Spectral metric and Einstein functionals for the Hodge–Dirac operator

Abstract

We examine the metric and Einstein bilinear functionals of differential forms introduced by Dąbrowski et al. (2023), for the Hodge–Dirac operator $d+\delta$ on an oriented, closed, even-dimensional Riemannian manifold. We show that they are equal (up to a numerical factor) to these functionals for the canonical Dirac operator on a spin manifold. Furthermore, we demonstrate that the spectral triple for the Hodge–Dirac operator is spectrally closed, which implies that it is torsion-free.