The noncommutative geometry of Yang-Mills fields

TL;DR

This paper extends the noncommutative geometric framework of Einstein-Yang-Mills systems to include topologically non-trivial gauge configurations, linking spectral triples with gauge theory and gravity.

Contribution

It introduces a method to incorporate topologically non-trivial gauge fields into noncommutative geometry using spectral triples and constructs a topological spectral action.

Findings

Constructed spectral triples for non-trivial gauge configurations

Linked spectral action to Yang-Mills action coupled with gravity

Defined a topological spectral action

Abstract

We generalize to topologically non-trivial gauge configurations the description of the Einstein-Yang-Mills system in terms of a noncommutative manifold, as was done previously by Chamseddine and Connes. Starting with an algebra bundle and a connection thereon, we obtain a spectral triple, a construction that can be related to the internal Kasparov product in unbounded KK-theory. In the case that the algebra bundle is an endomorphism bundle, we construct a PSU(N)-principal bundle for which it is an associated bundle. The so-called internal fluctuations of the spectral triple are parametrized by connections on this principal bundle and the spectral action gives the Yang-Mills action for these gauge fields, minimally coupled to gravity. Finally, we formulate a definition for a topological spectral action.