Anomalies of $(1+1)D$ categorical symmetries

TL;DR

This paper introduces a method to detect anomalies in (1+1)D categorical symmetries using Lagrangian algebras in the Drinfeld center, linking topological order boundaries to symmetry anomaly conditions.

Contribution

It provides a general framework for identifying anomalies in fusion category symmetries through the existence of specific Lagrangian algebras, with computable obstructions and applications to Tambara-Yamagami categories.

Findings

Existence of magnetic Lagrangian algebra indicates non-anomalous symmetry.

Obstructions to anomaly-freedom can be explicitly computed in certain cases.

Application to $ ext{Drinfeld}$ centers of $ ext{Z}_N imes ext{Z}_N$ categories recovers known results.

Abstract

We present a general approach for detecting when a fusion category symmetry is anomalous, based on the existence of a special kind of Lagrangian algebra of the corresponding Drinfeld center. The Drinfeld center of a fusion category $\mathcal{A}$ describes a $(2+1)D$ topological order whose gapped boundaries enumerate all $(1+1)D$ gapped phases with the fusion category symmetry, which may be spontaneously broken. There always exists a gapped boundary, given by the \emph{electric} Lagrangian algebra, that describes a phase with $\mathcal{A}$ fully spontaneously broken. The symmetry defects of this boundary can be identified with the objects in $\mathcal{A}$. We observe that if there exists a different gapped boundary, given by a \emph{magnetic} Lagrangian algebra, then there exists a gapped phase where $\mathcal{A}$ is not spontaneously broken at all, which means that $\mathcal{A}$ is not anomalous. In certain cases, we show that requiring the existence of such a magnetic Lagrangian algebra leads to highly computable obstructions to $\mathcal{A}$ being anomaly-free. As an application, we consider the Drinfeld centers of $\mathbb{Z}_N\times\mathbb{Z}_N$ Tambara-Yamagami fusion categories and recover known results from the study of fiber functors.