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

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.