Pith. sign in

REVIEW 1 major objections 5 minor 66 references

An MST-style series solution works for the reduced confluent Heun equation of 5D Schwarzschild-Tangherlini scalars, and matches the Seiberg-Witten value of the renormalized angular momentum.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · grok-4.5

2026-07-10 10:07 UTC pith:4VLLDE3J

load-bearing objection Solid original MST extension for the reduced confluent Heun equation in 5d ST, cross-checked against Seiberg-Witten, with useful geodesic and flux side results. the 1 major comments →

arxiv 2607.08280 v1 pith:4VLLDE3J submitted 2026-07-09 gr-qc

5d Schwarzschild-Tangherlini spacetime: MST-like formalism for a Reduced Confluent Heun Equation

Donato Bini , Giorgio Di Russo , Veronica Fantini This is my paper
classification gr-qc
keywords Schwarzschild-Tangherlinireduced confluent Heun equationMano-Suzuki-Takasugi formalismrenormalized angular momentumSeiberg-Witten curvescalar energy fluxscattering angleLyapunov exponent
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper studies the five-dimensional Schwarzschild-Tangherlini black hole both at the level of probe geodesics and at the level of massless scalar waves. For unbound equatorial geodesics it gives closed-form resummed expressions for the scattering angle and the radial action in terms of hypergeometric functions, and it extracts the Lyapunov exponent of the unstable photon sphere for an eikonal estimate of quasinormal frequencies. The central technical claim is that the radial wave equation, which is a reduced confluent Heun equation rather than the ordinary confluent Heun equation of four-dimensional Schwarzschild, admits a direct generalization of the Mano-Suzuki-Takasugi (MST) series construction. The resulting five-term recurrence for the expansion coefficients is solved for the renormalized angular momentum ν; the series for ν agrees, order by order in the small gauge coupling q = M^{2}ω^{2}/4, with an independent calculation that maps the same radial equation onto a quantum Seiberg-Witten curve. As a first application the authors compute the scalar energy flux from circular orbits through 2.5 post-Newtonian order. A reader who cares about higher-dimensional black-hole perturbation theory or about analytic methods for Heun-type equations now has an explicit, checkable MST-like toolkit for the five-dimensional case.

Core claim

The radial equation for massless scalar perturbations of five-dimensional Schwarzschild-Tangherlini is a reduced confluent Heun equation. The authors construct an original MST-like series solution for that equation, obtain the associated five-term recurrence, and show that the renormalized angular momentum ν extracted from the recurrence coincides with the fundamental cycle of the corresponding quantum Seiberg-Witten curve.

What carries the argument

MST-like series for the reduced confluent Heun equation: the radial function is expanded as a sum of shifted Gauss hypergeometric (ingoing) or Bessel (outgoing) functions whose coefficients satisfy a five-term recurrence that can be split into two identical three-term recurrences; the continuous-fraction condition on that recurrence determines ν.

Load-bearing premise

The five-term recurrence can be consistently reduced to its even sector alone (setting the first odd coefficient to zero) while still solving the original radial differential equation.

What would settle it

Compute the next few orders of ν from the continuous-fraction condition of the MST-like recurrence and from the quantum Seiberg-Witten cycle; any discrepancy that cannot be removed by redefinition of the gauge coupling would falsify the claimed agreement.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 5 minor

Summary. The paper studies the five-dimensional Schwarzschild-Tangherlini geometry, combining geodesic analysis with massless scalar perturbations. In the probe limit it obtains the scattering angle for unbound equatorial geodesics (PN and PM expansions, resummed via hypergeometric functions), the associated radial action, geodesic deviations and transport laws, and an eikonal estimate of QNMs from the Lyapunov exponent of the critical circular null orbit. The central technical contribution is an original MST-like construction for the reduced confluent Heun equation that governs the radial scalar wave equation: five-term recurrences for the in and up solutions are derived, reduced to three-term even/odd sectors, and used to extract the renormalized angular momentum ν. This ν is shown to agree order-by-order in q = M^{2}ω^{2}/4 with the independent quantum Seiberg-Witten cycle a (via ν = 2a - 1). As an application the scalar energy flux from circular equatorial orbits is computed through O(η^{10}) (2.5PN).

