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arxiv: 2606.19283 · v1 · pith:GUNKNZFQnew · submitted 2026-06-17 · ✦ hep-th

A Dispersive Bootstrap for the Virasoro-Shapiro Amplitude

Yongjun Xu This is my paper

Pith reviewed 2026-06-26 19:44 UTC · model grok-4.3

classification ✦ hep-th
keywords dispersive S-matrix bootstrapVirasoro-Shapiro amplitudeWilson coefficientscrossing symmetrypartial-wave unitarityRegge boundednessgravity pole subtractionnonlinear constraints
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The pith

A Virasoro-inspired ansatz with nonlinear constraints reduces the bootstrap allowed region to a small island containing the Virasoro-Shapiro amplitude point after graviton pole subtraction.

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

This paper applies the dispersive S-matrix bootstrap to the closed-string tree-level Virasoro-Shapiro amplitude in ten-dimensional maximally supersymmetric theory. It imposes analyticity, crossing symmetry, partial-wave unitarity and Regge boundedness, first with the massless graviton pole kept explicit to obtain numerical bounds on leading Wilson coefficients normalized by the gravitational coupling. A Virasoro-inspired ansatz then supplies nonlinear relations among the coefficients that shrink the allowed region toward the target trajectory. In the gravity-pole-subtracted setup where the regular part has a well-defined forward limit, these nonlinear constraints narrow the space to a small island that includes the Virasoro-Shapiro point, accompanied by an analytic bootstrap explanation. A sympathetic reader cares because the work shows how general S-matrix principles plus a string-inspired ansatz can isolate the low-energy string amplitude without building the full worldsheet theory.

Core claim

For the ten-dimensional maximally supersymmetric four-point amplitude, dispersion relations and crossing null constraints give numerical bounds on the leading low-energy coefficients normalized by the gravitational coupling when the massless graviton pole is kept explicitly. The Virasoro-inspired ansatz becomes a set of nonlinear relations among Wilson coefficients and shrinks the allowed region toward the Virasoro-Shapiro trajectory. In the gravity-pole-subtracted setup the nonlinear constraints reduce the allowed region to a small island containing the Virasoro-Shapiro point, for which an analytic bootstrap explanation is provided.

What carries the argument

Virasoro-inspired ansatz that supplies nonlinear relations among Wilson coefficients, used together with dispersion relations from analyticity, crossing symmetry and Regge boundedness.

If this is right

  • Dispersion relations with the graviton pole explicit produce numerical bounds on the leading low-energy coefficients.
  • The Virasoro-inspired ansatz generates nonlinear relations that shrink the allowed region toward the Virasoro-Shapiro trajectory.
  • In the gravity-pole-subtracted setup the nonlinear constraints reduce the space to a small island containing the Virasoro-Shapiro point.
  • An analytic bootstrap explanation accounts for the location of the Virasoro-Shapiro point inside that island.

Where Pith is reading between the lines

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

  • The isolation to a small island indicates that the combination of dispersion relations and the ansatz may fix higher Wilson coefficients to unique values under the stated assumptions.
  • The subtracted setup could be adapted to other tree-level string amplitudes that share the same Regge behavior and crossing properties.
  • Adding further physical inputs such as higher-spin partial-wave bounds might shrink the island further or eliminate it entirely.

Load-bearing premise

The Virasoro-inspired ansatz supplies a set of nonlinear relations among Wilson coefficients that are valid and independent of the target Virasoro-Shapiro trajectory itself.

What would settle it

A high-precision numerical bootstrap computation showing that the known low-energy expansion coefficients of the Virasoro-Shapiro amplitude lie outside the small island obtained after imposing the nonlinear constraints would falsify the central claim.

read the original abstract

We study the closed-string tree-level Virasoro-Shapiro amplitude using the dispersive S-matrix bootstrap. For the ten-dimensional maximally supersymmetric four-point amplitude, we impose analyticity, crossing symmetry, partial-wave unitarity, and Regge boundedness. With the massless graviton pole kept explicitly, the resulting dispersion relations and crossing null constraints give numerical bounds on the leading low-energy coefficients normalized by the gravitational coupling. We then introduce a Virasoro-inspired ansatz, which becomes a set of nonlinear relations among Wilson coefficients and shrinks the allowed region toward the Virasoro-Shapiro trajectory. Finally, we study a gravity-pole-subtracted setup, where the regular part of the amplitude has a well-defined forward limit. In this stripped problem, the nonlinear constraints reduce the allowed region to a small island containing the Virasoro-Shapiro point, for which we provide an analytic bootstrap explanation.

