The Inverse Laplacian: Traces in Infinite Dimensions

Bryce Morrow

REVIEW 3 major objections 4 minor 11 references

The Inverse Laplacian: Traces in Infinite Dimensions

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper proves exact trace formulas for powers of the inverse Laplacian on the circle and flat torus, and shows that a coupled operator built from the codifferential is trace class only above a power threshold, failing at n=2.

desk verdict Well-written expository paper with a small new computation that is currently undercut by an ambiguous d* definition and a false inequality in the n=2 divergence proof. read the letter →

arxiv 2411.17890 v1 pith:LLY7IEWA submitted 2024-11-26 math.SP math.APmath.DG

Bryce Morrow This is my paper · ORCID

classification math.SPmath.APmath.DG MSC 35P0547B10

keywords traceclassoperatorinverseLaplacianflattoruslatticesumRiemannzetafunctionDirichletbetacodifferentialLax–Milgramtheorem

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper sets out a self-contained path from the definition of trace class operators to exact trace computations for the inverse Laplacian on closed manifolds, aimed at an advanced undergraduate. On the flat torus it proves that $D^{-n}$ is trace class for every $n\ge 2$ with trace $4(-1)^n\zeta(n)\beta(n)$, and on the circle that $D^{-n}$ has trace $2(-1)^n\zeta(2n)$ for all $n$. It then couples the inverse Laplacian with the codifferential $d^*$ and proves that $P^n=(d^*D^{-1})^n$ is trace class for even $n>2$, with trace $i^n\sum_{(k,m)\ne(0,0)}(k+m)^n/(k^2+m^2)^n$, while $P^2$ is not trace class because its eigenvalue series diverges. The care taken with the inverse Laplacian construction and the threshold behavior matters because these traces are the kind of one-loop quantities that appear in topological quantum field theory.

What carries the argument

The load-bearing object is the inverse Laplacian $D^{-1}$, constructed on any closed manifold as the $L^2$-bounded operator sending $h$ to the unique $u\in K^\perp$ satisfying $\int_M \Delta u\,v = \int_M h v$ for all $v\in K^\perp$, with existence from Lax–Milgram and coercivity from the Poincaré inequality. On the torus it is diagonal in the exponential orthonormal basis, so every trace reduces to a sum over the eigenvalue sequence. The second piece of machinery is the lattice-sum identity $\sum_{(k,m)\ne(0,0)}(k^2+m^2)^{-n}=4\zeta(n)\beta(n)$, obtained from the Mellin transform of $\theta_3^2-1$, which evaluates the diagonal traces, and a binomial-comparison argument that decides absolute convergence of the coupled series $\sum (k+m)^n/(k^2+m^2)^n$.

What would settle it

A direct check of the Hodge adjoint on Fourier modes should confirm $d^*=\partial_{\theta_1}+\partial_{\theta_2}$ on the span $\Lambda$; if it does not, the eigenvalues in Section 4.3 are wrong, and likewise the partial sums of the $n=2$ series should grow without bound, contradicting Proposition 4.5 if they converge.

Watch

Extended reading notes

Core claim

The central discovery is that on the flat square torus $S^1\times S^1$ the inverse Laplacian $D^{-1}$, defined as the bounded solution operator of the weak equation $\int_M \Delta (D^{-1}h)\,v = \int_M h v$ on the complement of constants, is diagonalized by the Fourier basis with eigenvalues $-1/(k^2+m^2)$. Consequently $D^{-n}$ is trace class for $n\ge2$ with trace $4(-1)^n\zeta(n)\beta(n)$, where the lattice sum factorizes into the Riemann zeta and Dirichlet $\beta$ functions. Coupling $D^{-1}$ with the codifferential $d^*=\partial_{\theta_1}+\partial_{\theta_2}$ on the Fourier span produces $P^n$ with eigenvalues $(-i)^n(k+m)^n/(k^2+m^2)^n$; the paper proves the eigenvalue series converges absolutely exactly for even $n>2$, giving the stated trace, and diverges for $n=2$, so $P^2$ is outside the trace class. These computations extend and make explicit trace identities that arise in topological field theory.

Load-bearing premise

The coupled-operator computation rests entirely on the identified action of $d^*$ as $\partial_{\theta_1}+\partial_{\theta_2}$ on the Fourier span (the inverse Laplacian itself is also assumed to exist under the stated Poincaré and Lax–Milgram hypotheses).

