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

TL;DR

This work demonstrates that compactifying a subset of 6d ${ m N}=(1,0)$ theories on $T^2$ yields a natural class of 4d ${ m N}=2$ theories of class S of type $G$, specifically sphere theories with two full punctures and a simple puncture, realized as the 4d generalized bifundamental. The authors establish this via a multi-faceted program: a duality chain from M5 on ALE singularities, matching Coulomb-branch dimensions between 6d and 4d, and a Higgs-branch analysis that aligns with the ALE geometry; they also compute Seiberg-Witten curves and derive central charges using a new method for very Higgsable theories, finding agreement with known class S data. The central claim is supported by consistent cross-checks across 3d/4d pictures, anomaly inflow, and explicit examples (including $G={ m SU}(N)$, ${ m SO}(8)$, and $E_n$ cases), and the results provide a coherent framework linking 6d ${ m N}=(1,0)$ theories to class S constructions via $T^2$ compactification. The paper also outlines a program to extend these ideas to broader families of very Higgsable theories and to compactifications on more general Riemann surfaces, with implications for 4d SCFT landscapes and dualities.

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.