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On geometric spectral functionals

Arkadiusz Bochniak, Ludwik Dąbrowski, Andrzej Sitarz, Paweł Zalecki

TL;DR

This work develops a comprehensive framework that expresses geometric invariants of torsionful manifolds as spectral functionals defined via the Wodzicki residue. It systematically analyzes Dirac- and Laplace-type operators under general perturbations $D=D_0+B$ for both spin and Hodge realizations, deriving explicit local densities for the metric, Einstein, torsion, and scalar curvature functionals, and showing metric functionals are largely invariant under perturbations. It further introduces chiral (graded) spectral functionals using a grading operator $\chi$, and computes dimension-specific, nontrivial contributions to chiral invariants for both spin and Hodge Dirac cases, including Euler and Hodge gradings. Overall, the paper extends noncommutative geometric techniques to torsion geometries, yielding richer spectral-characterizations and potential applications in generalized gravity and spectral triple formulations.

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.

On geometric spectral functionals

TL;DR

This work develops a comprehensive framework that expresses geometric invariants of torsionful manifolds as spectral functionals defined via the Wodzicki residue. It systematically analyzes Dirac- and Laplace-type operators under general perturbations for both spin and Hodge realizations, deriving explicit local densities for the metric, Einstein, torsion, and scalar curvature functionals, and showing metric functionals are largely invariant under perturbations. It further introduces chiral (graded) spectral functionals using a grading operator , and computes dimension-specific, nontrivial contributions to chiral invariants for both spin and Hodge Dirac cases, including Euler and Hodge gradings. Overall, the paper extends noncommutative geometric techniques to torsion geometries, yielding richer spectral-characterizations and potential applications in generalized gravity and spectral triple formulations.

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.

Paper Structure

This paper contains 9 sections, 30 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Spectral functionals: preliminaries
  3. General results on spectral functionals
  4. The spin Dirac operator
  5. The Hodge-Dirac operator
  6. Chiral spectral functionals
  7. The spin Dirac operator
  8. The Hodge-Dirac operator
  9. The Euler and Hodge gradings

Key Result

Proposition 1

bochniak2025 For a second-order differential operator $O$, with a symbol expressed in normal coordinates around a point on $M$, where $F^{ab} = F^{ba}$, $G^a$, and $H$ are endomorphisms of the fiber $V$ at the point $x = 0$, we have: where $R$ is the scalar curvature and $\mathrm{Ric}$ is the Ricci tensor.

Theorems & Definitions (36)

  • Proposition 1
  • Corollary 1
  • Corollary 2
  • Definition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5: cf. dabrowski2023
  • Proposition 6
  • Corollary 3
  • ...and 26 more