# Fusion category symmetry. Part I. Anomaly in-flow and gapped phases

“Fusion category symmetry. Part I. Anomaly in-flow and gapped phases” (2024), by Ryan Thorngren et al. Mathematics research paper. Original abstract below, unchanged. A free PDF is available.

- Authors: Ryan Thorngren, Yifan Wang
- Published: April 24, 2024
- Field: Mathematics
- DOI: https://doi.org/10.1007/jhep04(2024)132
- Free PDF: https://link.springer.com/content/pdf/10.1007/JHEP04(2024)132.pdf

## Abstract

A bstract 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.

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