) that the specialisation of α is also
+an isomorphism.
+□
+References
+[1] Casarin, L. A Feigin Frenkel theorem with n singularities. preprint, 2022.
+[2] Feigin, B., and Frenkel, E. Affine Kac-Moody algebras at the critical level and Gelfand-
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+MIT, PhD thesis, 2013.
+
+SEMI-INFINITE COHOMOLOGY OF WEYL MODULES
+23
+[4] Fortuna, G., Lombardo, D., Maffei, A., and Melani, V. Local opers with two singularities:
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+E-mail addresses: [email protected], [email protected],
+[email protected], [email protected]
+

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