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The dressing field method for diffeomorphisms: a relational framework

Jordan T. Francois Andre

TL;DR

This work extends the dressing field method to theories with diffeomorphism symmetry, embedding the formalism in the bundle geometry of field space and clarifying the distinction between field-independent and field-dependent diffeomorphisms. It shows how to construct Diff(M)-invariant, relational observables and presymplectic data, and connects the BKSS extended bracket to the Frölicher-Nijenhuis bracket, enabling a clean interpretation of Noether charges and their brackets within covariant phase space. The formalism naturally yields a relational reformulation of general relativity, unifying edge modes, gravitational dressings, and dynamical reference frames under a single geometric umbrella. It also analyzes residual symmetries and the role of dressing-field ambiguities, highlighting when dressed quantities carry physical meaning and when they merely repackage gauge freedom. Overall, the paper provides a rigorous, geometry-driven pathway to read diffeomorphism-invariant physics in a relational language, with practical implications for gravity, edge modes, and the construction of observables.

Abstract

The dressing field method is a tool to reduce gauge symmetries. Here we extend it to cover the case of diffeomorphisms. The resulting framework is a systematic scheme to produce Diff(M)-invariant objects, which has a natural relational interpretation. Its precise formulation relies on a clear understanding of the bundle geometry of field space. By detailing it, among other things we stress the geometric nature of field-independent and field-dependent diffeomorphisms, and highlight that the heuristic "extended bracket" for field-dependent vector fields often featuring in the covariant phase space literature can be understood as arising from the Frölicher-Nijenhuis bracket. Furthermore, by articulating this bundle geometry with the covariant phase space approach, we give a streamlined account of the elementary objects of the (pre)symplectic structure of a Diff(M)-theory: Noether charges and their bracket, as induced by the standard prescription for the presymplectic potential and 2-form. We give conceptually transparent expressions allowing to read the integrability conditions and the circumstances under which the bracket of charge is Lie, and the resulting Poisson algebras of charges are central extensions of the Lie algebras of field-independent ($\mathfrak{diff}(M)$) and field-dependent vector fields. We show that, applying the dressing field method, one obtains a Diff(M)-invariant and manifestly relational formulation of a general relativistic field theory. Relying on results just mentioned, we easily derive the "dressed" (relational) presymplectic structure of the theory. This reproduces or extends results from the gravitational edge mode and gravitational dressing literature. In addition to simplified technical derivations, the conceptual clarity of the framework supplies several insights and allows us to dispel misconceptions.

The dressing field method for diffeomorphisms: a relational framework

TL;DR

This work extends the dressing field method to theories with diffeomorphism symmetry, embedding the formalism in the bundle geometry of field space and clarifying the distinction between field-independent and field-dependent diffeomorphisms. It shows how to construct Diff(M)-invariant, relational observables and presymplectic data, and connects the BKSS extended bracket to the Frölicher-Nijenhuis bracket, enabling a clean interpretation of Noether charges and their brackets within covariant phase space. The formalism naturally yields a relational reformulation of general relativity, unifying edge modes, gravitational dressings, and dynamical reference frames under a single geometric umbrella. It also analyzes residual symmetries and the role of dressing-field ambiguities, highlighting when dressed quantities carry physical meaning and when they merely repackage gauge freedom. Overall, the paper provides a rigorous, geometry-driven pathway to read diffeomorphism-invariant physics in a relational language, with practical implications for gravity, edge modes, and the construction of observables.

Abstract

The dressing field method is a tool to reduce gauge symmetries. Here we extend it to cover the case of diffeomorphisms. The resulting framework is a systematic scheme to produce Diff(M)-invariant objects, which has a natural relational interpretation. Its precise formulation relies on a clear understanding of the bundle geometry of field space. By detailing it, among other things we stress the geometric nature of field-independent and field-dependent diffeomorphisms, and highlight that the heuristic "extended bracket" for field-dependent vector fields often featuring in the covariant phase space literature can be understood as arising from the Frölicher-Nijenhuis bracket. Furthermore, by articulating this bundle geometry with the covariant phase space approach, we give a streamlined account of the elementary objects of the (pre)symplectic structure of a Diff(M)-theory: Noether charges and their bracket, as induced by the standard prescription for the presymplectic potential and 2-form. We give conceptually transparent expressions allowing to read the integrability conditions and the circumstances under which the bracket of charge is Lie, and the resulting Poisson algebras of charges are central extensions of the Lie algebras of field-independent () and field-dependent vector fields. We show that, applying the dressing field method, one obtains a Diff(M)-invariant and manifestly relational formulation of a general relativistic field theory. Relying on results just mentioned, we easily derive the "dressed" (relational) presymplectic structure of the theory. This reproduces or extends results from the gravitational edge mode and gravitational dressing literature. In addition to simplified technical derivations, the conceptual clarity of the framework supplies several insights and allows us to dispel misconceptions.
Paper Structure (39 sections, 2 theorems, 331 equations)

This paper contains 39 sections, 2 theorems, 331 equations.

Table of Contents

  1. Introduction
  2. Geometry of field space
  3. Field space as a principal bundle
  4. Natural transformation groups
  5. Differential structure
  6. Tangent bundle and subbundles
  7. Differential forms and their derivations
  8. Remarkable forms
  9. General vertical transformations, and gauge transformations
  10. Connections on field space
  11. Ehresmann connections
  12. Twisted connections
  13. Associated bundles, fondamental representation of $\mathop{\mathrm{Diff}}\nolimits(M)$, and integration on $M$
  14. Associated bundle of regions
  15. Integration map
  16. ...and 24 more sections

Key Result

Proposition 1

If $\ \boldsymbol u : \Phi \ \rightarrow\ \mathcal{D} r[M, M]\ $ is s.t. then $\boldsymbol\alpha^{\boldsymbol u} \in \Omega_{\text{tens}}(\Phi, \rho )$ and $\boldsymbol \omega^{\boldsymbol u} \in \mathcal{C}$. Therefore, their residual $C^\infty(\Phi; \mathop{\mathrm{Diff}}\nolimits_{\text{\scriptsize r}}(M))$-transformations are:

Theorems & Definitions (2)

  • Proposition 1
  • Proposition 2