Spectral torsion of the internal noncommutative geometry of the Standard Model

TL;DR

This paper analyzes the internal finite noncommutative geometry underlying the Standard Model to compute the nonvanishing spectral torsion and relate it to algebraic torsion via two second-order calculi. It demonstrates that the spectral torsion functional $ ext{Tr}(uvwD)$ is nonzero for nontrivial 1-forms and shows how a Mesland–Rennie-based differential calculus can reproduce this torsion by selecting a special idempotent and a corresponding connection. The work then contrasts algebraic and spectral functionals for metric, curvature, and torsion, highlighting a gap in scalar curvature due to the Majorana mass term $oldsymbol{ m Υ}_R$, and discusses how this gap is resolved (or not) in the Connes versus Mesland–Rennie formalisms. A key outcome is the uniqueness claim: in the Mesland–Rennie framework there exists a unique idempotent (Φ) yielding an algebraic connection matching the spectral torsion, with explicit formulas for the connection on the generating 1-forms. The findings suggest that internal quantum torsion could influence early-universe cosmology or dark-sector phenomena, while simultaneously motivating refinements to noncommutative geometric frameworks to better unify spectral and algebraic 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.