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Notes on generalized global symmetries in QFT

E. Sharpe

TL;DR

This work argues that generalized global symmetries, including $1$-form and higher-form symmetries, are naturally realized as higher group (2-group and beyond) actions in quantum field theory, unifying these symmetries under a common mathematical framework.It surveys concrete realizations across gauge theories, WZW models, boundary Dijkgraaf-Witten theory, current algebras, and generalized moduli spaces (stacks), showing how Wilson lines, defects, and moduli structures encode higher-group actions through objects like ${\bf B}G$, ${\bf B}_{\flat}G_{\rm conn}$, and their extensions.The paper proposes that certain anomalies can be interpreted as transmutations of ordinary group symmetries into higher-group symmetries in the quantum theory, linking this perspective to WZW models, bosonization, and fibered constructions, and extends these ideas to generalized moduli spaces and higher current algebras.By providing a unifying higher-categorical viewpoint for defects, moduli, and anomalies, the work suggests new directions for understanding dualities, decomposition phenomena, and potential connections to elliptic genera and moonshine, while highlighting the need for further development of higher-group representation theory.

Abstract

It was recently argued that quantum field theories possess one-form and higher-form symmetries, labelled `generalized global symmetries.' In this paper, we describe how those higher-form symmetries can be understood mathematically as special cases of more general 2-groups and higher groups, and discuss examples of quantum field theories admitting actions of more general higher groups than merely one-form and higher-form symmetries. We discuss analogues of topological defects for some of these higher symmetry groups, relating some of them to ordinary topological defects. We also discuss topological defects in cases in which the moduli `space' (technically, a stack) admits an action of a higher symmetry group. Finally, we outline a proposal for how certain anomalies might potentially be understood as describing a transmutation of an ordinary group symmetry of the classical theory into a 2-group or higher group symmetry of the quantum theory, which we link to WZW models and bosonization.

Notes on generalized global symmetries in QFT

TL;DR

This work argues that generalized global symmetries, including $1$-form and higher-form symmetries, are naturally realized as higher group (2-group and beyond) actions in quantum field theory, unifying these symmetries under a common mathematical framework.It surveys concrete realizations across gauge theories, WZW models, boundary Dijkgraaf-Witten theory, current algebras, and generalized moduli spaces (stacks), showing how Wilson lines, defects, and moduli structures encode higher-group actions through objects like ${\bf B}G$, ${\bf B}_{\flat}G_{\rm conn}$, and their extensions.The paper proposes that certain anomalies can be interpreted as transmutations of ordinary group symmetries into higher-group symmetries in the quantum theory, linking this perspective to WZW models, bosonization, and fibered constructions, and extends these ideas to generalized moduli spaces and higher current algebras.By providing a unifying higher-categorical viewpoint for defects, moduli, and anomalies, the work suggests new directions for understanding dualities, decomposition phenomena, and potential connections to elliptic genera and moonshine, while highlighting the need for further development of higher-group representation theory.

Abstract

It was recently argued that quantum field theories possess one-form and higher-form symmetries, labelled `generalized global symmetries.' In this paper, we describe how those higher-form symmetries can be understood mathematically as special cases of more general 2-groups and higher groups, and discuss examples of quantum field theories admitting actions of more general higher groups than merely one-form and higher-form symmetries. We discuss analogues of topological defects for some of these higher symmetry groups, relating some of them to ordinary topological defects. We also discuss topological defects in cases in which the moduli `space' (technically, a stack) admits an action of a higher symmetry group. Finally, we outline a proposal for how certain anomalies might potentially be understood as describing a transmutation of an ordinary group symmetry of the classical theory into a 2-group or higher group symmetry of the quantum theory, which we link to WZW models and bosonization.

Paper Structure

This paper contains 20 sections, 67 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Overview of 2-groups and higher groups
  3. Examples of higher group symmetries in QFT
  4. Review of $q$-form symmetries in gauge theories
  5. Gauge theories with subgroup-invariant massless matter
  6. Boundary structures in Dijkgraaf-Witten theory
  7. WZW models
  8. Current algebras
  9. Other Kac-Moody actions
  10. Cosmological defects and generalized moduli spaces
  11. Generalities on defects and ordinary moduli spaces
  12. Generalized moduli spaces
  13. Analogues of Goldstone's theorem and generalized moduli spaces
  14. Anomalies as transmutation
  15. Two-dimensional cases
  16. ...and 5 more sections

Figures (1)

  • Figure 1: A 3-simplex.