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REVIEW 2 major objections 5 minor 64 references

At nonzero spatial separation, a residual KMS ambiguity in holographic thermal two-point functions is fixed by a zero-frequency bulk wave equation solved with Heun functions, yielding spin-resolved double-trace coefficients and canceling un

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2026-07-31 13:06 UTC pith:7I2JOZKR

load-bearing objection Solid finite-x extension of the thermal bootstrap: Heun zero mode is clean, spin-resolved b_{m,J} are new, but higher coefficients sit on unexamined Padé-Borel resummation. the 2 major comments →

arxiv 2607.24499 v1 pith:7I2JOZKR submitted 2026-07-27 hep-th

Spin-resolved double-trace thermal coefficients in holography

Ilija Buri\'c , Ivan Gusev , Andrei Parnachev This is my paper
classification hep-th
keywords holographythermal bootstrapdouble-trace operatorsHeun functionsKMS conditionbulk-cone singularitiesthermal two-point functionAdS black hole
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.

Holographic thermal two-point functions were previously fixed at vanishing spatial separation by the stress-tensor OPE sector plus the KMS condition. Away from that limit, KMS alone leaves an undetermined function of spatial separation. This paper shows that residual piece is exactly the zero Matsubara mode, obtained by solving a two-variable bulk wave equation that reduces to a Heun equation in momentum space. The resulting full correlator produces thermal coefficients of double-trace operators resolved by spin, and it cancels complex bulk-cone singularities that appear in the stress-tensor sector alone at spacelike separation. A sympathetic reader cares because this turns an incomplete image-sum construction into an efficient, analytic route to previously inaccessible CFT data and clarifies which singularities are physical.

Core claim

The stress-tensor sector plus a regularized image sum determines every nonzero Matsubara mode of the holographic thermal two-point function; the leftover pure function of spatial separation is fixed by the zero-frequency bulk Klein-Gordon equation, whose exact momentum-space solution is a ratio of Heun connection data. That zero mode completes the correlator, supplies the spin-resolved double-trace thermal coefficients, and removes the complex bulk-cone singularities present in the stress-tensor sector by itself.

What carries the argument

The zero-frequency bulk radial equation, rewritten as a Heun equation whose connection coefficients A(k)/B(k) give the exact momentum-space zero mode êg^{(0)}(k) = −4π A(k)/B(k); inverse Fourier transform plus the image-sum projection then determine all double-trace coefficients b_{m,J}.

Load-bearing premise

That zeta-regularized, Padé-Borel-resummed image sums of the stress-tensor sector correctly reproduce the nonzero frequency modes of the true holographic correlator, including delicate cancellations at half-integer operator dimension.

What would settle it

An independent high-precision numerical solution of the full three-variable bulk wave equation at finite spatial separation whose extracted double-trace coefficients disagree with the reported b_{m,J} values, or that still exhibits the predicted complex bulk-cone singularities after the double-trace sector is included.

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

If this is right

  • Low-lying spin-resolved double-trace thermal coefficients (including several previously unknown) are now available for the Δ_φ = 3/2 holographic scalar.
  • Complex bulk-cone singularities seen in the stress-tensor sector at spacelike separation are artifacts canceled by the double-trace contribution in the full correlator.
  • The same zero-mode Heun solution encodes the asymptotic density of complex-momentum poles, controlled by the radial optical length tied to the complex-geodesic scale.
  • The construction extends in principle to generic Δ_φ, other boundary dimensions, and finite-volume black holes on S¹ × S³.

Where Pith is reading between the lines

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

  • Because the zero-mode problem is identical (after double Wick rotation) to the AdS-soliton glueball eigenvalue problem, the same Heun data should reproduce known large-n glueball spacings for the corresponding bulk mass.
  • Spin resolution opens a direct numerical check of light-cone and thermal-EFT scaling regimes for double-trace thermal coefficients that earlier spin-averaged data could not test.
  • If the Padé-Borel step misses non-perturbative sectors, the discrepancy should first appear in high-spin or high-twist b_{m,J} rather than in the already cross-checked b_{0,0}.

