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The spectral torsion for the Connes type operator

Jian Wang, Yong Wang

TL;DR

The work provides an explicit computation of the spectral torsion associated with Connes type operators on even-dimensional compact manifolds and extends to manifolds with boundary. It formulates the trilinear torsion functional via the noncommutative residue, and derives concrete residue densities for several perturbations, including a 3-form torsion and twisted fluctuations, across different dimensions; boundary phenomena are treated using Boutet de Monvel calculus. The results connect torsion data to trace densities of twisted Dirac-type operators, yielding dimension-dependent formulas that illuminate how torsion and boundary effects influence the spectral action. This advances the spectral-torsion program of Dabrowski–Szabo–Zajac and the KW-type gravitational action in a Connes-type setting, with potential implications for noncommutative geometry and geometric analysis on manifolds with boundary.

Abstract

This paper aims to provide an explicit computation of the spectral torsion associated with the Connes type operator on even dimension compact manifolds.And we also extend the spectral torsion for the Connes type operator to compact manifolds with boundary.

The spectral torsion for the Connes type operator

TL;DR

The work provides an explicit computation of the spectral torsion associated with Connes type operators on even-dimensional compact manifolds and extends to manifolds with boundary. It formulates the trilinear torsion functional via the noncommutative residue, and derives concrete residue densities for several perturbations, including a 3-form torsion and twisted fluctuations, across different dimensions; boundary phenomena are treated using Boutet de Monvel calculus. The results connect torsion data to trace densities of twisted Dirac-type operators, yielding dimension-dependent formulas that illuminate how torsion and boundary effects influence the spectral action. This advances the spectral-torsion program of Dabrowski–Szabo–Zajac and the KW-type gravitational action in a Connes-type setting, with potential implications for noncommutative geometry and geometric analysis on manifolds with boundary.

Abstract

This paper aims to provide an explicit computation of the spectral torsion associated with the Connes type operator on even dimension compact manifolds.And we also extend the spectral torsion for the Connes type operator to compact manifolds with boundary.
Paper Structure (7 sections, 11 theorems, 80 equations)

This paper contains 7 sections, 11 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Preliminaries for the Connes type operator
  3. Trilinear functional for the Connes type operator with torsion
  4. The computation of the spectral torsion for the Connes type operator
  5. Spectral torsion for the Connes type operator with 3-form
  6. Spectral torsion for the Connes type operator with twisted fluctuation as torsion
  7. The spectral torsion for manifold with boundary associated with $D_{F}+(c(T)+\sqrt{-1}c(Y)) \otimes \Phi$

Key Result

Lemma 4.1

In the normal coordinates $U$ of $x_0$ in $M$,

Theorems & Definitions (25)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • Definition 3.1
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 15 more