  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.
