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arxiv: 2511.19788 · v2 · submitted 2025-11-24 · ✦ hep-th

Recognition: no theorem link

Yangian Symmetry Escapes from the Fishnet

Niklas Beisert , Benedikt K\"onig
Authors on Pith no claims yet

Pith reviewed 2026-05-17 05:15 UTC · model grok-4.3

classification ✦ hep-th
keywords Yangian symmetryfishnet modelbi-scalar modelcorrelation functionsdual Coxeter numberplanar QFTFeynman graphsintegrability
0
0 comments X

The pith

Yangian symmetry is realized classically in bi-scalar and supersymmetric fishnet models under specific parameter patterns but does not hold for generic quantum correlation functions.

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

The paper checks whether Yangian symmetry survives from the classical equations of motion and action into the quantum regime of the bi-scalar and supersymmetric fishnet theories in four dimensions. It finds that the symmetry works classically when the evaluation parameters follow particular patterns. At the quantum level, however, the symmetry fails for generic correlation functions in the bi-scalar model, with explicit counterexamples constructed from sums of ordinary Feynman graphs and from graphs that contain octagon-shaped loops. This limitation points to the non-zero dual Coxeter number as a structural barrier that prevents full quantum Yangian invariance and may also block complete integrability in planar quantum field theories.

Core claim

Yangian symmetry is classically realised in these models subject to specific evaluation parameter patterns. Curiously, Yangian invariance does not extend to generic quantum correlation functions in the bi-scalar model beyond the well-established classes of Yangian invariant correlators. Several concrete counter-examples are given by sums of Feynman graphs and by bi-scalar graphs with octagon-shaped loops. This finding underlines the notion that a non-zero dual Coxeter number represents an obstacle towards quantum Yangian symmetry and possibly also for complete integrability in planar QFT models.

What carries the argument

Yangian invariance of the classical action and equations of motion under specific evaluation parameter patterns, tested against sums of Feynman graphs and octagon-loop graphs at the quantum level.

If this is right

  • Quantum Yangian symmetry remains confined to already-known special classes of correlators in the bi-scalar model.
  • A non-zero dual Coxeter number blocks the extension of Yangian invariance to generic quantum correlation functions.
  • Complete integrability of planar QFT models may be obstructed when the dual Coxeter number is non-zero.
  • The same pattern of classical realization followed by quantum breakdown could appear in related fishnet-type theories.

Where Pith is reading between the lines

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

  • Alternative quantum symmetries or deformations may be needed to restore integrability once Yangian invariance is lost.
  • The obstruction identified here could apply to other planar models that share a non-zero dual Coxeter number.
  • Systematic checks of additional loop topologies or higher-point functions would test how widespread the counterexamples are.

Load-bearing premise

That the presented sums of Feynman graphs and octagon-loop graphs are representative of generic quantum correlators and that the specific evaluation parameter patterns fully capture the conditions for classical realization.

What would settle it

A direct computation demonstrating that one of the generic bi-scalar correlators built from Feynman graphs or octagon loops actually preserves Yangian invariance after quantum corrections, or conversely, that none of them do.

Figures

Figures reproduced from arXiv: 2511.19788 by Benedikt K\"onig, Niklas Beisert.

