[Paper Review] Non-invertible Symmetries and Higher Representation Theory II

Thomas Bartsch, Mathew Bullimore+2 more|arXiv (Cornell University)|Dec 14, 2022

Homotopy and Cohomology in Algebraic Topology26 citations

TL;DR

This paper develops a higher group-theoretical framework for non-invertible symmetries arising from gauging finite higher groups and their subgroups, and provides concrete categorical descriptions in D=2,3,4.

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.

Motivation & Objective

• Motivate and formalize global categorical symmetries from gauging finite higher groups and higher subgroups with discrete torsion.
• Define higher group-theoretical fusion categories as symmetry categories arising from gauging anomaly-free subgroups of finite higher groups.
• Provide unified categorical descriptions of symmetry categories for 3D and 4D gauge theories related to so(N).
• Describe non-invertible symmetries from gauging 1-form symmetries with mixed ’t Hooft anomalies.
• Investigate the impact of discrete torsion on symmetry categories via obstruction theory and spectral sequences.

Proposed method

• Introduce the symmetry category (D-1)Vec^α(G) for a D-dimensional theory with finite group-like symmetry G and anomaly α ∈ Z^{D+1}(G,U(1)).
• Gauging an anomaly-free (D-1)-subgroup H ⊂ G with trivialisation ψ of α|_H, yielding the higher group-theoretical fusion category C(G,α|H,ψ).
• Describe simple objects, morphisms, and fusion rules as bimodules over an algebra object A(H,ψ) in Vec^α(G).
• Specialize to D=2,3,4 with detailed two-, three-, and four-dimensional case studies and discuss obstructions from Lyndon-Hochschild-Serre spectral sequence.
• Relate constructions to gapped boundary conditions of Dijkgraaf-Witten theories and to TQFT couplings with anomalous symmetries.

Experimental results

Research questions

• RQ1 How can non-invertible global symmetries in D>2 be captured by higher group-theoretical fusion categories?
• RQ2 How does gauging anomaly-free subgroups H of a finite group G, with trivialisation ψ, modify the symmetry category and its simple objects?
• RQ3 What is the role of projective (higher) representations in labeling simple objects after gauging?
• RQ4 How do discrete torsion and spectral sequence obstructions affect the existence and structure of symmetry categories?
• RQ5 What are the concrete categorical descriptions of symmetry categories for SO(N)-based gauge theories in D=3 and D=4, and how do they relate to TQFTs and anomaly cancellation?

Key findings

• Gauging an anomaly-free subgroup H of G with a trivialisation ψ yields a higher group-theoretical fusion category C(G,α|H,ψ).
• Simple objects in C(G,α|H,ψ) correspond to pairs (g, Φ_g) where g runs over double cosets Hackslash G/H and Φ_g is an irreducible projective representation of H_g = H ∩ gHg^{-1} with a specific 2-cocycle c_g built from α and ψ.
• Morphisms are intertwiners compatible with H-defect junctions, effectively giving Hom spaces as intertwiners between projective representations of H_g.
• Fusion is governed by the double coset ring Z[Hackslash G/H], with fusion rules preserving the defined support and computed from lifting double coset products to G.
• In D=4, the framework anticipates TQFT-valued fusion coefficients due to braiding of topological lines on 3D defects, and the approach links to gapped boundary conditions of DW theory with higher group data.
• Discrete torsion introduces obstructions captured by spectral sequence differentials, affecting the possible extensions and the resulting symmetry category structure.