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+1 
+ 
+Spectral Bandwidth Recovery of Optical Coherence 
+Tomography Images using Deep Learning 
+TIMOTHY T. YU1,4, DA MA2,1*, JAYDEN COLE1, MYEONG JIN JU3,4, MIRZA F. BEG1, AND 
+MARINKO V. SARUNIC1,5,6* 
+1Engineering Science, Simon Fraser University, Burnaby BC V5A1S6, Canada 
+2Wake Forest University School of Medicine, Winston-Salem, NC, 27151, USA 
+3Dept. of Ophthalmology and Visual Sciences, University of British Columbia, Vancouver, BC, V5Z 3N9, Canada 
+4School of Biomedical Engineering, University of British Columbia, Vancouver, BC, V5Z 3N9, Canada 
+5School of Biomedical Engineering, University of British Columbia, Vancouver, BC, V5Z 3N9, Canada 
+5Institute of Ophthalmology, University College London, London, UK 
+6Department of Medical Physics and Biomedical Engineering, University College London, United Kingdom 
+ 
+*[email protected] 
+*[email protected]  
+1. Abstract  
+Optical coherence tomography (OCT) captures cross-sectional data and is used for the screening, monitoring, 
+and treatment planning of retinal diseases. Technological developments to increase the speed of acquisition 
+often results in systems with a narrower spectral bandwidth, and hence a lower axial resolution. Traditionally, 
+image-processing-based techniques have been utilized to reconstruct subsampled OCT data and more 
+recently, deep-learning-based methods have been explored. In this study, we simulate reduced axial scan (A-
+scan) resolution by Gaussian windowing in the spectral domain and investigate the use of a learning-based 
+approach for image feature reconstruction. In anticipation of the reduced resolution that accompanies wide-
+field OCT systems, we build upon super-resolution techniques to explore methods to better aid clinicians in 
+their decision-making to improve patient outcomes, by reconstructing lost features using a pixel-to-pixel 
+approach with an altered super-resolution generative adversarial network (SRGAN) architecture. 
+2. Introduction 
+Optical coherence tomography (OCT) is a non-invasive imaging modality that allows for high-resolution 
+volumetric visualization of the retina, the light-sensitive tissue at the back of the eye. OCT is the gold standard 
+diagnostic for diseases such as age-related macular degeneration (AMD) and diabetic macular edema (DME), 
+but is not a widely accepted modality for diabetic retinopathy (DR) diagnosis and monitoring due to the 
+limited field of view (FOV) 1. OCT facilitates the characterization of retinal thickness changes and 
+abnormalities that are indicative of DR which if integrated as a secondary diagnostic modality, may benefit 
+the outcome of patients with DR. We have previously demonstrated that the vasculature outside the parafovea 
+contains features indicative of early changes from DR 2. Hence, technology is advancing towards wide-field 
+OCT systems to capture more details of the retina. 
+As the OCT hardware advancements move towards capturing a wider FOV on the retina, there is an 
+effort to minimize motion artefacts and patient discomfort that often accompany the longer acquisition from 
+an increased FOV. Increasing the speed of the acquisition system often results in engineering compromises 
+that reduce the spectral bandwidth of the OCT system, and hence lower the axial resolution. Methods like 
+montaging 3 and motion-tracking software 4 are some techniques that have been explored to minimize the 
+negative implications of wide-field OCT systems. In addition, machine learning has been explored for feature 
+reconstruction of OCT-angiography (OCTA) to improve the image quality for clinical utility 5.  
+In this study, we simulate the narrower spectral bandwidth on OCT volumes and investigate the use of 
+a generative adversarial network (GAN). For image-to-image generation, many implementations use a pixel-
+to-pixel (pix2pix) GAN, which has the encoder/decoder in the generator. We used a pix2pix approach 
+leveraging a modified super-resolution GAN (SRGAN) 6 architecture to recover high-resolution features in 
+the OCT B-scans. The SRGAN is comprised of a VGG-19 style discriminator and a generator with residual 
+blocks and subpixel convolutional layers. We have modified the SRGAN architecture by deepening the 
+discriminator and generator and removed the pixel up-sampling layers from the generator for our pix2pix 
+implementation.  
+
+2 
+ 
+We were limited by the size of our dataset and leveraged transfer learning from an open-source natural 
+flower dataset 7, referencing a similar SRGAN approach used to upscale radiographs 8, to facilitate the 
+convergence of our deep neural networks (DNNs). The flower dataset contains images with dense and well-
+defined edges from complex features of different flowers, petals, and seeds. These features exaggerate the 
+blurring process and may prove useful when used as initialized weights for our OCT-SRGAN. 
+The original architecture was designed to upscale images and reconstruct features lost from downsizing. 
+Rather than utilizing the OCT-SRGAN to reconstruct a larger image from a resized smaller image, we 
+investigate the use of an OCT-SRGAN to reconstruct reduced spectral resolution in the axial direction (A-
+scan blurring) through transfer learning and demonstrate the ability to reconstruct lost features using learning-
+based approaches. Our findings suggest that DNNs may benefit clinicians if developed in parallel with wide-
+field OCT systems by reconstructing features lost due to the reduced narrower spectral bandwidth that often 
+accompanies OCT systems with faster A-scan line rates. Our main contribution is to adapt the feature 
+recovery using an SRGAN for the super-resolution of OCT B-scans. We present our preliminary results while 
+we continue to develop our OCT-SRGAN and will branch into alternative learning-based methods like 
+feature pyramid networks. This experiment was done in a two-step approach as described in Figure 1: (1) 
+preliminary dataset consisting of 16 eyes to understand the impact of transfer learning; (2) utilize the full 
+dataset of 35 eyes to compare the performance of reconstruction in the spatial versus spectral domain. 
+ 
+ 
+Figure 1 
+Parallel schematics comparing the spatial and spectral super-resolution GAN processing pipeline. 
+3. Methods 
+OCT dataset and preparation 
+This study included 35 eyes (including pathological) 27 unique patients, each imaged on a ~2x2mm FOV 
+centered on the fovea using a 1060nm swept source OCT prototype adaptive optics system. We utilized B-
+scans that visualize the retinal cross-section. An initial subset of this dataset consisted of 16 eyes was initially 
+used to validate transfer learning from natural images. Each B-scan was cropped from 1024x400 to 240x400 
+(axial x lateral position) for the preliminary experiment and were cropped to 470x400 and processed as strips 
+of 20 A-scans for the spatial domain portion of the second experiment. The image dimensions for the spectral 
+domain GAN were unchanged before evaluation. The full dataset of 35 eyes consisted of 14,000 B-scan 
+samples or 5,600,000 independent A-scan samples. The dataset was split into 60%, 20%, 20% for training, 
+validation, and testing, respectively. 21 eyes were used for training (8,400 B-scans or 3,360,000 A-scans), 7 
+eyes for validation (2,800 B-scans or 1,120,000 A-scans), and 7 eyes (2,800 B-scans or 1,120,000 A-scans) 
+were allocated for testing. Adjacent B-scans contain similar information. Thus, every 8 B-scan was used to 
+
+Preprocessing
+SRGAN
+470x400
+.UNP Binary
+OCT
+Normalize
+Dataformat
+Processing
+[-1,1]
+Discriminator:VGG-base
+Gaussian
+2-D IFFT →>
+Generator:SRGAN
+Windowing
+abs()→crop
+Augmentations:Flips
+Loss: MSE
+1024x400
+Preprocessing
+SRGAN
+Postprocessing
+470x400
+.UNP Binary
+OCT
+Normalize
+Data format
+Processing
+[-1, 1]
+2-D IFFT >
+Gaussian
+abs()→crop
+Windowing
+1024x400
+Discriminator:VGG-base
+Generator:ResUNet-a
+Augmentations:None
+Loss: MSE3 
+ 
+allow enough spacing between acquisitions to minimize the chance of overfitting to repeating consecutive 
+scans. The images were shuffled within the training set prior to training. Special attention was made to ensure 
+that eyes from the same patient was used for either training, validation, or testing. Since the flower images 
+were 3-channel RGB images, when training the neural networks on the flower images, we selected the first 
+channel to form a 1-channel input to replicate the domain of OCT images.  
+The axial resolution in OCT images is related to the spectral characteristics of the light source. A 
+commonly used expression for the axial resolution is the coherence length, lc, given by 𝑙𝑐 = 𝜆𝑜
+2/Δ𝜆, where 
+𝜆𝑜 is the central wavelength, and Δ𝜆 is the spectral bandwidth. By using our swept source OCT prototype 
+system, we had access to data in the wavelength domain and performed Gaussian windowing on the spectrum 
+and hence reduced the axial resolution. We utilized the MATLAB function gausswin with α = 8, chosen 
+based on earlier trials of training and the appearance of the OCT B-scan image output, where the coefficients 
+of a gaussian window (w) are as described in1. This gaussian windowing mask was element-wise multiplied 
+with the spectral domain OCT data, as shown in Figure 2. 
+ 
+Figure 2 
+Windowing process of our OCT data in the spectral domain (A). A gaussian windowing mask (B) is used to reduce the 
+axial resolution with α = 4 to generate spectral domain gaussian windowed data (C). 
+Flower Dataset and Preparation 
+We used an open-sourced flower dataset compiled by the team at TensorFlow 7 to initialize the weights of 
+our OCT-SRGAN. The flower dataset is comprised of 3,670 flower images. The black borders surrounding 
+the floral images were removed and the images were reshaped to 240x400 and randomly shuffled before 
+being allocated for training (3303) and testing (367). 
+The artefacts introduced to the flower dataset should mimic the appearance of the reduced axial 
+resolution OCT B-scans. Hence, we generated the low-resolution images by convolving the original high-
+resolution (HR) inputs with a 1xn mean filter to smoothen each pixel vertically with n values of 3, 5, 7, 9, 
+and 11 pixels, as shown in Figure 3. 
+ 
+1 https://www.mathworks.com/help/signal/ref/gausswin.html 
+
+A
+B
+c4 
+ 
+ 
+Figure 3. Graphical representation of the windowing process of high-resolution flower images (A) using a vertical 1xn, in this case 
+n=11, mean filter (B) to generate mono-directional (vertical) smoothened images (C). 
+Data Augmentation and GAN Training Techniques 
+In reconstructing in the spatial domain, data augmentation was achieved through horizontal and vertical flip 
+using the ImgAug library to increase the effective dataset size and improve the generalizability. Since the 
+data experienced smoothening in A-scans (vertical direction) only, rotations were not considered. 
+Additionally, random noise was not introduced through augmentation to not interfere with the reconstruction 
+of the speckle pattern in OCT B-scans. In both domains, the data was augmented during preprocessing and 
+the did not undergo further augmentations during training. These augmentations performed on the spectral 
+domain fringe data included a random center and width of the Gaussian windowing filter, as demonstrated 
+in Figure 4. 
+ 
+
+A
+B
+CA
+B
+C
+D5 
+ 
+Figure 4 
+An illustration of the random augmentations performed during the preprocessing of the spectral domain fringe data: (A & 
+C) original Gaussian windowed fringe data; (B) center-shifted; (D) center-shifted and widened.  
+We refer to the previous literature 9–13 for neural network training procedures to improve our GAN 
+framework and encourage convergence. Soft labels are used in classification neural networks to decrease the 
+error rate and have been adapted to the adversarial component of a GAN 9–11. Instance noise is a technique 
+where the discriminator’s training labels are randomly flipped 12, 13. Both techniques were implemented to 
+improve the stability of GAN training by reducing the ambiguity between the generated and ground truth 
+samples which promotes training convergence. We implemented soft labelling by randomly distributing real 
+labels between (0-0.1) and generated labels from (0.9-1). In addition, instance noise was introduced by 
+randomly, with a 5 percent chance, providing the discriminator with an incorrect label. The discriminator 
+was trained on an entire batch of the real ground truth data and followed by a batch of the generated data. 
+This minibatch feature approach allowed the discriminator to compare an example of a minibatch of 
+generated samples to the real samples and allowed the discriminator to detect similarities across the 
+minibatches 9, 14. 
+Neural Network Design 
+Spatial-domain Feature Recovery Network 
+In the first step, we propose a variation of the SRGAN proposed in the Literature 6. Instead of to upscales the 
+smaller image to a larger image through subpixel convolutional layers (PixelShuffler x2), our implementation 
+of the SRGAN removes the subpixel resolution layers and instead, utilizes the SRGAN architecture for a 
+pix2pix application to preserve the B-scan resolution, as shown in Figure 5. 
+ 
+Figure 5 Super-resolution Generative Adversarial Network (SRGAN) architecture. Architecture (A) is the discriminator based off 
+VGG networks and architecture (B) is the generator comprised of residual of blocks. The loss functions are the discriminator ground 
+truth (lGT) and generated image (lGenerated) binary cross entropy loss and generated/ground truth mean squared error (lMSE). 
+Compared to the SRGAN 6, we used a deeper discriminator and generator because deeper networks have 
+been shown to yield better results with the tradeoff of being more difficult to train 15. The discriminator 
+consisted of convolutional blocks that contain 2D convolutional layers that utilizes more filters in deeper 
+layers followed by batch normalization and a Leaky Rectified Linear Unit (LeakyReLU). Batch 
+normalization has been found to improve the optimization of GANs 9, 16 and LeakyReLU has been found to 
+generate better results, especially for higher resolution implementations 16. In the generator, we used a 
+parametric ReLU which is a LeakyReLU with a learnable negative slope along with batch normalization and 
+skip connections through element-wise addition to form a residual block. Skip connections allow the network 
+to pass forward simple features that may be difficult to learn through convolutional filters 6, 17.  
+
+Convolution
+Flatten
+ReL
+Dense
+Sigmoid
+Dense
+eaky
+Batci
+Batch Normalization
+Batch Normalization
+Element-Wise Sum
+ReLU
+zation
+ReLU
+Sum
+ReLU
+Convolution
+Convolution
+Convolution
+Parametric
+Parametric
+Elemen
+Cor
+Batch
+e6 
+ 
+Spectral-Domain Feature Recovery Network 
+In the second step of the study comparing super-resolution in the spatial versus spectral domain, all the 
+components within the neural network including convolutional filters, pooling filters, among others, utilized 
+vertical 1-D filters. This ensured that the neural network would explicitly process the A-scans independently. 
+The method of Gaussian windowing was only performed on A-scans, and each was independent from the 
+others. 
+The spectral domain reconstruction processed of images with relevant information near the center of the 
+B-scan with Gaussian-distributed diminishing intensity towards the end of the image. The generator was 
+updated to account for the distance between pixels using dilation rates in the convolutional layer, which 
+increase the spacing of the convolutional filters. Referencing the published ResUnet-a architecture 18, a wide 
+range of dilation rates were leveraged in parallel where each block of batch normalization and activation 
+layers were followed by a 1D convolutional layer with different dilation rates. Upwards of 8 parallel blocks 
+were leveraged, each with different dilation rates, to allow the neural network to explore the relationship 
+between pixels that were higher in distance. The parallel convolutional blocks with different dilation rates 
+are combined through addition with an identity function through skip connections. Figure 6 graphically 
+outlines the different building blocks used including the application of dilation rates in parallel to construct 
+the ResUNet-a generator, as shown in Figure 7. 
+ 
+ 
+ 
+ 
+Figure 6  ResUNet-a building blocks as described 18. 
+
+Batch Normalization
+ParametricReLU
+d [1,3,15,31,51,71,121,251]
+Convolution
+n32k3s1
+Batch Normalization
+ParametricReLU
+Addition
+Convolution1X
+n;rk1s1d1
+Concatenate
+Convolution
+2X
+4X
+8x
+Max Pooling
+kx
+Upsampling
+kx
+ConvolutionConvolution
+n512k1s1d1
+ResUNet-a block
+Concatenate
+(n512k3s1,d=(1))
+ParametricReLU
+2x Upsampling7 
+ 
+ 
+Figure 7  ResUNet-a architecture used for reconstructing in the spectral domain. 
+Experimental Settings for Training Feature Recovery Network 
+DNN training for both experiments were implemented in a similar manner. The optimizer used was Adam 
+with β1 = 0.5. The optimal learning rate was found through quick training sessions with different learning 
+rate schedules. For all experiments, the optimal initial learning rates of 1x10-4 were used in both the 
+discriminator and generator. For the preliminary experiment, the SRGAN was trained for 200 epochs with a 
+batch size of 32. The second experiment was performed on images of size 1024x400 and was trained on 100 
+epochs with a batch size of 10. The SRGAN was trained and evaluated within 24 hours and 52 hours for the 
+preliminary experiment and second experiment, respectively.  
+The content loss (lMSE) was calculated through pixel-wise mean squared error (MSE). The 
+discriminator was trained separately on minibatches of ground truth and generated images and resulted in 
+two binary cross-entropy losses (lGT, lgenerated) which refers to the generator’s ability to fool the 
+discriminator. Models that improved with a lower generator lMSE loss, higher discriminator lGT, or higher 
+lgenerated loss, were saved. The preliminary experiment leveraged the SRGAN trained on the natural flower 
+dataset that was used to initialize the weights for OCT B-scan training; all the neural network parameters 
+were trainable. All DNNs were developed and evaluated in TensorFlow and the Keras API using Python 
+3.6.3 on Canadian supercomputer “Cedar” nodes powered by the NVIDIA Tesla V100-SXM2 GPU and 
+32GB RAM.  
+Evaluation 
+The evaluation of GANs is often qualitative as there are minor artefacts that may drastically impact evaluation 
+metrics like MSE, peak signal-to-noise ratio (PSNR), and structural similarity (SSIM) 19 among others. For 
+
+Convolution
+PSP
+Convolution
+Pooling
+ndino
+ndul
+ResUNet-a block
+(n32k3s1, d={1))
+ResUNet-ablock(n32k3s1,
+d={1,3,15,31,51,71,121,251))
+Convolution
+ResUNet-a block
+(n64k3s1, d=(1))
+ResUNet-a block (n64k3s1)
+d=(1,3,15,31,51,71,121,251))
+ResUNet-a block
+Convolution
+(n128k3s1,d=(1))
+ResUNet-a block (n128k3s1,
+d=(1,3,15,31,51))
+ResUNet-a block
+Convolution
+(n256k3s1,d=(1))
+ResUNet-a block
+(n256k3s1, d=(1,3,15)
+ResUNet-a block
+Convolution
+(n512k3s1,d=(1))
+ResUNet-a block
+(n512k3s1,d={1))
+Combine
+Convolution
+ResUNet-a block
+PSP Pooling
+ 2x Upsampling
+(n1024k3s1, d=(1))8 
+ 
+example, in our implementation of reconstructing features, we may weigh the exact intensity of an image 
+less than the sharpness of an OCT layer boundary and some evaluation metrics may weigh according to the 
+neural network’s function. Hence, one approach is to enlist human annotators to grade the performance of 
+the GAN, also called the mean opinion score (MOS), based on the function and objective of the study 8, 9. 
+To improve the qualitative evaluation of the generated outputs, we cropped regions of the images near 
+the retinal layers and enlarged the image using nearest-neighbour interpolation to preserve the pixel 
+resolution and qualitatively compared to the ground truth and windowed/mean filtered images to better 
+visualize and compare features. The models were quantitatively evaluated using MSE, PSNR, and SSIM 
+across the entire test set for the entire image of the preliminary experiment. RMSE and SSIM were leveraged 
+to evaluate the experiment on the full dataset. 
+4. Results – Preliminary Transfer Learning 
+A range of filter sizes (1x3, 1x5, 1x7, 1x11) was utilized to introduce vertical blurring to the flower neural 
+network. The SRGAN trained on the 1x11 smoothened flower images yielded the best results when used as 
+initialized weights for OCT B-scans. Therefore, in the transfer learning experiment, the neural network was 
+initialized on the 1x11 smoothened flower image.  
+Qualitative Evaluation 
+Our direct implementation of the OCT-SRGAN to reconstruct high-axial-resolution from low-axial-
+resolution B-scans successfully sharpened the features of the OCT image. As shown in Figure 9 (A - C), the 
+same region is cropped and enlarged using nearest-neighbor interpolation for qualitative evaluation.  
+The results of our flower SRGAN and the performance of reconstructing vertical smoothening using a 
+1x11 mean filter, as shown in Figure 8, shows promising utility in recovering features lost from smoothening.  
+ 
+Figure 8 
+Flower SRGAN comparison between (A) 1x11 mean filtered, (B) ground truth, and (C) generated images from the test set.  
+ 
+
+A
+B9 
+ 
+ 
+Figure 9 
+Results of SRGAN for OCT B-scan images. Samples from the generated images from the test set. Comparison between (A) 
+windowed, (B) ground truth, and (C) generated B-scans without transfer learning, and (D) generated B-scans with transfer learning.  
+Quantitative Evaluation 
+The whole OCT B-scan image was evaluated for similarities across the test set post-contrast adjustment 
+between the generated and ground truth image using MSE and SSIM, as shown in Table 1. This was compared 
+to the baseline comparison of the Gaussian windowed/vertically smoothened and ground truth images.  
+While it is difficult to quantitatively evaluate the performance of a generated output as the function or 
+goal is often subjective, by standardizing the image and enhancing the contrast of all images using the same 
+pipeline, the issues of differences in intensities were minimized. By doing so, we effectively catered our 
+metrics to weigh the features more heavily than the absolute intensity. Table 1 shows that across the 
+evaluation metrics, the generated B-scans more resembled the original high-resolution B-scans. 
+Table 1. 
+Quantitative evaluation of the generated images (GEN) comparing the OCT-only with the natural images transfer learning 
+approach. Mean values are shown with standard deviation in parentheses. This portion of the experiment was performed on a subset 
+(16 eyes) of the entire dataset (35 eyes). 
+ 
+ Best value between generated and windowed for the specific test is bolded. 
+ Mean squared error (MSE) and structural similarity (SSIM). 
+ Ground truth (GT) 
+5. Results – Reconstruction in the Spatial Versus Spectral Domain 
+The method of preprocessing significantly affected the comparison between the training on the preliminary 
+dataset compared to the full dataset since a different cropping algorithm was utilize. This section leverages 
+the entire 1024x400 B-scan; conversely, the preliminary experiment cropped the images into 470x400 B-
+
+B
+D10 
+ 
+scans. However, within this experiment, the method of evaluation was consistent between both the spatial 
+and spectral domain with minor differences in functions used for some final processing and evaluation steps. 
+Qualitative Evaluation 
+For the spatial SRGAN, the full set of 35 eyes from 27 unique patients were used without transfer learning 
+and cropped regions of two scans from the test set capturing the retinal layers are shown in Figure 10.  
+ 
+Figure 10.  Results of spatial SRGAN without transfer learning on the full OCT B-scan dataset (35 eyes). Samples from the generated 
+images from the test set. Comparison between (A & D) windowed, (B & E) ground truth, and (C & F) generated B-scans. The red 
+circles highlight a region of interest. 
+The full dataset reconstructed from the spectral domain was used to train a model and a cropped region 
+of an eye from the test set is shown in Figure 11. Another example from the test set can be seen in the Figure 
+12. The data is independent across A-scans and should be further explored as so. Figure 13 graphically 
+compares the intensity of the fringe data.  
+
+A
+B
+C
+D
+E
+F11 
+ 
+ 
+Figure 11.  Results of the spectral SRGAN trained on the spectral fringes of OCT B-scan images on the full dataset (35 eyes). Samples 
+from the generated images from the test set. Comparison between (A) windowed fringe data, (B) ground truth fringe data, (C) 
+generated fringe data, (D) windowed spatial domain data, (E) spatial domain data which is Fourier-transformed from the ground truth 
+spectral data, and (F) spatial domain data which is Fourier-transformed from the generated spectral data.  
+
+A
+B
+C12 
+ 
+ 
+Figure 12.  Potential failure case of the SRGAN trained on the spectral fringes of OCT B-scan images on the full dataset (35 eyes). 
+Samples from the generated images from the test set. Comparison between (A) windowed fringe data, (B) ground truth fringe data, (C) 
+generated fringe data, (D) windowed spatial domain data, (E) ground truth spatial domain data, (F) generated spatial domain data, (G) 
+cropped windowed, (H) cropped ground truth, and (I) cropped generated.  
+ 
+
+B
+c
+E
+G13 
+ 
+ 
+Figure 13.  Results of SRGAN trained on the spectral fringes of OCT B-scan images on the full dataset (35 eyes). Samples from the 
+generated images from the test set. The generated A-scans, in yellow, are compared to the ground truth (GT) in orange and windowed 
+fringe in blue. (A) presents the B-scan and the A-scan of interested highlighted by the red arrows. (B) shows the full A-scan in the 
+spectral domain. (C) examines the top half of the image to provide understanding on the generator’s performance nearing the edge of 
+the fringes (normally represent high-frequency signal and spectral noise). (D) examines the central 300 pixels. 
+Quantitative Evaluation 
+This experiment was performed on the entire dataset and was more comprehensive in the evaluation than the 
+preliminary study. The spatial versus spectral reconstruction was evaluated for PSNR and the MSE was 
+normalized, and square rooted to provide a more standardized range to compare models with different scales. 
+Table 2 quantitatively compares the performance of the reconstruction in the spectral against the spatial 
+domain.  
+
+150
+120
+A
+B
+Windowed
+C
+Windowed
+GT
+GT
+Generated
+Generated
+100
+100
+80
+Intensity
+Intensity
+60
+50
+40
+20
+0
+200
+400
+600
+800
+1000
+0
+100
+200
+300
+400
+500
+Vertical Position
+Vertical Position
+150
+D
+Windowed
+GT
+Generated
+100
+Intensity
+50
+400
+500
+600
+700
+Vertical Position14 
+ 
+Table 2. 
+Quantitative evaluation of the generative adversarial network (GAN) images comparing reconstructing in the spectral 
+domain versus in the spatial domain. The spectral domain images were transformed to the spatial domain before the evaluation. Mean 
+values are shown with standard deviation in parentheses.  
+ 
+Best value between generated and windowed for the specific test is bolded. Abbreviation: Normalized Root Mean 
+squared error (NRMSE), peak signal-to-noise ratio (PSNR), and structural similarity (SSIM). Ground truth (GT) 
+ 
+6. Discussion 
+As OCT hardware moves towards capturing larger field of view including more peripheral parts of the retina, 
+the axial resolution may be compromised to minimize the increase in acquisition time for patient comfort and 
+reduced motion artefacts. The resulting reduced bandwidth in the spectral domain hinders the micrometer-
+resolution which is one of the benefits of utilizing this modality. This study simulates the reduced axial 
+resolution and aims to reconstruct lost features. The contributions of this study are as follows: (1) the effect 
+of transfer learning from a natural dataset for initializing the OCT-SRGAN; and (2) the comparison between 
+reconstructing in the spectral versus spatial domain. 
+Published studies have investigated the use of an SRGAN on OCT data. GANs have been leveraged for 
+super-resolution through reconstructing features lost through downsampling in the spatial domain 20, speckle 
+removal 21–23, domain adaptation 24, 25, and synthesizing retinal diseases 26 or other imaging modalities 27 
+among other applications. Recently, groups have begun utilizing GANs for super-resolution. They have 
+simulated low axial resolution OCT data by windowing in the spectral domain and reconstructed in the spatial 
+domain 28–31. To the best of our knowledge, this study is the first to leverage a GAN to reconstruct a simulated 
+reduced spectral bandwidth in the spectral domain for ophthalmic OCT data. 
+The preliminary step of this experiment was performed to understand the effect of transfer learning on a 
+GAN-based reconstruction of OCT data. As shown in Figure 9, the SRGAN leveraging the pre-trained 
+weights on the flower dataset successfully reconstructed the OCT data. The features were sharpened 
+especially in the speckle texture of the retinal layers. However, when compared to the evident effect of the 
+GAN on reconstructing the smoothened features on the flower dataset, the effect is minor (comparing Figure 
+9 C and D). The quantitative evaluation mirrored this sentiment. . By leveraging transfer learning to initialize 
+the SRGAN on natural flower images, the generated images are slightly better in MSE but slightly worse in 
+SSIM than the OCT-only approach. However, all metrics are well within one standard deviation between the 
+two approaches. These results were promising and provided us the confidence to move forward with the 
+entire dataset.  
+As the preliminary trial progressed, different windowing alpha values were utilized when reducing the 
+spectral bandwidth. The alpha value of 8 was sufficient to visualize the effect of reduced axial resolution. 
+Transfer learning from the floral dataset was relatively successful and the flower-initialized neural network 
+was able to slightly reduce training time. However, the flower dataset was convolved by a mean filter to 
+simulate the effects of reduced axial resolution in the spatial OCT images. For a fair comparison, the spectral 
+domain neural network must also transfer knowledge from a similar domain. The differences in the two 
+approaches supported an approach without transfer learning. The trials that leveraged transfer learning had 
+superior mean values of the evaluation metrics. However, it also resulted in a higher standard deviation for 
+both MSE and SSIM. We decided to approach the second experiment without transfer learning due to the 
+lower variance of the performance and the flexibility it provided. If transfer learning was utilized, each 
+alteration to the neural network would require the same changes to be retrained on the flower dataset. 
+
+15 
+ 
+The floral dataset was selected as proposed in the Literature 8 for reconstructing reduced resolution in 
+radiographs. The complex textures and patterns in the flower petals provided high contrast edges. When 
+convolving the image with a vertical filter, the effect of blurring is more evident than one performed on a 
+more homogenous image. Further exploration into transfer learning from a more similar domain, such as 
+ultrasound, should be performed. This will allow us to understand the impact of transfer learning from a 
+similar domain compared to a dataset selected to exaggerate the desired effect. 
+The secondary experiment compared reconstruction in the spatial and spectral domain. Both experiments 
+allocated the same eyes for training, validation, and testing. The preprocessing and evaluation were 
+performed using the same functions. However, differences in rounding, saving formats, and the order of the 
+pipeline resulted in minor changes to the images, as shown in Table 2. The column setting the baseline of 
+evaluation metrics comparing the windowed and ground truth images ideally would be identical since the 
+GAN has no impact on either set of images. However, the data was saved as an image for training at different 
+stages in the spectral and spatial domains. Training also required standardization to [-1, 1] before training 
+and [0, 1] for evaluation. All of the differences in the order of standardization, rounding, and processing must 
+be considered when comparing the two domains. To replicate the changes that occur to the generated images, 
+the GT and windowed fringes were also subject to the same processing pipeline. 
+When evaluating the reconstruction qualitatively, both approaches were successful in reconstructing lost 
+features. In the spatial domain, as shown in Figure 10, the images appear sharper in both the speckle pattern 
+in the background and the retinal layers. However, slight intensity changes are visible even after intensity 
+normalization. When referencing Figure 10 (D-F), a vertical line in the choroidal region (highlighted by the 
+red circle) is approximately 3 pixels long in the ground truth image. In the corresponding windowed and 
+generated images, the feature appears to be 4-5 pixels and 3-4 pixels long, respectively. In the spectral 
+domain, as shown in Figure 11, the generated fringe data (C) is capable of reconstructing features towards 
+the tails of the Gaussian window. Figure 13 graphically shows a randomly selected A-scan highlighted by 
+the arrows in (A). The zoomed-in view of the center of an edge of the A-scan (C) and the center of the A-
+scan (D) confirms that the generator can reconstruct the features lost from Gaussian windowing. Near the 
+center presented in (D), the generated signal is capable of reconstructing patterns in the ground truth. By 
+examining (C), which is the top half of the signal, the signals appear to be similar up until 200 pixels from 
+the center whereas the windowed signal has nearly reached zero intensity. Thus, we conclude that the minor 
+signals remaining in the windowed image paired with the patterns found near the center of the image are 
+sufficient for the GAN to reconstruct features that have been suppressed through Gaussian windowing. 
+When evaluating the mean of the metrics, shown in Table 2, the spatial GAN outperforms the spectral 
+GAN in both NRMSE and PSNR and the spectral GAN is superior in SSIM. However, the trends are evident 
+in the right column comparing the windowed and ground truth images. The differences between the two 
+approaches when evaluating the generated images are also within one standard deviation. Thus, we conclude 
+that both approaches are comparable and effective for reconstructing features lost from the reduced spectral 
+bandwidth.  
+The spectral domain models were optimized within 15 epochs or approximately 100,000 training 
+iterations. A failure case can be seen in Figure 14 demonstrating the effect of overfitting at 19 epochs. The 
+fringe data contained cyclical vertical streaking patterns upon inspection and when converted into the spatial 
+domain, regions above the retina were removed. Conversely, the spatial GANs were able to train up to 
+approximately 80 epochs without overfitting. This issue should be further explored as it could be a result of 
+the similarities between fringe data or the necessity of revising the learning rate scheduler.  
+ 
+7. Limitations and Future Works 
+The SRGAN study was limited by the lack of comparison to a traditional ‘deconvolution’ method. We have 
+compared the performance of a model that leveraged transfer learning from a natural floral dataset, but lack 
+a fair comparison to other conventional programming reconstruction techniques. The study would also 
+benefit from exploring reconstruction in the spectral domain and investigating other methods of qualitative 
+evaluation. Some groups have adapted a qualitative evaluation into a quantitative evaluation through a mean 
+opinion score, where a group of experts are randomly polled to select the best between the generated, 
+windowed, and other alternative methods. This is a potential avenue of evaluating the benefit and 
+
+16 
+ 
+performance of the generated super-resolution images. However, this requires the opinion of experts, and the 
+process may be time-consuming. Another method of evaluation, inspired by my previous work 24, could be 
+through other processing tools by biasing a layer segmentation tool towards the high-resolution domain and 
+evaluating the generated super-resolution image by the performance of the segmentation.  
+A potential failure case was examined in Figure 12 where the fringe data contained vertical line artefacts. 
+The SRGAN was still able to reconstruct the features despite the visually unappealing lines. Upon visual 
+inspection, the bandwidth did not seem to increase as drastically as other examples. However, when we 
+transform the image into the spatial domain, it is evident that the GAN was still successful in reconstructing 
+the lost features. 
+ 
+Figure 14.  Results of an overfit SRGAN (19 epochs) trained on the spectral fringes of OCT B-scan images on the full dataset (35 
+eyes). Samples from the generated images from the test set. Comparison between (A) windowed fringe data, (B) ground truth fringe 
+data, (C) generated fringe data, (D) windowed spatial domain data, (E) ground truth spatial domain data, and (F) generated spatial 
+domain data.  
+Future works includes incorporating the spatial domain A-scans as part of the loss function in the spectral 
+domain GAN, and vice versa. Additional future works include combining GANs from both domains into one 
+processing pipeline, exploring transfer learning from a similar domain of data such as ultrasound or frequency 
+signals from music, and investigating the use of the ResUNet-a architecture on the spatial domain. 
+8. Conflict of Interest 
+MVS: Seymour Vision, Inc. (I). 
+ 
+9. References 
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+

