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

TL;DR

This paper analyzes the topological properties of gauge-invariant gauge fields in two-color QCD, introducing a winding number that remains invariant under small gauge transformations and exploring its implications for the field's topology.

Contribution

It provides a new expression for gauge-invariant gauge fields in two-color QCD and defines a novel winding number invariant under small gauge transformations.

Findings

Winding number is invariant under small gauge transformations.

Gauge-invariant gauge fields can have non-integer winding numbers.

Winding numbers are half-integers only when the original winding number is zero.

Abstract

We investigate solutions to a nonlinear integral equation which has a central role in implementing the non-Abelian Gauss's 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 field. We also show that when the winding number of the gauge field is an integer $\ell{\neq}0$, the gauge-invariant gauge field manifests winding numbers that are not integers, and are half-integers only when $\ell=0$.