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A Hidden Permutation Symmetry of Squared Amplitudes in ABJM Theory

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper claims that all planar ABJM squared-amplitude integrands with a fixed N = n + L are lightlike limits of a single S_N-symmetric generating function F_N, which can be bootstrapped graphically to yield new loop-level results.

desk verdict A genuinely new organizing structure for ABJM squared amplitudes, with real small-N checks, but the N=10 bootstrap rests on conjectural rules whose two fitted constants lose meaning once the overall normalization is dropped. read the letter →

arxiv 2508.03813 v1 pith:CD7THJSP submitted 2025-08-05 hep-th

Song He , Canxin Shi , Yichao Tang , Yao-Qi Zhang This is my paper
classification hep-th
keywords ABJMtheorysquaredamplitudesgeneratingfunctionpermutationsymmetryf-graphsbipartitegraphsgraphicalbootstrapdualconformalinvariance
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 claims that in planar ABJM theory, all n-point L-loop squared-amplitude integrands with a fixed value of N = n + L are not separate objects: they are lightlike limits of a single generating function F_N that is symmetric under permuting its N points. Working graphically, F_N can be expanded in weight-3 f-graphs, and in three dimensions it admits a second expansion in bipartite f-graphs that makes the absence of odd-multiplicity amplitudes manifest. The paper verifies this for every case with N = 4, 6, 8, then conjectures two graphical rules connecting F_N to F_{N-2}, uses them to uniquely determine F_10, and extracts new squared tree amplitudes and loop integrands from it. A sympathetic reader should care because, if correct, this gives a unified combinatorial description of ABJM squared amplitudes and points to a 'bipartite correlator' behind the theory, closely paralleling the amplitude/correlator duality of N = 4 super-Yang-Mills.

What carries the argument

The central object is the generating function F_N, expanded in weight-3 f-graphs: graphs whose vertices are dual points and whose solid or dashed edges denote denominator or numerator factors 1/$x_ij^{2}$ or $x_ij^{2}$, with every vertex of weight 3. A named identity doing the heavy lifting is the three-dimensional Gram identity, which equates certain linear combinations of planar f-graphs with combinations of bipartite f-graphs, so F_N lives in the intersection B∩P; the bipartite graphs contain only even cycles and make the absence of odd-particle amplitudes manifest. The argument is carried by two conjectured graphical rules: the quadrangle rule, shrinking a solid-line quadrangle in F_N to an edge in F_{N-2}, and the double-quadrangle rule, pinching a double-quadrangle to a cusp, each with one overall constant fitted to N = 6, 8 data. These rules, together with the small 120-element bipartite basis at N = 10, uniquely determine F_10.

What would settle it

Compute the 2-loop 8-point squared amplitude M_8^(2) independently, for instance from the known two-loop eight-point ABJM amplitude, and compare it with the lightlike limit of the bootstrapped F_10; a mismatch would falsify the graphical rules. Similarly, the four-loop six-point squared integrand extracted from F_10 could be tested on generalized unitarity cuts.

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Extended reading notes

Core claim

The central claim is that there exists a single object F_N, a function of N dual-space points, that packages every ABJM squared integrand M_n^(L) with N = n + L through the n-gon lightlike limit. F_N is invariant under the full symmetric group S_N, so loops and external legs are unified; after stripping the lightlike prefactor it expands into weight-3 f-graphs whose solid lines are denominators and dashed lines numerators. The paper's key structural observation is that F_N also lies in the intersection of the planar and bipartite f-graph vector spaces, and the bipartite representation makes the vanishing of all odd-n squared amplitudes immediate. Checking M_4^(2), M_6^(0), M_4^(4), M_6^(2), and M_8^(0) confirms F_6 and F_8. The authors then conjecture a quadrangle rule and a double-quadrangle rule, motivated by SYM's triangle and double-triangle rules, with two overall constants fixed against F_6 and F_8. These rules uniquely fix the bipartite coefficients of F_10 up to Gram identities, and the paper verifies the resulting 10-point tree squared amplitude against the Grassmannian formula and the extracted 6-loop 4-point amplitude against unitarity cuts.

Load-bearing premise

The whole N = 10 bootstrap rests on the conjecture that the quadrangle and double-quadrangle graphical rules, adapted from N = 4 super-Yang-Mills by analogy and with their overall constants fitted to just two low-order cases, actually hold in ABJM; if either rule is wrong or incomplete, the unique F_10 result would fail, even though its tree limit and one extracted amplitude pass independent checks.

Editorial extensions

If this is right

  • All squared amplitudes with a fixed N = n + L are determined by the single function F_N, so additional loop orders for a given n are not independent calculations.
  • The bipartite f-graph expansion makes the absence of odd-multiplicity ABJM amplitudes manifest through the absence of odd-length cycles.
  • The B∩P bootstrap reduces the ansatz dramatically, from 61 to 2 coefficients at N = 8, so higher-N generating functions can be fixed by symmetry plus a small set of rules.
  • The N = 10 result yields new explicit data: 10-point tree squared amplitudes, 2-loop 8-point and 4-loop 6-point squared integrands, and a 6-loop 4-point amplitude.
  • The consistency of F_10 supports the existence of a 'bipartite correlator' in ABJM theory that unifies all squared amplitudes, extending the amplitude/correlator duality story to three dimensions.

