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REVIEW 2 major objections 8 minor 75 references

Thermal OPE data and quasinormal modes are the same data, linked by an overlapping region of the complex-time plane.

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2026-07-31 06:06 UTC pith:BBDQP3IH

load-bearing objection Solid dictionary between thermal OPE and QNMs in mixed (t,k), with real new asymptotics and AdS5 checks; scope is holographic/free large-N, not generic large-N CFTs. the 2 major comments →

arxiv 2607.24909 v1 pith:BBDQP3IH submitted 2026-07-27 hep-th cond-mat.stat-mechcond-mat.str-elgr-qc

OPE = QNM

Paolo Arnaudo , Cristoforo Iossa , Robin Karlsson , Benjamin Withers This is my paper
classification hep-th cond-mat.stat-mechcond-mat.str-elgr-qc
keywords thermal OPEquasinormal modesretarded correlatorMellin transformthermal bootstrapbouncing singularitiesconformal collider boundsAdS black brane
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.

In a hot large-N conformal field theory the short-distance response of a two-point function is organised by a thermal operator product expansion, while the long-time response is organised by a sum of damped collective modes called quasinormal modes. This paper shows that the two expansions of the mixed time-and-momentum retarded correlator are valid in overlapping regions of the complex-time plane, so they must agree there. That agreement supplies an explicit dictionary: OPE coefficients become residues of a Mellin transform, while the quasinormal frequencies and residues appear as the spectral data of the same transform. Matching the two sides at singularities yields new asymptotic formulae for high-overtone modes; analytic continuation of a finite OPE sum recovers the lowest modes; and Mellin-space sum rules force every double-trace moment of the quasinormal data to vanish. The same dictionary applied at large spatial momentum implies that stress-tensor correlators ring down more slowly as the conformal collider bounds approach saturation. The result is a concrete thermal bootstrap in which ultraviolet OPE data and infrared quasinormal data constrain each other.

Core claim

The OPE and QNM representations of the mixed retarded correlator GR(t,k) share a nonempty open set of the complex-t plane on which they are identical. Consequently the defining OPE coefficients are in one-to-one correspondence with the quasinormal frequencies and residues, realised concretely by Mellin residues and by an infinite family of double-trace sum rules.

What carries the argument

The mixed retarded correlator GR(t,k) written once as a thermal OPE series and once as a quasinormal residue sum (Eq. 2.44). Their common domain of convergence, together with the Mellin transform that turns the QNM sum into a spectral zeta function whose poles are the OPE data, carries the entire dictionary.

Load-bearing premise

That the retarded correlator is meromorphic with only simple poles whose residue sum converges throughout a right half-plane or wedge, so the only singularities that limit the OPE disk are bouncing-type singularities.

What would settle it

Compute a high-overtone quasinormal frequency of the Schwarzschild-AdS5 black brane by independent bulk numerics and check whether it matches the analytic large-n expansion obtained from the OPE singularity matching (Eq. 4.26) to the predicted O(n^{-7/3}) accuracy.

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

If this is right

  • High-overtone QNM asymptotics, including new fractional powers, are completely fixed by the local expansion of the thermal OPE near its first singularity.
  • Low-lying QNMs can be extracted from a finite number of OPE coefficients by Padé continuation plus Prony fitting, without solving any bulk wave equation.
  • An infinite set of Mellin-space sum rules forces every double-trace moment of the QNM data to vanish, rigidly constraining any finite deformation of the spectrum.
  • At large spatial momentum the light-cone OPE implies that stress-tensor correlators thermalise more slowly as the conformal collider bounds approach saturation.
  • The same dictionary supplies a practical thermal bootstrap in which ultraviolet OPE data and infrared QNM data mutually constrain each other.

Where Pith is reading between the lines

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

  • Once the overlapping-region map is accepted, any independent bound on OPE coefficients (unitarity, collider bounds, etc.) immediately becomes a bound on allowed quasinormal spectra, and vice versa.
  • The same complex-time overlap should exist for charged or rotating black branes; the extra purely imaginary modes would only affect the OPE singularity, leaving the bouncing-singularity matching intact.
  • Non-holographic large-N models with finitely many normal modes (such as the critical O(N) vector model) furnish the simplest solvable points of the bootstrap, where the Mellin transform reduces to a finite exponential polynomial.

