Yang-Mills moduli spaces over an orientable closed surface via Fr\'echet reduction

TL;DR

This paper constructs a Fréchet manifold structure on the space of gauge equivalence classes of connections over a closed surface, and characterizes Yang-Mills connections using Wilson loops and topology without directly solving the Yang-Mills equation.

Contribution

It introduces a novel Fréchet slice analysis to study Yang-Mills moduli spaces, linking topology, gauge theory, and symplectic geometry in a new framework.

Findings

Fréchet manifold structure on gauge equivalence classes

Characterization of Yang-Mills connections via Wilson loops and topology

Stratified symplectic structure on unbased gauge classes

Abstract

Given a principal bundle on an orientable closed surface with compact connected structure group, we endow the space of based gauge equivalence classes of smooth connections relative to smooth based gauge transformations with the structure of a Fr\'echet manifold. Using Wilson loop holonomies and a certain characteristic class determined by the topology of the bundle, we then impose suitable constraints on that Fr\'echet manifold that single out the based gauge equivalence classes of central Yang-Mills connections but do not directly involve the Yang-Mills equation. We also explain how our theory yields the based and unbased gauge equivalence classes of all Yang-Mills connections and deduce the stratified symplectic structure on the space of unbased gauge equivalence classes of central Yang-Mills connections. The crucial new technical tool is a slice analysis in the Fr\'echet setting.