Significance. If the construction is correct, the work supplies the first explicit MST-type formalism for the reduced confluent Heun equation that appears in five-dimensional ST scalar perturbations and in several fuzzball geometries. The order-by-order match between the MST recurrence and the SW quantization condition (Eqs. (5.14) and (6.29)) is a non-trivial, parameter-free consistency check that strengthens both approaches. The geodesic results (scattering angle, radial action, Lyapunov exponent) are independently cross-checked and provide useful benchmarks. The flux formulae through 2.5PN give concrete data for future five-dimensional self-force studies. The paper is therefore a solid technical contribution to higher-dimensional black-hole perturbation theory.

major comments (1)
  1. After Eq. (6.30) the authors assert that only the even sector of the five-term recurrence is retained (a_{1} = 0) in order for the series (6.8) and (6.18) to solve the original radial ODE (4.6). This truncation is presented as an empirical consistency condition rather than derived from a general property of the RCHE. While the resulting ν series matches the SW cycle at the orders shown, a short argument (or explicit verification that the odd coefficients vanish identically when the recurrence coefficients of (6.10) are used) would make the construction fully rigorous and would clarify whether the same truncation applies to other RCHE problems (e.g., fuzzballs).
minor comments (5)
  1. Table I: the eikonal column for ℓ = 0 is listed as 0.536252 - 0.376968i while the analytic formula (3.46) is singular at ℓ = 0; a clarifying footnote would help.
  2. Eq. (6.31) proposes a dimensional guess ν = (d-3)(a-1/2). The text correctly flags it as unproven; either remove the guess or add a one-line check for d = 6 if available.
  3. Notation: the same symbol M is used for the ST mass parameter and (in the appendix) for the ADM mass; a brief remark distinguishing them would avoid confusion.
  4. Several long displayed expansions (e.g., (2.31), Table II) would benefit from being moved to an appendix or supplemental file to improve readability of the main text.
  5. Typographical: 'formal ism' in the title on page 1; 'updted' in Sec. V; 'recon-struction' hyphenation in Sec. VI C.

Circularity Check

0 steps flagged

No significant circularity: MST-like ν and SW cycle a are independent constructions on the same RCHE; agreement is a cross-check, not a definitional identity.

full rationale

The central claim is an original MST-style series solution for the reduced confluent Heun radial equation of STd5, validated by order-by-order agreement of the renormalized angular momentum ν with the independent quantum Seiberg-Witten fundamental cycle a (via ν = 2a - 1). Both routes start from the same differential equation (the RCHE obtained by separation of the massless scalar wave equation on the ST metric) but use distinct techniques: (i) a five-term recurrence from hypergeometric/Bessel ansätze reduced to a continued-fraction condition on ν, and (ii) the Nekrasov-Shatashvili quantization of the (0,2) SW curve with the dictionary m1 = -m2 = i√q, u = (ℓ + 1/2)^{2} - q, q = M^{2}ω^{2}/4. Neither method fits parameters to the other; the expansions (generic ℓ in (5.14), ℓ = 2 in (6.29)) are computed separately and then compared. Geodesic results (scattering angle, radial action, Lyapunov exponent) follow directly from the metric and Hamilton-Jacobi separation with no free parameters. The even-sector truncation a1 = 0 is an empirical consistency requirement that the series satisfy the original ODE; it does not force the SW match. Self-citations to related topological-star/fuzzball work are background, not load-bearing for the ν agreement. The derivation chain is therefore self-contained against external benchmarks and exhibits no self-definitional, fitted-prediction, or uniqueness-import circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 1 invented entities

The paper rests on the vacuum Einstein equations in five dimensions, the standard separation of the massless Klein-Gordon equation on a static spherically symmetric background, and the classical theory of three-term recurrences for hypergeometric-type series. No free parameters are fitted; the only new technical objects are the even/odd split of the five-term recurrence and the dictionary that maps the radial equation onto a (0,2) Seiberg-Witten curve.

axioms (3)
  • domain assumption The five-dimensional vacuum Einstein equations admit the Schwarzschild-Tangherlini metric as the unique static, spherically symmetric, asymptotically flat black-hole solution.
    Invoked from the outset (Sec. II) as the background geometry; standard uniqueness theorems are assumed.
  • standard math The massless Klein-Gordon equation separates in the coordinates (t,r,θ,φ,ψ) with angular eigenfunctions given by Wigner D-matrices on S³.
    Used to reduce the problem to a single radial ODE (Eq. (4.6)); follows from the isometries of the metric.
  • domain assumption The renormalized angular momentum ν is uniquely fixed by the requirement that the infinite continued fraction of the three-term recurrence vanish.
    Standard MST logic applied after the even/odd split (Eq. (6.28)); the uniqueness of the root is assumed rather than proved for the reduced equation.
invented entities (1)
  • MST-like five-term recurrence for the reduced confluent Heun equation independent evidence
    purpose: To generate analytic series solutions (in and up) that satisfy both horizon and infinity boundary conditions.
    Constructed in Sec. VI by rewriting the RCHE so that a source term proportional to the gauge coupling appears; the recurrence coefficients are new.

