Breaking bad theories of class $\mathcal S$

TL;DR

This paper investigates the structure of 4d $ ext{N}=2$ class $ ext{S}$ theories through their 3d mirror duals, introducing a new family of 'broken' bad theories to systematically analyze non-Lagrangian sectors and gauge groups.

Contribution

It introduces 'broken' bad theories in 3d $ ext{N}=4$ to analyze complex 4d class $ ext{S}$ theories and develops a method to identify non-Lagrangian sectors and gauge groups.

Findings

Identified a new family of 'broken' bad theories.

Developed an analytic method for non-Lagrangian sectors.

Explained emergence of weakly-coupled gauge groups via Higgs mechanisms.

Abstract

We study weakly-coupled descriptions/channel decompositions of the 4d $\mathcal{N}=2$ theories of class $\mathcal{S}$ of type $\mathfrak{su}(N)$, from the perspective of the 3d $\mathcal{N}=4$ mirror duals of their circle compactifications. This is a delicate problem when the channel decomposition produces pathological, or bad, 4d configurations that correspond to spheres with non-maximal punctures. The star-shaped quivers, describing the 3d mirrors associated with such bad 4d configurations, are bad 3d $\mathcal{N}=4$ theories. Leveraging recent results regarding 3d bad theories, we identify a new and interesting family of bad theories, which we coin \textit{broken} theories, that naturally arise in this context. Using these broken theories, we develop a systematic and analytic method that determines the generically non-Lagrangian matter sectors and the weakly-coupled gauge groups in such channel decompositions. We understand these weakly-coupled descriptions as emerging dynamically via Higgs mechanisms triggered by operators acquiring vacuum expectation values.