Pith. sign in

REVIEW 3 major objections 4 minor 2 cited by

Gravitational Effective Theories with Maximal Supersymmetry and a Peculiar Parity

T0 review · 3 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Maximal N=8 supersymmetry, a peculiar parity on (6,6) scalars, and positivity force the 4-point amplitude onto the Virasoro–Shapiro amplitude — once finitely many states sit at the lowest mass level.

desk verdict Conditional but genuinely new: nonlinear constraints from SUSY/factorization/peculiar parity select Virasoro–Shapiro if the all-orders exponentiation holds; worth serious refereeing. read the letter →

arxiv 2607.14230 v1 pith:LYPCXQFX submitted 2026-07-15 hep-th

Justin Berman , Simon Caron-Huot , Aditi V. Chandra , Henriette Elvang , Aidan Herderschee , Loki L. Lin , Roger Morales This is my paper · ORCID
classification hep-th PACS 04.65.+e11.25.-w11.55.-m
keywords N=8supergravitypeculiarparityVirasoro–ShapiroamplitudeS-matrixbootstrappositivityboundsWilsoncoefficientsscatteringamplitudesstringtheoryuniqueness
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks how much of string theory's rigidity can be derived from first principles rather than assumed, and answers: a great deal. For four-dimensional N=8 supergravity deformed by arbitrary higher-derivative corrections — keeping only maximal supersymmetry, SU(4)×SU(4) R-symmetry, tree-level factorization, and a 'peculiar parity' — the parity condition on a subset of six-scalar amplitudes generates nonlinear relations among the 4-point Wilson coefficients (κ²g₃ = ½g₀², κ²g₅ = g₂g₀, and so on). Combined with positivity, these relations confine the allowed Wilson coefficients to a non-convex region whose two sharp corners are the closed-superstring Virasoro–Shapiro amplitude and an infinite spin tower exchanging every spin at a single mass. Requiring only finitely many spins at the lowest mass level removes the tower, and the paper proves the surviving amplitude must have the linear spectrum mₙ² = n m₁² — exactly the Virasoro–Shapiro amplitude. If right, the closed string is not an input but an output of symmetry and consistency.

What carries the argument

The load-bearing object is peculiar parity: parity invariance — no Levi–Civita contractions of the momenta — imposed only on amplitudes of (6,6) scalars, e.g. the six-scalar amplitude Z_{1,1}. It is not definable on all states and loop effects necessarily violate it; the paper treats it as an approximate tree-level property. The machine is the 6-point NMHV superamplitude, built from a single S₂×S₄-symmetric function S₁ that factorizes into products of 4-point amplitudes; peculiar parity produces nonlinear relations among the 4-point Wilson coefficients, which resum into an exponential form. Meromorphy converts that exponential into a product over discrete masses, and residue positivity selec

What would settle it

Compute higher-order terms of the six-scalar amplitude Z_{1,1} under peculiar parity: the exponential form (6.9) — the bridge from the nonlinear constraints to the uniqueness theorem — is verified only through O(s¹³). If any coefficient at O(s¹⁴) or beyond fails to fit the exponential, or any new algebraic relation among the surviving g₂ₖ emerges, the analytic bootstrap collapses and the finite-spin uniqueness proof for Virasoro–Shapiro needs a new foundation. Complementary test: the theorem predicts the lowest-level residue of any allowed product-form amplitude must be a polynomial in t; any

Watch

Extended reading notes

Core claim

Central discovery: a parity property that cannot be defined on all states — 'peculiar parity', imposed only on amplitudes of (6,6) scalars — turns six-point factorization into a nonlinear constraint machine. Demanding Z_{1,1} = Z̄_{1,1} (no Levi–Civita contractions) fixes all 4-point Wilson coefficients except the forward-limit coefficients g₂ₖ, with relations like κ²g₃ = g₀²/2. These relations resum the 4-point amplitude into an explicit exponential form (verified to O(s¹³)), forcing a meromorphic product over discrete masses. Positivity of the residues then forces a linear spectrum mₙ² = n m₁² once only finitely many spins appear at the lowest mass — identically the Virasoro–Shapiro amplit

Load-bearing premise

Peculiar parity must be a property a genuine UV completion can possess: the argument goes through only if real theories can have tree-level (6,6)-scalar amplitudes free of Levi–Civita contractions, even though the paper itself shows the parity cannot be defined on all states and is unavoidably broken by fermion loops.