Reading between the lines

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

  • If F_N exists for all even N, the whole tower of planar ABJM squared amplitudes could be bootstrapped to arbitrarily high orders from finitely many graph coefficients, mirroring the SYM program; one testable sign would be whether the number of free coefficients in B∩P stays small at N = 12.
  • The hidden S_N symmetry suggests a geometric object, a three-dimensional squared amplituhedron or correlahedron, whose canonical form would be F_N, which could tie the f-graph expansion to the known ABJM amplituhedron.
  • Because the quantized Chern-Simons level blocks the standard Lagrangian-insertion mechanism, the physical correlator behind F_N may live in mass-deformed N = 2 ABJM theory, a prediction that could be tested by deriving F_N from such a correlator.
  • The double-quadrangle rule may encode a double-soft theorem of squared amplitudes; checking the rule against the known soft theorem would turn the conjecture into a derivation.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper defines squared-amplitude integrands M_n^(L) for planar ABJM theory via charge conjugation and proposes that they are packaged by a single S_N-symmetric generating function F_N with N=n+L through n-gon lightlike limits, Eq. (5). The construction is checked explicitly for N=4,6,8 using known amplitudes and is represented both in planar and bipartite f-graph bases connected by three-dimensional Gram identities. The authors then conjecture two "putative graphical rules"—the quadrangle rule, Eq. (27), and the double-quadrangle rule, Eq. (29)—calibrate their overall constants against F_6 and F_8, and use them to bootstrap a unique F_10. From F_10 they extract new results for the 10-point tree squared amplitude and integrands for (n,L)=(4,6) and (6,4), checking the tree limit against the Grassmannian formula and the extracted A_4^(6) against unitarity cuts. The paper concludes by conjecturing a "bipartite correlator" in ABJM that would unify all squared amplitudes.

Significance. If correct, the construction is significant: it would unify loop and external-leg data at fixed N, expose a hidden S_N symmetry in ABJM integrands, and provide a concrete bridge toward an amplitude/correlator duality in ABJM. The paper's strengths include explicit checks of F_4, F_6, and F_8 from existing amplitude data, the dramatic reduction of the N=8 ansatz from 61 planar f-graphs to a two-parameter bipartite space, the independent numerical check of the N=10 tree limit against the Grassmannian formula, and unitarity-cut checks on the extracted six-loop four-point amplitude. The main caveat is that the N=10 bootstrap rests on unproven graphical rules whose coefficients are fitted to lower-N data, and the most direct independent check, the M_8^(2) lightlike limit of F_10, is explicitly deferred. The N≤8 core is well supported, but the new N=10 predictions are, as presented, consequences of a conjecture.

major comments (3)
  1. [N=10: putative graphical rules, Eqs. (27),(29)] The N=10 bootstrap rests entirely on the quadrangle and double-quadrangle rules, which are introduced as "putative" graphical rules and whose constants α=1 and β=1/2 are fixed by matching the known F_6 and F_8 data (Eqs. (17),(24) and Fig. 1). No derivation from ABJM dynamics is given, and the most direct independent test—the lightlike limit of F_10 to the known two-loop eight-point integrand M_8^(2) of ref. [50]—is explicitly left to future work. The numerical check of M_10^(0) against the Grassmannian formula and the unitarity-cut check of A_4^(6) verify only two of the three N=10 components; the (6,4) and (8,2) outputs remain unverified. The N=10 claims should either be backed by the M_8^(2) check or clearly separated from the verified N≤8 results in the abstract and conclusion.
  2. [Squared amplitudes in ABJM, Eq. (5) and footnote 55] Equation (5) is stated as an exact equality between the lightlike limit of F_N and M_n^(L), but footnote 55 says the overall normalization of M_n^(L) is ignored and that the authors "believe there exists a good convention such that (5) holds exactly." The packaging property is therefore demonstrated only up to an unspecified normalization. This matters for the bootstrap: the values α=1 and β=1/2 in Eqs. (27),(29) are fixed using particular normalized versions of F_6 and F_8, and the paper does not show that these values are convention-independent. The normalization convention (or a proof that α and β are normalization-independent) should be stated before F_10 is presented as the unique output of the rules.
  3. [N=8: bipartite vs. planar; N=10: putative graphical rules] The bootstrap assumes F_N∈B∩P at N=10, but the bipartite/planar property is verified only for N=6 and N=8. Because the N=10 ansatz space is defined as B∩P, the uniqueness of the resulting F_10 cannot by construction provide evidence that F_10 lies in B∩P; that structural extension is an input, not an output. A successful check of the M_8^(2) lightlike limit would simultaneously test both the numerical coefficients of F_10 and the assumed B∩P property, and until such a check is performed the conclusion that "our results strongly suggest the existence of a bipartite correlator" should be phrased as conditional on the graphical rules.
minor comments (5)
  1. [Abstract] The abstract uses "square amplitudes" while the body and title use "squared amplitudes"; please harmonize the terminology.
  2. [Introduction] The first paragraph contains the typo "phenomenogically"; it should be "phenomenologically".
  3. [Fig. 1 caption] The caption uses coefficients such as c_2/8 and b_1/4 without defining c_i and b_j; please define them explicitly or refer to Eqs. (27) and (29) for the notation.
  4. [Supplemental Material, numerical computation of squared tree amplitudes] The authors note that the signs of J± have not been well documented in the literature, but the actual sign prescription used in the numerical M_10^(0) check is not given. Since the claimed tree-level verification depends on this choice, the sign convention should be specified precisely.
  5. [N=10: putative graphical rules] The paper references an ancillary file for F_10, but the ancillary data are not visible in the version under review; please ensure the file is included with the submission so the claimed uniqueness and the numerical checks are reproducible.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the N=10 bootstrap is calibrated on lower-N data but targets new coefficients and is checked against independent tree-level and unitarity data.

