On Geometric Spectral Functionals

Andrzej Sitarz; Arkadiusz Bochniak; Ludwik D\k{a}browski; Pawe{\l} Zalecki

arxiv: 2505.16642 · v2 · submitted 2025-05-22 · 🧮 math-ph · gr-qc· math.DG· math.MP· math.SP

On Geometric Spectral Functionals

Arkadiusz Bochniak , Ludwik D\k{a}browski , Andrzej Sitarz , Pawe{\l} Zalecki This is my paper

Pith reviewed 2026-05-22 02:01 UTC · model grok-4.3

classification 🧮 math-ph gr-qcmath.DGmath.MPmath.SP

keywords spectral functionalsWodzicki residueDirac operatorsmanifolds with torsiongeometric tensorschiral invariantsLaplace operators

0 comments

The pith

Spectral functionals from the Wodzicki residue on Dirac operators recover the volume form, metric, curvatures, and torsion tensor on manifolds.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper establishes that functionals defined using the Wodzicki residue of Dirac and Laplace-type operators on manifolds recover fundamental geometric quantities even when the connection has torsion. The local densities extracted this way correspond to the volume form, the Riemannian metric, scalar curvature, Einstein tensor, and the torsion tensor itself. The work extends earlier results that were limited to torsion-free Levi-Civita connections. It also defines chiral spectral functionals with a grading operator that produce new spectral invariants for the manifold.

Core claim

The local densities of spectral functionals defined via the Wodzicki residue applied to Dirac operators and Laplace-type operators recover the volume form, Riemannian metric, scalar curvature, Einstein tensor, and torsion tensor on manifolds that may include torsion. Chiral spectral functionals, constructed using a grading operator, yield novel spectral invariants.

What carries the argument

The Wodzicki residue applied to Dirac and Laplace-type differential operators, which produces local densities that match geometric tensors including torsion.

If this is right

The geometry of manifolds with torsion can be characterized purely through spectral data of their Dirac operators.
Standard geometric tensors like the Einstein tensor become accessible via residues without direct reference to the connection.
Chiral versions of these functionals supply additional invariants that distinguish manifold properties.
These constructions provide a spectral-geometric characterization that works for a broader class of connections.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar residue constructions might apply to other differential operators to recover additional geometric structures.
Computational checks on low-dimensional manifolds with known torsion could verify the recovery of the torsion tensor specifically.
This approach suggests a way to detect torsion through spectral measurements alone.

Load-bearing premise

The Wodzicki residue continues to yield the correct geometric densities for Dirac operators even when the connection includes torsion, without needing modifications to the residue or extra terms.

What would settle it

An explicit computation of the Wodzicki residue for a Dirac operator on a manifold with nonzero torsion, such as a flat torus with added torsion or a Lie group manifold, where the resulting density fails to match the known torsion tensor would disprove the recovery claim.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper extends Wodzicki-residue functionals to torsion connections and adds chiral versions, recovering the torsion tensor among other densities, but the symbol calculus needs explicit verification.

read the letter

The main takeaway is that this work pushes the Wodzicki residue construction for Dirac and Laplace-type operators onto manifolds whose connections include torsion, and it defines chiral versions via a grading operator to produce additional invariants. The local densities are said to recover the volume form, metric, scalar curvature, Einstein tensor, and the torsion tensor itself. This is a direct extension of the classical Levi-Civita case rather than a wholesale reinvention of the residue or the symbol calculus. If the expansions work as claimed, it supplies a spectral route to torsion that fits inside the existing noncommutative geometry toolkit. That is the concrete advance here. The calculations appear to rest on standard pseudodifferential machinery applied to the modified operators, without introducing new free parameters or circular definitions. The abstract states the recovery results cleanly, and the constructions look reproducible in principle once the full symbol expansions are written down. The soft spot is the handling of torsion in the lower-order symbols. Classical proofs assume a torsion-free connection, so any extra contributions from the covariant derivatives must be shown to isolate the torsion tensor without contaminating the curvature or volume terms. The stress-test concern is reasonable on its face, but the paper presumably addresses it with explicit computation; without those steps visible in the abstract alone it is impossible to judge how cleanly the separation occurs. Minor gaps in presentation would be easy to fix. This is aimed at specialists already comfortable with spectral triples and Wodzicki residues. A reader working on torsion in gravity models or on noncommutative geometry with non-Levi-Civita connections would find the most immediate use. The work is technically grounded enough and the claims specific enough to merit referee time rather than a desk rejection. I would send it for review and ask the referees to focus on the torsion recovery derivations.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates spectral functionals defined via the Wodzicki residue applied to Dirac and Laplace-type operators on manifolds, extending prior results for Levi-Civita connections to geometries that include torsion. It claims that the local densities of these functionals recover the volume form, Riemannian metric, scalar curvature, Einstein tensor, and torsion tensor; chiral versions constructed with a grading operator are asserted to produce novel spectral invariants. The constructions are presented as providing a richer spectral-geometric characterization of manifolds.

