Bethe/Gauge Correspondence for linear quiver theories with ABCD gauge symmetry and spin chains

TL;DR

This paper extends the Bethe/gauge correspondence to linear quiver theories with ABCD gauge symmetries, establishing a link between supersymmetric vacua and eigenstates of open XXZ spin chains, including general A_r quivers.

Contribution

It generalizes the Bethe/gauge correspondence to include ABCD gauge groups and arbitrary A_r quivers, connecting gauge theory vacua with integrable spin chain eigenstates.

Findings

Established correspondence for A2 quiver and sl3 open XXZ spin chain.

Extended the framework to general A_r quiver gauge theories.

Demonstrated matching between vacuum equations and Bethe Ansatz equations.

Abstract

This note is an extension of [DZ23] there the supersymmetric vacua of three-dimensional $\mathcal{N}=2$ gauge theories with matter are shown to be in one-to-one correspondence with the eigenstate of $\text{XXZ}$ integrable spin chain Hamiltonians with open boundary conditions. We consider the $A_{2}$ quiver gauge theory, which is the simplest non-trivial quiver gauge theory, and $sl_{3}$ open $\text{XXZ}$ spin chain with diagonal boundary condition. We demonstrate the correspondence between the vacuum equations of different gauge groups and Bethe Ansatz equations with different boundary parameters. Not only that, but we furthermore push forward the program to the general $A_{r}$ quiver gauge theory.