Singularities of massless scattering and cluster algebras

James Drummond; Lara Bossinger; \"Omer G\"urdo\u{g}an; Ross Glew; Rowan Wright

REVIEW 3 major objections 4 minor 2 cited by

Singularities of massless scattering and cluster algebras

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Cluster algebras of partial flag varieties encode much of the singularity alphabet of massless five- and six-point scattering, with the momentum-twistor embedding doing better at six points.

desk verdict A careful, honest evidence report; the six-point story is postdictive and rests on an unproven tropical truncation, but the framework, five-point checks, and ancillary files make it worth refereeing. read the letter →

arxiv 2507.01015 v1 pith:AKNQN7DT submitted 2025-07-01 hep-th math-phmath.MP

Lara Bossinger , James Drummond , Ross Glew , Ömer Gürdoğan , Rowan Wright This is my paper · ORCID

classification hep-thmath-phmath.MP

keywords clusteralgebraspartialflagvarietiesscatteringamplitudessymbolalphabetsmomentumtwistorsspinorhelicityadjacencytropicalgeometry

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

Massless scattering amplitudes and related observables have a set of possible singularities, captured in a symbol alphabet, and for planar maximally supersymmetric Yang-Mills theory that alphabet is known to come from Grassmannian cluster algebras. This paper asks whether the same is true for observables that lack dual conformal symmetry, and argues that partial flag varieties supply the right cluster algebras: F(2,4;n) for momentum-twistor kinematics and F(2,n-2;n) for spinor-helicity kinematics. At five points both embeddings reproduce the known two-loop pentagon alphabet after permutation completion, and the finite A4 flag explains observed adjacency triples of quadratic letters. At six points the momentum-twistor embedding of F(2,4;6) recovers a larger subset of the 289-letter planar hexagon alphabet than the spinor-helicity embedding, and the flag's limit rays reproduce the square-root letters. A sympathetic reader would care because this suggests a systematic way to predict singularity alphabets for observables without dual conformal symmetry, potentially extending symbol-bootstrap methods beyond the special amplitudes where they already work.

What carries the argument

The central objects are the cluster algebras of partial flag varieties $F(2,4;n)$ and $F(2,n-2;n)$, where Plücker coordinates serve as $A$-coordinates and are interpreted either as momentum twistors ($p_{ijkl}=\langle ijkl\rangle$, $p_{ij}=\langle ij\rangle$) or spinor-helicity variables ($p_{ij}=\langle ij\rangle$, $p_{\widehat{ij}}=\pm[ij]$). For six points the flag algebra $F(2,4;6)$ is embedded in $Gr(4,8)$ by requiring twistor labels 7 and 8 to appear together, and the infinite list of $A$-coordinates is truncated using a tropical construction to 77 flag-compatible variables. The square-root letters come from origin clusters whose repeated mutations obey $z_{n+2}z_n = b f_z F^n + z_{n+1}^2$, with discriminant $\Delta = P_z^2 - 4F_z$ defining the algebraic letters $\varphi_z$ and $\varphi_w$. The argument also relies on the cluster mutation relation, which makes the $X$-coordinate between mutation-related letters the unique rational function appearing between them in integrable symbol words.

What would settle it

Compute the untruncated flag cluster algebra of F(2,4;6) (or its Gr(4,8) embedding) far enough to search for a multiplicative combination of A-coordinates equal to a letter in a missing permutation class such as S6 or S9; the paper reports no such combination within the truncation, so an explicit counterexample would disprove the saturation claim. A second check is whether a letter like W138, which the paper compares to the five-point letter V31, appears in any finite two-loop observable; if it does, the dimensional-regularisation explanation for some missing classes would be weakened.

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Extended reading notes

Core claim

The central claim is that the cluster algebras of partial flag varieties contain the singularity data of non-dual-conformal massless scattering. For five points, both the spinor-helicity flag F(2,3;5) and the momentum-twistor flag F(2,4;5) reproduce the two-loop pentagon alphabet once all permutations are used; the momentum-twistor flag of type A4 also predicts that quadratic letters whose factors form a mutation pair appear one slot apart with a unique X-coordinate between them, matching pentagonal Wilson loop data. For six points both embeddings use the same flag F(2,4;6), realised as a subalgebra of Gr(4,8), and a tropical truncation to 77 flag-compatible cluster variables. The momentum-twistor embedding identifies 54 rational letters of the planar hexagon alphabet, eleven of nineteen permutation classes after permutation completion, whereas the spinor-helicity embedding recovers fewer letters and loses the permutation class S5. The algebraic letters are produced from sixteen origin clusters of F(2,4;6), which share one limit ray and yield the square roots r1,...,r5 after permutation completion. The paper also shows that specific observed triples in the hexagonal Wilson loop, such as W23 ⊗ (W1W7)/(W2W28) ⊗ W10, are realised as mutation pairs in the momentum-twistor embedding.

