pith. sign in

arxiv: 2512.07706 · v3 · submitted 2025-12-08 · ✦ hep-th

Bootstrapping non-unitary CFTs

Yu-tin Huang , Shao-Cheng Lee , Henry Liao , Justinas Rumbutis This is my paper

Pith reviewed 2026-05-17 00:50 UTC · model grok-4.3

classification ✦ hep-th
keywords non-unitary CFTsconformal bootstrapVirasoro blocksOPE coefficientscrossing symmetryminimal modelsnumerical bootstraptruncation error
0
0 comments X

The pith

Crossing symmetry can bootstrap non-unitary CFTs by treating the cross-ratio variation of extracted OPE coefficients as a direct measure of truncation error.

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

The paper introduces a numerical bootstrap strategy that searches for approximate solutions to crossing equations without assuming unitarity or positive norms. For any trial spectrum, crossing symmetry at several different cross-ratios produces OPE coefficients whose residual dependence on the cross-ratio position serves as a scalar error measure. Minimizing this measure yields spectra that satisfy the bootstrap constraints to a controllable accuracy. When applied to two-dimensional Virasoro blocks, the search recovers the known A-series minimal models, including those with negative central charge, and produces new candidate solutions for central charge greater than one whose crossing violation matches the level seen in minimal models.

Core claim

The central claim is that the residual cross-ratio dependence of OPE coefficients obtained from crossing at multiple points directly quantifies the truncation error in a trial spectrum, thereby defining a scalar objective function that can be minimized to find approximate solutions to the crossing equations without imposing unitarity constraints.

What carries the argument

The scalar objective function given by the cross-ratio variation of OPE coefficients, which quantifies how well a truncated spectrum satisfies crossing without reference to unitarity.

If this is right

  • The method recovers all known A-series minimal models in two-dimensional Virasoro CFTs, including the non-unitary members.
  • It produces candidate truncated spectra for central charge c>1 whose crossing violation is comparable in size to that of the minimal models.
  • The same stability objective provides a practical search route for bootstrap solutions outside the convex unitary regime.
  • The framework extends bootstrap searches to any setting where unitarity cannot be imposed or is not desired.

Where Pith is reading between the lines

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

  • The same stability criterion could be tested on higher-dimensional or supersymmetric algebras to locate additional non-unitary solutions.
  • The candidate c>1 spectra could be further vetted by imposing modular invariance or other global consistency conditions not used in the original search.
  • If the objective function remains convex or near-convex in practice, gradient-based or other efficient optimizers could scale the method to larger truncations.

Load-bearing premise

The assumption that the amount of cross-ratio dependence left in the OPE coefficients after imposing crossing at several points reliably quantifies the truncation error in a way that permits stable numerical optimization.

What would settle it

A direct numerical check of the full crossing equations for the reported c>1 candidate spectra at substantially higher truncation level, showing whether the violation remains comparable to that of minimal models.

Figures

Figures reproduced from arXiv: 2512.07706 by Henry Liao, Justinas Rumbutis, Shao-Cheng Lee, Yu-tin Huang.

Figure 1
Figure 1. Figure 1: FIG. 1: The reward distributions of minimal model [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Search result in [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Search results for non-unitary CFTs with [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Search result of different truncated space where [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Search results around Yang-Lee CFT region with [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Search results around Yang-Lee CFT region with [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
read the original abstract

We introduce a non-unitary-compatible numerical bootstrap strategy based on the statistical stability of OPE data inferred from crossing at multiple cross-ratios. For a trial spectrum, crossing determines OPE coefficients whose residual cross-ratio dependence directly measures the truncation error. This defines a scalar objective on the space of spectra, allowing bootstrap searches without imposing unitarity. Applied to two-dimensional Virasoro blocks, the method reproduces known A-series minimal models, including non-unitary examples, and yields candidate truncated solutions for c>1 with crossing violation comparable to that of minimal models. More generally, our framework provides a practical route to solving bootstrap constraints beyond the convex, unitary setting.

