The noncommutative geometry of Yang–Mills fields

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 has typical fiber MN(C)MN(C), we construct a PSU(N)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.