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 applies dispersive bootstrap methods to the tree-level Virasoro-Shapiro amplitude in ten-dimensional maximal supergravity. It imposes analyticity, crossing symmetry, partial-wave unitarity and Regge boundedness on the four-point amplitude, derives dispersion relations and crossing null constraints that bound leading low-energy Wilson coefficients (normalized to the gravitational coupling), introduces a Virasoro-inspired ansatz that generates nonlinear relations among those coefficients, and shows that in a gravity-pole-subtracted formulation the allowed region collapses to a small island containing the Virasoro-Shapiro point, which is then given an analytic bootstrap explanation.

Significance. If the nonlinear relations generated by the ansatz are shown to be independent of the target Virasoro-Shapiro trajectory, the result would constitute a non-trivial bootstrap derivation of the closed-string amplitude from S-matrix axioms alone. This would be a significant advance in the program of deriving string theory from consistency conditions, especially given the explicit numerical island and the analytic explanation provided for the subtracted setup.

major comments (2)
  1. [Section introducing the Virasoro-inspired ansatz (likely §4 or §5)] The central claim in the gravity-pole-subtracted setup (that the nonlinear constraints reduce the allowed region to an island containing the Virasoro-Shapiro point) rests on the independence of the relations generated by the Virasoro-inspired ansatz. The abstract states that the ansatz 'becomes a set of nonlinear relations' that shrink the region 'toward the Virasoro-Shapiro trajectory'; if the ansatz is constructed by importing the specific Regge trajectory, pole residues or low-energy expansion of the known VS amplitude rather than from crossing, analyticity or unitarity alone, the reduction follows by construction. An explicit demonstration that the relations remain valid when the ansatz is varied away from the VS form is required.
  2. [Section providing the analytic bootstrap explanation (likely §6)] The analytic bootstrap explanation for the island in the subtracted setup must be checked for hidden dependence on the same input used to motivate the ansatz. If the explanation relies on the same Regge or residue assumptions that define the ansatz, it does not constitute an independent verification of the numerical result.
minor comments (2)
  1. [Setup of the subtracted problem] Clarify the precise definition of the gravity-pole-subtracted amplitude and the forward-limit regularization used; the current description leaves ambiguous whether the subtracted amplitude remains crossing-symmetric in the same way as the original.
  2. [Numerical bounds section] The numerical bounds in the unsubtracted case should be presented with explicit comparison to the VS values of the Wilson coefficients so that the shrinkage induced by the ansatz can be quantified.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need to clarify the independence of the Virasoro-inspired ansatz and the analytic explanation. We address each major comment below and will incorporate clarifications and explicit checks in a revised manuscript.

read point-by-point responses

  1. Referee: [Section introducing the Virasoro-inspired ansatz (likely §4 or §5)] The central claim in the gravity-pole-subtracted setup (that the nonlinear constraints reduce the allowed region to an island containing the Virasoro-Shapiro point) rests on the independence of the relations generated by the Virasoro-inspired ansatz. The abstract states that the ansatz 'becomes a set of nonlinear relations' that shrink the region 'toward the Virasoro-Shapiro trajectory'; if the ansatz is constructed by importing the specific Regge trajectory, pole residues or low-energy expansion of the known VS amplitude rather than from crossing, analyticity or unitarity alone, the reduction follows by construction. An explicit demonstration that the relations remain valid when the ansatz is varied away from the VS form is required.

    Authors: The ansatz encodes the general Regge trajectory and pole structure expected for a string amplitude but is substituted into dispersion relations and crossing equations that are derived exclusively from analyticity, crossing symmetry, partial-wave unitarity and Regge boundedness. The resulting nonlinear relations among Wilson coefficients are therefore consequences of these axioms once the functional form is imposed. To demonstrate that the reduction is not by construction, we will add to the revised manuscript an explicit numerical check in which the trajectory slope and residue parameters are varied by O(10%) away from their Virasoro-Shapiro values; the nonlinear constraints continue to produce a significantly smaller allowed region (though the island may enlarge or shift), confirming that the shrinkage follows from consistency with the bootstrap equations rather than from exact matching to the target amplitude. revision: yes

  2. Referee: [Section providing the analytic bootstrap explanation (likely §6)] The analytic bootstrap explanation for the island in the subtracted setup must be checked for hidden dependence on the same input used to motivate the ansatz. If the explanation relies on the same Regge or residue assumptions that define the ansatz, it does not constitute an independent verification of the numerical result.

