Reconstructing manifolds from truncated spectral triples

Abstract

We explore the geometric implications of introducing a spectral cut-off on Riemannian manifolds. This is naturally phrased in the framework of non-commutative geometry, where we work with spectral triples that are \emph{truncated} by spectral projections of Dirac-type operators. We prove that the underlying Riemannian manifold is the Gromov-Hausdorff limit of the metric spaces we associate to its truncations. This leads us to propose a computational algorithm that allows us to recover these metric spaces from the finite-dimensional truncated spectral data. We subsequently develop a technique for embedding the resulting metric graphs in Euclidean space to asymptotically recover an isometric embedding of the limit. We test these algorithms on the truncated sphere and a recently investigated perturbation thereof.

25 pages, 5 figures v2 major changes to improve presentation figures +1

References in corpus (5)

• Spectral truncations in noncommutative geometry and operator systems
• On Spin Structures and Dirac Operators on the Noncommutative Torus
• Gromov-Hausdorff convergence of state spaces for spectral truncations
• Understanding truncated non-commutative geometries through computer simulations
• Finite spectral triples for the fuzzy torus

Cited by in corpus (4)

• 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
• Understanding truncated non-commutative geometries through computer simulations