Neumann scalar determinants on constant curvature disks

Soumyadeep Chaudhuri

REVIEW 2 major objections 5 minor 27 references

Neumann scalar determinants on constant curvature disks

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper shows that the zeta-regularized massive scalar determinant with Neumann boundary conditions on a constant-curvature round disk is given by a convergent infinite series for arbitrary mass, with exact closed forms at the special…

desk verdict A solid, mostly verifiable computation of Neumann determinants on constant-curvature disks, with one unproved analytic-continuation step that should be fixed before the negative-mass results are cited. read the letter →

arxiv 2507.05159 v1 pith:E6DRCFYG submitted 2025-07-07 hep-th math-phmath.MP

Soumyadeep Chaudhuri This is my paper · ORCID

classification hep-thmath-phmath.MP

keywords functionaldeterminantszeta-functionregularizationNeumannboundaryconditionsDirichlet-to-NeumannmapconstantcurvaturedisksBarnesG-functionSteklovproblem

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

This paper establishes that the logarithm of the massive scalar determinant $\det_N(\Delta_\eta + m^2)$, regularized by the zeta-function scheme, on a round disk of constant curvature $R=2\eta/L^2$ ($\eta=0,\pm1$) with Neumann boundary conditions can be written as a convergent infinite series for arbitrary mass $m^2$. The derivation uses a recently proven relation connecting the Neumann determinant to the corresponding Dirichlet determinant through the determinant of the Dirichlet-to-Neumann map on the disk boundary, so the new work reduces to computing that map's determinant and combining it with known Dirichlet results. For the curved disks ($\eta=\pm1$) and the special masses $m^2 = -\eta q(q+1)/L^2$ with $q\in\mathbb{N}$, the series collapse into exact expressions involving Legendre polynomials and Gamma functions. On the hemisphere the determinant is given in closed form in terms of Barnes $G$-functions and vanishes at exactly those special masses. These results matter because such determinants appear directly in the one-loop partition functions of free scalar fields and in two-dimensional gravity models, where exact formulas are rare.

What carries the argument

The load-bearing object is the BFK-type gluing identity (2.8), $\ln\det_N - \ln\det_D = b_0 + \ln\det(\mathcal{O}_{m^2})$, which converts the Neumann problem into the already-solved Dirichlet problem plus the determinant of the Dirichlet-to-Neumann map $\mathcal{O}_{m^2}$ (the operator sending boundary data to the normal derivative of the solution of $(\Delta+m^2)\Phi=0$). The paper's new computation is the determinant of $\mathcal{O}_{m^2}$, obtained from the explicit Steklov eigenvalues $v_p(m^2)$ (Bessel-function ratios for $\eta=0$, hypergeometric-function ratios for $\eta=\pm1$), together with a polynomial $Q_{q+1}(x;z_\eta)$ whose roots $\omega_k^{(q+1)}(z_\eta)$ encode the sum of the special-mass series. The massless value $b_0=\frac{k_\eta\ell_\eta}{2\pi}$ follows from Weyl-invariance arguments and matches the general formula of [22].

What would settle it

For the hemisphere with $L=1$ and a fixed $q\in\mathbb{N}$ (say $q=2$), compute the Neumann spectrum of $\Delta_+$ directly: the paper's Appendix D asserts that $q(q+1)$ is an eigenvalue, which would make the determinant at $m^2=-q(q+1)$ vanish. If direct spectral computation shows no such eigenvalue, the vanishing comes from the analytic continuation rather than the spectrum, contradicting the stated interpretation. Alternatively, numerically sum (4.4) for a specific hyperbolic disk at a positive mass and compare with a truncated eigenvalue product.

