Ricci Flow from the Renormalization of Nonlinear Sigma Models in the Framework of Euclidean Algebraic Quantum Field Theory

Renormalization Group and the Ricci flow

We discuss from a geometric point of view the connection between the renormalization group flow for non--linear sigma models and the Ricci flow. This offers new perspectives in providing a geometrical landscape for 2D quantum field theories. In particular we argue that the structure of Ricci flow singularities suggests a natural way for extending, beyond the weak coupling regime, the embedding of the Ricci flow into the renormalization group flow.

非紧等距Calabi-Yau流形上超对称非线性西格玛模型的构造

Construction of Supersymmetric Nonlinear Sigma Models on Noncompact Calabi-Yau Manifolds with Isometry

在非紧Ricci平坦Kahler流形上提出了一类N=2的超对称非线性西格玛模型，将其解释为Hermitian对称空间上的复线丛.利用超对称规范理论的技巧，构造了这些等距流形的Kahler势和Ricci平坦度规.每个度量都包含一个控制这些基流形大小的分辨率参数，当该参数消失时，就会出现圆锥奇异性。

A vanishing result for the supersymmetric nonlinear sigma model in higher dimensions

We prove a vanishing result for critical points of the supersymmetric nonlinear sigma model on complete non-compact Riemannian manifolds of positive Ricci curvature that admit an Euclidean type Sobolev inequality, assuming that the dimension of the domain is bigger than two and that a certain energy is sufficiently small.

无RG流的规范线性σ模型产生的里奇平坦度量

Ricci-flat metrics from gauged linear $σ$-models without RG flow

该研究利用带不定号的规范线性σ模型（GLSMs）引入影子坐标，构造出广义锥上的里奇平坦凯勒度量，给出凯勒势通用公式，为卡拉比-丘度量和环面几何研究提供新方向。

Dynamically Stable Poincaré Embeddings for Neural Manifolds

In a Riemannian manifold, the Ricci flow is a partial differential equation for evolving the metric to become more regular. We hope that topological structures from such metrics may be used to assist in the tasks of machine learning. However, this part of the work is still missing. In this paper, we propose Ricci flow assisted Eucl2Hyp2Eucl neural networks that bridge this gap between the Ricci flow and deep neural networks by mapping neural manifolds from the Euclidean space to the dynamically stable Poincaré ball and then back to the Euclidean space. As a result, we prove that, if initial me…

On dual description of the deformed $O(N)$ sigma model

We study dual strong coupling description of integrability-preserving deformation of the $O(N)$ sigma model. Dual theory is described by a coupled theory of Dirac fermions with four-fermion interaction and bosonic fields with exponential interactions. We claim that both theories share the same integrable structure and coincide as quantum field theories. We construct a solution of Ricci flow equation which behaves in the UV as a free theory perturbed by graviton operators and show that it coincides with the metric of the $η-$deformed $O(N)$ sigma-model after $T-$duality transformation.

A Simple Formula for Scalar Curvature of Level Sets in Euclidean Spaces

A simple formula is derived for the Ricci scalar curvature of any smooth level set ${ψ(x_0,x_1,...,x_n)=C}$ embedded in the Euclidean space $ \mathbb R^{n+1}$, in terms of the gradient $ \nablaψ$ and the Laplacian $ Δψ$. Some applications are given to the geometry of low-dimensional $p$-harmonic functions and high-dimensional harmonic functions.

Lorentzian Einstein-Ricci Flows

We study the Ricci flow for the Lorentzian Einstein-Hilbert action. We show that Einstein gravity emerges as a fixed point of the Einstein-Ricci flow equations and derive a renormalization group flow in Euclidean signature. By considering linearizations near the fixed point, the dynamics of the metric reveal that curvature deformations flow according to a forward heat equation with the stress-energy tensor acting as a source.

Equilibrium density matrices for the 2D black hole sigma models from an integrable spin chain

This work concerns the quantum Lorentzian and Euclidean black hole non-linear sigma models. For the Euclidean black hole sigma model an equilibrium density matrix is proposed, which reproduces the modular invariant partition function from the 2001 paper of Maldacena, Ooguri and Son. For the Lorentzian black hole sigma model, using its formulation as a gauged ${\rm SL}(2,\mathbb{R})$ WZW model, we describe the linear and Hermitian structure of its space of states and also propose an expression for the equilibrium density matrix. Our analysis is guided by the results of the study of a certain cr…

The Chern-Gauss-Bonnet Theorem via supersymmetric Euclidean field theories

We prove the Chern-Gauss-Bonnet Theorem using sigma models whose source supermanifolds have super dimension 0|2. Along the way we develop machinery for understanding manifold invariants encoded by families of 0|n-dimensional Euclidean field theories and their quantization.

非常平均曲率曲面的广义Weierstrass系统与非线性西格玛模型

The Generalized Weierstrass System for Nonconstant Mean Curvature Surfaces and the Nonlinear Sigma Model

研究了三维欧氏空间中诱导平均曲率曲面的广义Weierstrass系统。