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Thermal OPE data fix quasinormal modes through momentum-space inversion formulae and sum rules.

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 →

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2026-07-31 05:48 UTC pith:P2K6EO7H

load-bearing objection Solid analytic bridge from thermal OPE to momentum-space correlators and QNM data; the advertised sum rules work via zeta continuation, so the bootstrap problem is real but prescription-dependent.

arxiv 2607.24919 v1 pith:P2K6EO7H submitted 2026-07-27 hep-th

Analytic thermal bootstrap in momentum space: From thermal OPE to QNMs

Julien Barrat , Deniz N. Bozkurt , Enrico Marchetto , Alessio Miscioscia , Elli Pomoni This is my paper
classification hep-th
keywords thermal bootstrapthermal OPEquasinormal modesPolyakov blocksinversion formularetarded correlatorsKMS conditionfinite temperature CFT
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.

This paper starts a bootstrap that turns short-distance thermal operator-product data into long-distance infrared observables: low-frequency moments and the poles of retarded correlators known as quasinormal modes. The authors first complete each thermal OPE block into a KMS-symmetric object (a thermal Polyakov block), Fourier-transform it, and obtain an asymptotic expansion of the retarded correlator that holds at any spatial momentum. From that expansion they derive Euclidean inversion formulae that recover OPE coefficients from the correlator. When the retarded correlator is assumed meromorphic, the same formulae express those coefficients as residues of a spectral function built from the quasinormal frequencies and residues, producing infinite families of sum rules and asymptotic constraints on the mode spectrum at large momentum. The construction is checked in free theories, two-dimensional CFTs, the large-N and epsilon-expansion O(N) models, and the R-current of strongly coupled N=4 SYM; a byproduct gives spin-resolved asymptotics for heavy thermal OPE coefficients that match Monte Carlo data for the three-dimensional Ising model inside the OPE regime.

Core claim

Under the assumption that the retarded thermal correlator is meromorphic in complex frequency, the momentum-space inversion formulae write thermal OPE coefficients directly in terms of the quasinormal-mode frequencies and residues. The resulting spectral function Z(u) must vanish at negative integers (absent operators), develop poles fixed by non-integer OPE data, and encode the small-frequency Taylor coefficients at positive integers, thereby converting ultraviolet OPE data into infrared sum rules and large-momentum asymptotics for the mode spectrum.

What carries the argument

Thermal Polyakov blocks: KMS-symmetric completions of individual thermal OPE blocks, obtained by summing thermal images. Their Fourier transforms supply the asymptotic expansion of the retarded correlator at arbitrary spatial momentum; Mellin projection of that expansion yields the inversion formulae that relate OPE coefficients to the quasinormal spectral function Z(u).

Load-bearing premise

The retarded correlator is assumed to have only isolated simple poles in complex frequency, with no branch cuts; if continuous spectral density is present the discrete sum rules no longer close.

What would settle it

Compute or measure a retarded thermal correlator known to possess branch cuts (for example beyond large N in the O(N) model) and check whether the discrete Z(u) sum rules still hold; systematic violation would falsify the meromorphic inversion.

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

If this is right

  • Infinite vanishing conditions on the quasinormal spectrum follow whenever no primary sits at a negative-integer Mellin location.
  • Large-mode and large-momentum asymptotics of frequencies and residues are fixed by the lightest non-integer operators in the thermal OPE.
  • Heavy thermal OPE coefficients acquire universal spin-resolved formulae that improve truncated position-space correlators against Monte Carlo data.
  • The same inversion applies at finite spatial momentum, constraining the k-dependence of poles and residues once a large-k ansatz is supplied.

Where Pith is reading between the lines

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

  • The same blocks and inversion should extend to current and stress-tensor correlators, turning the method into a constraint on hydrodynamic transport coefficients.
  • Once branch-cut contributions are restored in Z(u), the formulae become a practical diagnostic for multiparticle continua in non-holographic critical theories.
  • Combining the large-k residue expansion with numerical quasinormal data from gravity duals could bootstrap unknown multi-stress-tensor OPE coefficients.

