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On Gauging Symmetry of Modular Categories

Shawn X. Cui, César Galindo, Julia Yael Plavnik, Zhenghan Wang

TL;DR

The paper provides a rigorous higher-category formulation of gauging global symmetries of two-dimensional topological orders modeled by unitary modular tensor categories. It actionably constructs gauged theories in two steps: extend a symmetry to a $G$-crossed braided extension and then perform equivariantization, while deriving an explicit $H^4$ obstruction formula and establishing a sequential gauging mechanism. The work yields concrete results via two nontrivial examples, including the $\mathbb{Z}_2$ gauging of $\mathrm{Fib}^{\boxtimes 2}$ (leading to $\mathrm{SU}(2)_8$) and the $S_3$ gauging of $SO(8)_1$, showcasing how symmetry and topological order interplay to produce new UMTCs. These insights advance the classification and construction of symmetry-enriched topological phases and their gauged counterparts, with implications for both mathematics and condensed matter physics.

Abstract

Topological order of a topological phase of matter in two spacial dimensions is encoded by a unitary modular (tensor) category (UMC). A group symmetry of the topological phase induces a group symmetry of its corresponding UMC. Gauging is a well-known theoretical tool to promote a global symmetry to a local gauge symmetry. We give a mathematical formulation of gauging in terms of higher category formalism. Roughly, given a UMC with a symmetry group $G$, gauging is a 2-step process: first extend the UMC to a $G$-crossed braided fusion category and then take the equivariantization of the resulting category. Gauging can tell whether or not two enriched topological phases of matter are different, and also provides a way to construct new UMCs out of old ones. We derive a formula for the $H^4$-obstruction, prove some properties of gauging, and carry out gauging for two concrete examples.

On Gauging Symmetry of Modular Categories

TL;DR

The paper provides a rigorous higher-category formulation of gauging global symmetries of two-dimensional topological orders modeled by unitary modular tensor categories. It actionably constructs gauged theories in two steps: extend a symmetry to a -crossed braided extension and then perform equivariantization, while deriving an explicit obstruction formula and establishing a sequential gauging mechanism. The work yields concrete results via two nontrivial examples, including the gauging of (leading to ) and the gauging of , showcasing how symmetry and topological order interplay to produce new UMTCs. These insights advance the classification and construction of symmetry-enriched topological phases and their gauged counterparts, with implications for both mathematics and condensed matter physics.

Abstract

Topological order of a topological phase of matter in two spacial dimensions is encoded by a unitary modular (tensor) category (UMC). A group symmetry of the topological phase induces a group symmetry of its corresponding UMC. Gauging is a well-known theoretical tool to promote a global symmetry to a local gauge symmetry. We give a mathematical formulation of gauging in terms of higher category formalism. Roughly, given a UMC with a symmetry group , gauging is a 2-step process: first extend the UMC to a -crossed braided fusion category and then take the equivariantization of the resulting category. Gauging can tell whether or not two enriched topological phases of matter are different, and also provides a way to construct new UMCs out of old ones. We derive a formula for the -obstruction, prove some properties of gauging, and carry out gauging for two concrete examples.

Paper Structure

This paper contains 28 sections, 18 theorems, 28 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Unitary fusion categories
  4. Unitary braided fusion category and the center construction
  5. Unitary modular categories and the symmetric center
  6. Unitary categorical actions and their equivariantizations
  7. Gauging a Global Symmetry
  8. Global symmetry of unitary modular categories
  9. Symmetry defects
  10. Definition of gauging
  11. Obstructions for gauging and gauging data
  12. General properties
  13. Sequentially Gauging
  14. Obstructions
  15. $H^3$ obstruction
  16. ...and 13 more sections

Key Result

Proposition 1

DGNO2 A unitary $G$-crossed braided extension ${\mathcal{B}}_{G}^{\times}$ of a UMC $\mathcal{B}$ is modular if and only if $\mathcal{B}$ is modular.

Figures (2)

  • Figure 1: coherence for tri-homomorphism
  • Figure 2: Symmetry for $\text{SU}(3)_3$

Theorems & Definitions (29)

  • Remark 1
  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 1
  • Definition 4
  • Theorem 1
  • Definition 5
  • Proposition 2
  • Definition 6
  • ...and 19 more