Editorial extensions

If this is right

  • With unbroken SU(8) R-symmetry the same machinery is destructive: peculiar parity (or vanishing single-soft scalar limits) plus positivity forces every higher-derivative correction to vanish, leaving pure N=8 supergravity as the only tree-level EFT.
  • The nonlinear constraints hold at every corner of the allowed region: they are satisfied by the Virasoro–Shapiro expansion (whose free g₂ₖ coefficients match the first occurrence of each odd zeta value), by the Infinite Spin Tower, and by the interpolating family M⁽ᴺ⁾₄ whose zeta values are replaced by generalized harmonic numbers.
  • Positivity combined with the non-convex constraints bifurcates the allowed (g₂/g₀, g₃/g₀) region: any amplitude on the line between the string and the tower, other than the string itself, must exchange infinitely many high spins at the mass gap — so a finite-spin spectrum at the gap is exactly what selects the string.
  • Peculiar parity automatically implies vanishing single-soft scalar limits for the (6,6) scalars (checked to O(s¹³)), but the converse fails: soft limits alone do not produce the nonlinear constraints.
  • The reconstructed 4-point amplitude agrees with the explicit closed-string amplitude at 5 points to O(s⁹) and at 6 points to O(s⁸), and the analysis explains why no algebraic constraint can go further: the first algebraically independent multi-zeta value, ζ₃,₃,₅, enters at O(s¹²).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The decisive open question the paper itself flags is whether the Infinite Spin Tower amplitude is a real theory or a mathematical model: if no complete unitary embedding exists, the finite-spin requirement is a physical fact rather than an extra assumption, and the uniqueness claim becomes unconditional; if one exists, the uniqueness is genuinely conditional on excluding towers.
  • The exponential resummation reconstructs the entire 4-point function from its forward limit, suggesting a template: in any SUSY theory with a discrete symmetry definable on only a bosonic subsector, higher-point factorization may pin lower-point data. A direct testable extension is the other scalar sectors — for (4,4) scalars and the axio-dilaton the same parity currently forces g₀ = 0 and hence n
  • The proof rests on two acknowledged soft spots: the exponential form verified only to O(s¹³), and the adoption of real analyticity in −s < t < 0 as an axiom of massless scattering, which the paper notes lacks a rigorous axiomatic treatment. Both are local: a counterexample to either would bound the theorem's reach even if the physics conclusion survives.
  • If an analogue of peculiar parity exists in large-N holographic correlators, the same logic could constrain AdS/CFT data away from the supergravity limit — the paper's own suggested direction, and one testable by existing bootstrap technology.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies weakly-coupled four-dimensional N=8 supergravity EFTs with SU(4)xSU(4) R-symmetry. It constructs the 6-point NMHV superamplitude bottom-up, imposing maximal SUSY, tree-level factorization, and a 'peculiar parity' condition on a subset of six-scalar amplitudes. From this it derives nonlinear constraints among the 4-point Wilson coefficients, Eq. (5.3), verified through O(s^13). These constraints are then combined with positivity/dispersion relations and a mass-gap assumption to argue numerically and analytically that the Virasoro-Shapiro amplitude is the unique UV completion with finitely many spins at the lowest mass level. The analytic proof in Section 6.4 relies on an all-orders exponentiated form Eq. (6.9), checked only up to O(s^13), and on a real-analyticity axiom in the physical region.

Significance. If the main theorem holds, this is a significant result: it would provide a bottom-up, non-string-theoretic derivation of the Virasoro-Shapiro amplitude from maximal SUSY, factorization, a discrete parity-like condition, and positivity, closely paralleling the N=4 SYM result. The paper contains substantial original technical work: a detailed 6-point ansatz with explicit spurious-pole checks, a systematic O(4)-character counting of S2 x S4-symmetric polynomials, and explicit tests against the Virasoro-Shapiro amplitude, the infinite spin tower, and generalized tower amplitudes. The authors are also commendably transparent about the conditional nature of the all-orders step and about the fact that peculiar parity cannot be defined on all states. The numerical bootstrap is coherent, and the identification of two corners (Virasoro-Shapiro and IST) is well supported by the displayed constraints. The main caveat is that the advertised uniqueness is conditional on an unproven all-orders exponentiation and on extra analyticity assumptions.

