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

TL;DR

This paper explores generalized discrete symmetries in 1+1D quantum field theories using fusion categories, classifies associated gapped phases, and develops techniques to analyze their anomalies and boundary conditions.

Contribution

It introduces a framework for understanding fusion category symmetries, classifies gapped phases, and provides methods to analyze anomalies and boundary conditions in 1+1D quantum field theories.

Findings

Classification of gapped phases stabilized by fusion category symmetries

Development of gauge theoretic techniques for Tambara-Yamagami categories

Examples of CFTs with fusion category symmetry from Kramers-Wannier dualities

Abstract

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. This is akin to gauging a discrete global symmetry at the boundary of Dijkgraaf-Witten theory. We describe how to "ungauge" the fusion category symmetry in these boundary conditions and separate the symmetry-preserving phases from the symmetry-breaking ones. For Tambara-Yamagami categories and their generalizations, which are associated with Kramers-Wannier-like self-dualities under orbifolding, we develop gauge theoretic techniques which simplify the analysis. We include some examples of CFTs with fusion category symmetry derived from Kramers-Wannier-like dualities as an appetizer for the Part II companion paper.