6d N = 1 , 0 $$ \mathcal{N}=\left(1,\;0\right) $$ theories on S 1 /T 2 and class S theories: part II

Abstract

We study the T 2 compactification of a class of 6d $$ \mathcal{N}=\left(1,\;0\right) $$ theories that is Higgsable to $$ \mathcal{N}=\left(2,\;0\right) $$ theories. We show that the resulting 4d $$ \mathcal{N}=2 $$ theory at the origin of the Coulomb branch and the parameter space is generically given by two superconformal matter sectors coupled by an infrared-free gauge multiplet and another conformal gauge multiplet. Our analysis utilizes the 5d theories obtained by putting the same class of 6d theories on S 1. Our class includes, among others, the 6d theories describing multiple M5 branes on an ALE singularity, and we analyze them in detail. The resulting 4d theory has manifestly both the $$ \mathrm{S}\mathrm{L}\left(2,\mathbb{Z}\right) $$ and the full flavor symmetry. We also discuss in detail the special cases of 6d theories where the infrared-free gauge multiplet is absent. In an appendix, we give a field-theoretical argument for an F-theoretic constraint that forbids a particular 6d anomaly-free matter content, as an application of our analysis.