Non-Abelian sigma models from Yang-Mills theory compactified on a circle

TL;DR

This paper explores how compactifying SU(N) Yang-Mills theory on a circle leads to a sigma model with a rich target space structure, revealing new dualities and geometric features in the low-energy limit.

Contribution

It demonstrates that the infrared limit of compactified Yang-Mills theory can produce sigma models with target spaces like SU(N)×SU(N)/Z_N, highlighting the role of gauge bundle framing choices.

Findings

Derivation of sigma models with complex target spaces from Yang-Mills theory.

Identification of the role of gauge bundle framing in low-energy limits.

Extension of results to general non-Abelian gauge groups.

Abstract

We consider SU($N$) Yang-Mills theory on ${\mathbb R}^{2,1}\times S^1$, where $S^1$ is a spatial circle. In the infrared limit of a small-circle radius the Yang-Mills action reduces to the action of a sigma model on ${\mathbb R}^{2,1}$ whose target space is a $2(N{-}1)$-dimensional torus modulo the Weyl-group action. We argue that there is freedom in the choice of the framing of the gauge bundles, which leads to more general options. In particular, we show that this low-energy limit can give rise to a target space SU$(N){\times}$SU$(N)/{\mathbb Z}_N$. The latter is the direct product of SU($N$) and its Langlands dual SU$(N)/{\mathbb Z}_N$, and it contains the above-mentioned torus as its maximal Abelian subgroup. An analogous result is obtained for any non-Abelian gauge group.