pith-pipeline@v1.1.0-grok45 · 39288 in / 2496 out tokens · 26123 ms · 2026-07-10T10:07:37.372959+00:00 · methodology

0 comments
read the original abstract

We study the five-dimensional Schwarzschild-Tangherlini solution, with particular attention to its geodesic structure and massless scalar perturbations. In the probe limit, we present two applications. First, we compute the scattering angle for unbound geodesics showing both post-Newtonian and post-Minkowskian type expansions, and succeeding in resumming the resulting series in terms of hypergeometric functions. Second, we derive the Lyapunov exponent for deviations from a critical circular orbit, which is relevant to the eikonal estimation of quasinormal modes. We then investigate the dynamics of massless scalar $(s=0)$ perturbations, for which the radial equation becomes a Reduced Confluent Heun equation. In this $d=5$ Schwarzschild-Tangherlini case we develop an original extension of the standard Mano-Suzuki-Takasugi (MST) formalism and validate the construction by computing the renormalized angular-momentum parameter $\nu$, whose value agrees with an independent determination based on the quantum Seiberg-Witten formalism. Finally, we analyze the energy flux from circular orbits, obtaining post-Newtonian results through 2.5PN order.

Figures

Figures reproduced from arXiv: 2607.08280 by Donato Bini, Giorgio Di Russo, Veronica Fantini.

Figure 1
Figure 1. Figure 1: FIG. 1: The Newton’s polygon of Eq. (6.4) at [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: The Newton’ polygon of Eq. (6.15) at [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

66 extracted references · 66 canonical work pages · 35 internal anchors

  1. [1]

    (5.1) In the SW perspective, the confluence corresponds to the decoupling of the hypermultiplet of mass m3

    − 2Q(1 − m1 − m2) + 4u 4y(1 +y) . (5.1) In the SW perspective, the confluence corresponds to the decoupling of the hypermultiplet of mass m3. Opera- tionally, this is implemented by taking the double scal- ing limit Q → 0 and m3 → ∞ , while keeping the product Qm3 as fixed, and identifying it with the new gauge cou- 11 pling q = −Qm3. The resulting curve is...

    work page

  2. [2]

    + 4u 4(1 +y) + − 1 + 2(m2 1 +m2

    work page

  3. [3]

    shift relation

    + 4(q − u) 4y , (5.2) through the introduction of the following dimensionless coordinate r =M √ 1 +y. (5.3) The dictionary for such a correspondence is the following m1 = −m2 =i√ q, u = ( ℓ + 1 2 ) 2 − q, (5.4) with the gauge theory coupling q = M 2ω 2 4 . (5.5) Some brief reminders are in order. In the non commutative Nekrasov-Shatashvili back- ground [4...

    work page

  4. [4]

    The orthogonality relation of the spherical harmonics on S3 ∫ S3 dΩ 3Yℓmφmψ Y ∗ ℓ′m′ φm′ ψ =δℓℓ′δmφm′ φ δmψ m′ ψ . (7.16)

    work page

  5. [5]

    The completeness relation for spherical harmonics on S3 δS3(χ,χ ′) = ∑ ℓmφmψ Yℓmφmψ (χ )Y ∗ ℓmφmψ (χ ′), (7.17) where χ = (θ,φ,ψ ) is a commonly used notation for all the angular variables. The non homogeneous equation (7.1) in presence of the source (7.2 ) can be made explicit as follows □Ψ = ∑ ℓmφmψ ∫ dω 2πe− iωtYℓmφmψ (θ,φ,ψ )LrRℓmφmψ ω (r) = − 4π ∫ dω...

    work page

  6. [6]

    eikonal limit

    Radial action To this end, let start from Eq. (A8) where the redshift factor reads f (r) = 1 − ( M r ) d− 3 . (A15) Assuming R(r) = r1− d 2 √ M 2 − Mdr3− dψ (r) (A16) one gets the following normal form ψ ′′(r) + QW, dψ (r) = 0 , (A17) where QW, d = 1 4r2 (r3Md − M 3rd)2 [ 4M 6r2(1+d)ω 2 + 2 ( d2 + 2d(ℓ − 3) + 2(ℓ − 3)ℓ + 8 ) Md+3rd+3 − M 6(d+2ℓ− 4)(d+2ℓ− ...

    work page

  7. [7]

    (A32) Hence, the Fox-Wright function is a genuine extension of the generalized hypergeometric function

