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

TL;DR

This paper investigates the compactification of 6d $ ext{(1,0)}$ theories to 4d $ ext{(2,0)}$ theories, revealing a structure of coupled superconformal sectors and gauge multiplets, with detailed analysis of theories including M5 branes on ALE singularities.

Contribution

It provides a detailed analysis of the $T^2$ compactification of 6d $ ext{(1,0)}$ theories, identifying their 4d structure as coupled superconformal sectors and gauge multiplets, including special cases and anomaly constraints.

Findings

4d theories are coupled superconformal matter sectors with gauge multiplets.

The resulting theories exhibit $ ext{SL}(2, ext{Z})$ and full flavor symmetry.

Analysis includes theories from multiple M5 branes on ALE singularities.

Abstract

We study the $T^2$ compactification of a class of 6d $\mathcal{N}{=}(1,0)$ theories that is Higgsable to $\mathcal{N}{=}(2,0)$ 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 $\mathrm{SL}(2,\mathbb{Z})$ 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.