    Authors: The analytic explanation derives from the subtracted dispersion relations, the existence of a well-defined forward limit, and the structure of the crossing null constraints; it does not invoke the specific functional form or parameter values of the ansatz. We will revise the relevant section to list the assumptions explicitly and to separate them from the ansatz motivation, thereby confirming that the analytic argument stands on the same S-matrix axioms used for the numerical bounds. revision: yes

Circularity Check

1 steps flagged

Virasoro-inspired ansatz renders nonlinear constraints dependent on target VS trajectory by construction

specific steps
  1. other [Abstract]
    "We then introduce a Virasoro-inspired ansatz, which becomes a set of nonlinear relations among Wilson coefficients and shrinks the allowed region toward the Virasoro-Shapiro trajectory. Finally, we study a gravity-pole-subtracted setup, where the regular part of the amplitude has a well-defined forward limit. In this stripped problem, the nonlinear constraints reduce the allowed region to a small island containing the Virasoro-Shapiro point, for which we provide an analytic bootstrap explanation."

    The ansatz is defined as 'Virasoro-inspired' and is stated to 'become' the nonlinear relations that then force the allowed region onto the VS trajectory. This makes the island result a direct consequence of importing the target amplitude's structure into the ansatz rather than an independent derivation from the imposed axioms alone.

full rationale

The central bootstrap result in the gravity-pole-subtracted setup relies on nonlinear constraints that originate from a Virasoro-inspired ansatz explicitly introduced to shrink the allowed region toward the Virasoro-Shapiro point. Because the ansatz is chosen to encode features of the target amplitude (Regge trajectory, pole structure), the resulting relations among Wilson coefficients are not independent external constraints; the reduction to a small island containing the VS point follows tautologically once the ansatz is imposed. This is the load-bearing step identified in the abstract, with no evidence in the provided text that the ansatz derives solely from crossing, analyticity, or unitarity without reference to the VS form.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The ledger is constructed from the abstract alone. The primary inputs are standard S-matrix axioms; the Virasoro-inspired ansatz functions as an additional modeling choice whose independence is not demonstrated in the given text.

axioms (1)
  • domain assumption The ten-dimensional maximally supersymmetric four-point amplitude satisfies analyticity, crossing symmetry, partial-wave unitarity, and Regge boundedness.
    These properties are imposed to generate the dispersion relations and crossing null constraints described in the abstract.

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discussion (0)

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Reference graph

Works this paper leans on

79 extracted references · 15 linked inside Pith

  1. [1]

    Adams, N

    A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis and R. Rattazzi,Causality, analyticity and an IR obstruction to UV completion,JHEP10(2006) 014 [hep-th/0602178]

    Pith/arXiv arXiv 2006

  2. [2]

    Tolley, Z.-Y

    A.J. Tolley, Z.-Y. Wang and S.-Y. Zhou,New positivity bounds from full crossing symmetry,JHEP05 (2021) 255 [2011.02400]

    arXiv 2021

  3. [3]

    Caron-Huot and V

    S. Caron-Huot and V. Van Duong,Extremal Effective Field Theories,JHEP05(2021) 280 [2011.02957]

    arXiv 2021

  4. [4]

    Paulos, J

    M.F. Paulos, J. Penedones, J. Toledo, B.C. van Rees and P. Vieira,The S-matrix bootstrap. Part III: higher dimensional amplitudes,JHEP12(2019) 040 [1708.06765]

    Pith/arXiv arXiv 2019

  5. [5]

    Sinha and A

    A. Sinha and A. Zahed,Crossing Symmetric Dispersion Relations in Quantum Field Theories,Phys. Rev. Lett.126(2021) 181601 [2012.04877]

    arXiv 2021

  6. [6]

    Guerrieri and A

    A. Guerrieri and A. Sever,Rigorous Bounds on the Analytic S Matrix,Phys. Rev. Lett.127(2021) 251601 [2106.10257]

    arXiv 2021

  7. [7]

    de Rham, A.J

    C. de Rham, A.J. Tolley, Z.-H. Wang and S.-Y. Zhou,Primal S-matrix bootstrap with dispersion relations,JHEP01(2026) 027 [2506.22546]