full rationale

The central derivation is not circular. F_N is introduced in eq. (5) as a generating object whose lightlike limits give squared amplitudes; F_6 and F_8 are built from and verified against independently computed integrands (refs. [33,48,49,52]), so the unification claim is a genuine cross-check rather than a restatement. The N=10 bootstrap uses the conjectural quadrangle and double-quadrangle rules (27) and (29), with alpha=1 and beta=1/2 fixed by matching the known F_6 and F_8 data. This is calibration to lower-N inputs, not circularity: the F_10 coefficients are not among the fitted data, and the paper reports independent checks (M_10^(0) against the Grassmannian formula, and unitarity cuts for A_4^(6)), while explicitly deferring the M_8^(2) check to future work. The manuscript itself flags the conjectural status of the rules and the ignored overall normalization (footnote 55, conclusion), which are limitations on certainty rather than evidence that the output is equivalent to the input. Self-citations ([26], [52], [64]) supply prior published results or methods; the load-bearing calculations are not reduced to a self-citation chain. Accordingly, no concrete circular step can be exhibited, and the appropriate score is low.

Assumptions & free parameters 3 free parameters · 4 assumptions · 2 invented entities

The central claim rests on the existence and symmetry of F_N, which is verified only for small N, and on the two conjectured graphical rules with fitted constants. The 'bipartite correlator' is an invented physical object without independent evidence, so it carries the highest burden.

free parameters (3)
  • alpha = 1
    Overall factor in the quadrangle rule (Eq. 27); chosen so that comparing Fig. 1(a) with the known F8 and F6 results gives the correct relative coefficient in F8.
  • beta = 1/2
    Overall factor in the double-quadrangle rule (Eq. 29); chosen consistent with beta = alpha/2 by matching the known F8 and F6 results.
  • Overall normalization of M_n^(L) = not specified
    Footnote [55] states the overall normalization of M_n^(L) is ignored and a convention is assumed to make Eq. (5) exact; the scale is not fixed in the paper.
assumptions (4)
  • domain assumption F_N exists and has an expansion in weight-3 planar f-graphs for all even N
    Checked for N=4,6,8 and assumed for N=10 in the bootstrap; underlies the entire f-graph method. Introduced in the section 'Squared amplitudes in ABJM' and used throughout.
  • ad hoc to paper The quadrangle and double-quadrangle rules hold for F_N
    Putative rules motivated by analogy with SYM, not derived from ABJM dynamics; used to fix F8 and to bootstrap F10. See section 'N=10: putative graphical rules'.
  • standard math The 3D conformal Gram determinant vanishes, det[x_ij^2] = 0 for six points
    Used to relate planar and bipartite f-graph representations of F6 (Eq. 18); cited to [56].
  • domain assumption The Grassmannian formula for ABJM tree amplitudes and the associated Jacobian computation are correct
    Used to numerically verify M_10^(0) in the Supplemental Material; the sign convention for J± is newly determined by the authors.
invented entities (2)
  • bipartite correlator
    purpose: Conjectured physical observable in ABJM whose lightlike limits yield all squared amplitudes, analog to the four-point half-BPS correlator in SYM
    No concrete definition or independent observable is identified; the evidence is the properties of F_N. The paper itself says it 'strongly suggests the existence' but leaves identification to future work.
  • generating function F_N independent evidence
    purpose: Mathematical object packaging all n-point L-loop squared integrands with N = n + L
    F_N is concretely defined by its lightlike limits to M_n^(L) and is checked against explicit small-N data, giving it a falsifiable handle.

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Pith. "Pith review of A Hidden Permutation Symmetry of Squared Amplitudes in ABJM Theory." pith.science (2026). https://pith.science/paper/CD7THJSP

@misc{pith2026250803813,
  author       = {Pith},
  title        = {Pith review of: A Hidden Permutation Symmetry of Squared Amplitudes in ABJM Theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CD7THJSP}},
  note         = {Machine review of arXiv:2508.03813}
}
abstract

We define the square amplitudes in planar Aharony-Bergman-Jafferis-Maldacena theory (ABJM), analogous to that in $\mathcal{N}{=}4$ super-Yang-Mills theory (SYM). Surprisingly, the $n$-point $L$-loop integrands with fixed $N{:=}n{+}L$ are unified in a single generating function. Similar to the SYM four-point half-BPS correlator integrand, the generating function enjoys a hidden $S_N$ permutation symmetry in the dual space, allowing us to write it as a linear combination of weight-3 planar $f$-graphs. Remarkably, through Gram identities it can also be represented as a linear combination of bipartite $f$-graphs which manifest the important property that no odd-multiplicity amplitude exists in the theory. The generating function and these properties are explicitly checked against squared amplitudes for all $n$ with $N{=}4,6,8$. By drawing analogies with SYM, we conjecture some graphical rules the generating function satisfy, and exploit them to bootstrap a unique $N{=}10$ result, which provides new results for $n{=}10$ squared tree amplitudes, as well as integrands for $(n,L){=}(4,6),(6,4)$. Our results strongly suggest the existence of a "bipartite correlator" in ABJM theory that unifies all squared amplitudes and satisfies physical constraints underlying these graphical rules.

Figures

Figures reproduced from arXiv: 2508.03813 by the authors.

Figure 1
Figure 1. FIG. 1: The (double-)quadrangle rule relating [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

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Cited by 1 Pith paper

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

  1. Loops and legs: ABJM amplitudes from $f$-graphs

    hep-th 2026-01 unverdicted novelty 7.0 of 10

    ABJM amplitudes of arbitrary multiplicity and loop order can be reconstructed from squared amplitudes encoded in a permutation-symmetric generating function of planar f-graphs.