Figure 1
Figure 1. Figure 1: A unique fishnet graph depicted with a suitable sequence of evaluation parameters along the perimeter curve which has been cut open between the two legs at the upper right corner. Let us first recollect the pertinent results for Yangian invariance of fishnet graphs, see [6, 8] for further details: For convenience, we label the four flavours of scalar fields (ϕ1, ϕ2, ϕ¯ 1, ϕ¯ 2) as the Z4-integers (1, 2, 3,… view at source ↗
Figure 2
Figure 2. Figure 2: A pair of non-unique non-fishnet graphs at one loop. 1 5 1 1 1 3 3 3 3 3 2 2 2 2 0 4 4 4 1 1 1 1 1 3 −1 3 3 3 2 2 2 2 0 4 4 4 [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: A pair of non-unique non-fishnet graphs at tree level. this sequence of flavours. We provide arguments towards uniqueness of fishnet and more general bi-scalar graphs in App. A. Importantly, not all contributions to bi-scalar correlators are of fishnet graph type: Such graphs have been described in [6] as “singular fishnet graphs”. They are cut out from a fishnet plane with a non-trivial structure of inter… view at source ↗
Figure 4
Figure 4. Figure 4: Octagon non-fishnet graph. breaks down for all higher polygons. Here, we present a planar octagon loop graph [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Non-fishnet graphs with equal perimeter and different interiors. Note that this formula computes the fishnet area enclosed by the fishnet shape which equivalently describes the total twist of the flavour configuration when viewed as a de￾formation of N = 4 SYM. Given V and the structure of vertices, we can deduce further useful characteristic numbers. The numbers of internal propagators of each flavour rea… view at source ↗
Figure 6
Figure 6. Figure 6: Basic reconfiguration move and corresponding adjustment to evalua￾tion parameter sequence. 9 1 2 2 3 3 4 4 4 5 5 6 6 6 7 7 8 8 9 1 2 2 3 3 4 4 4 5 5 6 6 6 7 7 8 8 [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Two graphs with equal evaluation parameter sequences (up to peri￾odicity), but inequivalent interiors. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The fishnet shape corresponding to [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: A globally overlapping fishnet shape with branch cut resolution. The example demonstrates that ∆s represents an independent characteristic for the graph which cannot be deduced from the values sk alone. In that sense, all the numbers dk de￾scribing the perimeter configuration in terms of consecutive propagators must be provided (together with the initial flavour f1). Let us formulate two maximal uniqueness… view at source ↗
Figure 10
Figure 10. Figure 10: Turns of the fishnet shape by +90◦ , 0 ◦ , −90◦ , −180◦ , −270◦ , −360◦ , . . . corresponding to dj = 0, 1, 2, 3, 4, 5, . . . adjacent perimeter propagators. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_10.png] view at source ↗
read the original abstract

We investigate Yangian symmetry for the equations of motion and the action of the classical bi-scalar and supersymmetric fishnet models in four spacetime dimensions, and we subsequently discuss its applicability to planar correlation functions. We argue that Yangian symmetry is classically realised in these models subject to specific evaluation parameter patterns. Curiously, Yangian invariance does not extend to generic quantum correlation functions in the bi-scalar model beyond the well-established classes of Yangian invariant correlators. We present several concrete counter-examples of bi-scalar correlators given by sums of Feynman graphs and of bi-scalar graphs with octagon-shaped loops. This finding underlines the notion that a non-zero dual Coxeter number represents an obstacle towards quantum Yangian symmetry and possibly also for complete integrability in planar QFT models.

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 examines Yangian symmetry for the equations of motion and action in the classical bi-scalar and supersymmetric fishnet models in four dimensions, claiming it is realized for specific evaluation parameter patterns. It then argues that this symmetry does not extend to generic quantum correlation functions in the bi-scalar model beyond known invariant classes, supported by explicit counterexamples consisting of sums of Feynman graphs and bi-scalar graphs containing octagon-shaped loops. The work concludes that a non-zero dual Coxeter number obstructs quantum Yangian symmetry and possibly complete integrability in planar QFT models.

Significance. If the counterexamples are shown to be representative of generic correlators rather than special cases tied to diagram topology or parameter choice, the result would be significant for the study of integrability and symmetries in planar quantum field theories. The explicit construction of counterexamples and the classical parameter patterns constitute a concrete strength, providing falsifiable tests that distinguish this work from purely formal discussions of Yangian invariance.

major comments (2)
  1. [Quantum correlation functions discussion] Quantum section (counterexamples with Feynman sums and octagon loops): the central claim that Yangian invariance fails for generic quantum correlators rests on these specific constructions, but the manuscript does not demonstrate that the chosen diagram topologies or evaluation-parameter patterns are representative rather than special cases where invariance fails due to topology. This is load-bearing for the broader conclusion regarding the dual Coxeter number as an obstacle.
  2. [Classical bi-scalar and fishnet models] Classical realization section: the statement that Yangian symmetry holds subject to specific evaluation parameter patterns is presented without a full derivation or exhaustive check of alternative patterns; if other patterns exist that preserve invariance at the quantum level, the separation between classical and quantum cases would need re-examination.
minor comments (2)
  1. [Abstract and introduction] Clarify the precise definition of 'generic quantum correlation functions' versus the 'well-established classes of Yangian invariant correlators' to avoid ambiguity in the scope of the claim.
  2. [Results on parameter patterns] Include a brief table or summary listing the specific evaluation parameter patterns that work classically and those that fail in the quantum examples for easier comparison.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and indicate the revisions we will incorporate.

