Computing the elliptic genus of higher rank E-strings from genus 0 GW invariants

We show that the elliptic genus of the higher rank E-strings can be computed based solely on the genus 0 Gromov-Witten invariants of the corresponding elliptic geometry. To set up our computation, we study the structure of the topological string free energy on elliptically fibered Calabi-Yau manifolds both in the unrefined and the refined case, determining the maximal amount of the modular structure of the partition function that can be salvaged. In the case of fibrations exhibiting only isolated fibral curves, we show that the principal parts of the topological string partition function at given base-wrapping can be computed from the knowledge of the genus 0 Gromov-Witten invariants at this base-wrapping, and the partition function at lower base-wrappings. For the class of geometries leading to the higher rank E-strings, this leads to the result stated in the opening sentence.Comment: 40 page

High-order accurate well-balanced energy stable adaptive moving mesh finite difference schemes for the shallow water equations with non-flat bottom topography

This paper proposes high-order accurate well-balanced (WB) energy stable (ES) adaptive moving mesh finite difference schemes for the shallow water equations (SWEs) with non-flat bottom topography. To enable the construction of the ES schemes on moving meshes, a reformulation of the SWEs is introduced, with the bottom topography as an additional conservative variable that evolves in time. The corresponding energy inequality is derived based on a modified energy function, then the reformulated SWEs and energy inequality are transformed into curvilinear coordinates. A two-point energy conservative (EC) flux is constructed, and high-order EC schemes based on such a flux are proved to be WB that they preserve the lake at rest. Then high-order ES schemes are derived by adding suitable dissipation terms to the EC schemes, which are newly designed to maintain the WB and ES properties simultaneously. The adaptive moving mesh strategy is performed by iteratively solving the Euler-Lagrangian equations of a mesh adaptation functional. The fully-discrete schemes are obtained by using the explicit strong-stability preserving third-order Runge-Kutta method. Several numerical tests are conducted to validate the accuracy, WB and ES properties, shock-capturing ability, and high efficiency of the schemes.Comment: 40 pages, 16 figure

ZN\mathbb{Z}_N Duality and Parafermions Revisited

Given a two-dimensional bosonic theory with a non-anomalous $\mathbb{Z}_2$ symmetry, the orbifolding and fermionization can be understood holographically using three-dimensional BF theory with level $2$. From a Hamiltonian perspective, the information of dualities is encoded in a topological boundary state which is defined as an eigenstate of certain Wilson loop operators (anyons) in the bulk. We generalize this story to two-dimensional theories with non-anomalous $\mathbb{Z}_N$ symmetry, focusing on parafermionization. We find the generic operators defining different topological boundary states including orbifolding and parafermionization with $\mathbb{Z}_N$ or subgroups of $\mathbb{Z}_N$, and discuss their algebraic properties as well as the $\mathbb{Z}_N$ duality web.Comment: 39 pages, 5 figure

High-order accurate well-balanced energy stable finite difference schemes for multi-layer shallow water equations on fixed and adaptive moving meshes

This paper develops high-order well-balanced (WB) energy stable (ES) finite difference schemes for multi-layer (the number of layers $M\geqslant 2$) shallow water equations (SWEs) on both fixed and adaptive moving meshes, extending our previous works [20,51]. To obtain an energy inequality, the convexity of an energy function for an arbitrary $M$ is proved by finding recurrence relations of the leading principal minors or the quadratic forms of the Hessian matrix of the energy function with respect to the conservative variables, which is more involved than the single-layer case due to the coupling between the layers in the energy function. An important ingredient in developing high-order semi-discrete ES schemes is the construction of a two-point energy conservative (EC) numerical flux. In pursuit of the WB property, a sufficient condition for such EC fluxes is given with compatible discretizations of the source terms similar to the single-layer case. It can be decoupled into $M$ identities individually for each layer, making it convenient to construct a two-point EC flux for the multi-layer system. To suppress possible oscillations near discontinuities, WENO-based dissipation terms are added to the high-order WB EC fluxes, which gives semi-discrete high-order WB ES schemes. Fully-discrete schemes are obtained by employing high-order explicit SSP-RK methods and proved to preserve the lake at rest. The schemes are further extended to moving meshes based on a modified energy function for a reformulated system, relying on the techniques proposed in [51]. Numerical experiments are conducted for some two- and three-layer cases to validate the high-order accuracy, WB and ES properties, and high efficiency of the schemes, with a suitable amount of dissipation chosen by estimating the maximal wave speed due to the lack of an analytical expression for the eigenstructure of the multi-layer system.Comment: 54 pages, 19 figure

Can the energy bound E ≥ 0 imply supersymmetry?

We utilize the integrality conjecture to show that the torus partition function of a fermionic rational conformal theory in the Ramond-Ramond sector becomes a constant when the bound hR ≥ c⁄24 is satisfied, where hR denotes the conformal weights of Ramond states and c is the central charge. The constant-valued Ramond-Ramond partition function strongly suggests the presence of supersymmetry unless a given theory has free fermions. The lower bound hR ≥ c⁄24 can then be identified with the unitarity bound of N = 1 supersymmetry. We thus propose that, for rational CFTs without free fermions, (hR − c/24) ≥ 0 can imply supersymmetry

Lossy Image Compression with Quantized Hierarchical VAEs

Recent research has shown a strong theoretical connection between variational autoencoders (VAEs) and the rate-distortion theory. Motivated by this, we consider the problem of lossy image compression from the perspective of generative modeling. Starting with ResNet VAEs, which are originally designed for data (image) distribution modeling, we redesign their latent variable model using a quantization-aware posterior and prior, enabling easy quantization and entropy coding at test time. Along with improved neural network architecture, we present a powerful and efficient model that outperforms previous methods on natural image lossy compression. Our model compresses images in a coarse-to-fine fashion and supports parallel encoding and decoding, leading to fast execution on GPUs. Code is available at https://github.com/duanzhiihao/lossy-vae.Comment: WACV 2023 Best Algorithms Paper Award, revised versio

On Classification of Fermionic Rational Conformal Field Theories

We systematically study how the integrality of the conformal characters shapes the space of fermionic rational conformal field theories in two dimensions. The integrality suggests that conformal characters on torus with a given choice of spin structures should be invariant under a principal congruence subgroup of $\mathrm{PSL}(2,\mathbb{Z})$. The invariance strongly constrains the possible values of the central charge as well as the conformal weights in both Neveu-Schwarz and Ramond sectors, which improves the conventional holomorphic modular bootstrap method in a significant manner. This allows us to make much progress on the classification of fermionic rational conformal field theories with the number of independent characters less than five.Comment: 36 pages, 1 figure; minor changes, published versio

An Improved Upper Bound on the Rate-Distortion Function of Images

Recent work has shown that Variational Autoencoders (VAEs) can be used to upper-bound the information rate-distortion (R-D) function of images, i.e., the fundamental limit of lossy image compression. In this paper, we report an improved upper bound on the R-D function of images implemented by (1) introducing a new VAE model architecture, (2) applying variable-rate compression techniques, and (3) proposing a novel \ourfunction{} to stabilize training. We demonstrate that at least 30\% BD-rate reduction w.r.t. the intra prediction mode in VVC codec is achievable, suggesting that there is still great potential for improving lossy image compression. Code is made publicly available at https://github.com/duanzhiihao/lossy-vae.Comment: Conference paper at ICIP 2023. The first two authors share equal contribution