Discrete torsion in gauging non-invertible symmetries

TL;DR

This paper extends the concept of discrete torsion to non-invertible symmetries in 2D quantum field theories, revealing two different generalizations and their implications for gauge actions and algebra twists.

Contribution

It introduces two complementary generalizations of discrete torsion for non-invertible symmetries, clarifies their mathematical structure, and connects them to gauge actions on B fields.

Findings

Two types of discrete torsion generalizations counted by H^2(G,U(1)

One generalization encodes actions on B fields, extending orbifold results

Generalization leads to new twists on gaugeable algebras and fiber functors

Abstract

In this paper we discuss generalizations of discrete torsion to noninvertible symmetries in 2d QFTs. One point of this paper is to explain that there are two complementary generalizations. Both generalizations are counted by $H^2(G,U(1))$ when one specializes to ordinary finite groups $G$. However, the counting is different for more general fusion categories. Furthermore, only one generalizes the picture of discrete torsion as differences in choices of gauge actions on B fields. Explaining this in detail, how one of the generalizations of discrete torsion to noninvertible cases encodes actions on B fields, is the other point of this paper. In particular, this generalizes old results in ordinary orbifolds that discrete torsion is a choice of group action on the B field. We also explain how this same generalization of discrete torsion gives rise to physically-sensible twists on gaugeable algebras and fiber functors.