  We present an intelligible review of recent results concerning cyclic
cocycles in the spectral action and one-loop quantization. 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 $V=\pi_D(A)$, 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.
