The dressing field method for diffeomorphisms: a relational framework

TL;DR

This paper extends the dressing field method to diffeomorphisms, providing a geometric, relational framework for constructing Diff(M)-invariant objects and analyzing the symplectic structure in gravitational theories.

Contribution

It introduces a systematic scheme to produce Diff(M)-invariant objects using the dressing field method, clarifies the geometric nature of diffeomorphisms, and connects this to the covariant phase space approach.

Findings

Provides a geometric understanding of field-dependent diffeomorphisms.

Derives the relational presymplectic structure for gravity.

Shows that the charge algebra forms a central extension of the Lie algebra.

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 Fr\"olicher-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.