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Spectral forms and de-Rham Hodge operator

Jian Wang, Yong Wang, Mingyu Liu

TL;DR

The paper develops a framework of spectral forms as multilinear functionals on differential forms within finitely summable regular spectral triples equipped with a noncommutative residue. It derives explicit density formulas for two-, three-, and four-linear torsion functionals associated with perturbed de-Rham Hodge operators, including cases with 3-form and 4-form torsion, and extends the analysis to manifolds with boundary, where boundary contributions are computed via Boutet de Monvel calculus. The results connect spectral torsion with classical geometric torsion (linear connection, raised to higher-form torsions) and provide concrete expressions that link noncommutative residue data to geometric invariants. Collectively, these spectral forms offer a robust operator-algebraic route to capture torsion-type geometric information and extend known Kastler-Kalau-Walze-type results to perturbed and boundary settings, with broader implications for noncommutative geometry and spectral action principles.

Abstract

Motivated by the trilinear functional of differential one-forms, spectral triple and spectral torsion for the Hodge-Dirac operator, we introduce a multilinear functional of differential one-forms for a finitely summable regular spectral triple with a noncommutative residue, which generalize the spectral torsion defined by Dabrowski-Sitarz-Zalecki. The main results of this paper recover two forms, torsion of the linear connection and four forms by the noncommutative residue and perturbed de-Rham Hodge operators, and provides an explicit computation of generalized spectral torsion associated with the perturbed de-Rham Hodge Dirac triple.

Spectral forms and de-Rham Hodge operator

TL;DR

The paper develops a framework of spectral forms as multilinear functionals on differential forms within finitely summable regular spectral triples equipped with a noncommutative residue. It derives explicit density formulas for two-, three-, and four-linear torsion functionals associated with perturbed de-Rham Hodge operators, including cases with 3-form and 4-form torsion, and extends the analysis to manifolds with boundary, where boundary contributions are computed via Boutet de Monvel calculus. The results connect spectral torsion with classical geometric torsion (linear connection, raised to higher-form torsions) and provide concrete expressions that link noncommutative residue data to geometric invariants. Collectively, these spectral forms offer a robust operator-algebraic route to capture torsion-type geometric information and extend known Kastler-Kalau-Walze-type results to perturbed and boundary settings, with broader implications for noncommutative geometry and spectral action principles.

Abstract

Motivated by the trilinear functional of differential one-forms, spectral triple and spectral torsion for the Hodge-Dirac operator, we introduce a multilinear functional of differential one-forms for a finitely summable regular spectral triple with a noncommutative residue, which generalize the spectral torsion defined by Dabrowski-Sitarz-Zalecki. The main results of this paper recover two forms, torsion of the linear connection and four forms by the noncommutative residue and perturbed de-Rham Hodge operators, and provides an explicit computation of generalized spectral torsion associated with the perturbed de-Rham Hodge Dirac triple.

Paper Structure

This paper contains 10 sections, 26 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. spectral two-forms for the de-Rham triple
  3. Trilinear functional for the de-Rham Hodge type operator with 3-form torsion
  4. The proof of Theorem 1.2
  5. The proof of Theorem 1.3
  6. The four-linear functional for the de-Rham Hodge type operator
  7. The proof of Theorem 1.4
  8. The proof of Theorem 1.5
  9. The spectral torsion for de-Rham Hodge operator on manifold with boundary
  10. spectral forms for spectral triples

Key Result

Theorem 1.1

For the de-Rham Hodge operator with the bilinear Clifford multiplication by functional of differential one-forms $\widehat{c}(u),\widehat{c}(v)$, the spectral torsion for $d+\delta+\sqrt{-1}T_{1}$ equals to where $T_{1}=\sum_{k<l }T(\widetilde{e_k},\widetilde{e_l}) \widehat{c}(\widetilde{e_j})\widehat{c}(\widetilde{e_l})$ is a two form.

Theorems & Definitions (51)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • ...and 41 more