major comments (3)
  1. [6.4, Eq. (6.9)] The central uniqueness theorem is built on an assumed all-orders exponentiated form. The text states that the nonlinear constraints are verified only up to O(s^13), and then says 'upon assuming that this exponentiated form holds to all orders, we prove...'. If at O(s^14) or beyond a new independent Wilson coefficient appears that is not determined by the forward-limit coefficients g_{2k}, Eqs. (6.11)-(6.13) and the finite-spin argument do not follow. This is a load-bearing gap. Please either prove (6.9) directly from the nonlinear constraints (5.3), or explicitly present the all-orders exponentiation as an assumption and adjust the abstract/conclusion claims accordingly. As written, the claim that the Virasoro-Shapiro amplitude is 'unique' is conditional.
  2. [2.2 / 5.1, Eq. (5.2)] Peculiar parity is a selective tree-level condition. The paper is explicit that it cannot be defined on all states and is necessarily broken by fermion loops. Since the nonlinear constraints (5.3) and everything that follows are derived from this condition, the physical scope of the uniqueness claim is limited to theories satisfying peculiar parity. This is not a circularity, but it is a correctness risk: a UV completion with the same SUSY, factorization, and positivity but without peculiar parity lies outside the analysis. A concrete test would be to construct or identify a tree-level completion satisfying all other axioms but violating (5.2); such an example would show where the assumption bites. At minimum, the paper should more sharply distinguish the conditional theorem from an unconditional statement about all N=8 SUGRA completions.
  3. [6.4, after Eq. (6.12)] The step from the exponential representation to the meromorphic product (6.13) relies on an additional real-analyticity assumption in the region -s < t < 0. The authors acknowledge this is not rigorously established and state they 'accept' it as a basic axiom. Because this assumption is essential for excluding continuous spectral densities and for the finite-spin uniqueness argument, it should be either derived from more standard principles or explicitly listed among the axioms of the theorem. The current wording is transparent, but it means the advertised uniqueness is not unconditional within the stated framework.
minor comments (4)
  1. [5.3.2, Eq. (5.15)] The substitution ζ_j -> 1 is formal; since odd zeta values are believed algebraically independent, it is not a well-defined algebraic map on the amplitude. The text already says this, but a sentence clarifying that this is only a heuristic connection to the IST would help avoid confusion.
  2. [Table 1 and Table 3] The counting of independent S2 x S4-symmetric polynomials is verified explicitly only up to order 8, but the tables extend to order 13. Please state how the higher-order counts are obtained or verified.
  3. [Eq. (5.2)] The notation Z1,1 = Z1,1, with bars on both the amplitude and the indices, is confusing when first introduced. A short verbal explanation of which conjugation acts on which SU(4) factor would improve readability.
  4. [Appendix E] The comparison to closed string amplitudes is reported only up to O(s^8) at six points, while the nonlinear constraints are checked to O(s^13). A brief comment on why the string comparison stops earlier, e.g. multi-zeta values, would be useful.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the nonlinear constraints are derived from an independent peculiar-parity input, and the uniqueness argument is conditional but not circular.

full rationale

I find no circular step. The central derivation is bottom-up: the 6-point NMHV superamplitude is constructed from SUSY Ward identities, tree-level factorization, and a systematic local-contact-term ansatz (Section 4), and the peculiar parity condition Z_{1,1} = Z_{1,1} (Eq. 5.2) is an extra input imposed on that amplitude. The resulting nonlinear constraints (5.3) are derived, not fitted, and the Virasoro–Shapiro amplitude and IST amplitude are then tested examples that satisfy these constraints, not inputs used to produce them. The bootstrap then combines positivity with the nonlinear constraints, and Section 6.4 explicitly states that the exponentiated form (6.9) has been verified only up to O(s^13) and is assumed to all orders. That is a transparent conditional assumption and an internal proof gap, but it is not circular: the exponential form still contains the free coefficients g_{2k} and does not itself encode the Virasoro–Shapiro spectrum; the meromorphy argument, product form (6.13), and finite-spin uniqueness argument are a subsequent logical chain that selects VS. Self-citations such as [11], [8], [22], and [49] provide context, motivation, and numerical bootstrap tools, but the load-bearing N=8 nonlinear constraints and uniqueness argument are re-derived in this paper rather than imported as conclusions from those references. No equation or parameter is renamed as a prediction, and no uniqueness theorem is invoked from the authors' prior work to forbid alternatives. The paper even flags its own limitation on the all-orders exponentiation, which supports the assessment that the derivation is genuinely conditional rather than circular.

Assumptions & free parameters 4 free parameters · 7 assumptions · 0 invented entities

The analysis introduces no new particles, forces, or dimensions. The Infinite Spin Tower and M^{(N)} amplitudes are constructed/bootstrap examples rather than physical inventions. The central claim rests on a combination of standard bootstrap assumptions, one deliberately peculiar symmetry input, and two acknowledged technical gaps: all-orders exponentiation and real analyticity in the unphysical region.

