Spectral torsion of the internal noncommutative geometry of the Standard Model

TL;DR

This paper investigates the spectral torsion in the noncommutative geometry framework of the Standard Model, revealing how modifications affect geometric functionals and the influence of physical parameters like Yukawa couplings.

Contribution

It introduces a modified differential calculus to compute spectral torsion and analyzes its effects on geometric invariants within the Standard Model's noncommutative geometry.

Findings

Spectral torsion functional computed for the Standard Model

Modified calculus aligns torsion with connection-based functional

Yukawa couplings and Majorana mass impact geometric invariants

Abstract

We compute the nonvanishing spectral torsion functional of the internal part of the noncommutative geometry behind the Standard Model. We show that with a suitable modification of the usual differential graded calculus it matches an analogous functional constructed in terms of the connection. We study also the impact of the torsion on the other spectral fuctionals, which correspond to geometric invariants such as volume integral, metric and Einstein tensors, and scalar curvature. We discuss the impact of the SM Yukawa couplings and the Majorana mass matrix on our results.