    Pith/arXiv arXiv 2026

  8. [8]

    Pham and T.N

    T.N. Pham and T.N. Truong,Evaluation of the Derivative Quartic Terms of the Meson Chiral Lagrangian From Forward Dispersion Relation,Phys. Rev. D31(1985) 3027

    1985

  9. [9]

    Pennington and J

    M.R. Pennington and J. Portoles,The Chiral Lagrangian parameters, l1, l2, are determined by the rho resonance,Phys. Lett. B344(1995) 399 [hep-ph/9409426]

    Pith/arXiv arXiv 1995

  10. [10]

    Nicolis, R

    A. Nicolis, R. Rattazzi and E. Trincherini,Energy’s and amplitudes’ positivity,JHEP05(2010) 095 [0912.4258]

    Pith/arXiv arXiv 2010

  11. [11]

    Komargodski and A

    Z. Komargodski and A. Schwimmer,On Renormalization Group Flows in Four Dimensions,JHEP12 (2011) 099 [1107.3987]. – 21 –

    Pith/arXiv arXiv 2011

  12. [12]

    Remmen and N.L

    G.N. Remmen and N.L. Rodd,Consistency of the Standard Model Effective Field Theory,JHEP12 (2019) 032 [1908.09845]

    arXiv 2019

  13. [13]

    Bellazzini, M

    B. Bellazzini, M. Lewandowski and J. Serra,Positivity of Amplitudes, Weak Gravity Conjecture, and Modified Gravity,Phys. Rev. Lett.123(2019) 251103 [1902.03250]

    arXiv 2019

  14. [14]

    Herrero-Valea, I

    M. Herrero-Valea, I. Timiryasov and A. Tokareva,To Positivity and Beyond, where Higgs-Dilaton Inflation has never gone before,JCAP11(2019) 042 [1905.08816]

    arXiv 2019

  15. [15]

    Bellazzini, J

    B. Bellazzini, J. Elias Mir´ o, R. Rattazzi, M. Riembau and F. Riva,Positive moments for scattering amplitudes,Phys. Rev. D104(2021) 036006 [2011.00037]

    arXiv 2021

  16. [16]

    Bellazzini, F

    B. Bellazzini, F. Riva, J. Serra and F. Sgarlata,Beyond Positivity Bounds and the Fate of Massive Gravity,Phys. Rev. Lett.120(2018) 161101 [1710.02539]

    Pith/arXiv arXiv 2018

  17. [17]

    Alberte, C

    L. Alberte, C. de Rham, S. Jaitly and A.J. Tolley,Positivity Bounds and the Massless Spin-2 Pole, Phys. Rev. D102(2020) 125023 [2007.12667]

    arXiv 2020

  18. [18]

    de Rham, S

    C. de Rham, S. Melville, A.J. Tolley and S.-Y. Zhou,Massive Galileon Positivity Bounds,JHEP09 (2017) 072 [1702.08577]

    Pith/arXiv arXiv 2017

  19. [19]

    Wang, F.-K

    Y.-J. Wang, F.-K. Guo, C. Zhang and S.-Y. Zhou,Generalized positivity bounds on chiral perturbation theory,JHEP07(2020) 214 [2004.03992]

    arXiv 2020

  20. [20]

    Alberte, C

    L. Alberte, C. de Rham, S. Jaitly and A.J. Tolley,QED positivity bounds,Phys. Rev. D103(2021) 125020 [2012.05798]

    arXiv 2021

  21. [21]

    Tokuda, K

    J. Tokuda, K. Aoki and S. Hirano,Gravitational positivity bounds,JHEP11(2020) 054 [2007.15009]

    arXiv 2020

  22. [22]

    X. Li, H. Xu, C. Yang, C. Zhang and S.-Y. Zhou,Positivity in Multifield Effective Field Theories,Phys. Rev. Lett.127(2021) 121601 [2101.01191]

    arXiv 2021

  23. [23]

    Caron-Huot, D

    S. Caron-Huot, D. Mazac, L. Rastelli and D. Simmons-Duffin,Sharp boundaries for the swampland, JHEP07(2021) 110 [2102.08951]

    arXiv 2021

  24. [24]

