Understanding truncated non-commutative geometries through computer simulations

TL;DR

This paper explores how to identify when a finite-dimensional non-commutative geometry model is a truncation of a classical spin manifold's Dirac spectral triple, using numerical methods to analyze spectral properties and topological invariants.

Contribution

It introduces a numerical approach to detect truncations of Dirac spectral triples consistent with non-commutative geometry principles.

Findings

Identified a bounded perturbation of the Dirac operator on the Riemann sphere that preserves the Chern class.

Demonstrated the applicability of the higher Heisenberg equation as a constraint for spectral truncations.

Provided a method to analyze partial spectral data in the context of non-commutative geometry.

Abstract

When aiming to apply mathematical results of non-commutative geometry to physical problems the question arises how they translate to a context in which only a part of the spectrum is known. In this article we aim to detect when a finite-dimensional triple is the truncation of the Dirac spectral triple of a spin manifold. To that end, we numerically investigate the restriction that the higher Heisenberg equation [A. H. Chamseddine, A. Connes, and V. Mukhanov, Journal of High Energy Physics, 98 (2014)] places on a truncated Dirac operator. We find a bounded perturbation of the Dirac operator on the Riemann sphere that induces the same Chern class.