Editorial analysis

A structured set of objections, weighed in public.

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

Circularity Check

1 steps flagged

No load-bearing circularity: inversion and QNM sum rules are derived and checked on independent correlators; self-citations supply prior tools/data, not the target claims.

specific steps
  1. self citation load bearing [Sec. 2.1 / intro to thermal Polyakov blocks; cf. abstract and Eq. (2.20)–(2.22)]
    "Starting from KMS-symmetric completions of individual thermal OPE blocks, which play the role of thermal Polyakov blocks... In [18], it was argued that the correlator can be expressed as a manifestly KMS-symmetric integral of its discontinuity... we will refer to these objects as thermal Polyakov blocks."

    The entire momentum-space and inversion program takes as its starting object the KMS-symmetric single-block completions constructed in the authors’ prior work [18]. This is ordinary cumulative self-citation of a toolkit, not a reduction of the new QNM sum rules to an unverified self-claim: [18] is used as input definition of the blocks, while the novel content (Fourier transform, inversion formulae, meromorphic Z(u) constraints) is derived and tested externally. Mild only; not load-bearing for the strongest claim.

full rationale

The central chain (KMS Polyakov blocks → Fourier/momentum-space OPE → Mellin/Gegenbauer inversion → meromorphic QNM representation Z(u) and sum rules) is a mathematical derivation, not a fit renamed as prediction. When the paper recovers OPE coefficients, it plugs independently known retarded correlators (free scalar, large-N O(N), ε-expansion, 2d Virasoro primary, holographic N=4 R-current, BTZ) into the inversion and matches literature values—consistency checks, not self-fulfilling definitions. Heavy-operator asymptotics (2.32) follow from expanding the image sum and are compared to Lorentzian inversion and Monte Carlo shapes; the only fit is an overall normalization constant for Ising plots. Self-citations ([18], [25], [16], [34]) provide the position-space dispersion toolkit and light Ising one-point inputs, but those are used as tools/inputs; the novel QNM sum rules and large-k residue scalings are not assumed in the citations. Convergence/zeta-regularization subtleties of Z(m)=0 affect well-definedness of the bootstrap problem, not circularity of the derivation. Score 1 only for routine reliance on the authors’ prior thermal-block construction as the starting object.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 1 invented entities

The central UV–IR map rests on standard CFT axioms plus a small set of domain assumptions (KMS, Regge boundedness, clustering, meromorphicity) that are standard in the thermal-bootstrap literature but not theorems. No free parameters are fitted to produce the main formulae; the Ising comparison uses previously published OPE coefficients. Thermal Polyakov blocks are a reorganization of existing OPE data, not a new physical entity.

axioms (6)
  • domain assumption KMS condition g(z,¯z)=g(1-z,1-¯z) for Euclidean thermal two-point functions
    Used from the outset to construct the image sum that defines thermal Polyakov blocks (Sec. 2.1).
  • domain assumption Regge boundedness: g does not grow faster than w^{J_*} at large w
    Invoked to drop arcs or control contour deformations (Sec. 2.1); stated as expected but not rigorously proved for all thermal CFTs.
  • domain assumption Clustering at large spatial or temporal separation
    Fixes the additive constant in the arc contribution (Sec. 2.1 and 2.3).
  • ad hoc to paper Meromorphicity of the retarded correlator in complex frequency (isolated simple poles only)
    Explicit working hypothesis that converts the inversion formula into discrete QNM sum rules (opening of Sec. 4); the paper later sketches the branch-cut generalization.
  • domain assumption Existence of an asymptotic large-n trajectory for QNM frequencies and residues of power-law form
    Used to extract large-n asymptotics from poles of Z(u) (Sec. 4.3); motivated by holographic examples but not derived.
  • standard math Standard CFT OPE convergence inside its radius and residual conformal symmetry fixing thermal blocks
    Background used throughout Sec. 2.
invented entities (1)
  • Thermal Polyakov blocks independent evidence
    purpose: KMS-symmetric completions of individual thermal OPE blocks that serve as globally defined building blocks for the correlator
    Defined by the image sum (Eq. 2.22) or the dispersion integral (Eq. 2.21); they reorganize existing OPE data rather than introduce new dynamical degrees of freedom. Independent checks in free and large-N theories confirm they reproduce known correlators once arcs are fixed.

