Fusion Category Symmetry I: Anomaly In-Flow and Gapped Phases

Understand

We study generalized discrete symmetries of quantum field theories in 1+1D generated by topological defect lines with no inverse.

• In particular, we describe 't Hooft anomalies and classify gapped phases stabilized by these symmetries, including new 1+1D topological phases.
• The algebra of these operators is not a group but rather is described by their fusion ring and crossing relations, captured algebraically as a fusion category.
• Such data defines a Turaev-Viro/Levin-Wen model in 2+1D, while a 1+1D system with this fusion category acting as a global symmetry defines a boundary condition.