Langlands Dualities through Bethe/Gauge Correspondence for 3d Gauge Theories

TL;DR

This paper explores the Bethe/Gauge correspondence for 3d and 2d gauge theories related to non-simple laced Lie algebras, revealing new dualities and a boundary-spin effect that connects gauge theories with spin chains.

Contribution

It introduces a novel Bethe/Gauge correspondence for non-simple laced Lie algebras and uncovers a boundary-spin effect related to Langlands duality.

Findings

Bethe/Gauge correspondence extended to non-simple laced Lie algebras.

Discovery of a boundary-spin effect reversing spins at boundaries.

Effective superpotentials exhibit Langlands duality.

Abstract

For non-simple laced Lie algebras, the $\text{B}_{N}$ and $\text{C}_{N}$ are Langlands dual to each other in mathematical. In this article, we give another Bethe/Gauge correspondence between 3d (or 2d) classical Lie group supersymmetry gauge theory with closed and open $\text{XXZ}$ (or $\text{XXX}$) spin chain. Here, the representations of the $\text{ADE}$ Lie algebras are self-dual, and while for the non-simple laced Lie algebras $\text{B}_{N}$ and $\text{C}_{N}$, their roles are exchanged in contrast with the results in \cite{DZ23a}. From Bethe/Gauge correspondence point of view, the two types of the effective superpotentials are Langlands duality to each other. For the $\text{B}_{N}$-type Lie algebra, a remarkable feature is that, to fix the spin sites by boundaries through Bethe/Gauge, the spins of the sites will be reversed. This is similarly to the so called electron-hole effect, we call this as a boundary-spin effect, a new kind of duality.