read point-by-point responses

  1. Referee: Quantum section (counterexamples with Feynman sums and octagon loops): the central claim that Yangian invariance fails for generic quantum correlators rests on these specific constructions, but the manuscript does not demonstrate that the chosen diagram topologies or evaluation-parameter patterns are representative rather than special cases where invariance fails due to topology. This is load-bearing for the broader conclusion regarding the dual Coxeter number as an obstacle.

    Authors: We agree that the counterexamples are specific constructions. They were selected precisely because they lie outside the known classes of Yangian-invariant correlators while still appearing in the perturbative expansion of generic planar correlators in the bi-scalar model. The octagon loops and sums of graphs are not isolated topological curiosities; they arise naturally when expanding correlators with more than four external legs. To make this explicit, we will add a short paragraph in the revised quantum section explaining the generic occurrence of these structures and why the failure of invariance in these cases supports the obstruction from a non-zero dual Coxeter number. This addresses the representativeness concern without claiming a complete classification of all diagrams. revision: partial

  2. Referee: Classical bi-scalar and fishnet models: the statement that Yangian symmetry holds subject to specific evaluation parameter patterns is presented without a full derivation or exhaustive check of alternative patterns; if other patterns exist that preserve invariance at the quantum level, the separation between classical and quantum cases would need re-examination.

    Authors: The evaluation-parameter patterns follow directly from requiring that the level-zero and level-one Yangian generators annihilate the classical equations of motion and leave the action invariant. We will insert a concise derivation of these conditions in the classical section, showing the algebraic steps that fix the allowed patterns. An exhaustive scan of every conceivable parameter choice lies outside the scope of the present work, but the patterns we retain are the ones compatible with the fishnet Lagrangian and the planar limit. We have not identified other patterns that would restore quantum invariance, and we will add a remark noting this as an open question for future investigation. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit counterexamples and direct classical analysis support the non-extension claim without reduction to inputs.

full rationale

The paper first examines the classical bi-scalar and supersymmetric fishnet models by direct inspection of their equations of motion and action, establishing Yangian realization only for specific evaluation parameter patterns. It then demonstrates the failure of extension to generic quantum correlators by constructing explicit counterexamples consisting of sums of Feynman graphs and bi-scalar graphs containing octagon-shaped loops. These constructions are presented as concrete instances rather than derived from fitted parameters or self-referential definitions. No load-bearing step reduces to a self-citation chain, an ansatz smuggled via prior work, or a renaming of known results; the central claim rests on the explicit failure in the chosen diagrams, rendering the derivation self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard properties of Yangian algebras and fishnet Feynman rules; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Yangian symmetry is classically realised subject to specific evaluation parameter patterns.
    Invoked to establish the classical case before contrasting with the quantum counterexamples.

pith-pipeline@v0.9.0 · 5424 in / 1164 out tokens · 54552 ms · 2026-05-17T05:15:03.297982+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages · 17 internal anchors

  1. [1]

    Review of AdS/CFT Integrability: An Overview

    N. Beisert et al.,“Review of AdS/CFT Integrability: An Overview”, Lett. Math. Phys. 99, 3 (2012),arxiv:1012.3982.•D. Bombardelli, A. Cagnazzo, R. Frassek, F. Levkovich-Maslyuk, F. Loebbert, S. Negro, I. M. Sz´ ecs´ enyi, A. Sfondrini, S. J. van Tongeren and A. Torrielli,“An integrability primer for the gauge-gravity correspondence: An introduction”,J. Phy...

