Non-invertible Symmetries and Higher Representation Theory II

TL;DR

This work extends the program of understanding global categorical symmetries in quantum field theory by systematically gauging finite higher groups and their subgroups across D=2,3,4. It introduces and leverages higher group-theoretical fusion categories, arising from projective higher representations, to capture non-invertible symmetry content and their fusion, morphism, and object structures. The paper provides concrete realizations in 2D, 3D, and 4D, including detailed case studies for groups such as Z_4, D_8, and D_4-related 2-groups, and connects these constructions to gapped boundaries of Dijkgraaf–Witten theories and to higher representation theory (2- and 3-representations). A central technical feature is the use of spectral sequences to organize obstructions to gauging and to track how discrete torsion and Postnikov data modify symmetry categories, leading to a rich landscape of non-invertible defects and duality phenomena. The results illuminate how higher-categorical symmetries encode both the topology of gauged theories and their potential boundary conditions, with applications to so(N) gauge theories and setups with mixed anomalies, thereby enriching the toolkit for classifying and constructing non-invertible symmetries in higher dimensions.

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.

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