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Works this paper leans on

87 extracted references · 10 canonical work pages · cited by 1 Pith paper

  1. [50]

    The two-loop eight-point amplitude in ABJM theory,

    S. He, Y.-t. Huang, C.-K. Kuo, and Z. Li, “The two-loop eight-point amplitude in ABJM theory,” JHEP 02 (2023) 065, arXiv:2211.01792 [hep-th]

    arXiv 2023

  2. [1]

    Review of AdS/CFT Integrability: An Overview,

    N. Beisert et al., “Review of AdS/CFT Integrability: An Overview,” Lett. Math. Phys.99 (2012) 3–32, arXiv:1012.3982 [hep-th]

    arXiv 2012

  3. [2]

    Structure Constants and Integrable Bootstrap in Planar N=4 SYM Theory,

    B. Basso, S. Komatsu, and P. Vieira, “Structure Constants and Integrable Bootstrap in Planar N=4 SYM Theory,” arXiv:1505.06745 [hep-th]

    arXiv

  4. [3]

    Perturbative four-point functions in planar N = 4 SYM from hexagonalization,

    F. Coronado, “Perturbative four-point functions in planar N = 4 SYM from hexagonalization,” JHEP 01 (2019) 056, arXiv:1811.00467 [hep-th]

    arXiv 2019

  5. [4]

    Octagon at finite coupling,

    A. V. Belitsky and G. P. Korchemsky, “Octagon at finite coupling,” JHEP 07 (2020) 219, arXiv:2003.01121 [hep-th]

    arXiv 2020

  6. [5]

    Crossing bridges with strong Szeg˝ o limit theorem,

    A. V. Belitsky and G. P. Korchemsky, “Crossing bridges with strong Szeg˝ o limit theorem,”JHEP 04 (2021) 257, arXiv:2006.01831 [hep-th]

    arXiv 2021

  7. [6]

    The SAGEX review on scattering amplitudes,

    G. Travaglini et al., “The SAGEX review on scattering amplitudes,” J. Phys. A55 no. 44, (2022) 443001, arXiv:2203.13011 [hep-th]

    arXiv 2022

  8. [7]

    Dual superconformal symmetry of scattering amplitudes in N=4 super-Yang-Mills theory,

    J. M. Drummond, J. Henn, G. P. Korchemsky, and E. Sokatchev, “Dual superconformal symmetry of scattering amplitudes in N=4 super-Yang-Mills theory,” Nucl. Phys. B828 (2010) 317–374, arXiv:0807.1095 [hep-th]

    arXiv 2010

Show all 87 references
  1. [8]

    A Note on dual superconformal symmetry of the N=4 super Yang-Mills S-matrix,

    A. Brandhuber, P. Heslop, and G. Travaglini, “A Note on dual superconformal symmetry of the N=4 super Yang-Mills S-matrix,” Phys. Rev. D78 (2008) 125005, arXiv:0807.4097 [hep-th]

    2008 arXiv

  2. [9]

    Generalized unitarity for N=4 super-amplitudes,

    J. M. Drummond, J. Henn, G. P. Korchemsky, and E. Sokatchev, “Generalized unitarity for N=4 super-amplitudes,” Nucl. Phys. B869 (2013) 452–492, arXiv:0808.0491 [hep-th]

    2013 arXiv

  3. [10]

    Yangian symmetry of scattering amplitudes in N=4 super Yang-Mills theory,

    J. M. Drummond, J. M. Henn, and J. Plefka, “Yangian symmetry of scattering amplitudes in N=4 super Yang-Mills theory,” JHEP 05 (2009) 046, arXiv:0902.2987 [hep-th]

    2009 arXiv

  4. [11]

    Exacting N=4 Superconformal Symmetry,

    T. Bargheer, N. Beisert, W. Galleas, F. Loebbert, and T. McLoughlin, “Exacting N=4 Superconformal Symmetry,” JHEP 11 (2009) 056, arXiv:0905.3738 [hep-th]

    2009 arXiv

  5. [12]

    Proof of the Dual Conformal Anomaly of One-Loop Amplitudes in N=4 SYM,

    A. Brandhuber, P. Heslop, and G. Travaglini, “Proof of the Dual Conformal Anomaly of One-Loop Amplitudes in N=4 SYM,” JHEP 10 (2009) 063, arXiv:0906.3552 [hep-th]

    2009 arXiv

  6. [13]

    One-Loop Superconformal and Yangian Symmetries of Scattering Amplitudes in N=4 Super Yang-Mills,

    N. Beisert, J. Henn, T. McLoughlin, and J. Plefka, “One-Loop Superconformal and Yangian Symmetries of Scattering Amplitudes in N=4 Super Yang-Mills,” JHEP 04 (2010) 085, arXiv:1002.1733 [hep-th]

    2010 arXiv

  7. [14]

    Jumpstarting the All-Loop S-Matrix of Planar N=4 Super Yang-Mills,

    S. Caron-Huot and S. He, “Jumpstarting the All-Loop S-Matrix of Planar N=4 Super Yang-Mills,” JHEP 07 (2012) 174, arXiv:1112.1060 [hep-th]

    2012 arXiv

  8. [15]

    Descent Equations for Superamplitudes,

    M. Bullimore and D. Skinner, “Descent Equations for Superamplitudes,” arXiv:1112.1056 [hep-th]

    arXiv

  9. [16]

    Arkani-Hamed, J

    N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo, A. B. Goncharov, A. Postnikov, and J. Trnka, Grassmannian Geometry of Scattering Amplitudes. Cambridge University Press, 4, 2016. arXiv:1212.5605 [hep-th]

    2016 arXiv

  10. [17]

