Reconstructing manifolds from truncations of spectral triples
| Autor: | Abel B. Stern, Lisa Glaser |
|---|---|
| Rok vydání: | 2020 |
| Předmět: |
Pure mathematics
Euclidean space Truncation Graph embedding 010102 general mathematics General Physics and Astronomy Spectral geometry 01 natural sciences Noncommutative geometry Manifold Metric space 0103 physical sciences Embedding 010307 mathematical physics Geometry and Topology 0101 mathematics Mathematical Physics Mathematics |
| Zdroj: | Journal of Geometry and Physics, 159, pp. 1-17 Journal of Geometry and Physics, 159, 1-17 |
| ISSN: | 0393-0440 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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