Cyclic cocycles in the spectral action

We show that the spectral action, when perturbed by a gauge potential, can be written as series of Chern–Simons actions and Yang–Mills all orders. In odd orders, generalized forms are integrated against an $(b,B)$-cocycle, whereas, in even powers curvature $(b,B)$-cocycles Hochschild cocycles well. both cases, cochains derived from Taylor expansion action $\\operatorname{Tr}(f(D+V))$ $V=\\pi_D(A)$, but unlike we expand increasing order $A$. This extends work Connes Chamseddine (2006), which computes only scale-invariant part works dimension at most 4, assumes vanishing tadpole hypothesis. our situation, obtain truly infinite $(b,B)$-cocycle. The analysis involved draws recent results multiple operator integration, also allows us to give conditions under this cocycle is entire, absolutely convergent. As consequence invariance $(b,B)$-cocycle pairs trivially with $K_1$.