    The SAGEX Review on Scattering Amplitudes, Chapter 8: Half BPS correlators,

    P. Heslop, “The SAGEX Review on Scattering Amplitudes, Chapter 8: Half BPS correlators,” J. Phys. A 55 no. 44, (2022) 443009, arXiv:2203.13019 [hep-th]

    2022 arXiv

  11. [18]

    Local integrands for the five-point amplitude in planar N=4 SYM up to five loops,

    R. G. Ambrosio, B. Eden, T. Goddard, P. Heslop, and C. Taylor, “Local integrands for the five-point amplitude in planar N=4 SYM up to five loops,” JHEP 01 (2015) 116, arXiv:1312.1163 [hep-th]

    2015 arXiv

  12. [19]

    Multi-particle amplitudes from the four-point correlator in planar N = 4 SYM,

    P. Heslop and V.-V. Tran, “Multi-particle amplitudes from the four-point correlator in planar N = 4 SYM,” JHEP 07 (2018) 068, arXiv:1803.11491 [hep-th]

    2018 arXiv

  13. [20]

    Bootstrapping form factor squared in N = 4 super-Yang-Mills,

    S. He, X. Li, J. Lin, J. Liu, and K. Yan, “Bootstrapping form factor squared in N = 4 super-Yang-Mills,” arXiv:2506.07796 [hep-th]

    arXiv

  14. [21]

    Energy Correlators: A Journey From Theory to Experiment,

    I. Moult and H. X. Zhu, “Energy Correlators: A Journey From Theory to Experiment,” arXiv:2506.09119 [hep-ph]

    arXiv

  15. [22]

    The super-correlator/super-amplitude duality: Part I,

    B. Eden, P. Heslop, G. P. Korchemsky, and E. Sokatchev, “The super-correlator/super-amplitude duality: Part I,” Nucl. Phys. B869 (2013) 329–377, arXiv:1103.3714 [hep-th]

    2013 arXiv

  16. [23]

    Hidden symmetry of four-point correlation functions and amplitudes in N=4 SYM,

    B. Eden, P. Heslop, G. P. Korchemsky, and E. Sokatchev, “Hidden symmetry of four-point correlation functions and amplitudes in N=4 SYM,” Nucl. Phys. B862 (2012) 193–231, arXiv:1108.3557 [hep-th]

    2012 arXiv

  17. [24]

    Perturbation Theory at Eight Loops: Novel Structures and the Breakdown of Manifest Conformality in N=4 Supersymmetric Yang-Mills Theory,

    J. L. Bourjaily, P. Heslop, and V.-V. Tran, “Perturbation Theory at Eight Loops: Novel Structures and the Breakdown of Manifest Conformality in N=4 Supersymmetric Yang-Mills Theory,” Phys. Rev. Lett. 116 no. 19, (2016) 191602, arXiv:1512.07912 [hep-th]

    2016 arXiv

  18. [25]

    Amplitudes and Correlators to Ten Loops Using Simple, Graphical Bootstraps,

    J. L. Bourjaily, P. Heslop, and V.-V. Tran, “Amplitudes and Correlators to Ten Loops Using Simple, Graphical Bootstraps,” JHEP 11 (2016) 125, arXiv:1609.00007 [hep-th]

    2016 arXiv

  19. [26]

    The cusp limit of correlators and: anew graphical bootstrap for correlators/amplitudes to eleven loops,

    S. He, C. Shi, Y. Tang, and Y.-Q. Zhang, “The cusp limit of correlators and: anew graphical bootstrap for correlators/amplitudes to eleven loops,” JHEP 03 (2025) 192, arXiv:2410.09859 [hep-th]

    2025 arXiv

  20. [27]

    The Four-Point Correlator of Planar sYM at Twelve Loops,

    J. L. Bourjaily, S. He, C. Shi, and Y. Tang, “The Four-Point Correlator of Planar sYM at Twelve Loops,” arXiv:2503.15593 [hep-th]

    arXiv

  21. [28]

    N = 6 Superconformal Chern-Simons-Matter Theories, M 2-Branes and Their Gravity Duals,

    O. Aharony, O. Bergman, D. L. Jafferis, and J. Maldacena, “ N = 6 Superconformal Chern-Simons-Matter Theories, M 2-Branes and Their Gravity Duals,” JHEP 0810 (2008) 091, arXiv:0806.1218 [hep-th]

    2008 arXiv

  22. [29]

    Review of AdS/CFT Integrability, Chapter IV.3: N=6 Chern-Simons and Strings on AdS4xCP3,

    T. Klose, “Review of AdS/CFT Integrability, Chapter IV.3: N=6 Chern-Simons and Strings on AdS4xCP3,” Lett. Math. Phys.99 (2012) 401–423, arXiv:1012.3999 [hep-th]

    2012 arXiv

  23. [30]

    Symmetries of Tree-level Scattering Amplitudes in N=6 Superconformal Chern-Simons Theory,

    T. Bargheer, F. Loebbert, and C. Meneghelli, “Symmetries of Tree-level Scattering Amplitudes in N=6 Superconformal Chern-Simons Theory,” Phys. Rev. D 82 (2010) 045016, arXiv:1003.6120 [hep-th]

    2010 arXiv

  24. [31]

    Dual Superconformal Symmetry of N=6 Chern-Simons Theory,

    Y.-t. Huang and A. E. Lipstein, “Dual Superconformal Symmetry of N=6 Chern-Simons Theory,” JHEP 11 (2010) 076, arXiv:1008.0041 [hep-th]

    2010 arXiv

  25. [32]

    Tree-level Recursion Relation and Dual Superconformal Symmetry of the ABJM Theory,

