arxiv: 2605.03960 · v1 · submitted 2026-05-05 · 🧮 math-ph · math.CA· math.MP· math.SP

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Two Regularized Determinants of Laplacian through Resurgence theory

Wen Shen , Shanzhong Sun

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classification 🧮 math-ph math.CAmath.MPmath.SP

keywords resurgence theoryregularized determinantLaplacianBorel-Laplace resummationtheta seriesRiemann manifoldSelberg trace formulaPoisson summation

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The pith

Resurgence theory yields closed formulas for two regularized determinants of the Laplacian.

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

The paper applies resurgence theory to two regularizations of the determinant of the Laplacian on Riemann manifolds. One version is the formal logarithmic derivative of the determinant and the other is an exponentially deformed version. Closed formulas for both are derived by Borel-Laplace resummation of a theta series built from the spectrum of the square root of the Laplacian, with explicit accounting for singularities that appear upon analytic continuation. The same method recovers the Poisson summation formula on the circle and the Selberg trace formula on compact surfaces of genus at least two. It further determines the 1-Gevrey asymptotics of the deformed regularization and proves that the two regularizations share the same derivative as the deformation parameter approaches zero.

Core claim

Under appropriate conditions, the closed formulas for both regularized determinants are established through Borel-Laplace resummation which takes into account the contribution of the singularities along the analytic continuation of Theta series hat Theta_{D_X}. The series resembles the trace of the heat kernel, but is defined via the spectrum of the square-root of the Laplacian.

What carries the argument

Borel-Laplace resummation of the theta series hat Theta_{D_X} that incorporates contributions from singularities in its analytic continuation.

Load-bearing premise

The manifold and spectrum must satisfy conditions that make the singularities of the analytically continued theta series of the precise type needed for the resummation to converge to the regularized determinant.

What would settle it

For the Laplacian on the circle S^1, evaluate the resummed expression and compare it directly with the explicit value given by the Poisson summation formula.

Figures

Figures reproduced from arXiv: 2605.03960 by Shanzhong Sun, Wen Shen.

**Figure 1.** Figure 1: The paths γ − t1,tk and γs,t1 corresponding to the alien operator ∆− ωk . Definition 2.18. For φˆ ∈ Rˆint Ωs , ωk := s + iτk ∈ Ω + s . For any t1, tk ∈ (0, 1), let γs,t1 be the straight line segment from 1 4 ω1 to s + it1τ1, and let γ − t1,tk be the path from s + it1τ1 to s+i(1−tk)τk−1 +itkτk along the vertical line, but circumventing each ωj (1 ≤ j ≤ k −1) from 3 In this paper, we relax the condition that… view at source ↗

**Figure 2.** Figure 2: Deformation of the Laplace integration paths along θ ± into Hankel contours Γωk,ϵ around singularities. can be divided into the sum of integrals along infinite Hankel contours Γωk,ϵ for any 0 < ϵ < δ with 2δ := infωk∈Ω + s d(ωk, ωk+1). From the fact that Ω + s is a closed and discrete set, we have δ > 0. Each contour Γωk,ϵ can be defined as follows: it starts from ωk + e iθ+ ∞ and goes to ωk + ϵeiθ+ along … view at source ↗

**Figure 3.** Figure 3: The holomorphic region Uδ and the singularities of (−t) dΘˆ DX (t). By (9), for − π 2 + ϵ < arg(ρ) < π 2 − ϵ, i m0 ^dm0 d(iρ)m0 logf Det gDX (iρ) + (−i) m0 ^dm0 d(−iρ)m0 logf Det gDX (−iρ) =i m0 Z e (− π 2 +ϵ)i∞ 0 (−t) m0−1Θˆ DX (t)e −iρtdt + (−i) m0 Z e ( π 2 −ϵ)i∞ 0 (−t) m0−1Θˆ DX (t)e iρtdt =i m0 view at source ↗

**Figure 4.** Figure 4: The holomorphic region Uδ (gray shaded) and the singularity distribution of Θˆ DX (t− s0). The yellow dots a and a0 denote the intersection points of the line <(t) = s0 with the rays {arg t = π − ϵ} and {arg t = π 2 + δ}, respectively. other hand, Θˆ DX (t−s0+a) ∈ N π 2 , π 2 + δ , 0 , which is due to the fact that Θˆ DX (t−s0+a0) ∈ N π 2 , π 2 + δ , 0 . Hence, we can apply the Lemma 2.20 to the la… view at source ↗

**Figure 5.** Figure 5: The integral contour γ. The approach used in [21, Theorem 2.4] and [3, §1.4] to derive the Poisson summation formula from (11) is summarized as follows: By multiplying both sides of (11) by a test function h(−iρ) satisfying Theorem 6.1 and integrating over the contour γ shown in view at source ↗

**Figure 6.** Figure 6: The relationship between two regularized determinants from the perspective of the singularities of Θˆ DX (t). Remark 8.2. In fact, the left hand side of Theorem 8.1 is uniformly convergent for any s0 < 0, and the theorem can be proved directly by sending the parameter s0 → 0. However, the reason for employing a more intricate proof strategy, specifically the Laplace transform approach presented here, is to… view at source ↗

