I have a background in theoretical physics and the first time I came across winding numbers was in the context of the vacuum of QCD.
By the way physicists treat this topic, I thought it had little to do with Winding numbers in complex analysis. Now I, came across some results of complex Brownian motion applied to proofs of Picard's theorems in complex analysis, related to winding and tangling of curves, and also to similarstochastic techniques applied to path integral formulation of quantum mechanics and related QFT topics. Thus, I am asking myself if winding numbers in QCD are related to closed loops on a certain bundle related to the SU(3) group. The paths could be the brownian motion that provides an equivalent measure to the vacuum state. I am very familiar with stochastic calculus applied to QFT, differential geometry and functional analysis, what I am lacking to answer the question in the title is the interpretation of the QCD winding numbers in this sense.
I have also been thinking about Berry phases for U(1) groups, and in that case the Berry phase gained through closed loops in the base of a fibre bundle. Is this related to an analogous question to the one of the winding numbers of QCD for this (way simpler) abelian group?
Thanks.
In Stochastic areas, Horizontal Brownian Motions, and Hypoelliptic Heat Kernels
they investigate the concept of winding functionals of Brownian motion over more general Lie groups like SU(2). Their construction might be more generalizable for SU(3) too since they look at general Lie groups.