Spectral continuity of almost commutative manifolds for the $C^1$ topology on Riemannian metrics

TL;DR

This paper proves the spectral continuity of Dirac operators in almost commutative models under $C^1$ topology changes of Riemannian metrics, ensuring stability of the physical and geometric content in these models.

Contribution

It introduces a novel spectral propinquity approach to establish Dirac spectrum continuity in almost commutative models with variable metrics.

Findings

Spectra of Dirac operators vary continuously with $C^1$ Riemannian metric changes.

The new method offers an alternative proof of classical spectral continuity results.

Application to non-commutative examples like quantum tori and solenoids demonstrates versatility.

Abstract

Almost commutative models provide a framework for Connes' work on the standard model of particle physics. These models are constructed as products of a the canonical spectral triple of a compact connected spin manifold with a finite dimensional spectral triple. Motivated by the fundamental question of the dependence of the spectra of Dirac operators under change of metrics, we prove the continuity of the spectra of Dirac operators for almost commutative models as functions of the underlying Riemannian metric. We allow both the Riemannian metric (in the $C^1$ topology) and the Dirac operator of the finite-dimensional factor to vary simultaneously. Since the physics of the system is fundamentally encoded in this spectrum, this result is a form of stability result regarding the geometry, or physical, content of these models. This result is based upon a novel approach to prove continuity of spectra of Dirac operators using the spectral propinquity. Notably, this method provides a new, different proof of the classical results as well. To illustrate the versatility of our new method, we also apply our results to completely non-commutative family of examples, including quantum tori and quantum solenoids.