Editorial extensions

If this is right

On the circle $S^1$, every power $D^{-n}$ is trace class with trace $2(-1)^n\zeta(2n)$.
On the flat torus, $D^{-n}$ is trace class for every $n\ge2$ with trace $4(-1)^n\zeta(n)\beta(n)$.
For even $n>2$, the coupled operator $P^n=(d^*D^{-1})^n$ is trace class, and its trace is the absolutely convergent lattice sum $i^n\sum_{(k,m)\ne(0,0)}(k+m)^n/(k^2+m^2)^n$.
At $n=2$ the coupled series diverges to infinity, so $P^2$ is not trace class despite $D^{-2}$ being trace class.
The torus formulas match and extend the trace identities appearing in one-dimensional Chern–Simons / BF-theory computations, giving them a self-contained spectral proof.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same power-counting predicts that on the $d$-dimensional flat torus the operator $(d^*D^{-1})^n$ is trace class only when $n>d/2$; the $d=2$ failure at $n=2$ is the first case of a general threshold.
The divergent $n=2$ series is a natural candidate for zeta regularization: analytic continuation would assign it a finite value, yielding a regularized trace for the non-trace-class operator $P^2$.
Repeating the computation on a torus with a non-square lattice should turn the lattice sums into Eisenstein series; the trace-class threshold should persist while the zeta–beta product is replaced by the appropriate modular object.
Because the traces depend only on the eigenvalue sequence, any TQFT computation of these one-loop quantities must reproduce the same lattice constants, making these formulas a checkable bridge between spectral geometry and field theory.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's my take. This is a well-organized expository paper that builds the trace class story from scratch, constructs D^{-1} via Lax-Milgram, and then computes traces on S1 and the torus. The first three sections are clean and the citations are appropriate; the S1 and torus D^{-n} traces are classical, but the presentation is careful and the construction of D^{-1} is genuinely useful for the intended readership. The new-looking part is Section 4.3 on P^n = (d* D^{-1})^n. Proposition 4.4's absolute convergence argument is correct and reasonably elegant. That is the real contribution.

The soft spots are in that same section. First, the definition of d* is not established. The paper says that on the flat square torus d* = ∂/∂θ1 + ∂/∂θ2 on the Fourier span and points to the unpublished [5]. That is not a trivial convention: d* normally maps 1-forms to functions, and to view it as a map C∞(T^2) → C∞(T^2) you have to identify Ω^1 with C∞. The eigenvalue formulas, and therefore the trace values in Propositions 4.4 and 4.5, depend on that identification. With the natural identification f ↦ f(dθ1+dθ2) and the standard sign convention, you get -(∂1+∂2), not +, which flips the sign of each eigenvalue and changes the trace by (-1)^n. A different identification gives a different series entirely. The paper does not supply the missing isomorphism or a derivation, so the central computation is not yet pinned down. Second, Proposition 4.5 contains a false bound: for k,m in [2^j, 2^{j+1}-1], the term is not ≥ 2^{-2j}; the denominator k^2+m^2 can be as large as roughly 2^{2j+3}, so the term can be about 2^{-2j-4}. The divergence conclusion is probably true, but this proof does not establish it.

The deferral of the series evaluation to [5] is also a bit unsatisfying, though not a flaw by itself. No fitted parameters or circular reasoning appear; the circularity burden is low. This looks like honest work with a repairable gap. Who is it for? Advanced undergraduates or beginning graduate students in spectral geometry, and anyone wanting the explicit torus trace formulas in the Grady-Gwilliam direction. I would not cite the P^n results until the d* issue and the n=2 proof are fixed. But the paper deserves a serious referee after revision; I would not desk reject it. My recommendation: send it out, and tell the author to derive d* carefully and repair Proposition 4.5.

Referee Report

3 major / 4 minor

Summary. The paper combines an expository development of trace class operator theory with original trace computations for powers of the inverse Laplacian. Sections 1–2 review finite-dimensional traces, Hilbert spaces, trace class operators, Lidskii’s theorem, and a diagonalizable-operator criterion (Proposition 2.18). Section 3 constructs the inverse Laplacian D^-1 on a closed Riemannian manifold via Lax–Milgram on the orthogonal complement of the kernel of the energy form. Section 4 computes traces on S^1 and on S^1×S^1: in particular, Corollary 4.3 gives Tr(D^-n) = 4(-1)^n ζ(n)β(n) for n≥2. The original part, Section 4.3, then introduces P^n = (d* ∘ D^-1)^n on the flat torus, claims that for even n>2 the operator is trace class with trace i^n times the displayed absolutely convergent series, and claims that P^2 is not trace class.