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

2 major / 5 minor

Summary. The authors extend their earlier thermal-bootstrap construction of holographic two-point functions from vanishing to nonzero spatial separation. The strategy has two steps: (i) the stress-tensor sector, known from the near-boundary bulk expansion of [17], is summed over thermal images using Hurwitz-ζ analytic continuation and Padé–Borel resummation of a Gevrey-4 asymptotic series, producing a KMS-invariant function that fixes all non-zero Matsubara modes; (ii) the residual zero-mode ambiguity, a function c(x) of the spatial separation only, is fixed by solving the zero-frequency bulk Klein–Gordon equation, which reduces to a Heun equation whose connection data give the exact momentum-space correlator g̃⁽⁰⁾(k) = −4πA(k)/B(k) (eq. 3.26). The 1/ε singularities at the half-integer value Δ_φ=3/2 are shown to cancel analytically between the zero-mode pieces (eqs. 2.46–2.49). From the full correlator, the authors extract spin-resolved double-trace thermal coefficients b_{m,J} (eq. 3.46), reproducing and extending results of [41], and show that the complex bulk-cone singularities present in the stress-tensor sector alone are canceled in the full correlator (Figs. 1–2).

Significance. If the numerical pipeline is sound, this is a solid and useful contribution. It delivers the first spin-resolved double-trace thermal coefficients in the planar holographic model, with several new values in (3.46) beyond the reach of previous bulk numerics; it provides an exact momentum-space zero mode in terms of Heun connection data — an explicit, checkable formula of independent interest for the Heun/holography connection; and it resolves a conceptual puzzle by showing that the complex bulk-cone singularities of the stress-tensor sector are canceled by double-trace contributions, with direct numerical support in Figs. 1–2. The agreement with the independent results of [41] for b_{0,0}, b_{1,0}, b_{0,2} (to more digits than previously available) is a genuine cross-check, and the WKB pole-density prediction (5.2) tied to the optical length L_opt is falsifiable using the exact Heun solution. The construction contains no free parameters; the only ad-hoc input is the ζ-regularization plus Padé–Borel resummation of the divergent image sums, whose control is the main open issue.

major comments (2)
  1. [§2.3, eqs. (2.32)–(2.34)] The monomial coefficients a_{k,l} are defined by the Laplace integral (2.33) over a diagonal Pade approximant of the truncated Borel transform (2.34), whose terms grow like (4i)! (eq. 2.32). Two load-bearing points are not established. (i) The singularity structure of the Borel transform is never examined: if Pade poles accumulate on or near the positive real t-axis, the prescription is effectively a lateral resummation and the a_{k,l} carry a non-perturbative ambiguity that propagates directly into the b_{m,J}. (ii) Convergence in the truncation order N (set by the available multi-stress-tensor coefficients a^i_{j,2i-j} from [17]) is not demonstrated for any coefficient. This matters most precisely where the paper's novelty lies: independent corroboration from [41] exists only for b_{0,0}, b_{1,0}, b_{0,2}, while the new entries in (3.46) (b_{3,0}=54.4327, b_{1,4}=-37.7555, etc.) are th
  2. [§3.5, eq. (3.46) and Table 1] The text claims to determine 'many more digits than were previously available' for the low-lying coefficients, and eq. (3.46) and Table 1 quote up to 14 significant digits. However, no uncertainty estimates accompany these numbers. The inputs combine (i) numerically integrated zero-mode coefficients gamma_k (eqs. 3.33, D.20, D.25), which involve cutoff extrapolations and finite-part prescriptions, and (ii) the Borel-resummed image contributions discussed above. Without a stated error budget (variation with cutoff Lambda, near-horizon epsilon, quadrature tolerance, Pade order, and truncation N), the quoted precision is not verifiable. A short table of estimated uncertainties, or a convergence plot for representative coefficients, would substantially strengthen the main result.
minor comments (5)
  1. [§2.1–2.3] The notation a_{k,i} (monomial coefficients of the double-trace sector, eq. 2.7) is easily confused with the multi-stress-tensor coefficients a^i_{j,2i-j} (eq. 2.11), especially since both appear in the same formulas (e.g., eq. 2.50). A distinguishing symbol or a summary table of notation would help.
  2. [§3.5, footnote 4] The beta=π to beta=1 rescaling that produces the factors π³ and π⁵ in eqs. (3.36) and (3.41) is explained only in footnote 4. Given that Table 1 and eq. (3.32) are at beta=π while (3.46) is at beta=1, the convention should be stated more prominently, e.g., directly above Table 1 and (3.46).
  3. [Fig. 2 caption] The caption states the correlator 'exhibits only the standard light-cone singularity, located at x=t=0'. For the equal-time correlator g_beta(0,x) the short-distance singularity is at x=0; the phrase 'light-cone' and 'x=t=0' is confusing here and should be reworded.
  4. [§2.1, below eq. (2.5)] Typo: 'containing “mboxes”' appears to be a stray artifact (presumably 'containing m boxes' or similar).
  5. [§2.2, end] The text states the construction is carried out for Δ_φ=3/2, 5/2, but only Δ_φ=3/2 results are presented. Either show the 5/2 coefficients or clarify that they are left to future work.

