On gauging Abelian extensions of finite and U(1) groups

TL;DR

This paper investigates the process of gauging Abelian symmetry extensions in quantum field theories, demonstrating equivalences in certain cases and exploring the topological and dual symmetry structures involved.

Contribution

It provides a detailed comparison of gauging procedures for Abelian extensions, especially for finite groups and U(1), and introduces differential cohomology to describe the resulting topological data.

Findings

Gauging the full extension is equivalent to sequential gauging for finite Abelian groups and U(1).

The dual symmetry in the continuous case fits into an extension related to magnetic U(1) symmetry.

Differential cohomology effectively describes the topological data of the gauged theories.

Abstract

We consider Abelian extensions of global symmetries of the form $A \to G \to K$, with $A$ finite (and similar higher-group structures). For a quantum field theory $\mathcal{T}$ with symmetry $G$, we compare gauging $G$ directly with gauging first $A$ and then $K$, and show that for finite Abelian groups and for $K \simeq U(1)$ the two procedures are equivalent as expected, $\mathcal{T}/G \simeq \mathcal{T}/A/K$. In the continuous case $K=U(1)$, after gauging the full extension, the dual symmetry $\widehat{\mathbb{Z}}_q^{(d-2)}$ fits into an extension characterizing the topological data of the magnetic $U(1)_m^{(d-3)}$ symmetry. This is better described using differential cohomology. We also briefly comment on the relation to symmetry fractionalization.