On Geometric Spectral Functionals
Mathematical Physics
2026-04-15 v2 General Relativity and Quantum Cosmology
Differential Geometry
math.MP
Spectral Theory
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.
Cite
@article{arxiv.2505.16642,
title = {On Geometric Spectral Functionals},
author = {Arkadiusz Bochniak and Ludwik Dąbrowski and Andrzej Sitarz and Paweł Zalecki},
journal= {arXiv preprint arXiv:2505.16642},
year = {2026}
}