Fusion Category Symmetry II: Categoriosities at c = 1 and Beyond

TL;DR

This paper investigates the complex fusion category symmetries in 1+1D quantum field theories, especially at central charge c=1, revealing continuous symmetries in theories with finite group-like symmetries and deriving new RG flow constraints.

Contribution

It introduces methods to identify fusion category symmetries in 1+1D CFTs and proves a Noether theorem linking these symmetries to non-local conserved currents.

Findings

Finite group-like symmetries can have continuous fusion category symmetries at c=1.

A Noether theorem relates fusion category symmetries to non-local conserved currents.

New constraints on RG flows between 1+1D CFTs are derived.

Abstract

We study generalized symmetries of quantum field theories in 1+1D generated by topological defect lines with no inverse. This paper follows our companion paper on gapped phases and anomalies associated with these symmetries. In the present work we focus on identifying fusion category symmetries, using both specialized 1+1D methods such as the modular bootstrap and (rational) conformal field theory (CFT), as well as general methods based on gauging finite symmetries, that extend to all dimensions. We apply these methods to $c = 1$ CFTs and uncover a rich structure. We find that even those $c = 1$ CFTs with only finite group-like symmetries can have continuous fusion category symmetries, and prove a Noether theorem that relates such symmetries in general to non-local conserved currents. We also use these symmetries to derive new constraints on RG flows between 1+1D CFTs.