Inner fluctuations of the spectral action

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 than or equal to four the obtained terms add up to a sum of a Yang–Mills action with a Chern–Simons action.