    Relation with the Generalized Hypergeometric Function When Ai = Bj = 1, the Fox-Wright function reduces to pΨ q = ∏p i=1 Γ(ai) ∏q j=1 Γ(bj) pFq(a1,...,a p;b1,...,b q;z). (A32) Hence, the Fox-Wright function is a genuine extension of the generalized hypergeometric function. 22

    work page

  8. [8]

    (A34) The above series (A27) converges for every complex value of z whenever D> 0

    Convergence Let us introduce the notation Ap = p∑ i=1 Ai, Bq = q∑ j=1 Bj, (A33) and let us denote D = 1 + Bq − A p. (A34) The above series (A27) converges for every complex value of z whenever D> 0. If D = 0 the series has a finite ra- dius of convergence depending on the parameters while when D < 0 the series generally diverges for z ̸= 0, al- though it m...

    work page

  9. [9]

    (A35) The Mittag-Leffler Function The two-parameter Mittag-Leffler function can be written as Eα,β (z) = ∞∑ k=0 zk Γ(αk +β ) = 1Ψ 1 [ (1, 1) (β,α );z ]

    Special Cases The Wright Function The Wright function is obtained as 0Ψ 1 [ − (β,α );z ] = ∞∑ k=0 zk k! Γ(αk +β ). (A35) The Mittag-Leffler Function The two-parameter Mittag-Leffler function can be written as Eα,β (z) = ∞∑ k=0 zk Γ(αk +β ) = 1Ψ 1 [ (1, 1) (β,α );z ] . (A36) The Generalized Hypergeometric Function When all step parameters are equal to one, one...

    work page

  10. [10]

    Relation with the Fox H-Function The Fox-Wright function can often be represented as a particular case of the Fox H-function, pΨ q(z) = H 1,p p,q +1(−z | · · ·), (A38) which places it within the hierarchy of generalized special functions together with the Meijer G-function and the generalized hypergeometric function. The Fox-Wright function therefore prov...

    work page

  11. [11]

    dimensional regularization

    PN-type solution to the radial equation (A8) We will not aim at discussing here in detail the generic d solution in comparison with the corresponding one in d = 4, since it deserves a specific treatment as well as ex- plicit examples in a context of dimensional regularization when d = 4 + ǫ. Dedicated studies on this topic are still in progress and will be...

    work page

  12. [12]

    (2.2)) ds2 =dθ2 + sin2θdφ 2 + cos2θdψ 2, (B1) 23 where the coordinates are xa = ( θ,φ,ψ ) vary in the ranges θ ∈ [0, π 2 ), φ,ψ ∈ [0, 2π )

    Metric and curvature Let us write the metric of S3 in the form (see Eq. (2.2)) ds2 =dθ2 + sin2θdφ 2 + cos2θdψ 2, (B1) 23 where the coordinates are xa = ( θ,φ,ψ ) vary in the ranges θ ∈ [0, π 2 ), φ,ψ ∈ [0, 2π ). The spacetime (B1) is an Einstein manifold because of the property Rab − 2gab = 0, R = 6, (B2) also implying Rab;c = 0. The independent Riemann t...

    work page

  13. [13]

    M. Picone

    Wigner matrices on S3 ≃ SU (2) SinceS3 is diffeomorphic to the compact group SU (2), the Wigner matrices Dℓ m1m2(g), with g ∈ SU (2), form an orthogonal basis of the space L2(SU (2)). Parametrizing g by the Euler angles, say ( φ,θ,ψ ), and introducing the generators of the rotations Ji g =e− iφJze− iθJye− iψJz, (B13) the matrix elements of the irreducible,...

    work page

  14. [14]

    Schwarzschild field in n dimensions and the dimensionality of space problem,

    F. R. Tangherlini, “Schwarzschild field in n dimensions and the dimensionality of space problem,” Nuovo Cim. 27, 636-651 (1963) doi:10.1007/BF02784569

    work page doi:10.1007/bf02784569 1963

  15. [15]

    Black strings and p-branes are unstable,

    R. Gregory and R. Laflamme, “Black strings and p-branes are unstable,” Phys. Rev. Lett. 70, 2837-2840 (1993) doi:10.1103/PhysRevLett.70.2837 [arXiv:hep- th/9301052 [hep-th]]

    work page internal anchor Pith review doi:10.1103/physrevlett.70.2837 1993

  16. [16]

    Black strings and p-branes,

    G. T. Horowitz and A. Strominger, “Black strings and p-branes,” Nuclear Physics B 360-1, 197-209 (1991) doi:10.1016/0550-3213(91)90440-9

    work page doi:10.1016/0550-3213(91)90440-9 1991

  17. [17]