0tE4T4oBgHgl3EQfzA1I/content/tmp_files/2301.05270v1.pdf.txt

@@ -0,0 +1,958 @@
+arXiv:2301.05270v1  [math.DG]  12 Jan 2023
+ON A PRE-ORDERING OF COMPACT PSC MANIFOLDS AND PSC
+RIEMANNIAN METRICS
+MOHAMMED LARBI LABBI
+Abstract. We study the 1-parameter family of nested curvature conditions Riemt(g) > 0 on a Rie-
+mannian n-manifold (M, g) that interpolates between positive scalar curvature (t = 0) and positive
+constant sectional curvature (t =
+�n
+2
+�
+). We define the metric invariant Riem(g) to be the supremum
+of parameters t for which the condition Riemt(g) > 0 is allowed on (M, g). The smooth invariant
+Riem(M) ∈ (0,
+�n
+2
+�
+)] is by definition the supremum of Riem(g) over the set of all psc Riemannian
+metrics g on M. This invariant allows in particular to (pre) order psc compact manifolds.
+We show that on the top, compact manifolds with Riem =
+�2
+n
+�
+are positive space forms. Next,
+Sn−1×S1, connected sums of these and connected sums of positive space forms have Riem =
+�n−1
+2
+�
+.
+For 1 ≤ p ≤ n − 2 ≤ 5, Sn−p × T p have the intermediate value of Riem =
+�n−p
+2
+�
+.
+From the bottom, we prove that simply connected (resp. 2-connected) compact manifolds of di-
+mension ≥ 5 (resp. ≥ 7) have Riem ≥ 1 (resp. ≥ 3.) The proof of the two last results is based
+on a result of this paper where we prove that the smooth Riem invariant doesn’t decrease after a
+surgery on the manifold with sufficiently high codimension.
+Keywords: PSC manifolds, Riem invariants, surgery theorem.
+Contents
+1.
+Introduction and Statement of the main results
+2
+1.1.
+Introduction
+2
+1.2.
+Statement of the main results of the paper
+4
+1.3.
+Plan of the paper
+6
+2.
+Relations with other curvature conditions: Proof of Theorems D and E
+6
+2.1.
+Proof of Theorem D
+7
+2.2.
+Positive Cp intermediate curvature condition: Proof of Theorem E
+8
+2.3.
+Proof of Theorem E
+9
+3.
+The Riem of total spaces of Riemannian submersions: Proof of Theorem A and
+Proposition 1.1
+9
+3.1.
+Proof of Proposition 1.1
+9
+2020 Mathematics Subject Classification. 53C21, 53C18 .
+1
+
+3.2.
+Proof of Theorem A
+9
+4.
+The Riem invariant and surgeries: Proof of Theorems B, B′ and Corollary B
+10
+4.1.
+Proof of Theorem B
+10
+4.2.
+Proof of Corollary B
+11
+4.3.
+Proof of Theorem B′
+12
+5.
+Vanishing theorems: Proof of Theorems C and C′
+12
+5.1.
+Proof of Theorem C
+12
+5.2.
+Proof of Theorem C′
+12
+6.
+Miscellaneous results
+13
+6.1.
+Positive Γ2(R) curvature
+13
+6.2.
+The small Riem invariant
+14
+6.3.
+The Riem invariant of a conformal class
+15
+6.4.
+Minimal vs. Maximal PSC compact manifolds: open questions
+15
+References
+16
+1. Introduction and Statement of the main results
+1.1. Introduction. Throughout this paper, (M, g) denotes a closed connected Riemannian mani-
+fold of dimension n. We denote by R and Scal respectively the Riemann curvature tensor and the
+scalar curvature of (M, g). We define a string of curvature tensors on (M, g) as follows
+(1)
+Riemt(g) := Scal g2
+2 − 2tR,
+Where t is a constant real number, and g2 is the square of g with respect to the Kulkarni-Nomizu
+product. These curvatures are defined in a similar way as the tensors Eink in [8]. We are interested
+in the positivity properties of these curvature tensors. In order to ensure that their positivity is
+stronger then positive scalar curvature, we restrict the parameter t to be less than n(n − 1)/2. In
+fact, one can easily show that the full trace of Riemt equals
+�
+(n(n − 1) − 2t
+�
+Scal.
+We remark that for a compact Riemannian n-manifold (M, g) with positive scalar curvature and
+for 0 < t < n(n − 1)/2, the tensor Riemt(g) is positive definite if and only if at each point of M
+one has
+(2)
+t < Scal
+2λmax
+,
+2
+
+where λmax denotes the maximal eigenvalue of the Riemann curvature operator at the correspond-
+ing point.
+A straightforward consequence of the above charecterisation of the positivity of the tensors
+Riemt(g) is the following descent positivity property
+(3)
+For 0 < t < s < n(n − 1)/2, Riems > 0 ⇒ Riemt > 0 ⇒ Scal > 0.
+We therefore define the metric invariant
+(4)
+Riem(g) := sup{t ∈ (0, n(n − 1)/2 : Riemt(g) > 0}.
+We set Riem(g) = 0 if the scalar curvature of g is not positive. An immediate consequence of
+property (2), is that for a metric g of positive scalar curvature one has
+(5)
+Riem(g) = inf
+M
+Scal(g)
+2λmax(g).
+The above formula shows in particular that the metric invariant Riem(g) is re-scale invariant, That
+is for any positive real number t, one has
+(6)
+Riem(tg) = Riem(g).
+Example 1.1.
+(1) For a compact Riemannian n-manifold (M, g) of positive constant sectional
+curvature, one has Riem(g) = n(n−1)
+2
+. It has the maximal possible value. Conversely, if
+Riem(g) = n(n−1)
+2
+then g has positive constant sectional curvature.
+(2) For the standard product metric g on the product Sn−p ×T p of a round sphere of curvature
+1 with a flat torus one has Riem(g) = (n−p)(n−p−1)
+2
+.
+(3) For the standard product metric g on the product Sn−p×Hp of a round sphere of curvature
+1 with a hyperbolic space of curvature −1 one has Riem(g) = (n−1)(n−2p)
+2
+.
+(4) For the standard Fubiny-Study metric g on CPn one has Scal(g) = 4n(n+1) and λmax(g) =
+2n + 2 [2], and consequently Riem(g) = n.
+(5) For the standard Fubiny-Study metric g on HPn one has Scal(g) = 16n(n+2) and λmax(g) =
+4n [2], and consequently Riem(g) = 2n + 4.
+The metric Riem invariant defines a pre-order on the set of Riemannian metrics on M as follows
+(7)
+g1 ⪯ g2 if Riem(g1) ≤ Riem(g2).
+This lead us naturally to the study of maximal metrics with respect to the above pre-order.
+We define then a smooth invariant Riem(M) to be
+(8)
+Riem(M) = sup{Riem(g): g ∈ M},
+where M denotes the space of all Riemannian metrics on M. We set Riem(M) = 0 if M has
+no psc metrics. For the seek to provide the reader a better feeling of this invariant, we provide
+(prematurely) the following examples whose proofs will be provided later in this paper.
+3
+
+Example 1.2.
+(1) For all n ≥ 2, It is easy to see that Riem(Sn) = Riem(RPn) =
+�n
+2
+�
+.
+(2) For r ≥ 2, connected sums of r-copies of (RPn) has Riem =
+�n−1
+2
+�
+.
+(3) For all n ≥ 2, Riem(Sn−1 × S1) =
+�n−1
+2
+�
+. Furthermore, the connected sum of arbitrary r
+copies of the previous manifold satisfies Riem(#r(Sn−1 × S1) =
+�n−1
+2
+�
+.
+(4) For 2 ≤ n ≤ 7, Riem(Sn−p × T p) =
+�n−p
+2
+�
+.
+In the literature one finds several notions of intermediate curvature conditions which are useful
+in different directions. The author beleives that the curvature conditions that are stable under
+surgeries, satisfy the ideal property with respect to the Cartesian product of manifolds and have a
+fibre bundle property (as below) are subtle and deserve a special attention. The point is that the
+above three properties are the keys to convert geometrical classification problems to topological
+ones. The typical example is of course positive scalar curvature condition and this approch was
+initiated by Gromov and Lawson [6].
+We will prove in the first part of this paper that the condition Riem > t satisfies all the three
+conditions. In the second part we shall show that the condition Riem > t imposes several restriction
+on the topology of the manifold.
+1.2. Statement of the main results of the paper. The first result shows that the condition
+Riem > t satisfies an ideal property with respect to Cartesian products
+Proposition 1.1. Let M1 and M2 be compact manifolds, then
+Riem(M1 × M2) ≥ max{Riem(M1), Riem(M2)}.
+The above property is still true for general Riemannian submersions. In particular, we show
+that the condition Riem > t satisfies a fibre bundle property
+Theorem A. Let π : M → B be a fibre bundle with fibre F of dimension p and structure group G.
+Suppose M and B are compact and that the fibre F admits a G-invariant metric ˆg with Riem(ˆg) > t
+for some t ∈ [0, p(p − 1)/2). Then Riem(M) > t.
+In particular, if compact manifolds M and N are respectively the total spaces of a CP2 bundle
+and an HP2 bundle with structure group the group of isometries of their Fubiny-Study metrics,
+then Riem(M) > 2 and Riem(N) > 8.
+The next theorem shows the stability under surgeries of the above condition
+Theorem B. Let p, n be integers such that 1 ≤ p ≤ n − 2. Let M be a compact n-manifold with
+p(p−1)
+2
+< Riem(M) ≤ p(p+1)
+2
+. If a compact manifold �
+M is obtained from M by surgeries of codi-
+mensions ≥ p + 2 then Riem(�
+M) ≥ Riem(M).
+One important consequence of the previous results is the following gap theorem
+4
+
+Theorem B′.
+(1) For a compact simply connected manifold M of dimension ≥ 5 one has:
+Riem(M) > 0 =⇒ Riem(M) ≥ 1.
+(2) For a compact 2-connected manifold M of dimension ≥ 7 one has:
+Riem(M) > 0 =⇒ Riem(M) ≥ 3.
+Another consequence of the surgery theorem is the stablity under connected sums of the condi-
+tion Riem(Mn) > t for the values of t between 0 and (n−1)(n−2)/2. The same theorem guarantees
+the stability under surgeries of codimensions n−1 or n of Riem(Mn) > t for the values of t between
+0 and (n − 2)(n − 3)/2. As a consequence of this last result, we prove that for t ∈ [0, (n−2)(n−3)
+2
+),
+the condition Riem(M) > t does not impose any restrictions on the fundamental group of M.
+Corollary B. Let π be a finitely presented group.
+Then for every n ≥ 4 and for every t ∈
+[o, (n−2)(n−3)
+2
+), there exists a compact n-manifold M with Riem(M) > t and π1(M) = π.
+It turns out that for values of t higher than (n−2)(n−3)
+2
+(that is the limit value of t that allows
+surgeries!), the condition Riem(Mn) > t imposes many restrictions on the topology of the manifold.
+The first obstruction result is a vanishing theorem of Betti numbers of the manifold
+Theorem C. Let M be a compact connected manfold with dimension n ≥ 3. Then one has
+Riem(M) > (n − 1)(n − 2)
+2
+=⇒ b1(M) = bn−1(M) = 0.
+Here b1(M) denotes the first Betti number of M. In particular, The standard metric of the product
+Sn−1 × S1 is maximal and one has
+a) Riem(Sn−1 × S1) = (n−1)(n−2)
+2
+.
+b) The connected sum of arbitrary r copies of Sn−1 × S1 satisfies
+Riem(#r(Sn−1 × S1) = (n − 1)(n − 2)
+2
+.
+In this paper we prove generalizations of the above theorem to all the higher Betti numbers.
+In particular, we prove that
+Theorem C′. Let M be a compact connected manfold with dimension n ≥ 4 and bk(M) denotes
+the k-th betti number of M. Then one has
+Riem(M) > n(n − 1)
+2
+− (n − 2) =⇒ bk(M) = 0, for 1 ≤ k ≤ n − 1.
+In particular, M is a homology sphere.
+The next theorem is a weaker version of a theorem due to B¨ohm and Wilking [3] and a second
+theorem due to Ni-Wu [14]. It shows in particular that no compact n-manifolds, n ≥ 3, can have
+their Riem within the interval (N − 2, N) where N = n(n − 1)/2.
+Theorem D. Let M be a compact manifold of dimension n ≥ 3.
+5
+
+(1) if Riem(M) > n(n−1)
+2
+− 2 then Riem(M) = n(n−1)
+2
+.
+Moreover, M has a metric g with constant sectionnal curvature and Riem(M) = Riem(g).
+(2) If Riem(M) = n(n−1)
+2
+− 2 = Riem(g) for some Riemannian metric g on M, then either M
+is locally symmetric or its universal cover is isometric to a product.
+The following theorem shows that taking a cartesian product with a torus does not increase the
+Riem. This result combined with the ideal property allowed us to compute the Riem invariant
+for products of spheres with tori. Precisely we have
+Theorem E. Let n ≥ 3 and 1 ≤ p ≤ n − 1 be two integers. Let N n−p be an arbitrary compact
+manifold of dimension n − p and T p be the p-dimentional torus. Then for n ≤ 7 one has
+Riem(N n−p × T p) ≤
+�n − p
+2
+�
+.
+In particular, if Sn−p is a positive spherical space form then one has
+Riem(Sn−p × T p) =
+�n − p
+2
+�
+.
+The proof of the above theorem is based on a recent result by Brendle-Hirsch-Johne [4].
+1.3. Plan of the paper. Theorem A is proved in section 3 and Theorems B and B′ in section 4.
+Theorems C and C′ are proved in section 5 and Theorems D and E in section 2.
+In the last section 6, we first show as a consequence of Theorem E that the product S2 × T p,
+for p ≤ 5, does not admit any Riemannian metric with both σ1(R) > 0 and σ2(R) > 0. Where
+σk(R) denotes the k-th elementary symmetric function in the eigenvalues of the Riemann curvature
+operator. Next, we introduce an analogous but weaker smooth invariant, namely riem invariant.
+We prove that it remains unchanged under surgeries in most of codimensions. Then we briefly
+define the Riem of a conformal class of metrics, and we determine the Riem of the conformal class
+of the product metric on the product of two space forms with opposite signs. At the end of this
+section we included some open related questions.
+2. Relations with other curvature conditions: Proof of Theorems D and E
+Let M be a compact manifold and g a Riemannian metric on M.
+If dim M = 2, then Riemt(g) = (1 − t)Scal g2
+2 is determined by the scalar curvature and Riem(M)
+is either 0 or 1. In 3 dimensions, it is determined by the Ricci curvature. It is easy to see that
+in three dimensions the eigenvalues of Riemt(g) are equal to (1 − t)Scal + 2tρi, for i = 1, 2, 3, and
+where ρi is an eigenvalue of Ricci curvature operator. In particular, Riem(M) > 1 =⇒ Ric > 0.
+Consequently, by the classification of PSC compact 3-dimensional manifolds, Riem(M) can take
+only the values 0, 1 and 3.
+In higher dimensions, still we don’t know whether Riem takes only integer values. As a first step in
+6
+
+this direction, we will show in this paper that there are gaps in the range of Riem in all dimensions.
+The next proposition shows a relation between our curvature condition and positive Ricci
+curvature condition and k-positive Riemann tensor.
+Proposition 2.1. Let g be a Riemannian metric on a compact n-manifold and let N = n(n−1)/2.
+(1) For any integer t ∈ (0, N), the tensor Riemt(g) is t-positive (resp. t-nonnegative) if and only
+if the Riemann curvature operator R of g is (N − t)-positive (resp. (N − t)-nonnegative).
+In particular,
+• The Riemann curvature operator R is positive (resp.
+nonnegative) if and only if
+Riem(N−1)(g) is (N − 1)-positive (resp. (N − 1)-nonnegative).
+• The Riemann curvature operator R is k-positive (resp. k-nonnegative) if and only if
+Riem(N−k)(g) is (N − k)-positive (resp. (N − k)-nonnegative).
+(2) Riem(g) > (n−1)(n−2)
+2
+=⇒ Ric > 0.
+(3) Riem(g) = (n−1)(n−2)
+2
+=⇒ Ric ≥ 0.
+Proof. To prove the first part, we take the sum of arbitrary t eigenvalues of Riemt(g) and get
+tScal − 2t
+�
+i∈I
+λi = 2t
+N
+�
+i=1
+λi − 2t
+�
+i∈I
+λi = 2t
+�
+i∈Ic
+λi,
+where N = n(n − 1)/2, I is a set of indices in {1, 2, ..., N} of length t and λi are the eigenvalues
+of the Riemann curvature operator. For the second part, let e1 be a given unit tangent vector and
+complete by {e2, ..., en} to get an orthonormal basis of the corresponding tangent space, then for
+t = (n−1)(n−2)
+2
+one has
+�
+i,j̸=1
+Riemt(ei, ej, ei, ej) = (n − 1)(n − 2)Scal − 2t
+��
+Scal − 2Ric(e1, e1)
+�
+= 4tRic(e1, e1).
+Finally, if Riem(g) = (n−1)(n−2)
+2
+then it is easy to show that Riemt(g) ≥ 0 for t = (n−1)(n−2)
+2
+and
+the last result follows.
+□
+2.1. Proof of Theorem D.
+Proof. Let N =
+n(n−1)
+2
+.
+By assumption there exists a Riemannian metric g on M such that
+Riem(g) > N − 2. Then RiemN−2(g) > 0, in particular it is (N − 2) − positive. Proposition
+2.1 shows then that the the Riemann curvature operator is 2-positive. The theorem follows from
+B¨ohm-Wilking Theorem 1 in [3].
+For the second part, let Riem(M) = n(n−1)
+2
+− 2 = Riem(g), for some Riemannian metric g on
+M. Consequently one has RiemN−2(g) ≥ 0. Proposition 2.1 shows that the Riemann curvature
+operator is 2-nonnegative. Then by a theorem of Ni-Wu [14], M is locally symmetric or its universal
+cover is isometric to a product.
+□
+7
+
+2.2. Positive Cp intermediate curvature condition: Proof of Theorem E. Brendle, Hirch
+and Johne defined in [4] the notion of p-intermediate curvature denoted Cp. It is defined on the
+Grassmannian of tangent p-planes of a Riemannian manifold as follows. For a tangent p-plane P,
+let {e1, ..., ep} be any orthonormal basis of P and set
+Cp(P) :=
+p
+�
+i=1
+n
+�
+j=i+1
+Sec(ei, ej),
+where Sec(ei, ej) denotes the sectional curvature of the plane spanned by {ei, ej}.
+A simple manipulation shows that
+Cp(P) =
+n
+�
+i=1
+n
+�
+j=i+1
+Sec(ei, ej) −
+n
+�
+i=p+1
+n
+�
+j=i+1
+Sec(ei, ej)
+=
+n
+�
+i,j=1
+i<j
+Sec(ei, ej) −
+n
+�
+i,j=p+1
+i<j
+Sec(ei, ej)
+=1
+2Scal − 1
+2sp(P),
+(9)
+where sp(P) is the p-curvature of the plane P [11].
+The next theorem shows the existence of a metric with positive Cp curvature on a given n-manifold
+M provided that Riem(M) > (n−p)(n−p−1)
+2
+.
+Theorem 2.2. Let M be a compact manifold of dimension n ≥ 3 and p be an integer such that
+1 ≤ p ≤ n − 1. If the manifold M has no Riemannian metrics with positive Cp curvature then
+Riem(M) ≤
+�n−p
+2
+�
+.
+Proof. Suppose that Riem(M) >
+�n−p
+2
+�
+. Then M has a Riemannian metric g with Riemt(g) > 0
+with t = (n−p)(n−p−1)
+2
+. We shall prove that the Cp curvature of g is positive.
+Recall that
+1
+2sp is the sectional curvature of the tensor ∗ gn−p−2
+(n−p−2)!R, here we are using double
+forms formalism [10], in particular ∗ is the double Hodge star operator acting on double forms.
+Consequently, the Cp curvature is the sectional curvature of the tensor
+1
+2Scalgp
+p! − ∗
+gn−p−2
+(n − p − 2)!R =1
+2Scal(∗ gn−p
+(n − p)!) − ∗
+gn−p−2
+(n − p − 2)!R
+= ∗
+�
+1
+(n − p)(n − p − 1)
+gn−p−2
+(n − p − 2)!
+�
+Scalg2
+2 − (n − p)(n − p − 1)R
+��
+=2
+t ∗
+�
+gn−p−2
+(n − p − 2)!Riemt(g)
+�
+=
+2
+t(n − p − 2)!
+�
+cn−p−2�
+∗ Riemt(g)
+�
+�
+.
+(10)
+Where t = (n−p)(n−p−1)
+2
+and c denotes the contraction map of double forms. In the last step we used
+the identity ∗(gkω) = ck(∗ω) which is valid for any symmetric double form ω, see [10]. In particular,
+8
+
+the Cp curvature is up to a positive factor the sectional curvature of the p-curvature tensor of Riemt.
+Now, it is easy to see that the positivity of Riemt implies the positivity of ∗Riemt and the positivity
+of all contractions of Riemt. The positivity of Cp follows as the sectionnal curvature of a positive
+symmetric double form. This completes the proof.
+□
+2.3. Proof of Theorem E.
+Proof. For n ≤ 7, the product N n−p × T p has no Riemannian metrics with positive Cp curvature
+by a recent result of Brendle-Hirch-Johne in [4]. The above theorem 2.2 in this paper implies that
+Riem(N n−p×T p) ≤ (n−p)(n−p−1)
+2
+. To prove the second part recall that the Riem of standard metric
+on Sn−p equals (n−p)(n−p−1)
+2
+. The ideal property of proposition 1.1 shows that Riem(Sn−p ×T p) ≥
+(n−p)(n−p−1)
+2
+. This completes the proof.
+□
+Remark.
+(1) The natural question whether the previous Theorem E remains true for all n ≥ 8,
+is an open question. This is clearly true for p = 1 and p = n − 1 by Theorem C.
+(2) The previous theorem shows clearly an interaction between Gromov’s macroscopic dimen-
+sion of the universal cover of the manifold and its Riem invariant.
+This suggests the
+following conjecture
+Let dimmc(M) denotes the macroscopic dimension of the universal cover of a compact man-
+ifold M and d be an integer such that 1 ≤ d ≤ n − 1. Then
+Riem(M) > (n − d)(n − d − 1)
+2
+=⇒ dimmc(M) < d.
+This is a generalization of Gromov’s conjecture obtained for d = n − 1.
+3. The Riem of total spaces of Riemannian submersions: Proof of Theorem A and
+Proposition 1.1
+3.1. Proof of Proposition 1.1.
+Proof. We will show that the Riem of the Cartesian product of two manifolds M1 and M2 cannot
+be less then the Riem of M1 or M2.
+If Riem(M1) = Riem(M2) = 0, the result is trivial. If Riem(M1) > 0, then M1 possesses a metric
+g1 with Riem(g1) > 0. Then one can amplify the metric g1 by multiplying it by t > 0 and use the
+compactness to show that Riem(M1 × M2) ≥ Riem(M1). The proof can be then completed easily
+after considering the cases Riem(M2) = 0 and Riem(M2) > 0.
+□
+3.2. Proof of Theorem A. Theorem A is a special case of the following more general theorem
+Theorem 3.1. Let M be compact and be the total space of a Riemannian submersion π : (M, g) →
+(B, ˇg) with totally geodesic fibers Fx, x ∈ B. Let p = dim Fx and k ∈ [0, p(p − 1)/2). Suppose that
+the induced metric on the fibers satisfy Riem(g|Fx) > k for all x ∈ B then Riem(M) > k.
+9
+
+Proof. We use the canonical variation gt of the metric g, that is the metric obtained by re-scaling
+g in the vertical directions by t2. We shall prove that there exists t > 0 such that Riem(gt) > k,
+and consequently one has Riem(M) > k as desired.
+Let U, V, W, W ′ be vertical vectors of gt-length 1 and X, Y, Z, Z′ be arbitrary forizontal vectors. We
+shall index by t all the invariants of the metric gt and put under a hat the invariants of the fibers
+with the induced metric. We omitt the index t in case t = 1. O’Neill’s formulas for Riemannian
+submersions show that, see for instance chapter 9 in [1] or chapter 2 in [12]
+Rt(U ∧ V, W ∧ W ′) = t2 ˆR(U ∧ V, W ∧ W ′),
+Rt(U ∧ V, W ∧ X) = 0,
+Rt(X ∧ U, Y ∧ V ) = O(1),
+Rt(U ∧ V, X ∧ Y ) = O(1),
+Rt(X ∧ Y, Z ∧ U) = o(t),
+Rt(X ∧ Y, Z ∧ Z′) = O(1).
+Here the Riemann tensor is seen as a (2, 2) double form, ˆR is the double form associated to the
+Riemann tensor of the fibre metric ˆg = g|Fx.
+Let now φ be any 2-form of gt-unit length in ∧2M and denote by ˆφ its pointwise ortogonal projection
+onto ∧2F, then the above formulas show that
+Rt(φ, φ) = t2 ˆR(ˆφ, ˆφ) + O(1).
+Suppose the maximum eigenvalue of the curvature operator Rt is λt
+max = Rt(φ, φ) for some gt-unit
+length 2-form φ. Then one has at each point of M the following
+λt
+max = 1
+t2 ˆR(t2 ˆφ, t2 ˆφ) + O(1) ≤ 1
+t2 ˆλmax + O(1).
+Here ˆλmax denotes the maximum eigenvalue of the curvature operator ˆR. In the last argument, we
+used the fact that ||t2 ˆφ|| = ||ˆφ||t ≤ ||φ||t = 1.
+To complete the proof just remark that
+Scal(gt)
+2λtmax
+≥ Scal(ˆg) + O(t2)
+2ˆλmax + O(t2)
+.
+Where we used the fact that Scal(gt) =
+Scal(ˆg)
+t2
++ O(1).
+Consequently, at each point of M, if
+k < Scal(ˆg)
+2ˆλmax then there exists t > 0 such k < Scal(gt)
+2λtmax . We conclude using the compactness of M.
+□
+4. The Riem invariant and surgeries: Proof of Theorems B, B′ and Corollary B
+4.1. Proof of Theorem B. The following theorem is a reformulation of Theorem B.
+Theorem 4.1. Let M be a compact n-manifold with 0 < Riem(M) ≤ (q−1)(q−2)
+2
+for some integer
+q such that 3 ≤ q ≤ n. If a compact manifold �
+M is obtained from M by surgeries of codimensions
+10
+
+≥ q then Riem(�
+M) ≥ Riem(M).
+Proof. The theorem follows directly from Hoelzel’s general surgery theorem [7]. We use the same
+notations as in [7]. Let CB(Rn) denote the vector space of algebraic curvature operators Λ2Rn →
+Λ2Rn satisfying the first Bianchi identity and endowed with the canonical inner product.
+For
+0 < t < n(n − 1)/2, let
+CRiemt>0 := {R ∈ CB(Rn) : Riemt(R) > 0},
+here Riemt(R) = Scal(R) − 2tR. The subset CRiemt>0 is clearly open, convex and it is an O(n)-
+invariant cone. Furthermore, it is easy to check that Riemt(Sq−1 × Rn−q+1) > 0 if t < (q−1)(q−2)
+2
+and q ≥ 3. This completes the proof.
+□
+We come now to the proof of Theorem B as follows.
+Proof. We use the same notations as in the proof of the above theorem. From one side the condition
+0 < 2t < (q − 1)(q − 2) is equivalent to 2q > 3 + √1 + 8t and q ≥ 3. From the other side, the
+condition p(p−1)
+2
+< Riem(M) ≤ p(p+1)
+2
+implies the existence of a Riemannian metric g such that
+Riemt(g) > 0 for all t such that p(p − 1)/2 ≤ t < Riem(g).These values of t satisfy in particular
+the following inequalities
+p(p − 1) ≤ 2t < p(p + 1),
+(2p − 1)2 ≤8t + 1 < (2p + 1)2,
+p + 1 ≤3 + √1 + 8t
+2
+< p + 2.
+So clearly the desired condition is q ≥ p + 2.
+□
+The following corollary follows directly from the above Theorem
+Corollary 4.2.
+a) Let M1 and M2 be two compact manifolds of dimensions n ≥ 3 and such
+that 0 < Riem(M1) ≤ (n−1)(n−2)
+2
+and Riem(M2) ≥ Riem(M1). Then their connected sum
+satisfies
+Riem(M1#M2) ≥ Riem(M1).
+b) Let M be a compact n-manifold with n ≥ 4 and 0 < Riem(M) ≤
+(n−3)(n−2)
+2
+.
+If a
+compact manifold �
+M is obtained from M by surgeries of codimensions ≥ n − 1 then
+Riem(�
+M) ≥ Riem(M).
+4.2. Proof of Corollary B.
+Proof. Corollary B results from the above corollary 4.2. The proof follows word by word the proof
+of an analogous result for the Ein invariant in [8]. We will not reproduce it here.
+□
+11
+
+4.3. Proof of Theorem B′.
+Proof. We prove the first part as follows. Let M be a compact simply connected spin (resp. non-
+spin) manifold of dimension ≥ 5. Gromov and Lawson [6] proved that if M is spin cobordant (resp.
+oriented cobordant) to a compact manifold M1 then M can be obtained from M1 by surgeries of
+codimensions ≥ 3.
+From another side, according to a result by F¨uhr [5], a closed oriented manifold of dimension ≥ 5 is
+always oriented cobordant to the total space M1 of CP2 bundle with structure group the isometry
+group of the Fubiny-Study metric of CP2. Theorem A shows that Riem(M1) ≥ 2. It follows from
+the surgery theorem B that Riem(M) ≥ 1. This completes the proof for the non-spin case.
+If M is spin with Riem > 0 then it has a positive scalar curvature metric. A theorem of Stolz [17]
+guarantees that M is spin cobordant to the total space M1 of an HP2 bundle with structure group
+the isometry group of the Fubiny-Study metric of HP2. Theorem A shows that Riem(M1) ≥ 8
+and then Riem(M) ≥ 1 by the surgery theorem B. This completes the proof of the first part.
+Next we prove the second part of the Theorem. Let M be compact and 2-conncted with Riem > 0
+and dimension ≥ 7. Then M has a canonical spin structure and has a metric with positive scalar
+curvature. The above mentioned theorem of Stolz shows that M is spin cobordant to a manifold
+M1 with Riem(M1) ≥ 8. On the other hand, a previous result of the author [11] asserts that M
+can be obtained from M1 by surgeries of codimension ≥ 4. The surgery theorem B shows that
+Riem(M) ≥ 3. This completes the proof of the Theorem.
+□
+5. Vanishing theorems: Proof of Theorems C and C′
+5.1. Proof of Theorem C.
+Proof. Let s = (n−1)(n−2)
+2
+. The condition Riem(M) > s guarantees the existence of a Riemannian
+metric g on M such that Riems(g) > 0. It follows from Proposition 2.1 that the Ricci curvature
+of g is positive and therefore the Betti numbers b1 and bn−1 of M vanish. As a consequence, one
+immediately has Riem(Sn−1 × S1) ≤ s. To prove equality, we recall that the standard product
+metric has Riem equal to s. This proves the a) part of Theorem C. We prove part b) in a similar
+way. From one side from corollary 4.2 we have Riem
+�
+#r(Sn−1 × S1)
+�
+≥ s. From another side it
+cannot be strictly higher than s because of the above positive Ricci curvature obstruction.
+□
+5.2. Proof of Theorem C′. We prove the following more general version of Theorem C′.
+Theorem 5.1. Let M be a compact connected manfold with dimension n ≥ 3 and let p be an
+integer such that 2 ≤ p ≤ n − 2. We denote by bk(M) as usual the k-th betti number of M, then
+one has
+Riem(M) > n(n − 1)
+2
+− p(n − p)
+2
+=⇒ bk(M) = 0, for p ≤ k ≤ n − p.
+12
+
+Proof. We shall use double forms formalism for the Weitzenb¨ock cuvature term [13]. The curvature
+term in Weitzenb¨ock formula once applied to p-forms takes the form
+Wp =
+gp−1
+(p − 1)!Ric − 2 gp−2
+(p − 2)!R.
+On the other hand we have
+gp−2
+(p − 2)!Riemt =
+gp−2
+(p − 2)!
+�
+Scalg2
+2 − 2tR
+�
+= p(p − 1)Scal
+2
+gp
+p! − 2t gp−2
+(p − 2)!R.
+Consequently, for t > 0 we get
+tWp =
+gp−2
+(p − 2)!Riemt +
+gp−1
+(p − 1)!
+�
+tRic − (p − 1)Scal
+2
+g
+�
+.
+We will show that under the theorem hypothesis where t > n(n−1)
+2
+− p(n−p)
+2
+the second term in
+the last sum is posiitve as well. This amounts to proving that the sum of the lowest p eigenvalues
+of tRic − (p−1)Scal
+2
+g is positive. Recall that the condition Riem(M) > t implies the existence of a
+Riemannian metric on M with Riemt > 0. After taking the trace, one can see that (n − 1)Scal g −
+2tRic > 0. Taking the sum of (n − p) eigenvalues of the later, we see that
+(n − 1)(n − p)Scal − 2t(Scal −
+�
+ı∈I
+ρi) > 0,
+where ρi denotes the eigenvalues of Ricci and I ⊂ {1, 2, ..., n} is any subset of indices of length p.
+Therefore, we get
+2t
+�
+ı∈I
+ρi − p(p − 1) > 2t − (n − 1)(n − p) − p(p − 1) > 0.
+This completes the proof of the theorem.
+□
+Remark. Let t = (n−1)(n−2)
+2
++ p. The condition Riem(M) > t implies the existence of a Rie-
+mannian metric on M with Riemt > 0. Proposition 2.1 implies then that the Riemann curvature
+tensor of g is (n(n−1)
+2
+− t)-positive, that is (n − (p + 1))-positive. Since p ≤ n−2
+2
+then p + 1 ≤ n/2
+and therefore a vanishing theorem of Petersen-Wink [16] shows the vanishing of Betti numbers
+bk(M) = 0 for 1 ≤ k ≤ p + 1 and n − p − 1 ≤ k ≤ n − 1.
+6. Miscellaneous results
+6.1. Positive Γ2(R) curvature. Let σ1(R) and σ2(R) be the first two elementary symmetric
+functions in the eigenvalues of the Riemann curvature operator. If σ1(R) > 0 and σ2(R) > 0, we
+shall write Γ2(R) > 0. Note that σ1(R) = Scal(R)
+2
+and 8σ2(R) = Scal2(R) − 4||R||2. In particular,
+Γ2(R) > 0 ⇐⇒ Scal(R) > 2||R||.
+The following theorem shows in particular that the product S2 × T p has positive scalar curvature
+but does not allow at the same time both Scal > 0 and σ2(R) > 0.
+13
+
+Theorem 6.1. If a Riemannian manifold (M, g) has positive Γ2(R) curvature then Riem(M) > 1.
+In particular, the product S2 × T p does not support any metric with positive Γ2(R) curvature (at
+least for p ≤ 5).
+Proof. We remark that the first Newton transformation of the curvature operator is t1(R) = Scal
+4 g2−
+R = 1
+2(Riem1(g). It is classic that if σ1 and σ2 of an operator are positive then its first Newton
+transformation is positive. Then Riem1(g) > 0, that is Riem(g) > 1. This cannot take place for
+the products S2 × T p with p ≤ 5 by Theorem E. This completes the proof.
+□
+Remark.
+• The author belives that Dirac operator techniques may help to prove that the
+positivity of Scal − 2||R|| is not allowed on products of S2 × T q, for all q, these are some
+how partially enlargeable manifolds.
+• The previous theorem is still true for products of a compact surface with a torus T p again
+by theorem E.
+6.2. The small Riem invariant. For a fixed Riemannian metric g on a compact n-manifold M
+and for s < t < 0, the tensors Riemt(g) enjoy the following descent propoerty
+Riems > 0 =⇒ Riemt > 0 =⇒ Scal > 0.
+We therefore define the metric invariant
+riem(g) := inf{t < 0 : Riemt(g) > 0}.
+We set it equal to −∞ if the above set is unbounded below and equal to zero if that set is empty.
+It is not difficult to see that riem(g) = −∞ if and only if the Riemann tensor R is nonnegative and
+with positive scalar curvature.
+We define the smooth invariant riem(M) := inf{riem(g): g ∈ M}, where M denotes the space
+of all Riemannian metrics on M. It is remarkable that this invariant remains unchanged after
+surgeries, precisely we have
+Theorem 6.2. Let M be a compact manifold of dimension n ≥ 4. If a compact manifold �
+M is
+obtained from M by surgeries of codimensions ≥ 3 but not equal to n−1 then riem(�
+M) = riem(M).
+Proof. Using Hoelzel’s surgery theorem [7], it is easy, as in the proof of Theorem B, to see that
+riem(�
+M) ≥ riem(M) as far as the codimension of the surgery is ≥ 3. To prove equality we apply
+a reversed surgery as in [15]. In fact one can recover the initial manifold M from the new manifold
+�
+M by applying a surgery of codimension n − q + 1. Consequently one gets riem(M) ≥ riem(�
+M)
+if the new codimension n − q + 1 ≥ 3. Consequently, the riem is unchanged if 3 ≤ q ≤ n − 2. The
+case of q = n can be ruled out, as in this cas �
+M is diffeomorphic to the connected sum of M with
+the product S1 × Sn−1. One can then recover M by killing the circle by the mean of a surgery of
+codimension only n − 1 ≥ 3. This completes the proof.
+□
+14
+
+One consequence of this theorem is that simply connected compact PSC manifolds of dimen-
+sions ≥ 5 have their riem equal to −∞.
+6.3. The Riem invariant of a conformal class. Let [g] denotes the conformal class of the metric
+g. We define the Riem of the conformal class [g] as
+Riem([g]) = sup{Riem(g) : g ∈ [g]}.
+As above, we set it equal to zero if the conformal class does not contain a psc metric. We prove
+the following vanishing theorem and a consequence of it which determines the conformal Riem for
+some conformally flat classes.
+Theorem 6.3. Let (M, g) be a compact oriented conformally flat n-manifold and p an integer such
+that 0 < p ≤ n/2. Then one has
+Riem([g]) > (n − 1)(n − 2p)
+2
+=⇒ bk(M) = 0, for p ≤ k ≤ n − p.
+In particular, for any n > 2p ≥ 2, let g0 denotes the product metric on the product of two space
+forms of opposite signs Sn−p × Hp, then one has
+Riem([g0]) = Riem(g0) = (n − 1)(n − 2p)
+2
+.
+Proof. In what follows, products of tensors are Kulkarni-Nomizu products. Recall tor a conformally
+flat metric g one has R = gA, where A is the Schouten tensor. Consequently, the Riemt tensor is
+determined by the Ricci tensor as follows
+(n − 2)Riemt(g) =
+�(n − 1)(n − 2) + 2t
+n − 1
+�
+Scalg2
+2 − 2tg Ric
+=
+�(n − 1)(n − 2) + 2t
+2(n − 1)
+�
+g EinT .
+(11)
+Here, EinT = Scalg − T Ric and T =
+4t(n−1)
+(n−1)(n−2)+2t. At this stage we use a vanishing theorem for
+the Ein invariant [9], which gurantees the vanishing of the bk(M) as in the Theorem that we are
+proving under the condition that T > (n−1)(n−2p)
+n−p−1
+. It is straighforward to see that this last condition
+is equivalent to t > (n−1)(n−2p)
+2
+. This proves the first part. For the second part, note that from one
+hand the p-th Betti number of the above product is not zero therefore Riem([g0]) ≤ (n−1)(n−2p)
+2
+. On
+the other hand the product metric satisfies Riem(g0) = (n−1)(n−2p)
+2
+. This completes the proof.
+□
+6.4. Minimal vs. Maximal PSC compact manifolds: open questions. The smooth Riem
+invariant defines a pre-order on the set of all compact PSC manifolds with a fixed dimenion n. The
+maximal manifolds are by B¨ohm-Wilking theorem (see Theorem D) the manifolds with constant
+positive sectional curvature (space forms). In the other extreme, one may ask the following ques-
+tions: What are the PSC compact n-manifolds with minimal Riem ?.
+What are the PSC compact
+simply connected (resp. 2-connected) manifolds with minimal Riem ?.
+For instance, these are more specefic questions:
+15
+
+�� Are there compact manifolds with 0 < Riem < 1? Is S2×T n−2 minimal among all compact
+PSC n-manifolds?
+• Are there compact simply connected manifolds with 0 < Riem < 2?
+The problem is very well understood in dimension 3. In fact, it results from the classification of
+compact PSC 3-manifolds that the minimal PSC manifolds are those with Riem = 1 and they are
+either S2 × S1 or connected sums of copies of the later with spherical space forms.
+It would be interesting as well to identify PSC manifolds with fixed Riem = k, for some fixed
+intermediate k ∈ (0, n(n − 1)/2).
+6.4.1. Best PSC Riemannian metrics on a given PSC manifold. Let M be a PSC compact n-
+manifold. We shall say that a Riemannian metric g on M is a best PSC metric if Riem(g) =
+Riem(M). We have seen in this paper several examples where the standard metrics are the best
+ones. The following questions seems to the author natural and legitimate to ask:
+• Which compact PSC manifolds have best PSC Riemannian metrics?
+• Is there any variational characterization of these best metrics?
+References
+[1] Besse A. L., Einstein Manifolds, Springer, Berlin-New York (1987).
+[2] J.-P. Bourguignon, H. Karcher, Curvature operators: pinching estimates and geometric examples, Ann. Sci. E.N.S.
+Paris, 11 (1978), 71–92.
+[3] B¨ohm C., Wilking B., Manifolds with positive curvature operators are space forms, Annals of Mathematics, 167,
+1079-1097, (2008).
+[4] Brendle S., Hirsch S., Johne F., A generalization of Geroch’s conjecture, arXiv:2207.08617 [math.DG], (2022).
+[5] Sven F¨uhring S., Bordism and projective space bundles, arXiv:2006.15394 [math.GT], (2020).
+[6] Gromov M. and Lawson H. B., The classification of simply connected manifolds of positive scalar curvature, Ann.
+of Math. 111, (1980), 423-434.
+[7] Hoelzel S., Surgery stable curvature conditions, Math. Ann. 365, 13-47 (2016).
+[8] Labbi M. L., On modified Einstein tensors and two smooth invariants of compact manifolds, Transactions of the
+American Mathematical Society, to appear, (2023).
+[9] Labbi M. L., On the conformal Ein invariants, arXiv:2009.11601v2 [math.DG] (2020)
+[10] Labbi M. L., Double forms, curvature structures and the (p, q)-curvatures. Transactions of the American Math-
+ematical Society 357 (10), 3971-3992, (2005).
+[11] Labbi M.L., Stability of the p-curvature positivity under surgeries and manifolds with positive Einstein tensor,
+Annals of Global analysis and geometry, 15: 299-312, 1997.
+[12] Labbi M. L., Vari´et´es riemanniennes `a p-courbure positive, PhD thesis, Montpellier University (1995).
+[13] Labbi M., On Weitzenbock curvature operators, Mathematische Nachrichten 288 (4), 402-411 (2015).
+[14] Ni L. and Wu B., Complete manifolds with nonnegative curvature operator, Proceedings of the American Math-
+ematical Society, 135, 9, 3021-3028, (2007).
+[15] Petean J., Computations of the Yamabe invariant, Mathematical Research Letters 5, 703–709 (1998).
+[16] Petersen P., Wink M., New curvature conditions for the Bochner Technique, Invent. math. 224, 33–54, (2021).
+[17] Stolz S., Simply connected manifolds of positive scalar curvature. Ann. of Math. (2) 136 (1992), no. 3, 511-540.
+16
+
+Department of Mathematics, College of Science, University of Bahrain, 32038, Bahrain.
+Email address: [email protected]
+17
+

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@@ -0,0 +1,445 @@
+Pion condensation at lower than physical quark masses
+Bastian B. Brandt,𝑎 Volodymyr Chelnokov,𝑏,∗ Francesca Cuteri𝑏 and Gergely
+Endrődi𝑎
+𝑎Institute for Theoretical Physics, University of Bielefeld,
+D-33615 Bielefeld, Germany
+𝑏Institut für Theoretische Physik, Goethe-Universität Frankfurt
+Max-von-Laue-Str. 1, 60438 Frankfurt am Main, Germany
+E-mail: [email protected], [email protected],
+[email protected], [email protected]
+In QCD at large enough isospin chemical potential Bose-Einstein Condensation (BEC) takes
+place, separated from the normal phase by a phase transition. From previous studies the location
+of the BEC line at the physical point is known. In the chiral limit the condensation happens
+already at infinitesimally small isospin chemical potential for zero temperature according to chiral
+perturbation theory. The thermal chiral transition at zero density might then be affected, depending
+on the shape of the BEC boundary, by its proximity. As a first step towards the chiral limit, we
+perform simulations of 2+1 flavors QCD at half the physical quark masses. The position of the
+BEC transition is then extracted and compared with the results at physical masses.
+The 39th International Symposium on Lattice Field Theory (Lattice2022),
+8-13 August, 2022
+Bonn, Germany
+∗Speaker
+© Copyright owned by the author(s) under the terms of the Creative Commons
+Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0).
+https://pos.sissa.it/
+arXiv:2301.08607v1  [hep-lat]  20 Jan 2023
+
+Pion condensation at lower than physical quark masses
+Volodymyr Chelnokov
+1.
+Introduction
+The phase structure of QCD continues to be an object of great interest for both experimental
+and theoretical studies. There is currently a wealth of numerical simulation results at zero quark
+density, from which we know, in particular, that the chiral symmetry restoration at physical quark
+masses is a continuous crossover [1]. Studying the theory in the chiral limit is complicated by the
+appearance of zero modes of the Dirac operator, and the need to go to the continuum limit to restore
+chiral symmetry with Wilson or staggered fermions, or to employ computationally much costlier
+chiral symmetric fermion discretizations. Nevertheless, there is a recent progress in this region
+which, in particular, suggests that in the limit 𝑚ud = 0 the chiral transition is of second order (for
+𝑁 𝑓 = 2, 3) and becomes a crossover at arbitrarily small nonzero quark masses [2].
+Extending the QCD phase diagram to nonzero densities is confounded by the sign problem
+– the Boltzmann weight of the lattice configuration becomes complex at nonzero 𝜇𝐵, preventing
+straightforward use of importance sampling. Different ways to overcome this problem were proposed
+(see [3, 4] for a review), but still the problem is far from being solved.
+Another possible extension of the QCD phase diagram is to the region of nonzero isospin
+density at 𝜇𝐼 ≠ 0, potentially relevant for early Universe cosmology [5], for instance. At pure
+isospin chemical potential 𝜇𝐼 ≠ 0, 𝜇𝐵 = 0, the theory is sign problem-free and the standard Monte-
+Carlo simulations can be performed [6]. The recent study [7] shows that the deconfinement transition
+remains a crossover for 𝜇𝐼 ≤ 𝑚 𝜋/2, until it intersects with the second order pion condensation line,
+which is found to be approximately vertical at small enough 𝑇 (see Figure 1). The 𝜇𝐼-𝑇 plane can
+also be used to check the methods targeted at studying QCD at finite 𝜇𝐵 – since we can compare
+Figure 1: The QCD phase diagram in 𝑇-𝜇𝐼 plane at physical quark masses, obtained in [7].
+2
+
+Pion condensation at lower than physical quark masses
+Volodymyr Chelnokov
+their results with the results of direct simulation, that is possible here [8]. Additionally, the ability
+to numerically sample the theory in the 𝜇𝐼-𝑇 plane can be exploited to perform the reweighting to
+nonzero 𝜇𝐵 [9].
+The phase diagram shown in Figure 1 has an interesting implication concerning the chiral
+phase diagram. At zero temperature, the pion condensation happens at 𝜇𝐼 = 𝑚 𝜋/2. When the
+light quark mass goes to zero, the pion mass also goes to zero as 𝑚 𝜋 ∼ √𝑚ud according to chiral
+perturbation theory. Thus at least at zero temperature in the chiral limit pion condensation happens
+already at arbitrarily small isospin chemical potential. From Figure 1 we see that at physical mass
+the isospin transition line remains vertical up until it meets the chiral crossover line. If that holds
+also at 𝑚ud = 0, then in the chiral limit the pion condensation line would lie on the 𝜇𝐼 = 0 axis up
+to 𝑇 = 𝑇𝑐 – i.e. up to the chiral phase transition temperature. In this scenario the pion condensate
+in the chiral limit at 𝑇 = 𝑇𝑐 exists at arbitrarily small isospin chemical potential, affecting the chiral
+phase transition at zero chemical potential (Figure 2, left). Alternatively, the phase transition line
+can start bending towards larger 𝜇𝐼 as the quark masses are reduced, resulting in a different phase
+structure (Figure 2, right). Previous studies using the Nambu-Jona-Lasinio model [10], as well as
+the functional renormalization group [11], support the first phase picture. To verify this, direct
+Monte Carlo QCD simulations are necessary.
+Figure 2: Two possibilities for the phase diagram in 𝑇-𝜇𝐼 plane at the chiral limit.
+As a first step, in this work we perform the simulation of QCD at nonzero isospin density for
+𝑚ud = 𝑚ud,phys/2, comparing the position of the pion condensation transition with the results at
+physical masses at a series of temperatures.
+2.
+Simulation setup
+We study 2 + 1-flavour QCD using staggered fermions on a 243 × 8 lattice. The setup is the
+same as the one used in [7], with the light quark masses changed to half their physical value. The
+partition function of the theory has the form
+Z =
+∫
+D𝑈𝜇 𝑒−𝛽𝑆𝐺 (det Mud)1/4 (det Ms)1/4 ,
+(1)
+3
+
+deconfinement
+T
+pion condensation
+μIdeconfinement
+pioncondensation
+μIPion condensation at lower than physical quark masses
+Volodymyr Chelnokov
+where 𝑆𝐺 is the tree-level Symanzik improved gauge action, Mud and Ms are, correspondingly,
+light and strange quark matrices
+Mud =
+�
+/𝐷(𝜇𝐼) + 𝑚ud
+𝜆𝜂5
+−𝜆𝜂5
+/𝐷(−𝜇𝐼) + 𝑚ud
+�
+,
+M𝑠 = /𝐷(0) + 𝑚𝑠 .
+(2)
+Here 𝜆 is a pion source term which explicitly breaks the 𝑈𝜏3(1) symmetry of the action, which both
+allows us to see the pion condensate on the finite volume lattice, and, at the same time, improves the
+condition number of the light quark matrix. To get physically meaningful results, the limit 𝜆 → 0
+must be taken.
+The Boltzmann weight defined by (1) is positive, which is obvious for the gauge action and
+the strange quark determinant. The positiveness of the light quark determinant follows from the
+generalized 𝜂5-hermiticity relation that the Dirac operator satisfies,
+𝜂5 /𝐷(−𝜇𝐼)𝜂5 = /𝐷(𝜇𝐼)† .
+(3)
+Now, taking the matrix 𝐵 = diag(1, 𝜂5) in flavour space, that has a unit determinant, we get
+𝐵Mud𝐵 =
+�
+/𝐷(𝜇𝐼) + 𝑚ud
+𝜆
+−𝜆
+� /𝐷(𝜇𝐼) + 𝑚ud
+�†
+�
+,
+det Mud = det (𝐵Mud𝐵) = det
+��� /𝐷(𝜇𝐼) + 𝑚ud
+��2 + 𝜆2�
+> 0 .
+(4)
+To locate the pion condensation onset, we measure the pion condensate
+Σ𝜋 = 𝑚𝑢𝑑
+𝑚2𝜋 𝑓 2𝜋
+�
+𝜋±�
+,
+(5)
+�
+𝜋±�
+= 𝑇
+𝑉
+𝜕 log Z
+𝜕𝜆
+= 𝑇
+2𝑉
+�
+Tr
+𝜆
+| /𝐷(𝜇𝐼) + 𝑚ud|2 + 𝜆2
+�
+,
+(6)
+where the trace can be calculated using noisy estimators. The multiplicative renormalization term
+in (5) depends on the light quark mass 𝑚𝑢𝑑, the pion mass 𝑚 𝜋 that corresponds to 𝑚𝑢𝑑, and the pion
+decay constant 𝑓𝜋. The chiral perturbation theory predicts that 𝑚 𝜋 ∼ √𝑚𝑢𝑑 when the light quark
+mass goes to zero, so assuming this relation holds for physical light quark mass we can approximate
+the renormalization (5) as
+Σ𝜋 = 𝑚𝑢𝑑,phys
+𝑚2
+𝜋,phys 𝑓 2𝜋
+�
+𝜋±�
+,
+(7)
+where 𝑚𝑢𝑑,phys and 𝑚 𝜋,phys are physical light quark and pion masses.
+The pion condensate estimated using Eq. (5) at 𝑇 = 114 MeV, for three different values of 𝜆
+can be seen in Figure 3. We see that even for the smallest value of 𝜆 we are still far from the limiting
+regime 𝜆 → 0. Unfortunately, the simulations at smaller 𝜆 values become prohibitively expensive
+due to very large condition numbers of the light quark matrix, that causes a sharp increase of
+iterations needed for matrix inversion. At even smaller 𝜆 values this results in a loss of convergence
+of the conjugate gradient method.
+4
+
+Pion condensation at lower than physical quark masses
+Volodymyr Chelnokov
+0.2
+0.4
+0.6
+0.8
+1.0
+I/m
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+/mud = 1.01
+/mud = 0.73
+/mud = 0.45
+/mud = 1.01, impr.
+/mud = 0.73, impr.
+/mud = 0.45, impr.
+Figure 3: Comparison of the improved and the unimproved pion condensate at 𝑇 = 114 MeV for three
+different values of 𝜆.
+3.
+Improved pion condensate
+We can improve the convergence to the 𝜆 → 0 limit by using the Banks-Casher type relation
+for the pion condensate (6), that was obtained in [7]
+�
+𝜋±�
+= 𝑇
+2𝑉
+�
+Tr
+𝜆
+| /𝐷(𝜇𝐼) + 𝑚ud|2 + 𝜆2
+�
+= 𝑇
+2𝑉
+�∑︁
+𝑛
+𝜆
+𝜉2𝑛 + 𝜆2
+�
+,
+(8)
+where 𝜉𝑛 is the 𝑛-th singular value of the Dirac operator:
+� /𝐷(𝜇𝐼) + 𝑚ud
+�† � /𝐷(𝜇𝐼) + 𝑚ud
+� 𝜙𝑛 = 𝜉2
+𝑛𝜙𝑛 .
+(9)
+In the infinite volume limit, the summation can be replaced by integration
+�
+𝜋±�
+= 𝜆
+2
+�∫
+d𝜉 𝜌(𝜉)(𝜉2 + 𝜆2)−1
+�
+,
+(10)
+and taking the limit 𝜆 → 0 we get
+�
+𝜋±�
+= 𝜋
+4 ⟨𝜌(0)⟩ .
+(11)
+Thus, the limit 𝜆 → 0 of the pion condensate is proportional to the singular value density at zero.
+Due to the finite size of the lattice the singular values form a discrete set, but we can approximate
+the density by calculating the fraction of the singular values lying in the interval [0, 𝜉], and then
+extrapolating to 𝜉 = 0
+𝜌(0) = 𝑇
+𝑉 lim
+𝜉→0
+𝑛(𝜉)
+𝜉
+,
+(12)
+The behavior of such “averaged densities” for three regimes – finite spectral gap at 𝜇𝐼/𝑚 𝜋 < 0.5,
+𝜌(0) ≈ 0 at 𝜇𝐼/𝑚 𝜋 ≈ 0.5, and 𝜌(0) > 0 at 𝜇𝐼/𝑚 𝜋 > 0.5, is shown on Figure 4.
+5
+
+Pion condensation at lower than physical quark masses
+Volodymyr Chelnokov
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+0.0
+0.5
+1.0
+T
+4V
+n( )
+/m = 0.38
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+0.0
+0.5
+1.0
+T
+4V
+n( )
+/m = 0.77
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+0.0
+0.5
+1.0
+T
+4V
+n( )
+/m = 0.57
+Figure 4: The integrated spectral density dependence on the integration region 𝜉 for 𝜇𝐼 /𝑚 𝜋 < 0.5 (left),
+𝜇𝐼 /𝑚 𝜋 > 0.5 (right), and 𝜇𝐼 /𝑚 𝜋 ≈ 0.5 (bottom) (𝑇 = 132 MeV).
+To perform the extrapolation to 𝜉 = 0 we perform a polynomial fit of the integrated spectral
+density in a given region, obtaining a value of ⟨𝜌(0)⟩ and then take a weighted median of all the
+fit results with the weight exp(−𝜒2
+𝑟), where 𝜒2
+𝑟 is the (correlated) chi-squared statistic of the fit per
+degree of freedom. The error estimate of ⟨𝜌(0)⟩ is then taken as a value Δ𝜌, for which the interval
+[⟨𝜌(0)⟩ − Δ𝜌, ⟨𝜌(0)⟩ + Δ𝜌] contains 0.68 of the total weight of the estimates.
+We can further improve the convergence of the improved pion condensate observable to the
+limit 𝜆 → 0 by approximating ⟨𝜌(0)⟩0 (the expectation value of 𝜌(0) with respect to the 𝜆 = 0
+partition function) using the leading order reweighting, expanding the Boltzmann weight of a given
+configuration in a Taylor series in 𝜆,
+⟨𝜌(0)⟩0 = ⟨𝜌(0)𝑊(𝜆)⟩𝜆
+⟨𝑊(𝜆)⟩𝜆
+,
+𝑊(𝜆) =
+� det M𝑢𝑑,0
+det M𝑢𝑑,𝜆
+�1/4
+≈ exp
+�
+−𝜆𝑉
+2𝑇 𝜋±
+�
+.
+(13)
+In Figure 5 we show the values of improved and unimproved pion condensate at the point close
+to 𝜇/𝑚 𝜋 = 0.5 together with an extrapolation of the improved condensate to 𝜆 = 0. We can see that
+the improved pion condensate value is much smaller and less sensitive to the 𝜆 than the unimproved
+condensate. This picture can be compared with the extrapolaton done in Figure 6 (top) in [7],
+which has a similar behavior for large 𝜆, but also provides enough data points at small 𝜆 to actually
+perform the extrapolation of both the unimproved and improved pion condensates and confirm that
+6
+
+Pion condensation at lower than physical quark masses
+Volodymyr Chelnokov
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+/mud
+0.2
+0.4
+0.6
+0.8
+improved
+unimproved
+Figure 5: Extrapolation of the improved and unimproved pion condensate to 𝜆 = 0 at 𝑇 = 114 MeV,
+𝜇/𝑚 𝜋 = 0.54.
+they reach the same value at 𝜆 = 0. Since in that study the linear extrapolation of the improved pion
+condensate worked well for 𝜆/𝑚𝑢𝑑 ≤ 1.3, we rely on it also in this study.
+4.
+Results and summary
+In Figure 6 we show our preliminary results on the value of the pion condensate and the location
+of the pion condensation line at four different values of the temperature. The interpolation was done
+using a cubic polynomial for the temperatures where we have enough points (𝑇 = 114 MeV, 132
+MeV), and a quadratic polynomial otherwise. Location of the transition point was determined as a
+point where the interpolation line intersects zero. In all cases all points giving a positive value of the
+pion condensate were included in the fit. The error estimates for the fit line and the transition point
+were obtained by taking the standard deviation of the fit result for 100 simulated sets of condensate
+values normally distributed around the “true” values with the known standard deviation.
+The results show that the vertical direction of the pion condensation line is preserved when
+going to smaller light quark masses, preferring the scenario shown in the left panel of the Figure 2:
+the location of the condensation line is compatible with the zero temperature location 𝜇 = 𝑚 𝜋/2.
+As mentioned earlier, these results are based on the linear extrapolation to 𝜆 = 0 from the data
+measured at nonzero values of the pion source parameter. To be able to check the validity of this
+extrapolation, we are currently performing further simulations at smaller values of 𝜆. Additionally,
+a 𝑇 scan at several values of 𝜇𝐼 > 𝑚 𝜋/2 is being performed in order to locate the “horizontal” part
+of the pion condensation line as well.
+7
+
+Pion condensation at lower than physical quark masses
+Volodymyr Chelnokov
+0.2
+0.4
+0.6
+0.8
+/m
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+T=114.37 MeV
+T=123.04 MeV
+T=132.24 MeV
+T=141.96 MeV
+0.3
+0.4
+0.5
+0.6
+0.7
+/m
+120
+130
+140
+T
+Figure 6: The value of the improved pion condensate (left) and the location of the pion condensation point
+(right) for four different values of temperature, obtained from the 243×8 lattice (no continuum extrapolation).
+Acknowledgments
+This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research
+Foundation) – project number 315477589 – TRR 211. The authors acknowledge the use of the
+Goethe-HLR cluster and thank the computing staff for their support.
+References
+[1] Y. Aoki, G. Endrődi, Z. Fodor, S. Katz, K. Szabo, The Order of the quantum chromodynam-
+ics transition predicted by the standard model of particle physics, Nature 443 (2006) 675
+[arXiv:hep-lat/0611014 ]
+[2] F. Cuteri, O. Philipsen, A. Sciarra, On the order of the QCD chiral phase transition for different
+numbers of quark flavours, JHEP 11 (2021) 141 [arXiv:2107.12739]
+[3] P.
+de
+Forcrand,
+Simulating
+QCD
+at
+finite
+density,
+PoS
+LAT2009
+(2009)
+010
+[arXiv:1005.0539]
+[4] K. Nagata, Finite-density lattice QCD and sign problem: Current status and open problems,
+Prog.Part.Nucl.Phys. 127 (2022) 103991 [arXiv:2108.12423]
+[5] V. Vovchenko, B. B. Brandt, F. Cuteri, G. Endrődi, F. Hajkarim and J. Schaffner-Bielich, Pion
+Condensation in the Early Universe at Nonvanishing Lepton Flavor Asymmetry and Its Grav-
+itational Wave Signatures, Phys. Rev. Lett. 126 (2021) no.1, 012701 [arXiv:2009.02309].
+[6] D. Son, M. Stephanov, QCD at finite isospin density, Phys.Rev.Lett. 86 (2001) 592
+[arXiv:hep-ph/0005225]
+[7] B. Brandt, G. Endrődi, S. Schmalzbauer, QCD phase diagram for nonzero isospin-asymmetry,
+Phys.Rev.D 97 (2018) 5, 054514 [arXiv:1712.08190]
+[8] B. Brandt, G. Endrődi, Reliability of Taylor expansions in QCD, Phys.Rev.D 99 (2019) 5,
+014518 [arXiv:1810.11045]
+8
+
+Pion condensation at lower than physical quark masses
+Volodymyr Chelnokov
+[9] B. Brandt, F. Cuteri, G. Endrődi, S. Schmalzbauer, Exploring the QCD phase dia-
+gram via reweighting from isospin chemical potential, PoS LATTICE2019 (2019) 189
+[arXiv:1911.12197]
+[10] L. He, M. Jin, P. Zhuang, Pion Superfluidity and Meson Properties at Finite Isospin Density,
+Phys.Rev.D 71 (2005) 1160001 [arXiv:hep-ph/0503272]
+[11] E. Svanes, J. Andersen, Functional renormalization group at finite density and Bose conden-
+sation, Nucl.Phys.A 857 (2011) 16 [arXiv:1009.0430]
+9
+

1NFAT4oBgHgl3EQfjx2F/content/tmp_files/load_file.txt

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+page_content='Pion condensation at lower than physical quark masses  Bastian B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content=' Brandt,𝑎 Volodymyr Chelnokov,𝑏,∗ Francesca Cuteri𝑏 and Gergely  Endrődi𝑎  𝑎Institute for Theoretical Physics, University of Bielefeld,  D-33615 Bielefeld, Germany  𝑏Institut für Theoretische Physik, Goethe-Universität Frankfurt  Max-von-Laue-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content=' 1, 60438 Frankfurt am Main, Germany  E-mail: brandt@physik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='uni-bielefeld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='de, chelnokov@itp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='uni-frankfurt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='de,  cuteri@itp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='uni-frankfurt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='de, endrodi@physik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='uni-bielefeld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='de  In QCD at large enough isospin chemical potential Bose-Einstein Condensation (BEC) takes  place, separated from the normal phase by a phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content=' From previous studies the location  of the BEC line at the physical point is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content=' In the chiral limit the condensation happens  already at infinitesimally small isospin chemical potential for zero temperature according to chiral  perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content=' The thermal chiral transition at zero density might then be affected, depending  on the shape of the BEC boundary, by its proximity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content=' As a first step towards the chiral limit, we  perform simulations of 2+1 flavors QCD at half the physical quark masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content=' The position of the  BEC transition is then extracted and compared with the results at physical masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='  The 39th International Symposium on Lattice Field Theory (Lattice2022),  8-13 August, 2022  Bonn, Germany  ∗Speaker  © Copyright owned by the author(s) under the terms of the Creative Commons  Attribution-NonCommercial-NoDerivatives 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='0 International License (CC BY-NC-ND 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='  https://pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='sissa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='it/  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='08607v1  [hep-lat]  20 Jan 2023  Pion condensation at lower than physical quark masses  Volodymyr Chelnokov  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='  Introduction  The phase structure of QCD continues to be an object of great interest for both experimental  and theoretical studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content=' There is currently a wealth of numerical simulation results at zero quark  density, from which we know, in particular, that the chiral symmetry restoration at physical quark  masses is a continuous crossover [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content=' Studying the theory in the chiral limit is complicated by the  appearance of zero modes of the Dirac operator, and the need to go to the continuum limit to restore  chiral symmetry with Wilson or staggered fermions, or to employ computationally much costlier  chiral symmetric fermion discretizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content=' Nevertheless, there is a recent progress in this region  which, in particular, suggests that in the limit 𝑚ud = 0 the chiral transition is of second order (for  𝑁 𝑓 = 2, 3) and becomes a crossover at arbitrarily small nonzero quark masses [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='  Extending the QCD phase diagram to nonzero densities is confounded by the sign problem  – the Boltzmann weight of the lattice configuration becomes complex at nonzero 𝜇𝐵, preventing  straightforward use of importance sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content=' Different ways to overcome this problem were proposed  (see [3, 4] for a review), but still the problem is far from being solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='  Another possible extension of the QCD phase diagram is to the region of nonzero isospin  density at 𝜇𝐼 ≠ 0, potentially relevant for early Universe cosmology [5], for instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content=' At pure  isospin chemical potential 𝜇𝐼 ≠ 0, 𝜇𝐵 = 0, the theory is sign problem-free and the standard Monte-  Carlo simulations can be performed [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content=' The recent study [7] shows that the deconfinement transition  remains a crossover for 𝜇𝐼 ≤ 𝑚 𝜋/2, until it intersects with the second order pion condensation line,  which is found to be approximately vertical at small enough 𝑇 (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content=' The 𝜇𝐼-𝑇 plane can  also be used to check the methods targeted at studying QCD at finite 𝜇𝐵 – since we can compare  Figure 1: The QCD phase diagram in 𝑇-𝜇𝐼 plane at physical quark masses, obtained in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='  2  Pion condensation at lower than physical quark masses  Volodymyr Chelnokov  their results with the results of direct simulation, that is possible here [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content=' Additionally, the ability  to numerically sample the theory in the 𝜇𝐼-𝑇 plane can be exploited to perform the reweighting to  nonzero 𝜇𝐵 [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='  The phase diagram shown in Figure 1 has an interesting implication concerning the chiral  phase diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content=' At zero temperature, the pion condensation happens at 𝜇𝐼 = 𝑚 𝜋/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content=' When the  light quark mass goes to zero, the pion mass also goes to zero as 𝑚 𝜋 ∼ √𝑚ud according to chiral  perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content=' Thus at least at zero temperature in the chiral limit pion condensation happens  already at arbitrarily small isospin chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content=' From Figure 1 we see that at physical mass  the isospin transition line remains vertical up until it meets the chiral crossover line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content=' If that holds  also at 𝑚ud = 0, then in the chiral limit the pion condensation line would lie on the 𝜇𝐼 = 0 axis up  to 𝑇 = 𝑇𝑐 – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content=' up to the chiral phase transition temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content=' In this scenario the pion condensate  in the chiral limit at 𝑇 = 𝑇𝑐 exists at arbitrarily small isospin chemical potential, affecting the chiral  phase transition at zero chemical potential (Figure 2, left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content=' Alternatively, the phase transition line  can start bending towards larger 𝜇𝐼 as the quark masses are reduced, resulting in a different phase  structure (Figure 2, right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content=' Previous studies using the Nambu-Jona-Lasinio model [10], as well as  the functional renormalization group [11], support the first phase picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content=' To verify this, direct  Monte Carlo QCD simulations are necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='  Figure 2: Two possibilities for the phase diagram in 𝑇-𝜇𝐼 plane at the chiral limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='  As a first step, in this work we perform the simulation of QCD at nonzero isospin density for  𝑚ud = 𝑚ud,phys/2, comparing the position of the pion condensation transition with the results at  physical masses at a series of temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='  Simulation setup  We study 2 + 1-flavour QCD using staggered fermions on a 243 × 8 lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content=' The setup is the  same as the one used in [7], with the light quark masses changed to half their physical value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content=' The  partition function of the theory has the form  Z =  ∫  D𝑈𝜇 𝑒−𝛽𝑆𝐺 (det Mud)1/4 (det Ms)1/4 ,  (1)  3  deconfinement  T  pion condensation  μIdeconfinement  pioncondensation  μIPion condensation at lower than physical quark masses  Volodymyr Chelnokov  where 𝑆𝐺 is the tree-level Symanzik improved gauge action, Mud and Ms are, correspondingly,  light and strange quark matrices  Mud =  �  /𝐷(𝜇𝐼) + 𝑚ud  𝜆𝜂5  −𝜆𝜂5  /𝐷(−𝜇𝐼) + 𝑚ud  �  ,  M𝑠 = /𝐷(0) + 𝑚𝑠 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='  (2)  Here 𝜆 is a pion source term which explicitly breaks the 𝑈𝜏3(1) symmetry of the action, which both  allows us to see the pion condensate on the finite volume lattice, and, at the same time, improves the  condition number of the light quark matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content=' To get physically meaningful results, the limit 𝜆 → 0  must be taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='  The Boltzmann weight defined by (1) is positive, which is obvious for the gauge action and  the strange quark determinant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content=' The positiveness of the light quark determinant follows from the  generalized 𝜂5-hermiticity relation that the Dirac operator satisfies,  𝜂5 /𝐷(−𝜇𝐼)𝜂5 = /𝐷(𝜇𝐼)† .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='  (3)  Now, taking the matrix 𝐵 = diag(1, 𝜂5) in flavour space, that has a unit determinant, we get  𝐵Mud𝐵 =  �  /𝐷(𝜇𝐼) + 𝑚ud  𝜆  −𝜆  � /𝐷(𝜇𝐼) + 𝑚ud  �†  �  ,  det Mud = det (𝐵Mud𝐵) = det  ��� /𝐷(𝜇𝐼) + ��ud  ��2 + 𝜆2�  > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='  (4)  To locate the pion condensation onset, we measure the pion condensate  Σ𝜋 = 𝑚𝑢𝑑  𝑚2𝜋 𝑓 2𝜋  �  𝜋±�  ,  (5)  �  𝜋±�  = 𝑇  𝑉  𝜕 log Z  𝜕𝜆  = 𝑇  2𝑉  �  Tr  𝜆  | /𝐷(𝜇𝐼) + 𝑚ud|2 + 𝜆2  �  ,  (6)  where the trace can be calculated using noisy estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content=' The multiplicative renormalization term  in (5) depends on the light quark mass 𝑚𝑢𝑑, the pion mass 𝑚 𝜋 that corresponds to 𝑚𝑢𝑑, and the pion  decay constant 𝑓𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content=' The chiral perturbation theory predicts that 𝑚 𝜋 ∼ √𝑚𝑢𝑑 when the light quark  mass goes to zero, so assuming this relation holds for physical light quark mass we can approximate  the renormalization (5) as  Σ𝜋 = 𝑚𝑢𝑑,phys  𝑚2  𝜋,phys 𝑓 2𝜋  �  𝜋±�  ,  (7)  where 𝑚𝑢𝑑,phys and 𝑚 𝜋,phys are physical light quark and pion masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='  The pion condensate estimated using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content=' (5) at 𝑇 = 114 MeV, for three different values of 𝜆  can be seen in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content=' We see that even for the smallest value of 𝜆 we are still far from the limiting  regime 𝜆 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content=' Unfortunately, the simulations at smaller 𝜆 values become prohibitively expensive  due to very large condition numbers of the light quark matrix, that causes a sharp increase of  iterations needed for matrix inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content=' At even smaller 𝜆 values this results in a loss of convergence  of the conjugate gradient method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='  4  Pion condensation at lower than physical quark masses  Volodymyr Chelnokov  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
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+page_content='0  I/m  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
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+page_content='0  /mud = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='01  /mud = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='73  /mud = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='45  /mud = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='01, impr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='  /mud = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='73, impr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='  /mud = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='45, impr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='  Figure 3: Comparison of the improved and the unimproved pion condensate at 𝑇 = 114 MeV for three  different values of 𝜆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='  Improved pion condensate  We can improve the convergence to the 𝜆 → 0 limit by using the Banks-Casher type relation  for the pion condensate (6), that was obtained in [7]  �  𝜋±�  = 𝑇  2𝑉  �  Tr  𝜆  | /𝐷(𝜇𝐼) + 𝑚ud|2 + 𝜆2  �  = 𝑇  2𝑉  �∑︁  𝑛  𝜆  𝜉2𝑛 + 𝜆2  �  ,  (8)  where 𝜉𝑛 is the 𝑛-th singular value of the Dirac operator:  � /𝐷(𝜇𝐼) + 𝑚ud  �† � /𝐷(𝜇𝐼) + 𝑚ud  � 𝜙𝑛 = 𝜉2  𝑛𝜙𝑛 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='  (9)  In the infinite volume limit, the summation can be replaced by integration  �  𝜋±�  = 𝜆  2  �∫  d𝜉 𝜌(𝜉)(𝜉2 + 𝜆2)−1  �  ,  (10)  and taking the limit 𝜆 → 0 we get  �  𝜋±�  = 𝜋  4 ⟨𝜌(0)⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='  (11)  Thus, the limit 𝜆 → 0 of the pion condensate is proportional to the singular value density at zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='  Due to the finite size of the lattice the singular values form a discrete set, but we can approximate  the density by calculating the fraction of the singular values lying in the interval [0, 𝜉], and then  extrapolating to 𝜉 = 0  𝜌(0) = 𝑇  𝑉 lim  𝜉→0  𝑛(𝜉)  𝜉  ,  (12)  The behavior of such “averaged densities” for three regimes – finite spectral gap at 𝜇𝐼/𝑚 𝜋 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='5,  𝜌(0) ≈ 0 at 𝜇𝐼/𝑚 𝜋 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='5, and 𝜌(0) > 0 at 𝜇𝐼/𝑚 𝜋 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='5, is shown on Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='  5  Pion condensation at lower than physical quark masses  Volodymyr Chelnokov  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
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+page_content='0  T  4V  n( )  /m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
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+page_content='0  T  4V  n( )  /m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='77  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
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+page_content='0  T  4V  n( )  /m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='57  Figure 4: The integrated spectral density dependence on the integration region 𝜉 for 𝜇𝐼 /𝑚 𝜋 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='5 (left),  𝜇𝐼 /𝑚 𝜋 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='5 (right), and 𝜇𝐼 /𝑚 𝜋 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='5 (bottom) (𝑇 = 132 MeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='  To perform the extrapolation to 𝜉 = 0 we perform a polynomial fit of the integrated spectral  density in a given region, obtaining a value of ⟨𝜌(0)⟩ and then take a weighted median of all the  fit results with the weight exp(−𝜒2  𝑟), where 𝜒2  𝑟 is the (correlated) chi-squared statistic of the fit per  degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content=' The error estimate of ⟨𝜌(0)⟩ is then taken as a value Δ𝜌, for which the interval  [⟨𝜌(0)⟩ − Δ𝜌, ⟨𝜌(0)⟩ + Δ𝜌] contains 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='68 of the total weight of the estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='  We can further improve the convergence of the improved pion condensate observable to the  limit 𝜆 → 0 by approximating ⟨𝜌(0)⟩0 (the expectation value of 𝜌(0) with respect to the 𝜆 = 0  partition function) using the leading order reweighting, expanding the Boltzmann weight of a given  configuration in a Taylor series in 𝜆,  ⟨𝜌(0)⟩0 = ⟨𝜌(0)𝑊(𝜆)⟩𝜆  ⟨𝑊(𝜆)⟩𝜆  ,  𝑊(𝜆) =  � det M𝑢𝑑,0  det M𝑢𝑑,𝜆  �1/4  ≈ exp  �  −𝜆𝑉  2𝑇 𝜋±  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='  (13)  In Figure 5 we show the values of improved and unimproved pion condensate at the point close  to 𝜇/𝑚 𝜋 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='5 together with an extrapolation of the improved condensate to 𝜆 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content=' We can see that  the improved pion condensate value is much smaller and less sensitive to the 𝜆 than the unimproved  condensate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content=' This picture can be compared with the extrapolaton done in Figure 6 (top) in [7],  which has a similar behavior for large 𝜆, but also provides enough data points at small 𝜆 to actually  perform the extrapolation of both the unimproved and improved pion condensates and confirm that  6  Pion condensation at lower than physical quark masses  Volodymyr Chelnokov  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
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+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='25  /mud  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='8  improved  unimproved  Figure 5: Extrapolation of the improved and unimproved pion condensate to 𝜆 = 0 at 𝑇 = 114 MeV,  𝜇/𝑚 𝜋 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='  they reach the same value at 𝜆 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content=' Since in that study the linear extrapolation of the improved pion  condensate worked well for 𝜆/𝑚𝑢𝑑 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='3, we rely on it also in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='  Results and summary  In Figure 6 we show our preliminary results on the value of the pion condensate and the location  of the pion condensation line at four different values of the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content=' The interpolation was done  using a cubic polynomial for the temperatures where we have enough points (𝑇 = 114 MeV, 132  MeV), and a quadratic polynomial otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content=' Location of the transition point was determined as a  point where the interpolation line intersects zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content=' In all cases all points giving a positive value of the  pion condensate were included in the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content=' The error estimates for the fit line and the transition point  were obtained by taking the standard deviation of the fit result for 100 simulated sets of condensate  values normally distributed around the “true” values with the known standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='  The results show that the vertical direction of the pion condensation line is preserved when  going to smaller light quark masses, preferring the scenario shown in the left panel of the Figure 2:  the location of the condensation line is compatible with the zero temperature location 𝜇 = 𝑚 𝜋/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='  As mentioned earlier, these results are based on the linear extrapolation to 𝜆 = 0 from the data  measured at nonzero values of the pion source parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content=' To be able to check the validity of this  extrapolation, we are currently performing further simulations at smaller values of 𝜆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content=' Additionally,  a 𝑇 scan at several values of 𝜇𝐼 > 𝑚 𝜋/2 is being performed in order to locate the “horizontal” part  of the pion condensation line as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='  7  Pion condensation at lower than physical quark masses  Volodymyr Chelnokov  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
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+page_content='8  /m  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
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+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='0  T=114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='37 MeV  T=123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='04 MeV  T=132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='24 MeV  T=141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='96 MeV  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='7  /m  120  130  140  T  Figure 6: The value of the improved pion condensate (left) and the location of the pion condensation point  (right) for four different values of temperature, obtained from the 243×8 lattice (no continuum extrapolation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content='  Acknowledgments  This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research  Foundation) – project number 315477589 – TRR 211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
+page_content=' The authors acknowledge the use of the  Goethe-HLR cluster and thank the computing staff for their support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfjx2F/content/2301.08607v1.pdf'}
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1NFKT4oBgHgl3EQfOi14/content/tmp_files/2301.11759v1.pdf.txt

