6d N = (1,0) theories on S¹ /T² and class S theories: part II

Abstract: We study the T² compactification of a class of 6d N=1,0$$ \mathcal{N}=\left(1,\;0\right) $$ theories that is Higgsable to N=2,0$$ \mathcal{N}=\left(2,\;0\right) $$ theories. We show that the resulting 4d N=2$$ \mathcal{N}=2 $$ 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 S¹. 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 SL2ℤ$$ \mathrm{S}\mathrm{L}\left(2,\mathbb{Z}\right) $$ 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.