free parameters (4)
  • g_{2k} (even Wilson coefficients)
    The nonlinear relations (5.3) leave the even-order coefficients g0, g2, g4, ... undetermined; they parameterize the exponential ansatz (6.9) and the allowed bootstrap region.
  • Mass-gap scale Λ^2 (or m_1^2)
    The scale of the first massive state enters the dispersion relations and the finite-spin cutoff in Eq. (6.7); the bootstrap constrains ratios of Wilson coefficients but not the absolute mass scale.
  • Coupling |λ0|^2 to the lightest scalar
    Introduced in Eq. (6.7) to model finitely many states at s=Λ^2; the bifurcation analysis allows this coupling to vary and is central to excluding the IST.
  • Higher-order 6-point contact-term coefficients
    Table 3 shows free local coefficients in the S1 ansatz at orders s8 and beyond; they are not fixed by current constraints but do not affect the 4-point central claim.
assumptions (7)
  • domain assumption N=8 supersymmetry with SU(4)×SU(4) R-symmetry
    Defines the theory class and the supermultiplet decomposition used throughout (Section 2.1).
  • domain assumption Weakly-coupled tree-level amplitudes with only tree factorization and no massless loops
    Assumed in the introduction and Section 3; restricts analysis to classical S-matrix of the EFT.
  • ad hoc to paper Peculiar parity: tree-level invariance of (6,6) scalar amplitudes, e.g. Z_{1,1}=conjugate
    Key extra assumption, imposed only on a subset of scalar amplitudes and known to be loop-anomalous (Sections 2.2, 5.1). It is the input that produces nonlinear constraints.
  • domain assumption Real analyticity in the physical region −s<t<0 for M4(zz̄z̄z)
    Used in Section 6.4 to derive meromorphy and the product form; the authors state no rigorous axiomatic treatment exists and adopt it as a basic axiom.
  • ad hoc to paper All-orders exponentiated form (6.9) of the amplitude
    Verified only up to O(s^13); the analytic uniqueness proof assumes this resummation to all orders.
  • domain assumption Dispersion-relation assumptions: unitarity, analyticity, polynomial boundedness (6.2), and a cutoff Λ
    Standard S-matrix bootstrap assumptions in Section 6.1; used for the positivity bounds and spectral representations.
  • ad hoc to paper Finite number of states at the lowest mass level
    Added in Sections 6.3–6.4 to exclude the IST; this is the decisive condition that forces the linear spectrum and selects Virasoro–Shapiro.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Gravitational Effective Theories with Maximal Supersymmetry and a Peculiar Parity." pith.science (2026). https://pith.science/paper/LYPCXQFX

@misc{pith2026260714230,
  author       = {Pith},
  title        = {Pith review of: Gravitational Effective Theories with Maximal Supersymmetry and a Peculiar Parity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LYPCXQFX}},
  note         = {Machine review of arXiv:2607.14230}
}
abstract

We study the space of four-dimensional ultraviolet completions for $\mathcal{N}=8$ supergravity that are described at low energies by weakly-coupled effective field theories (EFTs) with maximal supersymmetry and $\mathrm{SU}(4)\times\mathrm{SU}(4)$ R-symmetry. We show that tree-level factorization of the 4-, 5-, and 6-point EFT scattering amplitudes, together with a certain ``peculiar parity'' condition, leads to nonlinear constraints on the 4-point Wilson coefficients. This peculiar parity is a property that can only be imposed on a subset of scalar amplitudes. Combining the nonlinear constraints with positivity, we find that the allowed region of 4-point Wilson coefficients is reduced to a non-convex domain with two sharp corners: one being the closed superstring Virasoro--Shapiro amplitude, the other an infinite spin tower amplitude exchanging states of every spin at the same mass. We show both numerically and analytically that requiring a finite number of states near the first mass level leaves only the Virasoro--Shapiro amplitude.

Figures

Figures reproduced from arXiv: 2607.14230 by the authors.

Figure 1
Figure 1. Allowed region in the (g2/g0, g3/g0) plane. Blue corresponds to the region without nonlinear constraints imposed, yellow includes the first nonlinear constraint in (6.6), while the green region includes the first and second nonlinear constraints. In red is the scaling curve for the Virasoro–Shapiro amplitude (5.8), while in black is that of the IST amplitude (5.12). The purple line corresponds to the coefficients of… view at source ↗
Figure 2
Figure 2. Left: Full allowed region in the (g2/g0, g3/g0) plane. Blue corresponds to the region without nonlinear constraints imposed, yellow includes the first nonlinear constraint in (6.6) with a gap µc = 1.1, and green includes both the first and second nonlinear constraints with the same gap µc = 1.1. The orange region has µc = 1.1 and includes the first, second, and fourth constraints in (6.6), ignoring the third constra… view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

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

  1. Analytic Boundaries of Infinite-Spin-Tower Amplitudes from Hidden Zero

    hep-th 2026-07 conditional novelty 7.0 of 10

    The exact analytic boundary of unitary infinite-spin-tower amplitudes is derived and shown to be maximal unless energy poles accumulate.

  2. Unitary Dual-Resonance S-matrices

    hep-th 2026-07 conditional novelty 7.0 of 10

    Smearing Veneziano blocks over continuous Regge slopes yields local, analytic, crossing-symmetric S-matrices with full partial-wave unitarity and controllable second-sheet resonances.