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 / 8 minor

Summary. The paper studies the retarded thermal two-point function G_R(t,k) of large-N CFTs in mixed time/momentum variables. Its central claim is that the thermal OPE expansion (convergent for 0<|t|<t_c) and the quasinormal-mode sum (convergent for Re t>0 or in a wedge) share a nonempty open overlap in the complex t-plane (Eq. (2.44)), putting OPE data b_Δ̂,k and QNM data {r_n,ω_n} in one-to-one correspondence. Three constructive uses are developed: (1) Padé continuation of the OPE beyond t_c plus Prony fitting to extract low-lying QNMs of the Schwarzschild-AdS5 brane (Tables 1–2); (2) matching at the t=0 and first bouncing singularity to derive analytic large-n QNM asymptotics, including a new n^{−7/3} coefficient ((4.23)–(4.26), Fig. 7); (3) Mellin-space sum rules ((5.9)–(5.10)), with a rigidity result for the spectrum (§5.4), a convergent subtraction scheme tested numerically (§5.5), and a Carlson-theorem uniqueness argument for Mellin completion (§5.6). Supporting examples are worked out in BTZ, R-currents in N=4 SYM, the large-N O(N) model, and an ε-expansion counterexample (App. B.2). Section 6 relates large-k lightcone modes to conformal collider bounds and argues for slower thermalisation near saturation.

Significance. If the claims hold within their demonstrated domain (holographic and mean-field large-N theories), the paper is a significant contribution to the finite-temperature bootstrap: (i) new analytic QNM asymptotics — the n^{−7/3} coefficient in (4.25) and the n′-resummed tail (4.26) are new, confirmed numerically to the expected order in Fig. 7; (ii) a new set of exact constraints on holographic QNM data (the Mellin-space residue formula (5.9) and double-trace sum rules (5.10)), with a rigidity theorem (§5.4) and a convergent asymptotic-subtraction implementation tested quantitatively in (5.62)–(5.63); (iii) an explicit, quantitatively verified extraction of low-lying QNMs of the Schwarzschild-AdS5 brane from OPE data alone (Tables 1–2), with two complementary methods converging from opposite ends of the spectrum; (iv) a falsifiable application of the framework in §5.2, where a toy spectrum satisfying all sum rules is constructed and then ruled out by Ward identities — a genuinely bootstrap-style argument; and (v) a new lightcone connection between conformal collider bounds and thermalisation timescales of stress-tensor correlators (6.7), checked against Gauss-Bonnet WKB. The thermal Mel

major comments (2)
  1. [§2.2–2.3, App. B.2, Abstract] The abstract and §1 state the result for 'large-N CFTs' generically, but the overlap equation (2.44) rests on three simultaneous conditions: (i) meromorphy of G_R(ω,k) with simple poles (assumed in §2.2), (ii) convergence of the QNM residue sum in a region abutting t=0 (verified explicitly only for BTZ §2.3, the R-current (2.29)–(2.35), the holographic product formula of [36], and the large-N O(N) model App. B.1), and (iii) bouncing-type singularities as the only obstructions to the OPE disk. The manuscript's own App. B.2 shows that at the Wilson-Fisher fixed point the correlator develops branch points at ω=±k already at O(ε) (Eq. (B.21)), so the discrete QNM representation (1.2) — and hence (2.44) — fails in the generic interacting non-holographic large-N regime (in vector models, damping and cuts both appear at subleading 1/N). The closing sentence of App. B.2 ('the general relation is
  2. [§5.6, §1] The reverse direction of the claimed one-to-one correspondence (OPE→QNM) is constructive in the manuscript only through (a) Padé + Prony numerics on a truncated OPE (§3, Table 2), or (b) the Carlson completion of §5.6. For (b), uniqueness requires the normalised interpolation F(w) to have exponential type < π in the left half-plane, and the paper ties this growth bound to the extremal QNM angles (text after (5.91)). The suggestion that bouncing-singularity locations — i.e., large-order OPE data — can supply this bound non-circularly is demonstrated only for the R-current, where the full correlator is already known in closed form. As written, §5.6 does not establish that the Carlson-class condition can be verified from OPE data alone in a generic case, so the 'one-to-one' claim in §1 (and the statement that 'the defining set of data for the OPE is in one-to-one correspondence with the def
minor comments (8)
  1. [§4, Eq. (4.7)] The ansatz (4.7) mentions δ2 ('0 < δ2 − δ1 < 1') but δ2 is never defined or used; presumably only δ1 is intended.
  2. [§5.5, table (5.51)] The residuals of the raw zeta-regularised sum rules grow with truncation order for q=0 (−5.6×10⁻² → −7.8×10⁻²), which looks counterintuitive before the subtracted scheme (5.57) is introduced. A sentence explaining why including more asymptotic terms without subtraction need not improve the moment would help the reader.
  3. [§4.3, Table 3] Table 3 lists higher-order coefficients d_{8/3}, d_{11/3}, d_4, d_5 but gives no independent numerical cross-check (unlike Fig. 7 for the lower orders). If these are extracted by fitting, the fitting procedure and error estimate should be stated; if they are analytic predictions, the agreement criterion should be given.
  4. [§2.3, footnote 6] Footnote 6: 'This is is most clearly seen' — duplicated word.
  5. [§2.3, Eqs. (2.37)–(2.38)] The step from (2.37) to (2.38), discarding 'all perturbative terms at large ω while keeping all the non-perturbative terms', is load-bearing for the wedge-convergence picture but is justified only by citation to [19]. A brief explanation of what is dropped and why the singularity locations are unaffected would make §2.3 more self-contained.
  6. [§1, Eq. (1.6)] In (1.6) the sentence 'Then −7/3 prediction for the QNMs is new' appears garbled; presumably 'The n^{−7/3} prediction'.
  7. [§2.1, §4] Notation: ℓ and l are mixed in §2.1 ('taking equally many values for l = 0,1,...'); also the normalisation N is redefined between (3.4) and (4.2) without an explicit flag at the point of redefinition.
  8. [References] Reference [45] is listed with a trailing space in the title and no journal/arxiv identifier; please complete the entry.