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  2. [2]

    High energy QCD as a completely integrable model

    L. D. Faddeev and G. P. Korchemsky,“High-energy QCD as a completely integrable model”,Phys. Lett. B 342, 311 (1995),hep-th/9404173

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  3. [3]

    New integrable non-gauge 4D QFTs from strongly deformed planar N=4 SYM

    ¨O. G¨ urdo˘ gan and V. Kazakov,“New Integrable 4D Quantum Field Theories from Strongly Deformed Planar N=4 Supersymmetric Yang-Mills Theory”, Phys. Rev. Lett. 117, 201602 (2016),arxiv:1512.06704

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  4. [4]

    Chiral limit of N = 4 SYM and ABJM and integrable Feynman graphs

    J. Caetano, ¨O. G¨ urdo˘ gan and V. Kazakov,“Chiral limit of N=4 SYM and ABJM and integrable Feynman graphs”,JHEP 1803, 077 (2018),arxiv:1612.05895

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  5. [5]

    Gauge-string duality for superconformal deformations of N=4 Super Yang-Mills theory

    S. A. Frolov, R. Roiban and A. A. Tseytlin,“Gauge-string duality for superconformal deformations of N=4 super Yang-Mills theory”,JHEP 0507, 045 (2005),hep-th/0503192. •S. Frolov,“Lax pair for strings in Lunin-Maldacena background”, JHEP 0505, 069 (2005),hep-th/0503201.•S. A. Frolov, R. Roiban and A. A. Tseytlin, “Gauge-string duality for (non)supersymmetr...

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  6. [6]

    Yangian Symmetry for Bi-Scalar Loop Amplitudes

    D. Chicherin, V. Kazakov, F. Loebbert, D. M¨ uller and D.-l. Zhong,“Yangian Symmetry for Bi-Scalar Loop Amplitudes”,JHEP 1805, 003 (2018),arxiv:1704.01967

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  7. [7]

    Integrability of Conformal Fishnet Theory

    N. Gromov, V. Kazakov, G. Korchemsky, S. Negro and G. Sizov,“Integrability of Conformal Fishnet Theory”,JHEP 1801, 095 (2018),arxiv:1706.04167

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  8. [8]

    Yangian Symmetry for Fishnet Feynman Graphs

    D. Chicherin, V. Kazakov, F. Loebbert, D. M¨ uller and D.-l. Zhong,“Yangian Symmetry for Fishnet Feynman Graphs”,Phys. Rev. D 96, 121901 (2017),arxiv:1708.00007

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  9. [9]

    ‘Fishing-net’ diagrams as a completely integrable system

    A. B. Zamolodchikov,“‘Fishing-net’ diagrams as a completely integrable system”, Phys. Lett. B 97, 63 (1980)

    work page 1980

  10. [10]

    The One-Loop Spectral Problem of Strongly Twisted $\mathcal{N}$=4 Super Yang-Mills Theory

    A. C. Ipsen, M. Staudacher and L. Zippelius,“The one-loop spectral problem of strongly twisted N=4 Super Yang-Mills theory”,JHEP 1904, 044 (2019),arxiv:1812.08794

    work page internal anchor Pith review Pith/arXiv arXiv 1904

  11. [11]

    Quantum Spectral Curve of $\gamma$-twisted ${\cal N}=4$ SYM theory and fishnet CFT

    V. Kazakov,“Quantum Spectral Curve ofγ-twisted N=4 SYM theory and fishnet CFT”, arxiv:1802.02160. 22Some parts of the curve may run in parallel, see the case 180 ◦ in Fig. 10. Here, one should assume that there remains a small strip of intermediate area which is not enclosed by the curve. 23

    work page internal anchor Pith review Pith/arXiv arXiv

  12. [12]

    Hopf algebras and the quantum Yang–Baxter equation

    V. G. Drinfel’d,“Hopf algebras and the quantum Yang–Baxter equation”, Sov. Math. Dokl. 32, 254 (1985).•V. G. Drinfel’d,“Quantum groups”, J. Sov. Math. 41, 898 (1988)

    work page 1985

  13. [13]

    Integrable Feynman graphs and Yangian symmetry on the loom

    V. Kazakov, F. Levkovich-Maslyuk and V. Mishnyakov,“Integrable Feynman graphs and Yangian symmetry on the loom”,JHEP 2506, 104 (2025),arxiv:2304.04654

    work page arXiv 2025

  14. [14]

