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Spectral continuity of almost commutative manifolds for the $C^1$ topology on Riemannian metrics

Frederic Latremoliere

Abstract

Almost commutative models provide a framework for Connes' work on the standard model of particle physics. These models are constructed as products of a the canonical spectral triple of a compact connected spin manifold with a finite dimensional spectral triple. Motivated by the fundamental question of the dependence of the spectra of Dirac operators under change of metrics, we prove the continuity of the spectra of Dirac operators for almost commutative models as functions of the underlying Riemannian metric. We allow both the Riemannian metric (in the $C^1$ topology) and the Dirac operator of the finite-dimensional factor to vary simultaneously. Since the physics of the system is fundamentally encoded in this spectrum, this result is a form of stability result regarding the geometry, or physical, content of these models. This result is based upon a novel approach to prove continuity of spectra of Dirac operators using the spectral propinquity. Notably, this method provides a new, different proof of the classical results as well. To illustrate the versatility of our new method, we also apply our results to completely non-commutative family of examples, including quantum tori and quantum solenoids.

Spectral continuity of almost commutative manifolds for the $C^1$ topology on Riemannian metrics

Abstract

Almost commutative models provide a framework for Connes' work on the standard model of particle physics. These models are constructed as products of a the canonical spectral triple of a compact connected spin manifold with a finite dimensional spectral triple. Motivated by the fundamental question of the dependence of the spectra of Dirac operators under change of metrics, we prove the continuity of the spectra of Dirac operators for almost commutative models as functions of the underlying Riemannian metric. We allow both the Riemannian metric (in the topology) and the Dirac operator of the finite-dimensional factor to vary simultaneously. Since the physics of the system is fundamentally encoded in this spectrum, this result is a form of stability result regarding the geometry, or physical, content of these models. This result is based upon a novel approach to prove continuity of spectra of Dirac operators using the spectral propinquity. Notably, this method provides a new, different proof of the classical results as well. To illustrate the versatility of our new method, we also apply our results to completely non-commutative family of examples, including quantum tori and quantum solenoids.
Paper Structure (4 sections, 7 theorems, 178 equations)

This paper contains 4 sections, 7 theorems, 178 equations.

Table of Contents

  1. Background: The Spectral Propinquity
  2. A continuity result for the spectral propinquity
  3. Metric Spectral Triples from Compact Lie group actions
  4. Continuity from the $C^1$ topology on Riemannian metrics for the spectral propinquity

Key Result

Theorem 1.19

The spectral propinquity is a metric on the space of metric spectral triples, up to unitarily equivalence, i.e. for any two metric spectral triples $({\mathfrak{A}},{\mathscr{H}},{\mathfrak{D}})$ and $({\mathfrak{B}},{\mathscr{H}}{G},{\slashed{D}}{T})$, we have if, and only if, there exists a unitary $U : {\mathscr{H}}_{\mathfrak{A}}\rightarrow {\mathscr{H}}_{\mathfrak{B}}$ such that $a \in {\mat

Theorems & Definitions (27)

  • Definition 1.2: Latremoliere13Latremoliere13b
  • Remark 1.3
  • Remark 1.4
  • Definition 1.5: Connes89
  • Definition 1.6: Latremoliere18g
  • Definition 1.8
  • Definition 1.10: Latremoliere14
  • Definition 1.11: Latremoliere16cLatremoliere18d
  • Remark 1.12
  • Definition 1.13: Latremoliere16cLatremoliere18d
  • ...and 17 more