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Extended reading notes

Core claim

The paper's central claim is that, working in $\zeta$-function regularization, the Neumann determinants on constant-curvature disks obey the relation $\ln\det_N(\Delta_\eta+m^2) = \ln\det_D(\Delta_\eta+m^2) + b_0 + \ln\det(\mathcal{O}_{m^2})$, with $b_0 = k_\eta\ell_\eta/(2\pi)$ the mass-independent boundary term and $\mathcal{O}_{m^2}$ the Dirichlet-to-Neumann map of the massive Helmholtz operator. Using the Dirichlet determinants obtained previously and the Steklov-problem eigenvalues of $\mathcal{O}_{m^2}$ computed here (Bessel functions in the flat case, hypergeometric functions in the curved cases), the paper produces the infinite series (4.3) and (4.4) for $\ln\det_N(\Delta_\eta+m^2)$ valid for arbitrary $m^2$. For $\eta=\pm1$ and $m^2=-\eta q(q+1)/L^2$, $q\in\mathbb{N}$, the series is summed exactly, giving (5.20) with Legendre polynomials and the roots of an explicit polynomial $Q_{q+1}(x;z_\eta)$. On a hemisphere the arbitrary-mass determinant is evaluated in closed form as (D.18) in terms of Barnes $G$-functions, and this expression vanishes for $m^2=-q(q+1)/L^2$, $q\in\mathbb{N}$, because those masses are Neumann eigenvalues of the hemisphere Laplacian. An independent Sturm-Liouville computation in Appendix C reproduces the series.

Load-bearing premise

The main-text derivation hinges on the imported gluing formula (2.8) and on its constant term being independent of the mass, which the paper does not rederive from first principles.

Editorial extensions

If this is right

The Neumann determinants on flat, spherical, and hyperbolic disks are now known to the same precision as the Dirichlet determinants of [1], for every mass $m^2$.
At the special masses $m^2=-\eta q(q+1)/L^2$, both boundary-condition determinants have exact closed forms; on the hemisphere these masses coincide with Neumann eigenvalues, so the determinant vanishes.
The hemisphere result (D.18) agrees with the geodesic-boundary gluing computation of [17], providing a consistency check on the BFK-type relation in a case where both sides are explicit.
The explicit value of $b_0$ in (2.22) means the relation (2.8) is fully determined for these disks, removing any free normalization constant from the main-text derivation.
The independent Sturm-Liouville derivation of Appendix C confirms the series (4.4), so the result does not rest solely on the imported gluing formula.

Reading between the lines

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

A direct numerical check is immediate: for fixed $\eta$, $r_0$, and $m^2$, truncate the series (4.3)-(4.4) and compare with an eigenvalue product computed by Sturm-Liouville shooting; agreement would independently validate the zeta-regularized sums.
Because the only new ingredient is the Dirichlet-to-Neumann determinant, the same strategy should transfer to other axisymmetric two-dimensional domains (annuli, spherical caps, surfaces of revolution) once their Dirichlet determinants are known.
The hemisphere vanishing at $m^2=-q(q+1)/L^2$ suggests a spectral interpretation that could be checked directly: if $q(q+1)/L^2$ is an eigenvalue of the Neumann Laplacian for every $q\in\mathbb{N}$, the determinantal zero is forced; the paper proves this for the hemisphere in Appendix D.
For the hyperbolic disk ($\eta=-1$) the special masses are positive, and (5.24) predicts the $q=1$ determinant is positive everywhere; checking that positivity analytically in the general-$q$ expression would test the analytic-continuation prescription used for $\eta=+1$.

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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

The short version: this is a solid extension of the author's earlier Dirichlet work to Neumann boundary conditions, and the main series representations (4.3)-(4.4) look right—I spot-checked the flat-case algebra and it closes. The soft spot is Section 5: the special-mass formulas for η=+1 require analytic continuation to negative m², and the paper only asserts it.

What's actually new: the Neumann determinants for arbitrary mass via the BFK/DN-map relation, the exact special-mass formulas (5.20), and the hemisphere Barnes-G expression (D.18). The alternative Sturm-Liouville computation in Appendix C is a genuine cross-check, not padding, and it reproduces (4.4). The hemisphere limit matching [17] is a good sanity check.

