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 $M^4timesSigma^2=M^3timestilde{S}^1times S^1times{cal I}$ either over $S^1$ or over $tilde{S}^1$ leads to maximally supersymmetric 5d gauge theories on $M^4times{cal I}$ or on $M^3timesSigma^2$, 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 $M^4times{cal I}$ and on $M^3timesSigma^2$, where $M^4supset M^3=mathbb R_ttimes T_p^2$ with time $t$ and a punctured 2-torus, and ${cal I}subsetSigma^2$ 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.