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On gauging finite subgroups

Yuji Tachikawa

TL;DR

The paper develops a general framework for gauging a non-anomalous finite Abelian normal subgroup $A$ of a symmetry group $$ and analyzes how the remaining symmetry $G=/A$ combines with the dual higher-form symmetry $_{[D-2]}$ of the gauged theory. It identifies three possible outcomes depending on the anomaly structure: a direct product with a mixed $G$–$_{[D-2]}$ anomaly, an extension of $G$ by $_{[D-2]}$, or more exotic higher-categorical symmetry structures; it also connects these results to symmetry-localization obstructions and to Kapustin–Thorngren and Wang–Wen–Witten constructions. The work extends the discussion to higher-form extensions, incorporates ’t Hooft defects, and explores subtler effects in low and high dimensions, where the symmetry is described by fusion categories or higher categories rather than ordinary groups. Overall, the paper provides a cohesive, cohomology-based account of how gauging finite subgroups reshapes global symmetries, including non-invertible and higher-categorical aspects. It offers concrete prescriptions via the Lyndon–Hochschild–Serre spectral sequence and explicit examples, bridging field-theoretic and topological viewpoints.

Abstract

We study in general spacetime dimension the symmetry of the theory obtained by gauging a non-anomalous finite normal Abelian subgroup $A$ of a $Γ$-symmetric theory. Depending on how anomalous $Γ$ is, we find that the symmetry of the gauged theory can be i) a direct product of $G=Γ/A$ and a higher-form symmetry $\hat A$ with a mixed anomaly, where $\hat A$ is the Pontryagin dual of $A$; ii) an extension of the ordinary symmetry group $G$ by the higher-form symmetry $\hat A$; iii) or even more esoteric types of symmetries which are no longer groups. We also discuss the relations to the effect called the $H^3(G,\hat A)$ symmetry localization obstruction in the condensed-matter theory and to some of the constructions in the works of Kapustin-Thorngren and Wang-Wen-Witten.

On gauging finite subgroups

TL;DR

The paper develops a general framework for gauging a non-anomalous finite Abelian normal subgroup of a symmetry group and analyzes how the remaining symmetry combines with the dual higher-form symmetry of the gauged theory. It identifies three possible outcomes depending on the anomaly structure: a direct product with a mixed – anomaly, an extension of by , or more exotic higher-categorical symmetry structures; it also connects these results to symmetry-localization obstructions and to Kapustin–Thorngren and Wang–Wen–Witten constructions. The work extends the discussion to higher-form extensions, incorporates ’t Hooft defects, and explores subtler effects in low and high dimensions, where the symmetry is described by fusion categories or higher categories rather than ordinary groups. Overall, the paper provides a cohesive, cohomology-based account of how gauging finite subgroups reshapes global symmetries, including non-invertible and higher-categorical aspects. It offers concrete prescriptions via the Lyndon–Hochschild–Serre spectral sequence and explicit examples, bridging field-theoretic and topological viewpoints.

Abstract

We study in general spacetime dimension the symmetry of the theory obtained by gauging a non-anomalous finite normal Abelian subgroup of a -symmetric theory. Depending on how anomalous is, we find that the symmetry of the gauged theory can be i) a direct product of and a higher-form symmetry with a mixed anomaly, where is the Pontryagin dual of ; ii) an extension of the ordinary symmetry group by the higher-form symmetry ; iii) or even more esoteric types of symmetries which are no longer groups. We also discuss the relations to the effect called the symmetry localization obstruction in the condensed-matter theory and to some of the constructions in the works of Kapustin-Thorngren and Wang-Wen-Witten.

Paper Structure

This paper contains 27 sections, 80 equations, 4 figures.

Table of Contents

  1. Introduction and summary
  2. Organization of the note:
  3. Notations and conventions:
  4. Extension, anomaly and gauging
  5. Preliminaries: anomaly, topological defects and group cohomology
  6. From a nontrivial extension with a trivial anomaly
  7. From a trivial extension with a nontrivial anomaly
  8. Miscellaneous remarks
  9. From a nontrivial extension with a nontrivial anomaly
  10. Relation to other constructions
  11. Relation to Witten Witten:2016cio and Wang-Wen-Witten Wang:2017loc
  12. Relation to Kapustin-Thorngren Kapustin:2014lwaKapustin:2014zva
  13. When $\omega$ is in the image of $d_2$:
  14. When $\omega$ is in the image of $d_3$:
  15. Existence of finite Abelian extension where the anomaly trivializes
  16. ...and 12 more sections

Figures (4)

  • Figure 1: The anomalous phase is associated to the change in the topology of the domain walls on $X_D$ implementing the $G$ action.
  • Figure 2: The anomalous phase can be thought of as due to attaching a $(D+1)$-simplex of the bulk theory on $Y_{D+1}$.
  • Figure 3: Domain walls for $\Gamma$ as domain walls for $A\times G$ such that domain walls of $A$ can have boundaries as determined by the extension class $e$.
  • Figure 7: A curious skein relation in a 3d theory. Here a think line represents a $\mathbb{Z}_2$ line which is a boundary of a $\mathbb{Z}_2$ wall. When it protrudes across another $\mathbb{Z}_2$ wall, there is a minus sign.