One-loop corrections to the spectral action

TL;DR

This paper demonstrates the one-loop renormalizability of the spectral action in noncommutative geometry by analyzing perturbative expansions, path integrals, and Ward identities within a spectral framework.

Contribution

It establishes the one-loop renormalizability of the spectral action in noncommutative geometry using a spectral and perturbative approach, including higher Yang-Mills and Chern-Simons forms.

Findings

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

Ward identities enable a fully spectral formulation of the quantum theory.

The spectral action remains renormalizable at one loop within the noncommutative geometric framework.

Abstract

We analyze the perturbative quantization of the spectral action in noncommutative geometry and establish its one-loop renormalizability in a generalized sense, while staying within the spectral framework of noncommutative geometry. Our result is based on the perturbative expansion of the spectral action in terms of higher Yang-Mills and Chern-Simons forms. In the spirit of random noncommutative geometries, we consider the path integral over matrix fluctuations around a fixed noncommutative gauge background and show that the corresponding one-loop counterterms are of the same 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.