Circularity Check

0 steps flagged

No significant circularity: zero mode is an independent bulk Heun BVP; double-trace coefficients are outputs, not inputs.

full rationale

The derivation chain is linear and non-circular. Multi-stress-tensor coefficients a^i_{j,2i-j} are external bulk near-boundary data ([17]). The image sum plus ζ/Padé-Borel regularization produces g_images and leaves a residual pure function of x (eqs. 2.16–2.17), which the paper fixes by solving an independent zero-frequency bulk Klein–Gordon problem that reduces to a Heun ODE with explicit horizon and boundary conditions (eqs. 3.12–3.26); the connection ratio A(k)/B(k) yields g̃^{(0)}(k) and thence the monomial coefficients a_{k,0} via (2.50). Double-trace thermal coefficients b_{m,J} are then obtained by a linear change of basis (2.8)–(2.9)—they are extracted outputs, not fitted inputs or definitional ingredients. Self-citations [37, 38] only supply the prior x=0 bootstrap method being extended; they do not force the finite-x residual, the Heun solution, or the spin-resolved numbers in (3.46). Singularity cancellation in §4 is a numerical comparison of the constructed correlator against an independent bulk PDE/AAA continuation, not a tautology. Methodological risks (Borel-plane structure, truncation) are correctness issues, not circular reductions. No step equates a claimed prediction to its own defining input.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

The construction sits inside standard large-N holography for a free bulk scalar on the planar AdS black brane. No new particles or forces are postulated. The only non-standard technical inputs are the chosen regularization (ζ + Padé-Borel) of the image sum and the specialization to Δ_φ=3/2 for numerics; both are methodological, not free fits to the target coefficients.

axioms (6)
  • domain assumption Leading large-N holographic dictionary: thermal two-point function of a boundary scalar dual to a minimally coupled bulk scalar on the planar AdS-Schwarzschild black brane equals the boundary limit of the bulk-to-boundary propagator.
    Stated in §1 and §3.1; standard AdS/CFT assumption underlying the whole calculation.
  • domain assumption At this order the ϕ×ϕ OPE contains only multi-stress-tensor and double-trace families; no other single-trace primaries contribute.
    §2.1; standard large-N factorization for a generalized free field deformed by gravity.
  • domain assumption KMS periodicity plus the stress-tensor sector fix the correlator up to an additive function c(x) of spatial separation alone.
    §2.2 eqs. (2.16)–(2.17); the residual ambiguity the paper then fixes.
  • domain assumption Multi-stress-tensor coefficients a^i_{j,2i-j} (or λ_{n,J}) are correctly given by the near-boundary bulk expansion of Fitzpatrick-Huang and follow-ups.
    Input data from [17]; used throughout §2 without re-derivation.
  • ad hoc to paper Divergent image sums are defined by analytic continuation in Δ via Hurwitz ζ, followed by diagonal Padé-Borel resummation of the Gevrey-4 series for double-trace coefficients.
    §2.3 eqs. (2.33)–(2.34); the same scheme as [38], extended to finite x. Not a mathematical theorem that the resummed series equals the bulk correlator non-perturbatively.
  • standard math Standard properties of HeunG connection coefficients and of the Euclidean black-brane Klein-Gordon operator (regularity at horizon, normalizability/source at boundary).
    §3.3 and App. C–D; classical ODE theory plus standard holographic boundary conditions.

pith-pipeline@v1.2.0-grok45-kimik3 · 31043 in / 3671 out tokens · 65837 ms · 2026-07-31T13:06:38.334383+00:00 · methodology

