Global Form of Flavor Symmetry Groups in 4d N=2 Theories of Class S

TL;DR

This paper introduces a systematic method to determine the global form of flavor symmetry groups in 4d N=2 theories derived from 6d N=(2,0) SCFTs, enhancing understanding of their symmetry structures and dualities.

Contribution

The authors develop a method to deduce the global form of flavor symmetry groups in 4d N=2 theories from 6d origins, including cases with enhanced symmetries and dual descriptions.

Findings

Symmetry groups match with Lagrangian descriptions when available.

Method confirms consistency with known superconformal indices.

Identifies distinct global symmetry forms for similar SCFTs with identical invariants.

Abstract

We provide a systematic method to deduce the global form of flavor symmetry groups in 4d N=2 theories obtained by compactifying 6d N=(2,0) superconformal field theories (SCFTs) on a Riemann surface carrying regular punctures and possibly outer-automorphism twist lines. Apriori, this method only determines the group associated to the manifest part of the flavor symmetry algebra, but often this information is enough to determine the group associated to the full enhanced flavor symmetry algebra. Such cases include some interesting and well-studied 4d N=2 SCFTs like the Minahan-Nemeschansky theories. The symmetry groups obtained via this method match with the symmetry groups obtained using a Lagrangian description if such a description arises in some duality frame. Moreover, we check that the proposed symmetry groups are consistent with the superconformal indices available in the literature. As another application, our method finds distinct global forms of flavor symmetry group for pairs of interacting 4d N=2 SCFTs (recently pointed out in the literature) whose Coulomb branch dimensions, flavor algebras and levels coincide (along with other invariants), but nonetheless are distinct SCFTs.