Circularity Check

2 steps flagged

Mild bulk-input/bulk-check consistency and partial self-citation; the OPE↔QNM map itself is not circular by construction.

specific steps
  1. self citation load bearing [Sec. 3, Eqs. (3.4)–(3.10) and Table 2; input from [32],[24]]
    "we start with stress-tensor sector OPE data for scalar perturbations of Schwarzschild-AdS5 black brane, which are known exactly [32] ... a more efficient way to obtain them is to work directly at fixed k ... carried out in [24] ... the first 70 cm can be found in the ancillary file attached to [24]"

    OPE coefficients used to 'extract QNMs from OPE' are themselves computed from the bulk dual (including self-citation [24] by Arnaudo–Withers). Recovered frequencies are then compared to numerical bulk QNMs. This is a closed bulk→OPE→QNM→bulk consistency loop, not an independent CFT-only prediction. It does not make the field-theoretic map (2.18)/(2.44) definitional, so it is only a mild circularity of validation, not of the claimed correspondence.

  2. self citation load bearing [Sec. 2.3 Eqs. (2.36)–(2.41); Sec. 5.2 ruling out toy model via [36]]
    "in [36] a relation for the asymptotic QNM parameters was established as follows β=4πsinθ/r , 2ΔO−d=4scos(θ−ϕ)+2r/r ... the model is not consistent with the thermal product formula [36]. Indeed, if one takes only its pole locations and inserts them in the thermal product formula, the large n-scaling of the residues ... does not agree"

    Asymptotic residue scaling and the product-formula test that excludes the exact solvable toy spectrum of §5.2 rest on [36] (Dodelson–Iossa–Karlsson–Zhiboedov), which shares two authors with the present paper. The citation is load-bearing for those asymptotic identities and for one bootstrap-style exclusion, but the main OPE-matching derivation of (r,θ,d0,d4/3) in Sec. 4 is carried out independently from singularity structure; hence only partial, non-central self-citation weight.

full rationale

The load-bearing identification (2.44) equates two independently defined expansions of GR(t,k)—the thermal OPE after discontinuity and spatial Fourier transform, versus a residue sum of simple poles—on a claimed overlap in the complex-t plane. That equality is not a definitional tautology: OPE coefficients b_Δ̂,k are fixed by Euclidean thermal data (a_Δ,J and kinematics), while {rn,ωn} are poles/residues of the frequency-space retarded correlator. Mellin residues (5.8)–(5.9) and double-trace sum rules (5.10) follow from the absence of analytic terms in the discontinuity, not from fitting target frequencies. The AdS5 numerics take stress-tensor-sector OPE data from bulk methods ([32], [24]) and recover QNMs that match the same bulk dual (Tables 1–2); that is a consistency check of the map, not a prediction forced by construction. Large-n asymptotics (4.16)–(4.26) assume a two-line linear-spacing ansatz motivated by holography, then fix coefficients by matching OPE and first-bounce singularities whose local expansions come from large-order OPE data; the matching step is genuine, and the new n^{-7/3} term is not fitted to target ωn. Partial author overlap with the thermal product formula [36] supplies asymptotic relations and rules out the toy spectrum of §5.2, but is not used as an unverified uniqueness theorem that forces the central claim. Scope caveats (meromorphy fails in App. B.2 Wilson–Fisher) affect correctness/domain, not circularity. No step reduces Eq. X to Eq. Y by definition or by renaming a fit as a prediction.

