On gauging finite subgroups

TL;DR

This paper explores how gauging finite normal Abelian subgroups in theories with symmetry $\Gamma$ affects the resulting symmetry structure, revealing a variety of possible symmetry types including higher-form and non-group symmetries.

Contribution

It classifies the possible symmetry outcomes after gauging finite Abelian subgroups, including novel non-group symmetries, and relates these to known obstructions and constructions in condensed matter and quantum field theory.

Findings

Symmetry can be a direct product of $\Gamma/A$ and a higher-form symmetry with a mixed anomaly.

Symmetry can be an extension of $G$ by a higher-form symmetry $\hat A$.

Existence of non-group symmetries beyond traditional group structures.

Abstract

We study in general spacetime dimension the symmetry of the theory obtained by gauging a non-anomalous finite normal Abelian subgroup $A$ of a $\Gamma$-symmetric theory. Depending on how anomalous $\Gamma$ is, we find that the symmetry of the gauged theory can be i) a direct product of $G=\Gamma/A$ and a higher-form symmetry $\hat A$ with a mixed anomaly, where $\hat A$ is the Pontryagin dual of $A$; ii) an extension of the ordinary symmetry group $G$ by the higher-form symmetry $\hat A$; iii) or even more esoteric types of symmetries which are no longer groups. We also discuss the relations to the effect called the $H^3(G,\hat A)$ symmetry localization obstruction in the condensed-matter theory and to some of the constructions in the works of Kapustin-Thorngren and Wang-Wen-Witten.