Understanding truncated non-commutative geometries through computer simulations

Abstract

When aiming to apply mathematical results of non-commutative geometry to physical problems, the following 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 this end, we numerically investigate the restriction that the higher Heisenberg equation [A. H. Chamseddine et al., J. High Energy Phys. 2014, 98] 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.