Classification of Finite Spectral Triples

TL;DR

This paper classifies finite spectral triples in noncommutative geometry, linking their diagrammatic representations to gauge theories with spontaneous symmetry breaking, and extends the spin structure concept to finite geometries.

Contribution

It provides a complete classification of finite spectral triples using diagrams and connects them to physical gauge theories with symmetry breaking.

Findings

Finite spectral triples are fully described by matrices and diagrams.

Diagram vertices correspond to gauge multiplets of chiral fermions.

Links in diagrams represent Yukawa couplings.

Abstract

It is known that the spin structure on a Riemannian manifold can be extended to noncommutative geometry using the notion of a spectral triple. For finite geometries, the corresponding finite spectral triples are completely described in terms of matrices and classified using diagrams. When tensorized with the ordinary space-time geometry, finite spectral triples give rise to Yang-Mills theories with spontaneous symmetry breaking, whose characteristic features are given within the diagrammatic approach: vertices of the diagram correspond to gauge multiplets of chiral fermions and links to Yukawa couplings.