Spectral forms and de-Rham Hodge operator

TL;DR

This paper introduces a new multilinear functional for differential one-forms within spectral triples, generalizing spectral torsion, and provides explicit computations related to noncommutative geometry and de-Rham Hodge operators.

Contribution

It extends the concept of spectral torsion to a multilinear setting for spectral triples, incorporating noncommutative residues and perturbed de-Rham Hodge operators.

Findings

Recovered torsion of linear connection and four forms via noncommutative residue.

Explicit computation of generalized spectral torsion for perturbed de-Rham Hodge Dirac triple.

Generalized spectral torsion extends previous notions in noncommutative geometry.

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.