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Non-invertible Symmetries and Higher Representation Theory I

Thomas Bartsch, Mathew Bullimore, Andrea E. V. Ferrari, Jamie Pearson

TL;DR

The paper develops a unified higher-categorical framework for global symmetries arising from gauging finite higher groups in D≥3, showing that the full spectrum of topological defects is governed by (D−1)-representations. In D=3 it provides a detailed construction demonstrating that the symmetry category is the fusion 2-category 2Rep(G) for finite groups and split 2-groups G, with explicit classifications of objects, 1-morphisms, and fusion rules. It also connects these structures to condensation defects and non-invertible symmetries in gauge theories with disconnected gauge groups, and outlines generalisations to higher groups and Postnikov data. The results offer a concrete, representation-theoretic lens on non-invertible symmetries in higher dimensions and set the stage for broader explorations in subsequent work. The work has potential implications for understanding symmetry structures in 3D and higher gauge theories, including Langlands duality and anomaly inflow phenomena.

Abstract

The purpose of this paper is to investigate the global categorical symmetries that arise when gauging finite higher groups in three or more dimensions. The motivation is to provide a common perspective on constructions of non-invertible global symmetries in higher dimensions and a precise description of the associated symmetry categories. This paper focusses on gauging finite groups and split 2-groups in three dimensions. In addition to topological Wilson lines, we show that this generates a rich spectrum of topological surface defects labelled by 2-representations and explain their connection to condensation defects for Wilson lines. We derive various properties of the topological defects and show that the associated symmetry category is the fusion 2-category of 2-representations. This allows us to determine the full symmetry categories of certain gauge theories with disconnected gauge groups. A subsequent paper will examine gauging more general higher groups in higher dimensions.

Non-invertible Symmetries and Higher Representation Theory I

TL;DR

The paper develops a unified higher-categorical framework for global symmetries arising from gauging finite higher groups in D≥3, showing that the full spectrum of topological defects is governed by (D−1)-representations. In D=3 it provides a detailed construction demonstrating that the symmetry category is the fusion 2-category 2Rep(G) for finite groups and split 2-groups G, with explicit classifications of objects, 1-morphisms, and fusion rules. It also connects these structures to condensation defects and non-invertible symmetries in gauge theories with disconnected gauge groups, and outlines generalisations to higher groups and Postnikov data. The results offer a concrete, representation-theoretic lens on non-invertible symmetries in higher dimensions and set the stage for broader explorations in subsequent work. The work has potential implications for understanding symmetry structures in 3D and higher gauge theories, including Langlands duality and anomaly inflow phenomena.

Abstract

The purpose of this paper is to investigate the global categorical symmetries that arise when gauging finite higher groups in three or more dimensions. The motivation is to provide a common perspective on constructions of non-invertible global symmetries in higher dimensions and a precise description of the associated symmetry categories. This paper focusses on gauging finite groups and split 2-groups in three dimensions. In addition to topological Wilson lines, we show that this generates a rich spectrum of topological surface defects labelled by 2-representations and explain their connection to condensation defects for Wilson lines. We derive various properties of the topological defects and show that the associated symmetry category is the fusion 2-category of 2-representations. This allows us to determine the full symmetry categories of certain gauge theories with disconnected gauge groups. A subsequent paper will examine gauging more general higher groups in higher dimensions.
Paper Structure (85 sections, 28 theorems, 344 equations, 35 figures)

This paper contains 85 sections, 28 theorems, 344 equations, 35 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Summary of Results
  4. Groups
  5. 2-Groups
  6. Gauge Theories
  7. Outline
  8. Two dimensions: finite group
  9. Finite groups
  10. Gauging a finite group
  11. Discrete torsion
  12. Two dimensions: semi-direct product
  13. Semi-direct product
  14. Gauging a semi-direct product
  15. Objects
  16. ...and 70 more sections

Key Result

Proposition A.14

There is a 1:1-correspondence between 2-groups and crossed modules.

Figures (35)

  • Figure 1:
  • Figure 2:
  • Figure 3:
  • Figure 4:
  • Figure 5:
  • ...and 30 more figures

Theorems & Definitions (91)

  • Definition A.1
  • Definition A.2
  • Example A.3
  • Definition A.4
  • Definition A.5
  • Definition A.6
  • Definition A.7
  • Definition A.8
  • Definition A.9
  • Definition A.10
  • ...and 81 more