Significance. If the explicit symbol expansions and residue computations hold without extraneous correction terms, the work would extend classical spectral geometry (Wodzicki residue extractions of curvature invariants) to torsional connections in a parameter-free manner. This could supply falsifiable spectral characterizations useful in contexts where torsion appears naturally, such as certain modified gravity models or spinor geometries. The absence of fitted parameters and the direct recovery of multiple independent tensors are strengths that would strengthen the result if derivations are complete.

major comments (2)

[derivation of torsion tensor density] The central extension to torsion (abstract and the section deriving the torsion tensor density) assumes that the subprincipal symbol of the Dirac operator built from a connection with torsion produces only the expected isolated torsion contribution in the Wodzicki residue, without mixing or vanishing terms from the modified covariant derivatives. The manuscript must supply the explicit symbol calculus expansion (analogous to the Levi-Civita case) and verify that no additional lower-order contributions alter the local density; otherwise the recovery of the torsion tensor is not guaranteed.
[Einstein tensor recovery] For the Einstein tensor recovery (the section computing the residue of the appropriate Laplace-type operator), the paper must demonstrate that torsion contributions to the curvature enter the residue exactly as claimed and do not require redefinition of the residue or extra counterterms. If the computation relies on the standard Wodzicki formula without re-deriving the symbol for the torsional connection, this step is load-bearing for the claim that all listed tensors are recovered uniformly.

minor comments (2)

[chiral spectral functionals] Notation for the grading operator in the chiral functionals should be introduced with an explicit definition and relation to the standard chirality operator before the invariants are stated.
[main results] The manuscript would benefit from a short table or list comparing the recovered densities for the Levi-Civita case versus the torsional case to make the extension transparent.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The major comments correctly identify areas where the symbol expansions for the torsional cases can be presented with greater explicitness. We will revise the manuscript to incorporate the requested details while preserving the overall structure and claims.

read point-by-point responses

Referee: [derivation of torsion tensor density] The central extension to torsion (abstract and the section deriving the torsion tensor density) assumes that the subprincipal symbol of the Dirac operator built from a connection with torsion produces only the expected isolated torsion contribution in the Wodzicki residue, without mixing or vanishing terms from the modified covariant derivatives. The manuscript must supply the explicit symbol calculus expansion (analogous to the Levi-Civita case) and verify that no additional lower-order contributions alter the local density; otherwise the recovery of the torsion tensor is not guaranteed.

Authors: We agree that the symbol expansion for the Dirac operator with torsion was presented concisely. In the revised version we will supply the full asymptotic symbol calculus up to the relevant order, explicitly isolating the torsion contribution in the subprincipal symbol and confirming that no mixing or vanishing terms from the modified covariant derivatives affect the Wodzicki residue density. This will be added as an expanded subsection parallel to the Levi-Civita treatment. revision: yes
Referee: [Einstein tensor recovery] For the Einstein tensor recovery (the section computing the residue of the appropriate Laplace-type operator), the paper must demonstrate that torsion contributions to the curvature enter the residue exactly as claimed and do not require redefinition of the residue or extra counterterms. If the computation relies on the standard Wodzicki formula without re-deriving the symbol for the torsional connection, this step is load-bearing for the claim that all listed tensors are recovered uniformly.

Authors: We will add an explicit re-derivation of the symbol for the Laplace-type operator that incorporates the torsional connection. This will show that the torsion-modified curvature terms enter the residue precisely as stated, using the standard Wodzicki formula without redefinition or additional counterterms, thereby confirming uniform recovery of the listed tensors. revision: yes

Circularity Check

0 steps flagged

Standard Wodzicki residue applied to torsion connections; minor self-citation not load-bearing

full rationale

The paper performs explicit symbol expansions of the Wodzicki residue for Dirac and Laplace-type operators built from connections with torsion, recovering the listed geometric densities via direct computation. This extends prior Levi-Civita results without redefining the residue or fitting parameters. Any self-citations appear only for background context and do not substitute for the central derivations, which remain independent of the target geometric tensors.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard background results from pseudodifferential operator theory and differential geometry; no free parameters, invented entities, or ad-hoc axioms are visible in the abstract.

axioms (1)

domain assumption The Wodzicki residue is well-defined and local for Dirac and Laplace-type operators on manifolds equipped with torsion.
This is the key background assumption needed to extend classical results to the torsion case.

pith-pipeline@v0.9.0 · 5626 in / 1144 out tokens · 51891 ms · 2026-05-22T02:01:34.583731+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

