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

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.

Symplectic reduction of Yang-Mills theory with boundaries: from superselection sectors to edge modes, and back

“…In this regard, see [3,Sect.9] on the relation between flat connections, gauge fixings and dressings (cf. also [53]).…”

“…There is a relationship between gauge fixings, flat functional connections, and dressings of the charged matter fields [3, Sect.9] (on dressings, see [46][47][48][49][50][51][52][53]). As explained at the end of paragraph 2.1, flat connections correspond to integrable horizontal distribution, i.e.…”

“…depends on the choice of an origin. 53 In other words, the origin of these symmetry can be ultimately traced back to the fact that the projection from T * G to Lie(G) * fails to be canonical-even if the cotangent bundle T * G is trivial.…”

Report on scipost_202102_00016v1

“…In this regard, see [3,Sect.9] on the relation between flat connections, gauge fixings and dressings (cf. also [46]).…”

“…2.4 Flat functional connections, gauge fixings, and dressings There is a relationship between gauge fixings, flat functional connections, and dressings of the charged matter fields [3, Sect.9] (on dressings, see [39][40][41][42][43][44][45][46]). As explained at the end of paragraph 2.1, flat connections correspond to integrable horizontal distribution, i.e.…”