Bundle geometry of the connection space, covariant Hamiltonian formalism, the problem of boundaries in gauge theories, and the dressing field method

TL;DR

This paper explores the geometric structure of gauge theories using bundle geometry, addressing the boundary problem, and clarifies the dressing field method's role in understanding edge modes and presymplectic structures.

Contribution

It introduces a unified geometric framework for gauge theories' presymplectic structure, including non-invariant theories, and clarifies the dressing field method's significance.

Findings

General presymplectic structures are common in gauge theories with boundaries.

Twisted geometry naturally appears in non-invariant gauge theories.

The dressing field method clarifies and extends the edge modes strategy.

Abstract

We take advantage of the principal bundle geometry of the space of connections to obtain general results on the presymplectic structure of two classes of (pure) gauge theories: invariant theories, and non-invariant theories satisfying two restricting hypothesis. In particular, we derive the general field-dependent gauge transformations of the presymplectic potential and presymplectic 2-form in both cases. We point-out that a generalisation of the standard bundle geometry, called twisted geometry, arises naturally in the study of non-invariant gauge theories (e.g. non-Abelian Chern-Simons theory). These results prove that the well-known problem of associating a symplectic structure to a gauge theory over bounded regions is a generic feature of both classes. The edge modes strategy, recently introduced to address this issue, has been actively developed in various contexts by several authors. We draw attention to the dressing field method as the geometric framework underpinning, or rather encompassing, this strategy. The geometric insight afforded by the method both clarifies it and clearly delineates its potential shortcomings as well as its conditions of success. Applying our general framework to various examples allows to straightforwardly recover several results of the recent literature on edge modes and on the presymplectic structure of general relativity.