Inner fluctuations of the spectral action

TL;DR

This paper demonstrates that in noncommutative geometry, the inner fluctuations of the spectral action can be explicitly computed as residues, corresponding to counterterms in Feynman graphs, and relate to known physical actions in low dimensions.

Contribution

It provides a general method to compute inner fluctuations of the spectral action as residues and links them to physical actions like Yang-Mills and Chern-Simons in low-dimensional geometries.

Findings

Inner fluctuations can be computed as residues.

Counterterms correspond to Feynman graphs with fermionic lines.

In dimensions ≤4, the spectral action sums to Yang-Mills and Chern-Simons actions.

Abstract

We prove in the general framework of noncommutative geometry that the inner fluctuations of the spectral action can be computed as residues and give exactly the counterterms for the Feynman graphs with fermionic internal lines. We show that for geometries of dimension less or equal to four the obtained terms add up to a sum of a Yang-Mills action with a Chern-Simons action.