On geometric spectral functionals

2025/05/22 by Bochniak, Arkadiusz, Dąbrowski, Ludwik, Sitarz, Andrzej +1
#46L87 #58B34 #58J50 #83C65 #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Mathematical Physics (math-ph) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2505.16642

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.

Citations

• Algebraic Versus Spectral Torsion
• An impediment to torsion from spectral geometry
• Spectral forms and de-Rham Hodge operator
• On Gravity, Torsion and the Spectral Action Principle

Related

• The spectral Einstein functional for the Dirac operator with torsion
• Dirac Operator in Matrix Geometry
• The spectral Einstein functional and the noncommutative residue for manifolds with boundary
• Perturbations of Dirac operators, spectral Einstein functionals and the Noncommutative residue
• The spectrum of basic Dirac operators