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Collapse in Noncommutative Geometry and Spectral Continuity

Carla Farsi, Frederic Latremoliere

TL;DR

The paper develops a general collapse framework for metric spectral triples in noncommutative geometry, using the spectral propinquity to formalize convergence when a vertical fiber component is shrunk. Central to the approach is decomposing the Dirac operator into horizontal and vertical parts, $led{D}=led{D}_h+led{D}_v$, and proving that the rescaled family $led{D}_lvarepsilon=led{D}_h+ rac{1}{lvarepsilon}led{D}_v$ converges to a base spectral triple on $ rak{B}$ as $lvarepsilon o0^+$ under precise technical assumptions (including $0$ isolated in the spectrum and the existence of a conditional expectation $b E$). The results are instantiated in multiple settings: products with Abelian algebras, noncommutative principal $G$-bundles (e.g., crossed products), and commutative smooth projectable $U(1)$-bundles, yielding spectral convergence and, in some cases, a clean description of harmonic spinor content in the limit. This work bridges classical collapse phenomena in Riemannian geometry with noncommutative metric geometry, providing a toolkit for continuity of spectra and Dirac-type operators in collapsing noncommutative fiber bundles. The findings have potential implications for noncommutative physics models and the study of geometric invariants under degenerate fibrations, by demonstrating that spectra vary continuously with respect to the spectral propinquity in a broad range of noncommutative settings.

Abstract

If two compact quantum metric spaces are close in the metric sense, then how similar are they, as noncommutative spaces? In the classical realm of Riemannian geometry, informally, if two manifolds are close in the Gromov-Hausdorff distance, and belong to a class of manifolds with bounded curvature and diameter, then the spectra of their Laplacian or Dirac operators are also close under many scenari. Of particular interest is the case where a sequence of manifolds converge for the Gromov-Hausdorff distance to a manifold of lower dimension, and the question of the continuity, in some sense, of the spectra of geometrically relevant operators. In this paper, we initiate the study of the continuity of spectra and other properties of metric spectral triples under collapse in the noncommutative realm. As a first step in this study, we work with collapse for the spectral propinquity, an analogue of the Gromov-Hausdorff distance for spectral triples introduced by the second author, i.e. a form of metric for differential structures. Inspired by results from collapse in Riemannian geometry, we begin with the study of spectral triples which decompose, in some sense, in a vertical and a horizontal direction, and we collapse these spectral triples along the vertical direction. We obtain convergence results, and by the work of the second author, we conclude continuity results for the spectra of the Dirac operators of these spectral triples. Examples include collapse of product of spectral triples with one Abelian factor, $U(1)$ principal bundles over Riemannian spin manifolds, and noncommutative principal bundles, including C*-crossed-products and other noncommutative bundles.

Collapse in Noncommutative Geometry and Spectral Continuity

TL;DR

The paper develops a general collapse framework for metric spectral triples in noncommutative geometry, using the spectral propinquity to formalize convergence when a vertical fiber component is shrunk. Central to the approach is decomposing the Dirac operator into horizontal and vertical parts, , and proving that the rescaled family converges to a base spectral triple on as under precise technical assumptions (including isolated in the spectrum and the existence of a conditional expectation ). The results are instantiated in multiple settings: products with Abelian algebras, noncommutative principal -bundles (e.g., crossed products), and commutative smooth projectable -bundles, yielding spectral convergence and, in some cases, a clean description of harmonic spinor content in the limit. This work bridges classical collapse phenomena in Riemannian geometry with noncommutative metric geometry, providing a toolkit for continuity of spectra and Dirac-type operators in collapsing noncommutative fiber bundles. The findings have potential implications for noncommutative physics models and the study of geometric invariants under degenerate fibrations, by demonstrating that spectra vary continuously with respect to the spectral propinquity in a broad range of noncommutative settings.

Abstract

If two compact quantum metric spaces are close in the metric sense, then how similar are they, as noncommutative spaces? In the classical realm of Riemannian geometry, informally, if two manifolds are close in the Gromov-Hausdorff distance, and belong to a class of manifolds with bounded curvature and diameter, then the spectra of their Laplacian or Dirac operators are also close under many scenari. Of particular interest is the case where a sequence of manifolds converge for the Gromov-Hausdorff distance to a manifold of lower dimension, and the question of the continuity, in some sense, of the spectra of geometrically relevant operators. In this paper, we initiate the study of the continuity of spectra and other properties of metric spectral triples under collapse in the noncommutative realm. As a first step in this study, we work with collapse for the spectral propinquity, an analogue of the Gromov-Hausdorff distance for spectral triples introduced by the second author, i.e. a form of metric for differential structures. Inspired by results from collapse in Riemannian geometry, we begin with the study of spectral triples which decompose, in some sense, in a vertical and a horizontal direction, and we collapse these spectral triples along the vertical direction. We obtain convergence results, and by the work of the second author, we conclude continuity results for the spectra of the Dirac operators of these spectral triples. Examples include collapse of product of spectral triples with one Abelian factor, principal bundles over Riemannian spin manifolds, and noncommutative principal bundles, including C*-crossed-products and other noncommutative bundles.
Paper Structure (16 sections, 20 theorems, 199 equations, 1 figure)

This paper contains 16 sections, 20 theorems, 199 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Introductory Material
  3. A collapse result
  4. Examples
  5. Collapsing to a point
  6. Collapsing $C(X,{\mathfrak{A}})$
  7. Collapsing Noncommutative Principal $G$-Bundles
  8. Decomposition of ${\mathfrak{A}}$
  9. Free Actions
  10. The Representation of ${\mathfrak{A}}$ and $G$ on ${\mathscr{H}}_p$
  11. The Hilbert Space and Spectral Triple Operators
  12. The Noncommutative Principal $G$-Bundles Convergence Result
  13. Examples
  14. Collapsing Commutative Spin Principal $U(1)$-Bundles: the Smooth Projectable Case Ammann99Ammann-Bar-98
  15. Collapsing Commutative Smooth Projectable Spin $U(1)-$Bundles Ammann99Ammann-Bar-98
  16. ...and 1 more sections

Key Result

Theorem 1

(Theorem main-thm) Let $({\mathfrak{A}},{\mathscr{H}},{\slashed{D}})$ be a metric spectral triple, and let ${\mathfrak{B}} \subseteq{\mathfrak{A}}$ be a unital C*-subalgebra of ${\mathfrak{A}}$, and such that ${\slashed{D}}= {\slashed{D}}_h + {\slashed{D}}_v$, where ${\slashed{D}}_v$ is self-adjoint Then $({\mathfrak{A}},{\mathscr{H}},{\slashed{D}}_\varepsilon)$ is a metric spectral triple, and:

Figures (1)

  • Figure 1: A metrical tunnel between C*-correspondences associated to metric spectral triples. The extent of the above metrical tunnel is the max of the extents of the top and bottom tunnels. All the maps in the picture are quantum isometries.

Theorems & Definitions (57)

  • Theorem
  • Definition 2.1: Connes
  • Definition 2.3: Latremoliere13Latremoliere14Latremoliere15
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.9
  • Definition 2.10
  • Definition 2.11
  • Definition 2.12
  • ...and 47 more