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

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.

25 pages, one figure

Cited by in corpus (1)

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