    Black Holes in Higher Di- mensional Space-Times,

    R. C. Myers and M. J. Perry, “Black Holes in Higher Di- mensional Space-Times,” Annals Phys. 172, 304 (1986) doi:10.1016/0003-4916(86)90186-7

    work page doi:10.1016/0003-4916(86)90186-7 1986

  18. [18]

    Instability of non-supersymmetric smooth geometries

    V. Cardoso, O. J. C. Dias, J. L. Hovdebo and R. C. Myers, “Instability of non-supersymmetric smooth geometries,” Phys. Rev. D 73, 064031 (2006) doi:10.1103/PhysRevD.73.064031 [arXiv:hep-th/0512277 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.73.064031 2006

  19. [19]

    Bianchi, M

    M. Bianchi, M. Casolino and G. Rizzo, Nucl. Phys. B 954, 115010 (2020) doi:10.1016/j.nuclphysb.2020.115010 [arXiv:1904.01097 [hep-th]]

    work page doi:10.1016/j.nuclphysb.2020.115010 2020

  20. [20]

    2-charge circular fuzz-balls and their perturbations

    M. Bianchi and G. Di Russo, “2-charge circular fuzz- balls and their perturbations,” JHEP 08, 217 (2023) doi:10.1007/JHEP08(2023)217 [arXiv:2212.07504 [hep- th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep08(2023)217 2023

  21. [21]

    Charge instability of JMaRT geometries

    M. Bianchi, C. Di Benedetto, G. Di Russo and G. Sudano, “Charge instability of JMaRT geometries,” JHEP 09, 078 (2023) doi:10.1007/JHEP09(2023)078 [arXiv:2305.00865 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep09(2023)078 2023

  22. [22]

    Tidal resonances for fuzzballs

    G. Di Russo, F. Fucito and J. F. Morales, “Tidal resonances for fuzzballs,” JHEP 04, 149 (2024) doi:10.1007/JHEP04(2024)149 [arXiv:2402.06621 [hep- th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep04(2024)149 2024

  23. [23]

    Topological Stars, Black holes and Generalized Charged Weyl Solutions

    I. Bah and P. Heidmann, “Topological stars, black holes and generalized charged Weyl solutions,” JHEP 09, 147 (2021) doi:10.1007/JHEP09(2021)147 [arXiv:2012.13407 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep09(2021)147 2021

  24. [24]

    On the stability and deformability of top stars

    M. Bianchi, G. Di Russo, A. Grillo, J. F. Morales and G. Sudano, “On the stability and de- formability of top stars,” JHEP 12, 121 (2023) doi:10.1007/JHEP12(2023)121 [arXiv:2305.15105 [gr- qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep12(2023)121 2023

  25. [25]

    Cavity effect in the quasinormal mode spectrum of topological stars

    P. Heidmann, N. Speeney, E. Berti and I. Bah, “Cavity effect in the quasinormal mode spectrum of topolog- ical stars,” Phys. Rev. D 108, no.2, 024021 (2023) doi:10.1103/PhysRevD.108.024021 [arXiv:2305.14412 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.108.024021 2023

  26. [26]

    Charge (in)stability and superradiance of Topological Stars

    A. Cipriani, C. Di Benedetto, G. Di Russo, A. Grillo and G. Sudano, “Charge (in)stability and superra- diance of Topological Stars,” JHEP 07, 143 (2024) doi:10.1007/JHEP07(2024)143 [arXiv:2405.06566 [hep- th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep07(2024)143 2024

  27. [27]

    Scalar perturbations in a Top-Star spacetime

    M. Bianchi, D. Bini and G. Di Russo, “Scalar perturba- tions of topological-star spacetimes,” Phys. Rev. D 110, no.8, 084077 (2024) doi:10.1103/PhysRevD.110.084077 [arXiv:2407.10868 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.110.084077 2024

  28. [28]

    Scalar waves in a Topological Star spacetime: self-force and radiative losses

    M. Bianchi, D. Bini and G. Di Russo, “Scalar waves in a topological star spacetime: Self-force and radia- tive losses,” Phys. Rev. D 111, no.4, 044017 (2025) doi:10.1103/PhysRevD.111.044017 [arXiv:2411.19612 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.111.044017 2025

  29. [29]

    Scalar waves from unbound orbits in a TS spacetime: PN reconstruction of the field and radiation losses in a self-force approach

    G. Di Russo, M. Bianchi and D. Bini, “Scalar waves from unbound orbits in a topological star spacetime: PN re- construction of the field and radiation losses in a self- force approach,” Phys. Rev. D 112, no.2, 024002 (2025) doi:10.1103/sycf-brn1 [arXiv:2502.21040 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/sycf-brn1 2025