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.

Referee Report

2 major / 2 minor

Summary. The paper introduces a non-unitary-compatible numerical bootstrap strategy based on the statistical stability of OPE data inferred from crossing at multiple cross-ratios. For a trial spectrum, crossing determines OPE coefficients whose residual cross-ratio dependence directly measures the truncation error. This defines a scalar objective on the space of spectra, allowing bootstrap searches without imposing unitarity. Applied to two-dimensional Virasoro blocks, the method reproduces known A-series minimal models, including non-unitary examples, and yields candidate truncated solutions for c>1 with crossing violation comparable to that of minimal models.

Significance. If the central claims hold, this work would offer a valuable extension of the conformal bootstrap to non-unitary CFTs, which are relevant for a range of physical systems. The successful reproduction of known minimal models, including non-unitary ones, serves as an important validation of the approach. The framework provides a practical route to solving bootstrap constraints beyond the convex, unitary setting, potentially enabling new discoveries in the space of CFTs.

major comments (2)
  1. §3 (Method), Eq. (8) and surrounding discussion: the scalar objective is defined from residual cross-ratio variation of OPE coefficients extracted at several z values. The manuscript provides no separate diagnostic (e.g., norm of the crossing vector on a denser z-grid or comparison of objective value at the known spectrum versus nearby points) showing that this variation is dominated by truncation error rather than sign changes or convergence issues in non-unitary spectra.
  2. §4.3 (c>1 candidates): the reported crossing violation for the candidate solutions is stated to be comparable to that of minimal models, but without an explicit uniqueness analysis of the objective minimum or correlation with an independent error measure, it remains unclear whether these represent stable truncated solutions.
minor comments (2)
  1. Figure 2 caption: clarify the precise number of cross-ratio sampling points and the range of z values employed in the optimization.
  2. Notation for OPE coefficients: ensure consistent use of symbols (e.g., C_{ijk} versus lambda) across the method and results sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We address each major point below and have incorporated revisions to strengthen the presentation of the method and results.

read point-by-point responses

  1. Referee: §3 (Method), Eq. (8) and surrounding discussion: the scalar objective is defined from residual cross-ratio variation of OPE coefficients extracted at several z values. The manuscript provides no separate diagnostic (e.g., norm of the crossing vector on a denser z-grid or comparison of objective value at the known spectrum versus nearby points) showing that this variation is dominated by truncation error rather than sign changes or convergence issues in non-unitary spectra.

    Authors: We agree that an explicit diagnostic would help confirm the objective primarily reflects truncation error. In the revised manuscript we have added a new figure and accompanying discussion in §3 that computes the norm of the crossing vector on a denser z-grid for the reproduced A-series spectra (both unitary and non-unitary). This shows that the objective value is minimized precisely where the independent crossing measure is smallest. For non-unitary cases, the method extracts OPE coefficients without positivity assumptions; the empirical reproduction of the known non-unitary minimal models indicates that sign changes or convergence artifacts do not dominate the objective at the physical points. revision: yes

  2. Referee: §4.3 (c>1 candidates): the reported crossing violation for the candidate solutions is stated to be comparable to that of minimal models, but without an explicit uniqueness analysis of the objective minimum or correlation with an independent error measure, it remains unclear whether these represent stable truncated solutions.

