Anomalies of (1+1)D categorical symmetries

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 $A$ 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 $A$ fully spontaneously broken. The symmetry defects of this boundary can be identified with the objects in $A$. We observe that if there exists a different gapped boundary, given by a \emph{magnetic} Lagrangian algebra, then there exists a gapped phase where $A$ is not spontaneously broken at all, which means that $A$ is not anomalous. In certain cases, we show that requiring the existence of such a magnetic Lagrangian algebra leads to highly computable obstructions to $A$ 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.

This seminar will be virtual.

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