The main unresolved issue is the analytic continuation to m²<0. Section 4 derives (4.4) for positive m², where the DN-map eigenvalues are positive and the zeta definition is safe. Section 5 then evaluates at m² = -η q(q+1)/L²; for η=+1 that's negative, and the DN map has negative eigenvalues there, so neither the zeta determinant nor the infinite series is defined without an argument. The sentence saying the expressions can be continued is not a proof. The exact formulas might be right—the formal series evaluation is plausible and the hemisphere check works—but as written the derivation doesn't establish convergence or that the continuation of the series equals the continued determinant. For η=-1, m² is positive, so that half is solid. The BFK formula itself is imported from [19] and not rederived, but Appendix C gives independent support via Sturm-Liouville determinants, so I don't weight that heavily. There's also a likely sign typo in the Gamma product in (5.24); worth fixing.

Who's this for: anyone doing one-loop determinants on constant-curvature disks in JT/Liouville gravity, or working on BFK gluing formulas. It deserves a serious referee: the core computation is substantial and mostly verifiable, and the continuation gap is fixable by either proving convergence for m²<0 or stating the result as a meromorphic continuation/conjecture. I'd engage with it after that's addressed.

Referee Report

2 major / 5 minor

Summary. This paper computes the ζ-function-regularized determinant det_N(Δη+m²) for a massive scalar on round constant-curvature disks (η=0,±1) with Neumann boundary conditions. The strategy is to use the BFK-type gluing formula (2.8), which expresses the difference between the Neumann and Dirichlet determinants as b0 plus the determinant of the Dirichlet-to-Neumann map. The author computes det(O_m²) from Steklov eigenvalues in closed form (Section 3), combines it with the Dirichlet determinants from [1] to obtain infinite-series representations (4.3) and (4.4) for positive m², and then evaluates the curved-case series at the special masses m²=-η q(q+1)/L² to obtain exact expressions in terms of Gamma functions and Legendre polynomials (5.20). An independent Sturm-Liouville derivation (Appendix C) reproduces (4.4), and the hemisphere limit (Appendix D) yields a Barnes G-function expression that vanishes at the special negative masses.

Significance. If correct, this provides one of the few exact massive determinant formulas on disks with non-geodesic boundaries, extending [1] and relevant to Liouville/JT gravity and finite-volume QFT. The main-text derivation is algebraically consistent: combining (4.1), (3.18), and b0=1 reproduces (4.3), and the analogous curved combination gives (4.4); the independent derivation in Appendix C and the hemisphere check against [17] are genuine strengths. The main limitation is the analytic continuation of the η=+1 results from positive to negative m², which is asserted rather than proved; this affects the exact special-mass formulas for the positive-curvature disk.

major comments (2)

[Section 5, paragraph after Eq. (4.4)] The exact special-mass formulas for η=+1 are obtained by evaluating (4.4) at negative m², with the only justification being the sentence that the exponential of the series can be analytically continued to m²≤0. This is a nontrivial step: for m²<0 the operator Δ+m² has a negative eigenvalue (the constant mode), the series in (2.1) and (2.5) are not defined, and v_0(m²)=m²A/ℓ+O(m⁴) from Appendix B becomes negative, so the determinant det(O_m²) in (2.8) is not defined by (2.6) in that region. Moreover, individual logarithms in (4.4) have branch points where g_n^{(+)}(-m²L²) vanishes. Please provide a proof or a precise reference that the zeta-regularized determinant admits the required analytic continuation in m², and that the exponentiated series converges to that continuation on the needed domain; otherwise (5.20), (5.24), and the sign statements in §5.3 remain unsupported for general η=+1 caps.
[Section 5.1, Eqs. (5.2)–(5.17)] The derivation of ∑ W_n(q;zη) uses logarithms of expressions that can be negative or non-real for η=+1 and, as the paper itself notes in §5.3, for η=-1 when z- lies below about -0.077. The paper does not specify a branch of the logarithm and does not demonstrate that the imaginary parts cancel so that (5.20) is real and single-valued on the full allowed zη domain. The sign discussion for the determinant in (5.24) depends on this branch choice. Please specify the branch convention and verify the reality and single-valuedness of the final expression.