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+Reduction by invariants, stratifications, foliations,
+fibrations and relative equilibria, a short survey.
+J.C. van der Meer
+Faculteit Wiskunde en Informatica, Technische Universiteit Eindhoven,
+PObox 513, 5600 MB Eindhoven, The Netherlands.
+January 30, 2023
+Abstract
+In this note we will consider reduction techniques for Hamiltonian systems that are
+invariant under the action of a compact Lie group G acting by symplectic diffeo-
+morphisms and the related work on stability of relative equilibria. We will focus
+on reduction by invariants in which case it is possible to describe a reduced phase
+space within the orbit space by constructing an orbit map using a Hilbert basis of
+invariants for the symmetry group G. Results considering the stratification, folia-
+tion and fibration of the phase space and the orbit space are considered. Finally
+some remarks are made concerning relative equilibria and bifurcations of periodic
+solutions. We will combine results from a wide variety of papers.
+1
+Introduction
+In Hamiltonian systems the symmetry is usually given by a symmetry group acting on
+the phase space and leaving the system invariant. The symmetry group determines the
+geometry of the phase space and therefore also determines the behavior of systems having
+this symmetry. One of the tools to understand symmetric systems is reduction by dividing
+out the symmetry, which allows to study the dynamics on a lower dimensional space. The
+main questions are then how this reduced dynamics reconstructs to the original unreduced
+phase space and which part of the dynamics persists under non-symmetric perturbations.
+In this note we will consider reduction, especially singular reduction. After a short review
+we will focus on reduction by invariants. By constructing an orbit map also singular
+reduction can be considered. Given a symmetry group G, fulfilling the right conditions,
+one can consider the orbit type stratification of the phase space and orbit space, the
+1
+arXiv:2301.11759v1  [math.DS]  27 Jan 2023
+
+foliation by symplectic leaves of the orbit space, and the fibration of the phase space by
+group orbits. Given a Hamiltonian system with symmetry group G one can also consider
+relative equilibria. In general a relative equilibrium is an equilibrium position in a moving
+system. In the case of a symmetric system a relative equilibrium is an equilibrium that
+moves with the group action into a solution of the system and therefore corresponds to
+a stationary point of the reduced system. Relative equilibria for symmetric systems are
+of importance because, using singularity theoretic methods, it can be show that, under
+certain conditions, they persist under non-symmetric perturbation. As they are usually
+organised in families, they form an organising skeleton in the phase space of the symmetric
+system and perturbations thereof.
+In this note we will focus on reviewing the different concepts of reduction, the stratifica-
+tion, foliation and fibration of the phase space and orbit space, and on the concepts of
+relative equilibria and their stability.
+2
+Reduction
+The idea of reduction seems to go back on Reeb [53, 54] who studied perturbations of
+systems of differential equations only having periodic solutions. He considered the phase
+space as a fibred manifold or more precisely as an S1 fibre bundle. He constructed the
+reduced system by mapping to the base space of the fibre bundle. One could call this
+concept fibre bundle reduction. He considered Hamiltonian systems as a special case.
+Later the geometric reduction of Meyer [34] and Marsden and Weinstein [32] became the
+fundamental reference for reduction. Ideas on reduction can also be found in [4] and [62].
+The Meyer-Marsden-Weinstein reduction is formulated in [32] as follows.
+Consider a symplectic manifold M, on which we have the action by symplectic diffeomor-
+phisms of a Lie group G with Lie algebra g. For ξ ∈ g let ξM denote the corresponding
+infinitesimal generator or vector field on M. Let J : M → g∗ be an Ad∗ equivariant
+momentum map, that is, J ◦ ϕg = (Adg−1)∗ ◦ J , where ϕg is the action of g ∈ G, and
+(Adg−1)∗ is the co-adJoint action of G on g. Let µ ∈ g∗ be a regular value of J , and
+let Gµ be the isotropy subgroup of µ for the co-adjoint action, and assume that Gµ acts
+properly and freely on J −1(µ). Then Mµ = J −1(µ)/Gµ is the reduced phase space. If ω
+is the symplectic structure on M then there exists an unique symplectic structure ωµ on
+Mµ with π∗
+µωµ = i∗
+µω, with i∗
+µ the inclusion map of J −1(µ) in M, and π∗
+µ the projection
+map of J −1(µ) onto Mµ. If H is a G-invariant Hamiltonian function on M with respect
+to ω, then the flow of H induces a flow on Mµ which is Hamiltonian with respect to the
+symplectic form ωµ with reduced Hamiltonian function Hµ.
+Although this is a very general description it does not really allow us to construct the
+reduced phase space. Methods to construct the reduced phase space were given by Cush-
+man [11], Van der Meer [64, 65], Churchill, Kummer and Rod [10], Cushman an Rod
+2
+
+[12], Kummer [30]. They all use invariants for the group action to construct the reduced
+phase space.
+This method is formulated in a more general way using orbit spaces in
+[2, 65] were a theorem of Hilbert is used that provides a Hilbert basis of invariants for
+compact group actions. This makes it possible to define an orbit map ρ [52] under which
+all the G-orbits are mapped to points. Then ρ(J −1(µ)) is the reduced phase space, hence
+the name orbit space reduction or reduction by invariants. By a theorem of Schwarz
+G-invariant functions correspond to functions on the orbit space, that is, any G-invariant
+Hamiltonian function reduces to a function on the orbit space that naturally restricts to
+the reduced phase space. This method has the advantage that µ need not be a regular
+point, and that the action need not be free. The singularities of the orbit map reflect the
+fixed points, isotropy subgroups and orbit types. Note that singular reduction appears in
+[64, 65] for the non-semisimple 1:-1 resonance and in [30] for the k:l resonances. This was
+later formalized in the context of Marsden-Weinstein reduction in [3]. We will distinguish
+between the Meyer-Marsden-Weinstein reduction and the construction using orbit maps
+by calling the first momentum map reduction and the latter reduction by invariants. Note
+that when using reduction by invariants the reduced phase space need not be a manifold.
+In general it is defined as a semi-algebraic set by relations and inequalities for the invari-
+ants defining the orbit map. Cushman and Sniaticky studied these spaces in more detail
+including differential structures on them (see [13, 14, 63]) which allows to study more
+general vector fields on orbit spaces [7].
+The most general formulation for orbit map reduction does not start with a symplectic
+manifold but with a Poisson manifold.
+we consider C∞(M) together with a Poisson
+bracket { , } making C∞(M) into a Lie algebra and call (M, { , }) a Poisson manifold. If
+the Poisson structure is non-degenerate then M is a symplectic manifold. The Poisson
+structure on C∞(M) induces a Poisson structure on the orbit space.
+Kummer [29] gives a construction of the reduced phase space using principle G-bundles.
+In a neighborhood of regular values of the orbit map the reduced phase space is locally
+a G-bundle. That is, his ideas apply to ρ(J −1(µ)) without its critical set. The bundle
+structure of G-spaces is extensively studied in [16].
+Reduction by invariants through the orbit map should not be confused with orbit reduction
+as introduced by Ortega [45, 33]. Orbit reduction refers to the fact that the pre-image
+of the co-adjoint orbit in the image of the momentum map is reduced. Reduction by
+invariants should also not be confused with Poisson reduction. Although Poisson reduction
+considers reduction of Poisson manifolds it restricts to M = T ∗G and its momentum map.
+In orbit map reduction the orbit map is used to construct the reduced phase space and the
+reduced system. When using orbit map reduction one of the difficulties is to determine
+the invariants and the relations between the invariants. An extensive study of this is made
+in [19] (see also [56, 8]).
+3
+
+3
+Meyer-Marsden-Weinstein reduction, reduction by
+invariants and dual pairs
+Throughout this paper we will consider a connected, compact Lie group acting smoothly
+and properly on Rn which is assumed to be a symplectic space with the standard Poisson
+structure, that is it is a Poisson manifold, and G is assumed to act by Poisson (symplectic)
+diffeomorphisms. Thus we assume to be in the nicest possible situation where we can use
+the strongest possible results. However, in many applications this is the case. Many of
+the results stated below will also hold under weaker conditions, more generally when Rn
+is replaced by a connected compact Poisson manifold M.
+We will start by introducing reduction by invariants. In [24] Hilbert showed that the alge-
+bra of polynomials over C of degree d in n variables which are invariant under GL(n, C),
+acting by substitution of variables, is finitely generated. This was extended by Weyl in
+[69] who proved that the algebra of invariants is finitely generated for any representation
+of a compact Lie group or a complex semi-simple Lie group.
+Let R[x]G denote the space of G-invariant polynomials with coefficients in R. Consider a
+compact Lie group G acting linearly on Rn. Then there exist finitely many polynomials
+ρ1, · · · , ρk ∈ R[x]G which generate R[x]G as an R algebra. These generators can be chosen
+to be homogeneous of degree greater then zero. We call ρ1, · · · , ρk a Hilbert basis for
+R[x]G.
+Schwarz [59] proved that if ρ1, · · · , ρk is a Hilbert basis for R[x]G, and ρ : Rn → Rk; x →
+(ρ1(x), · · · , ρk(x)). Then ρ∗ : C∞(Rk, R) → C∞(Rn, R)G is surjective, with ρ∗ the pull-
+back of ρ. Thus all G-invariant smooth function can be written as smooth functions in
+the invariants.
+The following, showing that ρ is an orbit map, can be found in [52].
+The map ρ is
+proper and separates the orbits of G. Moreover the following diagram commutes, with ˜ρ
+a homomorphism
+Rn
+ρ
+−→ ρ(Rn)
+π ↘
+↙ ˜ρ
+Rn/G
+Here the orbit space Rn/G is the quotient space Rn/ ∼, where the equivalence relation is
+given by x ∼ y if x and y are in the same G-orbit. We can take ρ(Rn) as a model for the
+orbit space.
+Consider (R2n, ω) on which a Lie group G acts linearly and symplectically. Then (C∞(R2n, R), { , })
+is a Poisson algebra. If we consider on Rk the Poisson structure induced by ρ by taking
+as structure matrix Wij = {ρi, ρj} then (C∞(Rk, R), { , }W) is a Poisson algebra and ρ a
+Poisson map. We have a reduction of the Poisson manifold if we restrict the bracket on
+4
+
+Rk to ρ(R2n).
+In general there will be relations and inequalities determining the image of ρ. There-
+fore ρ(R2n) will in general be a real semi-algebraic subset of Rk, where a semi-algebraic
+subset of Rk is a finite union of sets of the form {x ∈ Rk|R1(x) = · · · = Rs(x) =
+0 , Rs+1(x), · · · , Rm(x) ⩾ 0} .
+Define C∞(ρ(R2n), R) = {F : ρ(R2n) → R|ρ∗(F) ∈
+C∞(R2n, R)}. This is a differential structure on ρ(R2n) and the orbit map is smooth (see
+[13, 14]). Note that the Ri, 1 ⩽ i ⩽ s, are Casimirs for the induced Poisson structure
+{ , }W.
+Let W be a real semi-algebraic variety in Rk. A point x ∈ W is nonsingular if there exists
+a neighborhood U ⊂ W of x such that for each y ∈ U the matrix ∂Ri
+∂xj (x) has maximal
+rank. A point x ∈ W is singular if the rank of ∂Ri
+∂xj (x) is strictly less than the maximal
+rank.
+Combining remarks in [33] and [60] we find that for any Poisson manifold (M, { , }) on
+which we have a compact Lie group G acting by Poisson maps, G has a Lie algebra g. To
+each element ξ ∈ g we may associate a Hamiltonian vector field XJξ = {Jξ, ·} on M with
+Hamiltonian function Jξ defined by J(z) = J (z) · ξ.
+Let ξi be a basis of g such that exp(ξi) generate G. Consider the corresponding functions
+Ji and consider the momentum map J(z) = (J1(z), · · · , Jr(z)). Furthermore consider the
+Hilbert basis of invariants ρi, i = 1, · · · , k, for the G-action. Obviously, {ρi, Jj} = 0 for
+all i, j. Thus the maps J and ρ form a dual pair. Note that without further conditions
+the images of J and ρ are at best semi-algebraic sets. Also the ρi need not span a Lie
+algebra. however, the ρi generate the Poisson algebra of G-invariant functions C∞(M)G.
+Let G′ denote the Lie group generated by the G-equivariant Poisson vector fields Xf,
+f ∈ C∞(M)G. Then the map M → M/G′ is called the optimal momentum map[45, 47].
+If the Ji are the invariants defining M/G′ the J is optimal. Consider C∞(J(M)), then
+C∞(M)G and C∞(J(M)) centralize each other in the Poisson algebra C∞(M) (see for
+instance [27]), i.e. we have a Howe dual pair [25]. Actually, as can be found in [46], the
+pair g ← M → ρ(M) is a Lie-Weinstein dual pair.
+As {ρi, Jj} = 0 it follows that
+Proposition 3.1 ker (dJ) is spanned by the Hamiltonian vector fields Xρi.
+Example 3.2 Consider SO(3) acting on R3. The lifted action on the co-tangent bundle
+T ∗R3 is the diagonal action of SO(3) on R6. Let (x, y) denote the coordinates on R6 =
+R3 × R3.
+The generators for the group are x1y2 − x2y1, x1y3 − x3y1, x2y3 − x3y2 (or
+the components of the cross product x × y), which span a Lie algebra isomorphic to
+so(3). A Hilbert basis for this action is |x|2, |y|2, < x, y >. The latter span a Lie algebra
+isomorphic to sl(2, R). According to [69] in general the full linear Lie algebra invariant
+5
+
+under a diagonal SO(n)-action is sl(2, R). Consequently the momentum map
+J : (x, y) → (x1y2 − x2y1, x1y3 − x3y1, x2y3 − x3y2) ,
+and the orbit map σ : (x, y) → (|x|2, |y|2, < x, y >) are a dual pair. The orbit space is
+defined by Lagrange identity |x|2|y|2− < x, y >2= |x × y|2 together with the inequalities
+|x|2 ⩾ 0, and |y|2 ⩾ 0. It is a solid cone. The reduced pase spaces are given by taking
+x × y constant, thus, provided |x × y| ̸= 0, the reduced phase space is one sheet of a two
+sheeted hyperboloid. In case |x × y| = 0 it is a cone.
+Example 3.3 Consider an integrable system on R2n, with n integrals in involution, that
+is, the group G generated by the integrals is a torus. The momentum map and the orbit
+map are the same, i.e. Ji = ρi. The orbit space is a polytope [5, 23]. A reduced phase
+space is a point. Regular points correspond to n-tori. The faces, edges and vertices of the
+polytope are the images of the singular points of the orbit map and correspond to lower
+dimensional tori.
+Example 3.4 Consider a group G which is the flow of a linear Hamiltonian system in
+two degrees of freedom in k : ℓ resonance, k ∈ N, ℓ ∈ Z, |k| ̸= ℓ). That is, the Hamiltonian
+generating the group G is H(x, y) = 1
+2 k(x2
+1 + y2
+1) + 1
+2 ℓ(x2
+2 + y2
+2). The action of G is an S1-
+action and a Hilbert basis for the invariants of G is given by polynomials I1(x, y), I2(x, y),
+R1(x, y) and R2(x, y), where I1, I2 are quadratic polynomials and R1, R2 are polynomials
+of order k+|ℓ|. When we choose I1(x, y) = H(x, y) and I2(x, y) = 1
+2 k(x2
+1+y2
+1)− 1
+2 l(x2
+2+y2
+2)
+we have bracket relations {I2, R1} = 2klR2, {I2, R2} = 2klR1, {I1, R1} = {I1, R2} = 0,
+and {R1, R2} = −2(I2
+1 − I2
+2) + (I1 + I2)2. Furthermore we have the following relation
+R2
+1 + R2
+2 = 1
+2 (I1 + I2)2(I1 − I2) (see [58]). In this case the invariants do not form a Lie
+subalgebra of C∞(R4)G. However, we do have an orbit map ρ : (x, y) → (I1, I2, R1, R2),
+where the image is determined by R2
+1 + R2
+2 = 1
+2 (I1 + I2)2(I1 − I2), I1 ⩾ 0. The reduced
+phase space is obtained by taking I1(x, y) = c, giving the reduced phase space given by
+the relation R2
+1 + R2
+2 = 1
+2 (c + I2)2(c − I2) in (I2, R1, R2)-space (cf [30]).
+We have a special situation if the Hilbert basis is a finite Lie subalgebra of C∞(Rn). In this
+case the orbit map can also be interpreted as a momentum map. We have a Lie-Weinstein
+dual pair of momentum maps.
+g∗
+1 ← M → g∗
+2 ,
+or in terms of generating functions
+(J1, · · · , Jr) ← M → (ρ1, · · · , ρk) .
+If the Ji generate G1 and the ρi generate G2 then G1 and G2 commute.
+Moreover
+C∞(M)G2 is the Lie algebra of smooth functions in the Ji. The center of g1 therefore
+consists of functions in C∞(M)G2.
+6
+
+This holds more generally. If we have a group G, with momentum map J, then the center
+of C∞(M)G in the Poisson algebra C∞(M) consists of smooth functions in the Ji, which
+can be considered as the universal enveloping algebra of g. As a consequence the Casimirs
+of C∞(M)G do belong to C∞(M)G and to the universal enveloping algebra of g. Thus
+the symplectic leaves for the Poisson structure on the image of the orbit map, which are
+obtained by setting the Casimirs equal to a constant [68], are given by ρ(J−1(µ)).
+When the invariants form a Lie algebra then the symplectic leaves are the co-adjoint
+orbits of G2 on g2 [68].
+4
+Stratifications, foliations and fibrations
+Orbit spaces were studied in connection to understanding the structure of G-spaces in
+the 1950’s [51]. Mainly to understand extrema of G-invariant functions [36, 37],[2] in
+connection to applications in solid state physics. More recent applications in quantum
+mechanics and molecular behavior that connect to the subject of this paper are [38], [70],
+[57]. In [67] a very nice overview of all the relevant theorems from the literature is given in
+connection to orbit spaces in the context of Meyer-Marsden-Weinstein reduction. When
+studying symmetric spaces and reduction one of the important issues is the orbit type
+stratification, see the above cited literature and [61], [16], [6] and the very accessible notes
+[17], [35].
+Consider a Hamiltonian G-action on a connected manifold M, G a compact Lie group
+acting properly and smoothly on M. Let G · x = {y ∈ M|y = g · x, g ∈ G} be the G-orbit
+in M through x ∈ M. Gx = {g ∈ G|g · x = x is the isotropy subgroup of G at x. Gx is
+a closed Lie subgroup of G. If H is a subgroup of G for which Hx = {g ∈ H|g · x = x},
+then we may call Hx also an isotropy subgroup of x. Obviously Hx ⊂ Gx and Gx is the
+maximal isotropy subgroup for x in G. We have
+dim(G) = dim(Gx) + dim(G · x) .
+We have Gg·x = gGxg−1, that is, the isotropy subgroups of points on the same orbit
+are each others conjugate and therefore isomorphic. For a subgroup H of G define the
+normalizer of H by
+NG(H) = {g ∈ G|gHg−1 = H} .
+NG(H) is a closed Lie subgroup of G and the largest subgroup of G containing H as a
+normal subgroup.
+Let H be a subgroup of G. Let MH = {x ∈ M|Gx = H}. MH is called the isotropy type
+of H in M.
+Lemma 4.1 [17] Let H be a closed Lie subgroup of G. Then the action of N(H) leaves
+MH invariant and induces a free action of N(H)/H on MH.
+7
+
+Denote by [H] the conjugacy class of H. Set M[H] = {m ∈ M|Gm = gHg−1 , g ∈ G}.
+M[H] is called the orbit type of [H] in M. Two points in M belong to the same orbit
+type if and only if their exists a G-equivariant bijection between their G-orbits [16]. M
+is partitioned into orbit types M[H] and each orbit type is partitioned into isotropy types
+MK, K ∈ [H] for conjugate subgroups of G. This induces a partition of ρ(M) into orbit
+types ρ(M[H]).
+Lemma 4.2 [17] ρ(MH) = ρ(M[H]) and because the G-action is proper we have that
+ρ|MH : MH → ρ(M[H]) is a principal fiber bundle with structure group N(H)/H.
+Note that if M is Rn and we have a faithful representation of G on Rn, with a subgroup
+H, then MH = Fix(H). On Fix(H) the action of G reduces to the action of N(H)/H.
+It is important to note that MH can have connected components with different dimensions.
+If the action is free then the orbit space M/G is a smooth manifold, there is only one
+orbit type, and the orbit map is a smooth fibration with structure group G. dim(M/G) =
+dim(M) − dim(G)
+For instance in [16] it can be found that the connected components of the orbit types
+form a Whitney stratification in M.
+There exists a partial ordering of isotropy and orbit types. For isotropy subgroups H and
+K we say that MH ⩽ MK, M[H] ⩽ M[K] if and only if H is conjugate to a subgroup of K.
+Theorem 4.3 [Principal orbit theorem] As before consider a group action by G on a
+connected differentiable manifold M.
+Then there exists a maximal orbit type.
+The
+maximal orbit type stratum Sm is open and dense in M. The orbit space Sm/G is open
+and dense and connected in M/G.
+The maximal orbit and its orbit type are also called the principal orbit and principal orbit
+type.
+Denote the orbit type strata by Si, that is, M = ∪iSi. When we have defined an orbit map
+by invariants we obtain an orbit type stratification of the orbit space ρ(M) = ∪iρ(Si).
+Note that in general our manifolds are embedded in Rn and thus separable. In [51] we
+find the following
+Corollary 4.4
+(i) dim(ρ(M[H])) = dim(M[H]) − dimG/H , H ⊆ G ,
+(ii) dim(M/G) = sup{dim(ρ(M[H])) − dim(G/H)|H ⊆ G} .
+8
+
+On the orbit space the partitioning given by the stratification consists of sets of points
+that have as a pre-image diffeomorphic orbits. A G-orbit is the pre-image ρ−1(ν) of a
+point ν on the orbit space. More precisely, if ν ∈ ρ(MH) the orbit through p ∈ ρ−1(ν)
+is N(H)/H · p. Its tangent space at a point p ∈ ρ−1(ν) is spanned by the Hamiltonian
+vectors XJi(p). We have
+dim(G.p) = rank(J)(p) = dim(N(H)/H · p) .
+The orbit space is given, as a subset of Rk, by the relations and inequalities for the
+invariants. If the action of G is free the principal orbit type corresponds to the isotropy
+subgroup I and we have by taking H = I in corollary 4.4, that the dim(Sm) =dim(ρ(M)),
+dim(G) + dim(ρ(M)) = dim(M) ,
+where dim(ρ(M)) equals the maximal rank of ρ. Moreover, the dimension of the orbit
+type stratum ρ(Si) for some G-orbit in M equals the rank of the orbit map at the points of
+this orbit. Consequently the orbit type stratification coincides withe the Thom-Boardman
+stratification [20] for the orbit map.The image of this stratification under the orbit map
+is of course the same as that of the orbit type stratification, but now it can be seen as the
+singular set stratification of the semi-algebraic set that is te image of the orbit map. The
+stratification of the image of the orbit map defines a partition of the semi-algebraic set
+into disjoint sets on which the rank of ρ is constant and on which each point corresponds
+to an orbit of the same type.
+Again let p ∈ ρ−1(ν). And let ν be in the orbit type stratum for MH. Following from
+lemma’s 4.1 and 4.2 we have that, p ∈ MH, dim(N(H)/H) = dim(G · p) = rank(J)(p).
+Furthermore
+dim(G) = dim(G · p) + dim(Gp) ,
+and
+dim(MH) = dim(G · p) + rank(ρ)(ν) .
+So far we have paid attention to the reconstruction of the orbit space in terms of G-orbits
+and orbit type strata. The orbit space is fibred into reduced phase spaces ρ(J−1(µ)).
+Each reduced phase space has a stratification into orbit type strata by considering the
+intersection of the reduced phase space with the orbit type strata, i.e. ρ(J−1(µ))∩ρ(M[H]).
+This is the symplectic stratification introduced in [61].
+On the other hand the Poisson structure on the orbit space allows us to obtain a foliation
+into symplectic leaves [68] of the orbit type strata. Each symplectic leaf is obtained by
+setting the Casimirs of the Poisson structure equal to a constant.
+Consequently, the
+symplectic leaves are subspaces of the reduced phase spaces. Besides that, the rank of
+the Poisson structure is constant along a symplectic leaf.
+9
+
+Proposition 4.5 If ˜W is the invertible structure matrix of the Poisson structure on the
+symplectic manifold M, then (dJ) ˜W(dJ)T is the induced structure matrix on the orbit
+space J(M), and the rank of the induced Poisson structure on the orbit space equals the
+rank of the orbit map, that is,
+rank((dJ) ˜W(dJ)T) = rank(dJ) .
+(1)
+The proof is a straightforward exercise in linear algebra. Now the stratification by rank of
+the orbit space coincides with the orbit type stratification, thus each orbit type stratum is
+foliated into symplectic manifolds of the same dimension on which the Poisson structure
+has the same rank. The same then also holds for the orbit type strata of the reduced phase
+spaces. Because the reduced phase space as well as the symplectic leaves are obtained
+by setting the Casimirs equal to a constant it follows that each orbit type stratum of the
+reduced phase space is a symplectic leaf.
+Thus conclusively
+Corollary 4.6 The orbit space is fibred into reduced phase spaces and each reduced
+phase space has a orbit type stratification, where each orbit type stratum is a symplectic
+leaf for the induced Poisson structure.
+Recall that each orbit type stratum is a principal fibre bundle 4.2.
+When we consider a symplectic manifold M which is embedded in Rn, and which is a
+symplectic leaf for the Poisson structure on Rn, then we may further reduce if a group G
+is acting on Rn by Poisson diffeomorphisms and leaving M invariant. This way we may
+reduce in stages [33].
+5
+Relative equilibria
+As before consider Rn with the standard non-degenerate Poisson structure (thus n is even)
+and a group action of a compact and connected Lie group G by symplectic (Poisson)
+diffeomorphisms. Let H ∈ C∞(Rn, R)G and consider the Hamiltonian (Poisson) vector
+field XH on Rn. XH has integrals Ji. Because H ∈ C∞(Rn, R)G there is a function ˜H ∈
+C∞(Rk, R) on the target space of the orbit map such that H = ˜H ◦ρ. The reduced vector
+field is now the Poisson vector field X ˜H with respect to the induced Poisson structure
+{ , }W.
+In [1] we find as a definition for relative equilibrium that a point x ∈ Rn is a relative
+equilibrium for XH if ρ(x) is a stationary point for the reduced vector field. Other ways
+of formulating this are that a relative equilibrium is a point x ∈ Rn such that the solution
+of Hamilton’s equations for XH with initial value x coincides with the orbit of a one
+10
+
+parameter sub-group of G [49] or, somewhat different, that relative equilibria are G group
+orbits which are invariant under the flow of XH [55].
+We will use the formulation from [60] stating that a point x ∈ Rn is a relative equilibrium
+for the Hamiltonian system XH if the trajectory γt of Hamilton’s equations for XH through
+x is given by
+γt(x) = exp(tXF)(x) ,
+whereXF with F = �r
+i=1 λiJi is an infinitesimal generator for an element of G and
+x ∈ J−1(µ). Thus XH(x) = XF(x). Thus a relative equilibrium is a critical point for
+the energy-momentum map H × J : Rn → Rr+1; x → (H(x), J1(x), · · · , Jr(x)) [62]. In
+[62] a reduction is performed by considering (H × J)−1(h, µ)/Gx. Obviously, as a relative
+equilibrium is contained in a G-orbit it maps to a point ρ(x) in the reduced phase space
+ρ(J−1(µ)) if x ∈ J−1(µ). Furthermore as the XH trajectory through x is a G-orbit it
+reduces to a stationary point ρ(x) for X ˜H.
+Of specific interest are of course the stability of relative equilibria and the persistence
+under change of the momentum. The latter results in families of relative equilibria that
+might be organized in manifolds. The study of these concepts has a long history. Some
+relevant references are [60], [49], [44], [42], [31], [50], [43]. As for the results concerning
+stability up till that moment a very nice discussion is given in the introduction of [50].
+In [60] the energy-momentum method is introduced to determine the formal stability of
+a relative equilibrium. To determine the relative equilibria we have to solve the Lagrange
+multiplier optimization problem of finding the critical points of H under the constraints
+J(x) = µ. To determine the stability one considers d2H(x). However, one has to restrict
+d2H(x) to some subspace S of kerdJ(x). To determine this subspace one has to remove
+the neutral directions from ker(dJ(x)). By Lemma 3.1 ker(dJ(x)) is spanned by the
+Xρi(x).
+However, there might be dependencies at x.
+These dependencies determine
+the neutral directions and are given by the Casimirs that determine the symplectic leaf
+through ρ(x). Note that these Casimirs can be expressed as smooth functions of the ρi
+and as smooth functions of the Ji. Suppose this set of Casimirs (independent at ρ(x)) is
+given by Ci, i = 1, · · · , s. Then the vectors XCi(x) determine the directions to be left
+out of ker(dJ(x)). The vector fields XCi are the infinitesimal generators for the isotropy
+subgroup Gµ of G, which is the group leaving µ fixed under the co-adjoint action of G on
+g∗. That is, S = ker(dJ(x))/Tx(Gµ · x) as in [60].
+Now the vectors Xρi(x) reduce to tangent vectors to the symplectic leaf in the reduced
+phase space at ρ(x). If one leaves out the dependencies given by the Casimirs we get that
+S maps to the tangent space to the symplectic leaf through ρ(x) at ρ(x). Thus
+Corollary 5.1 Formal stability of the relative equilibrium x as defined in [60] agrees with
+stability of ρ(x) on the reduced phase space.
+This is called relatively stable in [1].
+11
+
+The above becomes more clear when we consider the following to be found in [19]
+Theorem 5.2 Consider a Hilbert basis π1, · · · , πr for a faithful representation of a com-
+pact Lie group.
+Assume H is an isotropy subgroup of G in this representation with
+corresponding Fix(H). Then there exist invariants ˜π1, · · · , ˜πd, which are algebraically in-
+dependent polynomials in the πi, such that
+˜πi|Fix(H) ̸= 0 , i = 1, · · · , c , and ˜πi|Fix(H) = 0 , i = c + 1, · · · , d ′.
+furthermore ˜πi|Fix(H), i = 1, · · · , c are algebraically independent.
+That is, one can pick a set of invariants defining the orbit space such that the tangent
+space at a point p of the reduced phase space is spanned by the tangent vectors generated
+by ˜πi|Fix(H) ̸= 0 , i = 1, · · · , c, where H is the isotropy group of a point in ρ−1(p). (see
+also [28]).
+Thus the ˜πi|Fix(H) ̸= 0 , i = 1, · · · , c, can be considered as a set of invariants defining the
+orbit space for Fix(H), while ˜πi|Fix(H) ̸= 0 , i = c+1, · · · , d can be considered as Casimirs,
+that is, the set of Casimirs becomes larger when there is a non-trivial isotropy subgroup,
+and consequently the orbit space and the reduced phase spaces reduce in dimension.
+Example 5.3 Consider an integrable system, i.e. G is the torus group. The orbit map
+and the momentum map are the same thus the orbit space is the momentum polytope.
+The interior, faces, edges and vertices of this polytope correspond to the orbit type strata.
+The reduced phase spaces are points. Consequently each point is a relative equilibrium.
+As the pre-image of a point is a torus the trajectories of G-invariant vector fields are
+periodic orbits or quasi-periodic orbits. Quasi-periodic relative equilibria are considered
+in [17].
+In many examples the presentation of G is linear and explicitly known. In [39] we find
+the possible representations that can occur when we consider the linear symplectic action
+of a Lie group G on a vector space V .
+Theorem 5.4 ( [39]) Every symplectic representation V of G has a unique direct sum
+decomposition
+V = V1 ⊕ · · · ⊕ Vℓ ,
+where
+(a) The Vj are G-invariant subspaces of V ;
+12
+
+(b) Vj = K
+nj
+j ⊗Kj Wj, where W1, · · · , Wℓ are pairwise irreducible representations of G
+and HomG(Wj, Wj) ≈ Kj = R, C, H;
+(c) The action of G on Vj is the tensor product of the action Wj and the trivial action
+on K
+nj
+j
+Let SpG(R2n) denote the group of G-equivariant symplectic linear transformations on
+R2n. Then [39] SpG(R2n) ∼= S1 × · · · Sℓ, where each Sj is either Sp(m, R), U(p, q; C) or
+αU(r, H). Here Sp(m, R), U(p, q; C) and αU(r, H) are as defined in [39]. As we are dealing
+with linear symplectic maps the corresponding Lie algebra spG(R2n) is isomorphic to the
+Lie algebra under the Poisson bracket of G-invariant homogeneous quadratic polynomials.
+If these polynomials form a Hilbert basis then we are in the situation of a dual pair g,
+spG(R2n).
+6
+Bifurcations of periodic solutions
+When G is a symplectic S1 action the relative equilibria are periodic solutions. If fur-
+thermore the Hamiltonian depends on parameters one can study the bifurcation of these
+periodic solutions in dependence of the parameters. Here we have to distinguish between
+the parameters in the Hamiltonian, which are sometimes called distinguished parameters,
+or unfolding parameters, that usually are related to the physical system parameters, and
+parameters introduced by the value of the momentum map, i.e. introduced by the reduc-
+tion. When considering a G-invariant system in the neighbourhood of a stationary point
+one can, if the quadratic part of the Hamiltonian fulfills certain conditions, consider the
+additional S1 action by the semisimple part of this quadratic Hamiltonian. The bifurca-
+tion one wants to describe is then the bifurcation of periodic orbits with period close to
+the period of this S1 action. To this end one first uses Liapunov-Schmidt reduction, or a
+splitting theorem, to reduce the G-invariant system to a G × S1-invariant system. Then
+G × S1-equivariant singularity theory is used to reduce the power series of the G × S1-
+invariant Hamiltonian in the neighborhood of the stationary point to a finite part of the
+power series. Note that this depends on several non-degeneracy conditions that have to
+be fulfilled. When one finally has a G × S1 invariant polynomial system the parameter
+dependent equation for the stationary points of the S1-reduced system then gives the
+bifurcation equation. For S1-symmetric systems these ideas were introduced in [15], [65],
+and for G × S1 invariant systems in [39, 40, 41]. Also see [21, 22]. Note that on the S1
+bifurcation picture one still has the action of the group G, this leads to bifurcations with
+symmetry [66], [9].
+Following [39] we have for the representation of a linear symplectic G-action the decom-
+position V = V1 ⊕ · · · ⊕ Vℓ, i.e. G = G1 × · · · × Gell, where Gi acts on Vi. For any
+component Gi which is an S1-action we may find periodic solutions on the fixed point
+spaces of the subgroups of Gi using the ideas of [39]. Note that isomorphic subgroups
+13
+
+might have different fixed point spaces in terms of geometric place. Thus for each Gi
+which is an S1-action one can pass to the reduced phase space for this S1-action and find
+stationary points on all isotropy strata.
+Example 6.1 (see [18]) Consider on R8 with standard symplectic form and coordinates
+(q, Q) the Hamiltonian
+H(K, N, Ξ, L1, H2) =3
+4
+�
+3β2 − 2
+�
+K2H2 + (1 − β2)KΞL1 + 1
+2
+�
+4 − β2�
+NH2
++ (3
+2 + β2
+4 )H3
+2 − (β2
+2 + 1)H2
+2
+�
+L1
+2 + Ξ2�
+,
+with
+H2(q, Q) = 1
+2(Q2
+1 + Q2
+2 + Q2
+3 + Q2
+4) + 1
+2(q2
+1 + q2
+2 + q2
+3 + q2
+4) ,
+Ξ(q, Q) = q1Q2 − Q1q2 + q3Q4 − Q3q4 ,
+L1(q, Q) = q3Q4 − Q3q4 − q1Q2 + Q1q2 ,
+K(q, Q) = 1
+2(−(q2
+1 + Q2
+1) − (q2
+2 + Q2
+2) + (q2
+3 + Q2
+3) + (q2
+4 + Q2
+4)) .
+Furthermore
+N(q, Q) = 1
+2 (K2
+2 + K2
+3) − 1
+2 (L2
+2 + L2
+3) ,
+S(q, Q) = K2L3 − K3L2 ,
+with
+K2(q, Q) = (Q2Q3 + q2q3) − (Q1Q4 + q1q4) ,
+K3(q, Q) = −(Q1Q3 + q1q3) − (Q2Q4 + q1q4) ,
+L2(q, Q) = (q1Q3 − Q1q3) + (q2Q4 − Q2q4) ,
+L3(q, Q) = (q2Q3 − Q2q3) − (q1Q4 − Q1q4) .
+This Hamiltonian system has commuting integrals H2(q, Q), Ξ(q, Q), and L1(q, Q). That
+is, G = T3 generated by these three integrals. For G we have the orbit map
+ρ : (q, Q) → (H2(q, Q), Ξ(q, Q), L1(q, Q), N(q, Q), K(q, Q), S(q, Q)) .
+Setting H2(q, Q) = n, Ξ(q, Q) = ξ, L1(q, Q) = ℓ we obtain, after reduction with respect
+to the T3-action generated by H2(q, Q), Ξ(q, Q), and L1(q, Q), the reduced phase space
+(n2 + ξ2 − ℓ2 − K2)2 − 4(nξ − ℓK)2 = 4N 2 + 4S2 .
+in (N, K, S)-space. The T⊯ momentum map is
+J : R8 → (Ξ, L1, H2) ⊂ R3 ,
+14
+
+Figure 1: Different reduced phase space for the values of J , [18]
+.
+which is dual to the orbit map ρ. Hence we may classify the symplectic leaves in the orbit
+space by the values of the momentum map J , see fig. 1.
+Note that the image of this momentum map is not a polytope. It is related to a momentum
+map of deficiency 1 (see [26]). The image is an upside down pyramid with its diagonal
+planes
+Let G⟨F1,··· ,Fk⟩ denote the group generated by F1, · · · , Fk Consider the action of F1 =
+1
+2(L1 + Ξ). G⟨F1⟩ is a subgroup of G⟨H2,Ξ,L1⟩, and on R8 Fix(G⟨F1⟩) = {(q, Q) ∈ R8|q1 =
+Q1 = q2 = Q2 = 0} is an invariant space. Similarly for the action of F2 = 1
+2(Ξ − L1),
+G⟨F2⟩ is a subgroup of G⟨H2,Ξ,L1⟩, and Fix(G⟨F2⟩) = {(q, Q) ∈ R8|q3 = Q3 = q4 = Q4 = 0}
+is an invariant space.
+J (Fix(G⟨F1⟩)) is the restriction of the image of J to the plane Ξ = L1. J (Fix(G⟨F2⟩)) is
+the restriction of the image of J to the plane Ξ = −L1. In the image of the momentum
+map J the fibration in each diagonal plane is equivalent to the fibration of the energy-
+moment map for the harmonic oscillator. Points in the interior correspond to a fibre
+topologically equivalent to T 2, points on the edges correspond to a fibre topologically
+equivalent S1.
+A line with H2 = n corresponds to an invariant surface topologically
+equivalent to S3.
+The points in the interior of the diagonal planes correspond to the singular reduced phase
+15
+
+=1=0
+=1
+=-l
+S
+nElspaces. The fixed point spaces correspond to the isotropy type and orbit type strata on
+the reduced phase space which are the zero dimensional symplectic leaves of the final orbit
+space, that is, they are the cone-like singularities in the singular reduced phase spaces.
+The image J (Fix(G⟨Ξ⟩)) is given by the planes Ξ = ±H2. And J (Fix(G⟨L1⟩)) is given by
+the planes L1 = ±H2. Points in the interior correspond to a fibre topologically equivalent
+to T 2, points on the edges correspond to a fibre topologically equivalent S1.
+These points correspond to the isotropy strata given by the zero dimensional symplectic
+leaves on the final orbit space that correspond to the cases where the reduced phase space
+reduces to an isolated point.
+On these singular fibres one finds relative equilibria for all G-invariant systems.
+Stationary points for the reduced system XH other then the ones found sofar correspond
+to the points where the Hamiltonian is tangent to the reduced phase space. These points
+have as pre-image a T 3 on which one finds relative equilibria.
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+