**Figure 7.** Figure 7: Meromorphic extension of L (π−ε) Ä sinh s0 cosh t−cosh s0 ä (−iρ) by varying directions of the Laplace transform. Left (t-plane): Gray, red, and yellow rays represent transform directions π−ϵ, π/2, and −ϵ, respectively; the dashed gray rays and circles with arrows indicate the Hankel contours decomposed from the difference between the transforms along π/2 and π − ϵ. Right (ρ-plane): Shaded regions denote t… view at source ↗

**Figure 8.** Figure 8: The integral contour γN . which can in turn be simplified to X N∈Z h(n)e −s0n = lim N→∞ 1 2πi Z γN e is0ρh(−iρ) Å π tanh πρã dρ. On the horizontal segments of γN , where ρ = s ± i(N + 1 2 ) for s ∈ (−η, η), the term π tanh πρ is bounded. Given that h(−iρ) = O(|ρ| −1−δ ) as =(ρ) → −∞ and h(−iρ) = O(e δ ′ |ρ| ) as =(ρ) → +∞, the integrals along these segments satisfy view at source ↗

read the original abstract

We study two types of regularizations of the determinant of Laplacian on Riemann manifold from the viewpoint of resurgence theory. One is the formal logarithmic derivative of the determinant, and the other is its exponential deformation. Under appropriate conditions, the close formulas for both regularized determinant are established through Borel-Laplace resummation which takes into account the contribution of the singularities along the analytic continuation of Theta series $\hat{\Theta}_{D_X}$. The series resembles the trace of the heat kernel, but is defined via the spectrum of the square-root of the Laplacian. As applications, we revisit the well known formal logarithmic derivative of determinant on $S^1$ and compact Riemann surface with higher genus ($\geq2$) corresponding to the Poisson summation formula and Selberg trace formula respectively. Furthermore, the 1-Gevrey asymptotic behavior of the exponential deformation regularization at infinity is considered whose coefficients are determined by the trace of the heat kernel. In the end, we establish the relationship between the two regularized determinants. In fact, they have the same derivatives when the deformation parameter tends to $0$ in exponentially deformed regularization.

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

Resurgence recovers the usual Poisson and Selberg formulas for the two regularized determinants but rests on vague unverified conditions for the theta series singularities.

read the letter

The paper uses Borel-Laplace resummation on a theta series built from the spectrum of the square root of the Laplacian to produce closed forms for the logarithmic derivative of the determinant and for its exponential deformation. Under the stated appropriate conditions it recovers the Poisson summation formula on the circle and the Selberg trace formula on compact surfaces of genus at least 2. It also derives the 1-Gevrey asymptotics of the deformed version from the heat kernel trace and shows that the two regularizations agree in their derivatives as the deformation parameter approaches zero. These steps are laid out clearly and the consistency checks against classical results are useful to see in resurgence language. The explicit link drawn between the singularity data of the analytic continuation and the resummed expressions is the part that feels most concrete. The relation between the two regularizations is a clean observation that follows directly once the resummation is in place. The main limitation is that the argument depends entirely on those appropriate conditions on the manifold, the spectrum, and the analytic properties of the theta series, yet the paper does not list growth bounds, singularity locations, or Gevrey order requirements in enough detail to check them for the claimed cases. Without error estimates or a direct comparison showing that the resummed value equals the standard zeta-regularized determinant, the derivations read as formal rather than fully controlled. Because the final expressions match known formulas, there is no independent numerical or analytic test of whether the Stokes data is being handled correctly. This work is aimed at people already comfortable with resurgence and spectral geometry who might want to try the method on other manifolds. A reader who knows the trace formulas will follow the steps without trouble and can judge whether the conditions can be made explicit. The thinking is straightforward and the citations to prior trace formulas are appropriate. I would bring it to a reading group to walk through the resummation step. I would not cite it in the next year because it mainly re-derives existing results. It deserves peer review so that referees can ask for the missing bounds and a verification that the resummation converges to the determinant rather than merely resembling it.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a resurgence-theoretic approach to two regularized determinants of the Laplacian on Riemann manifolds. It defines a theta series from the spectrum of the square root of the Laplacian and uses Borel-Laplace resummation of its analytic continuation to derive closed formulas for the formal logarithmic derivative of the determinant and an exponentially deformed version, under appropriate conditions. Applications to the circle S^1 and compact Riemann surfaces of genus at least 2 recover the Poisson summation and Selberg trace formulas, respectively. The paper also examines the 1-Gevrey asymptotics of the deformed regularization and establishes that the two regularizations have the same derivatives as the deformation parameter tends to zero.

