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Lecture Notes on Symmetry Reduction via the Dressing Field Method

L. Ravera

Abstract

These notes - prepared for the conference school "Foundations of General-Relativistic Gauge Field Theory", held on March 17-19, 2026 at the Politecnico di Torino - present introductory material on symmetry reduction in general-relativistic Gauge Field Theory (gRGFT) via the Dressing Field Method (DFM). The DFM provides a systematic framework for extracting gauge- and diffeomorphism-invariant, manifestly relational, physical observables and degrees of freedom in gRGFT. A range of illustrative examples are discussed, spanning both Gauge Field Theory and general-relativistic settings. These include applications to non-Abelian Chern-Simons theory, Maxwell electromagnetism, the non-Abelian Higgs model, supersymmetric field theory, General Relativity, and scalar coordinatization.

Lecture Notes on Symmetry Reduction via the Dressing Field Method

Abstract

These notes - prepared for the conference school "Foundations of General-Relativistic Gauge Field Theory", held on March 17-19, 2026 at the Politecnico di Torino - present introductory material on symmetry reduction in general-relativistic Gauge Field Theory (gRGFT) via the Dressing Field Method (DFM). The DFM provides a systematic framework for extracting gauge- and diffeomorphism-invariant, manifestly relational, physical observables and degrees of freedom in gRGFT. A range of illustrative examples are discussed, spanning both Gauge Field Theory and general-relativistic settings. These include applications to non-Abelian Chern-Simons theory, Maxwell electromagnetism, the non-Abelian Higgs model, supersymmetric field theory, General Relativity, and scalar coordinatization.

Paper Structure

This paper contains 35 sections, 124 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Hole and point-coincidence arguments in a nutshell
  3. Dressing Field Method for gauge symmetries
  4. Dressed GFT kinematic
  5. Dressed dynamics in GFT
  6. Residual transformations of the 1st kind and transformations of the 2nd kind
  7. Dressing Field Method vs gauge-fixing
  8. Gauge-fixing
  9. Difference between dressing via the DFM and gauge-fixing
  10. Examples of applications of the DFM to gauge symmetries
  11. Three-dimensional non-Abelian Chern-Simons theory
  12. Maxwell electromagnetism and the "Lorenz gauge as a dressing" via the DFM
  13. The Abelian Higgs model without SSB
  14. Lorentz dressing in GR
  15. Dressing Field Method in supersymmetric field theory
  16. ...and 20 more sections

Figures (2)

  • Figure 1: Gauge-fixing in $\Phi$, i.e., a choice of local section $\sigma$ of $\Phi$. The gauge-fixing slice is the image of $\sigma$. Picture extracted from JTF-Ravera2025DFMSusyReviewMisu.
  • Figure 2: Difference between dressing as done via the DFM and gauge-fixing in field space $\Phi$. Picture extracted from JTF-Ravera2025DFMSusyReviewMisu.