Non-invertible Symmetries and Higher Representation Theory II

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.

56 pages + appendix, v2: clarifications and citations added

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• Gapped Phases with Non-Invertible Symmetries: (1+1)d
• Anomalies of Non-Invertible Symmetries in (3+1)d
• Non-invertible and higher-form symmetries in 2+1d lattice gauge theories
• The Club Sandwich: Gapless Phases and Phase Transitions with Non-Invertible Symmetries
• Gauging modulated symmetries: Kramers-Wannier dualities and non-invertible reflections
• Generalized Charges, Part II: Non-Invertible Symmetries and the Symmetry TFT
• Emergent generalized symmetries in ordered phases and applications to quantum disordering
• Non-invertible symmetry-protected topological order in a group-based cluster state
• Hasse Diagrams for Gapless SPT and SSB Phases with Non-Invertible Symmetries
• Categorical Symmetry of the Standard Model from Gravitational Anomaly
• Fermionic Non-Invertible Symmetries in (1+1)d: Gapped and Gapless Phases, Transitions, and Symmetry TFTs
• Boundary SymTFT
• Fiber 2-Functors and Tambara-Yamagami Fusion 2-Categories
• Generalized symmetries in singularity-free nonlinear $\sigma$ models and their disordered phases
• Symmetry fractionalization, mixed-anomalies and dualities in quantum spin models with generalized symmetries
• Symmetry fractionalization and duality defects in Maxwell theory
• The 3d $A$-model and generalised symmetries, Part I: bosonic Chern-Simons theories
• (2+1)d Lattice Models and Tensor Networks for Gapped Phases with Categorical Symmetry