Significance. If the resummation procedure rigorously recovers the regularized determinants under the stated conditions, the work would provide a valuable bridge between resurgence theory and spectral geometry. It offers a method to obtain closed expressions for determinants via Stokes data from the theta series, with the recovery of classical trace formulas serving as important consistency checks. The analysis of Gevrey asymptotics and the relationship between the two regularizations adds further insight into the analytic properties of these objects.

major comments (2)

[Abstract] Abstract: The central claim that closed formulas for both regularized determinants are established through Borel-Laplace resummation of the analytic continuation of hat Theta_{D_X} rests on unspecified 'appropriate conditions' on the Riemann manifold, the spectrum, and the analytic properties of the theta series (e.g., growth bounds, singularity locations, or Gevrey order). These conditions are not listed or verified for the applications to S^1 and genus >=2 surfaces, yet the entire derivation depends on the singularities permitting the required resummation to equal the regularized determinant.
[Applications] Applications to S^1 and genus >=2 surfaces: The recovery of the Poisson summation formula and Selberg trace formula provides consistency checks but does not include independent verification, error estimates, or direct comparison showing that the resummed series equals the regularized determinant. Since the central claim is that the resummation yields the closed formulas, the absence of such checks leaves the equality unconfirmed beyond known results.

minor comments (2)

In the abstract, 'close formulas' is a typographical error and should be 'closed formulas'.
The 1-Gevrey asymptotic behavior is mentioned but the precise definition of the exponential deformation and its relation to the heat kernel trace should be clarified for readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We agree that the appropriate conditions require explicit statement and will revise accordingly. We also address the verification in the applications by clarifying the role of the recovered formulas.

read point-by-point responses

Referee: [Abstract] Abstract: The central claim that closed formulas for both regularized determinants are established through Borel-Laplace resummation of the analytic continuation of hat Theta_{D_X} rests on unspecified 'appropriate conditions' on the Riemann manifold, the spectrum, and the analytic properties of the theta series (e.g., growth bounds, singularity locations, or Gevrey order). These conditions are not listed or verified for the applications to S^1 and genus >=2 surfaces, yet the entire derivation depends on the singularities permitting the required resummation to equal the regularized determinant.

Authors: We agree that the 'appropriate conditions' must be stated explicitly. In the revised manuscript we will add a new paragraph in the introduction (and update the abstract) that lists the required assumptions: the manifold is compact without boundary, the spectrum of the square-root Laplacian consists of discrete positive eigenvalues with Weyl-type growth, the theta series admits an analytic continuation to a sector with controlled growth at infinity, and the singularities are of exponential type permitting Borel-Laplace resummation to recover the regularized determinant. For the applications to S^1 and genus >=2 surfaces we will explicitly verify that these spectral and analytic properties hold, using the known eigenvalue asymptotics and the resulting trace formulas. revision: yes
Referee: [Applications] Applications to S^1 and genus >=2 surfaces: The recovery of the Poisson summation formula and Selberg trace formula provides consistency checks but does not include independent verification, error estimates, or direct comparison showing that the resummed series equals the regularized determinant. Since the central claim is that the resummation yields the closed formulas, the absence of such checks leaves the equality unconfirmed beyond known results.

Authors: The recovery of the Poisson summation and Selberg trace formulas is not merely a consistency check; it constitutes the verification that the resummation procedure produces the exact closed-form expressions for the regularized determinants. Because these formulas are independently established in the literature, exact matching demonstrates that the Borel-Laplace sum, incorporating the Stokes data from the theta series, equals the known regularized determinant. In the revision we will add a short explanatory paragraph after each application, recalling how the singularity contributions in the resummation directly reproduce the classical formulas. We maintain that further numerical error estimates or independent comparisons are not required for this theoretical derivation, as the equality is exact once the stated conditions are satisfied. revision: partial

Circularity Check

0 steps flagged

No circularity: resurgence resummation connects theta series to determinants without reducing to input by construction

full rationale

The derivation begins with the spectral definition of the theta series hat Theta_{D_X} (resembling a heat-kernel trace but using sqrt(Laplacian) eigenvalues) and applies Borel-Laplace resummation along its analytic continuation to obtain closed formulas for the logarithmic derivative and exponentially deformed regularized determinants. This step relies on the contribution of singularities under explicitly invoked 'appropriate conditions' rather than defining the output in terms of the input. Applications recover the Poisson summation and Selberg trace formulas as consistency checks on known cases (S^1 and genus >=2 surfaces), without the final expressions being tautological rewrites of the starting spectral data. No self-citation chains, fitted parameters renamed as predictions, ansatz smuggling, or uniqueness theorems imported from the authors' prior work appear as load-bearing steps. The relationship between the two regularizations (same derivatives as deformation parameter ->0) follows directly from the shared resummation construction without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard properties of resurgence and Borel summation together with domain assumptions about the analytic continuation of the spectral theta series on Riemann manifolds.

axioms (2)

standard math Borel-Laplace resummation recovers the regularized determinant when singularities of the theta series are properly accounted for
Invoked to obtain the closed formulas from the divergent series.
domain assumption The theta series hat Theta_{D_X} admits an analytic continuation with singularities whose contributions can be isolated
Required for the resummation procedure to produce explicit expressions.

pith-pipeline@v0.9.0 · 5494 in / 1430 out tokens · 52129 ms · 2026-05-07T12:45:56.872931+00:00 · methodology

discussion (0)

Reference graph

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