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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Author comments upon resubmission
In this paper, we are only considering 3-dimensional TQFTs of Turaev-Viro type when constructing topological symmetry defects in four dimensions, where the mathematical literature is less well-developed. Incorporating more general TQFTs would require the development of substantial new theoretical background, which is beyond the scope of this paper and is left to future work. As a result, our formalism only captures a subset of the topological defects constructed for example in 2111.01141. We clarified this point in the last paragraph of subsection 1.1 and the third paragraph of section 4.
List of changes
- Commented on the fact that we are only considering dressing with 3d TQFTs of Turaev-Viro type in D=4 in the last paragraph of subsection 1.1 and the third paragraph of section 4 , as incorporating more general TQFTs is beyond the scope of this paper.