Axiom & Free-Parameter Ledger

2 free parameters · 7 axioms · 0 invented entities

The central map rests on standard CFT/thermal axioms plus the large-N meromorphy and overlap hypotheses that the paper motivates by holography and the O(N) model. No new particles or forces are introduced; free parameters are limited to conventional normalisations and numerical continuation choices.

free parameters (2)
  • Overall correlator normalisation N = scheme-dependent (e.g. Eq. 4.2)
    Chosen to match conventions of [18] or [24]; drops out of frequency ratios but rescales residues.
  • Padé order and Prony fitting windows [ta,tb] = [1e-5,4]β/π and [1e-3,6]β/π with 300 samples
    Numerical analytic-continuation hyperparameters; stability under window change is used to reject spurious modes (Sec. 3).
axioms (7)
  • domain assumption Thermal OPE of the Euclidean two-point function converges for √(τ²+x²)<β and has the Gegenbauer block form (1.1)/(2.13).
    Standard large-N thermal CFT input from [2,4]; used throughout Sec. 2.1.
  • domain assumption Frequency-space retarded correlator is meromorphic with simple poles (generic large-N holographic behaviour); higher-order poles or cuts require only mild generalisation.
    Stated in Sec. 2.2; justified by scattering theory in holography [36], assumed for the QNM sum (2.22).
  • domain assumption OPE disk and QNM right-wedge (or half-plane) have nonempty overlap in complex t, limited by bouncing singularities in holographic examples.
    Core working hypothesis of Sec. 2.3; demonstrated for BTZ, R-currents, black branes, O(N); not proved for arbitrary large-N CFTs.
  • standard math Causality: GR(t,x)=0 outside the lightcone; eGR analytic in Im ω>|Im k|.
    Used to restrict the spatial Fourier integral (2.11)–(2.12) and to rule out finite-QNM spectra with h>2 (Sec. 5.3).
  • domain assumption Double-trace / analytic terms have vanishing discontinuity, hence vanish from the mixed OPE of GR (sin(π(Δ/2-ΔO)) factor).
    Standard; produces the double-trace sum rules (5.10) in holographic cases.
  • standard math Mellin transform and inverse with appropriate vertical contour; Carlson’s theorem uniqueness under exponential type <π.
    Sec. 5 and 5.6; used to complete Mellin data from OPE poles for the R-current example.
  • domain assumption No light scalars with twist τ<d-2 when discussing universal lightcone stress-tensor physics.
    Stated at the opening of Sec. 6 so that identity and stress tensor dominate the large-k lightcone OPE.

pith-pipeline@v1.2.0-grok45-kimik3 · 53675 in / 3650 out tokens · 78013 ms · 2026-07-31T06:06:34.773753+00:00 · methodology

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At microscopic scales the linear response of thermal states of large-$N$ CFTs is governed by a thermal operator product expansion (OPE), while at large scales response is governed by collective excitations known as quasinormal modes (QNM). We show that the OPE and QNM representations of the retarded correlator in mixed time and spatial momentum coordinates have an overlapping region of convergence in the complex time plane, giving a map between OPE and QNM data. We show that large-overtone QNM asymptotics are related to OPE singularities, while low-overtone QNM data appear in analytic continuation from short to large times. Using this approach we obtain new analytic results for QNM asymptotics, and numerically obtain low-overtone QNMs from OPE data for the Schwarzschild-AdS$_5$ black brane. We further show that the OPE spectrum is intimately related to QNM data through a set of sum rules which we derive in Mellin space. Finally, using the lightcone OPE, we argue that stress tensor correlators at large spatial momentum thermalise slower as the conformal collider bounds approach saturation. This work points to a new thermal bootstrap programme where OPE and QNM data constrain each other.

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Reference graph

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