    Z.-Z. Du, C. Zhang and S.-Y. Zhou,Triple crossing positivity bounds for multi-field theories,JHEP12 (2021) 115 [2111.01169]

    arXiv 2021

  25. [25]

    Z. Bern, D. Kosmopoulos and A. Zhiboedov,Gravitational effective field theory islands, low-spin dominance, and the four-graviton amplitude,J. Phys. A54(2021) 344002 [2103.12728]

    arXiv 2021

  26. [26]

    X. Li, K. Mimasu, K. Yamashita, C. Yang, C. Zhang and S.-Y. Zhou,Moments for positivity: using Drell-Yan data to test positivity bounds and reverse-engineer new physics,JHEP10(2022) 107 [2204.13121]

    arXiv 2022

  27. [27]

    Caron-Huot, Y.-Z

    S. Caron-Huot, Y.-Z. Li, J. Parra-Martinez and D. Simmons-Duffin,Causality constraints on corrections to Einstein gravity,JHEP05(2023) 122 [2201.06602]

    arXiv 2023

  28. [28]

    Saraswat,Weak gravity conjecture and effective field theory,Phys

    P. Saraswat,Weak gravity conjecture and effective field theory,Phys. Rev. D95(2017) 025013 [1608.06951]

    Pith/arXiv arXiv 2017

  29. [29]

    Arkani-Hamed, Y.-t

    N. Arkani-Hamed, Y.-t. Huang, J.-Y. Liu and G.N. Remmen,Causality, unitarity, and the weak gravity conjecture,JHEP03(2022) 083 [2109.13937]

    arXiv 2022

  30. [30]

    Herrero-Valea, R

    M. Herrero-Valea, R. Santos-Garcia and A. Tokareva,Massless positivity in graviton exchange,Phys. Rev. D104(2021) 085022 [2011.11652]

    arXiv 2021

  31. [31]

    Guerrieri, J

    A. Guerrieri, J. Penedones and P. Vieira,Where Is String Theory in the Space of Scattering Amplitudes?,Phys. Rev. Lett.127(2021) 081601 [2102.02847]

    arXiv 2021

  32. [32]

    Henriksson, B

    J. Henriksson, B. McPeak, F. Russo and A. Vichi,Rigorous bounds on light-by-light scattering,JHEP 06(2022) 158 [2107.13009]. – 22 –

    arXiv 2022

  33. [33]

    Elias Miro, A

    J. Elias Miro, A. Guerrieri and M.A. Gumus,Bridging positivity and S-matrix bootstrap bounds,JHEP 05(2023) 001 [2210.01502]

    arXiv 2023

  34. [34]

    Bellazzini, M

    B. Bellazzini, M. Riembau and F. Riva,IR side of positivity bounds,Phys. Rev. D106(2022) 105008 [2112.12561]

    arXiv 2022

  35. [35]

    Herrero-Valea, A.S

    M. Herrero-Valea, A.S. Koshelev and A. Tokareva,UV graviton scattering and positivity bounds from IR dispersion relations,Phys. Rev. D106(2022) 105002 [2205.13332]

    arXiv 2022

  36. [36]

    Hong, Z.-H

    D.-Y. Hong, Z.-H. Wang and S.-Y. Zhou,Causality bounds on scalar-tensor EFTs,JHEP10(2023) 135 [2304.01259]

    arXiv 2023

  37. [37]

    Chiang, Y.-t

    L.-Y. Chiang, Y.-t. Huang, W. Li, L. Rodina and H.-C. Weng,(Non)-projective bounds on gravitational EFT,2201.07177

    arXiv

  38. [38]

    Huang, J.-Y

    Y.-t. Huang, J.-Y. Liu, L. Rodina and Y. Wang,Carving out the Space of Open-String S-matrix,JHEP 04(2021) 195 [2008.02293]

    arXiv 2021

  39. [39]

    Noumi and J

    T. Noumi and J. Tokuda,Gravitational positivity bounds on scalar potentials,Phys. Rev. D104(2021) 066022 [2105.01436]

    arXiv 2021

  40. [40]

    Xu and S.-Y

    H. Xu and S.-Y. Zhou,Triple crossing positivity bounds, mass dependence and cosmological scalars: Horndeski theory and DHOST,JCAP11(2023) 076 [2306.06639]

    arXiv 2023

  41. [41]

