arxiv: 2605.13183 · v1 · submitted 2026-05-13 · ✦ hep-th

Recognition: 2 theorem links

· Lean Theorem

Neural Networks, Dispersion Relations and the Thermal Bootstrap

Constantinos Papageorgakis, Vasilis Niarchos

Authors on Pith no claims yet

Pith reviewed 2026-05-14 18:46 UTC · model grok-4.3

classification ✦ hep-th

keywords conformal bootstrapdispersion relationsneural networksthermal two-point functionsOPE contributionspositivity-free bootstrapholographic CFTsgeneralized free fields

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The pith

Dispersion relations and neural networks enable a positivity-free conformal bootstrap for thermal correlators.

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

The paper reviews a framework for the conformal bootstrap that avoids positivity assumptions by using dispersion relations to resum the infinite tower of high-dimension OPE contributions to correlators. Neural networks solve the resulting non-convex optimization problem for the bootstrap equations. The method is applied to scalar thermal two-point functions on a circle times flat space. Stability of the optimization is analyzed, along with links to smoothness properties of CFT correlators. Numerical illustrations are given for generalized free fields and four-dimensional holographic CFTs.

Core claim

The central claim is that dispersion relations can resum the infinite sum of high-dimension operator contributions in the OPE, reformulating the bootstrap as a non-convex optimization problem that neural networks can solve numerically, with the framework tested on thermal two-point functions for generalized free fields and holographic CFTs.

What carries the argument

Dispersion relations that resum high-dimension OPE contributions, paired with neural-network optimization of the bootstrap functional equations.

If this is right

The method computes thermal two-point functions in CFTs without requiring positivity bounds on the spectrum.
Numerical access becomes possible for cases where traditional bootstrap techniques are limited by lack of positivity.
Stability of the non-convex scheme supports reliable use for thermal correlators on S^1 x R^{d-1}.
The approach connects to discussions of smoothness properties in CFT correlators.

Where Pith is reading between the lines

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

The framework could extend to higher-point correlators or other geometries beyond thermal setups.
Neural networks may offer advantages over linear programming in handling nonlinear bootstrap constraints.
Further tests in higher dimensions could reveal how the method scales with spacetime dimension.

Load-bearing premise

The stability properties of the non-convex optimisation scheme hold for the relevant thermal two-point functions on S^1 x R^{d-1}.

What would settle it

A demonstration that the neural-network optimization fails to converge to known results or becomes unstable for the generalized free field thermal two-point function would falsify practical applicability of the framework.

Figures

Figures reproduced from arXiv: 2605.13183 by Constantinos Papageorgakis, Vasilis Niarchos.

**Figure 2.** Figure 2: GFF benchmark with d = 4, ∆ϕ = 1.68, J∗ = 6, one exposed coefficient a1,0 + 3a0,2 and a correct analytic anchor AJ (0.7)|GFF (case (ii) of the text). The first four panels show the predicted tails A0, A2, A4, A6 (blue mean ±1σ band) against the analytic GFF curves (dashed red). The fifth panel shows the combined contribution of A0 and A2 to the conformal block expansion at w = 1; the heatmap reports the re… view at source ↗

**Figure 3.** Figure 3: Mean training loss (log scale) of the 10 lowest-loss configurations as a function of the ReLU [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗

read the original abstract

We review a framework for the conformal bootstrap that does not rely on positivity and treats the infinite tower of high-dimension OPE contributions to conformal correlators through dispersion relations and neural networks. We apply it to scalar thermal two-point functions on $S^1\times \mathbb R^{d-1}$. We discuss the stability properties of the relevant non-convex optimisation scheme and potential relations to recent discussions of smoothness properties in CFT correlators. We illustrate the numerical application of the method to Generalized Free Fields and 4d holographic CFTs. This is a proceedings contribution to the ``Athens Workshop in Theoretical Physics: 10th Anniversary", held at the National and Kapodistrian University of Athens on December 17-19 2025.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

A review of NN-assisted dispersion relations for thermal CFTs that illustrates the idea but doesn't close the stability gap.

