Non-invertible Symmetries and Higher Representation Theory II

TL;DR

This paper advances the understanding of non-invertible symmetries in higher-dimensional gauge theories by proposing a higher group-theoretical framework and analyzing the effects of discrete torsion on symmetry categories.

Contribution

It introduces a unified categorical approach to non-invertible symmetries via higher group representations and applies it to gauge theories with discrete torsion and anomalies.

Findings

Symmetry categories can be described as higher group-theoretical fusion categories.

Gauging higher subgroups yields non-invertible symmetries characterized by projective higher representations.

Discrete torsion influences symmetry categories through spectral sequence obstructions.

Abstract

In this paper we continue our investigation of the global categorical symmetries that arise when gauging finite higher groups and their higher subgroups with discrete torsion. The motivation is to provide a common perspective on the construction of non-invertible global symmetries in higher dimensions and a precise description of the associated symmetry categories. We propose that the symmetry categories obtained by gauging higher subgroups may be defined as higher group-theoretical fusion categories, which are built from the projective higher representations of higher groups. As concrete applications we provide a unified description of the symmetry categories of gauge theories in three and four dimensions based on the Lie algebra $\mathfrak{so}(N)$, and a fully categorical description of non-invertible symmetries obtained by gauging a 1-form symmetry with a mixed 't Hooft anomaly. We also discuss the effect of discrete torsion on symmetry categories, based a series of obstructions determined by spectral sequence arguments.