Cyclic cocycles and one-loop corrections in the spectral action

TL;DR

This paper reviews recent advances in cyclic cocycles within the spectral action framework, demonstrating its perturbative quantization, one-loop renormalizability, and the structure of counterterms as Chern-Simons and Yang-Mills actions.

Contribution

It introduces a spectral formulation of one-loop quantum corrections in noncommutative geometry, showing renormalizability and explicit structure of counterterms.

Findings

Spectral action expanded as series of Chern-Simons and Yang-Mills actions.

One-loop counterterms are of the same form as the original action.

Establishment of one-loop renormalizability for the spectral action.

Abstract

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.