    Q. Chen, K. Mimasu, T.A. Wu, G.-D. Zhang and S.-Y. Zhou,Capping the positivity cone: dimension-8 Higgs operators in the SMEFT,JHEP03(2024) 180 [2309.15922]

    arXiv 2024

  42. [42]

    Noumi and J

    T. Noumi and J. Tokuda,Finite energy sum rules for gravitational Regge amplitudes,JHEP06(2023) 032 [2212.08001]

    arXiv 2023

  43. [43]

    de Rham, S

    C. de Rham, S. Kundu, M. Reece, A.J. Tolley and S.-Y. Zhou,Snowmass White Paper: UV Constraints on IR Physics, inSnowmass 2021, 3, 2022 [2203.06805]

    arXiv 2021

  44. [44]

    Hong, Z.-H

    D.-Y. Hong, Z.-H. Wang and S.-Y. Zhou,On Capped Higgs Positivity Cone,2404.04479

    arXiv

  45. [45]

    Z. Bern, E. Herrmann, D. Kosmopoulos and R. Roiban,Effective Field Theory islands from perturbative and nonperturbative four-graviton amplitudes,JHEP01(2023) 113 [2205.01655]

    arXiv 2023

  46. [46]

    T. Ma, A. Pomarol and F. Sciotti,Bootstrapping the chiral anomaly at large N c,JHEP11(2023) 176 [2307.04729]

    arXiv 2023

  47. [47]

    De Angelis and G

    S. De Angelis and G. Durieux,EFT matching from analyticity and unitarity,SciPost Phys.16(2024) 071 [2308.00035]

    arXiv 2024

  48. [48]

    Acanfora, A

    F. Acanfora, A. Guerrieri, K. H¨ aring and D. Karateev,Bounds on scattering of neutral Goldstones, JHEP03(2024) 028 [2310.06027]

    arXiv 2024

  49. [49]

    K. Aoki, T. Noumi, R. Saito, S. Sato, S. Shirai, J. Tokuda et al.,Gravitational positivity for phenomenologists: Dark gauge boson in the swampland,Phys. Rev. D110(2024) 016002 [2305.10058]

    arXiv 2024

  50. [50]

    Xu, D.-Y

    H. Xu, D.-Y. Hong, Z.-H. Wang and S.-Y. Zhou,Positivity Bounds on parity-violating scalar-tensor EFTs,2410.09794

    arXiv

  51. [51]

    Elias Miro, A.L

    J. Elias Miro, A.L. Guerrieri and M.A. Gumus,Extremal Higgs couplings,Phys. Rev. D110(2024) 016007 [2311.09283]

    arXiv 2024

  52. [52]

    McPeak, M

    B. McPeak, M. Venuti and A. Vichi,Adding subtractions: comparing the impact of different Regge behaviors,2310.06888

    arXiv

  53. [53]

    Riembau,Full Unitarity and the Moments of Scattering Amplitudes,2212.14056

    M. Riembau,Full Unitarity and the Moments of Scattering Amplitudes,2212.14056. – 23 –

    arXiv

  54. [54]

    Caron-Huot and J

    S. Caron-Huot and J. Tokuda,String loops and gravitational positivity bounds: imprint of light particles at high energies,JHEP11(2024) 055 [2406.07606]

    arXiv 2024

  55. [55]

    Caron-Huot and Y.-Z

    S. Caron-Huot and Y.-Z. Li,Gravity and a universal cutoff for field theory,JHEP02(2025) 115 [2408.06440]

    arXiv 2025

  56. [56]

    Wan and S.-Y

    S.-L. Wan and S.-Y. Zhou,Matrix moment approach to positivity bounds and UV reconstruction from IR,2411.11964

    arXiv

  57. [57]

    Berman and N

    J. Berman and N. Geiser,Analytic bootstrap bounds on masses and spins in gravitational and non-gravitational scalar theories,2412.17902

    arXiv

  58. [58]

    Beadle, G

    C. Beadle, G. Isabella, D. Perrone, S. Ricossa, F. Riva and F. Serra,Non-forward UV/IR relations, JHEP08(2025) 188 [2407.02346]

    arXiv 2025

  59. [59]

    Bellazzini, A

    B. Bellazzini, A. Pomarol, M. Romano and F. Sciotti,(Super) gravity from positivity,JHEP03(2026) 028 [2507.12535]

    arXiv 2026

  60. [60]

    Y. Ye, X. Cao, Y.-H. Wu and J. Gu,Positivity bounds in scalar-QED EFT at one-loop level, 2507.06302

    arXiv

  61. [61]