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@@ -0,0 +1,1817 @@
+arXiv:2301.03738v1  [math.NT]  10 Jan 2023
+Hyperbolic summations derived using the Jacobi functions dc and nc
+John M. Campbell
+Abstract
+We introduce a method that is based on Fourier series expansions related to Jacobi
+elliptic functions and that we apply to determine new identities for evaluating hyper-
+bolic infinite sums in terms of the complete elliptic integrals K and E. We apply our
+method to determine generalizations of a family of sech-sums given by Ramanujan and
+generalizations of a family of csch-sums given by Zucker. Our method has the advan-
+tage of producing evaluations for hyperbolic sums with sign functions that have not
+previously appeared in the literature on hyperbolic sums. We apply our method using
+the Jacobian elliptic functions dc and nc, together with the elliptic alpha function, to
+obtain new closed forms for q-digamma expressions, and new closed forms for series
+related to discoveries due to Ramanujan, Berndt, and others.
+Keywords: Jacobi elliptic function, elliptic integral, hyperbolic function, Γ-function, symbolic evaluation,
+elliptic alpha function
+Mathematics Subject Classification: 42A16, 33E05
+1
+Introduction
+The Jacobi elliptic functions are defined via the inversion of the elliptic integral
+u = F(φ, k) =
+� φ
+0
+dt
+�
+1 − k2 sin2 t
+(1)
+of the first kind, letting 0 < k2 < 1. The expression k = mod u is referred to as the elliptic
+modulus, and φ = am(u, k) = am(u) is referred to as the Jacobi amplitude. The Jacobi
+elliptic functions sn, cn, and dn may be defined as follows:
+sin φ = sn(u, k),
+(2)
+cos φ = cn(u, k),
+and
+(3)
+�
+1 − k2 sin2 φ = dn(u, k).
+(4)
+In this article, we apply Fourier series expansions related to Jacobi elliptic functions to build
+on the work on hyperbolic infinite sums due to Zucker [67], Yakubovich [64], and Xu [63]. The
+relevance of Zucker’s work in [67] (cf. [66]) within high-energy physics [17, 18, 23], nuclear
+physics [22], electrostatics [38], the study of lattice sums [16], number-theoretic subjects
+in the study of Fibonacci sums [24, 25, 26, 27, 28], and number-theoretic areas concerning
+Ramanujan’s series for 1
+π [4, 5, 15] and for constants such as Ap´ery’s constant [59] serve as
+a main source of motivation behind the techniques and results introduced in this article.
+1
+
+1.1
+Background and further preliminaries
+The complete elliptic integral of the first kind K is such that K(k) = F
+� π
+2, k
+�
+. We may also
+write K(k) = K and K′ = K′(k) = K(k′), with k′ =
+√
+1 − k2, and similarly with respect to
+the complete elliptic integral E(k) =
+� π
+2
+0
+�
+1 − k2 sin2 θ dθ of the second kind.
+Combinations involving the sn, cn, and dn functions give us the remaining Jacobi elliptic
+functions, as listed below:
+cd(u, k) = cn(u, k)
+dn(u, k),
+(5)
+cs(u, k) = cn(u, k)
+sn(u, k),
+dc(u, k) = dn(u, k)
+cn(u, k),
+(6)
+ds(u, k) = dn(u, k)
+sn(u, k) ,
+nc(u, k) =
+1
+cn(u, k),
+nd(u, k) =
+1
+dn(u, k),
+ns(u, k) =
+1
+sn(u, k),
+sc(u, k) = sn(u, k)
+cn(u, k),
+and
+sd(u, k) = sn(u, k)
+dn(u, k).
+The Γ-function [49, §8] is famous and ubiquitous as a special function, and is to frequently
+arise in our work. We recall that this special function is defined for ℜ(x) > 0 with the Euler
+integral Γ(x) =
+� ∞
+0 ux−1e−u du [49, §8]. A useful feature concerning our method in Section
+1.2 is given by how this method may be used to obtain new evaluations for expressions
+involving the q-digamma function ψq(z), which may be defined so that
+ψq(z) =
+1
+Γq(z)
+∂Γq(z)
+∂z
+,
+where Γq denotes the q-analogue of the Γ-function.
+Equivalently, we may define the q-
+digamma function so that
+ψq(z) = − ln(1 − q) + ln q
+∞
+�
+n=0
+qn+z
+1 − qn+z .
+(7)
+2
+
+In Ramanujan’s second notebook [12, §17] (cf. [8]), identities for evaluating
+∞
+�
+n=0
+(−1)n(2n + 1)ssech
+�2n + 1
+2
+K′
+K π
+�
+(8)
+in terms of K are given, such as the identity [12, p. 134]
+∞
+�
+n=0
+(−1)n(2n + 1)sech
+�2n + 1
+2
+K′
+K π
+�
+= 2kk′K2
+π2
+.
+(9)
+In Zucker’s seminal article on hyperbolic sums [67], an identity for evaluating (8) was also
+included, and [67] also provided identities for the sums given by replacing sech with csch in
+Ramanujan’s sums in (8):
+∞
+�
+n=0
+(−1)n(2n + 1)scsch
+�2n + 1
+2
+K′
+K π
+�
+= 2Js,
+(10)
+with, for example,
+4J0 =
+�2K
+π
+�
+k
+and
+4J2 =
+�2K
+π
+�3
+k(1 − k2).
+Zucker’s methods in [67] mainly relied on double series manipulations together with expan-
+sions such as
+Js(c) = Js =
+∞
+�
+n=1
+(−1)n+1(2n − 1)sqn−1/2
+1 − q2n−1
+,
+writing
+q = e−π K′
+K
+(11)
+to denote the nome for Jacobian elliptic functions. In this article, we provide a method that
+may be used to to evaluate the members of both of the families of generalizations of (8) and
+(10) indicated as follows:
+1. The sums obtained by replacing sech (resp. csch) with the higher power sech2 (resp.
+csch2), for all odd powers s, in (8) (resp. (10)); and
+2. The sums given by replacing the sign function (−1)n with the sign function (−1)⌊ n
+2⌋
+within the sums indicated in the preceding point.
+Our evaluation technique may be applied much more broadly, apart from the families of
+hyperbolic sums indicated above. For example, our method may also be applied to produce
+new summations that resemble and are related to the summations
+∞
+�
+n=1
+n
+e2πn − 1 = 1
+24 − 1
+8π
+and
+∞
+�
+n=1
+n13
+e2nπ − 1 = 1
+24
+3
+
+due to Ramanujan [51, pp. 326, xxvi]. These Ramanujan summations have been explored by
+authors such as Nanjundiah [44] and Sandham [53]. As in [44], we record that Ramanujan
+had given the first out of the above identities in his seminal article on modular equations
+and approximations to π [50].
+As in Yakubovich’s article [64], our work is a continuation of the methods due to Ling
+and Zucker [39, 40, 67]. Yakubovich’s symbolic forms as in
+Γ4 �1
+4
+�
+32π4 + Γ8 � 1
+4
+�
+512π6 −
+1
+8π2 =
+∞
+�
+n=1
+n2 cosh(πn)csch2(πn)
+for infinite series involving csch2 were highlighted as Corollaries in [64] and motivate our
+symbolic forms for csch2-sums as in Examples 7 and 8 below.
+1.2
+Main technique
+Our main technique may be summarized in the following manner, letting j(u, k) denote a
+Jacobi elliptic function.
+1. Start with the Fourier series expansion for j(u, k) or for some expression involving a
+Jacobi elliptic function, or some manipulation of such Fourier series expansions such
+as a series expansion obtained via term-by-term applications of a differential operator;
+2. In order to use built-in CAS algorithms for reducing, if possible, derivatives of q-powers
+with respect to the elliptic modulus and using hyperbolic functions, we need to rewrite
+the summand of the series indicated in the previous step so that any exponential
+expressions only appear in the denominator, and we need to simplify the powers for
+any such expressions;
+3. Enforce a substitution such as u �→ 2wK for a variable w;
+4. Argue, if possible, that if w were to be set to some special value, the series obtained
+from the third step would reduce to a closed-form evaluation; and
+5. If the resultant summand is non-vanishing, and if differentiating with respect to the
+elliptic modulus yields a summand that may be expressed in terms of hyperbolic func-
+tions, then simplify the resultant summand.
+1.3
+Organization of the article
+The above technique may be applied to many out of the 12 of the Jacobian functions among
+cd, cn, cs, dc, dn, ds, nc, nd, ns, sc, sd, and sn. However, we sometimes obtain equivalent
+results by applying our technique to different Jacobian elliptic functions. For example, if we
+apply this technique to ns, then the results that we would obtain would also be obtainable
+by applying our technique to dc. The results in this article in Sections 2 and 3 are devoted
+4
+
+to the application of our technique to dc and and nc, and if we were to attempt to apply
+the same method to Jacobi elliptic functions other than dc and and nc, this method would
+either not be applicable, or we would obtain the same results as in Sections 2 and 3 below.
+Our method may also be applied to Fourier series expansions for expressions other than
+the 12 Jacobi functions we have listed. For example, an equivalent formulation of the q-
+expansion
+�2K
+π
+�2
+ns2(2Kw, k) = 4K(K − E)
+π2
++ csc2(πw) − 8
+∞
+�
+n=1
+nq2n cos(2nπw)
+1 − q2n
+(12)
+for ns2 which, as recorded in [67], was included in the classic text [62, p. 535] and dates back
+to the work of Jacobi [32], may be applied in accordance with the technique in Section 1.2.
+This is briefly considered in Section 4.
+2
+Applications of the Jacobi elliptic function dc
+The connections between the Jacobi elliptic functions and the generalized Bessel functions,
+as explored in [20], serve as a further source of motivation concerning our applications related
+to series expansions given in [20]. To begin with, we record the Fourier series expansion
+dc(u, k) = π
+2K sec v + 2π
+K
+∞
+�
+n=0
+(−1)nq2n+1
+1 − q2n+1 cos((2n + 1)v)
+(13)
+given in [20] and in classic texts such as [61, pp. 511–512], writing v = πu
+2K and recalling (11).
+Following the evaluation technique in Section 1.2, we rewrite the above series expansion for
+dc as follows:
+dc(2Kw, k) = π
+2K
+�
+1
+cos(πw) + 4
+∞
+�
+n=0
+(−1)n
+e(2n+1)π K′
+K − 1
+cos((2n + 1)πw)
+�
+.
+(14)
+Setting w = 0, (14) this gives us that
+K
+2π − 1
+4 =
+∞
+�
+n=0
+(−1)n
+e(2n+1)π K′
+K − 1
+,
+(15)
+and the formula in (15) was also proved by Bagis in [3] (Corollary 1), using an identity due
+to Ramanujan [12, p. 174]. Applying term-by-term differentiation to (15) with respect to
+the elliptic modulus, we may obtain that
+2K2 (E + (k2 − 1) K)
+π2(K′(E − K) + E′K) =
+∞
+�
+n=0
+(−1)n(2n + 1)csch2
+�2n + 1
+2
+K′
+K π
+�
+.
+(16)
+We intend to generalize (16), by analogy with Zucker’s sum in (10).
+5
+
+2.1
+Generalizing Zucker’s sum
+To generalize Zucker’s sum in (10) by replacing csch with csch2 in (10), we apply our tech-
+nique indicated in Section 1.2, as in the below proof.
+Theorem 1. The equality
+1
+4 + 2 (k2 − 1) K3
+π3
+=
+∞
+�
+n=0
+(−1)n(2n + 1)2
+e(2n+1) K′
+K π − 1
+holds if the above series converges.
+Proof. We again begin with the Fourier series expansion for dc in (13). We rewrite the
+left-hand side of (13) according to (6), and then apply the operator
+d2
+du2 to both sides of
+the resultant equality. We then expand the left-hand according to the following differential
+equations:
+d snu
+du
+= cnu dnu,
+(17)
+d cn u
+du
+= −snu dnu,
+and
+(18)
+d dnu
+du
+= −k2snu cnu,
+(19)
+After much simplification and rearrangement using (17)–(19), we can show that
+dn(u, k)
+�
+dn2(u, k) − k2cn2(u, k)
+�
+(cn2(u, k) + 2sn2(u, k))
+cn3(u, k)
+(20)
+equals
+π3 �
+3 − cos
+�πu
+K
+��
+sec3 � πu
+2K
+�
+16K3
+− π3
+2K3
+∞
+�
+n=0
+(−1)nq2n+1(2n + 1)2 cos
+�
+(2n+1)πu
+2K
+�
+1 − q2n+1
+,
+(21)
+reversing the order of the limiting operations
+d2
+du2 and �∞
+n=0 ·. Following the steps in Section
+1.2, we set u �→ 2wK. Setting w = 0 then gives us an equivalent formulation of the desired
+result.
+We let elliptic singular values be denoted in the usual way [14, p. 298], writing
+K′(kr)
+K(kr) = √r,
+(22)
+where the elliptic lambda function is such that
+λ∗(r) = kr,
+(23)
+with expressions of the form K(kr) admitting explicit symbolic evaluations for natural num-
+bers r ∈ N. Although the result highlighted as Theorem 1 is to be applied to obtain new
+hyperbolic summations, we are to apply this same result using the relations in (22) and (23).
+6
+
+Example 1. Setting r = 1 in (22), from the symbolic forms
+λ∗(1) = k1 =
+√
+2
+2
+(24)
+and
+K(k1) = Γ2 � 1
+4
+�
+4√π ,
+(25)
+this gives us, from Theorem 1, that
+1
+4 − Γ6 � 1
+4
+�
+64π9/2 =
+∞
+�
+n=0
+(−1)n(2n + 1)2
+e(2n+1)π − 1
+.
+Example 2. Setting r = 4 in (22), the symbolic forms
+λ∗(4) = k4 = 3 − 2
+√
+2,
+(26)
+and
+K(k4) =
+�
+1 +
+√
+2
+�
+Γ2 � 1
+4
+�
+8
+√
+2π
+(27)
+give us, via Theorem 1, that
+1
+4 −
+�
+1 +
+√
+2
+�
+Γ6 � 1
+4
+�
+128π9/2
+=
+∞
+�
+n=0
+(−1)n(2n + 1)2
+e2(2n+1)π − 1 .
+Theorem 1 may be further applied so as to obtain the following evaluation for the sum
+given by setting s = 3 and replacing csch with csch2 in Zucker’s sum in (10).
+Theorem 2. The identity
+8 (k2 − 1) K4 (3E + (k2 − 3) K)
+π4(K′(E − K) + E′K)
+=
+∞
+�
+n=0
+(−1)n(2n + 1)3csch2
+�2n + 1
+2
+K′
+K π
+�
+holds if the above series converges.
+Proof. This follows in a direct way by applying term-by-term differentiation with respect to
+the elliptic modulus in Theorem 1.
+Example 3. Using the elliptic alpha function
+α(r) =
+π
+4K2(kr) + √r − E(kr)√r
+K(kr)
+(28)
+together with the elliptic function identity
+E′ = π
+4K + α(r)K
+(29)
+7
+
+and the known formula α(1) = 1
+2, we may obtain symbolic forms for E(k1) and E′(k1). So,
+in Theorem 2, we set k = k1 as the value in (24), so as to obtain that
+Γ10 � 1
+4
+�
+128π15/2 − 3Γ6 � 1
+4
+�
+32π11/2 =
+∞
+�
+n=0
+(−1)n(2n + 1)3csch2
+�2n + 1
+2
+π
+�
+.
+Example 4. Setting k = k4 as the value in (26), from the known valuation α(4) = 2(
+√
+2−1)2,
+this, together with the elliptic alpha function identities in (28) and (29), allows us to obtain,
+via Theorem 2, that
+�
+2 +
+√
+2
+�
+Γ10 � 1
+4
+�
+1024π15/2
+− 3
+�
+1 +
+√
+2
+�
+Γ6 � 1
+4
+�
+128π11/2
+=
+∞
+�
+n=0
+(−1)n(2n + 1)3csch2((2n + 1)π).
+Relative to our proof of Theorem 2, we may similarly evaluate
+∞
+�
+n=0
+(−1)n(2n + 1)scsch2
+�2n + 1
+2
+K′
+K π
+�
+(30)
+for odd natural numbers s ∈ N≥5. It would be desirable to explicitly evaluate (30) for odd
+s ∈ N. This is nontrivial, and may require combinatorial formulas for higher derivatives
+for dc(u, k) resulting from repeated applications of the differential equation system given by
+(17)–(19). We leave this, together with some other problems considered in Section 5, for a
+separate project.
+2.2
+A new sign function
+Our identities for alternating sums involving factors of the form
+csch2
+�2n + 1
+2
+K′
+K π
+�
+,
+(31)
+as in Theorem 4 below, are of interest in part because these identities cannot be derived
+from previously known results on sums involving expressions such as
+csch
+�2n + 1
+2
+K′
+K π
+�
+,
+such as the relation whereby
+kK
+π
+=
+∞
+�
+n=0
+(−1)ncsch
+�2n + 1
+2
+K′
+K π
+�
+given as (5.3.4.1) in [48] and employed in [64] via term-by-term differentiation with respect to
+the elliptic modulus. To obtain new sums involving (31), we are to interchange the limiting
+operations given by the application of
+d
+dk and the application of �∞
+n=0 · in the new result
+8
+
+highlighted as Theorem 3 below. To the best of our knowledge, hyperbolic sums involving
+sign functions such as n �→ (−1)⌊ n
+2⌋ have not previously appeared in the relevant literature
+on hyperbolic sums, which further motivates the research interest in results as in Theorems
+3 and 4 below.
+Theorem 3. The identity
+�√
+1 − k +
+√
+1 + k
+�
+K
+2π
+− 1
+2 =
+∞
+�
+n=0
+(−1)⌊ n
+2⌋
+e(2n+1) K′
+K π − 1
+holds if the above series converges.
+Proof. From the identity in (14), we find that the quotient
+dn(2Kw, k)
+cn(2Kw, k)
+admits the same expansion as in (14). So, setting w = 1
+4 and applying the half-K formulas
+cn
+�1
+2K, k
+�
+=
+√
+2
+4√
+1 − k2
+√
+1 + k +
+√
+1 − k
+(32)
+and
+dn
+�1
+2K, k
+�
+=
+4√
+1 − k2,
+(33)
+we obtain the expansion
+√
+1 − k +
+√
+k + 1
+√
+2
+= π
+2K
+�
+√
+2 + 4
+∞
+�
+n=0
+(−1)n cos
+�2n+1
+4 π
+�
+e(2n+1) K′
+K π − 1
+�
+,
+which is equivalent to the desired result.
+Example 5. For the elliptic singular value corresponding to the r = 1 case in Theorem 3,
+we obtain that
+�
+2 +
+√
+2Γ2 �1
+4
+�
+8π3/2
+− 1
+2 =
+∞
+�
+n=0
+(−1)⌊ n
+2⌋
+e(2n+1)π − 1.
+This is equivalent to the q-digamma evaluation shown below:
+4π +
+�
+2 +
+√
+2
+π
+Γ2
+�1
+4
+�
+= −ψe8π
+�1
+8
+�
+− ψe8π
+�3
+8
+�
++ ψe8π
+�5
+8
+�
++ ψe8π
+�7
+8
+�
+.
+Example 6. Setting k = k4 in Theorem 3, we may obtain that
+�
+3 + 2
+√
+2 + 2
+�
+4 + 3
+√
+2Γ2 �1
+4
+�
+16π3/2
+− 1
+2 =
+∞
+�
+n=0
+(−1)⌊ n
+2⌋
+e2(2n+1)π − 1.
+9
+
+This is equivalent to the q-digamma evaluation shown below:
+8π +
+�
+3 + 2
+√
+2 + 2
+�
+4 + 3
+√
+2
+π
+Γ2
+�1
+4
+�
+=
+− ψe16π
+�1
+8
+�
+− ψe16π
+�3
+8
+�
++ ψe16π
+�5
+8
+�
++ ψe16π
+�7
+8
+�
+.
+Theorem 4. The identity
+2k (k2 − 1) K2
+�
+(
+√1−k+√1+k)(E+(k2−1)K)
+k(k2−1)
+− 1
+2
+�
+1
+√1+k −
+1
+√1−k
+�
+K
+�
+π2(E′K + (E − K)K′)
+=
+∞
+�
+n=0
+(−1)⌊ n
+2⌋(2n + 1)csch2
+�2n + 1
+2
+K′
+K π
+�
+holds for suitably bounded k.
+Proof. This follows in a direct way by differentiating both sides of the identity in Theorem
+3 with respect to the elliptic modulus, and then reversing the order of differentiation and
+infinite summation, and then applying much simplification.
+Our series as in Example 8 are motivated by series evaluations of a similar appearance
+recorded in [7]. For example, the series evaluation
+∞
+�
+n=1
+(−1)n+1ncsch(πn) = 1
+4π
+recorded [7] was, as noted in [7], previously proved by many difference authors in [19, 35,
+44, 45, 52, 54, 67]. To obtain new closed forms from Theorem 4, we are to make use of the
+elliptic alpha function [14, §5]
+Example 7. Using the elliptic singular value k1, a special case of Theorem 4 gives us that
+�
+2 +
+√
+2Γ2 �1
+4
+�
+4π5/2
+−
+�
+2 −
+√
+2Γ6 � 1
+4
+�
+64π9/2
+=
+∞
+�
+n=0
+(−1)⌊ n
+2⌋(2n + 1)csch2
+�2n + 1
+2
+π
+�
+.
+Example 8. Using the elliptic singular value k4, a special case of Theorem 4 gives us that
+��
+1 +
+√
+2 +
+�
+2 +
+√
+2
+�
+Γ2 � 1
+4
+�
+16π5/2
+−
+�
+4 +
+√
+2 + 27/4Γ6 � 1
+4
+�
+256π9/2
+=
+∞
+�
+n=0
+(−1)⌊ n
+2⌋(2n + 1)csch2((2n + 1)π).
+10
+
+As in [13], we record that Nanjundiah’s formula
+∞
+�
+n=1
+csch2(nπ) = 1
+6 − 1
+2π
+has, subsequent to Nanjundiah 1951 proof [44], been proved by many different authors,
+including Berndt [9], Ling [40], Kiyek and Schmidt [33], Muckenhoupt [43], and Shafer [55].
+Nanjundiah’s formula, together with our new sums involving csch2 given in Examples 7 and
+8, inspire us to further apply our method in the evaluation of sums involving csch2.
+2.3
+Higher powers of 2n + 1
+We are to again make use of the equality of (20) and (21), to prove the following companion
+to Theorem 1.
+Theorem 5. The equality
+3
+2 + 4 (k2 (3 + k′) − 3 (1 + k′)) K3
+�√
+1 − k +
+√
+1 + k
+�
+π3
+=
+∞
+�
+n=0
+(−1)⌊ n
+2⌋(2n + 1)2
+e(2n+1) K′
+K π − 1
+holds if the above series converges.
+Proof. Starting with the equality of (20) and (21), we again set u �→ 2wK. Using the half-K
+formulas in (32) and (33), together with the half-K formula
+sn
+�K
+2 , k
+�
+=
+√
+2
+√
+1 + k +
+√
+1 − k,
+we may obtain that
+√
+2 (k′ + 2) (k′ + 1 − k2)
+√
+1 − k +
+√
+k + 1
+=
+3π3
+4
+√
+2K3 −
+π3
+2
+√
+2K3
+∞
+�
+n=0
+(2n + 1)2(−1)⌊ n
+2⌋
+e(2n+1) K′
+K π − 1
+,
+and this is equivalent to the desired result.
+Example 9. Theorem 5, together with the elliptic integral singular value k1, give us that
+3
+2 −
+�
+26 + 17
+√
+2Γ6 � 1
+4
+�
+64π9/2
+=
+∞
+�
+n=0
+(2n + 1)2(−1)⌊ n
+2⌋
+e(2n+1)π − 1
+.
+Example 10. Theorem 5, together with the elliptic integral singular value k4, give us that
+3
+2 −
+�
+54 + 37
+√
+2 + 4
+�
+352 + 249
+√
+2Γ6 �1
+4
+�
+128π9/2
+=
+∞
+�
+n=0
+(2n + 1)2(−1)⌊ n
+2⌋
+e2(2n+1)π − 1
+.
+11
+
+As below, Theorem 5 may be used to evaluate the s = 3 case for the sum obtained from
+Zucker’s sum in (10) by replacing (−1)n with (−1)⌊ n
+2⌋ and by replacing csch with csch2.
+Theorem 6. The series
+∞
+�
+n=0
+(−1)⌊ n
+2⌋(2n + 1)3csch2
+�2n + 1
+2
+K′
+K π
+�
+may be evaluated as
+�
+4k′K4�
+6
+�
+k
+�
+k
+�√
+1 − k +
+√
+1 + k
+�
+− 3
+√
+1 − k + 3
+√
+1 + k
+�
+− 6
+�√
+1 − k +
+√
+1 + k
+��
+E+
+�
+36
+�√
+1 − k +
+√
+1 + k
+�
++ k
+�
+18
+�√
+1 − k −
+√
+1 + k
+�
++ k
+�
+k
+�
+k
+�√
+1 − k +
+√
+1 + k
+�
+− 8
+√
+1 − k+
+8
+√
+1 + k
+�
+− 23
+�√
+1 − k +
+√
+1 + k
+����
+K
+��� �
+π4�
+k′ + 1
+�
+(K′(E − K) + E′K)
+�
+,
+if the above series is convergent.
+Proof. This may be proved by applying term-by-term differentiation, with respect to the
+elliptic modulus, in Theorem 5.
+Example 11. Setting k = k1, we obtain that
+�
+218 + 151
+√
+2Γ10 � 1
+4
+�
+512π15/2
+− 3
+�
+26 + 17
+√
+2Γ6 �1
+4
+�
+32π11/2
+=
+∞
+�
+n=0
+(−1)⌊ n
+2⌋(2n + 1)3csch2
+�2n + 1
+2
+π
+�
+.
+Example 12. Setting k = k4, we obtain that
+�
+402 + 287
+√
+2 + 4
+�
+20296 + 14358
+√
+2Γ10 � 1
+4
+�
+2048π15/2
+− 3
+�
+54 + 37
+√
+2 + 4
+�
+352 + 249
+√
+2Γ6 � 1
+4
+�
+128π11/2
+equals
+∞
+�
+n=0
+(−1)⌊ n
+2⌋(2n + 1)3csch2((2n + 1)π).
+We may mimic the above proof to evaluate
+∞
+�
+n=0
+(−1)⌊ n
+2⌋(2n + 1)scsch2
+�2n + 1
+2
+K′
+K π
+�
+for odd s.
+12
+
+3
+Applications of the Jacobi elliptic function nc
+The results introduced in this section on symbolic forms for summations involving the sech
+function are inspired by the formula
+Γ2 � 1
+4
+�
+4π3/2 − 1
+2 =
+∞
+�
+n=1
+sech(πn)
+due to Ramanujan, which is highlighted as an especially amazing formula in the Wolfram
+Mathworld entry on the hyperbolic secant [60]. A Ramanujan summation involving sech
+more closely related to our work is Ramanujan’s formula (cf. [9])
+∞
+�
+n=0
+(−1)n(2n + 1)4m−1sech
+�2n + 1
+2
+π
+�
+= 0.
+(34)
+Our applications of identities as in (15) are also inspired by well known Ramanujan formulas
+such as the following [11, §14]:
+∞
+�
+n=1
+n
+e2πn − 1 = 1
+24 − 1
+8π.
+We begin with the following Fourier series expansion [61, pp. 511–512]:
+nc(u, k) =
+π
+2Kk′ sec v − 2π
+Kk′
+∞
+�
+n=0
+(−1)nq2n+1
+1 + q2n+1 cos ((2n + 1)v) .
+Following the steps given in Section 1.2, we obtain that
+nc(2wK, k) =
+π
+2k′K sec(πw) − 2π
+k′K
+∞
+�
+n=0
+(−1)n
+e(2n+1) K′
+K π + 1
+cos((2n + 1)πw).
+Setting w = 0, we can show that the left-hand side of the above equality equals 1. This gives
+us an equivalent formulation of the following result:
+1
+4 −
+√
+1 − k2K
+2π
+=
+∞
+�
+n=0
+(−1)n
+e(2n+1) K′
+K π + 1
+.
+(35)
+An equivalent result is given as Theorem 6 by Bagis in [3]. From this previously known
+formula, by differentiating with respect to the elliptic modulus, we may obtain the identity
+2k′K2(K − E)
+π2(K′(E − K) + E′K) =
+∞
+�
+n=0
+(−1)n(2n + 1)sech2
+�2n + 1
+2
+K′
+K π
+�
+.
+(36)
+13
+
+By direct analogy with the material in Section 2.1, we may generalize (36) so as to evaluate
+∞
+�
+n=0
+(−1)n(2n + 1)ssech2
+�2n + 1
+2
+K′
+K π
+�
+(37)
+for odd s. For the sake of brevity, we omit a detailed examination of this, and we leave it
+do a separate project to obtain an explicit combinatorial formula for (37) based on repeated
+applications of the differential equations among (17)–(19).
+A direct application of the formula in (35) that we have proved is given by how this
+formula allows us to prove a stronger version of the formula
+2
+∞
+�
+n=0
+(−1)n
+e(2n+1)π + 1 +
+∞
+�
+n=0
+sech
+�2n + 1
+2
+π
+�
+= 1
+2
+(38)
+highlighted as Corollary 4.22 in [7]. This was proved using Euler polynomials in [7].
+Theorem 7. The following stronger version of Berndt’s identity in (38) holds:
+Γ2 � 1
+4
+�
+4
+√
+2π3/2 =
+∞
+�
+n=0
+sech
+�2n + 1
+2
+π
+�
+.
+Proof. Using the value for k1 in (24) together with the value for K(k1) shown in (25), the
+identity in (35) gives us that
+1
+4 − Γ2 � 1
+4
+�
+8
+√
+2π3/2 =
+∞
+�
+n=0
+(−1)n
+e(2n+1)π + 1,
+so that the desired result then follows from (38).
+Our strengthening of Corollary 4.22 in [7], as in Theorem 7, has not appeared in past
+publications influenced by [7], which have been based in areas such as number theory [2,
+21, 31, 34, 41, 42, 46, 47, 57, 65] mathematical physics [58], high energy physics [29], and
+nuclear physics [22]. The many areas in mathematics and physics related to the discoveries
+on hyperbolic infinite sums given by Berndt [7] inspire the development of further results as
+in Theorem 7. Since Berndt’s formula in (38) is highlighted as a Corollary in [7], we find it
+appropriate to highlight the following equivalent formulation of Theorem 7 as a Corollary.
+Corollary 1. The q-digamma evaluation
+ψeπ
+�1 − i
+2
+�
+− ψeπ
+�1 + i
+2
+�
+= −iπ − iΓ2 �1
+4
+�
+4
+√
+2π
+holds true.
+Proof. This is almost immediately equivalent to Theorem 7, from the definition in (7).
+14
+
+Many of our results, with a particular regard toward Theorem 7, are of a similar appear-
+ance relative to the formula
+∞
+�
+n=0
+sech2
+�2n + 1
+2
+π
+�
+= 1
+π
+recorded in [7]. As in [7], we record that the above formula has been proved by many different
+authors [33, 40, 44, 67].
+Theorem 8. The identity
+1
+2 −
+√
+k′ �√
+1 − k +
+√
+1 + k
+�
+K
+2π
+=
+∞
+�
+n=0
+(−1)⌊ n
+2⌋
+e(2n+1) K′
+K π + 1
+holds if the above series converges.
+Proof. Setting w =
+1
+4 in the above series expansion for nc(2wK, k), we find that
+1
+cn( k
+2 ,k)
+admits the following expansion:
+√
+2π
+√
+1 − k2K
+�
+1
+2 −
+∞
+�
+n=0
+(−1)⌊ n
+2⌋
+e(2n+1) K′
+K π + 1
+�
+.
+According to the half-K identity shown in (32), we find that
+√
+1 − k +
+√
+1 + k
+√
+2
+√
+k′
+admits the same expansion, and this gives us an equivalent version of the desired result.
+Example 13. The symbolic form
+1
+2 −
+�
+1 +
+√
+2Γ2 �1
+4
+�
+8π3/2
+=
+∞
+�
+n=0
+(−1)⌊ n
+2⌋
+e(2n+1)π + 1
+holds and may be proved using Theorem 8 together with the elliptic singular value indicated
+in (25). This is equivalent to the q-digamma evaluation
+−4π −
+�
+1 +
+√
+2
+π
+Γ2
+�1
+4
+�
+= ψe8π
+�1 − i
+8
+�
++ ψe8π
+�3 − i
+8
+�
+− ψe8π
+�5 − i
+8
+�
+− ψe8π
+�7 − i
+8
+�
+.
+Ramanujan’s closed forms for
+∞
+�
+n=0
+(−1)n(2n + 1)−4t−1sech
+�2n + 1
+2
+π
+�
+(39)
+have been considered, as in [10], as especially notable contributions out of the results given
+in Ramanujan’s notebooks. This further motivates our generalizations of or related to Ra-
+manujan’s sech-sums as in (8) and (39).
+15
+
+Example 14. The symbolic form
+1
+2 −
+�
+1
+2
+�
+2
+√
+2 +
+�
+4 + 3
+√
+2
+�
+Γ2 � 1
+4
+�
+8π3/2
+=
+∞
+�
+n=0
+(−1)⌊ n
+2⌋
+e2(2n+1)π + 1
+holds and may be proved using Theorem 8 together with the elliptic singular value indicated
+in (27). We may rewrite this a q-digamma evaluation in much the same way as before.
+Ramanujan’s many evaluations for infinite hyperbolic sums served as a main source of
+motivation behind Xu’s work in [63]. In particular, Ramanujan’s evaluations whereby
+∞
+�
+n=0
+(2n + 1)2sech
+�2n + 1
+2
+π
+�
+=
+Γ6 �1
+4
+�
+16
+√
+2π9/2
+and
+∞
+�
+n=0
+(2n + 1)2sech2
+�2n + 1
+2
+π
+�
+= Γ8 � 1
+4
+�
+192π6
+were highlighted as a main sources of motivation concerning the results introduced by Xu in
+[63].
+By applying term-by-term differentiation with respect to the elliptic modulus in Theorem
+8, we can show that the expression
+�
+K2√
+k′��
+k
+�√
+1 − k −
+√
+1 + k
+�
++ 2
+�√
+1 − k +
+√
+1 + k
+��
+K − 2
+�√
+1 − k +
+√
+1 + k
+�
+E
+���
+�
+π2(K′(E − K) + E′K)
+�
+equals
+∞
+�
+n=0
+(−1)⌊ n
+2⌋(2n + 1)sech2
+�2n + 1
+2
+K′
+K π
+�
+if the above series converges. By direct analogy with the material in Section 2.3, we can
+mimic our above derivation so as to evaluate
+∞
+�
+n=0
+(−1)⌊ n
+2⌋(2n + 1)ssech2
+�2n + 1
+2
+K′
+K π
+�
+for odd s.
+4
+Powers of Jacobi elliptic functions
+Using (12) together with the technique in Section 1.2, we can prove that
+−K2 (2EKk′ + E2 + K2 (k2(k′ + 1) − 2k′ − 1))
+4π3(K′(E − K) + E′K)
+16
+
+equals
+∞
+�
+n=1
+(−1)nn2csch2
+�2nπK′
+K
+�
+if the above series converges. For example, this gives us that
+−
+1
+32π2 −
+�
+1 +
+√
+2
+�
+Γ4 �1
+4
+�
+128π4
++
+�
+1 +
+√
+2
+�
+Γ8 �1
+4
+�
+2048π6
+=
+∞
+�
+n=1
+(−1)nn2csch2(2nπ).
+We leave it to a separate project to generalize this result and to investigate the use of our
+method together with Fourier series expansions apart from (12) that involve powers of Jacobi
+elliptic functions.
+5
+Further considerations
+The computation of the inverse functions among (2)–(4) often turns out to be difficult,
+even with state-of-the-art Computer Algebra Systems [6]. For example, there is no direct
+way of using built-in Mathematica commands such as JacobiSN, JacobiCN, or JacobiDN to
+compute the “actual” Jacobi elliptic functions defined via (2)–(4) relative to the Mathematica
+definition for the incomplete elliptic F-integral
+� φ
+0
+dt
+�
+1 − k sin2 t
+with arguments φ and k, which is in contrast to (1). This kind of practical computational
+problem motivates the development of new and efficient ways of expressing and applying the
+Jacobi elliptic functions in the form of series expansions involved in this article.
+Ramanujan’s identity in (9) was applied by Berndt in [8] to prove the following remarkable
+closed form:
+� ∞
+−∞
+dx
+cos √x + cosh √x = π
+4
+Γ2 � 1
+4
+�
+Γ2 � 3
+4
+�.
+How can we obtain similar results using our evaluations related to Ramanujan’s formula in
+(9), as in Section 3? We leave this for a separate project. Also, we leave it to a separate
+project to apply our technique using identities such as the third-K formula, as opposed to
+the half-K formula we have applied.
+Acknowledgements
+The author wants to express his sincere thanks to Alexey Kuznetsov for many useful discus-
+sions.
+17
+
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+John M. Campbell
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+[email protected]
+22
+

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+Temperature dependence of photoluminescence intensity and spin contrast in
+nitrogen-vacancy centers
+S. Ernst1,†, P. J. Scheidegger1,†, S. Diesch1, L. Lorenzelli1, and C. L. Degen1,2∗
+1Department of Physics, ETH Zurich, Otto Stern Weg 1, 8093 Zurich, Switzerland. and
+2Quantum Center, ETH Zurich, 8093 Zurich, Switzerland.
+(Dated: January 13, 2023)
+We report on measurements of the photoluminescence (PL) properties of single nitrogen-vacancy
+(NV) centers in diamond at temperatures between 4–300 K. We observe a strong reduction of the
+PL intensity and spin contrast between ca. 10–100 K that recovers to high levels below and above.
+Further, we find a rich dependence on magnetic bias field and crystal strain. We develop a com-
+prehensive model based on spin mixing and orbital hopping in the electronic excited state that
+quantitatively explains the observations. Beyond a more complete understanding of the excited-
+state dynamics, our work provides a novel approach for probing electron-phonon interactions and a
+predictive tool for optimizing experimental conditions for quantum applications.
+The long coherence time [1] and the ease of optical
+spin readout have made the negatively charged nitrogen-
+vacancy (NV) center in diamond a preferred qubit for
+applications in quantum metrology [2] and quantum in-
+formation [3]. Extraordinarily, the NV retains its quan-
+tum properties up to above room temperature, suggest-
+ing its use in both ambient and cryogenic environments.
+At room temperature, researchers have employed the
+NV’s spin as a sensor for magnetic [4, 5] and electric
+fields [6], and thermometry [7, 8]. Cooled to below 10 K,
+spin-dependent optical transitions [9] have facilitated the
+implementation of prototypical quantum networks [10]
+and multi-qubit quantum operations [11]. Additionally,
+cryogenic NV magnetometry has been performed at the
+micron- [12] and nanoscale [13, 14].
+While the photodynamics of NV centers at low temper-
+ature (below 10 K) and around room temperature have
+been studied in detail, the understanding in the interme-
+diate temperature range is incomplete. Initial studies of
+the photoluminescence (PL) emission intensity of NV en-
+sembles revealed a minimum around 25 K attributed to
+time-averaging in the electronic excited state (ES) [15].
+This averaging process is caused by phonon-mediated
+transitions between the two orbital branches [16, 17].
+A temperature-dependent reduction in PL intensity and
+spin contrast was also reported in connection with NV
+charge state instabilities [18].
+Further, spin mixing in
+the ES due to magnetic field [19, 20] or crystal strain [21]
+was identified as another mechanism for loss of PL. The
+strain-related spin mixing at low temperature was found
+to be partially mitigated by application of a large mag-
+netic bias field [15, 20, 22]. Because high PL intensity
+and spin contrast are essential for high-fidelity quantum
+readout and sensitive magnetometry, a complete picture
+of the NV photodynamics in the 10−100 K range is highly
+desirable.
+In this Letter, we report measurements of the PL in-
+tensity and spin contrast for single NV centers between
+4−300 K. We show that a combination of orbital hopping
+and spin mixing in the ES leads to a strong reduction of
+both quantities between 10 − 100 K. Based on measure-
+ments at varying magnetic field (0 − 200 mT) and intrin-
+sic strain (ES splitting 2 × (9 − 80) GHz), we develop
+a comprehensive theoretical model for the temperature-
+dependent dynamics of the ES. As a result, we are able
+to quantitatively describe the NV’s PL intensity and spin
+contrast over the complete parameter range of tempera-
+ture, magnetic field and strain, and find excellent agree-
+ment with experimental data.
+In our study, we investigate single NV centers situ-
+ated in nanostructured pillars, which serve to enhance
+the photon collection efficiency.
+Our samples include
+an isotopically pure diamond plate (NV-1 to NV-4, El-
+ementSix) and a scanning tip fabricated from natural-
+abundance material (NV-5, QZabre). Samples are mea-
+sured in a dry dilution refrigerator (Setup A) at tem-
+peratures between 4 − 100 K; an additional study down
+to 0.35 K did not show further variation in the NV be-
+havior [23].
+A second dry cryostat (Setup B) with a
+temperature range of 30 − 300 K is used to validate the
+aforementioned measurements and extend the range to
+room temperature [24]. Both setups operate in vacuum
+without addition of exchange gas (p < 5.5 · 10−5 mbar).
+Magnetic bias fields, when specified, are applied along
+the NV symmetry axis.
+The central experimental observation of this work is
+reported in Fig. 1, which plots the spin contrast as a
+function of temperature T = 4 − 300 K.
+The PL in-
+tensity follows a similar trend (see Fig. S7 [25]), but is
+more prone to experimental drift. We measure the con-
+trast by integrating the relative difference in PL between
+the mS = 0 and mS = −1 states (subsequently denoted
+by |0⟩ and |−1��) during the first 250 ns under excitation
+with a 520 nm diode laser (Fig. 1(a)). To initialize the
+spin state into |0⟩, we use a 2 µs laser pulse, followed by
+a state swap to |−1⟩ (when needed) using an adiabatic
+inversion microwave pulse [26]. Fig. 1(b) clearly reveals
+three temperature regimes: (I) Below ca. 10 K, the spin
+contrast is mostly constant. (II) Between ca. 10−100 K,
+the spin contrast is strongly reduced with a pronounced
+arXiv:2301.05091v1  [cond-mat.mes-hall]  12 Jan 2023
+
+2
+PL (norm.)
+(a)
+0
+200
+400
+0.0
+0.5
+1.0
+4K
+0
+200
+400
+Time (ns)
+32K
+0
+200
+400
+83K
+10
+100
+Temperature (K)
+0
+10
+20
+30
+40
+Contrast (%)
+3
+30
+300
+(b)
+NV-1
+NV-2
+NV-5
+I
+II
+III
+Figure 1.
+(a) Time-dependent PL traces for NV-2 during a
+laser pulse of the |0⟩ (open circles) and the |−1⟩ (filled circles)
+states at low, intermediate and high temperature. The spin
+contrast is given by the relative difference between the two
+curves (pink shading). Solid lines are fits to the PL dynamics.
+(b) Spin contrast versus temperature for three NV centers
+measured on Setup A (empty markers) and Setup B (filled
+marker). Solid lines show corresponding simulations for NV-
+1 and NV-2. A bias field of 3 mT is applied.
+minimum around 35 K and then recovers for higher tem-
+peratures. (III) Above ca. 100 K, the spin contrast re-
+mains approximately constant up to room temperature.
+In all measurements, the room temperature contrast ex-
+ceeds the low temperature limit.
+At even higher tem-
+peratures, the contrast slowly decrease until it vanishes
+around 700 K [27].
+Before providing a theoretical explanation for the be-
+havior seen in Fig. 1, we briefly recall the mechanism
+for contrast generation by looking at the spin-subspace
+of the NV given in Fig 2(a) [28]. After spin-conserving
+optical excitation from the ground state (GS) into the
+ES, a spin-selective intersystem crossing (ISC) leads to
+preferential population of the shelving state for |±1⟩. Be-
+cause the shelving state 1E has a relatively long lifetime,
+the average PL emission is lower for |±1⟩ compared to
+|0⟩, leading to spin contrast. The PL reduction is tem-
+porary and disappears due to optical re-pumping into |0⟩
+after a few hundred nanoseconds, see Fig 1(a). Crucially,
+this mechanism of contrast generation is effective only for
+as long as there are no transitions between the ES spin
+states.
+We next consider the orbital subspace of the NV ES
+(3E), which is a doublet shown in Fig. 2(b) [28]. In the
+presence of in-plane strain δ⊥ relative to the NV principal
+axis, the ES possesses two orbital branches, Ex and Ey,
+split by 2δ⊥ [29]. In the composite space of orbit and
+spin, each branch has three spin states, leading to a total
+of six energy levels (Fig. 2(c)). We now show that the
+contrast reduction and recovery can be explained by the
+interplay of two mechanisms: spin mixing and orbital
+branch hopping in the ES.
+First, we discuss the effects of spin mixing, meaning
+that the ES eigenstates are not pure spin eigenstates. As
+an example, we consider Fig. 2(c). Here, the |0⟩ state
+is in good approximation an eigenstate of the Ex branch
+but not the Ey branch, where it forms a superposition
+with the |−⟩ ∝ (|+1⟩ − |−1⟩) state. Consequently, op-
+tical excitation into the Ex branch is spin-conserving,
+while excitation into the Ey branch will lead to spin
+mixing.
+In general, the spin mixing amplitudes ϵi be-
+tween the six levels depend on the strain magnitude and
+direction [21], as well as magnetic field alignment [19]
+and magnitude [20]. Therefore, the spin contrast is both
+strain and field-dependent. Although the ϵi are typically
+small, they play a key role in the mechanism of spin re-
+laxation.
+Second, we consider the effects of orbital hopping,
+which refers to spin-conserving transitions between Ex
+and Ey driven by phonons.
+Fig. 2(b) schematically
+depicts the dominant contributions arising from one-
+phonon processes (rates k1) and two-phonon processes
+(rates k2) derived in Ref. [25][30, 31]. The one-phonon
+downward (Ex → Ey) hopping rate is given by
+k↓,1(T, δ⊥) ∝ ηδ3
+⊥ [n(2hδ⊥/kBT) + 1] ,
+(1)
+where η parametrizes the electron-phonon coupling and
+n is the Bose-Einstein distribution function. The rate of
+the two-phonon process is given by
+k↓,2(T) ∝ η2T 5I(T).
+(2)
+where
+I(T)
+is
+a
+mildly
+strain-
+and
+temperature-
+dependent integral over the phonon spectrum that we
+solve in the Debye approximation.
+The total hopping
+rates are the sums of the one- and two-phonon contribu-
+tions, k↓(↑) = k↓(↑),1 + k↓(↑),2. The upward (Ey → Ex)
+rate significantly differs from the downward rate only be-
+low 10 K, where it is reduced by the absence of sponta-
+neous emission (second term in Eq. 1). Fig. 2(d) plots the
+hopping rates for the parameters in Fig. 2(c) as a func-
+tion of temperature. It provides the key to explaining our
+experimental observations in the temperature regimes (I-
+III):
+(I) Below ca. 10 K, the orbital hopping is dominated
+by k↓,1 due to the spontaneous emission. Since k↓,1 is
+slower than the ES decay rate (T −1
+3E ≈ 108 s−1) for typical
+strain values δ⊥ ≲ 40 GHz, the ES spin states are mostly
+preserved (except for some small spin mixing ϵi) and the
+spin contrast is high.
+(II) Above 10 K, the two-phonon process starts to dom-
+inate. Once k > T −1
+3E , spin relaxation between |0⟩ and
+|±1⟩ is drastically amplified, because the time evolution
+under different Larmor precession in both branches be-
+comes randomized by the frequent hopping. As expected,
+
+3
+(c)
+(a)
+(b)
+3E
+Ex
+Ex
+Ey
+Ey
+1E
+3A2
+1A1
+ωx
+ωy
+|x〉|0〉
+|x〉|−〉
+|y〉(|−〉 − ε|0〉)
+|y〉(|0〉 − ε|−〉)
+|x〉
+|y〉
+|x〉|+〉
+|y〉|+〉
+|0〉
+|±1〉
+|0〉
+|±1〉
+2δ⟂
+k↑
+k↑
+(d)
+k↓,1
+k↓,2
+k↓,1
+k↓,2
+k↓
+k↓
+2ωx
+2ωy
+k
+3E
+T -1
+Figure 2.
+(a) Level diagram in the NV spin subspace Hspin
+of the electronic ground (3A2) and excited (3E) states, as well
+as the metastable (1A1, 1E) shelving states.
+The intersys-
+tem crossing (dotted) is spin selective, favoring decay out
+of |±1⟩.
+In (a-c), solid arrows mark spin conserving tran-
+sitions and curly arrows symbolize phonons.
+(b) Level di-
+agram in the orbital subspace Horbit of the NV ES (3E).
+Two orbital branches (Ex, Ey) split under strain δ⊥. One-
+and two-phonon processes cause hopping between branches
+at temperature-dependent rates k↓,1(2) (k↑,1(2) not shown).
+(c) Example of levels in the composite Hilbert space of orbit
+and spin Horbit ⊗ Hspin. Eigenstates are superpositions of |0⟩
+and |±⟩ ∝ (|+1⟩ ± |−1⟩). Spin-conserving, phonon-mediated
+transitions involving |0⟩ are depicted by gray arrows. ωx(y)
+are the Larmor frequencies of involved spin transitions. (d)
+Hopping rates as a function of temperature. Inverse optical
+lifetime T −1
+3E and Larmor frequencies (in MHz) are indicated
+by horizontal lines. For (c, d) we use δ⊥ = 40 GHz in the di-
+rection of a carbon bond at low magnetic field, yielding only
+one significant |ϵ|2 = 0.1.
+this relaxation mechanism is most efficient when a hop-
+ping event occurs every half of a Larmor precession pe-
+riod (2ωx(y))−1 (see Fig. S11 [25]). This occurs between
+ca. 30 − 40 K (gray shading in Fig. 2(d)) and coincides
+with the temperature where we observe the strongest sup-
+pression of the spin contrast.
+(III) As the temperature increases further, the orbital
+hopping rates become much faster than the spin dynam-
+ics and the two orbital states are time-averaged [15, 29].
+This effectively renders 3E an orbital singlet similar to
+the GS 3A2 [32] and leads to the commonly accepted
+room-temperature model appearing as in Fig. 2(a). Since
+|0⟩ and |±1⟩ are pure eigenstates of the time-averaged
+Hamiltonian, the highest spin contrast is observed in this
+regime.
+Armed with this theory, we implement a rate model to
+quantitatively reproduce the experimental observations
+by numerical simulations (details in Ref. [25]). We model
+the orbital hopping by spin-conserving Markovian tran-
+sitions between the two orbital branches. Since spin co-
+herences are maintained during the transitions, we use
+a Lindblad master equation rather than a classical rate
+model. We describe the ES in a composite Hilbert space
+of spin and orbit (HES = Horbit ⊗ Hspin) and formulate
+the spin-conserving jump operators as
+LES
+↓
+=
+�
+k↓,1 + k↓,2 |y⟩⟨x| ⊗ I3 ,
+(3)
+and likewise for LES
+↑ . We further introduce optical excita-
+tion, decay and ISC by classical jump operators. The re-
+sulting Liouville equation describes the time evolution of
+the 10-dimensional density matrix ρ(t), containing three
+GS levels, six ES levels and one combined shelving state.
+To simulate the behavior of a chosen NV center, we
+feed our model with values obtained from a simultane-
+ous fit of three sets of characterization measurements: (i)
+We use a measurement of the steady-state PL intensity
+as a function of magnetic field at base temperature (see
+Fig. 3(b, 4 K)) to obtain strain values and unintended
+misalignment of the bias field, fitting the minima in the
+PL at level anti-crossings [15, 20]. (ii) We pick a set of 24
+time-dependent PL traces (c.f. Fig. 1(a)), including two
+spin states (|0⟩, |−1⟩), six temperatures (4 − 100 K), and
+low and high bias field (3 mT, 200 mT). Fits to these PL
+traces then yield the optical decay and ISC rates, which
+are approximately temperature-independent [33] and are
+known to vary between NV centers [28, 34], as well as
+the coupling strength η. We determine the shelving state
+lifetime [34, 35], which has a mild, well-known tempera-
+ture dependence, in a separate calibration. (iii) For each
+time-dependent PL trace, we perform an optical satu-
+ration measurement to quantify drift in the background
+luminescence, optical alignment, and ratio of collection
+over excitation efficiency. Finally, we use literature val-
+ues for the NV fine structure [36, 37].
+As an important side result, our calibration yields
+values
+for
+the
+electron-phonon
+couplings
+η
+rang-
+ing from 176 µs−1 meV−3 (NV-2, used in Fig. 2(d))
+to 268 µs−1 meV−3 (NV-4), in good agreement with
+Refs. [17, 30, 31]. We note that these studies use dif-
+ferent phonon models in the evaluation of I(T). While
+our data does not allow validation of a particular model
+with certainty, our measurement approach provides com-
+plementary insight into I(T) [25].
+We are now ready to return to Fig. 1(b) and use
+our model and calibration to simulate the temperature-
+dependent PL and spin contrast (solid curves). Overall,
+we find an excellent agreement between experimental and
+simulated results. In particular, the model quantitatively
+reproduces all temperature regimes (I-III), including the
+minimum in contrast around 35 K and the recovery to-
+wards room temperature. Although the agreement is not
+perfect at elevated temperatures, which we attribute to
+setup instabilities and uncertainty in temperature cali-
+bration [25], our model successfully bridges the classi-
+
+4
+0
+50 100 150 200
+Magnetic Field (mT)
+E (arb. u.)
+4K
+(d)
+0
+50 100 150 200
+E (arb. u.)
+300K
+(c)
+0
+100
+200
+Magnetic Field (mT)
+20
+40
+60
+80
+100
+Temperature (K)
+(a)
+PL (norm.)
+0.6
+1.0
+90
+130
+100K
+(b)
+90
+130
+70K
+50
+90
+29K
+0
+100
+200
+Magnetic Field (mT)
+70
+110
+4K
+PL (kct/s)
+15
+30
+45
+(e)
+80GHz
+15
+30
+45
+Contrast (%)
+40GHz
+10
+5
+20
+40
+80
+Temperature (K)
+15
+30
+45
+= 9GHz
+Figure 3.
+(a) Simulation of the PL in dependence of the magnetic field and temperature. The PL is strongly reduced at avoided
+crossings of the excited state (symbols) and the ground state (103 mT) energy levels. The simulation is based on parameters
+fitted to NV-1. (b) Experimental PL curves for NV-1 measured as a function of magnetic field and temperature. Solid lines
+show the model prediction [25]. (c,d) Energy levels for the NV-1 ES at 300 K (c, time-averaged) and at 4 K (d). Symbols refer
+to (a). (e) Experimental spin contrast as a function of temperature for NV centers with high, intermediate and low intrinsic
+strain δ⊥. Measurements are taken at 3 mT (empty circles) and 200 mT (filled circles). Solid lines show the model prediction.
+cal rate models used in the the limits of low [20] and
+high [19, 34] temperatures.
+Next, we use our model to predict the PL proper-
+ties as a function of magnetic bias field. In Fig. 3(a,b),
+we plot the simulated PL intensity as a function of
+B = 0−200 mT and T = 0−100 K together with the ex-
+perimental results. The model successfully predicts the
+known minima in PL (indicated by symbols) at magnetic
+fields that correspond to level-anti-crossings in the ES,
+in both the low (Fig. 3(d)) and high temperature limit
+(Fig. 3(c), obtained from Fig. 3(d) by a partial trace
+over the orbital subspace). Our model also reveals that
+with increasing magnetic field, the PL minimum becomes
+less pronounced and shifts to higher temperatures. This
+behavior is readily explained by a lower degree of spin
+mixing in the eigenstates (smaller ϵi in Fig. 2(c)) and
+higher Larmor frequencies (larger ωx(y) in Fig. 2(d)) at
+high field. However, even at the highest field accessible
+in our experiment (200 mT), the PL minimum is still no-
+ticeable. Full recovery of the PL is expected for fields
+significantly above 1 T (Fig. S13 [25]).
+Finally, we examine the influence of crystal strain. In
+Fig. 3(e), we compare the temperature dependence of the
+spin contrast for NV centers with high (NV-4, 80 GHz),
+medium (NV-2, 40 GHz), and low (NV-3, 9 GHz) intrin-
+sic strain within our accessible range (NV-1 has 32 GHz).
+While all curves show the same qualitative behavior, we
+find that the most prominent feature is a decrease in
+the spin contrast at high strain δ⊥ already below 10 K.
+This feature can be understood through the factor δ3
+⊥
+in Eq. 1: k↑,1 is rapidly increasing as the required high
+energy phonon modes become thermally activated, ap-
+proaching k↓,1, which is generally high due to sponta-
+neous emission (Fig. S10 [25]).
+In conclusion, we developed a rate model that explains
+the NV center photo-physics over a broad range of tem-
+perature, magnetic bias field and crystal strain, and find
+excellent agreement with the experiment.
+In particu-
+lar, our model successfully predicts a minimum in the
+PL emission and spin contrast around 35 K due to rapid
+spin relaxation driven by an interplay of spin mixing and
+orbital hopping. This feature is of fundamental nature
+and thus universal to all NV centers, including NV cen-
+ters deep in the bulk that experience negligible crystal
+strain [29] (Fig. S13 [25]).
+Our work provides useful insight beyond giving a
+more complete picture of the NV excited-state dynam-
+ics.
+Firstly, our model can account for the observed
+temperature dependence by phonon-induced processes
+in the ES alone.
+Therefore, we conclude that charge-
+state switching between NV− and NV0 does not play a
+key role in explaining the spin contrast as a function of
+temperature.
+We also have not observed any signs of
+charge state instabilities on the few-minutes time scale
+of our measurements (see Fig. S2 [25]).
+Second, our
+work introduces a new measurement approach for prob-
+ing electron-phonon interactions and contributing modes,
+applicable in regimes where resonant laser PL excitation
+spectroscopy [30] or measurement of motional narrowing
+on ES ODMR lines [31] are unavailable. Third, we ex-
+amined the rich dependence on magnetic field, strain (or
+equivalently electric field [28]), and temperature. Here,
+our model offers a predictive tool for maximizing the PL
+intensity and spin contrast, which are the key quantities
+for achieving high spin readout fidelity and high metrol-
+ogy sensitivity in quantum applications.
+The authors thank Matthew Markham (ElementSix)
+for providing the 12C diamond, Jan Rhensius (QZabre)
+
+5
+for nanofabrication, and Erika Janitz, Fedor Jelezko, As-
+saf Hamo, Konstantin Herb, William Huxter, Patrick
+Maletinsky, Francesco Poggiali, Friedemann Reinhard,
+J¨org Wrachtrup and Jonathan Zopes for useful input and
+discussions. This work was supported by the European
+Research Council through ERC CoG 817720 (IMAG-
+INE), the Swiss National Science Foundation (SNSF)
+through Project Grant No. 200020 175600 and through
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+Research in Quantum Science and Technology, Grant
+No.
+51NF40-185902, and the Advancing Science and
+TEchnology thRough dIamond Quantum Sensing (AS-
+TERIQS) program, Grant No. 820394, of the European
+Commission.
+∗ [email protected]; †These authors contributed equally.
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+