    D. Gang, Y.-t. Huang, E. Koh, S. Lee, and A. E. Lipstein, “Tree-level Recursion Relation and Dual Superconformal Symmetry of the ABJM Theory,” JHEP 03 (2011) 116, arXiv:1012.5032 [hep-th]

    2011 arXiv

  26. [33]

    Dualities for Loop Amplitudes of N=6 Chern-Simons Matter Theory,

    W.-M. Chen and Y.-t. Huang, “Dualities for Loop Amplitudes of N=6 Chern-Simons Matter Theory,” 7 JHEP 11 (2011) 057, arXiv:1107.2710 [hep-th]

    2011 arXiv

  27. [34]

    Yangian Invariant Scattering Amplitudes in Supersymmetric Chern-Simons Theory,

    S. Lee, “Yangian Invariant Scattering Amplitudes in Supersymmetric Chern-Simons Theory,” Phys. Rev. Lett. 105 (2010) 151603, arXiv:1007.4772 [hep-th]

    2010 arXiv

  28. [35]

    The Positive orthogonal Grassmannian and loop amplitudes of ABJM,

    Y.-t. Huang, C. Wen, and D. Xie, “The Positive orthogonal Grassmannian and loop amplitudes of ABJM,” J. Phys. A47 no. 47, (2014) 474008, arXiv:1402.1479 [hep-th]

    2014 arXiv

  29. [36]

    On the ABJM four-point amplitude at three loops and BDS exponentiation,

    M. S. Bianchi and M. Leoni, “On the ABJM four-point amplitude at three loops and BDS exponentiation,” JHEP 11 (2014) 077, arXiv:1403.3398 [hep-th]

    2014 arXiv

  30. [37]

    Light-like polygonal Wilson loops in 3d Chern-Simons and ABJM theory,

    J. M. Henn, J. Plefka, and K. Wiegandt, “Light-like polygonal Wilson loops in 3d Chern-Simons and ABJM theory,” JHEP 08 (2010) 032, arXiv:1004.0226 [hep-th]. [Erratum: JHEP 11, 053 (2011)]

    2010 arXiv

  31. [38]

    From Correlators to Wilson Loops in Chern-Simons Matter Theories,

    M. S. Bianchi, M. Leoni, A. Mauri, S. Penati, C. Ratti, and A. Santambrogio, “From Correlators to Wilson Loops in Chern-Simons Matter Theories,” JHEP 06 (2011) 118, arXiv:1103.3675 [hep-th]

    2011 arXiv

  32. [39]

    Scattering Amplitudes/Wilson Loop Duality In ABJM Theory,

    M. S. Bianchi, M. Leoni, A. Mauri, S. Penati, and A. Santambrogio, “Scattering Amplitudes/Wilson Loop Duality In ABJM Theory,” JHEP 01 (2012) 056, arXiv:1107.3139 [hep-th]

    2012 arXiv

  33. [40]

    Conformal Anomaly for Amplitudes in N = 6 Superconformal Chern-Simons Theory,

    T. Bargheer, N. Beisert, F. Loebbert, and T. McLoughlin, “Conformal Anomaly for Amplitudes in N = 6 Superconformal Chern-Simons Theory,” J. Phys. A 45 (2012) 475402, arXiv:1204.4406 [hep-th]

    2012 arXiv

  34. [41]

    Fermionic T-Duality, Dual Superconformal Symmetry, and the Amplitude/Wilson Loop Connection,

    N. Berkovits and J. Maldacena, “Fermionic T-Duality, Dual Superconformal Symmetry, and the Amplitude/Wilson Loop Connection,” JHEP 09 (2008) 062, arXiv:0807.3196 [hep-th]

    2008 arXiv

  35. [42]

    A Requiem for AdS4 × CP 3 Fermionic self-T-duality,

    E. ´O. Colg´ ain and A. Pittelli, “A Requiem for AdS4 × CP 3 Fermionic self-T-duality,” Phys. Rev. D94 no. 10, (2016) 106006, arXiv:1609.03254 [hep-th]

    2016 arXiv

  36. [43]

    Wilson loops in N=6 superspace for ABJM theory,

    M. Rosso and C. Vergu, “Wilson loops in N=6 superspace for ABJM theory,” JHEP 06 (2014) 176, arXiv:1403.2336 [hep-th]

    2014 arXiv

  37. [44]

    Generalized unitarity and one-loop amplitudes in N=4 super-Yang-Mills,

    R. Britto, F. Cachazo, and B. Feng, “Generalized unitarity and one-loop amplitudes in N=4 super-Yang-Mills,” Nucl. Phys. B725 (2005) 275–305, arXiv:hep-th/0412103

    2005 arXiv

  38. [45]

    The Amplituhedron,

    N. Arkani-Hamed and J. Trnka, “The Amplituhedron,” JHEP 10 (2014) 030, arXiv:1312.2007 [hep-th]

    2014 arXiv

  39. [46]

    Nonperturbative negative geometries: amplitudes at strong coupling and the amplituhedron,

    N. Arkani-Hamed, J. Henn, and J. Trnka, “Nonperturbative negative geometries: amplitudes at strong coupling and the amplituhedron,” JHEP 03 (2022) 108, arXiv:2112.06956 [hep-th]

    2022 arXiv

  40. [47]

    One Loop Amplitudes In ABJM,

    M. S. Bianchi, M. Leoni, A. Mauri, S. Penati, and A. Santambrogio, “One Loop Amplitudes In ABJM,” JHEP 07 (2012) 029, arXiv:1204.4407 [hep-th]

    2012 arXiv

  41. [48]

    A note on amplitudes in N=6 superconformal Chern-Simons theory,

    A. Brandhuber, G. Travaglini, and C. Wen, “A note on amplitudes in N=6 superconformal Chern-Simons theory,” JHEP 07 (2012) 160, arXiv:1205.6705 [hep-th]

