Krajewski diagrams and the standard model

Abstract

This paper provides a complete list of Krajewski diagrams representing the standard model of particle physics. We will give the possible representations of the algebra and the anomaly free lifts which provide the representation of the standard model gauge group on the fermionic Hilbert space. The algebra representations following from the Krajewski diagrams are not complete in the sense that the corresponding spectral triples do not necessarily obey to the axiom of Poincaré duality. This defect may be repaired by adding new particles to the model, i.e., by building models beyond the standard model. The aim of this list of finite spectral triples (up to Poincaré duality) is therefore to provide a basis for model building beyond the standard model.