Talk at Corfu Summer Institute (September 20-26, 2021) [slides]

Abstract: We show that the spectral action, when perturbed by a gauge potential, can be written as a series of Chern-Simonsactions and Yang-Mills actions of all orders. In the odd orders, generalized Chern-Simons forms are integrated against anodd $(b,B)$-cocycle, whereas, in the even orders, powers of the curvature are integrated against $(b,B)$-cocycles that areHochschild cocycles as well. In both cases, the Hochschild cochains are derived from the Taylor series expansion of thespectral action $\tr(f(D+V))$ in powers of $V=\pi_D(A)$, but unlike the Taylor expansion we expand in increasing orderof the forms in $A$. We then analyze the perturbative quantization of the spectral action in noncommutative geometryand establish its one-loop renormalizability as a gauge theory. We show that the one-loop counterterms are of the sameChern-Simons-Yang-Mills form so that they can be safely subtracted from the spectral action. A crucial role will be playedby the appropriate Ward identities, allowing for a fully spectral formulation of the quantum theory at one loop.