    2012 arXiv

  42. [49]

    The two-loop six-point amplitude in ABJM theory,

    S. Caron-Huot and Y.-t. Huang, “The two-loop six-point amplitude in ABJM theory,” JHEP 03 (2013) 075, arXiv:1210.4226 [hep-th]

    2013 arXiv

  43. [51]

    The ABJM Amplituhedron,

    S. He, Y.-t. Huang, and C.-K. Kuo, “The ABJM Amplituhedron,” JHEP 09 (2023) 165, arXiv:2306.00951 [hep-th]. [Erratum: JHEP 04, 064 (2024)]

    2023 arXiv

  44. [52]

    All-Loop Four-Point Aharony-Bergman-Jafferis-Maldacena Amplitudes from Dimensional Reduction of the Amplituhedron,

    S. He, C.-K. Kuo, Z. Li, and Y.-Q. Zhang, “All-Loop Four-Point Aharony-Bergman-Jafferis-Maldacena Amplitudes from Dimensional Reduction of the Amplituhedron,” Phys. Rev. Lett.129 no. 22, (2022) 221604, arXiv:2204.08297 [hep-th]

    2022 arXiv

  45. [53]

    Elvang and Y.-t

    H. Elvang and Y.-t. Huang, Scattering Amplitudes in Gauge Theory and Gravity. Cambridge University Press, 4, 2015

    2015

  46. [54]

    See the discussion below eq.(11.51) of [53]

  47. [55]

    Therefore, we ignore the overall normalization of M (L) n in this Letter

    It is extremely difficult to align the conventions of different papers. Therefore, we ignore the overall normalization of M (L) n in this Letter. However, we believe there exists a good convention such that (5) holds exactly in ABJM

  48. [56]

    Constructing the correlation function of four stress-tensor multiplets and the four-particle amplitude in N=4 SYM,

    B. Eden, P. Heslop, G. P. Korchemsky, and E. Sokatchev, “Constructing the correlation function of four stress-tensor multiplets and the four-particle amplitude in N=4 SYM,” Nucl. Phys. B862 (2012) 450–503, arXiv:1201.5329 [hep-th]

    2012 arXiv

  49. [57]

    From M4= 1 2 [A(0) 4 ]2 exp log(A4(−a)A4(a)/[A(0) 4 ]2), we can express M (L) 4 in terms of ˜Ωℓ with even ℓ only, where ˜Ωℓ/ℓ! is the ℓ-loop integrand of log( A4(a)/A(0) 4 ) [52]

  50. [58]

    This is similar to SYM where F SYM N ≥6 are planar but F SYM 5 is nonplanar

    The lowest-point F4= 1 2 is an exception by being planar but not bipartite. This is similar to SYM where F SYM N ≥6 are planar but F SYM 5 is nonplanar

  51. [59]

    Fast Generation of Planar Graphs (expanded Version)

    G. Brinkmann and B. D. McKay, “Fast Generation of Planar Graphs (expanded Version).” http://cs.anu.edu.au/~bdm/papers/plantri-full.pdf

  52. [60]

    Practical graph isomorphism, ii,

    B. D. McKay and A. Piperno, “Practical graph isomorphism, ii,” Journal of Symbolic Computation60 (2014) 94–112. https://www.sciencedirect.com/ science/article/pii/S0747717113001193

    2014

  53. [61]

    From correlation functions to Wilson loops,

    L. F. Alday, B. Eden, G. P. Korchemsky, J. Maldacena, and E. Sokatchev, “From correlation functions to Wilson loops,” JHEP 09 (2011) 123, arXiv:1007.3243 [hep-th]

    2011 arXiv

  54. [62]

    ABJM Amplitudes in U-gauge and a Soft Theorem,

    S. Chin, S. Lee, and Y. Yun, “ABJM Amplitudes in U-gauge and a Soft Theorem,” JHEP 11 (2015) 088, arXiv:1508.07975 [hep-th]

    2015 arXiv

  55. [63]

    This is similar to SYM, where the double-triangle rule has a weaker version corresponding to the soft limit of the squared amplitude

  56. [64]

    Emergent unitarity, all-loop cuts and integrations from the ABJM amplituhedron,

    S. He, C.-K. Kuo, Z. Li, and Y.-Q. Zhang, “Emergent unitarity, all-loop cuts and integrations from the ABJM amplituhedron,” JHEP 07 (2023) 212, arXiv:2303.03035 [hep-th]

    2023 arXiv

  57. [65]

    Multiplets of Superconformal Symmetry in Diverse Dimensions,

    C. Cordova, T. T. Dumitrescu, and K. Intriligator, “Multiplets of Superconformal Symmetry in Diverse Dimensions,” JHEP 03 (2019) 163, arXiv:1612.00809 [hep-th]

    2019 arXiv

  58. [66]

    Superconformal Chern-Simons Theories and AdS(4)/CFT(3) Correspondence,

    M. Benna, I. Klebanov, T. Klose, and M. Smedback, “Superconformal Chern-Simons Theories and AdS(4)/CFT(3) Correspondence,” JHEP 09 (2008) 072, arXiv:0806.1519 [hep-th]

    2008 arXiv

  59. [67]

    The Correlahedron,

    B. Eden, P. Heslop, and L. Mason, “The Correlahedron,” JHEP 09 (2017) 156, arXiv:1701.00453 [hep-th]

    2017 arXiv

  60. [68]

    Amplituhedron-like geometries,

    G. Dian and P. Heslop, “Amplituhedron-like geometries,” JHEP 11 (2021) 074, arXiv:2106.09372 [hep-th]

    2021 arXiv

  61. [69]