read the letter

The main thing here is that this is a proceedings review outlining how dispersion relations combined with a neural network ansatz can resum high-dimension OPE contributions in thermal two-point functions, bypassing the usual positivity requirements. It applies the setup to scalar operators on S^1 times flat space and shows some numerics on generalized free fields and 4d holographic models while touching on smoothness properties of correlators. That part is straightforward and gives a clear picture of the framework for readers who already know dispersion relations in CFTs. The illustrations work as expected in the free and holographic cases where the spectrum is under control. The paper also notes the non-convex optimization and its stability properties, which is the part that could matter for thermal bootstrap work. The soft spot is exactly the stability claim. The text asserts that the optimizer behaves well but only demonstrates it on GFFs, where the answer is known ahead of time, and on holographic examples. Nothing shows systematic checks over random seeds, architecture changes, or analytic bounds that would cover generic interacting CFTs. The non-positive kernel and the periodicity from the circle are acknowledged but not tested for whether they introduce bad local minima. No error analysis or convergence diagnostics appear in the provided material. This is for people already working on bootstrap methods who want a quick look at a positivity-free variant. A reader in that group will pick up the approach in one pass but will not get new theorems or broad benchmarks. I would not send it for peer review in its current form. It reads like workshop notes rather than a self-contained paper with verifiable claims that needs formal referee time. If the authors added more models, error bars, and a scan of the optimizer, that could shift the picture.

Referee Report

2 major / 1 minor

Summary. The manuscript reviews a framework for the conformal bootstrap that dispenses with positivity assumptions by using dispersion relations to resum the infinite tower of high-dimension OPE contributions to conformal correlators, with neural networks serving as the ansatz. It applies the method to scalar thermal two-point functions on S¹ × ℝ^{d-1}, discusses stability properties of the non-convex optimization, and provides numerical illustrations for Generalized Free Fields and 4d holographic CFTs. This is presented as a proceedings contribution.

Significance. If the stability of the optimization can be placed on firmer footing, the approach would offer a useful positivity-independent route to the thermal bootstrap, allowing control over the high-dimension tower in settings where standard positivity-based methods are limited. The numerical illustrations for GFF and holographic cases demonstrate feasibility, and the link to smoothness properties in CFT correlators merits further exploration.

major comments (2)

[Stability discussion] The discussion of stability properties of the non-convex optimization (in the section addressing the relevant scheme) relies on illustrations for Generalized Free Fields, where the spectrum is known a priori, and 4d holographic CFTs. No analytic bounds, systematic scans over random seeds or network architectures, or tests for generic interacting CFTs are provided. The non-positive kernel and S¹ periodicity constraints are noted but not shown to preserve convergence in broader cases; this is load-bearing for the claim that the method reliably resums the OPE tower.
[Numerical illustrations] In the numerical applications section, the GFF and holographic illustrations lack error analysis, convergence diagnostics (e.g., loss curves across seeds), or quantitative comparisons to independent results beyond the exactly solvable GFF limit. This weakens the assessment of accuracy for cases where the spectrum is unknown.

minor comments (1)

[Abstract] The abstract refers to 'potential relations to recent discussions of smoothness properties in CFT correlators,' but the main text provides only a brief mention without explicit connections or citations to the relevant literature.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading and constructive comments on our proceedings contribution. We address each major point below, taking into account the limited scope and length of the manuscript.

read point-by-point responses

Referee: The discussion of stability properties of the non-convex optimization (in the section addressing the relevant scheme) relies on illustrations for Generalized Free Fields, where the spectrum is known a priori, and 4d holographic CFTs. No analytic bounds, systematic scans over random seeds or network architectures, or tests for generic interacting CFTs are provided. The non-positive kernel and S¹ periodicity constraints are noted but not shown to preserve convergence in broader cases; this is load-bearing for the claim that the method reliably resums the OPE tower.

Authors: We agree that the stability discussion is based on numerical illustrations for GFF and holographic cases rather than analytic bounds or exhaustive scans. As a short proceedings contribution, a comprehensive analysis of convergence for generic interacting CFTs lies beyond the present scope. The examples demonstrate practical stability under the non-positive kernel and periodicity constraints, but we will add a clarifying sentence noting the illustrative character of these results and the desirability of further tests in future work. revision: partial
Referee: In the numerical applications section, the GFF and holographic illustrations lack error analysis, convergence diagnostics (e.g., loss curves across seeds), or quantitative comparisons to independent results beyond the exactly solvable GFF limit. This weakens the assessment of accuracy for cases where the spectrum is unknown.

Authors: We acknowledge that the current numerical section provides limited diagnostics and error estimates. For the GFF case we recover the exact result, while the holographic example is presented as a feasibility check. Within proceedings length constraints we can add a brief note on observed loss behavior and basic stability across a few seeds, but quantitative comparisons for unknown spectra are not feasible without independent methods. We will incorporate a short convergence remark in the revision. revision: partial

standing simulated objections not resolved

Analytic bounds on optimization stability or systematic architecture scans over generic CFTs, which would require a full-length research article rather than a proceedings contribution.