    Nonlocal Symmetries of Planar Feynman Integrals

    F. Loebbert, L. R¨ uenaufer and S. F. Stawinski,“Nonlocal Symmetries of Planar Feynman Integrals”,Phys. Rev. Lett. 135, 151603 (2025),arxiv:2505.05550

    work page arXiv 2025

  15. [15]

    Massive Fishnets

    F. Loebbert and J. Miczajka,“Massive Fishnets”,JHEP 2012, 197 (2020), arxiv:2008.11739.•D. Chicherin and G. P. Korchemsky,“The SAGEX review on scattering amplitudes Chapter 9: Integrability of amplitudes in fishnet theories”, J. Phys. A 55, 443010 (2022),arxiv:2203.13020

    work page arXiv 2012

  16. [16]

    Yangian Bootstrap for Conformal Feynman Integrals

    F. Loebbert, D. M¨ uller and H. M¨ unkler,“Yangian Bootstrap for Conformal Feynman Integrals”,Phys. Rev. D 101, 066006 (2020),arxiv:1912.05561.•L. Corcoran, F. Loebbert, J. Miczajka and M. Staudacher,“Minkowski Box from Yangian Bootstrap”, JHEP 2104, 160 (2021),arxiv:2012.07852

    work page arXiv 2020

  17. [17]

    Yangian Symmetry and Integrability of Planar N=4 Super-Yang-Mills Theory

    N. Beisert, A. Garus and M. Rosso,“Yangian Symmetry and Integrability of Planar N=4 Supersymmetric Yang-Mills Theory”,Phys. Rev. Lett. 118, 141603 (2017), arxiv:1701.09162

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  18. [18]

    Yangian Symmetry for the Action of Planar N=4 Super Yang-Mills and N=6 Super Chern-Simons Theories

    N. Beisert, A. Garus and M. Rosso,“Yangian Symmetry for the Action of Planar N=4 Super Yang-Mills and N=6 Super Chern-Simons Theories”, Phys. Rev. D 98, 046006 (2018),arxiv:1803.06310

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  19. [19]

    Yangian Algebra and Correlation Functions in Planar Gauge Theories

    N. Beisert and A. Garus,“Yangian Algebra and Correlation Functions in Planar Gauge Theories”,SciPost Phys. 5, 018 (2018),arxiv:1804.09110

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  20. [20]

    Untwisting the symmetries of $\beta$-deformed Super-Yang--Mills

    A. Garus,“Untwisting the symmetries ofβ-deformed Super-Yang–Mills”, JHEP 1710, 007 (2017),arxiv:1707.04128

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  21. [21]

    Yangian form-alism for planar gauge theories

    N. Beisert and B. K¨ onig,“Yangian form-alism for planar gauge theories”, J. Math. Phys. 66, 062301 (2025),arxiv:2411.16176

    work page arXiv 2025

  22. [22]

    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 0905, 046 (2009),arxiv:0902.2987

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  23. [23]

    Integrable fishnet fromγ-deformed N=2 quivers

    A. Pittelli and M. Preti,“Integrable fishnet fromγ-deformed N=2 quivers”, Phys. Lett. B 798, 134971 (2019),arxiv:1906.03680

    work page arXiv 2019

  24. [24]

    On a CFT limit of planar $\gamma_i$-deformed $\mathcal{N}=4$ SYM theory

    C. Sieg and M. Wilhelm,“On a CFT limit of planarγ i-deformed N=4 SYM theory”, Phys. Lett. B 756, 118 (2016),arxiv:1602.05817.•D. Grabner, N. Gromov, V. Kazakov and G. Korchemsky,“Stronglyγ-Deformed N=4 Supersymmetric Yang-Mills Theory as an Integrable Conformal Field Theory”,Phys. Rev. Lett. 120, 111601 (2018), arxiv:1711.04786

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  25. [25]

    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.•B. Basso, J. Caetano and T. Fleury, “Hexagons and Correlators in the Fishnet Theory”,JHEP 1911, 172 (2019), arxiv:1812.09794.•E. Olivucci,“Hexagonalization of Fishnet integrals. Part I. Mirror excitations”,JHEP 2111, 204 (2021),arxi...

    work page internal anchor Pith review Pith/arXiv arXiv 1911