Anomalies and gauging of U(1) symmetries

TL;DR

This paper introduces the Symmetry TFT framework for $U(1)$ symmetries in various dimensions, capturing anomalies and topological features, and explores its applications including non-invertible symmetries.

Contribution

It constructs the Symmetry TFT as a BF theory for $U(1)$ and $ ext{R}$ groups, and proposes an operation to derive the TFT after gauging the $U(1)$ symmetry, with numerous examples.

Findings

Symmetry TFT formulated as a BF theory for $U(1)$ and $ ext{R}$ groups.

Operation to obtain TFT after gauging $U(1)$ symmetry.

Derived the TFT for non-invertible $ ext{Q}/ ext{Z}$ chiral symmetry in 4D.

Abstract

We propose the Symmetry TFT for theories with a $U(1)$ symmetry in arbitrary dimension. The Symmetry TFT describes the structure of the symmetry, its anomalies, and the possible topological manipulations. It is constructed as a BF theory of gauge fields for groups $U(1)$ and $\mathbb{R}$, and contains a continuum of topological operators. We also propose an operation that produces the Symmetry TFT for the theory obtained by dynamically gauging the $U(1)$ symmetry. We discuss many examples. As an interesting outcome, we obtain the Symmetry TFT for the non-invertible $\mathbb{Q}/\mathbb{Z}$ chiral symmetry in four dimensions.