Anomalies of (1+1) -dimensional 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) -dimensional topological order whose gapped boundaries enumerate all (1+1) -dimensional gapped phases with the fusion category symmetry, which may be spontaneously broken. There always exists a gapped boundary, given by the 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 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 Z₍Z₍ Tambara-Yamagami fusion categories and recover known results from the study of fiber functors.