Reconstructing manifolds from truncations of spectral triples

Abstract

We explore the geometric implications of introducing a spectral cut-off on compact Riemannian manifolds. This is naturally phrased in the framework of non-commutative geometry, where we work with spectral triples that are truncated by spectral projections of Dirac-type operators. We associate a metric space of ‘localized’ states to each truncation. The Gromov–Hausdorff limit of these spaces is then shown to equal the underlying manifold one started with. This leads us to propose a computational algorithm that allows us to approximate 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 truncation of sphere and a recently investigated perturbation thereof.