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

arxiv（2023）

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.

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