pith-pipeline@v1.2.0-grok45-kimik3 · 67943 in / 2927 out tokens · 56705 ms · 2026-07-31T05:48:27.630551+00:00 · methodology

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We initiate a bootstrap program that relates ultraviolet data, encoded in the thermal OPE, to infrared observables, namely, the low-frequency behavior and quasinormal modes. Starting from KMS-symmetric completions of individual thermal OPE blocks, which play the role of thermal Polyakov blocks, we construct their Fourier transform, yielding an asymptotic expansion of retarded thermal correlators valid at any spatial momentum. We use these results to derive inversion formulae and connect thermal OPE data to the analytic structure of retarded correlators in the complex frequency plane. Under the assumption of meromorphicity, the inversion formulae express OPE coefficients in terms of the quasinormal-mode frequencies, leading to nontrivial sum rules, constraints on the quasinormal spectrum, and its asymptotics at large spatial momentum. We illustrate these results in free theories, two-dimensional CFTs, the large-$N$ limit and $\varepsilon$-expansion of the $\mathrm{O}(N)$ model, and the $R$-current correlator of strongly coupled $\mathcal N = 4$ SYM at zero spatial momentum. As a byproduct, we derive universal asymptotic formulae for thermal OPE coefficients of heavy operators, resolving their dependence on spin and extending previous results at zero spatial separation. We test these formulae in the three-dimensional Ising CFT, finding good agreement between the resulting truncated correlators and Monte Carlo data.

Figures

Figures reproduced from arXiv: 2607.24919 by Alessio Miscioscia, Deniz N. Bozkurt, Elli Pomoni, Enrico Marchetto, Julien Barrat.