  30. [30]

    Asymptot- ically flat rotating topological stars,

    P. Heidmann, P. Pani and J. E. Santos, “Asymptot- ically flat rotating topological stars,” JHEP 03, 108 (2026) doi:10.1007/JHEP03(2026)108 [arXiv:2510.05200 [hep-th]]

    work page doi:10.1007/jhep03(2026)108 2026

  31. [31]

    Rotating Topological Stars

    M. Bianchi, G. Dibitetto, J. F. Morales and A. Ruip´ erez , “Rotating Topological Stars,” JHEP 01, 046 (2026) doi:10.1007/JHEP01(2026)046 [arXiv:2504.12235 [hep- th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep01(2026)046 2026

  32. [32]

    W-solitons as prototypical black hole microstates,

    A. Dima, P. Heidmann, M. Melis, P. Pani and G. Patashuri, “W-solitons as prototypical black hole microstates,” Phys. Rev. D 112, no.12, 124056 (2025) doi:10.1103/2wcq-4xny [arXiv:2509.18245 [gr-qc]]

    work page doi:10.1103/2wcq-4xny 2025

  33. [33]

    Analytic Solutions of the Regge-Wheeler Equation and the Post-Minkowskian Expansion

    S. Mano, H. Suzuki and E. Takasugi, “Analytic so- lutions of the Regge-Wheeler equation and the post- Minkowskian expansion,” Prog. Theor. Phys. 96, 549-566 (1996) doi:10.1143/PTP.96.549 [arXiv:gr-qc/9605057 [gr- qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1143/ptp.96.549 1996

  34. [34]

    Analytic Solutions of the Teukolsky Equation and their Low Frequency Expansions

    S. Mano, H. Suzuki and E. Takasugi, “Analytic solu- tions of the Teukolsky equation and their low frequency expansions,” Prog. Theor. Phys. 95, 1079-1096 (1996) doi:10.1143/PTP.95.1079 [arXiv:gr-qc/9603020 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1143/ptp.95.1079 1996

  35. [35]

    Analytic black hole per- turbation approach to gravitational radiation,

    M. Sasaki and H. Tagoshi, “Analytic black hole per- turbation approach to gravitational radiation,” Living Rev. Rel. 6, 6 (2003) doi:10.12942/lrr-2003-6 [arXiv:gr- qc/0306120 [gr-qc]]

    work page doi:10.12942/lrr-2003-6 2003

  36. [36]

    The mathematical theory of black holes,

    S. Chandrasekhar, “The mathematical theory of black holes,” Oxford Classic Texts in the Physical Sciences, Ed. revis. 1998, Oxford University Press, USA

    work page 1998

  37. [37]

    Notes on AdS-Schwarzschild eikonal phase

    A. Parnachev and K. Sen, “Notes on AdS- Schwarzschild eikonal phase,” JHEP 03, 289 (2021) doi:10.1007/JHEP03(2021)289 [arXiv:2011.06920 [hep- th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep03(2021)289 2021

  38. [38]

    Resummation of Universal Tails in Gravitational Waveforms

    M. M. Ivanov, Y. Z. Li, J. Parra-Martinez and Z. Zhou, “Resummation of Universal Tails in Gravitational Wave- forms,” Phys. Rev. Lett. 135, no.14, 141401 (2025) doi:10.1103/jzd1-qzkt [arXiv:2504.07862 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/jzd1-qzkt 2025

  39. [39]

    Topological Stars and scalar wave equation: Exact resummation of the renormalized angular momentum in the eikonal limit

    D. Bini and G. Di Russo, “Topological stars and scalar wave equation: Exact resummation of the renormal- ized angular momentum in the eikonal limit,” Phys. Rev. D 112, no.6, 064008 (2025) doi:10.1103/dw5y-4pv8 [arXiv:2506.14442 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/dw5y-4pv8 2025

  40. [40]

    Kerr spacetime and scalar wave equation: Exact resummation of the renormalized angular momentum in the eikonal limit

    D. Bini, G. Di Russo and A. Geralico, “Kerr space- time and scalar wave equation: Exact resumma- tion of the renormalized angular momentum in the eikonal limit,” Phys. Rev. D 112, no.6, 064077 (2025) doi:10.1103/mzqw-wbvf [arXiv:2508.12046 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/mzqw-wbvf 2025

  41. [41]

    Higher Order Relativistic Periastron Advances and Binary Pulsars,

    T. Damour and G. Schaefer, “Higher Order Relativistic Periastron Advances and Binary Pulsars,” Nuovo Cim. B 101, 127 (1988) doi:10.1007/BF02828697

    work page doi:10.1007/bf02828697 1988

  42. [42]

