Moduli Spaces of A-Connections of Yang–Mills Fields

Abstract

Our purpose in this chapter is to expose, in the abstract language that we employ throughout this treatise, the fundamentals of the classical theory indicated by the subject in the title. Roughly speaking, we want to put into perspective the classical and physically, yet mathematically, important (!) theme of the so-called geometry of Yang–Mills equations. This was first advocated by I.M. Singer [1] (see also, for instance, M.F. Atiyah [1: p. 2]). Equivalently, one considers the corresponding space of solutions of the said equations, thus, by definition (see Chapt. I, Definitions 4.1 and 4.2), the space of the Yang–Mills A-connections. However, in view of the physical significance of the “gauge invariant (A-)connections” (see Atiyah’s phrasing in the epigraph above), the same space is finally divided out by the corresponding “gauge group,” so that it is, in effect, the resulting quotient space (“moduli space,” or even “orbit space”) that is under consideration.