The spectral action principle, introduced by Chamseddine and Connes in ’97, successfully reproduces the Standard Model Lagrangian including the Higgs boson, massive neutrinos, and gravity. Mathematically, the spectral action is Trace(f(D+V)), where f is a test function, D is the Dirac operator of the spectral triple, and V is a bounded perturbation. In this talk, we will expand the spectral action in V and uncover a fascinating cyclic structure. We use this structure to define ribbon graphs and obtain one-loop renormalizability of the spectral action in a suitable sense. The open problem that lies at higher-loop shall be briefly touched upon, as well as the connections with work done on the Grosse-Wulkenhaar model.

Joint work with Walter van Suijlekom.