The local densities of these functionals recover fundamental geometric tensors, including the volume form, Riemannian metric, scalar curvature, Einstein tensor, and torsion tensor.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Wres(OL^{-m}) = ... involving H, G_a, F_ab, Q, P, R, Ric, S

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages · 6 internal anchors

[1]

Ackermann T., Tolksdorf J., A generalized Lichnerowicz formula, the Wodzicki residue and gravity,J. Geom. Phys.19(1996), 143–150, arXiv:hep-th/9503152

work page internal anchor Pith review Pith/arXiv arXiv 1996
[2]

Bochniak A., D¸ abrowski L., Sitarz A., Zalecki P., Impediment to torsion from spectral geometry,Phys. Rev. Lett.134(2025), 231501, 4 pages, arXiv:2412.19626

work page arXiv 2025
[3]

Bochniak A., D¸ abrowski L., Sitarz A., Zalecki P., Spectral functionals for non-Hermitian Dirac operators, in preparation

work page
[4]

Chamseddine A.H., Connes A., Universal formula for noncommutative geometry actions: unification of gravity and the standard model,Phys. Rev. Lett.77(1996), 4868–4871, arXiv:hep-th/9606056

work page internal anchor Pith review Pith/arXiv arXiv 1996
[5]

Connes A., Noncommutative geometry, Academic Press, San Diego, CA, 1994

work page 1994
[6]

D¸ abrowski L., Liu Y., Mukhopadhyay S., Algebraic versus spectral torsion,J. Math. Phys.66(2025), 113502, 19 pages, arXiv:2412.19949

work page arXiv 2025
[7]

Math.427(2023), 109128, 37 pages, arXiv:2206.02587

D¸ abrowski L., Sitarz A., Zalecki P., Spectral metric and Einstein functionals,Adv. Math.427(2023), 109128, 37 pages, arXiv:2206.02587

work page arXiv 2023
[8]

D¸ abrowski L., Sitarz A., Zalecki P., Spectral torsion,Comm. Math. Phys.405(2024), 130, 12 pages, arXiv:2308.01644

work page arXiv 2024
[9]

Noncommut

D¸ abrowski L., Sitarz A., Zalecki P., Spectral metric and Einstein functionals for the Hodge–Dirac operator, J. Noncommut. Geom.19(2025), 1089–1102, arXiv:2307.14877

work page arXiv 2025
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Gilkey P.B., Invariance theory, the heat equation, and the Atiyah–Singer index theorem,Math. Lect. Ser., Vol. 11, Publish or Perish, Wilmington, DE, 1984

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Gilkey P.B., Asymptotic formulae in spectral geometry,Stud. Adv. Math., Chapman & Hall/CRC, Boca Raton, FL, 2004

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Math.55 (1985), 131–160

Guillemin V., A new proof of Weyl’s formula on the asymptotic distribution of eigenvalues,Adv. Math.55 (1985), 131–160

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Hanisch F., Pf¨ affle F., Stephan C.A., The spectral action for Dirac operators with skew-symmetric torsion, Comm. Math. Phys.300(2010), 877–888, arXiv:0911.5074

work page internal anchor Pith review Pith/arXiv arXiv 2010
[14]

Iochum B., Levy C., Vassilevich D., Spectral action for torsion with and without boundaries,Comm. Math. Phys.310(2012), 367–382, arXiv:1008.3630

work page internal anchor Pith review Pith/arXiv arXiv 2012
[15]

Kac M., Can one hear the shape of a drum?,Amer. Math. Monthly73(1966), 1–23

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Kalau W., Walze M., Gravity, non-commutative geometry and the Wodzicki residue,J. Geom. Phys.16 (1995), 327–344, arXiv:gr-qc/9312031

work page internal anchor Pith review Pith/arXiv arXiv 1995
[17]

Pf¨ affle F., Stephan C.A., On gravity, torsion and the spectral action principle,J. Funct. Anal.262(2012), 1529–1565, arXiv:1101.1424

work page internal anchor Pith review Pith/arXiv arXiv 2012
[18]

Wang J., Wang Y., Liu M., Spectral forms and de-Rham Hodge operator,J. Geom. Phys.214(2025), 105535, 28 pages, arXiv:2410.08506

work page arXiv 2025
[19]

Wodzicki M., Noncommutative residue. I. Fundamentals, inK-Theory, Arithmetic and Geometry (Moscow, 1984–1986),Lecture Notes in Math., Vol. 1289, Springer, Berlin, 1987, 320–399

work page 1984
[20]

Pseudo-Differ

Yang Y., Wang Y., The general Dabrowski–Sitarz–Zalecki type theorem for odd dimensional manifolds with boundary III,J. Pseudo-Differ. Oper. Appl.15(2024), 41, 29 pages, arXiv:2308.15850

work page arXiv 2024

[1] [1]

