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Spectral (0,4)-tensor functionals and the noncommutative residue

Hongfeng Li, Yong Wang

TL;DR

The paper addresses constructing spectral (0,4)-tensor functionals from the Dirac operator using the noncommutative residue on even-dimensional spin manifolds without boundary. It advances the method by performing detailed symbol calculus to compute two concrete Wodzicki residues, revealing curvature-driven integral functionals that involve scalar curvature and the Ricci tensor, and then extends these functionals to general regular spectral triples. The explicit results for P_D and Q_D connect noncommutative geometric invariants to classical gravitational data, providing a bridge between spectral action principles and geometric analysis. This work broadens the toolkit for analyzing gravitational-type actions in noncommutative geometry and suggests avenues for applying these functionals to noncommutative spaces such as the noncommutative torus and almost-commutative models.

Abstract

In this paper, we derive some spectral (0,4)-tensor functionals by four one-forms and the Dirac operator and the noncommutative residue on even-dimensional compact spin manifolds without boundary. Then, we extend these spectral (0,4)-tensor functionals to a general spectral triple.

Spectral (0,4)-tensor functionals and the noncommutative residue

TL;DR

The paper addresses constructing spectral (0,4)-tensor functionals from the Dirac operator using the noncommutative residue on even-dimensional spin manifolds without boundary. It advances the method by performing detailed symbol calculus to compute two concrete Wodzicki residues, revealing curvature-driven integral functionals that involve scalar curvature and the Ricci tensor, and then extends these functionals to general regular spectral triples. The explicit results for P_D and Q_D connect noncommutative geometric invariants to classical gravitational data, providing a bridge between spectral action principles and geometric analysis. This work broadens the toolkit for analyzing gravitational-type actions in noncommutative geometry and suggests avenues for applying these functionals to noncommutative spaces such as the noncommutative torus and almost-commutative models.

Abstract

In this paper, we derive some spectral (0,4)-tensor functionals by four one-forms and the Dirac operator and the noncommutative residue on even-dimensional compact spin manifolds without boundary. Then, we extend these spectral (0,4)-tensor functionals to a general spectral triple.

Paper Structure

This paper contains 3 sections, 6 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. The Dirac operator and its symbols
  3. Spectral (0,4)-tensor functionals and the noncommutative residue

Key Result

Theorem 1.1

Let $M$ be an $n=2m$ dimensional ($n\geq 3$) Riemannian manifold, for the Dirac operator $D$, the following equalities hold where $g(u_i,u_j)=\sum_{a=1}^{n}u_{a}^i u_{a}^j$ and $c(u_i)=\sum_{a=1}^{n} u_a^i c(e_a)$$(i,j=1,2,3,4)$.

Theorems & Definitions (9)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Theorem 3.1
  • proof
  • Lemma 3.2
  • Lemma 3.3
  • Definition 3.4
  • Remark 3.5