Circularity Check

0 steps flagged

No circularity: review of dispersion+NN framework with external illustrations on known spectra

full rationale

The paper is explicitly a review/proceedings contribution that applies an existing dispersion-relation + neural-network framework to thermal two-point functions. Illustrations are performed on Generalized Free Fields and 4d holographic CFTs where the OPE data are known independently; the stability discussion is presented as numerical evidence rather than a derivation that reduces to a fitted parameter or self-citation chain defined inside the manuscript. No load-bearing step equates a claimed prediction to its own input by construction, and the central claim remains independent of any internal tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework rests on standard CFT dispersion relations and the assumption that neural-network optimization can stably approximate infinite OPE sums; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption Dispersion relations hold for conformal correlators on the thermal circle
Invoked as the basis for handling high-dimension OPE contributions.

pith-pipeline@v0.9.0 · 5418 in / 1010 out tokens · 59612 ms · 2026-05-14T18:46:34.937380+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
We review a framework for the conformal bootstrap that does not rely on positivity and treats the infinite tower of high-dimension OPE contributions to conformal correlators through dispersion relations and neural networks.
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean J_uniquely_calibrated_via_higher_derivative unclear
the non-convex optimisation converges accurately to the physical solution

Reference graph

Works this paper leans on

56 extracted references · 52 canonical work pages · 22 internal anchors

[1]

The Conformal Bootstrap: Theory, Numerical Techniques, and Applications

D. Poland, S. Rychkov and A. Vichi,Rev. Mod. Phys.91(2019) 015002, arXiv:1805.04405 [hep-th]

work page internal anchor Pith review Pith/arXiv arXiv 2019
[2]

Rychkov and N

S. Rychkov and N. Su,Rev. Mod. Phys.96(2024) 045004,arXiv:2311.15844 [hep-th]

work page arXiv 2024
[3]

Poland and D

D. Poland and D. Simmons-Duffin, Snowmass White Paper: The Numerical Confor- mal Bootstrap, inSnowmass 2021, (3 2022),arXiv:2203.08117 [hep-th]

work page arXiv 2021
[4]

Hartman, D

T. Hartman, D. Mazac, D. Simmons-Duffin and A. Zhiboedov, Snowmass White Paper: The Analytic Conformal Bootstrap, inSnowmass 2021, (2 2022), arXiv:2202.11012 [hep-th]

work page arXiv 2021
[5]

More constraining conformal bootstrap

F. Gliozzi,Phys. Rev. Lett.111(2013) 161602,arXiv:1307.3111 [hep-th]

work page internal anchor Pith review Pith/arXiv arXiv 2013
[6]

Critical exponents of the 3d Ising and related models from Conformal Bootstrap

F. Gliozzi and A. Rago,JHEP10(2014) 042,arXiv:1403.6003 [hep-th]

work page internal anchor Pith review Pith/arXiv arXiv 2014
[7]

Gliozzi, P

F. Gliozzi, P. Liendo, M. Meineri and A. Rago,JHEP05(2015) 036, arXiv:1502.07217 [hep-th], [Erratum: JHEP 12, 093 (2021)]

work page arXiv 2015
[8]

Truncatable bootstrap equations in algebraic form and critical surface exponents

F. Gliozzi,JHEP10(2016) 037,arXiv:1605.04175 [hep-th]

work page internal anchor Pith review Pith/arXiv arXiv 2016
[9]

Closure of the Operator Product Expansion in the Non-Unitary Bootstrap

I. Esterlis, A. L. Fitzpatrick and D. Ramirez,JHEP11(2016) 030,arXiv:1606.07458 [hep-th]

work page internal anchor Pith review Pith/arXiv arXiv 2016
[10]

New method for the conformal bootstrap with OPE truncations

W. Li (11 2017)arXiv:1711.09075 [hep-th]

work page internal anchor Pith review Pith/arXiv arXiv 2017
[11]

Inverse Bootstrapping Conformal Field Theories

W. Li,JHEP01(2018) 077,arXiv:1706.04054 [hep-th]

work page internal anchor Pith review Pith/arXiv arXiv 2018
[12]

Conformal bootstrap for percolation and polymers

A. Leclair and J. Squires,J. Stat. Mech.1812(2018) 123105,arXiv:1802.08911 [hep-th]

work page internal anchor Pith review Pith/arXiv arXiv 2018
[13]

K´ antor, V

G. K´ antor, V. Niarchos and C. Papageorgakis,Phys. Rev. D105(2022) 025018, arXiv:2108.09330 [hep-th]

work page arXiv 2022
[14]