    Authors: We appreciate the referee’s request for further validation. The candidates are identified as local minima of the objective, with reported crossing violations comparable to those of the minimal models. In the revised §4.3 we now include a local stability analysis: small perturbations of the candidate spectra are shown to increase the objective value, confirming they are local minima. We also add a plot correlating the objective value with an independent crossing-violation measure evaluated on a separate set of cross-ratios. A global uniqueness proof across the full parameter space lies beyond the numerical scope of the present work and is left for future investigation. revision: partial

Circularity Check

0 steps flagged

No significant circularity; objective derived directly from crossing equations

full rationale

The paper constructs a scalar objective from the residual cross-ratio dependence of OPE coefficients obtained by solving crossing equations at multiple points for a trial spectrum. This objective is minimized over the space of spectra to identify candidate solutions. The derivation chain is self-contained: the objective is defined from the crossing symmetry constraints themselves rather than from any fitted parameter or self-referential quantity. Reproduction of known A-series minimal models (including non-unitary cases) provides an independent external benchmark, and no load-bearing step reduces by construction to the inputs via self-definition, self-citation, or renaming. The method introduces a new optimization criterion without smuggling in prior ansatze or uniqueness theorems from the authors' own work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that residual cross-ratio dependence measures truncation error; no free parameters or invented entities are explicitly introduced in the abstract.

axioms (1)
  • domain assumption Residual cross-ratio dependence of OPE coefficients obtained from crossing equations directly measures truncation error.
    This assumption is invoked to turn the residual into a scalar objective function for spectrum search.

pith-pipeline@v0.9.0 · 5405 in / 1245 out tokens · 61903 ms · 2026-05-17T00:50:39.819330+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

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

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Neural Spectral Bias and Conformal Correlators I: Introduction and Applications

    hep-th 2026-04 unverdicted novelty 8.0

    Neural networks optimized solely on crossing symmetry reconstruct CFT correlators from minimal input data to few-percent accuracy across generalized free fields, minimal models, Ising, N=4 SYM, and AdS diagrams.

  2. Descending into the Modular Bootstrap

    hep-th 2026-04 unverdicted novelty 7.0

    Machine-learning optimization produces candidate truncated modular-invariant partition functions for 2d CFTs in the central-charge window 1 to 8/7, indicating a continuous solution space and a stricter spectral-gap bo...

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages · cited by 2 Pith papers · 9 internal anchors

  1. [1]

    Progress of Theoretical Bootstrap

    For these points eq.(4) can be organized as: NX k=1 Ck Gk,j +e j =v j, (5) whereC k = fϕϕk 2 ,G k,j =G ∆ϕ ∆k,sk(zj,¯zj)andv j = vac(zj,¯zj). For every choice of theN{zj,¯zj}points (we 1In practice, there will also be errors coming from the approximation for Virasoro blocks. denote them as{z}) we have a matrix equation, linear in the squared OPE coefficien...

    work page

  2. [2]

    Bounding scalar operator dimensions in 4D CFT

    R. Rattazzi, V. S. Rychkov, E. Tonni, and A. Vichi, JHEP12, 031 (2008), arXiv:0807.0004 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  3. [3]

    A. M. Polyakov, Zh. Eksp. Teor. Fiz.66, 23 (1974). b

    work page 1974

  4. [4]

    Poland, S

    D. Poland, S. Rychkov, and A. Vichi, Rev. Mod. Phys. 91, 015002 (2019)

    work page 2019

  5. [5]

    El-Showk, M

    S. El-Showk, M. F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin, and A. Vichi, Phys. Rev. D86, 025022 (2012)

    work page 2012

  6. [6]

    Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents

    S. El-Showk, M. F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin, and A. Vichi, J. Stat. Phys.157, 869 (2014), arXiv:1403.4545 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  7. [7]

    F. Kos, D. Poland, D. Simmons-Duffin, and A. Vichi, JHEP11, 106 (2015), arXiv:1504.07997 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  8. [8]

    A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov, Nucl. Phys. B241, 333 (1984)

    work page 1984

  9. [9]

    A. Laio, U. L. Valenzuela, and M. Serone, Phys. Rev. D 106, 025019 (2022), arXiv:2206.05193 [hep-th]

    work page arXiv 2022

  10. [10]

    Kántor, V

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

    work page arXiv 2022

  11. [11]

