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arxiv: 2511.19788 · v3 · pith:KQXXXTREnew · submitted 2025-11-24 · ✦ hep-th

Yangian Symmetry Escapes from the Fishnet

Niklas Beisert , Benedikt K\"onig This is my paper

Pith reviewed 2026-05-21 17:49 UTC · model grok-4.3

classification ✦ hep-th
keywords Yangian symmetryfishnet modelbi-scalar modelcorrelation functionsFeynman graphsdual Coxeter numberintegrabilityplanar QFT
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The pith

Yangian symmetry holds classically in bi-scalar fishnet models but fails 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, an infinite-dimensional extension of ordinary symmetries, survives from the classical equations of motion into the quantum theory for four-dimensional fishnet models. It establishes that the symmetry is realized at the classical level once evaluation parameters follow specific patterns. For the bi-scalar version the symmetry does not carry over to most quantum correlation functions, as shown by explicit sums of Feynman graphs and by graphs containing octagon loops. The results point to the non-zero dual Coxeter number as the feature that blocks full quantum invariance and may also limit complete integrability in planar quantum field theories.

Core claim

Yangian invariance is classically realised in the bi-scalar and supersymmetric fishnet 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. Concrete counter-examples are given by sums of Feynman graphs and by bi-scalar graphs with octagon-shaped loops. This finding underlines 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 algebra realized on the classical action and equations of motion via chosen evaluation parameters, obstructed at the quantum level by the non-zero dual Coxeter number, with octagon-loop graphs serving as explicit counter-examples.

If this is right

  • Only specific, well-established classes of correlators remain Yangian invariant at the quantum level.
  • The non-zero dual Coxeter number blocks quantum Yangian symmetry in the bi-scalar model.
  • Complete integrability may be obstructed in planar QFT models that share this feature.
  • The supersymmetric fishnet variant is not shown to escape the same limitation.

Where Pith is reading between the lines

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

  • Analogous obstructions may appear in other planar integrable models whose dual Coxeter number is non-zero.
  • Searches for quantum corrections that could restore invariance become a natural next step.
  • The result supplies a concrete diagnostic for testing candidate integrable deformations of four-dimensional theories.

Load-bearing premise

The evaluation-parameter patterns that produce classical Yangian symmetry do not survive into the quantum regime and no additional corrections cancel the effect of the non-zero dual Coxeter number.

What would settle it

Explicit computation of the action of the Yangian generators on a sum of bi-scalar Feynman graphs containing an octagon loop that returns a non-zero result instead of zero.

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 investigates Yangian symmetry for the equations of motion and action of the classical bi-scalar and supersymmetric fishnet models in four dimensions. It argues that the symmetry is realized subject to specific evaluation parameter patterns. The central claim is that Yangian invariance does not extend to generic quantum correlation functions in the bi-scalar model, supported by explicit counterexamples consisting of sums of Feynman graphs and bi-scalar graphs with octagon-shaped loops. This is tied to the non-zero dual Coxeter number as an obstacle to quantum Yangian symmetry and possibly complete integrability in planar QFTs.

Significance. If the central claim holds, the result is significant for clarifying the boundaries of Yangian symmetry in quantum regimes of models with known classical integrability features. It supplies concrete graph-based evidence that classical symmetries under restricted parameters need not survive quantization. The manuscript earns credit for its explicit symmetry checks on the classical action and equations of motion together with the construction of specific counterexamples via Feynman-graph sums.