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read the original abstract

It was previously shown that the stress-tensor sector of the OPE, together with the KMS condition, fixes holographic thermal two-point functions at vanishing spatial separation. We extend this construction to nonzero spatial separation, where the KMS condition leaves a residual ambiguity that depends only on the spatial separation. We show that this ambiguity is fixed by the zero-frequency bulk wave equation, which can be solved analytically in terms of Heun functions. This gives an efficient method for computing thermal coefficients of double-trace operators resolved by spin. We also study the Lorentzian analytic structure of the resulting correlator and show that complex bulk-cone singularities which appear at spacelike separation in the stress-tensor sector do not persist in the full two-point function; they are resolved by the double-trace contribution.

discussion (0)

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Reference graph

Works this paper leans on

64 extracted references · 44 linked inside Pith

  1. [1]

    Ferrara, A

    S. Ferrara, A. F. Grillo, and R. Gatto,Tensor representations of conformal algebra and conformally covariant operator product expansion,Annals Phys.76(1973) 161–188

    1973

  2. [2]

    A. M. Polyakov,Non-Hamiltonian approach to conformal quantum field theory,Zh. Eksp. Teor. Fiz.66(1974), no. 1 23–42

    1974

  3. [3]

    Mack,Convergence of Operator Product Expansions on the Vacuum in Conformal Invariant Quantum Field Theory,Commun

    G. Mack,Convergence of Operator Product Expansions on the Vacuum in Conformal Invariant Quantum Field Theory,Commun. Math. Phys.53(1977) 155

    1977

  4. [4]

    Rattazzi, V

    R. Rattazzi, V. S. Rychkov, E. Tonni, and A. Vichi,Bounding scalar operator dimensions in 4D CFT,JHEP12(2008) 031, [arXiv:0807.0004]

    Pith/arXiv arXiv 2008

  5. [5]

    El-Showk, M

    S. El-Showk, M. F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin, and A. Vichi,Solving the 3D Ising Model with the Conformal Bootstrap,Phys. Rev. D86 (2012) 025022, [arXiv:1203.6064]

    Pith/arXiv arXiv 2012

  6. [6]

    El-Showk, M

    S. El-Showk, M. F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin, and A. Vichi,Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents,J. Stat. Phys.157(2014) 869, [arXiv:1403.4545]

    Pith/arXiv arXiv 2014

  7. [7]

    F. Kos, D. Poland, D. Simmons-Duffin, and A. Vichi,Precision Islands in the Ising andO(N)Models,JHEP08(2016) 036, [arXiv:1603.04436]

    Pith/arXiv arXiv 2016

  8. [8]

    A. L. Fitzpatrick, J. Kaplan, D. Poland, and D. Simmons-Duffin,The Analytic Bootstrap and AdS Superhorizon Locality,JHEP12(2013) 004, [arXiv:1212.3616]. – 34 –

    Pith/arXiv arXiv 2013

  9. [9]

    Komargodski and A

    Z. Komargodski and A. Zhiboedov,Convexity and Liberation at Large Spin,JHEP 11(2013) 140, [arXiv:1212.4103]

    Pith/arXiv arXiv 2013

  10. [10]

    Simmons-Duffin,The Conformal Bootstrap, inTheoretical Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings, pp

    D. Simmons-Duffin,The Conformal Bootstrap, inTheoretical Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings, pp. 1–74, 2017.arXiv:1602.07982

    Pith/arXiv arXiv 2017

  11. [11]

    Poland, S

    D. Poland, S. Rychkov, and A. Vichi,The Conformal Bootstrap: Theory, Numerical Techniques, and Applications,Rev. Mod. Phys.91(2019) 015002, [arXiv:1805.04405]

    Pith/arXiv arXiv 2019

  12. [12]

    Bissi, A

    A. Bissi, A. Sinha, and X. Zhou,Selected topics in analytic conformal bootstrap: A guided journey,Phys. Rept.991(2022) 1–89, [arXiv:2202.08475]

    Pith/arXiv arXiv 2022

  13. [13]

    Rychkov and N

    S. Rychkov and N. Su,New developments in the numerical conformal bootstrap,Rev. Mod. Phys.96(2024), no. 4 045004, [arXiv:2311.15844]

    arXiv 2024

  14. [14]

    El-Showk and K

    S. El-Showk and K. Papadodimas,Emergent Spacetime and Holographic CFTs, JHEP10(2012) 106, [arXiv:1101.4163]

    Pith/arXiv arXiv 2012

  15. [15]