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+arXiv:2301.05325v1  [math.GR]  12 Jan 2023
+A NOTE ON PROPERLY DISCONTINUOUS ACTIONS
+MICHAEL KAPOVICH
+Abstract. We compare various notions of proper discontinuity for group actions. We
+also discuss fundamental domains and criteria for cocompactness.
+To the memory of Sasha Anan’in
+1. Introduction
+This note is meant to clarify the relation between different commonly used definitions
+of proper discontinuity without the local compactness assumption for the underlying topo-
+logical space. Much of the discussion applies to actions of nondiscrete topological groups,
+but, since my primary interest is geometric group theory, I will work only with discrete
+groups. All group actions are assumed to be continuous, in other words, these are homo-
+morphisms from abstract groups to groups of homeomorphisms of topological spaces. This
+combination of continuous and properly discontinuous, sadly, leads to the ugly terminology
+“a continuous properly discontinuous action.” A better terminology might be that of a
+properly discrete action, since it refers to proper actions of discrete groups.
+Throughout this note, I will be working only with topological spaces which are 1st count-
+able, since spaces most common in metric geometry, geometric topology, algebraic topology
+and geometric group theory satisfy this property. One advantage of this assumption is that
+if (xn) is a sequence converging to a point x ∈ X, then the subset {x} ∪ {xn : n ∈ N} is
+compact, which is not true if we work with nets instead of sequences. However, I will try
+to avoid the local compactness assumption whenever possible, since many spaces appear-
+ing in metric geometry and geometric group theory (e.g. asymptotic cones) and algebraic
+topology (e.g. CW complexes) are not locally compact. (Recall that topological space X
+is locally compact if every point has a basis of topology consisting of relatively compact
+subsets.) In the last two sections of the note I also discuss criteria for cocompactness of
+group actions and fundamental sets/domains of properly discontinuous actions.
+Acknowledgement. I am grateful to Boris Okun for pointing out several typos and
+the reference to [9].
+2. Group actions
+A topological group is a group G equipped with a topology such that the multiplication
+and inversion maps
+G × G → G, (g, h) �→ gh, G → G, g �→ g−1
+Date: January 16, 2023.
+
+2
+MICHAEL KAPOVICH
+are both continuous. A discrete group is a group with discrete topology. Every discrete
+group is clearly a topological group.
+A left continuous action of a topological group G on a topological space X is a continuous
+map
+λ : G × X → X
+satisfying
+1. λ(1G, x) = x for all x ∈ X.
+2. λ(gh, x) = λ(g, λ(h, x)), for all x ∈ X, g, h ∈ G.
+From this, it follows that the map ρ : G → Homeo(X)
+ρ(g)(x) = λ(g, x),
+is a group homomorphism, where the group operation φψ on Homeo(X) is the composition
+φ ◦ ψ.
+If G is discrete, then every homomorphism G → Homeo(X) defines a left continuous
+action of G on X.
+The shorthand for ρ(g)(x) is gx or g · x. Similarly, for a subset A ⊂ X, GA or G · A,
+denotes the orbit of A under the G-action:
+GA =
+�
+g∈G
+gA.
+The quotient space X/G (also frequently denoted G\X), of X by the G-action, is the
+set of G-orbits of points in X, equipped with the quotient topology: The elements of X/G
+are equivalence classes in X, where x ∼ y when Gx = Gy (equivalently, y ∈ Gx).
+The stabilizer of a point x ∈ X under the G-action is the subgroup Gx < G given by
+{g ∈ G : gx = x}.
+An action of G on X is called free if Gx = {1} for all x ∈ X. Assuming that X is Hausdorff,
+Gx is closed in G for every x ∈ X.
+Example 1. An example of a left action of G is the action of G on itself via left multipli-
+cation:
+λ(g, h) = gh.
+In this case, the common notation for ρ(g) is Lg. This action is free.
+3. Proper maps
+Properness of certain maps is the most common form of defining proper discontinuity;
+sadly, there are two competing notions of properness in the literature.
+A continuous map f : X → Y of topological spaces is proper in the sense of Bourbaki,
+or simply Bourbaki–proper (cf. [2, Ch. I, §10, Theorem 1]) if f is a closed map (images of
+closed subsets are closed) and point–preimages f −1(y), y ∈ Y , are compact. A continuous
+map f : X → Y is proper (and this is the most common definition) if for every compact
+subset K ⊂ X, f −1(K) is compact. It is noted in [2, Ch. I, §10; Prop. 7] that if X is
+Hausdorff and Y is locally compact then f is Bourbaki–proper if and only if f is proper.
+
+A NOTE ON PROPERLY DISCONTINUOUS ACTIONS
+3
+The advantage of the notion of Bourbaki-properness is that it applies in the case of
+Zariski topology, where spaces tend to be compact (every subset of a finite-dimensional
+affine space is Zariski-compact) and, hence, the standard notion of properness is useless.
+Since our goal is to trade local compactness for 1st countability, I will prove
+Lemma 2. If f : X → Y is proper, and X, Y are Hausdorff and 1st countable, then f is
+Bourbaki-proper.
+Proof. We only have to verify that f is closed. Suppose that A ⊂ X is a closed subset.
+Since Y is 1st countable, it suffices to show that for each sequence (xn) in A such that
+(f(xn)) converges to y ∈ Y , there is a subsequence (xnk) which converges to some x ∈ A
+such that f(x) = y. The subset C = {y} ∪ {f(xn) : n ∈ N} ⊂ Y is compact. Hence, by
+properness of f, K = f −1(C) is also compact. Since X is Hausdorff, and K is compact,
+follows that (xn) subconverges to a point x ∈ K. By continuity of f, f(x) = y. Since A is
+closed, x ∈ A.
+□
+Remark 3. In fact, for this lemma to hold, one does not need to assume that X is Haus-
+dorff and 1st countable, see
+The converse (each Bourbaki–proper map is proper) is proven in [2, Ch. I, §10; Prop.
+6] without any restrictions on X, Y . Hence:
+Corollary 4. For maps between 1st countable Hausdorff spaces, Bourbaki–properness is
+equivalent to properness.
+4. Proper discontinuity
+Suppose that X is a 1st countable Hausdorff topological space, G a discrete group and
+G × X → X a (continuous) action. We use the notation gn → ∞ in G to indicate that gn
+converges to ∞ in the 1-point compactification G ∪ {∞} of G, i.e. for every finite subset
+F ⊂ G,
+card({n : gn ∈ F}) < ∞.
+Given a group action G × X → X and two subsets A, B ⊂ X, the transporter subset
+(A|B)G is defined as
+(A|B)G := {g ∈ G : gA ∩ B ̸= ∅}.
+Properness of group actions is (typically) stated using certain transporter sets.
+Definition 5. Two points x, y ∈ X are said to be G-dynamically related if there is a
+sequence gn → ∞ in G and a sequence xn → x in X such that gnxn → y.
+A point x ∈ X is said to be a wandering point of the G-action if there is a neighborhood
+U of x such that (U|U)G is finite.
+Lemma 6. Suppose that the action G × X → X is wandering at a point x ∈ X. Then the
+G-action has a G-slice at x, i.e. a neighborhood Wx ⊂ U which is Gx-stable and for all
+g /∈ Gx, gWx ∩ Wx = ∅.
+
+4
+MICHAEL KAPOVICH
+Proof. For each g ∈ (U|U)G − Gx we pick a neighborhood Vg ⊂ U of x such that
+gVg ∩ Vg = ∅.
+Then the intersection
+V :=
+�
+g∈(U|U)G−Gx
+Vg
+satisfies the property that (V |V )G = Gx. Lastly, take
+Wx :=
+�
+g∈Gx
+V.
+□
+The next lemma is clear:
+Lemma 7. Assuming that X is Hausdorff and 1st countable, the action G × X → X is
+wandering at x if and only if x is not dynamically related to itself.
+Given a group action α : G × X → X, we have the natural map
+ˆα := α × idX : G × X → X × X
+where idX : (g, x) �→ x.
+Definition 8. An action α of a discrete group G on a topological space X is Bourbaki–
+proper if the map ˆα is Bourbaki-proper.
+Lemma 9. If the action α : G×X → X of a discrete group G on an Hausdorff topological
+space X is Bourbaki-proper then the quotient space X/G is Hausdorff.
+Proof. The quotient map X → X/G is an open map by the definition of the quotient
+topology on X/G. Since α is Bourbaki-proper, the image of the map ˆα is closed in X × X.
+This image is the equivalence relation on X ×X which use used to form the quotient X/G.
+Now, Hausdorffness of X/G follows from Theorem 7.7 in [12].
+□
+Definition 10. An action α of a discrete group G on a topological space X is properly
+discontinuous if the map ˆα is proper.
+We note that equivalence of (1) and (5) in the following theorem is proven in [2, Ch. III,
+§4.4, Proposition 7] without any assumptions on X.
+Theorem 11. Assuming that X is Hausdorff and 1st countable, the following are equiva-
+lent:
+(1) The action α : G × X → X is Bourbaki-proper.
+(2) For every compact subset K ⊂ X,
+card((K|K)G) < ∞.
+(3) The action α : G × X → X is proper, i.e. the map ˆα is proper.
+(4) For every compact subset K ⊂ X, there exists an open neighborhood U of K such
+that card((U|U)G) < ∞.
+
+A NOTE ON PROPERLY DISCONTINUOUS ACTIONS
+5
+(5) For any pair of points x, y ∈ X there is a pair of neighborhoods Ux, Vx (of x, y
+respectively) such that card((Ux|Vy)G)) < ∞.
+(6) There are no G-dynamically related points in X.
+(7) Assuming, that G is countable and X is completely metrizable1 : The G-stabilizer
+of every x ∈ X is finite and for any two points x ∈ X, y ∈ X − Gx, there exists a
+pair of neighborhoods Ux, Vy (of x, resp. y) such that ∀g ∈ G, gUx ∩ Vy = ∅.
+(8) Assuming that X is a metric space and the action G × X → X is equicontinuous2:
+There is no x ∈ X and a sequence hn → ∞ in G such that hnx → x.
+(9) Assuming that X is a metric space and the action G × X → X is equicontinuous:
+Every x ∈ X is a wandering point of the G-action.
+(10) Assuming that X is a CW complex and the action G × X → X is cellular: Every
+point of X is wandering.
+(11) Assuming that X is a CW complex the action G × X → X is cellular: Every cell
+in X has finite G-stabilizer.
+Proof. The action α is Bourbaki-proper if and only if the map ˆα is proper (see Corollary 4)
+which is equivalent to the statement that for each compact K ⊂ X, the subset (K|K)G×K
+is compact. Hence, (1) ⇐⇒ (2).
+The property (3) means that for each compact K ⊂ X, ˆα−1(K ×K) = {(g, x) ∈ G×K :
+x ∈ K, gx ∈ K} is compact. This subset is closed in G × X and projects onto (K|K)G in
+the first factor and to the subset
+(12)
+�
+g∈(K|K)G
+g−1(K).
+Hence, properness of α implies finiteness of (K|K)G.
+Conversely, if (K|K)G is finite,
+compactness of g−1(K) for every g ∈ G implies finiteness of the union (12). Thus, (2) ⇐⇒
+(3).
+In order to show (2)⇒(6), suppose that x, y are G-dynamically related points: There
+exists an sequence gn → ∞ in G and a sequence xn → x such that gn(xn) → y. The subset
+K = {x, y} ∪ {xn, gn(xn) : n ∈ N}
+is compact. However, yn ∈ gn(K) ∩ K for every n. A contradiction.
+(6)⇒(5): Suppose that the neighborhoods Ux, Vy do not exist. Let {Un}n∈N, {Vn}n∈N
+be countable bases at x, y respectively. Then for every n there exists gn ∈ G, such that
+gn(Un) ∩ Vn ̸= ∅ for infinitely many gn’s in G. After extraction, gn → ∞ in G. This yields
+points xn ∈ Un, yn = gn(xn) ∈ Vn. Hence, xn → x, yn → y. Thus, x is G-dynamically
+related to y. A contradiction.
+(5)⇒(4).
+Consider a compact K ⊂ X.
+Then for each x ∈ K, y ∈ K there exist
+neighborhoods Ux, Vy such that (Ux|Vy)G is finite. The product sets Ux × Vy, x, y ∈ K
+constitute an open cover of K2. By compactness of K2, there exist x1, ..., xn, y1, ..., ym ∈ K
+1It suffices to assume that X is hereditarily Baire: Every closed subset of X is Baire.
+2E.g. an isometric action.
+
+6
+MICHAEL KAPOVICH
+such that
+K ⊂ Ux1 ∪ ... ∪ Uxn
+K ⊂ Vy1 ∪ ... ∪ Vym
+and for each pair (xi, yj),
+card({g ∈ G : gUxi ∩ Vyj ̸= ∅}) < ∞.
+Setting
+W :=
+n�
+i=1
+Uxi, V :=
+m
+�
+j=1
+Vyj
+we see that
+card((W|V )G) < ∞.
+Taking U := V ∩ W yields the required subset U.
+The implication (4)⇒(2) is immediate.
+Thus, we concluded the proof of equivalence of the properties (1)—(6).
+(5)⇒(7): Finiteness of G-stabilizers of points in X is clear. Let x, y be points in distinct
+G-orbits. Let U ′
+x, V ′
+y be neighborhoods of x, y such that (U ′
+x|V ′
+y)G = {g1, ..., gn}. For each
+i, since X is Hausdorff, there are disjoint neighborhoods Vi of y and Wi of gi(xi). Now set
+Vy :=
+n�
+i=1
+Vi,
+Ux :=
+n�
+i=1
+g−1
+i
+(Wi).
+Then gUx ∩ Vy = ∅ for every g ∈ G.
+(7)⇒(6): It is clear that (7) implies that there are no dynamically related points with
+distinct G-orbits. In particular, every G-orbit in X is closed.
+Assume now that X is completely metrizable and G is countable. Suppose that a point
+x ∈ X is G-dynamically related to itself. Since the stabilizer Gx is finite, the point x is
+an accumulation point of Gx; moreover, Gx is closed in X. Hence, Gx is a closed perfect
+subset of X. Since X admits a complete metric, so does its closed subset Gx. Thus, for
+each g ∈ G, the complement Ug := Gx − {gx} is open and dense in Gx. By the Baire
+Category Theorem, the countable intersection
+�
+g∈G
+Ug
+is dense in Gx. However, this intersection is empty. A contradiction.
+It is clear that (6)⇒(8) (without any extra assumptions).
+(8)⇒(6). Suppose that X is a metric space and the G-action is equicontinuous. Equicon-
+tinuity implies that for each z ∈ X, a sequence zn → z and gn ∈ G,
+gnzn → gz.
+Suppose that there exist a pair of G-dynamically related points x, y ∈ X: ∃xn → x, gn ∈
+G, gnxn → y. By equicontinuity of the action, gnx → y.
+Since gn → ∞, there exist
+
+A NOTE ON PROPERLY DISCONTINUOUS ACTIONS
+7
+subsequences gni → ∞ and gmi → ∞ such that the products hi := g−1
+ni gmi are all distinct.
+Then, by equicontinuity,
+hix → x.
+A contradiction.
+The implications (5)⇒(9)⇒(8) and (5)⇒(10)⇒(11) are clear.
+Lastly, let us prove the implication (11)⇒(2). We first observe that every CW complex
+is Hausdorff and 1st countable. Furthermore, every compact K ⊂ X intersects only finitely
+many open cells eλ in X. (Otherwise, picking one point from each nonempty intersection
+K ∩ eλ we obtain an infinite closed discrete subset of K.) Thus, there exists a finite subset
+E := {eλ : λ ∈ Λ} of open cells in X such that for every g ∈ (K|K)G, gE ∩ E ̸= ∅. Now,
+finiteness of (K|K)G follows from finiteness of cell-stabilizers in G.
+□
+Unfortunately, the property that every point of X is a wandering point is frequently
+taken as the definition of proper discontinuity for G-actions, see e.g. [5]. Items (8) and
+(10) in the above theorem provide a (weak) justification for this abuse of terminology. I
+feel that the better name for such actions is wandering actions.
+Example 13. Consider the action of G = Z on the punctured affine plane X = R2 −
+{(0, 0)}, where the generator of Z acts via (x, y) �→ (2x, 1
+2y). Then for any p ∈ X, the
+G-orbit Gp has no accumulation points in X. However, any two points p = (x, 0), q =
+(0, y) ∈ X are dynamically related. Thus, the action of G is not properly discontinuous.
+This example shows that the quotient space of a wandering action need not be Hausdorff.
+Lemma 14. Suppose that G× X → X is a wandering action. Then each G-orbit is closed
+and discrete in X. In particular, the quotient space X/G is T1.
+Proof. Suppose that Gx accumulates at a point y. Then Gx ∩ Wy is nonempty, where Wy
+is a G-slice at y. It follows that all points of Gx ∩ Wy lie in the same Wy-orbit, which
+implies that Gx ∩ Wy = {y}.
+□
+There are several reasons to consider properly discontinuous actions; one reason is that
+such actions yield orbi-covering maps in the case of smooth group actions on manifolds:
+M → M/G is an orbi-covering provided that the action of G on M is smooth (or, at
+least, locally smoothable). Another reason is that for a properly discontinuous action on a
+Hausdorff space, G × X → X, the quotient X/G is again Hausdorff.
+Question 15. Suppose that G is a discrete group, G × X → X is a free continuous
+action on an n-dimensional topological manifold X such that the quotient space X/G is a
+(Hausdorff) n-dimensional topological manifold. Does it follow that the action of G on X
+is properly discontinuous?
+The answer to this question is negative if one merely assumes that X is a Hausdorff
+topological space and X/G is Hausdorff.
+
+8
+MICHAEL KAPOVICH
+5. Cocompactness
+There are two common notions of cocompactness for group actions:
+(1) G × X → X is cocompact if there exists a compact K ⊂ X such that G · K = X.
+(2) G × X → X is cocompact if X/G is compact.
+It is clear that (1)⇒(2), as the image of a compact under the continuous (quotient) map
+p : X → X/G is compact.
+Lemma 16. If X is locally compact then (2)⇒(1).
+Proof. For each x ∈ X let Ux denote a relatively compact neighborhood of x in X. Then
+Vx := p(Ux) = p(G · Ux),
+is compact since G· Ux is open in X. Thus, we obtain an open cover {Vx : x ∈ X} of X/G.
+Since X/G is compact, this open cover contains a finite subcover
+Vx1, ..., Vxn.
+It follows that
+p(
+n�
+i=1
+Uxi) = X/G.
+The set
+K =
+n�
+i=1
+Uxi
+is compact and p(K) = X/G. Hence, G · K = X.
+□
+Lemma 17. Suppose that X is normal, G × X → X is a proper action such that X/G is
+locally compact. Then X is locally compact.
+Proof. Pick x ∈ X. Let Wx be a slice for the G-action at x; then Wx/Gx → X/G is a
+topological embedding. Thus, our assumptions imply that Wx/Gx is compact for every
+x ∈ X.
+Let (xα) be a net in Wx. Since Wx/Gx is compact, the net (xα)/G contains
+a convergent subnet.
+Thus, after passing to a subnet, there exists g ∈ Gx such that
+(gxα) converges to some x ∈ Wx.
+Hence, (xα) subconverges to g−1(x).
+Thus, Wx is
+relatively compact. Since X is assumed to be normal, x admits a basis of relatively compact
+neighborhoods.
+□
+6. Fundamental sets
+Definition 18. A closed subset F ⊂ X is a fundamental set for the action of G on X if
+G·F = X and there exists an open neighborhood U = UF of F such that for every compact
+K ⊂ X, the transporter set (U|K)G is finite (the local finiteness condition).
+Fundamental sets appear naturally in the reduction theory of arithmetic groups (Siegel
+sets), see [10] and [1].
+There are several existence theorems for fundamental sets. The next proposition, proven
+in [8, Lemma 2], guarantees existence of fundamental sets under the paracompactness
+assumption on X/G.
+
+A NOTE ON PROPERLY DISCONTINUOUS ACTIONS
+9
+Proposition 19. Each properly discontinuous action G ↷ X with paracompact quotient
+X/G admits a fundamental set.
+One frequently encounters a sharper version of fundamental sets, called fundamental
+domains. A domain in a topological space X is an open subset U ⊂ X which equals the
+interior of its closure.
+Definition 20. Suppose that G × X → X is a properly discontinuous group action. A
+subset F in X is called a fundamental domain for an action G × X → X if the following
+hold:
+(1) F is a domain in X.
+(2) G · F = X.
+(3) gF ∩ F ̸= ∅ if and only if g = 1.
+(4) For every compact subset K ⊂ X, the transporter set (F|K)G is finite, i.e. the
+family {gF}g∈G of subsets in X is locally finite.
+Suppose that (X, d) is a proper geodesic metric space.
+Suppose, furthermore, that
+G × X → X is a properly discontinuous isometric action, x ∈ X is a point which is fixed
+only by the identity element.
+Remark 21. If G is countable and fixed point sets in X of nontrivial elements of G are
+nowhere dense, then Baire’s Theorem implies existence of such x.
+One defines the Dirichlet domain of the action as
+D = Dx = {y ∈ X : d(y, x) < d(y, gx)
+∀g ∈ G \ Gx}.
+Note that gDx = Dgx.
+Proposition 22. Each Dirichlet domain D is a fundamental domain for the G-action.
+Proof. 1. The closure D is contained in
+ˆD = ˆDx = {y ∈ X : d(y, x) ≤ d(y, gx)
+∀g ∈ G \ Gx}.
+As before, g ˆDx = ˆDgx. I claim that ˆD is the closure of D and D is the interior of ˆD; this
+will prove that D is a domain. Clearly, D is contained in the interior of ˆD and ˆD is closed.
+Hence, it suffices to prove that each point of ˆD is the limit of a sequence in D. Consider
+a point z ∈ ˆD \ D and let c : [0, T] → X be a geodesic connecting x to z. Then for each
+t ∈ [0, T) and g ∈ G \ {1},
+d(x, c(t)) < d(x, c(t)) + d(c(t), z) = d(x, z) ≤ d(z, gx),
+i.e. c(t) ∈ D. Thus, indeed, z lies in the closure of D, as claimed.
+2. Let us prove that g ˆD = X. For each y ∈ X the function g �→ d(z, gx) is a proper
+function on G, hence, it attains its minimum on some g ∈ G. Then, clearly, y ∈ ˆDgx,
+hence, y ∈ g ˆDx. Thus, gD = X.
+3. Suppose that g ∈ G \ {1} is such that gD = Dgx ∩ D ̸= ∅. Then each point y of
+intersection is closer to x than to gx (since y ∈ Dx) and also y is closer to gx than to
+g−1gx = x (since y ∈ Dgx). This is clearly impossible.
+
+10
+MICHAEL KAPOVICH
+4. Lastly, let us verify local finiteness. Consider a compact K ⊂ X. Then K ⊂ B =
+B(x, R) for some R. For every g ∈ G such that gB ∩ B ̸= ∅, d(x, gx) ≤ 2R. Since (X, d) is
+a proper metric space and the action of G on X is properly discontinuous, the set of such
+elements of G is finite.
+□
+I will now prove existence of fundamental domains for properly discontinuous group
+actions on a certain class of topological spaces.
+Theorem 23. Suppose that X is 2nd countable, connected and locally connected locally
+compact Hausdorff topological space. Suppose that G × X → X is a properly discontinuous
+action of a countable group such that the fixed-point set of each nontrivial element of G is
+nowhere dense in X. Then this action admits a fundamental domain.
+Proof. Our goal is to construct a G-invariant geodesic metric metrizing X. Then the result
+will follow from the proposition.
+Lemma 24. The quotient space Y = X/G is locally compact, connected, locally connected
+and metrizable.
+Proof. Local compactness and connectedness of Y follows from that of X. The 2nd count-
+ability of X implies the 2nd countability of Y . By Lemma 9, Y is Hausdorff. Since Y is
+locally compact and Hausdorff, its one-point compactification is compact and Hausdorff,
+hence, regular. It follows that Y itself is regular. In view of the 2nd countability of Y ,
+Urysohn’s metrization theorem implies that Y is metrizable.
+□
+Remark 25. Note that each locally compact metrizable space is also locally path-connected.
+It is proven in [11] that each locally compact, connected, locally connected metrizable
+space, such as Y , admits a complete geodesic metric which we fix from now on. Consider
+the projection p : X → Y . According to [3, Theorem 6.2], the map p satisfies the path-
+lifting property: Given any path c : [0, 1] → Y , a point x ∈ X satisfying p(x) = c(0), there
+exists a path ˜c : [0, 1] → X such that p ◦ ˜c = c. (This result is, of course, much easier if the
+G-action is free, i.e. p : X → Y is a covering map.) We let LX denote the set of paths in
+X which are lifts of rectifiable paths c : [0, 1] → Y . Clearly, the postcomposition of ˜c ∈ LX
+with an element of G is again in LX. Our next goal is to equip X with a G-invariant length
+structure using the family of paths LX. Such a structure is a function on LX with values
+in [0, ∞), satisfying certain axioms that can be found in [4, Section 2.1]. Verification of
+most of these axioms is straightforward, I will check only some (items 1, 2, 3 and 4 below).
+1. If ˜c ∈ LX is a lift of a a path c in Y , then we declare ℓ(˜c) to be equal to the length
+of c.
+2. If ˜ci, i = 1, 2, are paths in LX (which are lifts of the paths c1, c2 respectively) whose
+concatenation b = ˜c1 ⋆ ˜c2 is defined, then b is a lift of the concatenation c1 ⋆ c2. Clearly,
+ℓ(b) = ℓ(˜c1) + ℓ(˜c2).
+3. Let U be a neighborhood of some x ∈ X. We need to prove that
+(26)
+inf
+γ {ℓ(γ)} > 0,
+
+A NOTE ON PROPERLY DISCONTINUOUS ACTIONS
+11
+where the infimum is taken over all γ = ˜c ∈ LX connecting x to points of X \U. It suffices
+to prove this claim in the case when U is Gx-invariant, satisfies
+(27)
+U ∩ gU ̸= ∅ ⇐⇒ g ∈ Gx,
+and γ connects x to points of ∂U. Then V = p(U) is a neighborhood of y = p(x) in Y and
+the paths c = p ◦ γ connect y to points in ∂V . But the lengths of the paths c are clearly
+bounded away from zero and are equal to the lengths of their lifts ˜c. Thus, we obtain the
+required bound (26).
+4. Let us verify that any two points in X are connected by a path in LX. Since X is
+connected, it suffices to verify the claim locally. Let U is Gx-invariant neighborhood of x
+satisfying (27), such that V = p(U) is an open metric ball in Y centered at y = p(x). Take
+u ∈ U, v := p(u) ∈ V . Let c : [0, T] → V be a geodesic connecting v to y. Then there
+exists a lift ˜c : [0, T] → U of c with ˜c(0) = u. Since x ∈ U is the only point projecting to y,
+we get ˜c(T) = x. By taking concatenations of pairs of such radial paths in U, we conclude
+that any two points in U are connected by a path ˜c ∈ LX.
+Given a length structure on X, one defines a path-metric (metrizing the topology of X)
+by
+d(x1, x2) = inf
+�� {ℓ(γ)}
+where the infimum is taken over all γ ∈ LX connecting x1 to x2. Since X is locally compact,
+this path-metric is geodesic.
+Note that, by the construction, the length structure on X and, hence, the metric d, is
+G-invariant. This concludes the proof of the theorem.
+□
+For each fundamental set F we define its quotient space F/G as the quotient space of
+the equivalence relation x ∼ y ⇐⇒ ({x}|{y})G ̸= ∅. The following proposition explains
+why fundamental sets are useful: They allow one to describe quotient spaces of properly
+discontinuous group actions using less information than is contained in the description of
+that action.
+Proposition 28. Suppose that F is a fundamental set for a properly discontinuous action
+of G on a 1st countable and Hausdorff space. Then the natural projection map p : F/G →
+X/G is a homeomorphism.
+Proof. The map p is continuous by the definition of the quotient topology.
+It is also
+obviously a bijection. It remains to show that p is a closed map. Since F is closed, it
+suffices to show that the projection q : F → X/G is a closed map. Suppose that (xn)
+is a sequence in F such that q(xn) converges to some y ∈ X/G, y is represented by a
+point x ∈ F. Then there is a sequence gn ∈ G such that gn(xn) converges to x. Since
+{gn(xn) : n ∈ N}∪ {x} is compact which, without loss of generality is contained in UF , the
+local finiteness assumption implies that the sequence (gn) is finite. Hence, after extraction,
+gn = g for all n. The fact that F is closed then implies that x ∈ F. It follows that x is an
+accumulation point of (xn). Thus, q : F → F/G is a closed map.
+□
+
+12
+MICHAEL KAPOVICH
+References
+[1] A. Borel, L. Ji, “Compactifications of Symmetric and Locally Symmetric Spaces”, Birkhauser
+Verlag, Series “Mathematics: Theory and Applications”, 2005.
+[2] N. Bourbaki, “Elements of Mathematics. General Topology”, Parts I–IV, Hermann, Paris, 1966.
+[3] G. Bredon, “Introduction to Compact Transformation Groups,” Academic Press, 1972.
+[4] D. Burago, Y. Burago and S. Ivanov, “A course in metric geometry.” Graduate Studies in Math-
+ematics, vol. 33, American Mathematical Society, Providence, RI, 2001.
+[5] A. Hatcher, “Algebraic Topology”, Cambridge University Press, 2001.
+[6] http://mathoverflow.net/questions/50128/a-question-about-group-action-on-topological-space?rq=1
+[7] http://mathoverflow.net/questions/55726/properly-discontinuous-action?rq=1
+[8] J. L. Koszul, “Lectures on Groups of Transformations”, Tata Institute of Fundamental Research,
+Bombay, 1965.
+[9] R. Palais, When proper maps are closed, Proc. of AMS, 24 (1970), 835–836.
+[10] C. L. Siegel, Discontinuous groups, Ann. of Math. (2) 44 (1943), 674–689.
+[11] A. Tominaga and T. Tanaka, Convexification of locally connected generalized continua, J. Sci.
+Hiroshima Univ. Ser. A. 19 (1955), 301–306.
+[12] L. Tu, “An introduction to manifolds”, Springer Verlag, 2nd edition, 2010.
+Department of Mathematics, University of California, Davis, CA 95616
+Email address: [email protected]
+

CdE4T4oBgHgl3EQf5g79/content/tmp_files/load_file.txt

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+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='05325v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='GR]  12 Jan 2023  A NOTE ON PROPERLY DISCONTINUOUS ACTIONS  MICHAEL KAPOVICH  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' We compare various notions of proper discontinuity for group actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' We  also discuss fundamental domains and criteria for cocompactness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  To the memory of Sasha Anan’in  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Introduction  This note is meant to clarify the relation between different commonly used definitions  of proper discontinuity without the local compactness assumption for the underlying topo-  logical space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Much of the discussion applies to actions of nondiscrete topological groups,  but, since my primary interest is geometric group theory, I will work only with discrete  groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' All group actions are assumed to be continuous, in other words, these are homo-  morphisms from abstract groups to groups of homeomorphisms of topological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' This  combination of continuous and properly discontinuous, sadly, leads to the ugly terminology  “a continuous properly discontinuous action.” A better terminology might be that of a  properly discrete action, since it refers to proper actions of discrete groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Throughout this note, I will be working only with topological spaces which are 1st count-  able, since spaces most common in metric geometry, geometric topology, algebraic topology  and geometric group theory satisfy this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' One advantage of this assumption is that  if (xn) is a sequence converging to a point x ∈ X, then the subset {x} ∪ {xn : n ∈ N} is  compact, which is not true if we work with nets instead of sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' However, I will try  to avoid the local compactness assumption whenever possible, since many spaces appear-  ing in metric geometry and geometric group theory (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' asymptotic cones) and algebraic  topology (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' CW complexes) are not locally compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' (Recall that topological space X  is locally compact if every point has a basis of topology consisting of relatively compact  subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=') In the last two sections of the note I also discuss criteria for cocompactness of  group actions and fundamental sets/domains of properly discontinuous actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Acknowledgement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' I am grateful to Boris Okun for pointing out several typos and  the reference to [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Group actions  A topological group is a group G equipped with a topology such that the multiplication  and inversion maps  G × G → G, (g, h) �→ gh, G → G, g �→ g−1  Date: January 16, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  2  MICHAEL KAPOVICH  are both continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' A discrete group is a group with discrete topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Every discrete  group is clearly a topological group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  A left continuous action of a topological group G on a topological space X is a continuous  map  λ : G × X → X  satisfying  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' λ(1G, x) = x for all x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' λ(gh, x) = λ(g, λ(h, x)), for all x ∈ X, g, h ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  From this, it follows that the map ρ : G → Homeo(X)  ρ(g)(x) = λ(g, x),  is a group homomorphism, where the group operation φψ on Homeo(X) is the composition  φ ◦ ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  If G is discrete, then every homomorphism G → Homeo(X) defines a left continuous  action of G on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  The shorthand for ρ(g)(x) is gx or g · x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Similarly, for a subset A ⊂ X, GA or G · A,  denotes the orbit of A under the G-action:  GA =  �  g∈G  gA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  The quotient space X/G (also frequently denoted G\\X), of X by the G-action, is the  set of G-orbits of points in X, equipped with the quotient topology: The elements of X/G  are equivalence classes in X, where x ∼ y when Gx = Gy (equivalently, y ∈ Gx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  The stabilizer of a point x ∈ X under the G-action is the subgroup Gx < G given by  {g ∈ G : gx = x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  An action of G on X is called free if Gx = {1} for all x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Assuming that X is Hausdorff,  Gx is closed in G for every x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' An example of a left action of G is the action of G on itself via left multipli-  cation:  λ(g, h) = gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  In this case, the common notation for ρ(g) is Lg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' This action is free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Proper maps  Properness of certain maps is the most common form of defining proper discontinuity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  sadly, there are two competing notions of properness in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  A continuous map f : X → Y of topological spaces is proper in the sense of Bourbaki,  or simply Bourbaki–proper (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' [2, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' I, §10, Theorem 1]) if f is a closed map (images of  closed subsets are closed) and point–preimages f −1(y), y ∈ Y , are compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' A continuous  map f : X → Y is proper (and this is the most common definition) if for every compact  subset K ⊂ X, f −1(K) is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' It is noted in [2, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' I, §10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' 7] that if X is  Hausdorff and Y is locally compact then f is Bourbaki–proper if and only if f is proper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  A NOTE ON PROPERLY DISCONTINUOUS ACTIONS  3  The advantage of the notion of Bourbaki-properness is that it applies in the case of  Zariski topology, where spaces tend to be compact (every subset of a finite-dimensional  affine space is Zariski-compact) and, hence, the standard notion of properness is useless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Since our goal is to trade local compactness for 1st countability, I will prove  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' If f : X → Y is proper, and X, Y are Hausdorff and 1st countable, then f is  Bourbaki-proper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' We only have to verify that f is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Suppose that A ⊂ X is a closed subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Since Y is 1st countable, it suffices to show that for each sequence (xn) in A such that  (f(xn)) converges to y ∈ Y , there is a subsequence (xnk) which converges to some x ∈ A  such that f(x) = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' The subset C = {y} ∪ {f(xn) : n ∈ N} ⊂ Y is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Hence, by  properness of f, K = f −1(C) is also compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Since X is Hausdorff, and K is compact,  follows that (xn) subconverges to a point x ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' By continuity of f, f(x) = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Since A is  closed, x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  □  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' In fact, for this lemma to hold, one does not need to assume that X is Haus-  dorff and 1st countable, see  The converse (each Bourbaki–proper map is proper) is proven in [2, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' I, §10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  6] without any restrictions on X, Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Hence:  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' For maps between 1st countable Hausdorff spaces, Bourbaki–properness is  equivalent to properness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Proper discontinuity  Suppose that X is a 1st countable Hausdorff topological space, G a discrete group and  G × X → X a (continuous) action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' We use the notation gn → ∞ in G to indicate that gn  converges to ∞ in the 1-point compactification G ∪ {∞} of G, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' for every finite subset  F ⊂ G,  card({n : gn ∈ F}) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Given a group action G × X → X and two subsets A, B ⊂ X, the transporter subset  (A|B)G is defined as  (A|B)G := {g ∈ G : gA ∩ B ̸= ∅}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Properness of group actions is (typically) stated using certain transporter sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Two points x, y ∈ X are said to be G-dynamically related if there is a  sequence gn → ∞ in G and a sequence xn → x in X such that gnxn → y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  A point x ∈ X is said to be a wandering point of the G-action if there is a neighborhood  U of x such that (U|U)G is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Suppose that the action G × X → X is wandering at a point x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Then the  G-action has a G-slice at x, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' a neighborhood Wx ⊂ U which is Gx-stable and for all  g /∈ Gx, gWx ∩ Wx = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  4  MICHAEL KAPOVICH  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' For each g ∈ (U|U)G − Gx we pick a neighborhood Vg ⊂ U of x such that  gVg ∩ Vg = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Then the intersection  V :=  �  g∈(U|U)G−Gx  Vg  satisfies the property that (V |V )G = Gx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Lastly, take  Wx :=  �  g∈Gx  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  □  The next lemma is clear:  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Assuming that X is Hausdorff and 1st countable, the action G × X → X is  wandering at x if and only if x is not dynamically related to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Given a group action α : G × X → X, we have the natural map  ˆα := α × idX : G × X → X × X  where idX : (g, x) �→ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' An action α of a discrete group G on a topological space X is Bourbaki–  proper if the map ˆα is Bourbaki-proper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' If the action α : G×X → X of a discrete group G on an Hausdorff topological  space X is Bourbaki-proper then the quotient space X/G is Hausdorff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' The quotient map X → X/G is an open map by the definition of the quotient  topology on X/G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Since α is Bourbaki-proper, the image of the map ˆα is closed in X × X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  This image is the equivalence relation on X ×X which use used to form the quotient X/G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Now, Hausdorffness of X/G follows from Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='7 in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  □  Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' An action α of a discrete group G on a topological space X is properly  discontinuous if the map ˆα is proper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  We note that equivalence of (1) and (5) in the following theorem is proven in [2, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' III,  §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='4, Proposition 7] without any assumptions on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Assuming that X is Hausdorff and 1st countable, the following are equiva-  lent:  (1) The action α : G × X → X is Bourbaki-proper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  (2) For every compact subset K ⊂ X,  card((K|K)G) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  (3) The action α : G × X → X is proper, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' the map ˆα is proper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  (4) For every compact subset K ⊂ X, there exists an open neighborhood U of K such  that card((U|U)G) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  A NOTE ON PROPERLY DISCONTINUOUS ACTIONS  5  (5) For any pair of points x, y ∈ X there is a pair of neighborhoods Ux, Vx (of x, y  respectively) such that card((Ux|Vy)G)) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  (6) There are no G-dynamically related points in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  (7) Assuming, that G is countable and X is completely metrizable1 : The G-stabilizer  of every x ∈ X is finite and for any two points x ∈ X, y ∈ X − Gx, there exists a  pair of neighborhoods Ux, Vy (of x, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' y) such that ∀g ∈ G, gUx ∩ Vy = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  (8) Assuming that X is a metric space and the action G × X → X is equicontinuous2:  There is no x ∈ X and a sequence hn → ∞ in G such that hnx → x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  (9) Assuming that X is a metric space and the action G × X → X is equicontinuous:  Every x ∈ X is a wandering point of the G-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  (10) Assuming that X is a CW complex and the action G × X → X is cellular: Every  point of X is wandering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  (11) Assuming that X is a CW complex the action G × X → X is cellular: Every cell  in X has finite G-stabilizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' The action α is Bourbaki-proper if and only if the map ˆα is proper (see Corollary 4)  which is equivalent to the statement that for each compact K ⊂ X, the subset (K|K)G×K  is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Hence, (1) ⇐⇒ (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  The property (3) means that for each compact K ⊂ X, ˆα−1(K ×K) = {(g, x) ∈ G×K :  x ∈ K, gx ∈ K} is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' This subset is closed in G × X and projects onto (K|K)G in  the first factor and to the subset  (12)  �  g∈(K|K)G  g−1(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Hence, properness of α implies finiteness of (K|K)G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Conversely, if (K|K)G is finite,  compactness of g−1(K) for every g ∈ G implies finiteness of the union (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Thus, (2) ⇐⇒  (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  In order to show (2)⇒(6), suppose that x, y are G-dynamically related points: There  exists an sequence gn → ∞ in G and a sequence xn → x such that gn(xn) → y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' The subset  K = {x, y} ∪ {xn, gn(xn) : n ∈ N}  is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' However, yn ∈ gn(K) ∩ K for every n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' A contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  (6)⇒(5): Suppose that the neighborhoods Ux, Vy do not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Let {Un}n∈N, {Vn}n∈N  be countable bases at x, y respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Then for every n there exists gn ∈ G, such that  gn(Un) ∩ Vn ̸= ∅ for infinitely many gn’s in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' After extraction, gn → ∞ in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' This yields  points xn ∈ Un, yn = gn(xn) ∈ Vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Hence, xn → x, yn → y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Thus, x is G-dynamically  related to y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' A contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  (5)⇒(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Consider a compact K ⊂ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Then for each x ∈ K, y ∈ K there exist  neighborhoods Ux, Vy such that (Ux|Vy)G is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' The product sets Ux × Vy, x, y ∈ K  constitute an open cover of K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' By compactness of K2, there exist x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=', xn, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=', ym ∈ K  1It suffices to assume that X is hereditarily Baire: Every closed subset of X is Baire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  2E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' an isometric action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  6  MICHAEL KAPOVICH  such that  K ⊂ Ux1 ∪ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' ∪ Uxn  K ⊂ Vy1 ∪ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' ∪ Vym  and for each pair (xi, yj),  card({g ∈ G : gUxi ∩ Vyj ̸= ∅}) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Setting  W :=  n�  i=1  Uxi, V :=  m  �  j=1  Vyj  we see that  card((W|V )G) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Taking U := V ∩ W yields the required subset U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  The implication (4)⇒(2) is immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Thus, we concluded the proof of equivalence of the properties (1)—(6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  (5)⇒(7): Finiteness of G-stabilizers of points in X is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Let x, y be points in distinct  G-orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Let U ′  x, V ′  y be neighborhoods of x, y such that (U ′  x|V ′  y)G = {g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=', gn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' For each  i, since X is Hausdorff, there are disjoint neighborhoods Vi of y and Wi of gi(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Now set  Vy :=  n�  i=1  Vi,  Ux :=  n�  i=1  g−1  i  (Wi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Then gUx ∩ Vy = ∅ for every g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  (7)⇒(6): It is clear that (7) implies that there are no dynamically related points with  distinct G-orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' In particular, every G-orbit in X is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Assume now that X is completely metrizable and G is countable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Suppose that a point  x ∈ X is G-dynamically related to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Since the stabilizer Gx is finite, the point x is  an accumulation point of Gx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' moreover, Gx is closed in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Hence, Gx is a closed perfect  subset of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Since X admits a complete metric, so does its closed subset Gx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Thus, for  each g ∈ G, the complement Ug := Gx − {gx} is open and dense in Gx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' By the Baire  Category Theorem, the countable intersection  �  g∈G  Ug  is dense in Gx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' However, this intersection is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' A contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  It is clear that (6)⇒(8) (without any extra assumptions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  (8)⇒(6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Suppose that X is a metric space and the G-action is equicontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Equicon-  tinuity implies that for each z ∈ X, a sequence zn → z and gn ∈ G,  gnzn → gz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Suppose that there exist a pair of G-dynamically related points x, y ∈ X: ∃xn → x, gn ∈  G, gnxn → y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' By equicontinuity of the action, gnx → y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Since gn → ∞, there exist  A NOTE ON PROPERLY DISCONTINUOUS ACTIONS  7  subsequences gni → ∞ and gmi → ∞ such that the products hi := g−1  ni gmi are all distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Then, by equicontinuity,  hix → x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  A contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  The implications (5)⇒(9)⇒(8) and (5)⇒(10)⇒(11) are clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Lastly, let us prove the implication (11)⇒(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' We first observe that every CW complex  is Hausdorff and 1st countable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Furthermore, every compact K ⊂ X intersects only finitely  many open cells eλ in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' (Otherwise, picking one point from each nonempty intersection  K ∩ eλ we obtain an infinite closed discrete subset of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=') Thus, there exists a finite subset  E := {eλ : λ ∈ Λ} of open cells in X such that for every g ∈ (K|K)G, gE ∩ E ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Now,  finiteness of (K|K)G follows from finiteness of cell-stabilizers in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  □  Unfortunately, the property that every point of X is a wandering point is frequently  taken as the definition of proper discontinuity for G-actions, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Items (8) and  (10) in the above theorem provide a (weak) justification for this abuse of terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' I  feel that the better name for such actions is wandering actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Example 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Consider the action of G = Z on the punctured affine plane X = R2 −  {(0, 0)}, where the generator of Z acts via (x, y) �→ (2x, 1  2y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Then for any p ∈ X, the  G-orbit Gp has no accumulation points in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' However, any two points p = (x, 0), q =  (0, y) ∈ X are dynamically related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Thus, the action of G is not properly discontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  This example shows that the quotient space of a wandering action need not be Hausdorff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Lemma 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Suppose that G× X → X is a wandering action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Then each G-orbit is closed  and discrete in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' In particular, the quotient space X/G is T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Suppose that Gx accumulates at a point y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Then Gx ∩ Wy is nonempty, where Wy  is a G-slice at y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' It follows that all points of Gx ∩ Wy lie in the same Wy-orbit, which  implies that Gx ∩ Wy = {y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  □  There are several reasons to consider properly discontinuous actions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' one reason is that  such actions yield orbi-covering maps in the case of smooth group actions on manifolds:  M → M/G is an orbi-covering provided that the action of G on M is smooth (or, at  least, locally smoothable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Another reason is that for a properly discontinuous action on a  Hausdorff space, G × X → X, the quotient X/G is again Hausdorff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Question 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Suppose that G is a discrete group, G × X → X is a free continuous  action on an n-dimensional topological manifold X such that the quotient space X/G is a  (Hausdorff) n-dimensional topological manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Does it follow that the action of G on X  is properly discontinuous?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  The answer to this question is negative if one merely assumes that X is a Hausdorff  topological space and X/G is Hausdorff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  8  MICHAEL KAPOVICH  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Cocompactness  There are two common notions of cocompactness for group actions:  (1) G × X → X is cocompact if there exists a compact K ⊂ X such that G · K = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  (2) G × X → X is cocompact if X/G is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  It is clear that (1)⇒(2), as the image of a compact under the continuous (quotient) map  p : X → X/G is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Lemma 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' If X is locally compact then (2)⇒(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' For each x ∈ X let Ux denote a relatively compact neighborhood of x in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Then  Vx := p(Ux) = p(G · Ux),  is compact since G· Ux is open in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Thus, we obtain an open cover {Vx : x ∈ X} of X/G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Since X/G is compact, this open cover contains a finite subcover  Vx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=', Vxn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  It follows that  p(  n�  i=1  Uxi) = X/G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  The set  K =  n�  i=1  Uxi  is compact and p(K) = X/G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Hence, G · K = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  □  Lemma 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Suppose that X is normal, G × X → X is a proper action such that X/G is  locally compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Then X is locally compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Pick x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Let Wx be a slice for the G-action at x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' then Wx/Gx ��� X/G is a  topological embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Thus, our assumptions imply that Wx/Gx is compact for every  x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Let (xα) be a net in Wx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Since Wx/Gx is compact, the net (xα)/G contains  a convergent subnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Thus, after passing to a subnet, there exists g ∈ Gx such that  (gxα) converges to some x ∈ Wx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Hence, (xα) subconverges to g−1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Thus, Wx is  relatively compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Since X is assumed to be normal, x admits a basis of relatively compact  neighborhoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  □  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Fundamental sets  Definition 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' A closed subset F ⊂ X is a fundamental set for the action of G on X if  G·F = X and there exists an open neighborhood U = UF of F such that for every compact  K ⊂ X, the transporter set (U|K)G is finite (the local finiteness condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Fundamental sets appear naturally in the reduction theory of arithmetic groups (Siegel  sets), see [10] and [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  There are several existence theorems for fundamental sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' The next proposition, proven  in [8, Lemma 2], guarantees existence of fundamental sets under the paracompactness  assumption on X/G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  A NOTE ON PROPERLY DISCONTINUOUS ACTIONS  9  Proposition 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Each properly discontinuous action G ↷ X with paracompact quotient  X/G admits a fundamental set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  One frequently encounters a sharper version of fundamental sets, called fundamental  domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' A domain in a topological space X is an open subset U ⊂ X which equals the  interior of its closure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Definition 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Suppose that G × X → X is a properly discontinuous group action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' A  subset F in X is called a fundamental domain for an action G × X → X if the following  hold:  (1) F is a domain in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  (2) G · F = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  (3) gF ∩ F ̸= ∅ if and only if g = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  (4) For every compact subset K ⊂ X, the transporter set (F|K)G is finite, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' the  family {gF}g∈G of subsets in X is locally finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Suppose that (X, d) is a proper geodesic metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Suppose, furthermore, that  G × X → X is a properly discontinuous isometric action, x ∈ X is a point which is fixed  only by the identity element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Remark 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' If G is countable and fixed point sets in X of nontrivial elements of G are  nowhere dense, then Baire’s Theorem implies existence of such x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  One defines the Dirichlet domain of the action as  D = Dx = {y ∈ X : d(y, x) < d(y, gx)  ∀g ∈ G \\ Gx}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Note that gDx = Dgx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Proposition 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Each Dirichlet domain D is a fundamental domain for the G-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' The closure D is contained in  ˆD = ˆDx = {y ∈ X : d(y, x) ≤ d(y, gx)  ∀g ∈ G \\ Gx}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  As before, g ˆDx = ˆDgx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' I claim that ˆD is the closure of D and D is the interior of ˆD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' this  will prove that D is a domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Clearly, D is contained in the interior of ˆD and ˆD is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Hence, it suffices to prove that each point of ˆD is the limit of a sequence in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Consider  a point z ∈ ˆD \\ D and let c : [0, T] → X be a geodesic connecting x to z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Then for each  t ∈ [0, T) and g ∈ G \\ {1},  d(x, c(t)) < d(x, c(t)) + d(c(t), z) = d(x, z) ≤ d(z, gx),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' c(t) ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Thus, indeed, z lies in the closure of D, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Let us prove that g ˆD = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' For each y ∈ X the function g �→ d(z, gx) is a proper  function on G, hence, it attains its minimum on some g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Then, clearly, y ∈ ˆDgx,  hence, y ∈ g ˆDx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Thus, gD = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Suppose that g ∈ G \\ {1} is such that gD = Dgx ∩ D ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Then each point y of  intersection is closer to x than to gx (since y ∈ Dx) and also y is closer to gx than to  g−1gx = x (since y ∈ Dgx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' This is clearly impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  10  MICHAEL KAPOVICH  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Lastly, let us verify local finiteness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Consider a compact K ⊂ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Then K ⊂ B =  B(x, R) for some R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' For every g ∈ G such that gB ∩ B ̸= ∅, d(x, gx) ≤ 2R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Since (X, d) is  a proper metric space and the action of G on X is properly discontinuous, the set of such  elements of G is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  □  I will now prove existence of fundamental domains for properly discontinuous group  actions on a certain class of topological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Theorem 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Suppose that X is 2nd countable, connected and locally connected locally  compact Hausdorff topological space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Suppose that G × X → X is a properly discontinuous  action of a countable group such that the fixed-point set of each nontrivial element of G is  nowhere dense in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Then this action admits a fundamental domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Our goal is to construct a G-invariant geodesic metric metrizing X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Then the result  will follow from the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Lemma 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' The quotient space Y = X/G is locally compact, connected, locally connected  and metrizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Local compactness and connectedness of Y follows from that of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' The 2nd count-  ability of X implies the 2nd countability of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' By Lemma 9, Y is Hausdorff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Since Y is  locally compact and Hausdorff, its one-point compactification is compact and Hausdorff,  hence, regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' It follows that Y itself is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' In view of the 2nd countability of Y ,  Urysohn’s metrization theorem implies that Y is metrizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  □  Remark 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Note that each locally compact metrizable space is also locally path-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  It is proven in [11] that each locally compact, connected, locally connected metrizable  space, such as Y , admits a complete geodesic metric which we fix from now on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Consider  the projection p : X → Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' According to [3, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='2], the map p satisfies the path-  lifting property: Given any path c : [0, 1] → Y , a point x ∈ X satisfying p(x) = c(0), there  exists a path ˜c : [0, 1] → X such that p ◦ ˜c = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' (This result is, of course, much easier if the  G-action is free, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' p : X → Y is a covering map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=') We let LX denote the set of paths in  X which are lifts of rectifiable paths c : [0, 1] → Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Clearly, the postcomposition of ˜c ∈ LX  with an element of G is again in LX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Our next goal is to equip X with a G-invariant length  structure using the family of paths LX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Such a structure is a function on LX with values  in [0, ∞), satisfying certain axioms that can be found in [4, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Verification of  most of these axioms is straightforward, I will check only some (items 1, 2, 3 and 4 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' If ˜c ∈ LX is a lift of a a path c in Y , then we declare ℓ(˜c) to be equal to the length  of c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' If ˜ci, i = 1, 2, are paths in LX (which are lifts of the paths c1, c2 respectively) whose  concatenation b = ˜c1 ⋆ ˜c2 is defined, then b is a lift of the concatenation c1 ⋆ c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Clearly,  ℓ(b) = ℓ(˜c1) + ℓ(˜c2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Let U be a neighborhood of some x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' We need to prove that  (26)  inf  γ {ℓ(γ)} > 0,  A NOTE ON PROPERLY DISCONTINUOUS ACTIONS  11  where the infimum is taken over all γ = ˜c ∈ LX connecting x to points of X \\U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' It suffices  to prove this claim in the case when U is Gx-invariant, satisfies  (27)  U ∩ gU ̸= ∅ ⇐⇒ g ∈ Gx,  and γ connects x to points of ∂U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Then V = p(U) is a neighborhood of y = p(x) in Y and  the paths c = p ◦ γ connect y to points in ∂V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' But the lengths of the paths c are clearly  bounded away from zero and are equal to the lengths of their lifts ˜c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Thus, we obtain the  required bound (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Let us verify that any two points in X are connected by a path in LX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Since X is  connected, it suffices to verify the claim locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Let U is Gx-invariant neighborhood of x  satisfying (27), such that V = p(U) is an open metric ball in Y centered at y = p(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Take  u ∈ U, v := p(u) ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Let c : [0, T] → V be a geodesic connecting v to y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Then there  exists a lift ˜c : [0, T] → U of c with ˜c(0) = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Since x ∈ U is the only point projecting to y,  we get ˜c(T) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' By taking concatenations of pairs of such radial paths in U, we conclude  that any two points in U are connected by a path ˜c ∈ LX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Given a length structure on X, one defines a path-metric (metrizing the topology of X)  by  d(x1, x2) = inf  γ {ℓ(γ)}  where the infimum is taken over all γ ∈ LX connecting x1 to x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Since X is locally compact,  this path-metric is geodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Note that, by the construction, the length structure on X and, hence, the metric d, is  G-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' This concludes the proof of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  □  For each fundamental set F we define its quotient space F/G as the quotient space of  the equivalence relation x ∼ y ⇐⇒ ({x}|{y})G ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' The following proposition explains  why fundamental sets are useful: They allow one to describe quotient spaces of properly  discontinuous group actions using less information than is contained in the description of  that action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Proposition 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Suppose that F is a fundamental set for a properly discontinuous action  of G on a 1st countable and Hausdorff space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Then the natural projection map p : F/G →  X/G is a homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' The map p is continuous by the definition of the quotient topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  It is also  obviously a bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' It remains to show that p is a closed map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Since F is closed, it  suffices to show that the projection q : F → X/G is a closed map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Suppose that (xn)  is a sequence in F such that q(xn) converges to some y ∈ X/G, y is represented by a  point x ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Then there is a sequence gn ∈ G such that gn(xn) converges to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Since  {gn(xn) : n ∈ N}∪ {x} is compact which, without loss of generality is contained in UF , the  local finiteness assumption implies that the sequence (gn) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Hence, after extraction,  gn = g for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' The fact that F is closed then implies that x ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' It follows that x is an  accumulation point of (xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Thus, q : F → F/G is a closed map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  □  12  MICHAEL KAPOVICH  References  [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Borel, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Ji, “Compactifications of Symmetric and Locally Symmetric Spaces”, Birkhauser  Verlag, Series “Mathematics: Theory and Applications”, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  [2] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Bourbaki, “Elements of Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' General Topology”, Parts I–IV, Hermann, Paris, 1966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  [3] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Bredon, “Introduction to Compact Transformation Groups,” Academic Press, 1972.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  [4] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Burago, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Burago and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Ivanov, “A course in metric geometry.” Graduate Studies in Math-  ematics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' 33, American Mathematical Society, Providence, RI, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Hatcher, “Algebraic Topology”, Cambridge University Press, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  [6] http://mathoverflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='net/questions/50128/a-question-about-group-action-on-topological-space?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='rq=1  [7] http://mathoverflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='net/questions/55726/properly-discontinuous-action?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='rq=1  [8] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Koszul, “Lectures on Groups of Transformations”, Tata Institute of Fundamental Research,  Bombay, 1965.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  [9] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Palais, When proper maps are closed, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' of AMS, 24 (1970), 835–836.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  [10] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Siegel, Discontinuous groups, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' (2) 44 (1943), 674–689.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  [11] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Tominaga and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Tanaka, Convexification of locally connected generalized continua, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Hiroshima Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' 19 (1955), 301–306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  [12] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content=' Tu, “An introduction to manifolds”, Springer Verlag, 2nd edition, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='  Department of Mathematics, University of California, Davis, CA 95616  Email address: kapovich@ucdavis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}
+page_content='edu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQf5g79/content/2301.05325v1.pdf'}

