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

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.