Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Continuity for the spectral propinquity of the Dirac operators associated with an analytic path of Riemannian metrics

Carla Farsi, Frederic Latremoliere

TL;DR

This work investigates how the Dirac operator on a closed spin manifold depends on the ambient metric within Latrémolière's spectral propinquity framework. It proves a general result: continuous families of metric spectral triples with a structured, holomorphic-type perturbation (type-A) yield convergence in the spectral propinquity, tying spectral data continuity to propinquity continuity. Specializing to polynomial paths of smooth Riemannian metrics, the authors show that the associated metric spectral triples form a continuous path in the spectral propinquity, with Lipschitz control on Lip-norms that underscores a partial converse to prior results in noncommutative metric geometry. These findings bridge holomorphic perturbation theory (Kato-type families) with noncommutative geometry, ensuring stability of spectra and functional calculus under metric variations and offering a robust tool for analyzing geometric deformations in quantum metric spaces.

Abstract

We prove that a polynomial path of Riemannian metrics on a closed spin manifold induces a continuous field in the spectral propinquity of metric spectral triples.

Continuity for the spectral propinquity of the Dirac operators associated with an analytic path of Riemannian metrics

TL;DR

This work investigates how the Dirac operator on a closed spin manifold depends on the ambient metric within Latrémolière's spectral propinquity framework. It proves a general result: continuous families of metric spectral triples with a structured, holomorphic-type perturbation (type-A) yield convergence in the spectral propinquity, tying spectral data continuity to propinquity continuity. Specializing to polynomial paths of smooth Riemannian metrics, the authors show that the associated metric spectral triples form a continuous path in the spectral propinquity, with Lipschitz control on Lip-norms that underscores a partial converse to prior results in noncommutative metric geometry. These findings bridge holomorphic perturbation theory (Kato-type families) with noncommutative geometry, ensuring stability of spectra and functional calculus under metric variations and offering a robust tool for analyzing geometric deformations in quantum metric spaces.

Abstract

We prove that a polynomial path of Riemannian metrics on a closed spin manifold induces a continuous field in the spectral propinquity of metric spectral triples.

Paper Structure

This paper contains 4 sections, 10 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. The main results
  3. Families of spectral triples and proof of Theorem \ref{['main-thm']}
  4. Lipschitz convergence and proof of Theorem \ref{['Riemannian-cv-thm']}

Key Result

Theorem 1

(See Theorem Riemannian-cv-thm) Let $M$ be a closed connected spin manifold. If $t\in I \mapsto g(t)$ is a polynomial path of $C^\infty$ Riemannian metrics over $M$, then $t\in I\rightarrow (C(M),\Gamma^2 \mathrm{Spin}_{g(t)},{\slashed{D}}_t)$ is a continuous function for the spectral propinquity.

Theorems & Definitions (22)

  • Theorem
  • Theorem 2.1
  • Definition 2.3
  • Theorem 2.4
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 12 more