    Iliesiu, M

    L. Iliesiu, M. Kolo˘ glu, R. Mahajan, E. Perlmutter, and D. Simmons-Duffin,The Conformal Bootstrap at Finite Temperature,JHEP10(2018) 070, [arXiv:1802.10266]

    Pith/arXiv arXiv 2018

  16. [16]

    Marchetto, A

    E. Marchetto, A. Miscioscia, and E. Pomoni,Sum rules & Tauberian theorems at finite temperature,JHEP09(2024) 044, [arXiv:2312.13030]

    Pith/arXiv arXiv 2024

  17. [17]

    A. L. Fitzpatrick and K.-W. Huang,Universal Lowest-Twist in CFTs from Holography,JHEP08(2019) 138, [arXiv:1903.05306]

    Pith/arXiv arXiv 2019

  18. [18]

    Fidkowski, V

    L. Fidkowski, V. Hubeny, M. Kleban, and S. Shenker,The Black hole singularity in AdS / CFT,JHEP02(2004) 014, [hep-th/0306170]

    Pith/arXiv arXiv 2004

  19. [19]

    Festuccia and H

    G. Festuccia and H. Liu,Excursions beyond the horizon: Black hole singularities in Yang-Mills theories. I.,JHEP04(2006) 044, [hep-th/0506202]

    Pith/arXiv arXiv 2006

  20. [20]

    ˇCeplak, H

    N. ˇCeplak, H. Liu, A. Parnachev, and S. Valach,Black hole singularity from OPE, JHEP10(2024) 105, [arXiv:2404.17286]

    Pith/arXiv arXiv 2024

  21. [21]

    Parisini, K

    E. Parisini, K. Skenderis, and B. Withers,The ambient space formalism,JHEP05 (2024) 296, [arXiv:2312.03820]

    Pith/arXiv arXiv 2024

  22. [22]

    Afkhami-Jeddi, S

    N. Afkhami-Jeddi, S. Caron-Huot, J. Chakravarty, and A. Maloney,Imprint of the black hole singularity on thermal two-point functions,Phys. Rev. D114(2026), no. 2 025005, [arXiv:2510.21673]

    Pith/arXiv arXiv 2026

  23. [23]

    ˇCeplak, H

    N. ˇCeplak, H. Liu, A. Parnachev, and S. Valach,Fooling the censor: going beyond inner horizons with the OPE,JHEP05(2026) 001, [arXiv:2511.09638]

    arXiv 2026

  24. [24]

    Dodelson, C

    M. Dodelson, C. Iossa, and R. Karlsson,Bouncing off a stringy singularity, arXiv:2511.09616

    arXiv

  25. [25]

    H. F. Jia and M. Rangamani,Thermal spectral function asymptotics and black hole singularity in holography,arXiv:2512.15114. – 35 –

    arXiv

  26. [26]

    Ali Ahmad, A

    S. Ali Ahmad, A. Almheiri, and S. Lin,Continuing past the inner horizon using WKB,arXiv:2601.02354

    arXiv

  27. [27]

    Jia and M

    Y. Jia and M. Kulaxizi,Bulk Phase Shift and Singularity,arXiv:2602.06558

    arXiv

  28. [28]

    I. J. Araya, C. Esper, Y. Jia, M. Kulaxizi, and A. Parnachev,Bulkcone Singularities and Complex Geodesics,arXiv:2602.12893

    arXiv

  29. [29]

    Giombi, Y.-Z

    S. Giombi, Y.-Z. Li, and J. Shan,Bouncing singularities and thermal correlators on line defects,arXiv:2603.11012

    Pith/arXiv arXiv

  30. [30]

    Arnaudo and B

    P. Arnaudo and B. Withers,Analytic structure of holographic thermal correlators from Fourier series,JHEP06(2026) 205, [arXiv:2603.13469]

    arXiv 2026

  31. [31]

    Grozdanov, S

    S. Grozdanov, S. Valach, and M. Vrbica,Bouncing geodesics, black hole singularities, and singularities of thermal correlators,arXiv:2603.15598

    arXiv

  32. [32]

    H. F. Jia and M. Rangamani,Exact holographic thermal spectral functions: OPE, non-perturbative corrections, and black hole singularity,arXiv:2604.10803

    Pith/arXiv arXiv

  33. [33]