Significance. If the Section 4.3 claims are made rigorous, the paper would provide a clean, undergraduate-accessible derivation of classical trace identities and a new family of trace computations connected to the Grady–Gwilliam TQFT program. The exposition of Sections 1–3 is generally careful, and the use of Lidskii’s theorem, the Lax–Milgram construction, and the external lattice-sum evaluation from [1] are appropriate strengths. The central new results, however, currently depend on an unspecified identification of d* as an operator on functions and on an incorrect bound in the proof of Proposition 4.5; these issues are load-bearing for the claimed traces and for the trace-class threshold.

major comments (3)

[Section 4.3] The assertion "for the flat square torus, we have d* = ∂/∂θ1 + ∂/∂θ2 on Λ" is not derived. In the standard Hodge-theoretic definition, d* is the formal adjoint of d acting from Ω^0 to Ω^1, so d* maps Ω^1 to Ω^0; to regard d*∘D^-1 as an operator on L^2 functions one must specify an isomorphism Ω^1(T^2) ≅ C^∞(T^2), and no such isomorphism is given. Depending on that choice, the induced operator on functions can be +(∂1+∂2), −(∂1+∂2), or another first-order expression, changing the eigenvalues {−i(k+m)/(k²+m²)} and hence the traces in Propositions 4.4 and 4.5. The only citation for this identification is the unpublished companion paper [5], so the central claim of Section 4.3 is not yet established by the manuscript as written.
[Section 4.3, Proposition 4.5] The proof of Proposition 4.5 contains a false inequality. For 2^j ≤ k,m ≤ 2^{j+1}-1 the paper claims (k+m)^2/(k²+m²)^2 ≥ (2^j+2^j)^2/(2^{2j}+2^{2j})^2, but the replacement of the denominator by the smaller quantity (2^{2j}+2^{2j})² is not an upper bound for k²+m². For example, with j=1, k=2, m=3 the left side is 25/169 ≈ 0.148 while the right side is 16/64 = 0.25. Thus the displayed lower bound is reversed and the divergence argument as written is invalid. The statement may still be true, and a corrected dyadic bound (for instance, bounding k²+m² ≤ 2(2^{j+1})² on the block) can repair the proof, but the current manuscript does not prove Proposition 4.5.
[Section 4.3 and References] The evaluation of the series appearing in Proposition 4.4 is explicitly deferred to the companion paper [5], listed as "To Appear", and the same reference is the only support for the definition of d* used in the operator P^n. As a result, the paper’s advertised original trace computation is not self-contained at the point where the new mathematics occurs. The authors should either include the evaluation of the series, or state more narrowly that the theorem proves convergence and well-definedness of the trace and relegate the closed-form evaluation to the companion paper with a full derivation.

minor comments (4)

[Section 3.2] The word "compliment" is repeatedly used where "complement" is intended; this occurs in the definition of K^⊥ and in later references to it.
[Section 4.3] The claim that d*∘D^-1 extends by boundedness from Λ to L^2 is justified only by the phrase "Via an application of the Pythagorean identity"; displaying the inequality (k+m)² ≤ 2(k²+m²) would make the extension argument verifiable.
[Section 3.2, Proposition 3.11] The proof uses the notation ar{u} for the average value of u, which can be confused with complex conjugation; a distinct notation or an explicit definition would improve readability.
[References] Reference [10] (Taylor) does not appear to be cited in the text; it should either be cited where relevant or removed from the bibliography.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; the trace computations are derived from eigenbases and external lattice-sum identities, with only a non-load-bearing self-citation.

full rationale

The paper's central trace results are self-contained derivations rather than re-statements of inputs. In Section 4.1 and 4.2, D^{-1} is diagonalized using the standard Fourier orthonormal basis of L^2(S^1) and L^2(S^1×S^1), with eigenvalues obtained directly from the Laplacian eigenfunctions. The trace-class conclusions follow by applying Proposition 2.18, which is proved in the paper from the definition of trace class operators, to eigenvalue sums whose absolute convergence is established independently. The key lattice-sum evaluation in Proposition 4.2 is imported from Borwein et al. [1], an external reference, and is not derived from the target trace formulas. No fitted parameters appear, and no claimed prediction is statistically forced or defined in terms of the quantity it purports to compute. The only self-citation is [5], a companion paper by the same author group, cited for a 'more detailed description' of the operator P and for evaluation of the sum in Proposition 4.4. That citation is not load-bearing for the present derivation: Proposition 4.4 proves absolute convergence directly and states the trace as that convergent series without needing [5]'s evaluation. The Section 4.3 assertion that d* = ∂/∂θ1 + ∂/∂θ2 on Λ is indeed asserted rather than derived from the formal-adjoint definition on the Fourier span; this is an unproven identification that would require justification or a stated convention, but it is not circular because the eigenvalue formula is computed from that identification rather than being identical to a premise. Thus the paper has no significant circularity, though it does carry a correctness/justification gap in Section 4.3.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