major comments (2)
  1. [§5] §5 (quantum correlators and counterexamples): the counterexamples drawn from sums of Feynman graphs and octagon-loop bi-scalar graphs must be evaluated at the same evaluation-parameter patterns that realize classical Yangian symmetry (as defined earlier in the classical analysis). Without explicit confirmation that these patterns are used, the observed non-invariance does not demonstrate that the symmetry 'escapes' the quantum theory; it may simply reflect a parameter choice outside the classically invariant sector.
  2. [§5] §5: the claim that Yangian invariance fails for generic quantum correlators rests on selected counterexamples rather than a general argument or exhaustive classification. While the explicit graphs are valuable, a load-bearing gap remains in extending the conclusion from these cases to the generic statement without additional justification or a broader set of examples.
minor comments (2)
  1. A compact summary table collecting the allowed evaluation-parameter patterns for the classical case would improve readability and make cross-reference to the quantum counterexamples easier.
  2. [Introduction] Notation for the dual Coxeter number and its role could be introduced with a short dedicated paragraph in the introduction for readers less familiar with the algebraic context.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable comments on our manuscript. We have carefully considered the points raised regarding the quantum correlators and counterexamples in §5. Below, we provide point-by-point responses and indicate the revisions made to the manuscript.

read point-by-point responses

  1. Referee: §5 (quantum correlators and counterexamples): the counterexamples drawn from sums of Feynman graphs and octagon-loop bi-scalar graphs must be evaluated at the same evaluation-parameter patterns that realize classical Yangian symmetry (as defined earlier in the classical analysis). Without explicit confirmation that these patterns are used, the observed non-invariance does not demonstrate that the symmetry 'escapes' the quantum theory; it may simply reflect a parameter choice outside the classically invariant sector.

    Authors: We agree with the referee that it is crucial to use the same evaluation parameter patterns for the quantum counterexamples as those that preserve the classical Yangian symmetry. In the original manuscript, this was implicit in the construction of the examples, but we acknowledge that explicit confirmation is necessary. In the revised version, we have added a clear statement in §5 specifying that the counterexamples are computed with the restricted evaluation parameters identified in the classical analysis (specifically, the patterns that make the classical action and equations of motion Yangian invariant). This revision ensures that the non-invariance observed is a genuine quantum effect. revision: yes

  2. Referee: §5: the claim that Yangian invariance fails for generic quantum correlators rests on selected counterexamples rather than a general argument or exhaustive classification. While the explicit graphs are valuable, a load-bearing gap remains in extending the conclusion from these cases to the generic statement without additional justification or a broader set of examples.

    Authors: We appreciate this observation. Our conclusion that Yangian invariance does not extend to generic quantum correlators is indeed supported by explicit counterexamples rather than a general theorem. These counterexamples include sums of Feynman graphs and bi-scalar graphs with octagon-shaped loops, which are representative of the types of correlators that appear in the planar limit of the bi-scalar model. To strengthen the manuscript, we have revised §5 to include additional discussion explaining why the non-zero dual Coxeter number leads to the breaking of the symmetry in these generic cases, and why the known Yangian-invariant classes are special. While a complete classification of all possible correlators is beyond the scope of this work, we believe the provided examples and the underlying symmetry analysis provide sufficient evidence for the claim. revision: partial

Circularity Check

0 steps flagged

Explicit graph evaluations provide independent evidence; no reduction to inputs by construction.

full rationale

The paper conditions classical Yangian realization on specific evaluation parameter patterns and then presents direct counterexamples via explicit sums of Feynman graphs and bi-scalar graphs containing octagon loops to show that invariance fails for generic quantum correlators. These counterexamples are computed from the model Lagrangian and Feynman rules rather than being fitted or renamed from prior results. The dual-Coxeter-number obstruction is invoked as an explanatory remark but does not serve as a load-bearing derivation step that collapses the central claim into a self-citation or self-definition. No equation or argument reduces the claimed non-extension to its own inputs by construction; the derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard properties of Yangian algebras and classical field theory equations; no new free parameters or invented entities are introduced beyond the established fishnet model definitions.

axioms (2)
  • domain assumption Yangian algebra acts on the equations of motion and action of the fishnet models when evaluation parameters follow specific patterns.
    Invoked in the classical analysis section to establish realization of the symmetry.
  • domain assumption Non-zero dual Coxeter number prevents extension of Yangian invariance to generic quantum correlators.
    Used to interpret the counterexamples and link to integrability obstacles.

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Reference graph

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