Topology of the gauge-invariant gauge field in two-color QCD

We investigate solutions to a nonlinear integral equation which has a central role in implementing the non-Abelian Gauss law and in constructing gauge-invariant quark and gluon fields. Here we concern ourselves with solutions to this same equation that are not operator valued, but are, functions of spatial variables and carry spatial and SU(2) indices. We obtain an expression for the gauge-invariant gauge field in two-color QCD, define an index that we will refer to as the "winding number'' that characterizes it, and show that this winding number is invariant to a small gauge transformation of the gauge field on which our construction of the gauge-invariant gauge field is based. We discuss the role of this gauge field in determining the winding number of the gauge-invariant gauge held. We also show that when the winding number of the gauge field is an integer l not equal 0, the gauge-invariant gauge field manifests winding numbers that are not integers, and are half integers only when l = 0. [S0556-2821(99)05622-2].