    Characterizing geodesic deviations in a Topological Star spacetime: massive, charged, spinning and stringy-like objects

    D. Bini and G. Di Russo, “Characterizing geodesic deviations in a topological star spacetime: Massive, charged, spinning, and stringylike objects,” Phys. Rev. D 112, no.2, 024021 (2025) doi:10.1103/6xc4-wm2f [arXiv:2505.13020 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/6xc4-wm2f 2025

  43. [43]

    Geometric transport along circular orbits in stationary axisymmetric spacetimes

    D. Bini, C. Cherubini, G. Cruciani and R. T. Jantzen, “Geometric transport along circular orbits in station- 26 ary axisymmetric spacetimes,” Int. J. Mod. Phys. D 13, 1771-1804 (2004) doi:10.1142/S0218271804005237 [arXiv:gr-qc/0407004 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1142/s0218271804005237 2004

  44. [44]

    Geodesic stability, Lyapunov exponents and quasinormal modes

    V. Cardoso, A. S. Miranda, E. Berti, H. Witek and V. T. Zanchin, “Geodesic stability, Lyapunov expo- nents and quasinormal modes,” Phys. Rev. D 79, no.6, 064016 (2009) doi:10.1103/PhysRevD.79.064016 [arXiv:0812.1806 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.79.064016 2009

  45. [45]

    More on the SW-QNM correspondence

    M. Bianchi, D. Consoli, A. Grillo and J. F. Morales, “More on the SW-QNM correspondence,” JHEP 01, 024 (2022) doi:10.1007/JHEP01(2022)024 [arXiv:2109.09804 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep01(2022)024 2022

  46. [46]

    Breit-Wigner resonances and the quasinormal modes of anti-de Sitter black holes

    E. Berti, V. Cardoso and P. Pani, “Breit-Wigner res- onances and the quasinormal modes of anti-de Sit- ter black holes,” Phys. Rev. D 79, 101501 (2009) doi:10.1103/PhysRevD.79.101501 [arXiv:0903.5311 [gr- qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.79.101501 2009

  47. [47]

    Advanced Methods in Black-Hole Perturbation Theory

    P. Pani, “Advanced Methods in Black-Hole Perturba- tion Theory,” Int. J. Mod. Phys. A 28, 1340018 (2013) doi:10.1142/S0217751X13400186 [arXiv:1305.6759 [gr- qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1142/s0217751x13400186 2013

  48. [48]

    Light rings as observational evidence for event horizons: long-lived modes, ergoregions and nonlinear instabilities of ultracompact objects

    V. Cardoso, L. C. B. Crispino, C. F. B. Macedo, H. Okawa and P. Pani, “Light rings as observa- tional evidence for event horizons: long-lived modes, ergoregions and nonlinear instabilities of ultracom- pact objects,” Phys. Rev. D 90, no.4, 044069 (2014) doi:10.1103/PhysRevD.90.044069 [arXiv:1406.5510 [gr- qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.90.044069 2014

  49. [49]

    Eigenvalues and eigenfunctions of spin-weighted spheroidal harmonics in four and higher dimensions

    E. Berti, V. Cardoso and M. Casals, “Eigenvalues and eigenfunctions of spin-weighted spheroidal harmonics in four and higher dimensions,” Phys. Rev. D 73, 024013 (2006) [erratum: Phys. Rev. D 73, 109902 (2006)] doi:10.1103/PhysRevD.73.109902 [arXiv:gr-qc/0511111 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.73.109902 2006

  50. [50]

    Irregular Liouville correlators and connection formulae for Heun functions

    G. Bonelli, C. Iossa, D. Panea Lichtig and A. Tanzini, “Irregular Liouville Correlators and Connection For- mulae for Heun Functions,” Commun. Math. Phys. 397, no.2, 635-727 (2023) doi:10.1007/s00220-022-04497- 5 [arXiv:2201.04491 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/s00220-022-04497- 2023

  51. [51]

    CFT description of BH's and ECO's: QNMs, superradiance, echoes and tidal responses

    D. Consoli, F. Fucito, J. F. Morales and R. Poghossian, “CFT description of BH’s and ECO’s: QNMs, super- radiance, echoes and tidal responses,” JHEP 12, 115 (2022) doi:10.1007/JHEP12(2022)115 [arXiv:2206.09437 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep12(2022)115 2022

  52. [52]

