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Representation theory for categorical symmetries

Thomas Bartsch, Mathew Bullimore, Andrea Grigoletto

TL;DR

The paper develops a higher-categorical representation theory for non-invertible (categorical) symmetries acting on extended operators in quantum field theory. It introduces higher tube categories and higher tube algebras to encode how twisted-sector operators transform under actions of topological symmetry defects across dimensions, and ties these representations to bulk theories via the sandwich construction and factorization homology, including TV_C and its Drinfeld center. Through detailed 2D and 3D examples—group and Ising-like symmetries, 2-groups, and braided lines—it demonstrates that genuine and twisted sectors organize into representations of higher tube algebras, with explicit classifying data given by centralizers, cohomology, and condensations. The framework unifies invertible and non-invertible cases, clarifies the role of the Müger center and braided module categories, and provides a robust mathematical toolkit for analyzing how extended operators transform under higher symmetries in D ≥ 2.

Abstract

This paper addresses the question of how categorical symmetries act on extended operators in quantum field theory. Building on recent results in two dimensions, we introduce higher tube categories and algebras associated to higher fusion category symmetries. We show that twisted sector extended operators transform in higher representations of higher tube algebras and interpret this result from the perspective of the sandwich construction of finite symmetries via the Drinfeld center. Focusing on three dimensions, we discuss a variety of examples to illustrate the general constructions. In the case of invertible symmetries, we show that higher tube algebras are higher analogues of twisted Drinfeld doubles of finite groups, generalising known constructions in two dimensions. Building on this foundation, we discuss non-invertible Ising-like symmetry categories obtained by gauging finite subgroups. We also consider non-invertible topological symmetry lines described by braided fusion categories and discuss connections to the Müger center and braided module categories.

Representation theory for categorical symmetries

TL;DR

The paper develops a higher-categorical representation theory for non-invertible (categorical) symmetries acting on extended operators in quantum field theory. It introduces higher tube categories and higher tube algebras to encode how twisted-sector operators transform under actions of topological symmetry defects across dimensions, and ties these representations to bulk theories via the sandwich construction and factorization homology, including TV_C and its Drinfeld center. Through detailed 2D and 3D examples—group and Ising-like symmetries, 2-groups, and braided lines—it demonstrates that genuine and twisted sectors organize into representations of higher tube algebras, with explicit classifying data given by centralizers, cohomology, and condensations. The framework unifies invertible and non-invertible cases, clarifies the role of the Müger center and braided module categories, and provides a robust mathematical toolkit for analyzing how extended operators transform under higher symmetries in D ≥ 2.

Abstract

This paper addresses the question of how categorical symmetries act on extended operators in quantum field theory. Building on recent results in two dimensions, we introduce higher tube categories and algebras associated to higher fusion category symmetries. We show that twisted sector extended operators transform in higher representations of higher tube algebras and interpret this result from the perspective of the sandwich construction of finite symmetries via the Drinfeld center. Focusing on three dimensions, we discuss a variety of examples to illustrate the general constructions. In the case of invertible symmetries, we show that higher tube algebras are higher analogues of twisted Drinfeld doubles of finite groups, generalising known constructions in two dimensions. Building on this foundation, we discuss non-invertible Ising-like symmetry categories obtained by gauging finite subgroups. We also consider non-invertible topological symmetry lines described by braided fusion categories and discuss connections to the Müger center and braided module categories.
Paper Structure (70 sections, 187 equations, 59 figures, 1 table)

This paper contains 70 sections, 187 equations, 59 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Two dimensions
  4. Extended operators
  5. Synthesis
  6. Summary
  7. Higher tube categories
  8. Higher tube algebras
  9. Sandwich construction
  10. Group-theoretical symmetries
  11. Outline
  12. Two dimensions
  13. Fusion category symmetry
  14. Connected components
  15. Twisted sector operators
  16. ...and 55 more sections

Figures (59)

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  • ...and 54 more figures