Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 22 (2026), 035, 18 pages      arXiv:2505.16642      https://doi.org/10.3842/SIGMA.2026.035
Contribution to the Special Issue on Asymptotics, Randomness and Noncommutativity

On Geometric Spectral Functionals

Arkadiusz Bochniak ab, Ludwik Dąbrowski c, Andrzej Sitarz d and Paweł Zalecki d
a) Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, Garching, 85748, Germany
b) Munich Center for Quantum Science and Technology, Schellingstraße 4, München, 80799, Germany
c) Scuola Internazionale Superiore di Studi Avanzati, Via Bonomea 265, Trieste, 34136, Italy
d) Institute of Theoretical Physics, Jagiellonian University, Łojasiewicza 11, Kraków, 30-348, Poland

Received May 23, 2025, in final form March 24, 2026; Published online April 14, 2026

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.

Key words: spectral geometry; Wodzicki residue; noncommutative geometry; torsion.

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