    Arnaudo and B

    P. Arnaudo and B. Withers,Bouncing singularities in Schwarzschild: a geometric origin of the QNM convergence region,arXiv:2605.16489

    Pith/arXiv arXiv

  34. [34]

    Grozdanov, V

    S. Grozdanov, V. Movrin, and S. Valach,Bouncing Geodesics, Singularities, and the Cavity Thermal Product Formula in Asymptotically Flat and de Sitter Black Holes, arXiv:2606.11297

    Pith/arXiv arXiv

  35. [35]

    Dodelson, O

    M. Dodelson, O. Gupta, M. Mezei, and D. Wang,On the temperature dependence of quasinormal modes in SYK and holography,arXiv:2606.22679

    Pith/arXiv arXiv

  36. [36]

    Buri´ c, C.-M

    I. Buri´ c, C.-M. Chang, I. Gusev, E. Helfenberger, A. Parnachev, and M. Rangamani, Thermal two-point functions in SYK and complex-time singularities, arXiv:2607.05258

    Pith/arXiv arXiv

  37. [37]

    Buri´ c, I

    I. Buri´ c, I. Gusev, and A. Parnachev,Thermal holographic correlators and KMS condition,arXiv:2505.10277

    Pith/arXiv arXiv

  38. [38]

    Buri´ c, I

    I. Buri´ c, I. Gusev, and A. Parnachev,Holographic Correlators from Thermal Bootstrap,arXiv:2508.08373

    arXiv

  39. [39]

    Niarchos, C

    V. Niarchos, C. Papageorgakis, A. Stratoudakis, and M. Woolley,Deep finite temperature bootstrap,Phys. Rev. D112(2025), no. 12 126012, [arXiv:2508.08560]

    arXiv 2025

  40. [40]

    Barrat, D

    J. Barrat, D. N. Bozkurt, E. Marchetto, A. Miscioscia, and E. Pomoni,Analytic thermal bootstrap meets holography,arXiv:2510.20894

    arXiv

  41. [41]

    Niarchos, C

    V. Niarchos, C. Papageorgakis, and A. Stratoudakis,Thermal Double-Twist Data in Holography,arXiv:2606.30806

    Pith/arXiv arXiv

  42. [42]

    Iliesiu, M

    L. Iliesiu, M. Kolo˘ glu, and D. Simmons-Duffin,Bootstrapping the 3d Ising model at finite temperature,JHEP12(2019) 072, [arXiv:1811.05451]

    Pith/arXiv arXiv 2019

  43. [43]

    Barrat, E

    J. Barrat, E. Marchetto, A. Miscioscia, and E. Pomoni,Thermal Bootstrap for the Critical O(N) Model,Phys. Rev. Lett.134(2025), no. 21 211604, [arXiv:2411.00978]. – 36 –

    Pith/arXiv arXiv 2025

  44. [44]

    Barrat, D

    J. Barrat, D. N. Bozkurt, E. Marchetto, A. Miscioscia, and E. Pomoni,The analytic bootstrap at finite temperature,JHEP05(2026) 104, [arXiv:2506.06422]

    Pith/arXiv arXiv 2026

  45. [45]

    Aminov, A

    G. Aminov, A. Grassi, and Y. Hatsuda,Black Hole Quasinormal Modes and Seiberg–Witten Theory,Annales Henri Poincare23(2022), no. 6 1951–1977, [arXiv:2006.06111]

    Pith/arXiv arXiv 2022

  46. [46]

    Dodelson, A

    M. Dodelson, A. Grassi, C. Iossa, D. Panea Lichtig, and A. Zhiboedov,Holographic thermal correlators from supersymmetric instantons,SciPost Phys.14(2023), no. 5 116, [arXiv:2206.07720]

    Pith/arXiv arXiv 2023

  47. [47]

    Aminov, P

    G. Aminov, P. Arnaudo, G. Bonelli, A. Grassi, and A. Tanzini,Black hole perturbation theory and multiple polylogarithms,JHEP11(2023) 059, [arXiv:2307.10141]

    Pith/arXiv arXiv 2023

  48. [48]

    Arnaudo, G

    P. Arnaudo, G. Bonelli, and A. Tanzini,One loop effective actions in Kerr-(A)dS black holes,Phys. Rev. D110(2024), no. 10 106006, [arXiv:2405.13830]

