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The spectral Einstein functional for the Witten Deformation

Tong Wu, Yong Wang

TL;DR

The paper computes the spectral Einstein functional for the Witten deformation on even-dimensional closed manifolds without boundary by employing noncommutative residue techniques and pseudodifferential symbol calculus for the operator $\widetilde{D}_v=d+d^*+\hat{c}(V)$. It develops a Clifford-action framework, derives square and symbol expansions for the Witten-deformed Laplacian, and carries out detailed residue computations to obtain explicit formulas for the metric functional and the spectral Einstein functional. The main results express these functionals in curvature terms and the norm of the deformation field, namely $\mathscr{M}_{\widetilde{D}_v}=-2^{2m}\frac{2\pi^{m}}{\Gamma(m)}\int_M g(u,w)\,dVol_M$ and $\mathscr{S}_{\widetilde{D}_v}=2^{2m}\frac{2\pi^{m}}{\Gamma(m)}\int_M\left(-\frac{1}{6}\mathbb{G}(u,w)+|V|^2 g(u,w)\right)\,dVol_M$, where $\mathbb{G}(u,w)=\mathrm{Ric}(u,w)-\frac{1}{2}s g(u,w)$. These results extend Kastler-Kalau-Walze-type formulas to the Witten deformation on manifolds without boundary and connect spectral invariants to intrinsic geometry via curvature data and the deformation vector field $V$.

Abstract

In the paper, given two vector fields and the Witten deformation, we compute the spectral Einstein functional for the Witten deformation on even-dimensional spin manifolds without boundary.

The spectral Einstein functional for the Witten Deformation

TL;DR

The paper computes the spectral Einstein functional for the Witten deformation on even-dimensional closed manifolds without boundary by employing noncommutative residue techniques and pseudodifferential symbol calculus for the operator . It develops a Clifford-action framework, derives square and symbol expansions for the Witten-deformed Laplacian, and carries out detailed residue computations to obtain explicit formulas for the metric functional and the spectral Einstein functional. The main results express these functionals in curvature terms and the norm of the deformation field, namely and , where . These results extend Kastler-Kalau-Walze-type formulas to the Witten deformation on manifolds without boundary and connect spectral invariants to intrinsic geometry via curvature data and the deformation vector field .

Abstract

In the paper, given two vector fields and the Witten deformation, we compute the spectral Einstein functional for the Witten deformation on even-dimensional spin manifolds without boundary.
Paper Structure (3 sections, 6 theorems, 71 equations)

This paper contains 3 sections, 6 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Clifford Actions and the Witten Deformation
  3. The spectral Einstein functional for the Witten Deformation

Key Result

Theorem 1.1

Let $M$ be an $n=2m$ dimensional ($n\geq 3$) Riemannian manifold, for the Witten deformation $\widetilde{D}_v$, the metric functional $\mathscr{M}_{\widetilde{D}_v}$ and the spectral Einstein functional $\mathscr{S}_{\widetilde{D}_v}$ are equal to where $g(u,w)=\sum_{a,b=1}^{n}u_{a} w_{b}$ and $\mathbb{G}(u,w)=\operatorname{Ric}(u,w)-\frac{1}{2} s g(u,w)$, $c(u)=\sum_{\eta=1}^{n} u_{\eta}c(e_{\et

Theorems & Definitions (9)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 3.1
  • proof
  • Lemma 3.2
  • Lemma 3.3