Reference graph

Works this paper leans on

89 extracted references · 73 linked inside Pith · cited by 2 Pith papers

  1. [11]

    Elvang, A

    H. Elvang, A. Herderschee and R. Morales,String Theory from Maximal Supersymmetry,JHEP 07(2026) 105 [2601.11705]

    arXiv 2026

  2. [1]

    Huang, J.-Y

    Y.-t. Huang, J.-Y. Liu, L. Rodina and Y. Wang,Carving out the Space of Open-String S-matrix,JHEP04(2021) 195 [2008.02293]

    arXiv 2021

  3. [2]

    Chiang, Y.-t

    L.-Y. Chiang, Y.-t. Huang and H.-C. Weng,Bootstrapping string theory EFT,JHEP05(2024) 289 [2310.10710]

    arXiv 2024

  4. [3]

    Arkani-Hamed, C

    N. Arkani-Hamed, C. Cheung, C. Figueiredo and G.N. Remmen,Multiparticle Factorization and the Rigidity of String Theory,Phys. Rev. Lett.132(2024) 091601 [2312.07652]

    arXiv 2024

  5. [4]

    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

  6. [5]

    Berman and H

    J. Berman and H. Elvang,Corners and islands in the S-matrix bootstrap of the open superstring,JHEP09(2024) 076 [2406.03543]

    arXiv 2024

  7. [6]

    Cheung, A

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

    arXiv 2025

  8. [7]

    F. Bhat, D. Chowdhury, A.P. Saha and A. Sinha,Bootstrapping string models with entanglement minimization and machine learning,Phys. Rev. D111(2025) 066013 [2409.18259]

    arXiv 2025

Show all 89 references
  1. [8]

    Berman, H

    J. Berman, H. Elvang and C. Figueiredo,Splitting regions and shrinking islands from higher point constraints,JHEP10(2025) 226 [2506.22538]. – 45 –

    2025 arXiv

  2. [9]

    Cheung, G.N

    C. Cheung, G.N. Remmen, F. Sciotti and M. Tarquini,Strings from Almost Nothing,Phys. Rev. Lett.136(2026) 251601 [2508.09246]

    2026 arXiv

  3. [10]

    Eckner, F

    C. Eckner, F. Figueroa, S. Metayer and P. Tourkine,Regge trajectories for UV completions of graviton scattering from polynomial boundedness,2512.17828

  4. [12]

    Basile, G.N

    I. Basile, G.N. Remmen and G. Staudt,Higher-spin and higher-point constraints on stringy amplitudes,Phys. Rev. D114(2026) 026001 [2603.04485]

    2026 arXiv

  5. [13]

    Wan and S.-Y

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

    arXiv

  6. [14]

    Xu,A Dispersive Bootstrap for the Virasoro-Shapiro Amplitude,2606.19283

    Y. Xu,A Dispersive Bootstrap for the Virasoro-Shapiro Amplitude,2606.19283

    arXiv

  7. [15]

    Cremmer and B

    E. Cremmer and B. Julia,The N=8 Supergravity Theory. 1. The Lagrangian,Phys. Lett. B80 (1978) 48

    1978

  8. [16]

    Cremmer and B

    E. Cremmer and B. Julia,The SO(8) Supergravity,Nucl. Phys. B159(1979) 141

    1979

  9. [17]

    Plahte,Symmetry properties of dual tree-graph n-point amplitudes,Nuovo Cim

    E. Plahte,Symmetry properties of dual tree-graph n-point amplitudes,Nuovo Cim. A66(1970) 713

    1970

  10. [18]

    Stieberger,Open & Closed vs

    S. Stieberger,Open & Closed vs. Pure Open String Disk Amplitudes,0907.2211

    arXiv

  11. [19]

    Bjerrum-Bohr, P.H

    N.E.J. Bjerrum-Bohr, P.H. Damgaard and P. Vanhove,Minimal Basis for Gauge Theory Amplitudes,Phys. Rev. Lett.103(2009) 161602 [0907.1425]

    2009 arXiv

  12. [20]

    Bjerrum-Bohr, P.H

    N.E.J. Bjerrum-Bohr, P.H. Damgaard, T. Sondergaard and P. Vanhove,Monodromy and Jacobi-like Relations for Color-Ordered Amplitudes,JHEP06(2010) 003 [1003.2403]

    2010 arXiv

  13. [21]

    Bjerrum-Bohr, P.H

    N.E.J. Bjerrum-Bohr, P.H. Damgaard, T. Sondergaard and P. Vanhove,The Momentum Kernel of Gauge and Gravity Theories,JHEP01(2011) 001 [1010.3933]

    2011 arXiv

  14. [22]

    Berman, H

    J. Berman, H. Elvang and A. Herderschee,Flattening of the EFT-hedron: supersymmetric positivity bounds and the search for string theory,JHEP03(2024) 021 [2310.10729]

    2024 arXiv

  15. [23]

    Grisaru, H.N

    M.T. Grisaru, H.N. Pendleton and P. van Nieuwenhuizen,Supergravity and the S Matrix,Phys. Rev. D15(1977) 996