    All-loop geometry for four-point correlation functions,

    S. He, Y.-t. Huang, and C.-K. Kuo, “All-loop geometry for four-point correlation functions,” Phys. Rev. D110 8 no. 8, (2024) L081701, arXiv:2405.20292 [hep-th]

    2024 arXiv

  62. [70]

    Leading singularities and chambers of Correlahedron,

    S. He, Y.-t. Huang, and C.-K. Kuo, “Leading singularities and chambers of Correlahedron,” arXiv:2505.09808 [hep-th]

    arXiv

  63. [71]

    From squared amplitudes to energy correlators,

    S. He, X. Jiang, Q. Yang, and Y.-Q. Zhang, “From squared amplitudes to energy correlators,” arXiv:2408.04222 [hep-th]

    arXiv

  64. [72]

    Algorithm for symbol integrations for loop integrals,

    S. He and Y. Tang, “Algorithm for symbol integrations for loop integrals,” Phys. Rev. D108 no. 4, (2023) L041702, arXiv:2304.01776 [hep-th]

    2023 arXiv

  65. [73]

    Graphical functions in even dimensions,

    M. Borinsky and O. Schnetz, “Graphical functions in even dimensions,” Commun. Num. Theor. Phys.16 no. 3, (2022) 515–614, arXiv:2105.05015 [hep-th]

    2022 arXiv

  66. [74]

    Knots and numbers in Phi**4 theory to 7 loops and beyond,

    D. J. Broadhurst and D. Kreimer, “Knots and numbers in Phi**4 theory to 7 loops and beyond,” Int. J. Mod. Phys. C 6 (1995) 519–524, arXiv:hep-ph/9504352

    1995 arXiv

  67. [75]

    Quantum periods: A Census of phi**4-transcendentals,

    O. Schnetz, “Quantum periods: A Census of phi**4-transcendentals,” Commun. Num. Theor. Phys. 4 (2010) 1–48, arXiv:0801.2856 [hep-th]

    2010 arXiv

  68. [76]

    On the periods of some Feynman integrals,

    F. C. S. Brown, “On the periods of some Feynman integrals,” arXiv:0910.0114 [math.AG]

    arXiv

  69. [77]

    Graphical functions and single-valued multiple polylogarithms,

    O. Schnetz, “Graphical functions and single-valued multiple polylogarithms,” Commun. Num. Theor. Phys. 08 (2014) 589–675, arXiv:1302.6445 [math.NT]

    2014 arXiv

  70. [78]

    The Galois coaction on ϕ4 periods,

    E. Panzer and O. Schnetz, “The Galois coaction on ϕ4 periods,” Commun. Num. Theor. Phys.11 (2017) 657–705, arXiv:1603.04289 [hep-th]

    2017 arXiv

  71. [79]

    Notes on conformal integrals: Coulomb branch amplitudes, magic identities and bootstrap,

    S. He, X. Jiang, J. Liu, and Y.-Q. Zhang, “Notes on conformal integrals: Coulomb branch amplitudes, magic identities and bootstrap,” arXiv:2502.08871 [hep-th]

  72. [80]

    Solving Infinite Families of Dual Conformal Integrals and Periods,

    S. He and X. Jiang, “Solving Infinite Families of Dual Conformal Integrals and Periods,” arXiv:2506.20095 [hep-th]

    arXiv

  73. [81]

    Integrated correlators in N = 4 super Yang-Mills and periods,

    C. Wen and S.-Q. Zhang, “Integrated correlators in N = 4 super Yang-Mills and periods,” JHEP 05 (2022) 126, arXiv:2203.01890 [hep-th]

    2022 arXiv

  74. [82]

    Integrated correlators in N = 4 SYM beyond localisation,

    A. Brown, P. Heslop, C. Wen, and H. Xie, “Integrated correlators in N = 4 SYM beyond localisation,” Phys. Rev. Lett.132 no. 10, (2024) 101602, arXiv:2308.07219 [hep-th]

    2024 arXiv

  75. [83]

    Nonplanar integrated correlator in N=4 SYM,

    S.-Q. Zhang, “Nonplanar integrated correlator in N=4 SYM,” Phys. Rev. D110 no. 2, (2024) 025003, arXiv:2404.18900 [hep-th]. Supplemental Material Summary of graphical rules in SYM We summarize the constraints on FN and the corresponding graphical rules in SYM. This will make i...

    2024 arXiv

  76. [84]

    This split has to be the same for all σ±

    Split the k2 variables c±αβ into two subsets: {c′} with k(k+1) 2 + (k−2)(k−3) 2 elements, and {c′′} with (2 k − 3) elements. This split has to be the same for all σ±

  77. [85]

    Compute the Jacobian determinant J ′ = | ∂(constraints) ∂c′ | relating the constraints (namely, the k(k+1) 2 delta func- tion arguments and 1 2 (k − 2)(k − 3) vanishing minors) to the variables {c′}

  78. [86]

    Compute the Jacobian determinant J ′′ = | ∂c′′ ∂τs | relating the variables {c′′} and the parametrization {τs}

  79. [87]

    The next step is to localize the (2 k − 3)-form (42) using δk×2(C± · λT ) in (39) to obtain a distribution supported by momentum conservation δ3(λ · λT )

    The total Jacobian is given by J± Q 1+τ 2 s τs(1−τ 2s ) = J ′′/J ′. The next step is to localize the (2 k − 3)-form (42) using δk×2(C± · λT ) in (39) to obtain a distribution supported by momentum conservation δ3(λ · λT ). The solutions τ ∗ s can be easily obtained numerically...

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.