On a Classification of Irreducible Almost-Commutative Geometries IV

TL;DR

This paper classifies finite spectral triples with KO-dimension six, demonstrating that the Standard Model fits within this framework under specific minimal conditions, and shows all such triples are S0-real.

Contribution

It provides a detailed classification of finite spectral triples in KO-dimension six, revealing the Standard Model's compatibility and the automatic S0-reality of these triples.

Findings

Standard Model fits KO-dimension six spectral triples

All finite spectral triples in KO-dimension six are S0-real

Classification includes up to four summands in matrix algebra

Abstract

In this paper we will classify the finite spectral triples with KO-dimension six, following the classification found in [1,2,3,4], with up to four summands in the matrix algebra. Again, heavy use is made of Kra jewski diagrams [5]. Furthermore we will show that any real finite spectral triple in KO-dimension 6 is automatically S 0 -real. This work has been inspired by the recent paper by Alain Connes [6] and John Barrett [7]. In the classification we find that the standard model of particle physics in its minimal version fits the axioms of noncommutative geometry in the case of KO-dimension six. By minimal version it is meant that at least one neutrino has to be massless and mass-terms mixing particles and antiparticles are prohibited