No free parameters or invented entities appear. The central trace formulas depend on standard theorems (Lidskii, Lax-Milgram, Poincaré, lattice sum identities) and on one paper-specific assumption: the scalar operator form of d^* on the flat torus. All are listed above.

assumptions (8)

standard math Square Root Lemma: every positive semidefinite bounded operator has a unique positive semidefinite square root
Used to define |A|; cited to Reed-Simon [8], Section 2.2.
standard math Lidskii's theorem: trace of a trace class operator equals sum of eigenvalues with algebraic multiplicity
Used in Section 2.2; cited to Simon [9].
standard math Lax-Milgram theorem for coercive bilinear forms on Hilbert spaces
Used to construct D^{-1} in Section 3.2; cited to Evans [3].
standard math Poincaré inequality on H^1(M) for closed M
Used in coercivity proof in Proposition 3.11; cited to Hebey [7] with a note for dim M=1.
standard math Spectral theorem and Friedrich extension: Laplacian on a closed manifold has an orthonormal eigenbasis in L2 with smooth eigenfunctions
Used to diagonalize D^{-1}; cited to [2], Section 3.2.
standard math Classical lattice sum identity: sum over nonzero lattice points of (k^2+m^2)^{-s} equals 4 ζ(s)β(s) for s≥2
Used in Proposition 4.2 to evaluate the torus trace; cited to Borwein et al. [1], pages 31-33.
standard math Exponential functions form an orthonormal basis for L2 on S1 and S1×S1
Used in Section 4 to read off eigenvalues; justified by Stone-Weierstrass and density.
ad hoc to paper Codifferential on the flat torus is identified as d^* = ∂/∂θ1 + ∂/∂θ2 on the span of Fourier modes
Assumed in Section 4.3 without derivation from the Hodge inner product; the trace computation for P^n depends on this identification.

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Cite this review

Pith. "Pith review of The Inverse Laplacian: Traces in Infinite Dimensions." pith.science (2026). https://pith.science/paper/LLY7IEWA

@misc{pith2026241117890,
  author       = {Pith},
  title        = {Pith review of: The Inverse Laplacian: Traces in Infinite Dimensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LLY7IEWA}},
  note         = {Machine review of arXiv:2411.17890}
}

read the original abstract

In this paper, we present a concise development of the well-studied theory of trace class operators on infinite dimensional (separable) Hilbert spaces suitable for an advanced undergraduate, as well as a construction of the inverse Laplacian on closed manifolds. With these developments acting as prerequisite, we present original trace computations involving the inverse Laplacian on the (flat) torus, generalizing computations done by R. Grady and O. Gwilliam within the context of topological quantum field theory.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [5]

Toral Gene ra from BF Theory

Grady, Ryan; Morrow, Bryce; Williams, Brian. Toral Gene ra from BF Theory. To Appear

work page

[2] [1]

M.; et al

Borwein, J. M.; et al. Lattice Sums Then and Now. Cambridg e University Press, 2013

work page 2013

[3] [2]

Old and New Aspects in Spectral Geometry

Craioveanu, Mircea; Puta, Mircea; Rassias, Themistocl es M. Old and New Aspects in Spectral Geometry. 1st ed., Springer, 2001. THE INVERSE LAPLACIAN: TRACES IN INFINITE DIMENSIONS 31

work page 2001

[4] [3]

Partial Diﬀerential Equations

Evans, Lawrence C. Partial Diﬀerential Equations. 2nd e d., American Mathematical Society, 2010

work page 2010

[5] [4]

Folland, G. B. Real Analysis: Modern, Techniques and The ir Applications. 2nd ed., Wiley, 1999

work page 1999

[6] [6]

One-dimensional Chern-Si mons theory and the ˆA genus

Gwilliam, Owen; Grady, Ryan. One-dimensional Chern-Si mons theory and the ˆA genus. Algebr. Geom. Topol. 14 (2014), no. 4, 2299–2377

work page 2014

[7] [7]

Nonlinear Analysis on Manifolds: Sobo lev Spaces and Inequalities

Hebey, Emmanuel. Nonlinear Analysis on Manifolds: Sobo lev Spaces and Inequalities. Courant Institute of Mathematical Sciences, 2000

work page 2000

[8] [8]

Methods of Modern Mathemat ical Physics, 1: Functional Anal- ysis, Academic Press, London, 1980

Reed, Michael; Simon, Barry. Methods of Modern Mathemat ical Physics, 1: Functional Anal- ysis, Academic Press, London, 1980

work page 1980