1-form Symmetries of 4d N=2 Class S Theories

TL;DR

This paper determines the 1-form symmetry groups of 4d N=2 class S theories derived from 6d (2,0) SCFTs on Riemann surfaces, using both 6d and Type IIB perspectives, including cases with various punctures.

Contribution

It provides a comprehensive method to compute 1-form symmetries for class S theories with arbitrary punctures, linking 6d surface operators to 4d line operators and including Type IIB analysis.

Findings

Explicit formulas for 1-form symmetry groups in class S theories

Extension of results to irregular punctures

Dual description via Type IIB string theory

Abstract

We determine the 1-form symmetry group for any 4d N = 2 class S theory constructed by compactifying a 6d N=(2,0) SCFT on a Riemann surface with arbitrary regular untwisted and twisted punctures. The 6d theory has a group of mutually non-local dimension-2 surface operators, modulo screening. Compactifying these surface operators leads to a group of mutually non-local line operators in 4d, modulo screening and flavor charges. Complete specification of a 4d theory arising from such a compactification requires a choice of a maximal subgroup of mutually local line operators, and the 1-form symmetry group of the chosen 4d theory is identified as the Pontryagin dual of this maximal subgroup. We also comment on how to generalize our results to compactifications involving irregular punctures. Finally, to complement the analysis from 6d, we derive the 1-form symmetry from a Type IIB realization of class S theories.