On Geometric Spectral Functionals

TL;DR

This paper explores spectral functionals linked to Dirac and Laplace operators on manifolds, revealing their connection to fundamental geometric tensors and introducing new spectral invariants via chiral functionals.

Contribution

It extends classical spectral results to geometries with torsion and introduces chiral spectral functionals that provide new spectral invariants.

Findings

Local densities recover volume form, metric, scalar curvature, Einstein tensor, and torsion tensor.

Chiral spectral functionals yield novel spectral invariants.

Spectral functionals offer a richer geometric characterization of manifolds.

Abstract

We investigate spectral functionals associated with Dirac and Laplace-type differential operators on manifolds, defined via the Wodzicki residue, extending classical results for Dirac operators derived from the Levi-Civita connection to geometries with torsion. The local densities of these functionals recover fundamental geometric tensors, including the volume form, Riemannian metric, scalar curvature, Einstein tensor, and torsion tensor. Additionally, we introduce chiral spectral functionals using a grading operator, which yields novel spectral invariants. These constructions offer a richer spectral-geometric characterization of manifolds.