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The spectral torsion for the one form rescaled Dirac operator

Jian Wang, Yong Wang

TL;DR

This work extends the notion of spectral torsion to the one-form rescaled Dirac operator $\widetilde{D}=c(V)(D+\sqrt{-1}c(X))c(V)$ on even-dimensional closed spin manifolds, using the noncommutative residue framework of spectral triples. It derives a Lichnerowicz-type formula for $\widetilde{D}^2$ and develops the necessary symbol calculus in normal coordinates to compute the inverse-symbol expansion. The main result is an explicit local formula for the spectral torsion $\mathscr{S}_{\widetilde{D}}(c(\widetilde{u}),c(\widetilde{v}),c(\widetilde{w}))$, expressed as a scaled integral over $M$ of a combination involving $\|V\|$, the inner products of $\widetilde{u},\widetilde{v},\widetilde{w}$, and their actions on $\|V\|^{2}$. This provides a concrete geometric quantity linking spectral data to the rescaling data $(V,X)$ in noncommutative-geometry language and lays groundwork for further exploration in twisted spectral triples and inner fluctuations.

Abstract

The spectral torsion is defined by three vector fields and Dirac operators and the noncommutative residue. Motivated by the spectral torsion and the one form rescaled Dirac operator, we give some new spectral torsion which is the extension of spectral torsion for Dirac operators, and compute the spectral torsion for the one form rescaled Dirac operator on even-dimensional spin manifolds without boundary.

The spectral torsion for the one form rescaled Dirac operator

TL;DR

This work extends the notion of spectral torsion to the one-form rescaled Dirac operator on even-dimensional closed spin manifolds, using the noncommutative residue framework of spectral triples. It derives a Lichnerowicz-type formula for and develops the necessary symbol calculus in normal coordinates to compute the inverse-symbol expansion. The main result is an explicit local formula for the spectral torsion , expressed as a scaled integral over of a combination involving , the inner products of , and their actions on . This provides a concrete geometric quantity linking spectral data to the rescaling data in noncommutative-geometry language and lays groundwork for further exploration in twisted spectral triples and inner fluctuations.

Abstract

The spectral torsion is defined by three vector fields and Dirac operators and the noncommutative residue. Motivated by the spectral torsion and the one form rescaled Dirac operator, we give some new spectral torsion which is the extension of spectral torsion for Dirac operators, and compute the spectral torsion for the one form rescaled Dirac operator on even-dimensional spin manifolds without boundary.

Paper Structure

This paper contains 8 sections, 9 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on the spectral torsion for Dirac operator with one form rescaled
  3. The Lichnerowicz formula for Dirac operator with one form rescaled
  4. The algebra representation of symbols for the Dirac operator with one form rescaled
  5. The spectral torsion for Dirac operator with one form rescaled
  6. Torsion functional for Dirac operator with one form rescaled
  7. The spectral torsion for Dirac operator with one form rescaled
  8. Conclusions and outlook

Key Result

Theorem 1.1

Let $M$ be an $n=2m$ dimensional ($n\geq 3$) oriented compact spin Riemannian manifold, with the trilinear Clifford multiplication by functional of differential one-forms $c(u),c(v),c(w),$ the spectral torsion $\mathscr{S}_{\widetilde{D}}$ for one-forms rescaled Dirac operator $\widetilde{D}=c(V)(D+

Theorems & Definitions (13)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Definition 3.1
  • Lemma 3.2
  • proof
  • ...and 3 more