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An Introduction to Higher-Form Symmetries

Pedro R. S. Gomes

TL;DR

The notes develop higher-form and generalized symmetries, showing how extended objects such as lines and surfaces, rather than local operators, can carry charges and be topologically linked to symmetry generators. Starting from ordinary Noetherian symmetries, they extend to $1$-form and $q$-form symmetries, illustrating with Maxwell, Chern-Simons, and non-Abelian gauge theories how Wilson lines, ’t Hooft lines, and their algebras encode symmetry data, anomalies, and phase structure. A central thread is the topological interpretation of charges via linking numbers and the emergence of Goldstone-like modes (e.g., the photon) in spontaneously broken continuous higher-form symmetries, along with applications to confinement and dualities in 3D and 4D. The framework connects to fracton physics and topological order, offering a unified language for constraints, dualities, and the spectrum of gauge theories across dimensions. Overall, the work highlights how generalized symmetries refine our understanding of IR dynamics, phase structure, and the role of topology in quantum field theory.

Abstract

These notes are intended to be a pedagogical introduction to higher-form symmetries, which are symmetries whose charged objects are extended operators supported on lines, surfaces, and etc. This subject has been one of the most popular and effervescent topics of theoretical physics in recent years. Gauge theories are central in the study of higher-form symmetries, with Wilson and 't Hooft operators corresponding to the charged objects. Along these notes, we discuss in detail some basic aspects, including Abelian Maxwell and Chern-Simons theories, and $SU(N)$ non-Abelian gauge theories. We also discuss spontaneous breaking of higher-form symmetries.

An Introduction to Higher-Form Symmetries

TL;DR

The notes develop higher-form and generalized symmetries, showing how extended objects such as lines and surfaces, rather than local operators, can carry charges and be topologically linked to symmetry generators. Starting from ordinary Noetherian symmetries, they extend to -form and -form symmetries, illustrating with Maxwell, Chern-Simons, and non-Abelian gauge theories how Wilson lines, ’t Hooft lines, and their algebras encode symmetry data, anomalies, and phase structure. A central thread is the topological interpretation of charges via linking numbers and the emergence of Goldstone-like modes (e.g., the photon) in spontaneously broken continuous higher-form symmetries, along with applications to confinement and dualities in 3D and 4D. The framework connects to fracton physics and topological order, offering a unified language for constraints, dualities, and the spectrum of gauge theories across dimensions. Overall, the work highlights how generalized symmetries refine our understanding of IR dynamics, phase structure, and the role of topology in quantum field theory.

Abstract

These notes are intended to be a pedagogical introduction to higher-form symmetries, which are symmetries whose charged objects are extended operators supported on lines, surfaces, and etc. This subject has been one of the most popular and effervescent topics of theoretical physics in recent years. Gauge theories are central in the study of higher-form symmetries, with Wilson and 't Hooft operators corresponding to the charged objects. Along these notes, we discuss in detail some basic aspects, including Abelian Maxwell and Chern-Simons theories, and non-Abelian gauge theories. We also discuss spontaneous breaking of higher-form symmetries.
Paper Structure (43 sections, 369 equations, 13 figures)

This paper contains 43 sections, 369 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Aspects of Ordinary Symmetries
  3. Symmetries in Classical Field Theory
  4. Symmetries in Quantum Field Theory
  5. Canonical Formalism
  6. Path Integral and Ward Identities
  7. Spontaneous Symmetry Breaking and Goldstone Excitations
  8. Rephrasing Ordinary Symmetries in Terms of Topology
  9. Assigning Topological Meaning to Charges
  10. Higher-Form Symmetries
  11. Charged Operators
  12. Case Study: 1-Form Symmetry in $D=2$
  13. Case Study: 1-Form Symmetry in $D=3$
  14. Case Study: 1-Form Symmetry in $D=4$
  15. Generalization: $q$-form Symmetries
  16. ...and 28 more sections

Figures (13)

  • Figure 1: Link of spheres $S^{D-1}$ and a point.
  • Figure 2: Sequence of deformations to associate a closed surface $\Sigma$ to $U(g,\Sigma)$.
  • Figure 3: Intersection of the line $\mathcal{C}$ with the spatial point $x^1$.
  • Figure 4: Link of the curve $\mathcal{C}$ with the $S^0$ sphere.
  • Figure 5: Topological meaning of $\text{Link}(S^0,\mathcal{C})$.
  • ...and 8 more figures