A reconstruction theorem for almost-commutative spectral triples

TL;DR

This paper extends the definition of almost-commutative spectral triples to include non-trivial fibrations and stability under metric fluctuations, proving a reconstruction theorem that generalizes Connes's original result.

Contribution

It introduces a broader definition of almost-commutative spectral triples and establishes a reconstruction theorem under this new framework, weakening previous orientability assumptions.

Findings

Reconstruction theorem for the new class of spectral triples

Stability of spectral triple properties under Dirac operator perturbations

Weakened orientability hypothesis in the commutative case

Abstract

We propose an expansion of the definition of almost-commutative spectral triple that accommodates non-trivial fibrations and is stable under inner fluctuation of the metric, and then prove a reconstruction theorem for almost-commutative spectral triples under this definition as a simple consequence of Connes's reconstruction theorem for commutative spectral triples. Along the way, we weaken the orientability hypothesis in the reconstruction theorem for commutative spectral triples, and following Chakraborty and Mathai, prove a number of results concerning the stability of properties of spectral triples under suitable perturbation of the Dirac operator.