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+TarViS: A Unified Approach for Target-based Video Segmentation
+Ali Athar1
+Alexander Hermans1
+Jonathon Luiten1,2
+Deva Ramanan2
+Bastian Leibe1
+1RWTH Aachen University, Germany
+2Carnegie Mellon University, USA
+{athar,hermans,luiten,leibe}@vision.rwth-aachen.de
+[email protected]
+Abstract
+The general domain of video segmentation is currently
+fragmented into different tasks spanning multiple bench-
+marks. Despite rapid progress in the state-of-the-art, cur-
+rent methods are overwhelmingly task-specific and cannot
+conceptually generalize to other tasks. Inspired by recent
+approaches with multi-task capability, we propose TarViS:
+a novel, unified network architecture that can be applied to
+any task that requires segmenting a set of arbitrarily de-
+fined ‘targets’ in video. Our approach is flexible with re-
+spect to how tasks define these targets, since it models the
+latter as abstract ‘queries’ which are then used to predict
+pixel-precise target masks. A single TarViS model can be
+trained jointly on a collection of datasets spanning differ-
+ent tasks, and can hot-swap between tasks during infer-
+ence without any task-specific retraining. To demonstrate
+its effectiveness, we apply TarViS to four different tasks,
+namely Video Instance Segmentation (VIS), Video Panoptic
+Segmentation (VPS), Video Object Segmentation (VOS) and
+Point Exemplar-guided Tracking (PET). Our unified, jointly
+trained model achieves state-of-the-art performance on 5/7
+benchmarks spanning these four tasks, and competitive per-
+formance on the remaining two. Code will be made public
+upon acceptance.
+1. Introduction
+The ability to understand video scenes has been a long-
+standing goal of computer vision research because of wide-
+ranging applications in intelligent vehicles and robots.
+Early approaches tackled simpler tasks involving contour-
+based [33,40] and box-level tracking [21,25,41,53], back-
+ground subtraction [20, 62], and motion segmentation [8,
+50]. The deep learning boom then revolutionized the land-
+scape by enabling methods to perform pixel-precise seg-
+mentation on challenging, real-world videos. In the past
+few years, a number of benchmarks have emerged, which
+evaluate how well methods can perform video segmenta-
+tion according to various task formulations.
+Over time,
+VPS
+VIS
+VOS
+PET
+TarViS
+BEFORE
+Task-specific models
+NOW
+Task-specific targets
+VIS
+VPS
+VOS/PET
+Figure 1. Predicted results from a jointly trained TarViS model for
+four different video segmentation tasks.
+these tasks/benchmarks have ballooned into separate re-
+search sub-communities.
+Although existing methods are
+rapidly improving the state-of-the-art for these benchmarks,
+each of them typically tackles only one narrowly-defined
+task, and generalizing them is non-trivial since the task def-
+inition is baked into the core approach.
+We argue that this fragmentation is unnecessary be-
+cause video target segmentation tasks all require the same
+high-level capability, namely that of identifying, localizing
+and tracking rich semantic concepts.
+Meanwhile, recent
+progress on Transformer networks has enabled the wider
+AI research community to move towards unified, multi-task
+architectures [1, 30, 31, 39, 59], because the attention op-
+eration [63] is well-suited for processing feature sets with
+arbitrary structure and data modality. These developments
+give us the opportunity to unify the fractured landscape of
+target-based video segmentation. In this paper, we propose
+TarViS: a novel architecture which enables a single, unified
+1
+arXiv:2301.02657v1  [cs.CV]  6 Jan 2023
+
+hmodel to be jointly trained for multiple video segmentation
+tasks. During inference, the same model can perform differ-
+ent tasks at runtime by specifying the segmentation target.
+The core idea is that TarViS tackles the generic task of
+segmenting a set of arbitrary targets in video (defined as
+semantic classes or as specific objects). These targets are
+encoded as queries which, together with the video features,
+are input to a Transformer-based model. The model iter-
+atively refines these queries and produces a pixel-precise
+mask for each target entity. This formulation conceptually
+fuses all video segmentation tasks [3, 55, 67, 73] which fall
+under the umbrella of the above-mentioned generic task, be-
+cause they differ only in how the targets are defined. During
+both training and inference, TarViS can hot-swap between
+tasks at run-time by providing the desired target query set.
+To demonstrate our generalization capability, we tackle
+four different tasks:
+(1) Video Instance Segmenta-
+tion (VIS) [55, 73], (2) Video Panoptic Segmentation
+(VPS) [35], (3) Video Object Segmentation [54], and (4)
+Point Exemplar-guided Tracking [3] (PET). For VIS, the
+segmentation targets are all objects in the video belong-
+ing to a pre-defined set of classes.
+The target set for
+VPS includes that for VIS, and additionally, a set of non-
+instantiable stuff semantic classes. For VOS, the targets are
+a specific set of objects for which the first-frame ground-
+truth mask is provided. PET is a more constrained version
+of VOS which only provides the location of a single point
+inside the object, rather than the full object mask.
+Existing methods for these tasks lack generalization
+capability because task-specific assumptions are typically
+baked into the approach (see Sec. 2 and 3 for details). In
+contrast, TarViS can tackle all four tasks with a unified
+model because we encode the task-specific targets as a set
+of queries, thus decoupling the network architecture from
+the task definition. Moreover, our approach can theoreti-
+cally generalize further, e.g., one could potentially define
+the target set as all objects described by a given text prompt,
+though this is beyond the scope of this paper.
+To summarize, our contributions are as follows: we pro-
+pose TarViS, a novel architecture that can perform any task
+requiring segmentation of a set of targets from video. For
+the first time, we are able to jointly train and infer a single
+model on a collection of datasets spanning the four afore-
+mentioned tasks (VIS, VPS, VOS, PET). Our experimental
+results show that TarViS performs competitively for VOS,
+and achieves state-of-the-art results for VIS, VPS and PET.
+2. Related Work
+Multi-task Models.
+Multi-task learning has a long his-
+tory [11] with several architectures and training strate-
+gies [24,36,38,52,60,77]. Earlier approaches mostly con-
+sist of a shared backbone with fixed task-specific heads,
+whereas we design a more general architecture for video
+segmentation with task-specific targets to specify what to
+segment. Our approach is inspired by recent attention-based
+models, e.g., PerceiverIO [30,31], which can be trained on
+diverse data modalities and task-specific heads are replaced
+with output queries. UViM [39] follows a similar direction
+by creating a unified architecture for diverse dense predic-
+tion tasks. However, both of these models are trained sep-
+arately for different tasks. Recent, powerful multi-task vi-
+sion language models such as Flamingo [1] and GATO [59]
+tackle a multitude of tasks by requiring a sequence of task-
+specific input-output examples to prime the model. This is
+conceptually similar to our task-specific targets, however,
+our model does not require per-task priming. Moreover,
+our targets are not modeled as sequence prompts, and we
+aim for a video segmentation model which is several orders
+of magnitude smaller. In the realm of video tracking and
+segmentation, the recently proposed UNICORN [72] model
+tackles multiple object tracking-related tasks with a unified
+architecture. Unlike TarViS, however, UNICORN follows
+the task-specific output head approach and is generally ori-
+ented towards box-level tracking tasks [22,46,48,76], thus
+requiring non-trivial modifications to tackle VPS or PET.
+Query-based Transformer Architectures.
+Several
+works [2, 10, 13, 30, 31, 47, 66, 79] use query-based Trans-
+former architectures for various tasks.
+The fundamental
+workhorse for task learning here is the iterative application
+of self- and cross-attention, where a set of query vectors
+(e.g., representing objects) are refined by interacting with
+each other, and with the input data sample (e.g., an image).
+Unlike existing methods which use queries in a task-specific
+context, TarViS adopts a query-based Transformer archi-
+tecture in which the queries serve as a mechanism for de-
+coupling the task definition from the architecture, i.e., our
+model can learn to tackle different tasks while being ag-
+nostic to their definition because the latter is effectively ab-
+stracted behind a set of queries.
+Task-specific Video Segmentation Methods.
+Current
+Video Instance Segmentation (VIS) approaches broadly
+work by predicting object tracks from the input video, fol-
+lowed by classification into a pre-defined set of categories.
+Several approaches [6,9,23,28,34,55,65,70,73] are based
+on the tracking-by-detection paradigm, some model video
+as a joint spatio-temporal volume [4, 5], whereas many re-
+cent works [12, 26, 29, 66, 69] adopt Transformer-based ar-
+chitectures (comparison to our approach in Sec. 3.1).
+For
+Video
+Panoptic
+Segmentation
+(VPS),
+meth-
+ods [35, 56, 67] generally extend image-level panoptic
+approaches [14] by employing multi-head network ar-
+chitectures for semantic segmentation and instance mask
+regression, classification, and temporal association. In the
+Video Object Segmentation (VOS) community, state-of-
+2
+
+Qsem
+Qinst
+Qbg
+VIS or VPS
+EncObj
+F
+Qobj
+Qbg
+VOS or PET
+Backbone
+Temporal
+Neck
+Transformer Decoder
+Layer 1
+Masked Cross-Attention
+Self-Attention
+FFN
+Layer 2
+Layer L
+Qin
+F
+F4
+Qout
+×
+Q′
+inst
+F4
+×
+Q′
+inst, Q′
+sem
+F4
+×
+Classification
+Q′
+sem, Q′
+bg
+Q′
+inst
+×
+Q′
+obj
+F4
+Figure 2. TarViS Architecture. Segmentation targets for different tasks are represented by a set of abstract target queries Qin. The core
+network (in green) is agnostic to the task definitions. The inner product between the output queries Qout and video feature F4 yields
+segmentation masks as required by the task.
+the-art methods are broadly based on the seminal work of
+Oh et al. [51], which learns space-time correspondences
+between pixels in different video frames, and then uses
+these to propagate the first-frame masks across the video.
+Subsequent methods [15–17,61,64,71,74,74,75] have sig-
+nificantly improved the performance and efficiency of this
+approach. Point Exemplar-guided Tracking (PET) [3,27] is
+a relatively new task for which the current best approach [3]
+involves regressing a pseudo-ground-truth mask from the
+given point coordinates, and then applies a state-of-the-art
+VOS method [17] to this mask.
+The above methods thus incorporate task-specific as-
+sumptions into their core approach. This can be beneficial
+for per-task performance, but makes it difficult for them to
+generalize across tasks. By contrast, TarViS can tackle all
+four aforementioned tasks, and generally any target-based
+video segmentation task, with a single, unified, jointly
+trained model.
+3. Method
+TarViS can segment arbitrary targets in video since the
+architecture is flexible with respect to how these targets
+are defined, thus enabling us to conceptually unify and
+jointly tackle the four aforementioned tasks (VIS, VPS,
+VOS, PET). The architecture is illustrated in Fig. 2.
+For all tasks, the common input to the network is an RGB
+video clip of length T denoted by V ∈ RH×W ×T ×3. This
+is input to a 2D backbone network which produces image-
+level feature maps, followed by a Temporal Neck, which
+enables feature interaction across time and outputs a set of
+temporally consistent, multi-scale, D-dimensional feature
+maps F = {F32, F16, F8, F4} where Fs ∈ R
+H
+s × W
+s ×T ×D.
+The feature maps F are then fed to our Transformer de-
+coder, together with a set of queries Qin which represent the
+segmentation targets. The decoder applies successive layers
+of self- and masked cross-attention wherein the queries are
+iteratively refined by attending to each other, and to the fea-
+ture maps, respectively. The refined queries output by the
+decoder are denoted with Qout. The following subsections
+explain how TarViS tackles each task in detail.
+3.1. Video Instance Segmentation
+VIS defines the segmentation target set as all objects be-
+longing to a set of predefined classes. Accordingly, the in-
+put query set Qin for VIS contains three types of queries:
+(1) semantic queries denoted by Qsem ∈ RC×D where C is
+the number of classes defined by the dataset, i.e., each D-
+dimensional vector in Qsem represents a particular seman-
+tic class. (2) instance queries denoted by Qinst ∈ RI×D
+where I is assumed to be an upper bound on the number of
+instances in the video clip, and (3) a background query de-
+noted by Qbg ∈ R1×D to capture inactive instance queries.
+The three query sets are concatenated, i.e., Qin
+=
+concat(Qsem, Qinst, Qbg), and input to the Transformer de-
+coder, which refines their feature representation through
+successive attention layers and outputs a set of queries
+Qout = concat(Q′
+sem, Q′
+inst, Q′
+bg). These are then used to
+produce temporally consistent instance mask logits by com-
+puting the inner product ⟨F4, Q′
+inst⟩ ∈ RH×W ×T ×I. To
+obtain classification logits, we compute the inner product
+⟨Q′
+inst, concat(Q′
+sem, Q′
+bg)⟩ ∈ RI×(C+1).
+The three types of queries are initialized randomly at
+the start of training and optimized thereafter. The instance
+queries Qinst enable us to segment a varying number of ob-
+jects from the input clip. During training, we apply Hun-
+garian matching between the predicted and ground-truth in-
+stance masks to assign instance queries to video instances,
+and then supervise their predicted masks and classification
+3
+
+logits accordingly. When training on multiple datasets with
+heterogeneous classes, the semantic query sets are sepa-
+rately initialized per dataset, but Qinst and Qbg are shared.
+Comparison to Instance Segmentation Methods.
+Sev-
+eral Transformer-based methods [10, 13, 66, 79] for im-
+age/video instance segmentation also use queries to seg-
+ment a variable number of input instances. The key dif-
+ference to our approach is the handling of object classes:
+existing works employ only instance queries which are in-
+put to a fully-connected layer with a fan-out of C + 1 to
+obtain classification (and background) scores. The notion
+of class-guided instance segmentation is thus baked into
+the approach. By contrast, TarViS is agnostic to the task-
+specific notion of object classes because it models them
+as arbitrary queries which are dynamic inputs to the net-
+work. The semantic representation for these queries is thus
+decoupled from the core architecture and is only learned via
+loss supervision. An important enabler for this approach
+is the background query Qbg, which serves as a ‘catch-all’
+class to represent everything that is not in Qsem. It is used to
+classify non-active instance queries, and its mask logits are
+supervised to segment all non-object input pixels.
+3.2. Video Panoptic Segmentation
+VPS defines the segmentation targets as all objects be-
+longing to a set of thing classes (e.g., ‘person’, ‘car’), and
+additionally, a set of non-instantiable stuff classes (e.g.,
+‘sky’, ‘grass’) which cover all non-object pixels. TarViS
+can tackle VPS with virtually no modification to the work-
+flow in Sec. 3.1. We can compute semantic segmentation
+masks for the input clip by simply taking the inner prod-
+uct between Qsem and the video features: ⟨F4, Q′
+sem⟩ ∈
+RH×W ×T ×C. Note that here, Qsem contains queries rep-
+resenting both thing and stuff classes.
+Comparison to VPS Methods. Current VPS datasets [35,
+67] involve driving scene videos captured from moving ve-
+hicles.
+Methods tackling this task [35, 56] are based on
+earlier image panoptic segmentation approaches [14] which
+involve multi-head networks for semantic and instance seg-
+mentation prediction. In terms of image-level panoptic seg-
+mentation, Mask2Former [13] uses a Transformer-based ar-
+chitecture, but it models stuff classes as instances which are
+Hungarian-matched to the ground-truth target during train-
+ing, whereas TarViS models semantic classes and instances
+using separate, designated queries.
+3.3.
+Video
+Object
+Segmentation
+and
+Point
+Exemplar-guided Tracking
+VOS and PET can be seen as instantiations of a gen-
+eral task where the segmentation targets are a set of O
+objects for which some ground-truth cue G is given. For
+VOS, G is provided in the form of first-frame object masks
+Mobj ∈ RO×H×W , whereas for PET, G is provided as the
+(x, y) coordinates Pobj ∈ RO×2 of a point inside each of
+the objects. TarViS jointly tackles these tasks by adopting a
+generalized approach in which the O target objects are en-
+coded into a set of object queries Qobj. Thus, both VOS and
+PET boil down to designing a function EncodeObjects(·)
+which regresses Qobj from the given ground-truth cues G
+and feature maps F:
+Qobj ←− EncodeObjects(G, F).
+(1)
+Note that Qobj is conceptually analogous to Qsem and
+Qinst used for VIS in that all three are abstract representa-
+tions for their respective task-specific segmentation targets.
+Video
+Object
+Segmentation.
+To
+implement
+EncodeObjects for VOS, we seek inspiration from
+HODOR [2], a recent method for weakly-supervised VOS,
+which encodes objects into concise descriptors as follows:
+the descriptors are initialized by average pooling the image
+features inside the object masks, followed by an iterative
+refinement where the descriptors attend to each other
+(self-attention) and to their respective soft-masked image
+features (cross-attention).
+For TarViS, we employ a lightweight Object Encoder
+with a similar workflow to encode the objects as a set of
+queries Qobj, but with two differences to HODOR [2]: in-
+stead of cross-attending to the entire image feature map
+(H · W points) with soft-masked attention, we apply hard-
+masked cross-attention to at most pmax feature points per ob-
+ject, where pmax ≪ H · W. Object masks containing more
+than pmax points are sub-sampled accordingly. This signifi-
+cantly improves the memory/run-time overhead of our Ob-
+ject Encoder. Secondly, we note that the process of distill-
+ing object features into a single descriptor involves a loss
+of object appearance information, which degrades perfor-
+mance. We therefore model each object with qo queries (in-
+stead of one) by spatially dividing each object mask into qo
+segments, i.e., Qobj ∈ RO×qo×D (we use qo = 4).
+In addition to Qobj, we initialize a set of background
+queries Qbg ∈ RB×D to model the non-target pixels in
+the reference frame. Following HODOR [2], we employ
+multiple background queries, which are initialized dynam-
+ically by dividing the video frame containing the ground-
+truth masks Mobj into a 4 × 4 grid and average pooling
+the non-object pixels in each grid cell.
+The Object En-
+coder jointly refines the background and object queries to
+yield Qin = concat(Qobj, Qbg). During training, the mask
+logits for the multiple background queries are aggregated
+per-pixel by applying max(·) and supervised to segment all
+pixels not part of the target object set.
+The remaining workflow follows that for VIS and VPS:
+Qin is input to the Transformer decoder together with the
+video features F.
+The refined output query set Qout =
+4
+
+concat(Q′
+obj, Q′
+bg) is then used to compute the inner prod-
+uct ⟨F4, Q′
+obj⟩ ∈ RH×W ×T ×O×qo. Subsequently, max(·) is
+applied on the qo-sized dimension to obtain the final mask
+logits for the O target objects.
+Point Exemplar-guided Tracking.
+For PET we imple-
+ment EncodeObjects in the exactly same way as VOS: the
+given point coordinates Pobj are converted into a mask with
+just one non-zero pixel, followed by iterative refinement by
+the Object Encoder (with shared weights for VOS and PET).
+The only difference is that here we represent each of the O
+objects with just one query, i.e., Qobj ∈ RO×D (qo = 1).
+The subsequent workflow is also identical to that for VOS:
+the queries are refined by the Transformer decoder followed
+by an inner product with F4 to obtain object mask logits.
+Comparison to VOS and PET Methods. Current state-of-
+the-art VOS methods are largely based on STM [51]. It in-
+volves learning pixel-to-pixel correspondences across video
+frames, which are then used to propagate the given object
+mask across the video.
+This approach is effective since
+every pixel in the given mask can be individually mapped
+to future frames, thus preserving fine-grained object de-
+tails. The core approach is, however, task-specific since
+it assumes the availability of first-frame object masks, and
+does not generalize to the PET (see Sec. 4.2). PET can be
+viewed as a more constrained version of VOS, where only
+a single object point is provided instead of the full mask.
+Consequently, PET [3] is currently tackled by casting it as
+a VOS problem by using an image instance segmentation
+network [13] to regress pseudo-ground-truth object masks
+from the given point coordinates Pobj.
+On the other hand, our approach of encoding objects as
+concise queries causes loss of fine-grained object appear-
+ance information, but it has the advantage of being agnostic
+to how G is defined. As evident from the unified work-
+flow for VOS and PET, any variation of these tasks with
+arbitrary ground-truth cues G can be seamlessly fused into
+our architecture as long as we can implement an effective
+EncodeObjects function to regress Qobj from the given G.
+3.4. Network Architecture
+Temporal Neck.
+As explained earlier, TarViS produces
+target masks by computing the inner product between Qout
+and the video feature map F4. For this to be effective, the
+per-pixel features F must be aligned for the same, and dis-
+similar for different targets. Some image instance segmen-
+tation methods [13,79] apply Deformable Attention [79] to
+the backbone feature maps to efficiently learn consistent,
+multi-scale image features. For TarViS, however, the fea-
+tures must also be temporally consistent across the entire
+input video clip. To achieve this, we propose a novel Tempo-
+ral Neck architecture which is based on the work of Berta-
+sius et al. [7] for video action classification. We enable ef-
+x
+t
+y
+Deformable
+Attention
+F32
+F16
+F8
+Temporal
+Attention
+Figure 3. Temporal Neck Layer. Colored regions denote the at-
+tention field w.r.t the selected pixel (darkened). Deformable At-
+tention is spatially unrestricted but temporally limited to a single
+frame, whereas Temporal Attention is spatially localized, but tem-
+porally unrestricted. F8 is inactive for the temporal attention.
+ficient spatio-temporal feature interaction by applying two
+types of self-attention in an alternating fashion: the first is
+spatially global and temporally localized, whereas the sec-
+ond is spatially localized and temporally global. The first
+operation is implemented with Deformable Attention, fol-
+lowing existing work [12,79]. The second operation, Tem-
+poral Attention, involves dividing the input space-time vol-
+ume into a grid along the spatial axes, and then applying
+self-attention to the space-time feature volume inside each
+grid cell. Both operations allow feature interaction across
+multiple scales. Both attention operations are illustrated in
+Fig. 3. We exclude F8 from temporal attention since we
+found this to be more memory-efficient without negatively
+impacting prediction quality.
+Transformer Decoder.
+The decoder architecture follows
+that of Mask2Former [13]: the input queries are iteratively
+refined over multiple layers. In each layer, the queries first
+cross-attend to their respective masked video features, then
+self-attend to each other, followed by feed-forward layers.
+3.5. Inference
+To infer on videos with arbitrary length, we split the in-
+put video into clips of length Tclip with an overlap of Tov be-
+tween successive clips. Object tracks are associated across
+clips based on their mask IoU in the overlapping frames.
+For the VOS tasks, the object queries for an intermediate
+clip are initialized by using the predicted masks in the over-
+lapping frames from the previous clip as a pseudo-ground-
+truth. For VPS, we average the semantic segmentation log-
+its in the overlapping frames. Our approach is thus near-
+online because the time delay in obtaining the output for a
+given frame is at most Tclip − Tov − 1 (except for the very
+first clip in the video).
+5
+
+Table 1. Results for Video Instance Segmentation (VIS) on the YouTube-VIS 2021 [73] and OVIS [55] validation sets.
+Method
+Backbone
+Shared
+Model
+YouTube-VIS 2021
+OVIS
+AP
+AP50
+AP75
+AR1
+AR10
+AP
+AP50
+AP75
+AR1
+AR10
+Mask2Former-VIS [12]
+R-50
+
+40.6
+60.9
+41.8
+-
+-
+-
+-
+-
+-
+-
+IDOL [70]
+R-50
+
+43.9
+68.0
+49.6
+38.0
+50.9
+30.2
+51.3
+30.0
+15.0
+37.5
+MinVIS [28]
+R-50
+
+44.2
+66.0
+48.1
+39.2
+51.7
+25.0
+45.5
+24.0
+13.9
+29.7
+VITA [26]
+R-50
+
+45.7
+67.4
+49.5
+40.9
+53.6
+19.6
+41.2
+17.4
+11.7
+26.0
+Ours (TarViS)
+R-50
+
+48.3
+69.6
+53.2
+40.5
+55.9
+31.1
+52.5
+30.4
+15.9
+39.9
+Mask2Former-VIS [12]
+Swin-T
+
+45.9
+68.7
+50.7
+-
+-
+-
+-
+-
+-
+-
+Ours (TarViS)
+Swin-T
+
+50.9
+71.6
+56.6
+42.2
+57.2
+34.0
+55.0
+34.4
+16.1
+40.9
+IDOL [70]
+Swin-L
+
+56.1
+80.8
+63.5
+45.0
+60.1
+42.6
+65.7
+45.2
+17.9
+49.6
+VITA [26]
+Swin-L
+
+57.5
+80.6
+61.0
+47.7
+62.6
+27.7
+51.9
+24.9
+14.9
+33.0
+Ours (TarViS)
+Swin-L
+
+60.2
+81.4
+67.6
+47.6
+64.8
+43.2
+67.8
+44.6
+18.0
+50.4
+Table 2. Video Panoptic Segmentation (VPS) results for validation sets of KITTI-STEP [67], CityscapesVPS [35] and VIPSeg [45].
+Method
+Shared
+Model
+KITTI-STEP
+CityscapesVPS
+VIPSeg
+STQ
+AQ
+SQ
+VPQ
+VPQTh
+VPQSt
+VPQ
+VPQTh
+VPQSt
+STQ
+Mask Propagation [67]
+
+0.67
+0.63
+0.71
+-
+-
+-
+-
+-
+-
+Track [35]
+
+-
+-
+-
+55.9
+43.7
+64.8
+-
+-
+-
+VPSNet [35]
+
+0.56
+0.52
+0.61
+57.0
+44.7
+66.0
+14.0
+14.0
+14.2
+20.8
+VPSNet-SiamTrack [68]
+
+-
+-
+-
+57.3
+44.7
+66.4
+17.2
+17.3
+17.3
+21.1
+VIP-Deeplab [56]
+
+-
+-
+-
+63.1
+49.5
+73.0
+16.0
+12.3
+18.2
+22.0
+Clip-PanoFCN [45]
+
+-
+-
+-
+-
+-
+-
+22.9
+25.0
+20.8
+31.5
+Ours (TarViS - R-50)
+
+0.70
+0.70
+0.69
+53.3
+35.9
+66.0
+33.5
+39.2
+28.5
+43.1
+Ours (TarViS - Swin-T)
+
+0.71
+0.71
+0.70
+58.0
+42.9
+69.0
+35.8
+42.7
+29.7
+45.3
+Ours (TarViS - Swin-L)
+
+0.72
+0.72
+0.73
+58.9
+43.7
+69.9
+48.0
+58.2
+39.0
+52.9
+4. Experiments
+4.1. Implementation Details
+Our Temporal Neck contains 6 layers of Deformable
+and Temporal Attention.
+We pretrain our model for
+500k iterations with batch size 32 on pseudo-video clips
+generated by applying on-the-fly augmentations to im-
+ages from COCO [43], ADE20k [78], Mapillary [49] and
+Cityscapes [18]. These samples are either trained for VPS,
+VIS, VOS or PET. This is followed by fine-tuning for
+90k iterations jointly on samples from YouTube-VIS [73],
+OVIS [55],
+KITTI-STEP [67],
+CityscapesVPS [35],
+VIPSeg [45], DAVIS [54] and BURST [3]. For each of the
+four query types (Qsem, Qinst, Qobj, Qbg) discussed in Sec. 3,
+we employ a learned query embedding, which is used when
+computing the KeyT Query affinity matrix for multi-head
+attention inside the decoder. We refer to the supplementary
+for more details.
+4.2. Benchmark Results
+All results are computed with a single, jointly trained
+model which performs different tasks by simply providing
+the corresponding query set at run-time.
+Video Instance Segmentation (VIS). We evaluate on two
+benchmarks: (1) YouTube-VIS 2021 [73] which covers
+40 object classes and contains 2985/421 videos for train-
+ing/validation, and (2) OVIS [55] which covers 25 object
+classes. It contains 607/140 videos for training/validation
+which are comparatively longer and more occluded. The
+AP scores for both are reported in Tab. 1. For all three
+backbones, TarViS achieves state-of-the-art results for both
+benchmarks even though other methods are trained sepa-
+rately per benchmark whereas we use a single model. On
+YouTube-VIS, TarViS achieves 48.3 AP with a ResNet-50
+backbone compared to the 45.7 achieved by VITA [26].
+With Swin-L, we achieve 60.2 AP which is also higher than
+the 57.5 by VITA. On OVIS with ResNet-50, our 31.1 AP
+is higher than the 30.2 for IDOL [70], and with Swin-L,
+TarViS (43.2 AP) out-performs the current state-of-the-art
+IDOL (42.6 AP).
+Video Panoptic Segmentation (VPS). We evaluate VPS
+on three datasets: (1) KITTI-STEP [67], which contains
+12/9 lengthy driving scene videos for training/validation
+with 19 semantic classes (2 thing and 17 stuff classes), (2)
+CityscapesVPS [35], which contains 50 short driving scene
+clips, each with 6 annotated frames, and (3) VIPSeg [45],
+which is a larger dataset with 2806/343 in-the-wild videos
+for training/validation and 124 semantic classes. The results
+6
+
+are reported in Tab. 2. For KITTI-STEP, TarViS achieves
+70% STQ with a ResNet-50 backbone which is better than
+all existing approaches. The performance further improves
+to 72% with Swin-L. For CityscapesVPS, TarViS achieves
+58.9 VPQ which is higher than all other methods except
+VIP-Deeplab [56] (63.1). However, VIP-Deeplab performs
+monocular depth estimation for additional guidance, and
+therefore requires ground-truth depth-maps for training.
+For VIPSeg, TarViS out-performs existing approaches
+by a significant margin.
+With a ResNet-50 backbone,
+our 33.5 VPQ is 10.6% higher than the 22.9 by Clip-
+PanoFCN [45].
+With a Swin-Large backbone, TarViS
+achieves 48.0 VPQ which is more than double that of Clip-
+PanoFCN (22.9). Note that VIP-Deeplab performs signifi-
+cantly worse for VIPSeg (16.0 VPQ), showing that TarViS
+generalizes better across benchmarks. Finally, we note that
+larger backbones results in significant performance gains
+for datasets with in-the-wild internet videos as in VIPSeg,
+but for specialized driving scene datasets (e.g. KITTI-STEP
+and Cityscapes-VPS), the improvements are much smaller.
+Video Object Segmentation (VOS).
+We evaluate VOS
+on the DAVIS 2017 [54] dataset, which contains 60/30
+YouTube videos for training/validation.
+The results in
+Tab. 3 show that TarViS achieves 85.3 J &F which is
+higher than all existing methods except STCN [17] (85.4)
+and XMem [15] (86.2). As mentioned in Sec. 3.3, encod-
+ing objects as queries incurs a loss of fine-grained infor-
+mation, which is detrimental to performance. On the other
+hand, space-time correspondence (STC) based approaches
+learn pixel-to-pixel affinities between frames, which en-
+ables them to propagate fine-grained object appearance in-
+formation. We note, however, that TarViS is the first method
+not based on the STC paradigm which achieves this level
+is performance (85.3 J &F), out-performing several STC-
+based methods as well as all non-STC based methods e.g.
+HODOR [2] (81.5) and UNICORN [72] (70.6).
+Point
+Exemplar-guided
+Tracking
+(PET).
+PET
+is
+evaluated on the recently introduced BURST bench-
+mark [3] which contains 500/1000 diverse videos for train-
+ing/validation with indoor, outdoor, driving and scripted
+movie scenes. It is a constrained version of VOS which
+only provides the point coordinates of the object mask cen-
+troid instead of the full mask.
+Tab. 3 shows that exist-
+ing methods can only tackle either VOS or PET. To verify
+this, we tried adapting STCN [17] for PET by training it
+with point masks, but the training did not converge. By
+contrast, TarViS encodes objects into queries, which en-
+ables it to tackle both tasks with a single model since the
+object guidance (point or mask) is abstracted behind the
+EncodeObjects(·) function.
+TarViS achieves 37.5 HOTAall which is significantly
+better than the 24.4 achieved by the best performing base-
+Table 3. Results for Mask-guided VOS on DAVIS [54] and Point-
+guided VOS on BURST [3] (‘H’ denotes ‘HOTA’ [44]).
+Method
+DAVIS (M-VOS)
+BURST (P-VOS)
+J &F J
+F
+Hall
+Hcom Hunc
+UNICORN∗ [72]
+70.6 66.1 75.0
+-
+-
+-
+HODOR [2]
+81.3 78.4 83.9
+-
+-
+-
+STM [51]
+81.8 79.2 84.3
+-
+-
+-
+CFBI [74]
+81.9 79.1 84.6
+-
+-
+-
+RMNet [71]
+83.5 81.0 86.0
+-
+-
+-
+HMMN [61]
+84.7 81.9 87.5
+-
+-
+-
+MiVOS [16]
+84.5 81.7 87.4
+-
+-
+-
+AOT [75]
+84.9 82.3 87.5
+-
+-
+-
+STCN [17]
+85.4 82.2 88.6
+-
+-
+-
+XMem [15]
+86.2 82.9 89.5
+-
+-
+-
+Box Tracker [32]
+-
+-
+-
+12.7 31.7
+7.9
+STCN+M2F [13,17]
+-
+-
+-
+24.4 44.0 19.5
+Ours (TarViS - R-50)
+82.0 78.7 87.0
+30.9 43.2 27.8
+Ours (TarViS - Swin-T)
+82.8 79.6 86.0
+36.0 47.7 33.0
+Ours (TarViS - Swin-L)
+85.3 81.7 88.5
+37.5 51.7 34.0
+VIS
+VPS
+VOS
+PET
+Figure 4. Qualitative results from a single TarViS model for all
+four tasks. Further results are shown in the supplementary.
+line method which casts PET as a VOS problem by regress-
+ing a pseudo-ground-truth mask from the given point, fol-
+lowed by applying a VOS approach (STCN [17]).
+4.3. Ablations
+We ablate several aspects of our architecture/training un-
+der a reduced training setup with batch size 16. Pre-training
+is done only with COCO for 380k iterations, followed by
+fine-tuning on a smaller set of video datasets for 80k itera-
+tions. The results are presented in Table 4.
+Task-specific Training (row 1-2).
+The first two rows
+show results for task-specific models.
+Since the KITTI-
+STEP [67] dataset is quite small, it was not possible to train
+a VPS-only variant. Moreover, we train a single model for
+VOS and PET since both tasks are closely related. We note
+that the VIS only model performs noticeably worse than the
+7
+
+RDTable 4. Ablation experiment results. A Swin-T backbone is used for all settings.
+Training Data
+VIS
+VPS
+VOS
+PET
+Setting
+YTVIS
+OVIS
+KITTI
+DAVIS
+BURST
+YTVIS
+OVIS
+KITTI-STEP
+DAVIS
+BURST
+(mAP)
+(mAP)
+(STQ)
+(J &F)
+(HOTAall)
+1. VIS
+ 
+49.8
+30.0
+-
+-
+-
+2. VOS + PET
+ 
+-
+-
+-
+81.4
+32.8
+3. VIS (FC for CLS)
+ 
+50.5
+31.0
+-
+-
+-
+4. No Temporal Neck
+    
+46.8
+25.4
+0.67
+78.0
+29.7
+5. No Object Encoder
+    
+50.1
+32.6
+0.66
+75.6
+25.9
+Final
+    
+51.1
+31.7
+0.69
+81.5
+29.2
+final setting (49.8 vs. 51.1 mAP on YouTube-VIS), which
+indicates that the combination of additional training data
+and multi-task supervision in the final model is beneficial
+for VIS. For VOS on DAVIS [54], the task-specific model
+achieves similar performance to the final setting (81.4 vs.
+81.5), but for PET the task-specific variant performs no-
+ticeably better (32.8 vs. 29.2). We conclude that although
+the task-specific model performs better on PET, multi-task
+training generally improves performance across tasks.
+Semantic Queries for VIS (row 3).
+As mentioned in
+Sec. 3.1, TarViS represents object classes as dynamic query
+inputs to the network (Qsem). We ablate this by modifying
+our network to work for only VIS in the same way as ex-
+isting instance segmentation methods [13,66], i.e. using in-
+stance queries Qinst in conjunction with a linear layer (sep-
+arate for each dataset) for classification. The results show
+that this task-specific architecture is better suited for VIS
+since it achieves 50.5 mAP on YouTube-VIS vs. 49.8 for
+the setting in row 1 which is trained on similar data. How-
+ever, our final multi-task model still performs slightly better
+than the current setting on both YouTube-VIS and OVIS.
+Temporal Neck (row 4).
+We validate the effectiveness
+of our novel Temporal Neck (Sec. 3.4) by training a model
+with a simpler neck that contains only Deformable Atten-
+tion layers [79], similar to Mask2Former [13], i.e. there
+is no feature interaction across frames. The results show
+significant performance degradation for YouTube-VIS (46.8
+bs. 51.1), OVIS (25.4 vs. 31.7) and DAVIS (78.0 vs. 81.5).
+KITTI-STEP is impacted comparatively less (0.67 vs 0.69),
+whereas performance for PET actually shows a slight in-
+crease (29.7 vs. 29.2) Overall, however, we conclude that
+inter-frame feature interactions enabled by the Temporal
+Neck are beneficial for down-stream tasks.
+Object Encoder for VOS/PET (row 5).
+As discussed
+in Sec. 3.3, we encode objects from their given first-frame
+mask/point as object queries Qobj using an Object Encoder.
+We validate the efficacy of this module by training a simpler
+model which initializes object queries by average pooling
+the image features inside the mask for VOS, and indexing
+Figure 5. TarViS performing VIS and VOS in a single forward
+pass. We provide the mask for the dragon on the left, and the
+semantic query for the ‘person’ class.
+the feature map at the given point coordinates for PET. This
+model performs significantly worse than the final setting on
+both DAVIS (75.6 vs. 81.5) and BURST (25.9 vs. 29.2),
+indicating that the quality of the encoded object query has a
+profound impact on performance.
+5. Discussion
+Limitations.
+As noted above, training on multiple
+datasets/tasks does not necessarily improve performance on
+all benchmarks. For VOS, the model exhibits class bias
+since it sometimes fails to track unusual objects which were
+not seen during training.
+Future Outlook.
+We jointly trained TarViS for four dif-
+ferent tasks to validate its generalization capability. The ar-
+chitecture can, however, tackle any video segmentation task
+for which the targets can be encoded as queries. The re-
+cent emergence of joint language-vision models [42,57,58]
+thus makes it possible to perform multi-object segmenta-
+tion based on a text prompt if the latter can be encoded as a
+target query using a language encoder [19]. Another inter-
+esting possibility is that TarViS could be applied to multiple
+tasks in the same forward pass by simply concatenating the
+task-specific queries. Fig. 5 offers a promising outlook for
+this; it shows our model’s output for a video clip from a
+8
+
+popular TV series where we perform VIS and VOS simul-
+taneously by providing the semantic query for the ‘person’
+class (from YouTube-VIS [73]), and the VOS-based object
+queries for the dragon by annotating its first frame mask,
+i.e. Qin = concat(Qsem, Qinst, Qobj, Qbg). TarViS success-
+fully segments all four persons in the scene (VIS) and the
+dragon (VOS), even though our model was never trained to
+simultaneously tackle both tasks in a single forward pass.
+6. Conclusion
+We presented TarViS: a novel, unified approach for tack-
+ling any task requiring pixel-precise segmentation of a set
+of targets in video. We adopt a generalized paradigm where
+the task-specific targets are encoded into a set of queries
+which are then input to our network together with the video
+features. The network is trained to produce segmentation
+masks for each target entity, but is inherently agnostic to
+the task-specific definition of these targets. To demonstrate
+the effectiveness of our approach, we applied it to four dif-
+ferent video segmentation tasks (VIS, VPS, VOS, PET). We
+showed that a single TarViS model can be jointly trained for
+all tasks, and during inference can hot-swap between tasks
+without any task-specific fine-tuning. Our model achieved
+state-of-the-art performance on five benchmarks (YouTube-
+VIS, OVIS, KITTI-STEP, VIPSeg and BURST) and has
+multiple, promising directions for future work.
+Acknowledgments.
+This project was partially funded
+by ERC Consolidator Grant DeeVise (ERC-2017-COG-
+773161). We thank Istvan Sarandi, Christian Schmidt and
+Alexey Nekarsov for constructive feedback. Compute re-
+sources were granted by RWTH Aachen under project ID
+supp0003, and by the Gauss Centre for Supercomputing
+e.V. through the John von Neumann Institute for Computing
+on the GCS Supercomputer JUWELS at J¨ulich Supercom-
+puting Centre.
+9
+
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+Supplementary Material
+S1. Extended VOS/PET Ablations
+Extended ablation results are given in Table S1 and dis-
+cussed below. Note that similar to the ablations in Sec. 4 of
+the main text, we use a smaller training schedule with fewer
+datasets for these experiments.
+Table S1. Extended ablation results for VOS and PET tasks on
+DAVIS [54] and BURST [3] benchmarks, respectively.
+Setting
+VOS (J &F)
+PET (HOTAall)
+Without Qbg
+78.0
+25.9
+|Qobj| = q0 = 1
+80.6
+28.2
+Final
+81.5
+29.2
+Background Queries (row 1). We stated in the main text
+that we model the non-object pixels in the input video using
+background queries for the VOS and PET task. We ablate
+this design decision by training TarViS without this sort of
+background modeling, i.e. for both VOS and PET tasks, the
+input set of queries contains only the object queries Qobj.
+This reduces the J &F score for VOS from 81.5 to 78.0,
+and the HOTAall score for PET from 29.2 to 25.9. Thus,
+we conclude that background modelling has a noticeable,
+positive impact on prediction quality.
+Number of Object Queries (row 2).
+We mentioned in
+the main text that we modify the approach adopted by
+HODOR [2] for VOS by using multiple (q0) object queries
+to represent a single target object. We ablate this by training
+our model using q0 = 1 (in the final setting we use q0 = 4).
+We see that this causes the performance on DAVIS to re-
+duce from 81.5 to 80.6, and that on BURST from 29.2 to
+28.2. Note that q0 = 1 for PET even for the final setting, but
+because PET inference over lengthy videos involves VOS-
+style mask-guidance, the choice of q0 for VOS effects per-
+formance for PET as well.
+S2. Implementation Details
+Several details related to the training and inference setup
+which were omitted from the main paper are given below.
+Setup. We train our models on 32 Nvidia A100 GPUs with
+a batch size of 32 with clips of 3 frames. The pretraining
+takes 2-3 days (depending on the backbone) and the fine-
+tuning takes 10-16 hours. An AdamW optimizer is used,
+and the learning rate is 104 at the start of training followed
+by two step decays with a factor of 0.1 each. Inference is
+performed on a single RTX 3090 and runs at approximately
+10fps with a Swin-T backbone (there is some variation for
+different datasets due to the varying input image resolution).
+Table S2. Loss functions used for mask prediction for different tar-
+gets. BCE: Binary cross-entropy, MCE: Multi-class cross-entropy,
+DICE: soft IoU loss
+Target Type
+Task
+Loss
+Sparse
+Instance
+VIS
+DICE + BCE
+
+Semantic Class
+MCE
+
+Instance
+VPS
+DICE + BCE
+
+Semantic Class
+MCE
+
+Object
+VOS/PET
+DICE + BCE
+
+The clip length during inference is usually 12 with a frame
+overlap of 6 frames.
+Loss Supervision.
+Table S2 shows the type of loss func-
+tion applied for mask regression for different tasks. Gener-
+ally, the supervision signal is a combination of DICE and
+cross-entropy losses. For instances/objects we apply per-
+pixel binary cross-entropy whereas for semantic segmenta-
+tion (where multiple classes compete for each pixel), we
+apply a multi-class cross-entropy loss. ‘Sparse’ means that
+the loss is not applied to every pixel in the mask, but rather
+only to a subset of sampled pixels which contain a cer-
+tain fraction of hard negatives and other randomly sampled
+points. This type of supervision strategy was proposed by
+Kirillov et al. [37].
+Pretraining.
+We pretrain on synthetic video samples
+generated by applying random, on-the-fly augmentations
+from the following image-level datasets:
+COCO [43],
+ADE20k [78], Mapillary [49], Cityscapes [18]. Since these
+datasets provide panoptic annotations, we can train the
+data samples as any of the four target tasks (VPS, VIS,
+VOS, PET) e.g. to train for VOS/PET, we assume that the
+first-frame mask/point is available for a random sub-set of
+ground-truth objects and ignore the class labels. The task
+weights for pretraining are given in Table S3.
+Table S3. Task weights during pretraining stage.
+Task
+VPS
+VIS
+VOS
+PET
+Weight
+0.3
+0.3
+0.28
+0.12
+Video Finetuning. The finetuning is done on actual video
+datasets for all four tasks. The sampling weights for each
+dataset/task are given in Table S4. Note that data samples
+from DAVIS [54] and BURST [3] can be trained for both
+VOS and PET.
+Point Exemplar-guided Tracking Inference.
+As men-
+tioned in Sec. 3 of the main text, the PET task is tackled
+using the same workflow as for VOS i.e. the target objects
+1
+
+Table S4. Dataset weightage during video finetuning.
+Dataset
+Task
+Weight
+KITTI-STEP [67]
+VPS
+0.075
+CityscapesVPS [35]
+VPS
+0.075
+VIPSeg [45]
+VPS
+0.15
+YouTube-VIS [73]
+VIS
+0.225
+OVIS [55]
+VIS
+0.225
+DAVIS [54]
+VOS/PET
+0.05
+BURST [3]
+VOS/PET
+0.2
+are encoded as object queries using the Object Encoder. An
+additional detail about inference on arbitrary length video
+sequences which is not mentioned in the main text is as fol-
+lows: the point −→ object query regression is only used for
+the first clip in which the object appears. For subsequent
+clips, we have access to the dense mask predictions for that
+object from our model. Hence, for subsequent clips, we
+regress the object query from the previous mask predictions
+(as we do for VOS).
+S3. Query Visualization
+To gain some insight into the feature representation
+learned by TarViS for different targets, we provide visu-
+alizations of the target queries for various tasks and in-
+put video clips in Fig. S1,S2,S3. The setup is as follows:
+for each video clip, we run inference twice: (1) as VIS
+where the targets are all instances belonging to the 40 ob-
+ject classes from YouTube-VIS [73], and (2) as VOS by
+providing the first-frame mask for the objects. We delib-
+erately used videos where the set of set of ground-truth ob-
+jects would be the same for both tasks. The plot on the
+right visualizes the union of the target query set for both
+runs by projecting them from 256 dimensions down to 2
+using PCA. The image tile on the left shows our model’s
+predicted masks for the target objects (the prediction qual-
+ity for these video is very good for both VIS and VOS, so
+we choose one set of results arbitrarily).
+For ease of understanding, we use fixed colors for se-
+mantic and background queries (as indicated in the plot leg-
+end). For the object queries (VOS) and instance queries
+(VIS), the color of the query point is consistent with the
+color of the object mask in the image tile. Note that for
+VOS we used qo = 4 object queries per target, hence there
+are 4 hollow diamond shaped points per object.
+Though not all aspects of these plots are intuitively ex-
+plainable, we offer some speculative intuition as listed be-
+low:
+• The internal representation for a given object is gener-
+ally consistent across tasks. As an example, consider
+the horse and person targets in Fig. S1: we note that
+the green query points (person) are close to each other
+for both VIS and VOS. Likewise the blue query points
+(horse) follow the same behavior.
+• The network devotes a large portion of the feature
+space for instances/objects, and relatively less for the
+various semantic classes. As seen in all three plots,
+the semantic queries are tightly clustered together,
+whereas the instance/object queries are spread out over
+a larger span of the feature space.
+Iterative Evolution of Feature Representation.
+Fig. S4
+shows a side-by-side visualization of how the query feature
+representation evolves inside the transformer decoder as it
+iteratively refined the queries using multiple attention lay-
+ers. The plot on the left shows the queries at the ‘zeroth’
+layer (i.e. prior to any interaction with the video features),
+and the plot on the right shows the final output queries from
+the last layer (these are identical to the plot in Fig. S1 except
+for the axes range). We note that the distance between the
+queries for the two objects increases after refinement, and
+that the semantic queries are also slightly more spaced out
+after refinement.
+S4. Qualitative Results
+The following figures show qualitative results for the
+different tasks.
+VIS on YouTube-VIS (Fig. S5,S6,S7)
+and
+OVIS
+(Fig.
+S8,S9,S10),
+VPS
+on
+KITTI-STEP
+(Fig. S11,S12,S13), VOS on DAVIS (Fig. S14,S15,S16),
+and PET on BURST (Fig. S17,S18,S19). One can see that
+TarViS is able to segment a broad range of objects depend-
+ing on the target queries and overall is good at assigning
+consistent IDs. Fig. S20 shows an example of a failure case
+with several ID switches. Given that we run inference on
+short overlapping clips, once an ID switch has been made,
+we cannot recover the original ID. In the example, it seems
+that TarViS is not able to track the elephant while they are
+turning around, even though before and after the turn they
+are assigned consistent IDs. Given that we also train on
+similar short clips, it is not surprising that TarViS struggles
+here and we could potentially improve this by looking into
+other training schemes that span longer clips.
+2
+
+airplane
+bear
+bird
+boat
+car
+cat
+cow
+deer
+dog
+duck
+earless_seal
+elephant
+fish
+flying_disc
+fox
+frog
+giant_panda
+giraffe
+leopard
+lizard
+monkey
+motorbike
+mouse
+parrot
+rabbit
+shark
+skateboard
+snake
+snowboard
+squirrel
+surfboard
+tennis_racket
+tiger
+train
+truck
+turtle
+whale
+zebra
+3
+2
+1
+0
+1
+2
+3
+4
+1
+0
+1
+2
+3
+horse
+person
+VIS (background)
+VIS (instance)
+VIS (semantic)
+VOS (background)
+VOS (object)
+Figure S1. Target query visualization for the ‘horsejump-high’ sequence in DAVIS.
+airplane
+bear
+bird
+boat
+car
+cat
+cow
+deer
+dog
+duck
+earless_seal
+elephant
+fish
+flying_disc
+fox
+frog
+giant_panda
+giraffe
+horse
+leopard
+lizard
+monkey
+mouse
+parrot
+rabbit
+shark
+skateboard
+snake
+snowboard
+squirrel
+surfboard
+tennis_racket
+tiger
+train
+truck
+turtle
+whale
+zebra
+3
+2
+1
+0
+1
+2
+3
+2
+1
+0
+1
+2
+3
+motorbike
+person
+VIS (background)
+VIS (instance)
+VIS (semantic)
+VOS (background)
+VOS (object)
+Figure S2. Target query visualization for the ‘mbike-trick’ sequence in DAVIS.
+airplane
+bear
+bird
+boat
+car
+cat
+cow
+deer
+dog
+duck
+earless_seal
+elephant
+fish
+flying_disc
+fox
+frog
+giant_panda
+giraffe
+horse
+leopard
+lizard
+monkey
+motorbike
+mouse
+parrot
+rabbit
+shark
+skateboard
+snake
+snowboard
+squirrel
+tennis_racket
+tiger
+train
+truck
+turtle
+whale
+zebra
+2
+0
+2
+4
+6
+8
+10
+4
+3
+2
+1
+0
+1
+2
+3
+4
+person
+surfboard
+VIS (background)
+VIS (instance)
+VIS (semantic)
+VOS (background)
+VOS (object)
+Figure S3. Target query visualization for the ‘kitesurf’ sequence in DAVIS.
+3
+
+AAAAAAairplane
+bear
+birdboat
+car
+cat
+cow
+deer
+dog
+duck
+earless_seal
+elephant
+fish
+flying_disc
+fox
+frog
+giant_panda
+giraffe
+leopard
+lizard
+monkey
+motorbike
+mouse
+parrot
+rabbit
+shark
+skateboard
+snake
+snowboard
+squirrel
+surfboard
+tennis_racket
+tiger train
+truck
+turtle
+whale
+zebra
+4
+3
+2
+1
+0
+1
+2
+3
+4
+5
+3
+2
+1
+0
+1
+2
+3
+4
+horse
+person
+VIS (background)
+VIS (instance)
+VIS (semantic)
+VOS (background)
+VOS (object)
+(a) First layer queries.
+airplane
+bear
+bird
+boat
+car
+cat
+cow
+deer
+dog
+duck
+earless_seal
+elephant
+fish
+flying_disc
+fox
+frog
+giant_panda
+giraffe
+leopard
+lizard
+monkey
+motorbike
+mouse
+parrot
+rabbit
+shark
+skateboard
+snake
+snowboard
+squirrel
+surfboard
+tennis_racket
+tiger
+train
+truck
+turtle
+whale
+zebra
+4
+3
+2
+1
+0
+1
+2
+3
+4
+5
+3
+2
+1
+0
+1
+2
+3
+4
+horse
+person
+VIS (background)
+VIS (instance)
+VIS (semantic)
+VOS (background)
+VOS (object)
+(b) Last layer queries.
+Figure S4. Evolution of the different queries from the first layer to the last layer of the transformer decoder. Queries correspond to the
+‘horsejump-high’ video from DAVIS as shown in Figure S1
+Figure S5. VIS on a YTVIS sequence showing a cat and a dog.
+4
+
+Figure S6. VIS on a YTVIS sequence showing a turtle.
+Figure S7. VIS on a YTVIS sequence showing a man and a lizard.
+Figure S8. VIS on an OVIS sequence showing an aquarium with fish.
+5
+
+BBCBBCBBCBBOBBCBBC人Figure S9. VIS on an OVIS sequence showing several sheep.
+Figure S10. VIS on an OVIS sequence showing three cats.
+Figure S11. VPS on a KITTI STEP sequence showing a busy intersection.
+Figure S12. VPS on a KITTI STEP sequence showing how a car is followed for a while.
+6
+
+AFigure S13. VPS on a KITTI STEP sequence showing a busy pedestrian crossing.
+Figure S14. VOS on a DAVIS sequence of a dancer.
+Figure S15. VOS on a DAVIS sequence showing several gold fish.
+7
+
+tFigure S16. VOS on DAVIS sequence an action movie scene.
+Figure S17. PET on a BURST sequence showing three men and a gun.
+Figure S18. PET on a BURST sequence showing two bears fight, note there is no ID switch.
+8
+
+16
+TYPHDTYPHD
+16TVPHD
+16TVPHDTIVPHD16
+TVPHDFigure S19. PET on a BURST sequence showing several cars on a street.
+Figure S20. VIS on an OVIS sequence of several elephants and their trainers. This sequence shows that TarVis sometimes has issues with
+ID switches, especially when the appearance of objects changes, e.g. here the elephants are not tracked consistently while turning around..
+9
+
+chs

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+Bottomonium suppression in the quark-gluon plasma –
+From effective field theories to non-unitary quantum
+evolution∗
+Michael Strickland
+Department of Physics, Kent State University, Kent, OH 44242, USA
+Received January 4, 2023
+In this proceedings contribution I review recent work which computes
+the suppression of bottomonium production in heavy-ion collisions using
+open quantum systems methods applied within the potential non-relativistic
+quantum chromodynamics (pNRQCD) effective field theory. I discuss how
+the computation of bottomonium suppression can be reduced to solving
+a Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) quantum master equa-
+tion for the evolution of the b¯b reduced density matrix. The open quantum
+systems approach used allows one to take into account the non-equilibrium
+dynamics and decoherence of bottomonium in the quark-gluon plasma. Fi-
+nally, I present comparisons of phenomenological predictions obtained us-
+ing a recently obtained next-to-leading-order GKSL equation with ALICE,
+ATLAS, and CMS experimental data for bottomonium suppression and
+elliptic flow.
+1. Introduction
+Heavy-ion collisions have been used to produce and study the properties
+of the quark-gluon plasma (QGP), a state of matter thought to have existed
+in the early universe and being created terrestrially in relativistic heavy-ion
+collisions. The suppression of bottomonium production in such collisions
+is considered strong evidence for the creation of a deconfined QGP [1–10].
+In the past, it was proposed that this suppression was due to the Debye
+screening of chromoelectric fields in the QGP, which modified the potential
+between heavy quarks and resulted in a reduction of heavy-quarkonium
+production [11, 12].
+However, more recent studies have shown that, in
+addition to the real part of the potential being modified by Debye screening,
+there is also an imaginary contribution to the potential caused by processes
+such as Landau damping and singlet-to-octet transitions [13–20].
+These
+∗ Presented at Excited QCD 2022, Giardini Naxos, Sicily, October 2022
+(1)
+arXiv:2301.01031v1  [hep-ph]  3 Jan 2023
+
+2
+strickland
+printed on January 4, 2023
+processes result in large in-medium widths for heavy-quarkonium bound
+states.
+In the past decade, there has been significant progress in the use of
+open quantum systems (OQS) methods to study heavy-quarkonium sup-
+pression in the QGP [21–37]. In particular, recent works have applied OQS
+methods within the framework of the potential non-relativistic QCD (pN-
+RQCD) effective field theory [14, 16, 18–20, 38–40]. The pNRQCD EFT is
+applicable to systems with a large separation between energy scales. This
+naturally occurs when the velocity of the heavy quark relative to the center
+of mass is small (v ≪ 1). In Refs. [24, 26, 30], the authors considered the
+scale hierarchy relevant for small bound states in a high-temperature QGP,
+1/r ∼ Mv ≫ mD ∼ πT ≫ E, where r is the typical size of the state, M is
+the heavy quark mass, mD is the Debye mass, T is the temperature, and E
+is the binding energy.
+With this scale ordering the environment’s relaxation timescale is much
+shorter than both the system’s internal timescales and the system’s own
+relaxation timescale.
+This makes the quantum evolution Markovian.
+In
+Ref. [30] a Markovian Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equa-
+tion [41, 42] was derived for the heavy-quarkonium reduced density matrix,
+which was implemented in the open-source QTraj code of Ref. [43] to make
+predictions for heavy-ion collision bottomonium observables [44–47]. This
+was done by coupling the GKSL solver to 3+1D viscous hydrodynamics
+code using smooth Glauber initial conditions [48–50] and, most recently,
+fluctuating hydrodynamical backgrounds [47]. The formalism used in the
+most recent works [46, 47] is accurate to next-to-leading order (NLO) in
+the binding energy over temperature, which allows it to be used at lower
+temperatures than the original leading-order formalism.
+2. Results
+For details concerning the theoretical and numerical methods employed,
+I refer the reader to Refs. [46, 47]. The results obtained depend on two co-
+efficients ˆκ and ˆγ, which were extracted directly and indirectly from lattice
+QCD calculations [30, 51–55]. In Figs. 1 and 2, we present our NLO predic-
+tions for RAA as a function of Npart and pT , respectively. For these results
+we did not include the effect of dynamical quantum jumps. In the left panel
+we show the variation of ˆκ in the range ˆκ ∈ {ˆκL(T), ˆκC(T), ˆκU(T)} while
+holding ˆγ = −2.6. This value of ˆγ was chosen as to best reproduce the
+RAA[Υ(1S)]. In the right panel we show the variation of ˆγ in the range
+−3.5 ≤ ˆγ ≤ 0 with ˆκ(T) = ˆκC(T). The solid line corresponds to ˆγ = −2.6.
+As this figure demonstrates, our NLO predictions without quantum jumps
+are in quite good agreement with the experimental data for RAA[1S] and
+
+strickland
+printed on January 4, 2023
+3
+QTraj - Υ(1S)
+QTraj - Υ(2S)
+QTraj - Υ(3S)
+ALICE - Y(1S)
+ATLAS - Y(1S)
+CMS - Y(1S)
+ALICE - Y(2S)
+ATLAS - Y(2S)
+CMS - Y(2S)
+CMS - Y(3S)
+0
+100
+200
+300
+400
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Npart
+RAA
+κ ∈ {κL(T),κC(T),κU(T)}, γ = -2.6, τmed = 0.6 fm, NLO
+5.02 TeV Pb-Pb
+ALICE: pT < 15 GeV and 2.5 < y < 4
+ATLAS: pT < 15 GeV and |y| < 1.5
+CMS: pT < 30 GeV and |y| < 2.4
+QTraj: pT < 30 GeV and y=0
+QTraj - Υ(1S)
+QTraj - Υ(2S)
+QTraj - Υ(3S)
+ALICE - Y(1S)
+ATLAS - Y(1S)
+CMS - Y(1S)
+ALICE - Y(2S)
+ATLAS - Y(2S)
+CMS - Y(2S)
+CMS - Y(3S)
+0
+100
+200
+300
+400
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Npart
+RAA
+κ = κC(T), γ ∈ {-3.5,-2.6,0}, τmed = 0.6 fm, NLO
+5.02 TeV Pb-Pb
+ALICE: pT < 15 GeV and 2.5 < y < 4
+ATLAS: pT < 15 GeV and |y| < 1.5
+CMS: pT < 30 GeV and |y| < 2.4
+QTraj: pT < 30 GeV and y=0
+Fig. 1.
+The nuclear suppression, RAA[Υ(1S, 2S, 3S)], as a function of the number
+of participants, Npart. The left panel shows variation of ˆκ and the right panel shows
+variation of ˆγ. The experimental results shown are from the ALICE [7], ATLAS [8],
+and CMS [5, 10] collaborations.
+QTraj - Υ(1S)
+QTraj - Υ(2S)
+QTraj - Υ(3S)
+ALICE - Y(1S)
+ATLAS - Y(1S)
+CMS - Y(1S)
+ATLAS - Y(2S)
+CMS - Y(2S)
+CMS - Y(3S)
+0
+5
+10
+15
+20
+25
+30
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+pT [GeV]
+RAA
+κ ∈ {κL(T),κC(T),κU(T)}, γ = -2.6, τmed = 0.6 fm, NLO
+5.02 TeV Pb-Pb
+CMS: 0-100% and |y| < 2.4
+ALICE: 0-90% and 2.5 < y < 4
+ATLAS: 0-80% and |y| < 1.5
+QTraj: 0-100% and y=0
+QTraj - Υ(1S)
+QTraj - Υ(2S)
+QTraj - Υ(3S)
+ALICE - Y(1S)
+ATLAS - Y(1S)
+CMS - Y(1S)
+ATLAS - Y(2S)
+CMS - Y(2S)
+CMS - Y(3S)
+0
+5
+10
+15
+20
+25
+30
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+pT [GeV]
+RAA
+κ = κC(T), γ ∈ {-3.5,-2.6,0}, τmed = 0.6 fm, NLO
+5.02 TeV Pb-Pb
+CMS: 0-100% and |y| < 2.4
+ALICE: 0-90% and 2.5 < y < 4
+ATLAS: 0-80% and |y| < 1.5
+QTraj: 0-100% and y=0
+Fig. 2.
+The nuclear suppression factor, RAA, for Υ(1S, 2S, 3S) as a function of
+the transverse momentum, pT . The bands, etc. are the same as Fig. 1.
+RAA[3S]. However, for the 2S excited state, our NLO predictions without
+quantum jumps are somewhat lower than the experimental results, partic-
+ularly for the most central collisions.
+Recently, we computed the NLO bottomonium RAA and v2 using both
+smooth and fluctuating initial conditions for the hydrodynamic evolution [47].
+In Ref. [47] in was demonstrated that the results for RAA obtained using fluc-
+tuating and smooth initial conditions were nearly identical, indicating that
+initial state fluctuations do not play an important role in this observable. In
+Fig. 3, I present the OQS+pNRQCD+IP-Glasma predictions for v2[1S] as
+
+4
+strickland
+printed on January 4, 2023
+CMS
+QTraj-γ
+QTraj-κ
+10
+30
+50
+90
+-0.06
+-0.04
+-0.02
+0.00
+0.02
+0.04
+0.06
+Centrality (%)
+v2[Υ(1S)]
+τmed = 0.6 fm, NLO
+5.02 TeV Pb-Pb
+pT < 50 GeV
+CMS: |y| < 2.4
+QTraj: y=0
+-0.03
+-0.02
+-0.01
+0.00
+0.01
+0.02
+0.03
+10-90%
+QTraj-γ
+QTraj-κ
+ALICE
+CMS
+0
+5
+10
+15
+-0.05
+0.00
+0.05
+0.10
+0.15
+pT [GeV]
+v2[Υ(1S)]
+τmed= 0.6 fm, NLO
+5.02 TeV Pb-Pb
+5-60% Centrality
+CMS: |y| < 2.4
+ALICE: 2.5 < y < 4
+QTraj: y=0
+Fig. 3. The anisotropic flow coefficient v2[1S] as a function of centrality (left) and
+transverse momentum (right) obtained with fluctuating initial conditions. We show
+the ˆγ variation in blue and the ˆκ variation in red.
+a function of centrality (left panel) and transverse momentum (right panel)
+compared with experimental data from the ALICE and CMS collaborations
+[6, 9]. From this figure we see that the NLO OQS+pNRQCD+IP-Glasma
+framework predicts a rather flat dependence on centrality, with the max-
+imum v2[1S] being on the order of 1%.
+In the right portion of the left
+panel, we present the results integrated over centrality as two points that
+include the observed variations with ˆκ and ˆγ, respectively.1 The size of the
+error bars reflects the statistical uncertainty associated with the double av-
+erage over initial conditions and physical trajectories [47]. The red and blue
+shaded regions correspond to the uncertainty associated with the variation
+of ˆκ and ˆγ, respectively. Finally, in the right panel of Fig. 3, I present the
+dependence of v2[1S] on transverse momentum.
+3. Conclusions
+In this proceedings contribution, I focused on recent research that uses
+an OQS framework applied within the pNRQCD effective field theory. I
+presented predictions for the nuclear suppression factor (RAA) and elliptic
+flow coefficient (v2) based on smooth and fluctuating hydrodynamical initial
+conditions. We found that the impact of fluctuating initial conditions was
+small when considering RAA, but a larger, though still within statistical un-
+certainties, effect was observed for v2. For RAA[1S], RAA[3S], and v2[1S],
+we found good agreement between the NLO OQS+pNRQCD framework
+and experimental data. However, we found that the amount of Υ(2S) sup-
+pression was slightly overestimated regardless of the hydrodynamic initial
+conditions used.
+1 The scale of the right portion of the left panel is different from the left portion of this
+panel in order to make it more readable.
+
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+