minor comments (5)

[Abstract and Introduction] The phrase 'for arbitrary m²' is stronger than what is actually derived: the infinite series representations in Section 4 are established for positive m², while the continuation to m²≤0 is only asserted in Section 5. Please adjust the wording to avoid overstating the domain of validity.
[Eq. (C.64)] There is a typographical error: 'F(1,1,|p|+2,, ηAη/(4πL²))' contains a double comma before the last argument.
[Section 3.1, Eq. (3.1)] The Steklov problem is stated for 'some v≥0', but the later discussion of negative m² would require eigenvalues v that are not necessarily nonnegative. Please state 'for some real v' and explain the restriction separately.
[Section 5.3, after Eq. (5.24)] The statement that the determinant is always positive for η=-1 'as should be expected since m²=2/L² is positive' would be clearer if it explicitly noted that all eigenvalues of Δη+m² are positive in this case; positivity of m² alone does not by itself determine the sign of a regularized determinant.
[References, [19]] Reference [19] is a preprint. If a published version is available, it should be cited; if not, a short statement on the status of the BFK-type formula would help, since the flat-case result (4.3) relies on it and Appendix C independently verifies only the curved cases.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Neumann-determinant derivation is built from an external BFK-type relation, an independent Dirichlet-to-Neumann computation, and quoted Dirichlet results from prior work; the special-mass evaluation is a direct series calculation.

full rationale

The paper's main derivation chain is not circular. The load-bearing identity (2.8), relating Neumann and Dirichlet determinants through the Dirichlet-to-Neumann determinant, is imported from the external work [19] and is not the target of the paper. The coefficient b0 is fixed independently from Weyl-invariance arguments and known flat-disk determinants from [15,16,23], not fitted to the Neumann determinant being computed. The Dirichlet-to-Neumann determinant is calculated from explicit Steklov eigenfunctions in Section 3, with convergence checks and a direct zeta-function verification of the massless determinant. The Dirichlet determinants quoted from [1] are prior, parameter-free results that do not presuppose the Neumann determinant; although [1] shares an author, it is independent support rather than a self-referential input. Section 4 combines these inputs by simple algebra, and Appendix C provides a separate Sturm-Liouville derivation that reproduces (4.4), while Appendix D checks the hemisphere case against the known Dirichlet-to-Neumann result of [17]. The special-mass reduction in Section 5 is a direct evaluation of the already-derived series using a polynomial-root lemma; it does not fit or rename the answer. The main caveats are that the BFK relation from [19] is not rederived and that the analytic continuation to negative m^2 in Section 5 is asserted rather than proved. These are correctness or rigor concerns, not circularity, because the final expressions are not assumed in their own derivation. No step reduces by construction to its inputs, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The derivation imports four types of prior input: the BFK gluing formula from [19], the Weyl-invariance and massless DN-map results from [15,16], the Dirichlet determinants from [1], and an analytic-continuation assumption for negative m^2. None of these is fitted to the target result. No new physical entities are introduced.

assumptions (4)

domain assumption BFK-type gluing formula: ln det_N(Δ+m^2) - ln det_D(Δ+m^2) = b0 + ln det(O_m2) for d=2 (eq. 2.8), proven in [19].
Imported from a cited preprint, not rederived in this paper; used in Section 4 to build the Neumann determinant from the Dirichlet one.
domain assumption Weyl-invariance relations (2.13)-(2.14) from [15] and det'(O0)=ln ℓ_η from [16].
Used in Section 2.2 to fix the mass-independent constant b0 in (2.22).
domain assumption Dirichlet determinant expressions (4.1) and (4.2) from [1].
The Neumann determinant is obtained by adding b0 + ln det(O_m2) to these; [1] is by the same author and F. Ferrari.
ad hoc to paper Analytic continuation of the infinite series to m^2 ≤ 0 for η=+1.
Section 5 uses negative m^2 for positive-curvature disks; the paper asserts the series extends analytically but gives no general proof, only a hemisphere check.

