Talk in Research seminar Dirk Kreimer, Von Humboldt University, Berlin. (June 21, 2021) [slides]

**Abstract:** After a short but gentle recap of noncommutative geometry 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 $A$. As a consequence of our expansion and of the gauge invariance of the spectral action, we show that the odd $(b,B)$-cocycle pairs trivially with $K_1$.