Bethe/Gauge correspodence: a short review on an aspect of the integrability nature of supersymmetric gauge theories

TL;DR

本文综述了 Bethe/Gauge 对应在超对称规范场论中的量子可积性，重点展示如何将 2d N=(2,2) 规范理论中的有效扭曲超势与 1d 自旋链的 Bethe 拟设联系起来，以及如何通过对比四维瞬子和二维涡旋分配函数揭示两者的映射关系。核心方法包括在 Nekrasov–Shatashvili 极限下的降维与 Omega 背景的局域化，以及 Nakajima 矢量空间与量子代数在表示论中的几何实现。研究结果揭示了从 4d 到 2d 的瞬子/涡旋对应、从矩阵簇到 R 矩阵的代数结构，以及通过冻结极限将短程可积模型转化为长程自旋链的路径。这些工作为理解量子可积性在场论中的起源提供了几何与代数的统一框架，并为将来在更广泛的规范群和边界条件下扩展 Bethe/Gauge 对应提供线索。

Abstract

In this article, we provide a short review (written in Chinese) on the Bethe/Gauge correspondence. We first explain the basic idea in an explicit example of the correspondence between XXX spin chains and 2d $\mathcal{N}=(2,2)$ gauge theories. The connection between 4d and 2d will then be explored by comparing the instanton and vortex partition functions. We conclude this article by briefly mentioning the similarity in the integrability structure of 4d gauge theories and 2d ones from an algebraic aspect, and a potential relation between two different integrable systems.