    1977

  16. [24]

    Grisaru and H.N

    M.T. Grisaru and H.N. Pendleton,Some Properties of Scattering Amplitudes in Supersymmetric Theories,Nucl. Phys. B124(1977) 81

    1977

  17. [25]

    Bianchi, H

    M. Bianchi, H. Elvang and D.Z. Freedman,Generating Tree Amplitudes in N=4 SYM and N = 8 SG,JHEP09(2008) 063 [0805.0757]

    2008 arXiv

  18. [26]

    Elvang and Y.-t

    H. Elvang and Y.-t. Huang,Scattering Amplitudes,1308.1697

    arXiv

  19. [27]

    Virasoro,Alternative constructions of crossing-symmetric amplitudes with regge behavior, Phys

    M.A. Virasoro,Alternative constructions of crossing-symmetric amplitudes with regge behavior, Phys. Rev.177(1969) 2309

    1969

  20. [28]

    Shapiro,Electrostatic analog for the virasoro model,Phys

    J.A. Shapiro,Electrostatic analog for the virasoro model,Phys. Lett. B33(1970) 361

    1970

  21. [29]

    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]. – 46 –

    2006 arXiv

  22. [30]

    Caron-Huot, Z

    S. Caron-Huot, Z. Komargodski, A. Sever and A. Zhiboedov,Strings from Massive Higher Spins: The Asymptotic Uniqueness of the Veneziano Amplitude,JHEP10(2017) 026 [1607.04253]

    2017 arXiv

  23. [31]

    Caron-Huot and V

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

    2021 arXiv

  24. [32]

    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]

    2021 arXiv

  25. [33]

    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]

    2021 arXiv

  26. [34]

    Arkani-Hamed, T.-C

    N. Arkani-Hamed, T.-C. Huang and Y.-t. Huang,The EFT-Hedron,JHEP05(2021) 259 [2012.15849]

    2021 arXiv

  27. [35]

    Albert and L

    J. Albert and L. Rastelli,Bootstrapping pions at large N,JHEP08(2022) 151 [2203.11950]

    2022 arXiv

  28. [36]

    Hull and P.K

    C.M. Hull and P.K. Townsend,Unity of superstring dualities,Nucl. Phys. B438(1995) 109 [hep-th/9410167]

    1995 arXiv

  29. [37]

    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

  30. [38]

    Friedan, E.J

    D. Friedan, E.J. Martinec and S.H. Shenker,Conformal invariance, supersymmetry and string theory,Nucl. Phys. B271(1986) 93

    1986

  31. [39]

    Witten,Notes On Super Riemann Surfaces And Their Moduli,Pure Appl

    E. Witten,Notes On Super Riemann Surfaces And Their Moduli,Pure Appl. Math. Quart.15 (2019) 57 [1209.2459]

    2019 arXiv

  32. [40]

    Seiberg and S.-H

    N. Seiberg and S.-H. Shao,Majorana chain and Ising model - (non-invertible) translations, anomalies, and emanant symmetries,SciPost Phys.16(2024) 064 [2307.02534]

    2024 arXiv

  33. [41]

    Elvang and M

    H. Elvang and M. Kiermaier,Stringy KLT relations, global symmetries, andE 7(7) violation, JHEP10(2010) 108 [1007.4813]

    2010 arXiv

  34. [42]

    Brink, J.H

    L. Brink, J.H. Schwarz and J. Scherk,Supersymmetric Yang-Mills Theories,Nucl. Phys. B121 (1977) 77

    1977

  35. [43]

    Elvang, D.Z

    H. Elvang, D.Z. Freedman and M. Kiermaier,Recursion Relations, Generating Functions, and Unitarity Sums in N=4 SYM Theory,JHEP04(2009) 009 [0808.1720]

    2009 arXiv

  36. [44]

    Elvang and Y.-t

    H. Elvang and Y.-t. Huang,Scattering amplitudes in gauge theory and gravity, Cambridge University Press (2015)

    2015

  37. [45]

    grassmann.m: A package that teaches Mathematica how to manipulate Grassmann variables

    M. Headrick, “grassmann.m: A package that teaches Mathematica how to manipulate Grassmann variables.”https://sites.google.com/view/matthew-[]headrick/mathematica

  38. [46]

    Elvang, D.Z

    H. Elvang, D.Z. Freedman and M. Kiermaier,Solution to the Ward Identities for Superamplitudes,JHEP10(2010) 103 [0911.3169]

    2010 arXiv

  39. [47]

    Elvang, D.Z

    H. Elvang, D.Z. Freedman and M. Kiermaier,SUSY Ward identities, Superamplitudes, and Counterterms,J. Phys. A44(2011) 454009 [1012.3401]

