Talk at “Cyclic Cohomology at 40: achievements and future prospects”. Fields Institute (September 27, 2021). [slides]
Abstract: We show that the spectral action, when perturbed by a gauge potential, can be written as a series of Chern-Simons actions and Yang-Mills actions of all orders. In the odd orders, generalized Chern-Simons forms are integrated against an odd (b,B)-cocycle, whereas, in the even orders, powers of the curvature are integrated against (b,B)-cocycles that are Hochschild cocycles as well. In both cases, the Hochschild cochains are derived from the Taylor series expansion of the spectral action tr(f(D+V)) in powers of V=πD(A), but unlike the Taylor expansion we expand in increasing order of the forms in A.
We then analyze the perturbative quantization of the spectral action in noncommutative geometry and establish its one-loop renormalizability as a gauge theory. We show that the one-loop counterterms are of the same Chern-Simons-Yang-Mills form so that they can be safely subtracted from the spectral action. A crucial role will be played by the appropriate Ward identities, allowing for a fully spectral formulation of the quantum theory at one loop.