Spectral Metric and Einstein Functionals for Hodge-Dirac operator

TL;DR

This paper investigates the spectral and Einstein functionals related to the Hodge-Dirac operator on Riemannian manifolds, revealing their relation to canonical Dirac functionals and the spectral triple's properties.

Contribution

It establishes the connection between the spectral and Einstein functionals of the Hodge-Dirac operator and those of the canonical Dirac operator, and shows the spectral triple is spectrally closed and torsion-free.

Findings

Functionals reproduce those of the canonical Dirac operator up to a factor

Spectral triple associated is spectrally closed

Spectral triple is torsion-free

Abstract

We examine the metric and Einstein bilinear functionals of differential forms introduced in Adv.Math.,Vol.427,(2023)1091286, for Hodge-Dirac operator $d+\delta$ on an oriented even-dimensional Riemannian manifold. We show that they reproduce these functionals for the canonical Dirac operator on a spin manifold up to a numerical factor. Furthermore, we demonstrate that the associated spectral triple is spectrally closed, which implies that it is torsion-free.