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Cite this review

Pith. "Pith review of Neumann scalar determinants on constant curvature disks." pith.science (2026). https://pith.science/paper/E6DRCFYG

@misc{pith2026250705159,
  author       = {Pith},
  title        = {Pith review of: Neumann scalar determinants on constant curvature disks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E6DRCFYG}},
  note         = {Machine review of arXiv:2507.05159}
}

abstract

Working in the $\zeta$-function regularisation scheme, we find certain infinite series representations of the logarithms of massive scalar determinants, $\det(\Delta+m^{2})$ for arbitrary $m^2$, on finite round disks of constant curvature ($R=\frac{2\eta}{L^2}, \eta=0,\pm1$) with Neumann boundary conditions. The derivation of these representations relies on a relation between the Neumann determinants on the disks and the corresponding Dirichlet determinants via the determinants of the Dirichlet-to-Neumann maps on the boundaries of the disks. We corroborate the results in an appendix by computing the Neumann determinants in an alternative way. In the cases of disks with nonzero curvatures, we show that the infinite series representations reduce to exact expressions for some specific values of $m^2$, viz. $m^2=-\frac{\eta}{L^2}q(q+1)$ with $q\in \mathbb{N}$. Our analysis uses and extends the results obtained in arXiv:2405.14958 for similar Dirichlet determinants on constant curvature disks.

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [17]

Two-dimensional perturbative scalar QFT and Atiyah-Segal gluing

S. Kandel, P. Mnev and K. Wernli, “Two-dimensional perturbative scalar QFT and Atiyah–Segal gluing,”Adv.Theor.Math.Phys.25 (2021) 7, 1847-1952, arXiv: 1912.11202 [math-ph]

work page Pith review arXiv 2021

[2] [19]

The BFK type gluing formula of zeta-determinants for the Robin Boundary condition

K. Kirsten and Y. Lee, “The BFK type gluing formula of zeta-determinants for the Robin Boundary condition,” arXiv:2306.17572

work page Pith review arXiv

[3] [1]

Dirichlet Scalar Determinants On Two- Dimensional Constant Curvature Disks,

S. Chaudhuri and F. Ferrari, “Dirichlet Scalar Determinants On Two- Dimensional Constant Curvature Disks,” Ann. Henri Poincaré (2025), arXiv:2405.14958

arXiv 2025

[4] [2]

Functional determinants in quantum field theory,

G. V. Dunne, “Functional determinants in quantum field theory,” J.Phys.A41 (2008) 304006, arXiv: 0711.1178. 8In fact, 1/Γ(p−q 2 ) vanishes whenever(q−p) is a non-negative even integer. 42

arXiv 2008

[5] [3]

Multiloop Amplitudes for the Bosonic Polyakov String

E. D’Hoker and D. H. Phong, “Multiloop Amplitudes for the Bosonic Polyakov String”,Nucl. Phys.B 269(1986) 205-234

work page 1986

[6] [4]

R-torsion and the Laplacian on Riemannian man- ifolds,

D. B. Ray and I. M. Singer, “R-torsion and the Laplacian on Riemannian man- ifolds,” Advances in Mathematics, 7 (1971) 145-210

work page 1971

[7] [5]

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J. S. Dowker and R. Critchley, “Effective Lagrangian and Energy Momentum Tensor in de Sitter Space,”Phys. Rev.D 13(1976) 3224

work page 1976

[8] [6]

S. W. Hawking,Zeta function regularization of path integrals in curved space- time, Comm. Math. Phys.55 (1977) 133

work page 1977