Dual infrared limits of 6d $\cal N$=(2,0) theory

Compactifying type $A_{N-1}$ 6d ${\cal N}{=}(2,0)$ supersymmetric CFT on a product manifold $M^4\times\Sigma^2=M^3\times\tilde{S}^1\times S^1\times{\cal I}$ either over $S^1$ or over $\tilde{S}^1$ leads to maximally supersymmetric 5d gauge theories on $M^4\times{\cal I}$ or on $M^3\times\Sigma^2$, respectively. Choosing the radii of $S^1$ and $\tilde{S}^1$ inversely proportional to each other, these 5d gauge theories are dual to one another since their coupling constants $e^2$ and $\tilde{e}^2$ are proportional to those radii respectively. We consider their non-Abelian but non-supersymmetric extensions, i.e. SU($N$) Yang-Mills theories on $M^4\times{\cal I}$ and on $M^3\times\Sigma^2$, where $M^4\supset M^3=\mathbb R_t\times T_p^2$ with time $t$ and a punctured 2-torus, and ${\cal I}\subset\Sigma^2$ is an interval. In the first case, shrinking ${\cal I}$ to a point reduces to Yang-Mills theory or to the Skyrme model on $M^4$, depending on the method chosen for the low-energy reduction. In the second case, scaling down the metric on $M^3$ and employing the adiabatic method, we derive in the infrared limit a non-linear SU($N$) sigma model with a baby-Skyrme-type term on $\Sigma^2$, which can be reduced further to $A_{N-1}$ Toda theory.