    Scalar self- force effects in neutral W-soliton backgrounds,

    M. Bianchi, D. Bini and G. Di Russo, “Scalar self- force effects in neutral W-soliton backgrounds,” Phys. Rev. D 113, no.4, 044028 (2026) doi:10.1103/rf8g-yl2d [arXiv:2511.01402 [gr-qc]]

    work page doi:10.1103/rf8g-yl2d 2026

  53. [53]

    Regular and Floquet bases for gauge and gravity theories: a non perturbative approach

    D. Fioravanti and M. Rossi, “Regular and Floquet bases for gauge and gravity theories: a non perturbative ap- proach,” [arXiv:2508.19960 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv

  54. [54]

    Quantization of Integrable Systems and Four Dimensional Gauge Theories

    N. A. Nekrasov and S. L. Shatashvili, “Quantization of Integrable Systems and Four Dimensional Gauge Theories,” Contribution to: ICMP09, 265-289 (2009) doi:10.1142/9789814304634 0015 [arXiv:0908.4052 [hep- th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1142/9789814304634 2009

  55. [55]

    Divergent Series, Summa- bility and Resurgence I,

    C. Mitschi and D. Sauzin, “Divergent Series, Summa- bility and Resurgence I,” Lect. Notes Math. 2153, pp. xxi+298, Springer, 2016, doi:10.1007/978-3-319-28736-2

    work page doi:10.1007/978-3-319-28736-2 2016

  56. [56]

    Les fonctions r´ esurgentes. Tome I,

    J ´Ecalle, “Les fonctions r´ esurgentes. Tome I,” Publi- cations Math´ ematiques d’Orsay 81 [Mathematical Pub- lications of Orsay 81] , 5, Universit´ e de Paris-Sud, D´ epartement de Math´ ematiques, Orsay, 247 (1981)

    work page 1981

  57. [57]

    S´ eries divergentes et th´ eories asympto- tiques,

    J. P. Ramis “S´ eries divergentes et th´ eories asympto- tiques,” Bull. Soc. Math. France, 121,74 (1993)

    work page 1993

  58. [58]

    Divergent series, summability and resurgence. II

    M. Loday-Richaud, “Divergent series, summability and resurgence. II”, Lect. Notes Math., 2154, pp. xxiii+272, Springer, (2016) doi:10.1007/978-3-319-29075-1

    work page doi:10.1007/978-3-319-29075-1 2016

  59. [59]

    Scalar field self-force effects on a particle orbiting a Reissner-Nordstrom black hole

    D. Bini, G. Carvalho and A. Geralico, “Scalar field self- force effects on a particle orbiting a Reissner-Nordstr¨ om black hole,” Phys. Rev. D 94, no.12, 124028 (2016) doi:10.1103/PhysRevD.94.124028 [arXiv:1610.02235 [gr- qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.94.124028 2016

  60. [60]

    Turning black-holes and D-branes inside out their photon-spheres

    M. Bianchi and G. Di Russo, “Turning black holes and D-branes inside out of their photon spheres,” Phys. Rev. D 105, no.12, 126007 (2022) doi:10.1103/PhysRevD.105.126007 [arXiv:2110.09579 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.105.126007 2022

  61. [61]

    Turning rotating D-branes and BHs inside out their photon-halo

    M. Bianchi and G. Di Russo, “Turning rotating D-branes and black holes inside out their photon- halo,” Phys. Rev. D 106, no.8, 086009 (2022) doi:10.1103/PhysRevD.106.086009 [arXiv:2203.14900 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.106.086009 2022

  62. [62]

    Couch-Torrence conformal inversion, supersymmetry and conserved charges for D3-branes

    M. Akhond, M. Bianchi, A. Cristofaro and F. Riccioni, “Couch-Torrence conformal inversion, supersymmetry and conserved charges for D3-branes,” [arXiv:2604.27815 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv

  63. [63]

    Reduction of a class of Fox-Wright psi functions for certain rational pa- rameters,

    A. R. Miller, I. S. Moskowitz, “Reduction of a class of Fox-Wright psi functions for certain rational pa- rameters,” Computers & Mathematics with Applica- tions, 30, Issue 11, 73-82, (1995) ISSN 0898-1221, https://doi.org/10.1016/0898-1221(95)00165-U

    work page doi:10.1016/0898-1221(95)00165-u 1995

  64. [64]

    Wolfram Research, FoxH, Wolfram Language Documen- tation, https://reference.wolfram.com/language/ ref/FoxH.html

    work page

  65. [65]

    Usseglio, personal communication

    D. Usseglio, personal communication

    work page

  66. [66]

    The supplemental material included both in the arxiv version and in the published version of the present paper contains all data that support the findings of this article

    work page