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Higher-form symmetries as higher automorphism bundles

Alonso Perez-Lona

TL;DR

The paper develops a cohesive mathematical framework for global higher-form symmetries using higher automorphism bundles in a cohesive ∞-topos, enabling a systematic treatment of symmetry actions and gauging for field theories valued in higher stacks. By constructing explicit symmetry actions for principal 2-bundles and specializing to G = U(1) and B fields, it clarifies how center and outer automorphism symmetries arise and how gaugings operate in this setting. The work applies these ideas to string 2-groups, including SU(n) and U(1) cases, and discusses the interplay between connections, holonomies, and gauging, with applications to instanton restrictions and 5d supergravity. It provides a unified viewpoint that recovers known results and reveals subtle distinctions between symmetries of bundles versus bundles with connection, highlighting the need for smooth higher-stack refinements in gauging. Overall, the framework offers a versatile, rigorous language for higher-form symmetries in gauge theories and their higher-categorical generalizations, with broad implications for mathematical physics and future explorations of connective and higher-gauge phenomena.

Abstract

The notion of global higher-form symmetries has received much attention, but leaves room for a more systematic mathematical formulation. In this article, we highlight the concept of higher automorphism bundles from the field of higher categorical differential geometry and higher gauge theory, and we demonstrate that this neatly reproduces and clarifies many examples and phenomena discussed in the literature. We rigorously construct the higher-form symmetries of pure gauge theory of a general strict Lie $2$-group, featuring center higher-form symmetries. We then apply this explicitly to several physically-relevant examples, such as $U(1)$ bundles, bundle gerbes, and certain string $2$-groups related to $SU(n)$ instanton restriction, and $5d$ supergravity. We elaborate on the nontrivial interplay between global higher-form symmetries, connection data, and symmetry gauging.

Higher-form symmetries as higher automorphism bundles

TL;DR

The paper develops a cohesive mathematical framework for global higher-form symmetries using higher automorphism bundles in a cohesive ∞-topos, enabling a systematic treatment of symmetry actions and gauging for field theories valued in higher stacks. By constructing explicit symmetry actions for principal 2-bundles and specializing to G = U(1) and B fields, it clarifies how center and outer automorphism symmetries arise and how gaugings operate in this setting. The work applies these ideas to string 2-groups, including SU(n) and U(1) cases, and discusses the interplay between connections, holonomies, and gauging, with applications to instanton restrictions and 5d supergravity. It provides a unified viewpoint that recovers known results and reveals subtle distinctions between symmetries of bundles versus bundles with connection, highlighting the need for smooth higher-stack refinements in gauging. Overall, the framework offers a versatile, rigorous language for higher-form symmetries in gauge theories and their higher-categorical generalizations, with broad implications for mathematical physics and future explorations of connective and higher-gauge phenomena.

Abstract

The notion of global higher-form symmetries has received much attention, but leaves room for a more systematic mathematical formulation. In this article, we highlight the concept of higher automorphism bundles from the field of higher categorical differential geometry and higher gauge theory, and we demonstrate that this neatly reproduces and clarifies many examples and phenomena discussed in the literature. We rigorously construct the higher-form symmetries of pure gauge theory of a general strict Lie -group, featuring center higher-form symmetries. We then apply this explicitly to several physically-relevant examples, such as bundles, bundle gerbes, and certain string -groups related to instanton restriction, and supergravity. We elaborate on the nontrivial interplay between global higher-form symmetries, connection data, and symmetry gauging.

Paper Structure

This paper contains 47 sections, 5 theorems, 363 equations.

Table of Contents

  1. Introduction
  2. General statement
  3. $\sigma$-models and symmetries.
  4. $\sigma$-models of smooth higher stacks.
  5. Higher gauge theory.
  6. Explicit construction for principal $2$-bundles
  7. Pure $G$ gauge theory
  8. Symmetries of principal $G$ bundles
  9. Automorphism $2$-groupoid.
  10. Symmetry action.
  11. $G=U(1)$
  12. Symmetry $2$-group.
  13. Action on Wilson loops and holonomies.
  14. Gauging.
  15. Non-faithfully acting ${\mathbb R}$ $1$-form symmetry.
  16. ...and 32 more sections

Key Result

Proposition 3.1

Let $\textbf{Sym}(\Sigma, \textbf{B}{\cal G})$ be the mapping $2$-groupoid from Definition def:sym2grpd. It is a Gray monoid under which all objects are invertible.

Theorems & Definitions (25)

  • Definition 3.1: Automorphism $2$-groupoid of a crossed module
  • Remark 3.1: Delooping of Drinfeld center $\cal Z(G)$ of $\cal G$
  • Definition 3.2: $2$-groupoid of principal $\cal G$ bundles (cf. Appendix \ref{['app:adjbundles']}, e.g. Rist:2022hci)
  • Definition 3.3: Symmetry $2$-groupoid $\textbf{Sym}\left(\Sigma,\textbf{B}{\cal G}\right)$
  • Remark 3.2: Flat center principal $\cal Z(G)$ bundles
  • Proposition 3.1: $\textbf{Sym}(\Sigma, \textbf{B}{\cal G})$ is a Gray monoid
  • proof
  • Proposition 3.2: Fields is a $\textbf{Sym}\left(\Sigma,\textbf{B}{\cal G}\right)$-module
  • proof
  • Remark 3.3: Center higher-form symmetry
  • ...and 15 more