    Kántor, V

    G. Kántor, V. Niarchos, and C. Papageorgakis, Physical Review D105(2022), 10.1103/physrevd.105.025018

    work page doi:10.1103/physrevd.105.025018 2022

  12. [12]

    Kántor, V

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

    work page arXiv 2023

  13. [13]

    The CMA Evolution Strategy: A Tutorial

    N. Hansen, arXiv preprint arXiv:1604.00772 (2016)

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  14. [14]

    Belavin, A

    A. Belavin, A. Polyakov, and A. Zamolodchikov, Nuclear Physics B241, 333 (1984)

    work page 1984

  15. [15]

    F. A. Dolan and H. Osborn, Nucl. Phys. B599, 459 (2001), arXiv:hep-th/0011040

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  16. [16]

    OPE Convergence in Conformal Field Theory

    D. Pappadopulo, S. Rychkov, J. Espin, and R. Rattazzi, Phys. Rev. D86, 105043 (2012), arXiv:1208.6449 [hep- th]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  17. [17]

    G. C. R. Douglas C. Montgomery,Applied Statistics and Probability for Engineers, 7th Edition(Wiley, 2018)

    work page 2018

  18. [18]

    H. Chen, C. Hussong, J. Kaplan, and D. Li, Journal of High Energy Physics2017(2017), 10.1007/jhep09(2017)102

    work page doi:10.1007/jhep09(2017)102 2017

  19. [19]

    Github link to our implementation (2025)

    work page 2025

  20. [20]

    The Lightcone Bootstrap and the Spectrum of the 3d Ising CFT

    D. Simmons-Duffin, JHEP03, 086 (2017), arXiv:1612.08471 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  21. [21]

    Light Cone Bootstrap in General 2D CFTs and Entanglement from Light Cone Singularity

    Y. Kusuki, JHEP01, 025 (2019), arXiv:1810.01335 [hep- th]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  22. [22]

    Shala, A

    G. Shala, A. Biedenkapp, N. Awad, S. Adriaensen, M. Lindauer, and F. Hutter, inParallel Problem Solving from Nature – PPSN XVI, Lecture Notes in Computer Science, Vol. 12269 (Springer, Cham, 2020) pp. 691–706

    work page 2020

  23. [23]

    T. V. Vishnu, P. Malhotra, J. Narwariya, L. Vig, and G. Shroff, arXiv e-prints , arXiv:1907.06901 (2019), arXiv:1907.06901 [cs.LG]

    work page arXiv 1907

  24. [24]

    R. T. Lange, T. Schaul, Y. Chen, T. Zahavy, V. Dallibard, C. Lu, S. Singh, and S. Flennerhag, in Proceedings of the International Conference on Learning Representations (ICLR)(2023) arXiv:2211.11260

    work page arXiv 2023

  25. [25]

    T. Chen, X. Chen, W. Chen, H. Heaton, J. Liu, Z. Wang, and W. Yin, Journal of Machine Learning Research23, 1 (2022)

    work page 2022

  26. [26]

    Learning to Optimize

    K. Li and J. Malik, arXiv e-prints , arXiv:1606.01885 (2016), arXiv:1606.01885 [cs.LG]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  27. [27]

    Strominger,The dS/CFT Correspondence, JHEP10, 034 (2001), arXiv:hep- th/0106113

    A. Strominger, JHEP10, 034 (2001), arXiv:hep- th/0106113

    work page arXiv 2001

  28. [28]

    Poland, V

    D. Poland, V. Prilepina, and P. Tadić, JHEP10, 153 (2023), arXiv:2305.08914 [hep-th]. c FIG. 5: Search results around Yang-Lee CFT region with2and3internal operators in projection to (O,hint,2) whereO∈ {h int,1, hext, c}. The line of minimal model represents minimal models withp, q≤200. FIG. 6: Search results around Yang-Lee CFT region with3internal opera...

    work page arXiv 2023