On the finite spectral triple of an almost-commutative geometry

TL;DR

This paper investigates how the signature of space-time metrics influences the construction of spectral triples in noncommutative geometry, providing insights into the emergence of gauge groups like SU(2) and U(1) in the standard model.

Contribution

It offers new arguments on the role of space-time signature in spectral triples and its impact on deriving standard model gauge groups.

Findings

Signature affects the product of spectral triples.

Supports the emergence of SU(2) and U(1) gauge groups.

Highlights importance of metric signature in noncommutative geometry.

Abstract

In this short communication, we examine the relevance of the signature of the space-time metric in the construction of the product of a pseudo-Riemannian spectral triple with a finite triple describing the internal geometry. We obtain arguments favouring the appearance of SU(2) and U(1) as gauge groups in the standard model.