Understanding truncated non-commutative geometries through computer simulations

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.

arXiv commentThis article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in Journal of Mathematical Physics 61, 033507 (2020) and may be found at https://doi.org/10.1063/1.5131864

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Cited by in corpus (12)

• Spectral truncations in noncommutative geometry and operator systems
• Gromov-Hausdorff convergence of state spaces for spectral truncations
• On multimatrix models motivated by random Noncommutative Geometry I: the Functional Renormalization Group as a flow in the free algebra
• On multimatrix models motivated by random noncommutative geometry II: A Yang-Mills-Higgs matrix model
• From Noncommutative Geometry to Random Matrix Theory
• One-loop corrections to the spectral action
• Convergence of Fourier truncations for compact quantum groups and finitely generated groups
• Tolerance Relations and Quantization
• Computing the spectral action for fuzzy geometries: from random noncommutative geometry to bi-tracial multimatrix models
• Reconstructing manifolds from truncated spectral triples
• Tensors and Algebras: An Algebraic Spacetime Interpretation for Tensor Models
• A noncommutative integral on spectrally truncated spectral triples, and a link with quantum ergodicity