    2011 arXiv

  40. [48]

    Albert, W

    J. Albert, W. Knop and L. Rastelli,Where is tree-level string theory?,JHEP02(2025) 157 [2406.12959]. – 47 –

    2025 arXiv

  41. [49]

    Berman, H

    J. Berman, H. Elvang, N. Geiser and L.L. Lin,Bootstrapping extremal scalar amplitudes with and without supersymmetry,JHEP05(2026) 149 [2412.13368]

    2026 arXiv

  42. [50]

    H¨ aring and A

    K. H¨ aring and A. Zhiboedov,The stringy S-matrix bootstrap: maximal spin and superpolynomial softness,JHEP10(2024) 075 [2311.13631]

    2024 arXiv

  43. [51]

    Britto, F

    R. Britto, F. Cachazo and B. Feng,New recursion relations for tree amplitudes of gluons,Nucl. Phys. B715(2005) 499 [hep-th/0412308]

    2005 arXiv

  44. [52]

    Britto, F

    R. Britto, F. Cachazo, B. Feng and E. Witten,Direct proof of tree-level recursion relation in Yang-Mills theory,Phys. Rev. Lett.94(2005) 181602 [hep-th/0501052]

    2005 arXiv

  45. [53]

    Di Vecchia, R

    P. Di Vecchia, R. Marotta and M. Mojaza,Soft theorem for the graviton, dilaton and the Kalb-Ramond field in the bosonic string,JHEP05(2015) 137 [1502.05258]

    2015 arXiv

  46. [54]

    Di Vecchia, R

    P. Di Vecchia, R. Marotta and M. Mojaza,The B-field soft theorem and its unification with the graviton and dilaton,JHEP10(2017) 017 [1706.02961]

    2017 arXiv

  47. [55]

    Schlotterer and S

    O. Schlotterer and S. Stieberger,Motivic Multiple Zeta Values and Superstring Amplitudes,J. Phys. A46(2013) 475401 [1205.1516]

    2013 arXiv

  48. [56]

    Calisto and C

    F. Calisto and C. Cheung,An Ultraviolet Finite Theory of Scalars,2606.26450

    arXiv

  49. [57]

    Simmons-Duffin,A Semidefinite Program Solver for the Conformal Bootstrap,JHEP06 (2015) 174 [1502.02033]

    D. Simmons-Duffin,A Semidefinite Program Solver for the Conformal Bootstrap,JHEP06 (2015) 174 [1502.02033]

    2015 arXiv

  50. [58]

    Caron-Huot, D

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

    2021 arXiv

  51. [59]

    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]

    2023 arXiv

  52. [60]

    H¨ aring and A

    K. H¨ aring and A. Zhiboedov,Gravitational Regge bounds,SciPost Phys.16(2024) 034 [2202.08280]

    2024 arXiv

  53. [61]

    Bellazzini, J

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

    arXiv

  54. [62]

    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]

    2024 arXiv

  55. [63]

    Beadle, G

    C. Beadle, G. Isabella, D. Perrone, S. Ricossa, F. Riva and F. Serra,The EFT bootstrap at finite M P L,JHEP06(2025) 209 [2501.18465]

    2025 arXiv

  56. [64]

    Chang and J

    C.-H. Chang and J. Parra-Martinez,Graviton loops and negativity,JHEP08(2025) 175 [2501.17949]

    2025 arXiv

  57. [65]

    Pasiecznik,Bootstrapping Gravity with Crossing Symmetric Dispersion Relations, 2506.09884

    C. Pasiecznik,Bootstrapping Gravity with Crossing Symmetric Dispersion Relations, 2506.09884

    arXiv

  58. [66]

    Arkani-Hamed, Q

    N. Arkani-Hamed, Q. Cao, J. Dong, C. Figueiredo and S. He,Hidden zeros for particle/string amplitudes and the unity of colored scalars, pions and gluons,JHEP10(2024) 231 [2312.16282]

    2024 arXiv

  59. [67]

    Berman and N

    J. Berman and N. Geiser,Analytic bootstrap bounds on masses and spins in gravitational and non-gravitational scalar theories,JHEP06(2025) 168 [2412.17902]. – 48 –

    2025 arXiv

  60. [68]

    Correia, A

    M. Correia, A. Sever and A. Zhiboedov,An analytical toolkit for the S-matrix bootstrap,JHEP 03(2021) 013 [2006.08221]

    2021 arXiv

  61. [69]

    Hagiwara, R.D

    K. Hagiwara, R.D. Peccei, D. Zeppenfeld and K. Hikasa,Probing the Weak Boson Sector in e+ e- —>W+ W-,Nucl. Phys. B282(1987) 253

    1987

  62. [70]

    Gaiotto, A

    D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett,Generalized Global Symmetries,JHEP02 (2015) 172 [1412.5148]

