Renormalizability conditions for almost commutative manifolds

TL;DR

This paper establishes graph-theoretic conditions for the renormalizability of the spectral action in almost commutative manifolds, extending previous results to include the Standard Model within a geometric noncommutative framework.

Contribution

It introduces new graph-based criteria for renormalizability of spectral actions, broadening the class of models, including the Standard Model, that can be analyzed within noncommutative geometry.

Findings

Conditions for renormalizability are expressed via Krajewski diagrams.

The spectral action for the Standard Model is shown to be renormalizable.

Generalizes previous superrenormalizability results to a wider class of models.

Abstract

We formulate conditions under which the asymptotically expanded spectral action on an almost commutative manifold is renormalizable as a higher-derivative gauge theory. These conditions are of graph theoretical nature, involving the Krajewski diagrams that classify such manifolds. This generalizes our previous result on (super)renormalizability of the asymptotically expanded Yang-Mills spectral action to a more general class of particle physics models that can be described geometrically in terms of a noncommutative space. In particular, it shows that the asymptotically expanded spectral action which at lowest order gives the Standard Model of elementary particles is renormalizable.