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 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.

Gromov-Hausdorff convergence of state spaces for spectral truncations

“…Let us now proceed to show that there is a C 1 -approximate order isomorphism (σ, σ) for the sequence of spectral triples defined in (11) and the spectral triple of (9). As a consequence, we thus rederive the main conclusion of [21,Theorem 3.2] that the fuzzy sphere converges to the two-sphere in Gromov-Hausdorff distance as n → ∞, though this time formulated in terms of the above spectral triples.…”

“…We note that other convergence results on the distance function on quantum spaces are obtained for quantum tori in [16], for coherent states on the Moyal plane in [10]. More generally, in [11] certain sets of states have been identified for which the Connes' distance formula has good convergence properties with respect to a given metric on a Riemannian manifold.…”

“…A Dirac operator on the fuzzy sphere was introduced in [13] (see also [2]). It is based on the following spectral triple (11) (…”

“…Previous results in the literature on the distance function for spectral truncations have been reported in [9,12,11]. However, in these works the distance function on states of the truncated system was only computed after pulling back these states to the original metric geometry.…”

On multimatrix models motivated by random Noncommutative Geometry I: the Functional Renormalization Group as a flow in the free algebra

“…In this paper, the model of space(time) we focus on is an abstraction of fuzzy spaces [DO03,DHMO08,SchSt13], whose elements were later assembled into a spectral triple (the spin geometry object in NCG) called fuzzy geometry [Bar15,BG19]. For the future, in a broader NCG context, it would be desirable to relate the FRGE to the newly investigated truncations in the spectral NCG formalism [GS19a,GS19b,CvS20], but for initial investigations fuzzy geometries are interesting enough and also in line with them, e.g. [vS20].…”