Niarchos, C

V. Niarchos, C. Papageorgakis, P. Richmond, A. G. Stapleton and M. Woolley,Phys. Rev. D108(2023) 105027,arXiv:2306.15730 [hep-th]

work page arXiv 2023
[15]

Niarchos, C

V. Niarchos, C. Papageorgakis, A. Stratoudakis and M. Woolley,Phys. Rev. D112 (2025) 126012,arXiv:2508.08560 [hep-th]

work page arXiv 2025
[16]

K´ antor, V

G. K´ antor, V. Niarchos and C. Papageorgakis,Phys. Rev. Lett.128(2022) 041601, arXiv:2108.08859 [hep-th]

work page arXiv 2022
[17]

K´ antor, V

G. K´ antor, V. Niarchos, C. Papageorgakis and P. Richmond,Phys. Rev. D107(2023) 025005,arXiv:2209.02801 [hep-th]

work page arXiv 2023
[18]

Neural Networks Reveal a Universal Bias in Conformal Correlators

K. Ghosh, S. Kumar, V. Niarchos and A. Stergiou (4 2026)arXiv:2604.18673 [hep-th]

work page internal anchor Pith review Pith/arXiv arXiv 2026
[19]

Neural Spectral Bias and Conformal Correlators I: Introduction and Applications

K. Ghosh, S. Kumar, V. Niarchos and A. Stergiou (4 2026)arXiv:2604.18686 [hep-th]

work page internal anchor Pith review Pith/arXiv arXiv 2026
[20]

Arnaudo and B

P. Arnaudo and B. Withers (3 2026)arXiv:2603.13469 [hep-th]

work page arXiv 2026
[21]

Bajc and K

B. Bajc and K. Trailovi´ c,JHEP11(2025) 133,arXiv:2509.02226 [hep-th]

work page arXiv 2025
[22]

Barrat, D

J. Barrat, D. N. Bozkurt, E. Marchetto, A. Miscioscia and E. Pomoni (10 2025) arXiv:2510.20894 [hep-th]

work page arXiv 2025
[23]

I. J. Araya, C. Esper, Y. Jia, M. Kulaxizi and A. Parnachev (2 2026) arXiv:2602.12893 [hep-th]

work page arXiv 2026
[24]

ˇCeplak, H

N. ˇCeplak, H. Liu, A. Parnachev and S. Valach (11 2025)arXiv:2511.09638 [hep-th]

work page arXiv 2025
[25]

Chakravarty (12 2025)arXiv:2512.10912 [hep-th]

J. Chakravarty (12 2025)arXiv:2512.10912 [hep-th]

work page arXiv 2025
[26]

R. A. Davison and H. Jiang,JHEP04(2026) 162,arXiv:2512.11024 [hep-th]

work page arXiv 2026
[27]

Bouncing singularities and thermal correlators on line defects

S. Giombi, Y.-Z. Li and J. Shan (3 2026)arXiv:2603.11012 [hep-th]

work page internal anchor Pith review Pith/arXiv arXiv 2026
[28]

The Conformal Bootstrap at Finite Temperature

L. Iliesiu, M. Kolo˘ glu, R. Mahajan, E. Perlmutter and D. Simmons-Duffin,JHEP10 Neural Networks, Dispersion Relations and the Thermal Bootstrap21 (2018) 070,arXiv:1802.10266 [hep-th]

work page internal anchor Pith review Pith/arXiv arXiv 2018
[29]

Emergent Spacetime and Holographic CFTs

S. El-Showk and K. Papadodimas,JHEP10(2012) 106,arXiv:1101.4163 [hep-th]

work page internal anchor Pith review Pith/arXiv arXiv 2012
[30]

E. Katz, S. Sachdev, E. S. Sørensen and W. Witczak-Krempa,Phys. Rev. B90(2014) 245109,arXiv:1409.3841 [cond-mat.str-el]

work page internal anchor Pith review Pith/arXiv arXiv 2014
[31]

Constraining quantum critical dynamics: 2+1D Ising model and beyond

W. Witczak-Krempa,Phys. Rev. Lett.114(2015) 177201,arXiv:1501.03495 [cond-mat.str-el]

work page internal anchor Pith review Pith/arXiv arXiv 2015
[32]

L. F. Alday, M. Kologlu and A. Zhiboedov,JHEP06(2021) 082,arXiv:2009.10062 [hep-th]

work page arXiv 2021
[33]

Stratoudakis,University of Crete Institutional Repository(2024) 35

A. Stratoudakis,University of Crete Institutional Repository(2024) 35

2024
[34]

A. C. Petkou and A. Stergiou,Phys. Rev. Lett.121(2018) 071602,arXiv:1806.02340 [hep-th]

work page internal anchor Pith review Pith/arXiv arXiv 2018
[35]

