[Paper Review] Reconstructing manifolds from truncated spectral triples

Lisa Glaser, Abel B. Stern|arXiv (Cornell University)|Dec 19, 2019

Topological and Geometric Data Analysis15 references4 citations

TL;DR

This paper proposes a method to reconstruct Riemannian manifolds from truncated spectral triples using localized states and metric reconstruction. By defining a metric space of localized states via a spectral cutoff and applying the PointForge algorithm, the Gromov-Hausdorff limit of these spaces converges to the original manifold, enabling asymptotically isometric embeddings in Euclidean space, as validated on the sphere and its perturbation.

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.

Motivation & Objective

• To establish a geometric framework for truncated spectral triples that preserves the continuum limit of Riemannian manifolds.
• To address the lack of a natural metric link between finite-dimensional noncommutative geometries and their classical counterparts.
• To develop a computational algorithm capable of approximating finite metric spaces from truncated spectral data.
• To enable visualization and comparison of truncated geometries through asymptotically isometric embeddings in Euclidean space.
• To test the robustness of the framework on the 2-sphere and a perturbed spectral triple from prior work.

Proposed method

• Define localized states on truncated Hilbert spaces using a spectral projection of a Dirac-type operator, ensuring concentration near points on the manifold.
• Introduce a dispersion functional and a φ-barycenter to characterize the localization of states and their convergence to manifold points.
• Construct a finite metric space from the truncated spectral triple using the PointForge algorithm, which computes distances via the Connes metric on localized states.
• Apply stress-minimization techniques to embed the resulting metric graph into Euclidean space, aiming for local isometry.
• Utilize the higher Heisenberg equation as a guiding principle for identifying suitable operators in perturbed spectral triples.
• Leverage the Gromov-Hausdorff convergence of metric spaces of localized states to prove that the limit recovers the original Riemannian manifold.

Experimental results

Research questions

• RQ1 Can the Gromov-Hausdorff limit of metric spaces constructed from truncated spectral triples recover the original Riemannian manifold?
• RQ2 How can one algorithmically approximate the metric structure of a manifold from finite-dimensional truncated spectral data?
• RQ3 To what extent can finite metric graphs derived from truncated spectral triples be embedded isometrically into Euclidean space?
• RQ4 How do metric properties of a perturbed spectral triple (e.g., $D_{S^2} + cB$) differ from those of the unperturbed case?
• RQ5 Can the framework be generalized to noncommutative spectral triples beyond commutative ones?

Key findings

• The space of localized states on a truncated spectral triple, equipped with the pullback Connes metric, converges in the Gromov-Hausdorff sense to the original Riemannian manifold.
• The PointForge algorithm successfully generates a finite metric graph from a truncated spectral triple, enabling numerical reconstruction of the underlying geometry.
• For the 2-sphere with $ heta = 5$, the algorithm produced 35 localized states, with radii of embedded points ranging from 1.06 to 1.12 (mean 1.09) for $D_{S^2}$, and 0.94 to 0.98 (mean 0.96) for the perturbed $D_{S^2} + cB$.
• The locally isometric embedding reveals distinct geometric signatures: points from $D_{S^2}$ lie outside the reference sphere, while those from $D_{S^2} + cB$ lie inside, indicating a detectable metric difference.
• The framework is robust enough to distinguish between the unperturbed and perturbed spectral triples based on embedding geometry, even at finite truncation scale.
• The method is generalizable to arbitrary operator system spectral triples, provided a suitable embedding element $ heta$ is available, suggesting applicability to noncommutative and fuzzy geometries.