Figure 1
Figure 1. Figure 1: Analytic structure of the correlator g(z, z¯) in the w-plane. The Euclidean cor￾relator is located at |w| = 1, here represented by a blue circle. ⋆ Clustering condition. Thermal correlators are expected to cluster at large spatial distance, i.e., limx→∞ g(z, z¯) = ⟨ ϕ ⟩ 2 β . (2.12) Similarly, we can take the large-separation limit along the real time direction: lim τ→i∞ g(z, z¯) = ⟨ ϕ ⟩ 2 β . (2.13) Clust… view at source ↗
Figure 2
Figure 2. Figure 2: This plot illustrates the contribution of heavy operators for a given thermal Polyakov block FO(z, z¯), here the one associated with the identity operator O = 1. We set x = 0 for clarity. The pink curve shows the behavior of the thermal block in the s-channel, i.e., around z, z¯ ∼ 0 or τ ∼ 0. The green curve displays the thermal Polyakov block defined in (2.22). We see that an infinite number of heavy oper… view at source ↗
Figure 3
Figure 3. Figure 3: Convergence of garcs(x), expressed as a Taylor series as in Equation (2.45), to the asymptotic behavior (2.46) in the large-x limit. Combining the two expressions (2.42), (2.43), we fully resum the thermal Polyakov blocks decomposition of the thermal two-point function (2.39): X∞ m=0 a[σm] ζEH 1 2 − m + ε, x, τ = X ℓ∈Z e −mth√ x2+(τ+ℓ) 2 p x 2 + (τ + ℓ) 2 +  1 ε + 2γE + 2 log mth I0(mthx) − 2F(mthx) . … view at source ↗
Figure 4
Figure 4. Figure 4: Three-dimensional plot of g(τ, x) = ⟨ σ(τ, x)σ(0, 0)⟩β in the OPE regime τ 2 + x 2 < β2 with β = 40. The continuous surface represents the truncated expansion in thermal Polyakov blocks, while the red dots represent the Monte Carlo results. We observe very good agreement between the results, further supported by the plot of the discrepancy in [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Left: Plot of the Monte Carlo simulations (blue points with orange error bands) against the truncated expansion in thermal Polyakov blocks (black solid line) for the cor￾relator g(x) = ⟨ σ(0, x)σ(0, 0)⟩β in the spatial direction x for τ = 0. We observe a very good agreement in most of the OPE regime x 2 < β2 . As expected the two approaches start to disagree close to x 2 ∼ β 2 . Right: Discrepancy between … view at source ↗
Figure 6
Figure 6. Figure 6: Left: Plot of the Monte Carlo simulations (blue points with orange error bands) against the truncated expansion in thermal Polyakov blocks (black solid line) for the cor￾relator g(x) = ⟨ ϵ(0, x)ϵ(0, 0)⟩β in the spatial direction x for τ = 0. Right: Discrepancy between the analytic prediction of the truncated expansion in thermal Polyakov blocks and the Monte Carlo data for the correlator ⟨ ϵ(τ, x)ϵ(0, 0)⟩β… view at source ↗
Figure 7
Figure 7. Figure 7: The same Mellin–Barnes contour in the complex u-plane can be deformed in two different ways. The red dots are the kinematic poles of π/ sin(πu) at integer u. Closing to the right gives the Taylor expansion around ω = 0, while closing to the left gives the large-ω asymptotic expansion and, when present, the OPE poles coming from Z(u). The latter expression can be conveniently rewritten in terms of a Mellin–… view at source ↗
Figure 8
Figure 8. Figure 8: Real and imaginary part of the deep-IR Padé approximation gIR (5.51) (blue line) and of the exact microscopic retarded correlator gR(ω, k)−µ 0 0 (5.48) (orange line). We also plotted the absolute (black dashed) and relative (gray dashed) errors. The plots show the low-momentum limit L → ∞, where k = 1/L, ω = λ/L2 , and λ is big enough so that |D/λ| ≪ 1. Here we chose λ = 5, so that |D/λ| ≈ 0.027. which rep… view at source ↗
Figure 9
Figure 9. Figure 9: Translation of the choices of analytic structure from the z¯-plane to the w-plane. The first choice of branch cut, z¯ ∈ (−∞, 0) results in a cut crossing the Euclidean correlator (here represented by the cyan line and circle) in the w-plane. The second choice z¯ ∈ (1, ∞) produces a consistent cut for the w-plane. where we use the variables: z = rw , z¯ = r w . We wish to understand the analytic structure i… view at source ↗
Figure 10
Figure 10. Figure 10: Choice of analytic structure of the correlator when restricted on the main strip 0 < r < β. The symmetry w ←→ −w is restored by summing over images. B Evaluation of the inversion formula (5.52) In this appendix we compute explicitly the inversion formula. The object to evaluate reads: a˜ 2d O χDα = 1 π 2 1 + δJ,0 Res α=−∆ Z ∞ Λ dp p−α Z 1 −1 dη η 2 (1 − η 2 ) − 1 2 η + p(1 − η 2) TJ (η) . (B.1) The integr… view at source ↗
Figure 11
Figure 11. Figure 11: The analytic structure in the η plane of the integrand in the 2d inversion formula (B.1). The pole η (−) always lie on the integration path: to perform the computation it is necessary to use the +i0 prescription. This generates an additional residue which provides the imaginary part for the extracted OPE coefficients (see (5.53)). The principal value can be computed by recursion. For this purpose, we swit… view at source ↗

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