    2015 arXiv

  63. [71]

    Chang, Y.-H

    C.-M. Chang, Y.-H. Lin, S.-H. Shao, Y. Wang and X. Yin,Topological Defect Lines and Renormalization Group Flows in Two Dimensions,JHEP01(2019) 026 [1802.04445]

    2019 arXiv

  64. [72]

    Maldacena,The Large N limit of superconformal field theories and supergravity,Adv

    J.M. Maldacena,The Large N limit of superconformal field theories and supergravity,Adv. Theor. Math. Phys.2(1998) 231 [hep-th/9711200]

    1998 arXiv

  65. [73]

    Gubser, I.R

    S.S. Gubser, I.R. Klebanov and A.M. Polyakov,Gauge theory correlators from noncritical string theory,Phys. Lett. B428(1998) 105 [hep-th/9802109]

    1998 arXiv

  66. [74]

    Witten,Anti-de Sitter space and holography,Adv

    E. Witten,Anti-de Sitter space and holography,Adv. Theor. Math. Phys.2(1998) 253 [hep-th/9802150]

    1998 arXiv

  67. [75]

    Belitsky, S

    A.V. Belitsky, S. Hohenegger, G.P. Korchemsky and E. Sokatchev,N=4 superconformal Ward identities for correlation functions,Nucl. Phys. B904(2016) 176 [1409.2502]

    2016 arXiv

  68. [76]

    Chowdhury, A

    S.D. Chowdhury, A. Gadde, T. Gopalka, I. Halder, L. Janagal and S. Minwalla,Classifying and constraining local four photon and four graviton S-matrices,JHEP02(2020) 114 [1910.14392]

    2020 arXiv

  69. [77]

    Dolan,Counting BPS operators in N=4 SYM,Nucl

    F.A. Dolan,Counting BPS operators in N=4 SYM,Nucl. Phys. B790(2008) 432 [0704.1038]

    2008 arXiv

  70. [78]

    J. Gray, A. Hanany, Y.-H. He, V. Jejjala and N. Mekareeya,SQCD: A Geometric Apercu, JHEP05(2008) 099 [0803.4257]

    2008 arXiv

  71. [79]

    Lehman and A

    L. Lehman and A. Martin,Hilbert Series for Constructing Lagrangians: expanding the phenomenologist’s toolbox,Phys. Rev. D91(2015) 105014 [1503.07537]

    2015 arXiv

  72. [80]

    Henning, X

    B. Henning, X. Lu, T. Melia and H. Murayama,Hilbert series and operator bases with derivatives in effective field theories,Commun. Math. Phys.347(2016) 363 [1507.07240]

    2016 arXiv

  73. [81]

    Fulton and J

    W. Fulton and J. Harris,Representation theory. A first course, vol. 129 ofGrad. Texts Math., New York etc.: Springer-Verlag (1991)

    1991

  74. [82]

    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

  75. [83]

    Mafra, O

    C.R. Mafra, O. Schlotterer and S. Stieberger,Complete N-Point Superstring Disk Amplitude I. Pure Spinor Computation,Nucl. Phys. B873(2013) 419 [1106.2645]

    2013 arXiv

  76. [84]

    Mafra, O

    C.R. Mafra, O. Schlotterer and S. Stieberger,Complete N-Point Superstring Disk Amplitude II. Amplitude and Hypergeometric Function Structure,Nucl. Phys. B873(2013) 461 [1106.2646]

    2013 arXiv

  77. [85]

    Stieberger and T.R

    S. Stieberger and T.R. Taylor,Amplitude for N-Gluon Superstring Scattering,Phys. Rev. Lett. 97(2006) 211601 [hep-th/0607184]

    2006 arXiv

  78. [86]

    Stieberger and T.R

    S. Stieberger and T.R. Taylor,Multi-Gluon Scattering in Open Superstring Theory,Phys. Rev. D74(2006) 126007 [hep-th/0609175]

    2006 arXiv

  79. [87]

    Broedel and L.J

    J. Broedel and L.J. Dixon,R**4 counterterm and E(7)(7) symmetry in maximal supergravity, JHEP05(2010) 003 [0911.5704]. – 49 –

    2010 arXiv

  80. [88]

    Broedel, O

    J. Broedel, O. Schlotterer, S. Stieberger and T. Terasoma,All orderα ′-expansion of superstring trees from the Drinfeld associator,Phys. Rev. D89(2014) 066014 [1304.7304]. [89]https://wwwth.mpp.mpg.de/members/stieberg/mzv/index.html

    2014 arXiv

  81. [90]

    Mizera,Inverse of the String Theory KLT Kernel,JHEP06(2017) 084 [1610.04230]

    S. Mizera,Inverse of the String Theory KLT Kernel,JHEP06(2017) 084 [1610.04230]. – 50 –

    2017 arXiv

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.