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The general Kastler-Kalau-Walze type theorem for the J-twist DJ of the Dirac operator

Siyao Liu, Yong Wang

Abstract

In [21] and [22], we proved the Kastler-Kalau-Walze type theorem for the J-twist DJ of the Dirac operator on 3-dimensional, 4-dimensional and 6-dimensional almost product Riemannian spin manifold with boundary. In this paper, we generalize our previous conclusions and establish the proof of the general Kastler-Kalau-Walze type theorem for the J-twist DJ of the Dirac operator on even-dimensional almost product Riemannian spin manifold with boundary.

The general Kastler-Kalau-Walze type theorem for the J-twist DJ of the Dirac operator

Abstract

In [21] and [22], we proved the Kastler-Kalau-Walze type theorem for the J-twist DJ of the Dirac operator on 3-dimensional, 4-dimensional and 6-dimensional almost product Riemannian spin manifold with boundary. In this paper, we generalize our previous conclusions and establish the proof of the general Kastler-Kalau-Walze type theorem for the J-twist DJ of the Dirac operator on even-dimensional almost product Riemannian spin manifold with boundary.

Paper Structure

This paper contains 4 sections, 11 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Boutet de Monvel's calculus and the definition of the noncommutative residue
  3. The noncommutative residue $\widetilde{{\rm Wres}}[\pi^+(D_{J}^{-1})\circ\pi^+(D_{J}^{-n+3})]$ on even-dimensional manifolds with boundary
  4. Declarations

Key Result

Theorem 1.1

Let $M$ be an $n$-dimensional almost product Riemannian spin manifold with the boundary $\partial M$, then we get the following equality: where $n$ is the even number, $s$ is the scalar curvature, $Vol(S^{n-2})$ is the canonical volume of $S^{n-2}$ and $K$ are defined in section $3$.

Theorems & Definitions (12)

  • Theorem 1.1
  • Theorem 2.1
  • Definition 2.2
  • Theorem 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 3.6
  • Lemma 3.7
  • ...and 2 more