Note on the bundle geometry of field space, variational connections, the dressing field method, & presymplectic structures of gauge theories over bounded regions

TL;DR

This paper explores the geometric structure of field space in gauge theories, analyzing presymplectic forms, boundary issues, and comparing edge modes with variational connections, with applications to Yang-Mills and General Relativity.

Contribution

It provides a unified geometric framework for understanding boundary issues in gauge theories, linking edge modes and dressing fields, and generalizing recent results.

Findings

Derived the generic form of Noether charges and their Poisson brackets.

Compared edge modes and variational connections as boundary strategies.

Applied the framework to Yang-Mills and General Relativity, extending existing results.

Abstract

In this note, we consider how the bundle geometry of field space interplays with the covariant phase space methods so as to allow to write results of some generality on the presymplectic structure of invariant gauge theories coupled to matter. We obtain in particular the generic form of Noether charges associated with field-independent and field-dependent gauge parameters, as well as their Poisson bracket. We also provide the general field-dependent gauge transformations of the presymplectic potential and 2-form, which clearly highlight the problem posed by boundaries in generic situations. We then conduct a comparative analysis of two strategies recently considered to evade the boundary problem and associate a modified symplectic structure to a gauge theory over a bounded regions: namely the use of edge modes on the one hand, and of variational connections on the other. To do so, we first try to give the clearest geometric account of both, showing in particular that edge modes are a special case of differential geometric tool of gauge symmetry reduction known as the "dressing field method". Applications to Yang-Mills theory and General Relativity reproduce or generalise several results of the recent literature.