6d $\mathcal{N}=(1,0)$ theories on $T^2$ and class S theories: part I

Abstract: We show that the $\mathcal{N}=(1,0)$ superconformal theory on a single M5 brane on the ALE space of type $G=A_n, D_n, E_n$, when compactified on $T^2$, becomes a class S theory of type $G$ on a sphere with two full punctures and a simple puncture. We study this relation from various viewpoints. Along the way, we develop a new method to study the 4d SCFT arising from the $T^2$ compactification of a class of 6d $\mathcal{N}=(1,0)$ theories we call very Higgsable.