    Pith/arXiv arXiv 2024

  49. [49]

    Arnaudo and B

    P. Arnaudo and B. Withers,Exact low-temperature Green’s functions in AdS/CFT: From the Heun equation to the confluent Heun equation,Phys. Rev. D111(2025), no. 12 L121903, [arXiv:2412.01923]

    Pith/arXiv arXiv 2025

  50. [50]

    Bajc and K

    B. Bajc and K. Trailovi´ c,Holographic thermal propagator from modularity,JHEP11 (2025) 133, [arXiv:2509.02226]

    arXiv 2025

  51. [51]

    A. C. Antoulas and B. D. Q. Anderson,On the scalar rational interpolation problem, IMA Journal of Mathematical Control and Information3(1986), no. 2-3 61–88

    1986

  52. [52]

    Nakatsukasa, O

    Y. Nakatsukasa, O. S` ete, and L. N. Trefethen,The aaa algorithm for rational approximation,SIAM Journal on Scientific Computing40(Jan., 2018) A1494–A1522

    2018

  53. [53]

    Gobeil, A

    Y. Gobeil, A. Maloney, G. S. Ng, and J.-q. Wu,Thermal Conformal Blocks,SciPost Phys.7(2019), no. 2 015, [arXiv:1802.10537]

    Pith/arXiv arXiv 2019

  54. [54]

    Buric, F

    I. Buric, F. Russo, V. Schomerus, and A. Vichi,Thermal one-point functions and their partial wave decomposition,JHEP12(2024) 021, [arXiv:2408.02747]

    Pith/arXiv arXiv 2024

  55. [55]

    Benjamin, J

    N. Benjamin, J. Lee, H. Ooguri, and D. Simmons-Duffin,Universal asymptotics for high energy CFT data,JHEP03(2024) 115, [arXiv:2306.08031]

    Pith/arXiv arXiv 2024

  56. [56]

    Benjamin, J

    N. Benjamin, J. Lee, S. Pal, D. Simmons-Duffin, and Y. Xu,Angular fractals in thermal QFT,JHEP11(2024) 134, [arXiv:2405.17562]

    Pith/arXiv arXiv 2024

  57. [57]

    Buri´ c, F

    I. Buri´ c, F. Mangialardi, F. Russo, V. Schomerus, and A. Vichi,Heavy-heavy-light asymptotics from thermal correlators,JHEP04(2026) 027, [arXiv:2506.21671]

    Pith/arXiv arXiv 2026

  58. [58]

    Buri´ c, F

    I. Buri´ c, F. Mangialardi, F. Russo, V. Schomerus, and A. Vichi,Thermal One-point Functions and Asymptotic CFT Data: QFT in AdS,arXiv:2606.17167

    Pith/arXiv arXiv

  59. [59]

    Anand, N

    H. Anand, N. Benjamin, V. Kumar, S. Minwalla, J. Mukherjee, S. Pal, and A. Rahaman,Semi-universality of CFT d entropy at large spin,arXiv:2512.00158. – 37 –

    arXiv

  60. [60]

    Komargodski, A

    Z. Komargodski, A. Miscioscia, and F. K. Popov,Regge’s Inferno, arXiv:2603.10197

    arXiv

  61. [61]

    Festuccia and H

    G. Festuccia and H. Liu,A Bohr-Sommerfeld quantization formula for quasinormal frequencies of AdS black holes,Adv. Sci. Lett.2(2009) 221–235, [arXiv:0811.1033]

    Pith/arXiv arXiv 2009

  62. [62]

    Csaki, H

    C. Csaki, H. Ooguri, Y. Oz, and J. Terning,Glueball mass spectrum from supergravity,JHEP01(1999) 017, [hep-th/9806021]

    Pith/arXiv arXiv 1999

  63. [63]

    J. A. Minahan,Glueball Mass Spectra and Other Issues for Supergravity Duals of QCD Models,JHEP01(1999) 020, [hep-th/9811156]

    Pith/arXiv arXiv 1999

  64. [64]

    Bonelli, C

    G. Bonelli, C. Iossa, D. Panea Lichtig, and A. Tanzini,Irregular Liouville Correlators and Connection Formulae for Heun Functions,Commun. Math. Phys. 397(2023), no. 2 635–727, [arXiv:2201.04491]. – 38 –

    Pith/arXiv arXiv 2023