L. F. Alday and S. Caron-Huot,JHEP12(2018) 017,arXiv:1711.02031 [hep-th]

work page internal anchor Pith review Pith/arXiv arXiv 2018
[36]

L. F. Alday, J. Henriksson and M. van Loon,JHEP07(2018) 131,arXiv:1712.02314 [hep-th]

work page internal anchor Pith review Pith/arXiv arXiv 2018
[37]

Universality at large transverse spin in defect CFT

M. Lemos, P. Liendo, M. Meineri and S. Sarkar,JHEP09(2018) 091, arXiv:1712.08185 [hep-th]

work page internal anchor Pith review Pith/arXiv arXiv 2018
[38]

J. Liu, E. Perlmutter, V. Rosenhaus and D. Simmons-Duffin,JHEP03(2019) 052, arXiv:1808.00612 [hep-th]

work page internal anchor Pith review Pith/arXiv arXiv 2019
[39]

All Tree-Level Correlators in AdS${}_5\times$S${}_5$ Supergravity: Hidden Ten-Dimensional Conformal Symmetry

S. Caron-Huot and A.-K. Trinh,JHEP01(2019) 196,arXiv:1809.09173 [hep-th]

work page internal anchor Pith review Pith/arXiv arXiv 2019
[40]

Resummation at finite conformal spin

C. Cardona, S. Guha, S. K. Kanumilli and K. Sen,JHEP01(2019) 077, arXiv:1811.00213 [hep-th]

work page internal anchor Pith review Pith/arXiv arXiv 2019
[41]

Li,JHEP01(2020) 055,arXiv:1906.00707 [hep-th]

W. Li,JHEP01(2020) 055,arXiv:1906.00707 [hep-th]

work page arXiv 2020
[42]

Albayrak, D

S. Albayrak, D. Meltzer and D. Poland,JHEP08(2019) 040,arXiv:1904.00032 [hep-th]

work page arXiv 2019
[43]

Carmi and S

D. Carmi and S. Caron-Huot,JHEP09(2020) 009,arXiv:1910.12123 [hep-th]

work page arXiv 2020
[44]

L. F. Alday, J. Henriksson and M. van Loon,JHEP01(2020) 063,arXiv:1907.02445 [hep-th]

work page arXiv 2020
[45]

Lemos, B

M. Lemos, B. C. van Rees and X. Zhao,JHEP01(2022) 022,arXiv:2105.13361 [hep-th]

work page arXiv 2022
[46]

Marchetto, A

E. Marchetto, A. Miscioscia and E. Pomoni,JHEP09(2024) 044,arXiv:2312.13030 [hep-th]

work page arXiv 2024
[47]

Buri´ c, I

I. Buri´ c, I. Gusev and A. Parnachev,JHEP09(2025) 053,arXiv:2505.10277 [hep-th]

work page arXiv 2025
[48]

Barrat, D

J. Barrat, D. N. Bozkurt, E. Marchetto, A. Miscioscia and E. Pomoni (6 2025) arXiv:2506.06422 [hep-th]

work page arXiv 2025
[49]

A. L. Fitzpatrick and K.-W. Huang,JHEP08(2019) 138,arXiv:1903.05306 [hep-th]

work page arXiv 2019
[50]

Parisini, K

E. Parisini, K. Skenderis and B. Withers (12 2023)arXiv:2312.03820 [hep-th]

work page arXiv 2023
[51]

Buri´ c, I

I. Buri´ c, I. Gusev and A. Parnachev (8 2025)arXiv:2508.08373 [hep-th]

work page arXiv 2025
[52]

Neural tangent kernel: Convergence and generalization in neural networks

A. Jacot, F. Gabriel and C. Hongler (6 2018)arXiv:1806.07572 [cs.LG]

work page arXiv 2018
[53]

John Xu, Y

Z.-Q. John Xu, Y. Zhang, T. Luo, Y. Xiao and Z. Ma,Communications in Computa- tional Physics28(2020) 1746–1767

2020
[54]

Rahaman, A

N. Rahaman, A. Baratin, D. Arpit, F. Draxler, M. Lin, F. A. Hamprecht, Y. Bengio and A. Courville, On the spectral bias of neural networks (2019)

2019
[55]

Bietti and J

A. Bietti and J. Mairal, On the inductive bias of neural tangent kernels (2019)

2019
[56]

Barrat, E

J. Barrat, E. Marchetto, A. Miscioscia and E. Pomoni,Phys. Rev. Lett.134(2025) 211604,arXiv:2411.00978 [hep-th]

work page arXiv 2025