Hamiltonian facets of classical gauge theories on $E$-manifolds

TL;DR

This paper extends classical gauge theories to $E$-manifolds, revealing Hamiltonian structures and universal models for particles interacting with gauge fields in singular and constrained configuration spaces.

Contribution

It introduces an $E$-category framework for gauge theories, generalizing phase space descriptions, minimal coupling, and Hamiltonian dynamics to singular and constrained manifolds.

Findings

Existence of a universal model for particles with $E$-gauge fields

Generalization of phase space as $E$-Poisson manifolds

Hamiltonian formulation of Wong's equations in the $E$-setting

Abstract

Manifolds with boundary, with corners, $b$-manifolds and foliations model configuration spaces for particles moving under constraints and can be described as $E$-manifolds. $E$-manifolds were introduced in [NT01] and investigated in depth in [MS20]. In this article we explore their physical facets by extending gauge theories to the $E$-category. Singularities in the configuration space of a classical particle can be described in several new scenarios unveiling their Hamiltonian aspects on an $E$-symplectic manifold. Following the scheme inaugurated in [Wei78], we show the existence of a universal model for a particle interacting with an $E$-gauge field. In addition, we generalize the description of phase spaces in Yang-Mills theory as Poisson manifolds and their minimal coupling procedure, as shown in [Mon86], for base manifolds endowed with an $E$-structure. In particular, the reduction at coadjoint orbits and the shifting trick are extended to this framework. We show that Wong's equations, which describe the interaction of a particle with a Yang-Mills field, become Hamiltonian in the $E$-setting. We formulate the electromagnetic gauge in a Minkowski space relating it to the proper time foliation and we see that our main theorem describes the minimal coupling in physical models such as the compactified black hole.