DNAzT4oBgHgl3EQfGfv5/content/tmp_files/load_file.txt

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+page_content='Bottomonium suppression in the quark-gluon plasma –  From effective field theories to non-unitary quantum  evolution∗  Michael Strickland  Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' Kent State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' Kent,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' OH 44242,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' USA  Received January 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' 2023  In this proceedings contribution I review recent work which computes  the suppression of bottomonium production in heavy-ion collisions using  open quantum systems methods applied within the potential non-relativistic  quantum chromodynamics (pNRQCD) effective field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' I discuss how  the computation of bottomonium suppression can be reduced to solving  a Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) quantum master equa-  tion for the evolution of the b¯b reduced density matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' The open quantum  systems approach used allows one to take into account the non-equilibrium  dynamics and decoherence of bottomonium in the quark-gluon plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' Fi-  nally, I present comparisons of phenomenological predictions obtained us-  ing a recently obtained next-to-leading-order GKSL equation with ALICE,  ATLAS, and CMS experimental data for bottomonium suppression and  elliptic flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' Introduction  Heavy-ion collisions have been used to produce and study the properties  of the quark-gluon plasma (QGP), a state of matter thought to have existed  in the early universe and being created terrestrially in relativistic heavy-ion  collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' The suppression of bottomonium production in such collisions  is considered strong evidence for the creation of a deconfined QGP [1–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='  In the past, it was proposed that this suppression was due to the Debye  screening of chromoelectric fields in the QGP, which modified the potential  between heavy quarks and resulted in a reduction of heavy-quarkonium  production [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='  However, more recent studies have shown that, in  addition to the real part of the potential being modified by Debye screening,  there is also an imaginary contribution to the potential caused by processes  such as Landau damping and singlet-to-octet transitions [13–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='  These  ∗ Presented at Excited QCD 2022, Giardini Naxos, Sicily, October 2022  (1)  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='01031v1  [hep-ph]  3 Jan 2023  2  strickland  printed on January 4, 2023  processes result in large in-medium widths for heavy-quarkonium bound  states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='  In the past decade, there has been significant progress in the use of  open quantum systems (OQS) methods to study heavy-quarkonium sup-  pression in the QGP [21–37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' In particular, recent works have applied OQS  methods within the framework of the potential non-relativistic QCD (pN-  RQCD) effective field theory [14, 16, 18–20, 38–40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' The pNRQCD EFT is  applicable to systems with a large separation between energy scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' This  naturally occurs when the velocity of the heavy quark relative to the center  of mass is small (v ≪ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' In Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' [24, 26, 30], the authors considered the  scale hierarchy relevant for small bound states in a high-temperature QGP,  1/r ∼ Mv ≫ mD ∼ πT ≫ E, where r is the typical size of the state, M is  the heavy quark mass, mD is the Debye mass, T is the temperature, and E  is the binding energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='  With this scale ordering the environment’s relaxation timescale is much  shorter than both the system’s internal timescales and the system’s own  relaxation timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='  This makes the quantum evolution Markovian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='  In  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' [30] a Markovian Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equa-  tion [41, 42] was derived for the heavy-quarkonium reduced density matrix,  which was implemented in the open-source QTraj code of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' [43] to make  predictions for heavy-ion collision bottomonium observables [44–47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' This  was done by coupling the GKSL solver to 3+1D viscous hydrodynamics  code using smooth Glauber initial conditions [48–50] and, most recently,  fluctuating hydrodynamical backgrounds [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' The formalism used in the  most recent works [46, 47] is accurate to next-to-leading order (NLO) in  the binding energy over temperature, which allows it to be used at lower  temperatures than the original leading-order formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' Results  For details concerning the theoretical and numerical methods employed,  I refer the reader to Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' [46, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' The results obtained depend on two co-  efficients ˆκ and ˆγ, which were extracted directly and indirectly from lattice  QCD calculations [30, 51–55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' 1 and 2, we present our NLO predic-  tions for RAA as a function of Npart and pT , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' For these results  we did not include the effect of dynamical quantum jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' In the left panel  we show the variation of ˆκ in the range ˆκ ∈ {ˆκL(T), ˆκC(T), ˆκU(T)} while  holding ˆγ = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' This value of ˆγ was chosen as to best reproduce the  RAA[Υ(1S)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' In the right panel we show the variation of ˆγ in the range  −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='5 ≤ ˆγ ≤ 0 with ˆκ(T) = ˆκC(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' The solid line corresponds to ˆγ = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='  As this figure demonstrates, our NLO predictions without quantum jumps  are in quite good agreement with the experimental data for RAA[1S] and  strickland  printed on January 4, 2023  3  QTraj - Υ(1S)  QTraj - Υ(2S)  QTraj - Υ(3S)  ALICE - Y(1S)  ATLAS - Y(1S)  CMS - Y(1S)  ALICE - Y(2S)  ATLAS - Y(2S)  CMS - Y(2S)  CMS - Y(3S)  0  100  200  300  400  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='0  Npart  RAA  κ\uf111 ∈ {κ\uf111L(T),κ\uf111C(T),κ\uf111U(T)}, γ\uf111 = -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='6, τmed = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='6 fm, NLO  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='02 TeV Pb-Pb  ALICE: pT < 15 GeV and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='5 < y < 4  ATLAS: pT < 15 GeV and |y| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='5  CMS: pT < 30 GeV and |y| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='4  QTraj: pT < 30 GeV and y=0  QTraj - Υ(1S)  QTraj - Υ(2S)  QTraj - Υ(3S)  ALICE - Y(1S)  ATLAS - Y(1S)  CMS - Y(1S)  ALICE - Y(2S)  ATLAS - Y(2S)  CMS - Y(2S)  CMS - Y(3S)  0  100  200  300  400  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='0  Npart  RAA  κ\uf111 = κ\uf111C(T), γ\uf111 ∈ {-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='5,-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='6,0}, τmed = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='6 fm, NLO  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='02 TeV Pb-Pb  ALICE: pT < 15 GeV and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='5 < y < 4  ATLAS: pT < 15 GeV and |y| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='5  CMS: pT < 30 GeV and |y| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='4  QTraj: pT < 30 GeV and y=0  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='  The nuclear suppression, RAA[Υ(1S, 2S, 3S)], as a function of the number  of participants, Npart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' The left panel shows variation of ˆκ and the right panel shows  variation of ˆγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' The experimental results shown are from the ALICE [7], ATLAS [8],  and CMS [5, 10] collaborations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='  QTraj - Υ(1S)  QTraj - Υ(2S)  QTraj - Υ(3S)  ALICE - Y(1S)  ATLAS - Y(1S)  CMS - Y(1S)  ATLAS - Y(2S)  CMS - Y(2S)  CMS - Y(3S)  0  5  10  15  20  25  30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='0  pT [GeV]  RAA  κ\uf111 ∈ {κ\uf111L(T),κ\uf111C(T),κ\uf111U(T)}, γ\uf111 = -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='6, τmed = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='6 fm, NLO  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='02 TeV Pb-Pb  CMS: 0-100% and |y| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='4  ALICE: 0-90% and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='5 < y < 4  ATLAS: 0-80% and |y| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='5  QTraj: 0-100% and y=0  QTraj - Υ(1S)  QTraj - Υ(2S)  QTraj - Υ(3S)  ALICE - Y(1S)  ATLAS - Y(1S)  CMS - Y(1S)  ATLAS - Y(2S)  CMS - Y(2S)  CMS - Y(3S)  0  5  10  15  20  25  30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='0  pT [GeV]  RAA  κ\uf111 = κ\uf111C(T), γ\uf111 ∈ {-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='5,-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='6,0}, τmed = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='6 fm, NLO  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='02 TeV Pb-Pb  CMS: 0-100% and |y| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='4  ALICE: 0-90% and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='5 < y < 4  ATLAS: 0-80% and |y| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='5  QTraj: 0-100% and y=0  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='  The nuclear suppression factor, RAA, for Υ(1S, 2S, 3S) as a function of  the transverse momentum, pT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' The bands, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' are the same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='  RAA[3S].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' However, for the 2S excited state, our NLO predictions without  quantum jumps are somewhat lower than the experimental results, partic-  ularly for the most central collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='  Recently, we computed the NLO bottomonium RAA and v2 using both  smooth and fluctuating initial conditions for the hydrodynamic evolution [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='  In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' [47] in was demonstrated that the results for RAA obtained using fluc-  tuating and smooth initial conditions were nearly identical, indicating that  initial state fluctuations do not play an important role in this observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' In  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' 3, I present the OQS+pNRQCD+IP-Glasma predictions for v2[1S] as  4  strickland  printed on January 4, 2023  CMS  QTraj-γ\uf111  QTraj-κ\uf111  10  30  50  90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='06  Centrality (%)  v2[Υ(1S)]  τmed = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='6 fm, NLO  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='02 TeV Pb-Pb  pT < 50 GeV  CMS: |y| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='4  QTraj: y=0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
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+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='03  10-90%  QTraj-γ\uf111  QTraj-κ\uf111  ALICE  CMS  0  5  10  15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='15  pT [GeV]  v2[Υ(1S)]  τmed= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='6 fm, NLO  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='02 TeV Pb-Pb  5-60% Centrality  CMS: |y| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='4  ALICE: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='5 < y < 4  QTraj: y=0  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' The anisotropic flow coefficient v2[1S] as a function of centrality (left) and  transverse momentum (right) obtained with fluctuating initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' We show  the ˆγ variation in blue and the ˆκ variation in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='  a function of centrality (left panel) and transverse momentum (right panel)  compared with experimental data from the ALICE and CMS collaborations  [6, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' From this figure we see that the NLO OQS+pNRQCD+IP-Glasma  framework predicts a rather flat dependence on centrality, with the max-  imum v2[1S] being on the order of 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='  In the right portion of the left  panel, we present the results integrated over centrality as two points that  include the observed variations with ˆκ and ˆγ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='1 The size of the  error bars reflects the statistical uncertainty associated with the double av-  erage over initial conditions and physical trajectories [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' The red and blue  shaded regions correspond to the uncertainty associated with the variation  of ˆκ and ˆγ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' Finally, in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' 3, I present the  dependence of v2[1S] on transverse momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' Conclusions  In this proceedings contribution, I focused on recent research that uses  an OQS framework applied within the pNRQCD effective field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' I  presented predictions for the nuclear suppression factor (RAA) and elliptic  flow coefficient (v2) based on smooth and fluctuating hydrodynamical initial  conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' We found that the impact of fluctuating initial conditions was  small when considering RAA, but a larger, though still within statistical un-  certainties, effect was observed for v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' For RAA[1S], RAA[3S], and v2[1S],  we found good agreement between the NLO OQS+pNRQCD framework  and experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content=' However, we found that the amount of Υ(2S) sup-  pression was slightly overestimated regardless of the hydrodynamic initial  conditions used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
+page_content='  1 The scale of the right portion of the left panel is different from the left portion of this  panel in order to make it more readable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfGfv5/content/2301.01031v1.pdf'}
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+arXiv:2301.00098v1  [math.GT]  31 Dec 2022
+KIAS-P22086
+THE DESCENDANTS OF THE 3D-INDEX
+ZHIHAO DUAN, STAVROS GAROUFALIDIS, AND JIE GU
+Abstract. In the study of 3d-3d correspondence occurs a natural q-Weyl algebra associated
+to an ideal triangulation of a 3-manifold with torus boundary components, and a module of
+it. We study the action of this module on the (rotated) 3d-index of Dimofte–Gaiotto–Gukov
+and we conjecture some structural properties: bilinear factorization in terms of holomorphic
+blocks, pair of linear q-difference equations, the determination of the 3d-index in terms of
+a finite size matrix of rational functions and the asymptotic expansion of the q-series as q
+tends to 1 to all orders. We illustrate our conjectures with computations for the case of the
+three simplest hyperbolic knots.
+Contents
+1.
+Introduction
+2
+1.1.
+The 3D-index and the state-integral
+2
+1.2.
+Descendants
+2
+1.3.
+Our conjectures
+3
+2.
+Algebras of 3-dimensional ideal triangulations
+4
+3.
+The rotated 3D-index and its descendants
+4
+3.1.
+Definition
+4
+3.2.
+Factorization and holomorphic blocks
+5
+3.3.
+Descendants
+6
+3.4.
+Asymptotics
+8
+4.
+Examples
+8
+4.1.
+The 41 knot and its rotated 3D-index
+8
+4.2.
+Factorization
+9
+4.3.
+Defects
+9
+4.4.
+The 52 knot and its rotated 3D-index
+11
+4.5.
+Factorization
+11
+4.6.
+Defects
+12
+4.7.
+The (−2, 3, 7)-pretzel knot
+12
+Acknowledgements
+14
+Appendix A.
+The holomorphic blocks of the 41 knot
+14
+Appendix B.
+The holomorphic blocks of the 52 knot
+14
+Appendix C.
+NZ matrices and the 3D-index
+15
+References
+16
+Date: 1 January 2023.
+Key words and phrases: linear q-difference equations, q-holonomic functions, knots, hyperbolic knots, 3-
+manifolds, Neumann–Zagier matrices, 3d-index, supersymmetric index, BPS counts, line operators, defects,
+descendants, holomorphic blocks, factorization.
+1
+
+2
+ZHIHAO DUAN, STAVROS GAROUFALIDIS, AND JIE GU
+1. Introduction
+1.1. The 3D-index and the state-integral. Topological invariants of ideally triangu-
+lated 3-manifolds appeared in mathematical physics in relation to complex Chern–Simons
+theory [2] and its extension in the 3d-3d correspondence [5, 11]. Two of the best-known such
+invariants are the state-integrals of Andersen–Kashaev [2], which are analytic functions on
+C \ (−∞, 0], and the 3D-index of Dimofte–Gaiotto–Gukov [8, 9], which is a collection of
+q-series with integer coefficients parametrized by the integer homology of the boundary of a
+3-manifold. Although the state-integrals and the 3D-index are different looking functions,
+they are closely related on the mathematics side through the theory of holomorphic quantum
+modular forms developed by Zagier and the second author [22, 21], and on the physics side
+through the above mentioned 3d-3d correspondence.
+The state-integrals and the 3D-index share many common features, stemming from the fact
+that on the physics side, under the 3d-3d correspondence [10, 26, 9, 7] (see [6] for a review)
+become the invariants of the dual 3d N = 2 superconformal field theory on respectively S3
+and S1 × S2, both of which can be obtained by gluing two copies of D2 × S1 together.
+On the mathematics side, both invariants are defined using combinatorial data of ideal
+triangulations of 3-manifolds whose local weights (namely the Faddeev quantum diloga-
+rithm function, and the tetrahedron index, respectively) satisfy the same linear q-difference
+equations, whereas the invariants themselves are given by an integration/summation over
+variables associated to each tetrahedron.
+A common feature to both invariants is their conjectured bilinear factorization in terms
+of the same holomorphic blocks H(q), the latter being q-hypergeometric series defined for
+|q| ̸= 1. This leads to bilinear expressions for the state-integral in terms of H(q) times H(˜q)
+(where q = e2πiτ and ˜q = e−2πi/τ) and bilinear expressions for the 3D-index in terms of H(q)
+times H(q−1). This factorization is well-known in the physics literature [3] and interpreted
+as partition function of the dual 3d superconformal field theory on D2 × S1. They are also
+partially known for some examples of 3-manifolds in [17, 21]. We emphasize, however, that
+the bilinear factorization of state-integrals and of the 3D-index is conjectural, and so is the
+existence of the suitably normalized holomorphic blocks.
+Another common feature to state-integrals and the 3D-index is that they are given by
+integrals/lattice sums where the integrand/summand has a common annihilating ideal. This
+implies that both state-integrals and the rotated 3D-index satisfy a pair of linear q-difference
+equations which are in fact conjectured to be identical, and equal to the homogeneous part
+of the linear q-difference equation for the colored Jones polynomial of a knot [19].
+The
+conjectured common linear q-difference equations for state-integrals and for the 3D-index
+would also be a consequence of their common holomorphic block factorization. In physics
+these linear q-difference equations are interpreted as Ward identities of Wilson-’t Hooft line
+operators in the dual 3d superconformal field theory [8, 9].
+1.2. Descendants. Descendants appeared recently as computable, exponentially small cor-
+rections to the asymptotics of the Kashaev invariant of a knot, refining the Volume Con-
+jecture to all orders in perturbation theory to a Quantum Modularity Conjecture [22]. One
+of the discoveries was that the Kashaev invariant of a knot is a distinguished (σ0, σ1)-entry
+
+THE DESCENDANTS OF THE 3D-INDEX
+3
+in a square matrix of knot invariants at roots of unity. The rows and columns of the ma-
+trix are parametrized with boundary-parabolic PSL2(C)-representations, with σ0 denoting
+the trivial representation and σ1 denoting the geometric representation of a hyperbolic knot
+complement. The above mentioned matrix has remarkable algebraic, analytic and arithmetic
+properties explained in detail in Section 5 of [22], and given explicitly for the 41 and 52 knots
+in Sections 7.1 and 7.2 of i.b.i.d. The rows of the matrix are supposed to be
+Q(q1/2)-linear
+combinations of fundamental solutions to a linear q-difference equation (homogeneous for all
+but the first row), thus the elements in each row are supposed to be descendants of each
+other. Although the existence of such a matrix is conjectured, its top row was defined in [16]
+for all knots in terms of the descendant Kashaev invariants of a knot.
+The above mentioned matrix has three known realizations, one as functions at roots of
+unity mentioned above, a second as a matrix of Borel summable asymptotic series and a
+third as a matrix of q1/2-series. The idea of descendants can be extended to the matrix of
+asymptotic series (whose first column are simply the vector of asymptotic series of the pertur-
+bative Chern–Simons theory at a PSL2(C)-flat connection, and the remaining columns being
+descendants of the first column) as well as to a matrix of q-series. This extension was done
+for the case of the 41 and 52 knots by Mari˜no and two of the authors [13, Eqn.(13),App.A],
+with the later addition of the trivial PSL2(C)-representation in [14, Sec.2.2,Sec.4.1].
+To summarize, descendants are supposed to be the
+Q(q1/2)-span of a fundamental solution
+to a linear q-difference equation associated to the quantum invariants. It is becoming clear
+that this span is a fundamental quantum invariant of 3-manifolds, and we want to present
+further evidence for this using as an example an important quantum invariant, namely the
+3D-index.
+1.3. Our conjectures. A detailed study of the 3D-index of a 3-manifold with torus bound-
+ary and its structural properties, namely holomorphic block factorization, linear q-difference
+equations, computations and asymptotics was recently done in [20].
+The goal of the present paper is to extend the properties of the 3D-index by allowing
+observables, line operators, defects, descendants, all being synonymous names for the same
+object. On the topological side, an observable is a knot L in a 3-manifold M, where in the
+case of interest to us, M = S3 \ K is the complement of a knot in S3. On the algebra
+side, the conjectural 3d-quantum trace map sends a knot L ⊂ S3 \ K to an element O
+of a module over a q-Weyl algebra associated to an ideal triangulation T of M. We will
+postpone the description of the 3d-quantum trace map to a subsequent publication. Now O
+acts on the integrand/summand of the state-integral/3D-index, and by integrating/summing
+one obtains a state-integral/3D-index with insertion O. On the physics side, O becomes a
+line-operator supported on a line γ in the dual 3d N = 2 superconformal field theory T2[M]
+under the 3d-3d correspondence [10, 26, 9, 7]. The 3d-3d correspondence can be understood
+as a consequence of compactifying 6d N = 2 A1 superconformal field theory on the three
+manifold M and on
+R3 with topological twist along M. The 6d theory has surface operators
+which can be supported on L × γ, giving rise to the correspondence between the defect L
+in M and the line-operator on γ ⊂
+R3 in T2[M] [8, 9]. Our goal is to study the structural
+properties of the rotated, inserted, 3D-index Irot
+T ,O(q). Although this is a
+Z ×
+Z matrix, we
+will see that it is determined from the uninserted rotated 3D-index Irot
+T (q) in terms of a pair
+
+4
+ZHIHAO DUAN, STAVROS GAROUFALIDIS, AND JIE GU
+of linear q-difference equations and a finite size invertible matrix with coefficients in the field
+Q(q1/2); see Conjectures 3.3 and 3.6 below, illustrated by examples in Section 4.
+We emphasize that our paper concerns conjectural structural properties of topological
+invariants, such as the rotated inserted 3D-index, and not mathematical proofs. Nevertheless
+the structure of these invariants is rich, and leads to startling predictions and numerical
+conformations (see eg. Equation (36) below).
+2. Algebras of 3-dimensional ideal triangulations
+We recall here a q-Weyl algebra associated to an ideal triangulation T which was first
+considered by Dimofte on the context of the 3d-3d correspondence, and it was introduced as
+an attempt to quantize the SL2(C)-character variety of an ideally triangulated 3-manifold M
+using the symplectic structure of the Neumann–Zagier matrices, and following the ideas of
+Hamiltonian reduction of symplectic phase-spaces [5]. Similar ideas appeared in subsequent
+work [11].
+We fix an ideal triangulation T of M with N ideal tetrahedra. This defines a q-Weyl
+algebra
+Wq(T ) =
+Q(q)⟨�zj, �z′
+j | j = 1, . . . , N⟩ of Laurent variables �zj, �z′
+j that commute except
+in the following instance �zj �z′
+j = q�z′
+j �zj for j = 1, . . . , N. A more symmetric way is to introduce
+three invertible variables �z, �z′, �z
+′′ which satisfy the relations
+�z�z′ = q�z′�z,
+�z′�z′′ = q�z′′ �z′,
+�z′′ �z = q�z�z′′,
+�z�z′�z′′ = −q
+(1)
+(hence �z�z′�z′′ is in the center and it is invariant under cyclic permutations), and then
+Wq(T )
+is simply the tensor product of one algebra per tetrahedron. The combinatorics of the edge-
+gluing equations of M have symplectic properties discovered by Neumann–Zagier [24, 23].
+Using those properties, Dimofte [5] and later Gang et al [11] (see also [1, Eqn.(10)]) consider
+the quotient
+M(T ) =
+Wq(T )/(Wq(T )(Lagrangians) + (edge equations)Wq(T ))
+(2)
+of
+Wq(T ) by the left
+Wq(T )-ideal generated by the Lagrangian equations
+�z′−1 + �z − 1 = 0,
+(�z
+′′)−1 + �z′ − 1 = 0,
+�z−1 + �z
+′′ − 1 = 0
+(3)
+(one per each tetrahedron) plus the right ideal generated by the edge equations. This strange
+quotient M(T ), which is no longer a module over a q-Weyl algebra, but only a
+Q(q1/2)-vector
+space is a natural object that indeed annihilates the rotated 3D-index as we will see shortly.
+3. The rotated 3D-index and its descendants
+3.1. Definition. For simplicity, in the paper we will focus on the action of the quantum
+torus
+Wq(T ) on the 3D-index IT , and in fact in its rotated form Irot
+T
+explained to us by Tudor
+Dimofte and studied extensively in [20]. To begin with, we fix an ideal triangulation T with
+N tetrahedra of a 3-manifold M whose torus boundary is marked by a pair of a meridian and
+longitude. The building block of the 3D-index is the tetrahedron index I∆(m, e)(q) ∈
+Z[[q1/2]]
+defined by
+I∆(m, e)(q) =
+∞
+�
+n=(−e)+
+(−1)nq
+1
+2n(n+1)−(n+ 1
+2 e)m
+(q; q)n(q; q)n+e
+(4)
+
+THE DESCENDANTS OF THE 3D-INDEX
+5
+where e+ = max{0, e} and (q; q)n =
+�n
+i=1(1 − qi). If we wish, we can sum in the above
+equation over the integers, with the understanding that 1/(q; q)n = 0 for n < 0.
+The rotated 3D-index is given by
+Irot
+T (n, n′)(q) =
+�
+k∈ZN
+ST (k, n, n′)(q)
+(5)
+where
+ST (k, n, n′)(q) = (−q1/2)ν·k−(n−n′)νλqkN(n+n′)/2
+N
+�
+j=1
+I∆(λ′′
+j (n−n′)−bj ·k, −λj(n−n′)+aj ·k)(q) (6)
+is assembled out of a product of tetrahedra indicies I∆ evaluated to linear forms that depend
+on the Neumann–Zagier matrices (A|B) of T . The detailed definition of the Neumann–Zagier
+matrices is given in Appendix C.
+Note that the degree δ(I∆(m, e)) of the tetrahedron index is a nonnegative piecewise
+quadratic function of (m, e)
+δ(I∆(m, e)) = 1
+2 (m+(m + e)+ + (−m)+e+ + (−e)+(−e − m)+ + max{0, m, −e}) .
+(7)
+It follows that for 1-efficient triangulations (see [15]) the degree of the summand in (5) is
+bounded below by a positive constant times max{|k1|, |k2|, . . ., |kN|}, thus the sum in (5) is
+a well-defined element of
+Z((q1/2)).
+The topological invariance of the 3D-index is a bit subtle, since the definition requires
+1-efficient ideal triangulations, and the latter are not known to be connected under 2–3
+Pachner moves.
+Nonetheless, in [15], it was shown that the 3D-index (and likewise, its
+rotated version) is a topological invariant of cusped hyperbolic 3-manifolds. An alternative
+proof of this fact was given in [18], where the rotated 3D-index was reformulated in terms
+of a meromorphic function of two variables.
+3.2. Factorization and holomorphic blocks. From its very definition as a sum of proper
+q-hypergeometric series, it follows that Irot
+T (n, n′)(q) is a q-holonomic function of n and
+n′ [27, 25]. But more is true. The rotated 3D-index factorizes into a sum of a product of
+pairs of colored holomorphic blocks. This holomorphic block factorization is a well-known
+phenomenon explained in [3], and most recently in [20] whose presentation we will follow.
+Let us recall how this works. We can assemble the collection Irot
+T (n, n′)(q) of q-series indexed
+by pairs of integers into a
+Z ×
+Z matrix Irot
+T (q) whose (n, n′) entry is Irot
+T (n, n′)(q). Then,
+in [20] we explained the origin of the following conjecture for the rotated 3D-index.
+Conjecture 3.1. For every 1-efficient triangulation T there exists a palindromic linear q-
+difference operator �AT of order r with a fundamental solution
+Z × r matrix HT (q) and a
+symmetric, invertible r × r matrix BT with rational entries such that
+Irot
+T (q) = HT (q)BT HT (q−1)t .
+(8)
+When the triangulation is fixed and clear, we will drop it from the notation. If we denote
+the (n, α) entry of HT (q) whose (n, α) entry by h(α)
+n (q), these functions are the so-called
+
+6
+ZHIHAO DUAN, STAVROS GAROUFALIDIS, AND JIE GU
+colored holomorphic blocks.
+It follows that the matrix H(q) is a (properly normalized)
+fundamental solution to a pair of q-difference equations
+�AT (M+, L+)H(q) = 0,
+�AT (M−, L−)H(q−1) = 0,
+(9)
+where the operators act respectively by
+M+h(α)
+n (q) = qnh(α)
+n (q),
+L+h(α)
+n (q) = h(α)
+n+1(q)
+M−h(α)
+n (q−1) = q−nh(α)
+n (q−1),
+L−h(α)
+n (q−1) = h(α)
+n+1(q−1).
+(10)
+Consequently the rotated 3D-index satisfies a pair of (left and right) linear q-difference
+equations
+�AT (M+, L+)Irot
+T
+= �AT (M−, L−)Irot
+T
+= 0
+(11)
+acting in a decoupled way on each of the rows and columns of Irot
+T .
+The factorization (8) of the rotated 3D-index and the left and right linear q-difference
+equations (11) imply the following.
+Corollary 3.2. (of Conjecture 3.1) The rotated 3D-index Irot
+T (q) is uniquely determined by
+(1) the r × r matrix Irot
+T (q)[r] and
+(2) the pair of linear q-difference equations (11).
+Here, Irot
+T (q)[r] denotes the r × r matrix (Irot
+T (n, n′)(q)) for 0 ≤ n, n′ ≤ r − 1.
+The holomorphic blocks satisfy the symmetry
+h(α)
+T ,n(q) = h(α)
+T ,−n(q)
+(12)
+for all α and all integers n, which together with Equation (8) implies the symmetries
+Irot
+T (n, n′)(q) = Irot
+T (n, −n′)(q) = Irot
+T (−n, n′)(q) = Irot
+T (−n, −n′)(q) ,
+(13)
+and
+Irot
+T (n, n′)(q−1) = Irot
+T (n′, n)(q) ,
+(14)
+for the rotated 3D-index.
+Let us finally mention that the colored holomorphic blocks can be computed by the limit
+as x → 1
+Irot
+T (n, n′)(q) = lim
+x→1
+�
+α
+B(α)
+T (q−n′x−1; q−1)B(α)
+T (qnx; q) .
+(15)
+of the x-deformed holomorphic blocks B(α)
+T (x; q) and the latter can be determined from a
+factorization of an appropriate state-integral.
+3.3. Descendants. There is an important
+Q(q)-linear action of
+Wq(T ) on the set of func-
+tions ST (k, n, n′)(q) giving rise to a map
+Wq(T ) →
+Z((q1/2))
+ZN×Z2
+(16)
+which descends to a push-forward
+Q(q1/2)-linear map
+M(T ) →
+Z((q1/2))
+Z2,
+O �→ Irot
+T ,O .
+(17)
+
+THE DESCENDANTS OF THE 3D-INDEX
+7
+Concretely, when O =
+�N
+j=1 �z
+αj
+j (�z
+′′
+j )βj, we have
+Irot
+T ,O(n, n′)(q) =
+�
+k∈ZN
+(O ◦ ST )(k, n, n′)(q) ,
+(18)
+where
+(O ◦ ST )(k, n, n′)(q) = (−q1/2)ν·k−(n−n′)νλqkN(n+n′)/2+LO(n,n′,k)
+×
+N
+�
+j=1
+I∆(λ′′
+j (n − n′) − bj · k + βj, −λj(n − n′) + aj · k − αj)(q) ,
+(19)
+LO(n, n′, k) = 1
+2
+N
+�
+j=1
+�αj(λ′′
+j n − λ′′
+jn′ − bj · k) + βj(−λjn + λjn′ + aj · k) − αjβj
+�
+(20)
+This action was written down explicitly in [1, Eqn.(104)]. The symmetries of the tetrahe-
+dron index [8, Eqns.(136)] imply that the three Lagrangian operators given in Equation (3)
+annihilate ST (k, n, n′)(q), and thus the sum Irot
+T (n, n′)(q). In addition, the insertion Ei cor-
+responding to the i-th edge (for i = 1, . . ., N − 1) when quantized as in [5] satisfies
+(Ei ◦ ST )(k, n, n′) = qST (k − ei, n, n′)
+(21)
+Summing over k, this implies that Ei − q annihilates Irot
+T (n, n′)(q). Thus, Irot
+T ,O(n, n′)(q) is
+well-defined for all O ∈ M(T ), justifying the strange quotient given in Equation (2). Note
+that the action of the edge operators considered in [5] differs by factor of q from that of [1,
+Eqn.(130)].
+Our conjecture relates the colored holomorphic blocks and the rotated 3D-index of T
+to those of (T , O). Simply put, it asserts that inserting O simply changes the invariants
+(Z((q1/2))-series) by multiplication of a matrix of rational functions, and changes the left
+q-difference equation whereas it preserves the right one. This implies that the
+Q(q1/2)-span
+of the collection {Irot
+T ,O(q) | O ∈ M(T )} is a finite dimensional
+Q(q1/2)-vector space.
+Fix a 1-efficient ideal triangulation T of a 1-cusped 3-manifold M.
+Conjecture 3.3. For every O ∈ M(T )
+(a) there exists a linear q-difference operator �AT ,O with a fundamental solution matrix
+HT ,O(q) such that
+Irot
+T ,O(q) = HT ,O(q)BT HT (q−1)t ,
+(22)
+(b) there exists QT ,O(q) ∈ GLr(Q(q1/2)) such that
+Irot
+T ,O[r] = QT ,OIrot[r],
+HT ,O[r] = QT ,OH[r] .
+(23)
+The above conjecture implies the following.
+Corollary 3.4. (of Conjecture 3.3) The rotated 3D-index Irot
+T ,O(q) is uniquely determined by
+(1) the r × r matrices Irot
+T (q)[r] and QT ,O(q)
+(2) the pair of linear q-difference equations �AT ,O and �AT .
+
+8
+ZHIHAO DUAN, STAVROS GAROUFALIDIS, AND JIE GU
+Another corollary of the above conjecture concerns the descendants of the rotated 3D-
+index, analogous to the descendants of the colored Jones polynomial of a knot defined in [16]
+and the descendants of the holomorphic blocks defined in [13, Eqn.(13), App.A]. To phrase
+it, let
+DIrot
+T
+= Span
+Q(q1/2){Irot
+T (n, n′)(q) | n, n′ ∈
+Z}
+(24)
+denote the
+Q(q1/2)-span of the elements Irot
+T (n, n′) of the ring
+Q((q1/2)). Note that DIrot
+T
+is
+a finite dimensional vector space over the field
+Q(q1/2). Likewise, one defines Irot
+T ,O. The next
+corollary justifies the title of the paper.
+Corollary 3.5. (of Conjecture 3.3) We have:
+∪O∈M(T ) DIrot
+T ,O = DIrot
+T .
+(25)
+In other words, the descendants DIrot
+T ,O of the rotated 3D-index DIrot
+T
+are expressed effec-
+tively by a finite-size matrix with entries in
+Q(q1/2).
+We now formulate a relative version of the AJ-Conjecture. Let �A(M, L)|q=1 = A(M, L)
+denote the classical limit of a linear q-difference equation. The AJ-Conjecture [12] relates
+the classical limit of the �A-polynomial with the A-polynomial of a knot given in [4].
+Conjecture 3.6. For every O ∈ M(T ), we have
+AT ,O(M, L) =M AT (M, L)
+(26)
+where =M means equality up to multiplication by a nonzero function of M.
+3.4. Asymptotics. A consequence of Conjecture (3.3) (and Equation (22)) is that the all-
+order asymptotics of the colored holomorphic blocks h(α)
+T ,O,n(q) and the Irot
+T ,O(n, n′)(q) are a
+Q(q)-linear combination of those of h(α)
+T ,n(q) and Irot
+T (n, n′)(q), respectively. The asymptotics
+of the latter were studied in detail in [20].
+A corollary of this and Conjecture 3.6 is a
+resolution and an explanation from first principles, of the quantum length conjecture of [1].
+4. Examples
+In this section we illustrate our conjectures with the case of the three simplest hyperbolic
+knots, the 41 (figure eight) knot, the 52 knot and the (−2, 3, 7) pretzel knot.
+4.1. The 41 knot and its rotated 3D-index. The complement of the 41 knot has an ideal
+triangulation with two tetrahedra. Using the gluing equation matrices
+G =
+
+
+
+
+
+2
+2
+0
+0
+1
+0
+1
+1
+
+
+
+
+ ,
+G′ =
+
+
+
+
+
+1
+1
+1
+1
+0
+0
+1
+−1
+
+
+
+
+ ,
+G′′ =
+
+
+
+
+
+0
+0
+2
+2
+0
+−1
+1
+−3
+
+
+
+
+ ,
+(27)
+with the conventions explained in Appendix C, we obtain the matrices
+A =
+�
+1
+1
+1
+0
+�
+, B =
+�
+−1
+−1
+0
+−1
+�
+, ν =
+�
+0
+0
+�
+(28)
+
+THE DESCENDANTS OF THE 3D-INDEX
+9
+in terms of which, the rotated 3D-index is given by
+Irot
+41 (n, n′)(q) =
+�
+k1,k2∈Z
+qk2(n+n′)/2I∆(k1, k1 + k2)(q)I∆(k1 + k2 − n + n′, k1 − n + n′)(q) (29)
+where I∆ is the tetrahedron index given in (4). (The above formula agrees with [1, Eqn.(108)]
+after a shift k1 �→ k1 − k2). Using Equation (7), it follows that the degree of the summand
+in (29) is bounded below by a positive constant times max{|k1|, |k2|}, thus the sum in (29)
+is a well-defined element of
+Z((q1/2)).
+4.2. Factorization. In this section we briefly summarize the properties of the rotated 3D-
+index of the 41 knot following [20], namely its factorization in terms of colored holomorphic
+blocks, the linear q-difference equation, their symmetries and their asymptotics. All the
+functions in this section involve the knot 41, which we suppress from the notation.
+The rotated 3D-index is given by [20, Prop.9]
+Irot
+41 (n, n′)(q) = −1
+2h(1)
+41,n′(q−1)h(0)
+41,n(q) + 1
+2h(0)
+41,n′(q−1)h(1)
+41,n(q)
+(n, n′ ∈
+Z)
+(30)
+with the colored holomorphic blocks h(0)
+41,n(q) and h(1)
+41n(q) given in the Appendix A.
+The colored holomorphic blocks satisfy the symmetries
+h(0)
+41,n(q−1) = h(0)
+41,n(q),
+h(1)
+41,n(q−1) = −h(1)
+41,n(q) ,
+(31)
+and
+h(α)
+41,−n(q) = h(α)
+41,n(q),
+α = 0, 1 ,
+(32)
+and the linear q-difference equation [20, Eqn.(63)]
+P41,0(qn, q)h(α)
+n (q) + P41,1(qn, q)h(α)
+n+1(q) + P41,2(qn, q)h(α)
+n+2(q) = 0
+(α = 0, 1, n ∈
+Z) (33)
+where
+P41,0(x, q) = q2x2(q3x2 − 1) ,
+P41,1(x, q) = −q1/2(1 − q2x2)(1 − qx − qx2 − q3x2 − q3x3 + q4x4) ,
+P41,2(x, q) = q3x2(−1 + qx2) .
+(34)
+We denote the corresponding operator of the q-difference equation (33) by
+�A41(x, σ, q) =
+�2
+j=0 P41,j(x, q)σj.
+4.3. Defects. We now consider two defects. The first one is the element
+O = −�y−1 − �z−1 + �y−1�z−1 ∈ M(T )
+(35)
+from [1, Eqn.(81)]. Computing the values of Irot
+41 (n, n′)(q) and Irot
+41,O(n, n′)(q) for 0 ≤ n, n′ ≤ 1
+up to O(q121), we find out that the 2 × 2 matrices
+Irot
+41 (q)[2] =
+�
+1 − 8q − 9q2 + 18q3 + 46q4 + 90q5 + 62q6 + 10q7 + . . .
+−q−1/2 + q1/2 − q3/2 + 6q5/2 + 20q7/2 + 29q9/2 + 25q11/2 + . . .
+−q−1/2 + q1/2 − q3/2 + 6q5/2 + 20q7/2 + 29q9/2 + . . .
+2q + 2q2 + 7q3 + 8q4 + 3q5 − 22q6 − 67q7 + . . .
+�
+
+10
+ZHIHAO DUAN, STAVROS GAROUFALIDIS, AND JIE GU
+and
+Irot
+41,O(q)[2] =
+�
+−3 + 15q + 24q2 − 15q3 − 69q4 − 174q5 − 183q6 − 165q7 + . . .
+2q−1/2 − q1/2 + 4q3/2 − 7q5/2 − 34q7/2 − 64q9/2 + . . .
+q−3/2 − q−1/2 − q1/2 + q3/2 − 5q5/2 − 26q7/2 − 48q9/2 + . . .
+−1 − 2q − 4q2 − 9q3 − 17q4 − 13q5 + 10q6 + 77q7 + . . .
+�
+satisfy
+(q − 1)Irot
+41,O(q)[2](Irot(q)41[2])−1 =
+�
+2 − q
+−q1/2
+q1/2
+−q − 1 + q−1
+�
++ O(q121)
+(36)
+illustrating the dramatic collapse of the q-series into short rational functions of q1/2. This
+implies that the matrix Q41,O(q) is given by
+Q41,O(q) =
+1
+q − 1
+�
+2 − q
+−q1/2
+q1/2
+−q − 1 + q−1
+�
+(37)
+with det(Q41,O)(q) = 1 + 2q−1.
+After computing the values of Irot
+41,O(n, 0)(q)+O(q120) for n = 0, . . ., 10 and finding a short
+linear recursion among three consecutive values, and further interpolating for all n, we found
+out that the left
+�A-polynomial of Irot
+41,O(q) is given by
+�A41,O(x, σ, q) = �2
+j=0 P41,O,j(x, q)σj
+where
+P41,O,0(x, q) = q3/2x2(−1 + q3x2)(1 + qx + q3x2) ,
+P41,O,1(x, q) = (−1 + qx)(1 + qx)
+(1 + x − qx − qx2 − q3x2 − qx3 − 2q3x3 − q5x3 − q3x4 − q5x4 + q4x5 − q5x5 + q6x6) ,
+P41,O,2(x, q) = q7/2x2(−1 + qx2)(1 + x + qx2) .
+(38)
+The �A41,O polynomial is palindromic, and together with the skew-symmetry of the Q41,O(q)
+matrix, it follows that the colored holomorphic blocks h(0)
+41,O,n(q) and h(1)
+41,O,n(q) satisfy the
+symmetries (31) and (32).
+When we set q = 1, we obtain
+�A41,O(x, σ, 1) = 2(x2 − 1)(x2 + x + 1) �A41(x, σ, 1)
+(39)
+confirming Conjecture 3.6.
+Equation (37) and the recursion (38) imply that for all integers n and n′, Irot
+41,O(n, n′)(q) is
+a
+Q(q1/2)-linear combination of the three q-series Irot
+41 (0, 0)(q), Irot
+41 (0, 1)(q) and Irot
+41 (1, 0)(q).
+For instance, Equation (23) implies that
+Irot
+41,O(0, 0)(q) =
+1
+q−1((2 − q)Irot
+41 (0, 0)(q) − q
+1
+2Irot
+41 (0, 1)(q))
+(40)
+and likewise for other values of Irot
+41,O(n, n′)(q). This reduces the problem of the asymptotic
+expansion of Irot
+41,O(n, n′)(q) for q = e2πiτ to all orders in τ as τ tends to zero in a ray
+(nearly vertically, horizontally, or otherwise) to the problem of the asymptotics of colored
+holomorphic blocks and of the rotated 3D-index. This problem was studied in detail and
+solved in the work of Wheeler and the second author [20, Sec.5.7,5.8] for the 41 knot.
+As a second example, consider the element
+O2 = �y−1 ∈ M(T ) .
+(41)
+
+THE DESCENDANTS OF THE 3D-INDEX
+11
+Repeating the above computations, we find out that the matrix Q41,O2(q) is given by
+Q41,O2(q) =
+1
+q − 1
+�
+−1
+q1/2
+−q1/2
+−q2 + 2q + 1 − q−1
+�
+(42)
+with det(Q41,O2)(q) = 1 + q−1, and that the left
+�A-polynomial of Irot
+41,O2(q) is given by
+�A41,O2(x, σ, q) = �2
+j=0 P41,O2,j(x, q)σj where
+P41,O2,0(x, q) = q3/2x2(−1 + q2x)(1 + q2x) ,
+P41,O2,1(x, q) = (−1 + q3x2)(1 − qx − q2x2 − q4x2 − q4x3 + q6x4) ,
+P41,O2,2(x, q) = q7/2x2(−1 + qx)(1 + qx) .
+(43)
+In this case, we lose the Weyl-invariance symmetry of the colored holomorphic blocks, but
+we retain the AJ Conjecture 3.6 since
+�A41,O2(x, σ, 1) = (x2 − 1) �A41(x, σ, 1) .
+(44)
+4.4. The 52 knot and its rotated 3D-index. The complement of the 52 knot has an ideal
+triangulation with three tetrahedra. Using the gluing equation matrices
+G =
+
+
+
+
+
+
+
+1
+1
+1
+0
+0
+0
+1
+1
+1
+−1
+0
+0
+3
+2
+1
+
+
+
+
+
+
+
+,
+G′ =
+
+
+
+
+
+
+
+0
+2
+0
+1
+0
+1
+1
+0
+1
+0
+0
+0
+1
+2
+1
+
+
+
+
+
+
+
+,
+G′′ =
+
+
+
+
+
+
+
+1
+0
+1
+1
+2
+1
+0
+0
+0
+0
+1
+0
+−1
+0
+3
+
+
+
+
+
+
+
+,
+(45)
+with the conventions explained in Appendix C, we obtain the matrices
+A =
+
+
+
+1
+−1
+1
+−1
+0
+−1
+−1
+0
+0
+
+
+ ,
+B =
+
+
+
+1
+−2
+1
+0
+2
+0
+0
+1
+0
+
+
+ ,
+ν =
+
+
+
+0
+0
+0
+
+
+ .
+(46)
+The rotated 3D-index is given by
+Irot
+52 (n, n′)(q) =
+�
+k1,k2,k3∈Z
+qk3(n+n′)/2I∆(k1 − k2, k3 + k2 + n − n′)
+× I∆(−k1 + 2k2 − n + n′, k3 + 2k1 − 2k2 + n − n′)I∆(k3 + k1 − k2 + n − n′, k2 − 2n + 2n′) .
+(47)
+Equation (7) implies that the degree of the summand in (47) is bounded below by a positive
+constant times max{|k1|, |k2|, |k3|}, thus the sum in (47) is a well-defined element of
+Z((q1/2)).
+4.5. Factorization. The 52 knot has three colored holomorphic blocks h(α)
+n (q) for α = 0, 1, 2,
+n an integer and q a complex number |q| ̸= 1, whose definition in terms of q-hypergeometric
+series was given in [20, App.A] and reproduced for the convenience of the reader in Appendix
+B. The rotated 3D-index is given by [20, Prop.13]
+Irot
+52 (n, n′)(q) = −1
+2h(0)
+52,n′(q−1)h(2)
+52,n(q) − h(1)
+52,n′(q−1)h(1)
+52,n(q) − 1
+2h(2)
+52,n′(q−1)h(0)
+52,n(q) .
+(48)
+The colored holomorphic blocks satisfy the symmetries
+h(α)
+52,−n(q) = h(α)
+52,n(q),
+α = 0, 1, 2 .
+(49)
+
+12
+ZHIHAO DUAN, STAVROS GAROUFALIDIS, AND JIE GU
+and the linear q-difference equation [20, Eqn.(63)]
+P52,0(qn, q)h(α)
+n (q) + P52,1(qn, q)h(α)
+n−1(q) + P52,2(qn, q)h(α)
+n−2(q) + P52,3(qn, q)h(α)
+n−3(q) = 0 ,
+(50)
+for all α = 0, 1, 2 and all integers n, where [20, Eqn.(126)]
+P52,0(x, q) = −q−2x2(1 − q−2x)(1 + q−2x)(1 − q−5x2) ,
+P52,1(x, q) = q3/2x−3(1 − q−1x)(1 + q−1x)(1 − q−5x2)
+· (1 − q−1x − q−1x2 − q−4x2 + q−2x2 + q−3x2 + q−2x3 + q−5x3 + q−5x4 + q−5x4 − q−6x5) ,
+P52,2(x, q) = q5x−5(1 − q−2x)(1 + q−2x)(1 − q−1x2)
+· (1 − q−2x − q−2x − q−2x2 − q−5x2 + q−4x3 + q−7x3 − q−5x3 − q−6x3 + q−7x4 − q−9x5) ,
+P52,3(x, q) = q
+11
+2 x−5(1 − q−1x)(1 + q−1x)(1 − q−1x2) .
+(51)
+4.6. Defects. We now consider two defects O1 and O2 given by
+O1 = �z1
+O2 = �z1 + �z3 .
+(52)
+Computing the 3×3 matrix of the rotated 3D-index with and without insertion up to O(q81),
+and dividing one matrix by another, we found out that the corresponding 3 × 3 matrices
+QOj(q) + O(q81) for j = 1, 2 are given by
+Irot
+52,O1(q)[3](Irot
+52 (q)[3])−1 =
+1
+(1 − q2)(1 − q3)·
+�
+−q2 − q3 − q4
+q1/2 − q3/2 + q7/2 + 2q9/2 + 2q11/2 − q13/2
+−q7
+−q3/2 − q5/2 − q7/2
+1 − q + q3 + 2q4 + 2q5 − q6
+−q13/2
+−1 − q−2 − q−1
+−q−5/2 + 2q−3/2 + 2q−1/2 + q1/2 − q5/2 + q7/2
+−q3
+�
++ O(q81)
+(53)
+and
+Irot
+52,O2(q)[3](Irot
+52 (q)[3])−1 =
+1
+(1 − q2)(1 − q3)·
+�
+−q − 2q2 − q3 + q5
+q1/2 + q5/2 + q7/2 + q9/2 + q11/2 − q13/2
+−q7
+−q−1/2 − q3/2 − q7/2
+4 − q−1 − q − q2 − q3 + q4 + 5q5 − 2q6
+q11/2 − 2q13/2
+−2 + q−4 + q−3 − q−2 − 2q−1
+q−9/2 − 2q−7/2 − 4q−5/2 + 2q−3/2 + 4q−1/2 + 4q1/2 − q3/2 − 2q5/2 + 2q7/2
+1 + q − q2 − 2q3
+�
++O(q81)
+(54)
+illustrating Corollary 3.5 of Conjecture 3.3.
+4.7. The (−2, 3, 7)-pretzel knot. As a final experiment, we studied the rotated 3D-index
+of the (−2, 3, 7) pretzel-knot. This knot is interesting in several ways, and exhibits behavior
+of general hyperbolic knots. The complement of the (−2, 3, 7)-pretzel knot is geometrically
+similar to that of the 52 knot, i.e., both are obtained by the gluing of three three ideal
+tetrahedra, only put together in a combinatorially different way. Thus, the 52 and (−2, 3, 7)
+pretzel knots have the same cubic trace field, and the same real volume. But the similarities
+end there. The 52 knots has three boundary parabolic PSL2(C)-representations, all Galois
+conjugate to the geometric one.
+On the other hand, one knows from [22] and [21] that
+the (−2, 3, 7)-pretzel knot has 6 colored holomorphic blocks, corresponding to the fact that
+the (−2, 3, 7)-pretzel knot has 6 boundary parabolic representations, three coming from the
+Galois orbit of the geometric PSL2(C)-representation (defined over the cubic trace field of
+discriminant −23) and three more coming from the Galois orbit of a PSL2(C)-representation
+
+THE DESCENDANTS OF THE 3D-INDEX
+13
+defined over the totally real abelian field
+Q(cos(2π/7)). Although [21] gives explicit expres-
+sions for the 6 × 6 matrices of the holomorphic blocks (inside and outside the unit disk), the
+colored holomorphic blocks have not been computed, partly due to the complexity of the
+calculation.
+Going back to the 3D-index of the (−2, 3, 7) knot, the gluing equation matrices are
+G =
+
+
+
+
+
+
+
+1
+1
+1
+1
+0
+0
+0
+1
+1
+0
+0
+−1
+−1
+1
+−18
+
+
+
+
+
+
+
+,
+G′ =
+
+
+
+
+
+
+
+1
+0
+0
+0
+2
+2
+1
+0
+0
+0
+0
+0
+1
+−1
+−2
+
+
+
+
+
+
+
+,
+G′′ =
+
+
+
+
+
+
+
+0
+1
+0
+2
+1
+0
+0
+0
+2
+2
+0
+0
+35
+1
+0
+
+
+
+
+
+
+
+,
+(55)
+with the conventions explained in Appendix C, from which we obtain that
+A =
+
+
+
+0
+1
+1
+1
+−2
+−2
+0
+0
+−1
+
+
+ ,
+B =
+
+
+
+−1
+1
+0
+2
+−1
+−2
+2
+0
+0
+
+
+ ,
+ν =
+
+
+
+1
+−2
+0
+
+
+ .
+(56)
+The rotated 3D-index is given by
+Irot
+(−2,3,7)(n, n′)(q) =
+�
+k1,k2,k3∈ Z
+(−q1/2)k1−2k2−n+n′qk3(n+n′)/2I∆(k1 − 2k2 − 2k3 + 17n − 17n′, k2 + n − n′)
+× I∆(−k1 + k2 + n − n′, k1 − 2k2 − n + n′)I∆(2k2 + n − n′, k1 − 2k2 − k3 + 8n − 8n′) .
+(57)
+So, in our final experiment we computed the rotated 3D-index of the (−2, 3, 7) pretzel-
+knot, and more precisely the 6×6 matrix Irot
+(−2,3,7)(q)[6]. To give an idea of what this involves,
+the leading term of the above matrix is
+Irot
+(−2,3,7)(q)[6] =
+
+
+
+
+
+
+1
+−q−9/2
+q−19
+−q−87/2
+q−78
+−q−245/2
+−q9/2
+6q2
+−q−27/2
+q−38
+−q−145/2
+q−117
+q17
+−q27/2
+q
+−q−45/2
+q57
+−q−203/2
+−q75/2
+q34
+−q45/2
+q4
+−q−63/2
+q−76
+q66
+−q125/2
+q51
+−q63/2
+q2
+−q−81/2
+−q205/2
+q99
+−q175/2
+q68
+−q81/2
+q6
+
+
+
+
+
+
+(58)
+and this alone required an internal truncation of the summand of (57) up to O(q103). For
+safety, we computed up to O(q160) and we found out that the last computed coefficients of
+Irot
+(−2,3,7)(q)[6] were given by
+�
+3099301802486871q158
+15368338814987064q315/2
+39577501827964202q158
+−717771103116611523q315/2
+−7908419005020915850q158
+1907856058463675359575q315/2
+−2510483414752309q315/2
+3797180920247821q158
+46280099948395184q315/2
+−661349858819489021q158
+6373738664932074312q315/2
+1164148757149541167314q158
+−830392595916755q158
+−1589679235709546q315/2
+5002197250330240q158
+−59052244117713785q315/2
+4279809698340893447q158
+−25447538708964750026q315/2
+21883932028960q315/2
+52039830772006q158
+−208430252255007q315/2
+5021231467477637q158
+−203334247925102214q315/2
+−14980307260595602909q158
+68212497673q158
+−14703374329q315/2
+−986065940989q158
+1182082042782q315/2
+3294633659679268q158
+225454885754595400q315/2
+7690268q315/2
+27909767q158
+−486018210q315/2
+−12829067397q158
+3046756706011q315/2
+1068804228132263q158
+�
+On the other hand, the determinant of Irot
+(−2,3,7)(q)[6] to that precision was given by
+det(Irot
+(−2,3,7)(q)[6]) = q−15(1 − q)2(1 − q2)4(1 − q3)4(1 − q4)2 + O(q160) .
+(59)
+But more reassuring was the fact that repeating the computation of Irot
+(−2,3,7),O(q)[6] for the
+insertion �z2 (corresponding to the second shape), we found out that the new matrix had
+equally big coefficients of q-series, but the quotient
+Q(−2,3,7),O(q) = Irot
+(−2,3,7),O(q)[6](Irot
+(−2,3,7)(q)[6])−1
+
+14
+ZHIHAO DUAN, STAVROS GAROUFALIDIS, AND JIE GU
+had entries short rational functions
+Q(−2,3,7),O (q) + O(q160) =
+1
+(1 − q3)(1 − q4)
+·
+
+
+
+
+
+
+
+
+
+0
+q−1/2(q2 + 1)
+q−19(q2 − 1)2(q2 + 1)
+q−77/2(−q4 − q2 − 1)
+q−74(q4 − 1)2
+q−221/2
+q15/2(q4 + q3 + 2q2 + q + 1)
+(q − 1)2(q4 + q3 + 2q2 + q + 1)
+−q−23/2(q + 1)(q2 + 1)2
+q−37(q2 + 1)(q3 − 1)2
+q−131/2(q2 + 1)
+0
+0
+q37/2(q2 + 1)
+(q2 − 1)2(q2 + 1)
+q−39/2(−q4 − q2 − 1)
+q−55(q4 − 1)2
+q−183/2
+q89/2(q4 + q3 + 2q2 + q + 1)
+(q − 1)2q39(q4 + q3 + 2q2 + q + 1)
+−q51/2(q + 1)(q2 + 1)2
+(q2 + 1)(q3 − 1)2
+q−57/2(q2 + 1)
+0
+0
+q147/2(q2 + 1)
+q58(q2 − 1)2(q2 + 1)
+−q71/2(q4 + q2 + 1)
+(q4 − 1)2
+q−73/2
+q235/2(q4 + q3 + 2q2 + q + 1)
+(q − 1)2q112(q4 + q3 + 2q2 + q + 1)
+−q197/2(q + 1)(q2 + 1)2
+q75(q2 + 1)(q3 − 1)2
+q89/2(q2 + 1)
+0
+
+
+
+
+
+
+
+
+
+Surely this cancellation is not an accident, and it is a confirmation that our computational
+method and Corollary 3.5 of Conjecture 3.3 are correct.
+Incidentally, the 3 × 3 matrices Irot
+(−2,3,7),O(q)[3] and Irot
+(−2,3,7),O(q)[3] obey no rationality
+property similar to Equation (4.7), as one would not expect.
+Acknowledgements. The authors wish to thank Tudor Dimofte, Rinat Kashaev, Marcos
+Mari˜no, Campbell Wheeler and Don Zagier for many enlightening conversations. ZD would
+like to thank International Center for Mathematics, SUSTech for hospitality where this work
+was initiated. ZD is supported by KIAS individual Grant PG076902.
+Appendix A. The holomorphic blocks of the 41 knot
+The 41 knot has two colored holomorphic blocks of the 41 knot given by q-hypergeometric
+formulas in [20, Prop.8] as follows:
+h(0)
+41,n(q) = (−1)nq|n|(2|n|+1)/2
+∞
+�
+k=0
+(−1)k
+qk(k+1)/2+|n|k
+(q; q)k(q; q)k+2|n|
+,
+(60)
+and
+h(1)
+41,n(q) = (−1)nq|n|(2|n|+1)/2
+∞
+�
+k=0
+
+−4E1(q) +
+k+2|n|
+�
+ℓ=1
+1 + qℓ
+1 − qℓ +
+k
+�
+ℓ=1
+1 + qℓ
+1 − qℓ
+
+ (−1)k
+qk(k+1)/2+|n|k
+(q; q)k(q, q)k+2|n|
+− 2(−1)nq|n|(2|n|−1)/2
+2|n|−1
+�
+k=0
+(−1)k qk(k+1)/2−|n|k(q−1, q−1)2|n|−1−k
+(q; q)k
+,
+(61)
+for |q| ̸= 1. Here, for a positive integer ℓ, we define Eℓ(q) = ζ(1−ℓ)
+2
++ �∞
+s=1 sℓ−1
+qs
+1−qs , (where
+ζ(s) is the Riemann zeta function), analytic for |q| < 1 and extended to |q| > 1 by the
+symmetry Eℓ(q−1) = −Eℓ(q).
+Appendix B. The holomorphic blocks of the 52 knot
+The 52 knot has three colored holomorphic blocks h(α)
+52,n(q) for α = 0, 1, 2. They were given
+explicitly in [20, Lem.12], and we copy the answer for the benefit of the reader. Using the
+q-harmonic functions
+Hn(q) =
+n
+�
+j=1
+qj
+1 − qj ,
+H(2)
+n (q) =
+n
+�
+j=1
+qj
+(1 − qj)2
+(62)
+
+THE DESCENDANTS OF THE 3D-INDEX
+15
+we have:
+h(0)
+52,n(q) = (−1)nq|n|/2
+∞
+�
+k=0
+q|n|k
+(q−1; q−1)k(q; q)k+2|n|(q; q)k+|n|
+,
+(63)
+h(1)
+52,n(q) = −(−1)nq|n|/2
+∞
+�
+k=0
+q|n|k
+(q; q)k+2|n|(q−1; q−1)k(q; q)k+|n|
+×
+�
+k + |n| − 1
+4 − 3E1(q) + Hk(q) + Hk+|n|(q) + Hk+2|n|(q)
+�
++ q−n2/2
+|n|−1
+�
+k=0
+(q−1, q−1)|n|−1−k
+(q−1, q−1)k(q; q)k+|n|
+,
+(64)
+and
+h(2)
+52,n(q) = (−1)nq|n|/2
+∞
+�
+k=0
+q|n|k
+(q−1; q−1)k(q; q)k+|n|(q; q)k+2|n|
+×
+
+E2(q) + 1
+8 − H(2)
+k (q) − H(2)
+k+|n|(q) − H(2)
+k+2|n|(q)
+−
+�
+k + |n| − 1
+4 − 3E1(q) + Hk(q) + Hk+|n|(q) + Hk+2|n|(q)
+�2
+
++ 2q−n2/2
+|n|−1
+�
+k=0
+(q−1, q−1)|n|−1−k
+(q−1, q−1)k(q; q)k+|n|
+×
+
+|n| − 3
+4 − 3E1(q) + Hk(q) + Hk+|n|(q) + H|n|−k−1(q)
+
+
+− 2(−1)nq−|n|/2
+|n|−1
+�
+k=0
+q−|n|k(q−1; q−1)2|n|−k−1(q−1; q−1)|n|−k−1
+(q−1; q−1)k
+,
+(65)
+for |q| ̸= 1.
+Appendix C. NZ matrices and the 3D-index
+Since there are various formulas for the 3D-index in the literature, let us present our
+conventions briefly.
+Let T be an ideal triangulation with N tetrahedra of a 1-cusped hyperbolic 3-manifold M
+equipped with a symplectic basis µ and λ of H1(∂M,
+Z) and such that λ is the homological
+longitude. Then the edge gluing equations together with the peripheral equations are encoded
+by three (N + 2) × N matrices G, G′ and G′′ whose rows are indexed by the edges, the
+meridian and the longitude and the columns indexed by tetrahedra. The gluing equations
+in logarithmic form are given by
+N
+�
+j=1
+�
+Gij log zj + G′
+ij log z′
+j + G′′
+ij log z′′
+j
+�
+= πi ηi,
+i = 1, . . ., N + 2
+(66)
+where η = (2, . . . , 2, 0, 0)t ∈
+ZN+2.
+
+16
+ZHIHAO DUAN, STAVROS GAROUFALIDIS, AND JIE GU
+If we eliminate the variable z′ in each tetrahedron using zz′z′′ = −1, we obtain the matrices
+A = G−G′, B = G′′ −G′ and the vector ν = (2, . . ., 2, 0, 0)t −G′(1, . . . , 1)t, and the gluing
+equations take the form
+N
+�
+j=1
+�
+Aij log zj + Bij log z′′
+j
+�
+= πi νi,
+i = 1, . . . , N + 2 .
+(67)
+Let aj and bj denote the j-th column of A and B, respectively. For integers m and e,
+consider the vector k = (k1, . . . , kN−1, 0, e, −m/2). Then, the 3D-index of [8] is given by [8]
+(see also [15, Sec.4.5])
+IT (m, e)(q) =
+�
+k1,...,kN−1∈Z
+(−q1/2)ν·k
+N
+�
+j=1
+I∆(−bj · k, aj · k)(q)
+(68)
+and the rotated 3D-index is given by [20, Sec.2.1]
+Irot
+T (n, n′)(q) =
+�
+e∈Z
+IT (n − n′, e)(q)qe(n+n′)/2 .
+(69)
+Let us define the N × N matrices A and B obtained by removing the N and N + 2 rows
+of A and B, respectively. In other words, the rows of A and B correspond to the first N − 1
+edge gluing equations and the meridian gluing equation, respectively. Let (λ1, . . . , λN) and
+(λ′′
+1, . . . , λ′′
+N) denote half the last row of A and B respectively. We assume that these are
+vectors of integers and this can be arranged by adding, if necessary, an integer multiple of
+some of the first N rows of A and B to the last row. Let aj and bj denote the j-th column
+of A and B, respectively, and let k = (k1, . . . , kN). Let ν ∈
+ZN be obtained from ν ∈
+ZN+2
+by removing the N-th and the N + 2 entry of it, and let νλ denote half of the last entry of
+ν.
+Then, combining (68) and (69) (where we rename its summation variable from e to kN)
+we obtain that
+Irot
+T (n, n′)(q) =
+�
+k∈ZN
+(−q1/2)ν·k−(n−n′)νλqkN(n+n′)/2
+N
+�
+j=1
+I∆(λ′′
+j (n−n′)−bj ·k, −λj(n−n′)+aj ·k)(q) .
+(70)
+References
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+17
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+ories. JHEP, 11:151, 2016.
+[8] Tudor Dimofte, Davide Gaiotto, and Sergei Gukov. 3-manifolds and 3d indices. Adv. Theor. Math.
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+[12] Stavros Garoufalidis. On the characteristic and deformation varieties of a knot. In Proceedings of the
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+Coventry, 2004.
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+theory at the trivial flat connection. Preprint 2021, arXiv:2111.04763.
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+invariant. Preprint 2022, arXiv:2209.02843.
+[21] Stavros Garoufalidis and Don Zagier. Knots and their related q-series. Preprint 2021.
+[22] Stavros Garoufalidis and Don Zagier. Knots, perturbative series and quantum modularity. Preprint
+2021, arXiv:2111.06645.
+[23] Walter Neumann. Combinatorics of triangulations and the Chern-Simons invariant for hyperbolic 3-
+manifolds. In Topology ’90 (Columbus, OH, 1990), volume 1 of Ohio State Univ. Math. Res. Inst. Publ.,
+pages 243–271. de Gruyter, Berlin, 1992.
+[24] Walter Neumann and Don Zagier. Volumes of hyperbolic three-manifolds. Topology, 24(3):307–332, 1985.
+[25] Marko Petkovˇsek, Herbert S. Wilf, and Doron Zeilberger. A = B. A K Peters, Ltd., Wellesley, MA,
+1996. With a foreword by Donald E. Knuth, With a separately available computer disk.
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+“q”) multisum/integral identities. Invent. Math., 108(3):575–633, 1992.
+
+18
+ZHIHAO DUAN, STAVROS GAROUFALIDIS, AND JIE GU
+Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul 02455, Korea
+Email address: [email protected]
+International Center for Mathematics, Department of Mathematics, Southern Univer-
+sity of Science and Technology, Shenzhen, China
+http://people.mpim-bonn.mpg.de/stavros
+Email address: [email protected]
+School of Physics and Shing-Tung Yau Center, Southeast University, Nanjing 210096,
+China
+Email address: [email protected]
+