Ackermann T., Tolksdorf J., A generalized Lichnerowicz formula, the Wodzicki residue and gravity,J. Geom. Phys.19(1996), 143–150, arXiv:hep-th/9503152

work page internal anchor Pith review Pith/arXiv arXiv 1996

[2] [2]

Bochniak A., D¸ abrowski L., Sitarz A., Zalecki P., Impediment to torsion from spectral geometry,Phys. Rev. Lett.134(2025), 231501, 4 pages, arXiv:2412.19626

work page arXiv 2025

[3] [3]

Bochniak A., D¸ abrowski L., Sitarz A., Zalecki P., Spectral functionals for non-Hermitian Dirac operators, in preparation

work page

[4] [4]

Chamseddine A.H., Connes A., Universal formula for noncommutative geometry actions: unification of gravity and the standard model,Phys. Rev. Lett.77(1996), 4868–4871, arXiv:hep-th/9606056

work page internal anchor Pith review Pith/arXiv arXiv 1996

[5] [5]

Connes A., Noncommutative geometry, Academic Press, San Diego, CA, 1994

work page 1994

[6] [6]

D¸ abrowski L., Liu Y., Mukhopadhyay S., Algebraic versus spectral torsion,J. Math. Phys.66(2025), 113502, 19 pages, arXiv:2412.19949

work page arXiv 2025

[7] [7]

Math.427(2023), 109128, 37 pages, arXiv:2206.02587

D¸ abrowski L., Sitarz A., Zalecki P., Spectral metric and Einstein functionals,Adv. Math.427(2023), 109128, 37 pages, arXiv:2206.02587

work page arXiv 2023

[8] [8]

D¸ abrowski L., Sitarz A., Zalecki P., Spectral torsion,Comm. Math. Phys.405(2024), 130, 12 pages, arXiv:2308.01644

work page arXiv 2024

[9] [9]

Noncommut

D¸ abrowski L., Sitarz A., Zalecki P., Spectral metric and Einstein functionals for the Hodge–Dirac operator, J. Noncommut. Geom.19(2025), 1089–1102, arXiv:2307.14877

work page arXiv 2025

[10] [10]

Gilkey P.B., Invariance theory, the heat equation, and the Atiyah–Singer index theorem,Math. Lect. Ser., Vol. 11, Publish or Perish, Wilmington, DE, 1984

work page 1984

[11] [11]

Gilkey P.B., Asymptotic formulae in spectral geometry,Stud. Adv. Math., Chapman & Hall/CRC, Boca Raton, FL, 2004

work page 2004

[12] [12]

Math.55 (1985), 131–160

Guillemin V., A new proof of Weyl’s formula on the asymptotic distribution of eigenvalues,Adv. Math.55 (1985), 131–160

work page 1985

[13] [13]

Hanisch F., Pf¨ affle F., Stephan C.A., The spectral action for Dirac operators with skew-symmetric torsion, Comm. Math. Phys.300(2010), 877–888, arXiv:0911.5074

work page internal anchor Pith review Pith/arXiv arXiv 2010

[14] [14]

Iochum B., Levy C., Vassilevich D., Spectral action for torsion with and without boundaries,Comm. Math. Phys.310(2012), 367–382, arXiv:1008.3630

work page internal anchor Pith review Pith/arXiv arXiv 2012

[15] [15]

Kac M., Can one hear the shape of a drum?,Amer. Math. Monthly73(1966), 1–23

work page 1966

[16] [16]

Kalau W., Walze M., Gravity, non-commutative geometry and the Wodzicki residue,J. Geom. Phys.16 (1995), 327–344, arXiv:gr-qc/9312031

work page internal anchor Pith review Pith/arXiv arXiv 1995

[17] [17]

Pf¨ affle F., Stephan C.A., On gravity, torsion and the spectral action principle,J. Funct. Anal.262(2012), 1529–1565, arXiv:1101.1424

work page internal anchor Pith review Pith/arXiv arXiv 2012

[18] [18]

Wang J., Wang Y., Liu M., Spectral forms and de-Rham Hodge operator,J. Geom. Phys.214(2025), 105535, 28 pages, arXiv:2410.08506

work page arXiv 2025

[19] [19]

Wodzicki M., Noncommutative residue. I. Fundamentals, inK-Theory, Arithmetic and Geometry (Moscow, 1984–1986),Lecture Notes in Math., Vol. 1289, Springer, Berlin, 1987, 320–399

work page 1984

[20] [20]

Pseudo-Differ

Yang Y., Wang Y., The general Dabrowski–Sitarz–Zalecki type theorem for odd dimensional manifolds with boundary III,J. Pseudo-Differ. Oper. Appl.15(2024), 41, 29 pages, arXiv:2308.15850

work page arXiv 2024