    Bonnefoy, V

    Q. Bonnefoy, V. Cort´ es, E. Gendy, C. Grojean, K.R. von Merkl and P.N. Pilatus,Geometry of effective field theory positivity cones,2508.18165

    arXiv

  62. [62]

    Bellazzini, J

    B. Bellazzini, J. Berman, G. Isabella, F. Riva, M. Romano and F. Sciotti,Positivity with Long-Range Interactions,2512.13780

    Pith/arXiv arXiv

  63. [63]

    Gumus, S

    M.A. Gumus, S. Metayer and P. Tourkine,Tracking S-matrix bounds across dimensions,2512.24474

    arXiv

  64. [64]

    Fernandez, M

    J. Fernandez, M. Ruhdorfer and J. Serra,Negative running of gravitational positivity,2603.15755

    Pith/arXiv arXiv

  65. [65]

    Veneziano,Construction of a crossing - symmetric, Regge behaved amplitude for linearly rising trajectories,Nuovo Cim

    G. Veneziano,Construction of a crossing - symmetric, Regge behaved amplitude for linearly rising trajectories,Nuovo Cim. A57(1968) 190

    1968

  66. [66]

    Berman and H

    J. Berman and H. Elvang,Corners and islands in the S-matrix bootstrap of the open superstring,JHEP 09(2024) 076 [2406.03543]

    arXiv 2024

  67. [67]

    Berman, H

    J. Berman, H. Elvang and C. Figueiredo,Splitting regions and shrinking islands from higher point constraints,JHEP10(2025) 226 [2506.22538]

    arXiv 2025

  68. [68]

    Wan and S.-Y

    S.-L. Wan and S.-Y. Zhou,Analytic Bootstrap of the Veneziano Amplitude,2605.11084

    Pith/arXiv arXiv

  69. [69]

    Geiser and L.W

    N. Geiser and L.W. Lindwasser,Properties of infinite product amplitudes: Veneziano, virasoro, and coon,JHEP12(2022) 112 [2207.08855]

    arXiv 2022

  70. [70]

    Cheung and G.N

    C. Cheung and G.N. Remmen,Veneziano variations: how unique are string amplitudes?,JHEP01 (2023) 122 [2210.12163]

    arXiv 2023

  71. [71]

    Cheung and G.N

    C. Cheung and G.N. Remmen,Stringy dynamics from an amplitudes bootstrap,Phys. Rev. D108 (2023) 026011 [2302.12263]

    arXiv 2023

  72. [72]

    Cheung, A

    C. Cheung, A. Hillman and G.N. Remmen,Bootstrap Principle for the Spectrum and Scattering of Strings,Phys. Rev. Lett.133(2024) 251601 [2406.02665]

    arXiv 2024

  73. [73]

    G. Peng, L. Rodina, A. Tokareva and Y. Xu,Sampling the Graviton Pole and Deprojecting the Swampland,2604.15235

    Pith/arXiv arXiv

  74. [74]

    Albert, W

    J. Albert, W. Knop and L. Rastelli,Where is tree-level string theory?,JHEP02(2025) 157 [2406.12959]

    arXiv 2025

  75. [75]

    Cheung, A

    C. Cheung, A. Hillman and G.N. Remmen,Uniqueness criteria for the Virasoro-Shapiro amplitude, Phys. Rev. D111(2025) 086034 [2408.03362]. – 24 –

    arXiv 2025

  76. [76]

    Cheung, G.N

    C. Cheung, G.N. Remmen, F. Sciotti and M. Tarquini,Strings from Almost Nothing,2508.09246

    Pith/arXiv arXiv

  77. [77]

    Elvang, A

    H. Elvang, A. Herderschee and R. Morales,String theory from maximal supersymmetry,2601.11705

    Pith/arXiv arXiv

  78. [78]

    Kawai, D.C

    H. Kawai, D.C. Lewellen and S.H.H. Tye,A Relation Between Tree Amplitudes of Closed and Open Strings,Nucl. Phys. B269(1986) 1

    1986

  79. [79]

    Alday, R.S

    L.F. Alday, R.S. Pitombo and A. Str¨ omholm Sangar´ e,Monodromy relations for string amplitudes on AdS space,Phys. Rev. D113(2026) L021903 [2509.02719]. – 25 –

    arXiv 2026