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+arXiv:2301.02447v1  [econ.TH]  6 Jan 2023
+Regret theory, Allais’ Paradox, and Savage’s omelet
+V.G. Bardakhchyan1,2) and A.E. Allahverdyan1,2)
+1)Alikhanian National Laboratory (Yerevan Physics Institute),
+Alikhanian Brothers Street 2, Yerevan 0036, Armenia,
+2)Yerevan State University, 1 A. Manoogian street, Yerevan 0025, Armenia
+(Dated: January 9, 2023)
+Abstract
+We study a sufficiently general regret criterion for choosing between two probabilistic lotteries.
+For independent lotteries, the criterion is consistent with stochastic dominance and can be made
+transitive by a unique choice of the regret function.
+Together with additional (and intuitively
+meaningful) super-additivity property, the regret criterion resolves the Allais’ paradox including
+the cases were the paradox disappears, and the choices agree with the expected utility. This super-
+additivity property is also employed for establishing consistency between regret and stochastic
+dominance for dependent lotteries. Furthermore, we demonstrate how the regret criterion can be
+used in Savage’s omelet, a classical decision problem in which the lottery outcomes are not fully
+resolved. The expected utility cannot be used in such situations, as it discards important aspects
+of lotteries.
+Keywords: Regret theory, Allais’ paradox, stochastic dominance, transitive regret.
+JEL Classification: D81.
+1
+
+I.
+INTRODUCTION
+The history of expected utility theory (EUT) started with Bernoulli’s work resolving the
+St. Petersburg paradox [1]. Several axiomatic schemes for EUT are known [2, 3]. Currently,
+EUT has applications in a wide range of fields, including economics [4], psychology [5],
+evolutionary game theory [6], and general artificial intelligence [7].
+EUT shows how to choose between two lotteries [2–4]:
+(x, p) =
+
+x1 x2 ... xn
+p1 p2 ... pn
+
+ ,
+(y, q) =
+
+y1 y2 ... yn
+q1 q2 ... qn
+
+ ,
+(1)
+�n
+k=1pk =
+�n
+k=1qk = 1,
+(2)
+where (p1, ..., pn) and (q1, ..., qn) are (resp.)
+the probabilities of monetary outcomes
+(x1, ..., xn) and (y1, ..., yn) within each lottery. EUT proposes the following functional for
+each lottery [2–4]:
+V (x, p) =
+�n
+i=1u(xi)pi,
+(3)
+where u(xi) is the utility of the monetary value xi. EUT recommends choosing in (1) the
+first lottery, if V (x, p) > V (y, q).
+Experiments revealed problems with EUT and its axiomatic foundations. In particular,
+several classic experiments cannot be explained by EU for any choice of the utility function
+u(.) in (3) [8, 9]. People generally choose in contradiction to EUT, violating the independence
+axiom, one of four axioms of the von Neumann-Morgenstern formulation of EUT [2]. The
+most prominent example of this is Allais’s paradox [8], where each human subject chooses
+between two lotteries.
+The prospect theory [10, 11], and rank-dependent utility theory
+[12, 13] discarded the independence axiom, and proposed functionals similar to V (x, p) in
+(3), where instead of probabilities pi one employs weights πi that generally depend both on
+(p1, ..., pn) and (x1, ..., xn). Refs. [4, 5, 9] discuss these and other alternatives to EUT.
+There are also other situations where EUT does not apply. EUT cannot be used directly
+when the lottery outcome remains uncertain even after the lottery choice has been made. A
+good example of this situation is the decision problem known as Savage’s omelet [3]. In our
+knowledge, this problem has never been studied from a viewpoint of EUT’s inapplicability.
+As we show below, both Allais’ paradox and Savage’s omelet can be resolved by the regret
+theory (RT), which is one of the alternatives of EUT. The main difference of RT compared
+2
+
+to EUT is that RT does not operate with a value functional for a single lottery. Instead it
+counter-factually compares two lotteries. RT has an intuitive emotional appeal, and it is
+also related to cognitive aspects of decision making [14]. RT was first proposed by Savage in
+minimax form [3] [see [15] for an update of this approach], and later brought to its current
+form in [16, 17]; see [9, 18] for a review. Ref. [16] extended the regret to independent lotteries
+and noted its potential in explaining Allais’ paradox. Ref. [16] also analyzed transitivity,
+common ration effect, and preference reversals.
+Functional forms involving two lotteries
+were given axiomatic foundation in [19]. An axiomatic formulation of regret was attempted
+in [20].
+This work has three purposes. First, we want to show how Allais’ paradox is solved by a
+transitive and super-additive RT. People mentioned regret in the context of Allais’ paradox
+[see e.g. [5, 14, 16]], but so far no systematic and complete solution of this paradox was pro-
+vided. Our solution is rather complete, because it also predicts conditions under which the
+paradox does not hold. Both transitivity and super-additivity have transparent meaning for
+regret theories in general. We do clarify their applicability range. This is especially impor-
+tant for transitivity, because generally regret theories do not lead to transitive predictions
+[21].
+Second, we prove that the transitive and super-additive regret theory is consistent with
+the stochastic dominance criterion [2]. Stochastic dominance is a useful tool, but it does not
+apply to comparing any pair of lotteries. The previous literature in this direction is mostly
+negative showing that regret-based approaches violate first order stochastic dominance [22,
+23] 1. Third, we demonstrate—using as an example Savage’s omlet problem—that RT can
+recommend choosing between lotteries with not resolved outcomes, a task which cannot be
+consistently addressed by EUT.
+The paper is organized as follows. Section II is devoted to regret functional for indepen-
+dent lotteries and some of its properties related to the expected utility. In decision making
+theory the functional form is frequently derived from axiomatic foundation. In contrast,
+here we first introduce the functional considered, then derive its properties. Section III is
+devoted to Allais’ paradox and its relations to other concepts. Stochastic dominance abid-
+1 Ref. [22] analyzed relations between RT and stochastic dominance for a specific case. This analysis is based
+on more general formulation of first order stochastic dominance that compares cumulative distribution
+functions. Here we focus on the simplest version of stochastic dominance.
+3
+
+ance is considered in section IV. Section V analyzes Savage’s omelet problem, identifies an
+aspect that prevents the applicability of the expected utility theory, and solves this problem
+via the regret. We summarize in the last section.
+II.
+REGRET AND ITS FEATURES
+A.
+Axioms of Expected Utility Theory (EUT)
+We remind the four axioms of EUT (3)—completeness, transitivity, continuity, indepen-
+dence—since they will motivate our further consideration.
+First of all one introduces a
+preference relation ⪰, and indifference relation ∼ between the lotteries (1), where ∼ means
+that both ⪰ and ⪯ hold. When comparing two lotteries in (1) we sometimes assume (with-
+out loss of generality) the same outcomes: {xk = yk}n
+k=1. If they are initially different, we
+can introduce suitable zero-probability events and make them identical.
+1. The completeness axiom states that any pair of lotteries in (1) can be compared:
+(x, p) ⪰ (x, q)
+or
+(x, q) ⪰ (x, p)
+or
+(x, q) ∼ (x, p),
+(4)
+where (x, p) ⪰ (x, q) means that lottery (x, q) is not preferred to (x, p).
+2. The transitivity axiom states:
+(x, p) ⪰ (x, q) ⪰ (x, r)
+means
+(x, p) ⪰ (x, r),
+(5)
+3. The continuity axiom states for any three lotteries
+(x, p) ⪰ (x, q) ⪰ (x, r)
+implies
+(x, q) ∼ (x, αp + (1 − α)r),
+(6)
+for some α ∈ [0, 1]. This axiom implies continuity of the value function to be deduced from
+the four axioms.
+4. The independence axiom—also known independence of irrelevant alternatives or the
+sure-thing principle—claims that combining each of two lotteries with any fixed one will not
+alter the preferences [5, 24]:
+(x, p) ⪰ (x, q)
+means
+(x, αp + (1 − α)r) ⪰ (x, αq + (1 − α)r),
+(7)
+where the irrelevant alternative is (x, r). Eq. (7) is among the most controversial axioms in
+decision theory and has triggered many debates [5, 24]; see in this context also Appendix
+4
+
+B, where we explain why specifically the meaning of (7) can be ambiguous. Ref. [25] briefly
+reviews its current status with counter-examples. Experimental studies showed violations
+of (7), with some concerns on whether these violations are systematic [5].
+B.
+Definition of regret
+The regret defines a counterfactual outcome-wise comparison between the lotteries (1)
+using certain ideas of EUT. Hence for particular cases it would coincide with the decision
+criterion of EUT. The utility function u(x) is assumed to exist beforehand and known to
+the decision-maker [5].
+Assume that (y, q) is chosen and its outcome yj is found. The decision-maker compares
+this outcome with what would be found if (x, p) would be taken and defines:
+R(x, p; yj) ≡
+�n
+i=1f(u(xi) − u(yj))pi,
+(8)
+where u(x) is the utility function, and f(x) is a function holding
+f(x ≥ 0) ≥ 0,
+f(x ≤ 0) ≤ 0,
+f(0) = 0.
+(9)
+In particular, R(x, p; yj) > 0 (positive regret), if xi > yj. Generally, f(x) accounts for both
+regret and appreciation. We get a pure regret (appreciation), if f(x ≤ 0) = 0 (f(x ≥ 0) = 0).
+Since (x, p) was not actually chosen, its outcomes are not known; hence the averaging in
+(8). Moreover, once the decision-maker keeps on choosing (y, q) and explores all its outcomes
+according to their probabilities, the average of (8) reads:
+R(x, p; y, q) ≡
+�n
+j=1qjR(x, p; yj) =
+�n
+i,j=1f(u(xi) − u(yj))piqj,
+(10)
+where (10) already assumed that the events (yj, xi) are independent, i.e. their joint prob-
+ability is qjpi. This additional information is to be provided for unambiguous definition of
+lotteries in (1).
+Note that (8, 10) are asymmetric with respect to the lotteries (1), because (y, q) is actually
+chosen, while (x, p) is reasoned counter-factually given this choice. The regret preference
+⪰reg is defined as [9, 16–18]
+(x, p) ⪰reg (y, q)
+iff
+R(y, q; x, p) − R(x, p; y, q) =
+�n
+i,j=1g(u(yj) − u(xi))piqj ≤ 0, (11)
+5
+
+where
+g(x) ≡ f(x) − f(−x),
+(12)
+is anti-symmetric and monotonic:
+g(x) = −g(−x),
+(13)
+g(x) ≥ g(y)
+for
+x ≥ y.
+(14)
+The meaning of R(y, q; x, p) − R(x, p; y, q) ≤ 0 is that (x, p) is preferred if its leads to a
+smaller average regret. For a particular case
+g(x) = ax,
+a > 0,
+(15)
+where a is a constant, we revert from (11) to the expected utility. Note that (15) is achieved
+for various functions f(x); e.g. f(x) = ax/2 or f(x) = a max[x, 0].
+The above definition generalizes for a non-trivial joint probability P(xi, yj) of (xi, yj) with
+�n
+i=1P(xi, yj) = qj,
+�n
+j=1P(xi, yj) = pi.
+(16)
+Now pi in (8) should be replaced by conditional probability P(xi|yj), which is reasonable for
+a counter-factual reasoning, and instead of (8–11) we have
+R(x, p; yj) ≡
+�n
+i=1f(u(xi) − u(yj))P(xi|yj),
+(17)
+R(x, p; y, q) ≡
+�n
+j=1qjR(x, p; yj) =
+�n
+i,j=1f(u(xi) − u(yj))P(xi, yj),
+(18)
+(x, p) ⪰reg (y, q)
+iff
+�n
+i,j=1g(u(yj) − u(xi))P(xi, yj) ≤ 0.
+(19)
+In particular, the outcomes in (16) can refer to the same states of nature [2, 20, 24]. This
+implies
+P(xi, yj) = piδij,
+i, j = 1, ..., n,
+(20)
+where δij is the Kroenecker delta, and where {pi = qi}n
+i=1 are the probabilities for those
+unknown states of nature; see section V for details.
+C.
+Two propositions about regret
+Note that for the regret preference relation (11) we can take lotteries (1) to have the same
+outcomes, xk = yk, using the same argument as before (4). Now the completeness axiom
+(4) obviously holds for ⪰reg. The continuity axiom is valid as well.
+6
+
+Proposition 1. For the regret preference relation (11)
+(x, p) ⪰reg (x, q) ⪰reg (x, r)
+implies
+(x, q) ∼reg (x, αp + (1 − α)r),
+(21)
+for some α ∈ [0, 1]. Working out the last relation in (21) we find
+α = B/(A + B) ∈ [0, 1],
+(22)
+A =
+�n
+i,j=1piqjg(u(xi) − u(xj)) ≥ 0,
+B =
+�n
+i,j=1riqjg(u(xj) − u(xi)) ≥ 0, (23)
+where (23) follows from first and second relations in (21).
+It is known that ⪰reg violates transitivity for a general choice of f(x) [21]. In particular,
+the transitivity is violated under (20) [26]; e.g. for the same states of nature. Transitivity
+violation is not necessarily a drawback, since there are arguments for involving non-transitive
+choices even in normative choices [27]. Ref. [28] shows that for the most general form of
+regret there exist models not violating transitivity.
+Let us now provide a sufficiently complete solution for the transitivity of ⪰reg. First, we
+show that ⪰reg will be transitive for a particular choice of f(x) in (11). Define
+f(x) = b(ax − 1),
+(24)
+where a > 0 and b > 0. Eq. (9) holds. Now (x, p) ⪰reg (x, q) amounts to
+v(p)w(q) ≥ v(q)w(p),
+(25)
+v(p) ≡
+�n
+i=1au(xi)pi > 0,
+w(q) ≡
+�n
+i=1a−u(xi)qi > 0.
+(26)
+Eqs. (25, 26) imply that with the choice (24), ⪰reg is transitive.
+Fisburn’s theorem on
+transitivity [29] shows that (24) is also necessary for transitivity.
+Proposition 2. The regret preference relation ⪰reg given by (11) preserves transitivity
+iff (24) holds.
+Returning to (4–7) we see that only the independence axiom can be violated by ⪰reg; see
+below for more details.
+III.
+SOLVING ALLAIS’ PARADOX WITH REGRET
+There was a great deal of attention focused on Allais’ paradox as one of the major
+systematic violations of EUT [5, 8–11, 30]. Regret theory is mentioned in the context of
+7
+
+Allais’s paradox [5, 14, 16], but no systematic solution of the paradox via the regret theory
+was so far provided. We show below that this solution can be achieved by respecting the
+transitivity and that it does provide an important constraint on the form of g(x) in (11, 12).
+Consider the standard formulation of the Allais’ paradox [5, 8].
+A decision make is
+choosing between the following two lotteries [cf. (1)]:
+I ≡
+
+1
+1
+
+ ,
+II ≡
+
+ 0
+1
+5
+0.01 0.89 0.1
+
+ ,
+(27)
+and then between
+III ≡
+
+ 0
+1
+0.89 0.11
+
+ ,
+IV ≡
+
+ 0
+5
+0.9 0.1
+
+ ,
+(28)
+where the monetary outcomes in (27, 28) are normally given in millions of $.
+There are 4 possible outcomes here: (I, III), (I, IV), (II, III), (II, IV), where (I, III) means
+choosing I in (27) and III in (28). Choosing (I, III) or (II, IV) is consistent with the EUT;
+e.g. (I, III) is achieved if u(1) < u(5) and u(1) ≈ u(5). In contrast, most of people take
+(I, IV) thereby violating the expected utility theory (EUT) [5].
+Applying preference relation (11) to the choice (I, IV), we will find an important and
+intuitive condition for function g(x). Now I ⪰reg II reads from (11):
+0.01 · g(u(0) − u(1)) + 0.1 · g(u(5) − u(1)) < 0.
+(29)
+Since g(x) is an increasing function [cf. (14)], (29) implies
+u(5) − u(1) < u(1) − u(0).
+(30)
+Thus (30)—which can be realized with a concave function u(x) and hence relates to risk-
+aversion—is a necessary condition for (11) to explain Allais’ paradox. Likewise, demanding
+IV ⪰reg III in (28) we get
+0.089 · g(u(5) − u(0)) − 0.099 · g(u(1) − u(0)) + 0.011 · g(u(5) − u(1)) > 0
+(31)
+Taking the difference of (31) and (29) we get
+−0.089 · g(u(5) − u(0)) + 0.089 · g(u(1) − u(0)) + 0.089 · g(u(5) − u(1)) < 0,
+yielding
+g(u(5) − u(0)) > g(u(1) − u(0)) + g(u(5) − u(1)).
+(32)
+8
+
+Now (32) is the second necessary condition for solving Allais’s paradox. Taking (32) and
+(29) together is necessary and sufficient for solving the paradox. It is intuitively clear what
+(32) means. The decision maker is more impressed (i.e. experiences more regret) with the
+difference u(5) − u(0), than with this difference u(5) − u(0) = u(1) − u(0) + u(5) − u(1)
+coming in two separate pieces: u(1) − u(0) and u(5) − u(1). We rewrite (32) as a more
+general condition:
+g(x + y) ≥ g(x) + g(y),
+x ≥ 0,
+y ≥ 0,
+(33)
+which is the super-additivity (in positive domain) for g(x). Noting from (13) that g(0) = 0,
+we recall that any convex function g(x) with g(0) = 0 is super-additive 2. A simple example
+of a function that is easily shown to be super-additive, but is not convex is g(x) = x e−x−2
+[31].
+Indeed,
+d2
+dx2g(x) = 2 e−x−2 x−5(2 − x2), i.e.
+g(x) is concave (convex) for x >
+√
+2
+(
+√
+2 > x > 0) 3. We formulate our results as follows.
+Proposition 3. Allais’s paradox can be explained by regret, if and only if function g(x)
+in (11) is strongly super-additive for some values in positive domain.
+Example. We take the transitive regret and logarithmic utility [cf. (13, 24)]
+g(x) = sinh
+�x
+β
+�
+,
+u(x) = ln
+�x
+γ + 1
+�
+,
+(34)
+where β > 0 and γ > 0 are positive parameters that characterize the decision maker. Here
+γ > 0 defines the threshold of the concave (risk-averse) utility u(x) (u(0) = 0), because only
+for x
+γ ≪ 1 we have u(x) ≃ 0. In a sense, γ defines the initial money, since only for x
+γ ≳ 1 the
+decision maker will care about money. Likewise, β has a similar meaning of threshold, but
+for the regret function: if x
+β ≪ 1, then g(x) = sinh( x
+β) ≃ x
+β is effectively in the regime EUT.
+Now g(x) in (34) holds super-additivity condition (33), since sinh(0)
+=
+0 and
+d2
+dx2 sinh(x) = sinh(x) ≥ 0 for x ≥ 0; hence (32) holds.
+For solving Allais’ paradox we
+need to look at condition (29), which from (34) amounts to
+γ < ζ(β),
+(35)
+ζ(β → ∞) = 5−10,
+ζ(1) = 0.021,
+ζ(β → 0) = 1/3.
+(36)
+2 This fact should be known, but let us present its short proof. First note that g(tx) ≤ tg(x) for 0 < t < 1 due
+to g(t(x)+(1−t)·0) ≤ tg(x)+(1−t)g(0) = tg(x). Next, g(x)+g(y) = g
+�
+(x + y)
+x
+x+y
+�
++g
+�
+(x + y)
+y
+x+y
+�
+≤
+x
+x+yg(x + y) +
+y
+x+yg(x + y) = g(x + y).
+3 Ref. [20] mentioned the super-additivity condition in the context of regret. Ref. [16] employed convexity
+(concavity) features of regret functional, but without any definite reason.
+9
+
+Hence ζ(β) changes from 5−10 to 1/3, when β moves from ∞ to 0. Let us focus on γ < 0.021
+in (36). We know that (27, 28) are to be given in millions of $. Hence we multiply both
+x and γ in u(x) = ln( x
+γ + 1) by 106, and reach the following conclusion: starting from the
+initial money ≥ 21000 $ the decision maker will behave according to the expected utility
+and choose lotteries (II, IV) in (27, 28). The interpretation of the other two values of ζ(β)
+in (36) is similar. Note in this context that 5−10 is equivalent to 5−10 × 108 ≃ 10 cents.
+It is reported that with smaller outcomes—not millions of $ in (27, 28)—Allais’ paradox
+need not hold [32–34]. Other authors note that when shifting all outcomes in (27, 28) with
+the same substantial positive amount, Allais’ paradox will not hold (aversion of ”0” outcome)
+[35]. The scheme given by (34) handles both experimental results.
+Remark 1. The super-additivity (33) of g(x) (and its ensuing relations with convexity)
+does not relate to risk-aversion and risk-seeking, as defined via utility u(x). To understand
+this, compare the following two lotteries:
+
+x
+1
+
+
+and
+
+x − ǫ x + ǫ
+0.5
+0.5
+
+ ,
+ǫ > 0.
+(37)
+Now the first (certain) lottery is regret-preferable compared with the second (uncertain)
+lottery if g(u(x)−u(x−ǫ)) > −g(u(x)−u(x+ǫ)), which is achieved due to a monotonically
+increasing g(x), and concavity of u(x); i.e.
+the risk-aversion at the level of the utility.
+Likewise, the convexity of u(x) (risk-seeking utility) will lead to preferring the uncertain
+lottery.
+Remark 2. Note that the regret is invariant with respect to u(x) → u(x) + a, where a
+is arbitrary, but it is not invariant with respect to u(x) → bu(x), where b > 0; see e.g. the
+very example (34). After transformation u(x) → bu(x), one can redefine gb(x) = g(bx) such
+that the regret stays invariant. This redefinition respects transitivity and super-additivity
+of g(x).
+Remark 3. Recall that the independence axiom (7) (or the axiom of irrelevant alter-
+natives) is the main axiom of EUT violated by the regret theory. Allais’ paradox can be
+reformulated in such a way that the presence of this axiom is made obvious. To this end
+10
+
+one writes (27, 28) as
+I =
+
+ 1
+1
+1
+0.01 0.1 0.89
+
+ ,
+III =
+
+ 1
+1
+0
+0.01 0.1 0.89
+
+ ,
+(38)
+II =
+
+ 0
+5
+1
+0.01 0.1 0.89
+
+ ,
+IV =
+
+ 0
+5
+0
+0.01 0.1 0.89
+
+ .
+(39)
+We emphasize that I and II in (38, 39) (as well as III and IV) refer to independent events.
+It is seen that I and II have the common last column (
+1
+0.89), while for III and IV the
+common last column is (
+0
+0.89). These last columns (i.e. the corresponding outcomes with
+their probabilities) plays the role of independent alternatives. If they are deemed to be
+irrelevant, e.g. (
+1
+0.89) is irrelevant when deciding between I and II, then I becomes equivalent
+to III, and II is equivalent to IV. Hence one takes either (I, III) or (II, IV). Note that this
+reasoning is more general than appealing directly to the axiom (7), since this mathematical
+axiom does not specify the interpretation of the mixture model αp + (1 − α)r; see Appendix
+B for details.
+If experimental subjects are presented Allais’ lotteries in the form (38, 39), then majority
+of them behave according to EUT than for (27, 28) [5]. Naturally, for the regret (11) the
+difference between (38, 39) and (27, 28) is absent. Hence these subjects did not use the
+regret theory in their decision making.
+IV.
+REGRET AND STOCHASTIC DOMINANCE
+For lotteries (1) with independent probabilities, a clear-cut definition of superiority is
+provided by the stochastic dominance ⪰sto [2].
+Recall its definition: we assume 4 that
+xk = yk in (1) hold with
+xi < xj
+for
+i < j.
+(40)
+Now define [2]
+(x, p) ⪰sto (x, q)
+iff
+�k
+i=1pi ≤
+�k
+i=1qi
+for
+k = 1, .., n.
+(41)
+4 This assumption of identical outcomes is not necessary, since the stochastic dominance can be formulated
+more generally. We do not focus on this general definition, since it is equivalent to the situation, when
+the outcomes are made the same by increasing their number via adding zero-probability events; cf. the
+discussion before (4).
+11
+
+Recall that the utility u(x) in (11) is an increasing function of x. Stochastic dominance does
+not depend on a specific form of the utility u(x) in (11) provided that it is an increasing
+function of x, as we assume. This is an advantage of stochastic dominance. Its weakness
+is that it clearly does not apply to all lotteries, i.e. the completeness axiom (4) is violated.
+Indeed, it is sufficient to violate (41) for one value of k, and this will make ⪰sto inapplicable.
+A related weakness is that its applicability is not stable with respect to small variations of
+outcomes. To see this, assume that (40, 41) hold and perturb y1 = x1 → y′
+1 < x1. Even a
+small variation of this type violates condition (41) for k = 1.
+Regret and stochastic dominance do not contradict each other, as the following proposi-
+tion shows.
+Proposition 4. (x, p) ⪰sto (x, q) implies (x, p) ⪰reg (x, q) defined from (11). The proof
+is given in Appendix A.
+Note that Proposition 4 does not require any specific feature of g(x) apart of (13, 14).
+However, it does require independent probabilities for the lotteries, as implied by (11).
+Lotteries with independent probabilities have vast but still limited range of applications.
+Even within the framework of initially independent lotteries, one can envisage new dependent
+lotteries for which the regret is given via (19). For dependent lotteries the relation between
+regret and stochastic dominance is partially explained by the following proposition.
+Proposition 5. For the joint probability P(xi, xj) given by (16), let us define the
+marginal probabilities {pi}n
+i=1 and {qj}n
+j=1, as well as deviation of P(xi, xj) from piqj:
+pi :=
+�n
+j=1P(xi, xj),
+qj :=
+�n
+i=1P(xi, xj),
+(42)
+θi,j := P(xi, xj) − piqj,
+(43)
+�n
+i=1θi,j =
+�n
+j=1θi,j = 0,
+|θi,j| ≤ piqj.
+(44)
+Then if g(x) is super-additive on positive domain [see (33)] and if
+θi,j ≥ θj,i,
+for
+i > j,
+(45)
+one has that (x, p) ⪰sto (x, q) defined via (40, 41) leads to (x, p) ⪰reg (x, q) in the sense of
+(19).
+Thus the super-additivity of g(x) plus condition (45) make the regret consistent with the
+stochastic dominance. The proof of Proposition 5 is given in Appendix C.
+12
+
+V.
+SAVAGE’S OMELET IS SOLVED VIA THE REGRET THEORY
+Eq. (1) with {pk = qk}n
+k=1 can refer to the to the decision model which assumes that at
+the moment of action-taking there is an uncertain state of nature (environment) Sk to be
+realized from {Sk}n
+k=1 with probabilities {pk}n
+k=1, which are known to the decision maker
+[2, 24]. Sk are called states of nature, since their future realization is independent from
+the action taken, but an action A (B) in a state Sk leads to consequences with monetary
+outcome xk (yk) and utilities u(xk) (u(yk)) [2, 24]; cf. (1, 20).
+The following classic decision problem is described in [3]: A decision maker has to finish
+making an omelet began by his wife, who has already broken into a bowl five good eggs. A
+sixth unbroken egg is lying on the table, and it must be either used in making the omelet,
+or discarded. There are two states of the nature: good (the sixth egg is good) and rotten
+(the sixth egg is rotten), which do not depend on the actions A1, A2 and A3 of the decision
+maker.
+A1: break the sixth egg into the bowl.
+A2: discard the sixth egg.
+A3: break the sixth egg into a saucer; add it to the five eggs if it is good, discard it if it
+is rotten.
+The consequences of the acts can be written as lotteries:
+A1 =
+
+u−5
+u6
+p
+1 − p
+
+ ,
+A2 =
+
+u5 u5 + z
+p
+1 − p
+
+ ,
+A3 =
+
+u5 + w u6 + w
+p
+1 − p
+
+ ,
+(46)
+where p (1 − p) is the objective probability for the sixth egg to be rotten (good), u6 (u5) is
+the utility of the six-egg (five-egg) omelet, u−5 < 0 is the utility of five spoiled eggs and no
+omelet whatsoever, w < 0 is the utility of washing the saucer, and z < 0 is the utility of the
+good egg being lost. 5
+Now looking at the consequences of A2, we see that—in contrast to A1 and A3—acting A2
+does not resolve the uncertain state of nature: once the egg is discarded, the decision maker
+will not know (without additional actions), whether it was rotten or good. Put differently,
+utility z is not obtained after acting A2, and cannot be obtained without additional actions.
+5 The concrete utilities of washing the saucer may differ depending on the state of the sixth egg. We,
+however, neglect this difference. Also, for simplicity w was simply added to u5 and u6.
+13
+
+Calculating the expecting utility of A2 in the usual way as pu5 + (1 − p)(u5 + z) does not
+apply, because it disregards this aspect A2. It is natural to take the expected utility as
+pu5 + (1 − p)u5 = u5 (i.e. once z is not obtained, it is not included), but then comparing
+with expected utilities of A1 and A3, we see that the parameter z will appear nowhere.
+Hence, we suggest that the expected utility does not apply to comparing A2 with the other
+two actions.
+Employing in (46) the reasoning of regret [cf. (8, 18, 20)] does take into account the
+difference between A2 and the other two actions. Let us for example calculate the regret
+about not taking A1 once A2 has been taken:
+R(A1, A2) = pf(u−5 − u5) + (1 − p)f(u6 − u5).
+(47)
+This expression does not contain z, since the uncertain state of nature was not resolved after
+acting A2, i.e. after acting A2 the obtained utility is u5.
+On the other hand, acting A1 resolves the uncertainty about the state of nature. Hence
+the regret of not taking A2, once A1 was acted reads:
+R(A2, A1) = pf(u5 − u−5) + (1 − p)f(u5 + z − u6),
+(48)
+i.e. once A1 is taken and the egg is rotten (good), then the decision maker already knows
+that if A2 would be taken, then the egg will turn out rotten (good). It is seen that (48)
+contains z (the utility of discarding a good egg), while (47) does not. Now
+A1 ⪰reg A2
+iff
+R(A2, A1) ≤ R(A1, A2),
+(49)
+where R(A2, A1) − R(A1, A2) does feel the parameter z. As an example of (49) consider
+f(x) = x [cf. the discussion after (15)]:
+p(u5 − u−5) < (1 − p)(u6 − u5 − z
+2).
+(50)
+where we recall that u6 > u5 > u−5 and z < 0. We can naturally assume u5−u−5 > u6−u5 >
+0 under which (50) is non-trivial even for p = 1/2. Note that the formal application of the
+expected utility will claim that A1 is preferred over A2 for p(u5 −u−5) < (1−p)(u6 −u5 −z),
+which is clearly different from (50). This is not just a different outcome; rather, the expected
+utility does not apply.
+14
+
+VI.
+SUMMARY
+This paper studied regret functionals over the utility differences of two probabilistic lot-
+teries; see section II. There are various types of lotteries, from independent to fully dependent
+that refer to the same state of nature. The regret functional compares the lotteries counter-
+factually taking notice of their probabilities. For particular cases, the regret reverts to the
+expected utility. More generally, it does not satisfy the independence (from irrelevant al-
+ternatives) axiom of the expected utility, also known as the sure thing principle. This not
+satisfying is by itself non-trivial and is explored in Appendix B. In contrast to the expected
+utility, the regret is also generally not invariant with respect to multiplying the utility by
+a positive number. It is also due to these two differences compared to the expected utility
+that the regret is efficient for explaining and resolving Allais’s paradox; see section III. The
+resolution demands a non-trivial features of the regret functional, viz. its super-additivity,
+which does make an intuitive sense. We show that the regret functional can be chosen such
+that the regret-ordering holds transitivity. In particular, Allais’s paradox can be resolved via
+a transitive regret, and this resolution provides a consistent account of changes in monetary
+outcomes.
+We devoted a special attention to relations between (the first-order) stochastic dominance
+and the regret-preference; see section IV. The former ordering is normatively appealing, but
+it is incomplete, since not every two lotteries can be compared with each other. We show
+that for independent lotteries the stochastic dominance implies the regret-preference. For
+dependent lotteries the relations between the two are more complex. Here we proposed a
+sufficient condition for the implication stochastic dominance → regret-preference, which,
+interestingly is also based on the super-additivity of the regret; see Proposition 5.
+Finally, we show in section V how the considered regret theory can be useful in those
+situations, where actions of the decision maker do not resolve the uncertain situation. The
+expected utility theory does not apply to such a situation in the sense that there is an
+important information about the lotteries that it simply discards. In the regret, this infor-
+mation is employed, since the regret compares the unresolved uncertainty with the resolved
+uncertainty.
+Our results show that though the concept of regret was initially deduced from certain
+emotional features of decision makers, it does have many features one intuitively expects
+15
+
+from rationality. Hence we envisage its further applications in e.g. reinforcement learning.
+Acknowledgements
+This work was supported by State Science Committee of Armenia, grants No. 21AG-
+1C038. We thank Andranik Khachatryan for useful remarks and for participating in initial
+stages of this work.
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+Appendix A: Proof of Proposition 4
+.
+We are to show that
+n
+�
+i=1,j=1
+piqjg(u(xj) − u(xi)) ≤ 0.
+(A1)
+It can be written in the following form
+n
+�
+i=1
+i
+�
+j=1
+(piqj − pjqi)g(u(xj) − u(xi)).
+(A2)
+Now g’s in (A2) are strictly negative (recall (40) and monotonicity of g(x) and u(x)).
+18
+
+We show that the following holds
+n
+�
+i=1
+i
+�
+j=1
+piqjg(u(xj) − u(xi)) ≤
+n
+�
+i=1
+i
+�
+j=1
+pipjg(u(xj) − u(xi))
+≤
+n
+�
+i=1
+i
+�
+j=1
+pjqig(u(xj) − u(xi)),
+(A3)
+from which the (A2) will follow.
+To do it, we make use of the following lemma.
+Lemma.
+Given sequences pi, qi as above, then for any increasing sequence of negative
+numbers gi, it is true that
+k
+�
+i=1
+pigi ≥
+k
+�
+i=1
+qigi,
+for
+k = 1, ..., n.
+(A4)
+A simple induction will help. Obviously p1g1 ≥ q1g1. Supposing it is true for some k we will
+have:
+k
+�
+i=1
+pigi ≥
+k
+�
+i=1
+qigi
+(A5)
+Let us now subtract gk+1 from each component gi. (As we are speaking about any increasing
+sequence, we haven’t specified any concrete g-s yet. So, having that all components are less
+gk+1, subtract, without change of sign in inequality)
+We will have the following:
+k
+�
+i=1
+pi(gi − gk+1) ≥
+k
+�
+i=1
+qi(gi − gk+1)
+(A6)
+Now by (41)
+k+1
+�
+i=1
+pi ≤
+k+1
+�
+i=1
+qi
+(A7)
+multiplying by gk+1, we get
+k+1
+�
+i=1
+pigk+1 ≥
+k+1
+�
+i=1
+qigk+1.
+(A8)
+Summing up (A6) and (A8), the desired result is obtained:
+k+1
+�
+i=1
+pigi ≥
+k+1
+�
+i=1
+qigi.
+(A9)
+19
+
+Using the lemma we go back and consider first part of (A3)
+n
+�
+i=1
+i
+�
+j=1
+piqjg(u(xj) − u(xi)) ≤
+n
+�
+i=1
+i
+�
+j=1
+pipjg(u(xj) − u(xi)).
+(A10)
+Obviously
+n
+�
+i=1
+i
+�
+j=1
+piqjg(u(xj) − u(xi)) =
+n
+�
+i=1
+pi
+i
+�
+j=1
+qjg(u(xj) − u(xi))
+≤
+n
+�
+i=1
+pi
+i
+�
+j=1
+pjg(u(xj) − u(xi)),
+(A11)
+The last part is implied by lemma.
+Returning to the second part of (A3):
+n
+�
+i=1
+i
+�
+j=1
+pjqig(u(xj) − u(xi)) ≥
+n
+�
+i=1
+i
+�
+j=1
+pipjg(u(xj) − u(xi)).
+(A12)
+and changing the order of summation, we get
+n
+�
+i=1
+i
+�
+j=1
+pjqig(u(xj) − u(xi)) =
+n
+�
+j=1
+n
+�
+i=j
+pjqig(u(xj) − u(xi))
+=
+n
+�
+j=1
+pj
+n
+�
+i=j
+qig(u(xj) − u(xi)) ≥
+n
+�
+j=1
+pj
+n
+�
+i=j
+pig(u(xj) − u(xi))
+=
+n
+�
+i=1
+i
+�
+j=1
+pipjg(u(xj) − u(xi)),
+(A13)
+where the inequality is the inverse of the one used in lemma. The proof is complete.
+Appendix B: Regret and the independence axiom (7)
+We already emphasized around (38, 39) that regret preference ⪰reg defined in (11) must
+violate the independence axiom for solving Allais’ paradox. Now we provide clarifications
+regarding the form (7) of this axiom. We note that (6, 7) do not define how precisely the
+mixing of the two probabilities with weights α is implemented.
+Below we discuss three
+interesting possibilities for implementing the set-up of (7). So we are given three lotteries
+20
+
+[cf. (1)]
+(x, p) =
+
+x1 x2 ... xn
+p1 p2 ... pn
+
+ ,
+(y, q) =
+
+y1 y2 ... yn
+q1 q2 ... qn
+
+ ,
+(z, r) =
+
+z1 z2 ... zn
+r1 r2 ... rn
+
+ ,
+(B1)
+�n
+k=1pk =
+�n
+k=1qk =
+�n
+k=1rk = 1,
+(B2)
+where (p1, ..., pn), (q1, ..., qn), and (r1, ..., rn) are (resp.) the probabilities of monetary out-
+comes (x1, ..., xn) and (y1, ..., yn), (z1, ..., zn) within each lottery.
+1. Here one chooses between two composite lotteries A = {(1 − α)(x, p) + α(z, r)} and
+B = {(1 − α)(y, q) + α(z, r)}. If A is taken, then a binary random variable SA is realized
+that takes values SA = 0 and SA = 1 with probabilities 1 − α and α, respectively. For
+SA = 0 or SA = 1 one faces lottery (x, p) or (z, r), respectively. If B is taken, then a binary
+random variable SB (independent from SA) is realized that takes values SB = 0 and SB = 1
+with probabilities 1 − α and α, respectively. For SB = 0 or SB = 1 one faces lottery (y, q)
+or (z, r), respectively. Let us now assume that (x, p) is independent from (y, q), but the
+lottery (z, r) in both options is the same. Using definition of regret (11), we end up with
+the following preference relation:
+A ⪰reg,1 B
+iff
+(1 − α)2�n
+i,j=1g(u(yj) − u(xi))piqj + (1 − α)α
+�n
+i,j=1g(u(zj) − u(xi))pirj
++(1 − α)α
+�n
+i,j=1g(u(yj) − u(zi))qjri ≤ 0.
+(B3)
+Note that (B3) does not contain terms with g(u(zj) − u(zi)), because the decision maker
+does not expect to find different outcomes zj and zi within options A and B.
+2. Now we have the situation of 1, but S = SA = SB; e.g. one can assume that S is
+realized beforehand, but the result is not known to the decision maker at the time of decision
+making. Now the regret is different [cf. (B3)]:
+A ⪰reg,2 B
+iff
+(1 − α)2�n
+i,j=1g(u(yj) − u(xi))piqj ≤ 0,
+(B4)
+where (z, r) does not enter to regret comparison (B4), which is formally consistent with
+axiom (7).
+21
+
+3. We have the situation of 1, but (z, r) in option A and (z, r) in option B are two
+different lotteries with independent probabilities:
+A ⪰reg,3 B
+iff
+(1 − α)2�n
+i,j=1g(u(yj) − u(xi))piqj + (1 − α)α
+�n
+i,j=1g(u(zj) − u(xi))pirj
++(1 − α)α
+�n
+i,j=1g(u(yj) − u(zi))qjri + α2�n
+i,j=1g(u(zj) − u(zi))rjri ≤ 0.
+(B5)
+The standard interpretation of the independence axiom within the expected utility the-
+ory hints at 3. We however emphasized that this situation is not unique. Note that all
+possibilities (B3, B4, B5) agree with the expected utility theory, where g(x) = x.
+Appendix C: Proof of Proposition 5
+It is known that n × n matrices θi,j from (43) form vector space of (n − 1) × (n − 1)
+dimension. So for example any such 3 × 3 matrix can be rewritten
+
+
+
+
+
+θ1,1 θ1,2 θ1,3
+θ2,1 θ2,2 θ2,3
+θ3,1 θ3,2 θ3,3
+
+
+
+
+ = θ1,1
+
+
+
+
+
+1
+0 −1
+0
+0
+0
+−1 0
+1
+
+
+
+
++θ1,2
+
+
+
+
+
+0
+1
+−1
+0
+0
+0
+0 −1
+1
+
+
+
+
++θ2,1
+
+
+
+
+
+0
+0
+0
+1
+0 −1
+−1 0
+1
+
+
+
+
++θ2,2
+
+
+
+
+
+0 0
+0
+0 1 −1
+0 1
+1
+
+
+
+
+
+(C1)
+Denoting the basis matrices by Mi,j we have that any matrix Θ of thetas can be rewritten
+as
+Θ =
+n−1
+�
+i,j=1
+θi,jMi,j,
+(C2)
+where Mi,j is the matrix whose (i, j)-th and (n, n)-th elements are 1, the (i, n)-th and (n, j)-
+th elements are −1.
+Let us compute the regret in this case
+R =
+n
+�
+i=1
+n
+�
+j=1
+P(xi, xj)g(u(xj)−u(xi)) =
+n
+�
+i=1
+n
+�
+j=1
+piqjg(u(xj)−u(xi))+
+n
+�
+i=1
+n
+�
+j=1
+θi,jg(u(xj)−u(xi))
+(C3)
+We already know that the first term is negative ((A1) and Proposition 4). So it remains
+to show that second part is also negative. Denoting by G the matrix whose elements are
+G(i, j) = g(u(xj) − u(xi)), we can rewrite
+n
+�
+i=1
+n
+�
+j=1
+θi,jg(u(xj) − u(xi)) = ||Θ ⊙ G||,
+(C4)
+22
+
+Where under ||.|| we understand sum of all elements and ⊙ is Hadamard’s (element-wise)
+product.
+Note that
+n
+�
+i=1
+n
+�
+j=1
+θi,jg(u(xj) − u(xi)) = ||Θ ⊙ G|| =
+n−1
+�
+i=1
+n−1
+�
+j=1
+θi,j||Mi,j ⊙ G||.
+(C5)
+We have
+||Mi,j ⊙ G|| = g(u(xi) − u(xj)) − g(u(xn) − u(xj)) − g(u(xi) − u(xn)).
+(C6)
+Now consider
+θi,j||Mi,j ⊙ G|| + θj,i||Mi,j ⊙ G||
+= θi,j(g(u(xi) − u(xj)) − g(u(xn) − u(xj)) − g(u(xi) − u(xn)))
++ θj,i(g(u(xj) − u(xi)) − g(u(xn) − u(xi)) − g(u(xj) − u(xn)))
+= (θi,j − θj,i)(g(u(xi) − u(xj)) + g(u(xn) − u(xi)) − g(u(xn) − u(xj))
+(C7)
+Note that while i > j we have by super-additivity that term in second parenthesis is negative.
+So toghether with θi,j ≥ θj,i we conclude that
+θi,j||Mi,j ⊙ G|| + θj,i||Mi,j ⊙ G|| ≤ 0
+(C8)
+Rewriting
+n
+�
+i=1
+n
+�
+j=1
+θi,jg(u(xj)−u(xi)) = ||Θ⊙G|| =
+n−1
+�
+i=1
+n−1
+�
+j=1;j̸=i
+θi,j||Mi,j ⊙G||+
+n−1
+�
+i=1
+θi,i||Mi,i⊙G||. (C9)
+The second sum in (C9) is obviously 0, as g(x) is antisymmetric, and Mi,i-s are symmetric
+matrices.
+So
+n
+�
+i=1
+n
+�
+j=1
+θi,jg(u(xj) − u(xi)) =
+n−1
+�
+i=1
+n−1
+�
+j=1;j̸=i
+θi,j||Mi,j ⊙ G|| ≤ 0.
+(C10)
+23
+

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