Load-bearing premise

The six-point comparison assumes that the tropically truncated list of 77 flag-compatible cluster variables is exhaustive: the saturation is a computational observation, not a proof, so a missing cluster variable could change which permutation classes appear recoverable.

Editorial extensions

If this is right

At five points, both $F(2,3;5)$ and $F(2,4;5)$ yield the full two-loop pentagon symbol alphabet after permutation completion, and $F(2,4;5)$ explains the observed one-slot separation of quadratic letters in planar and non-planar data.
At six points, the momentum-twistor embedding of $F(2,4;6)$ reproduces 54 rational letters of the planar hexagon alphabet and, after permutation completion, eleven of nineteen permutation classes; the spinor-helicity embedding reproduces a strictly smaller subset and loses class $S_5$.
All five square roots of the six-point alphabet arise from the single limit ray of $F(2,4;6)$ after permutation completion, and the algebraic letters built from the sixteen origin clusters span the observed algebraic letters involving $r_1$ in one embedding or $r_2$ in the other.
Cluster adjacency in the flag algebras constrains symbol words: quadratic letters that are cluster-incompatible and not mutation partners cannot appear adjacent or one slot apart, and the unique letters between them are $X$-coordinates of mutations.
The $A_4$ cluster-adjacent weight-four polylogarithm basis built from $F(2,4;5)$ is sufficient to express the leading-singularity symbols of the pentagonal Wilson loop with Lagrangian insertion.

Reading between the lines

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

The failure of both embeddings to recover classes $S_6$, $S_7$, $S_9$, $S_{10}$, $S_{11}$, $S_{12}$, $S_{15}$, and $S_{17}$ suggests those letters may require a different cluster algebra or embedding; the exchange-type relation $W_{26}W_{22} = W_1 W_{31} - W_{61}$ noted for one missing triple is a hint that such a structure exists.
A testable extension is that the $D_4$-type prediction for five points, namely that quadratic pairs of Type 3 appear only when separated by two slots, should become visible at weight six or higher, exactly in the unique weight-two function the paper identifies from $F(2,3;5)$.
The observation that the four-particle form factor needs $Gr(4,8)$ variables beyond the 272-variable truncation suggests the tropical truncation is observable-dependent; one could test this by checking whether any higher-loop or non-planar observable requires letters outside the truncated list.
If the limit-ray dictionary is correct, the unrecovered three-loop pentagon letters mentioned in the paper might be identifiable as limit rays of $F(2,4;5)$ or $F(2,4;6)$, extending the flag-cluster prescription to the next loop order.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Punchline: the new content here is the systematic six-point comparison of the two flag embeddings and the observation that certain triples of symbol letters are explained by cluster mutation relations. The structural setup — spinor-helicity and momentum twistor cluster structures on F(2,4;n) and F(2,n−2;n) — was already in Bossinger–Li, Pokraka et al., and Bossinger–Drummond–Glew; the authors are honest about that. What they add is concrete: explicit five-point recoveries, the 19 permutation classes of the 244 rational six-point letters, recovery of 11 classes via the momentum-twistor embedding after permutation completion, the algebraic letters from F(2,4;6) origin clusters matching the square-root letters, and a form-factor test. The ancillary files with per-letter expressions are a real plus.

Credit where due: the cluster algebra is independently grounded in geometry; there are no fitted parameters. The five-point formulas are explicit and checkable, and the A4 adjacency analysis — the 339 vs 370 word count — is a genuine quantitative observation. The paper is well organized and careful about what is evidence versus what is theorem.

Soft spots, in order of severity:

1. The six-point negative statements rest on the saturation claim for the tropically truncated list of 272 Gr(4,8) coordinates. That claim is a computational observation, not a proof. The paper's own 'Comment on unused flag variables' shows the recovery set depends on accidental presence of cofactors: p1378 is unused while its cyclic image p2478 is used, because the partner variable is missing from the truncated list. Worse, the form-factor section shows that in a closely related setup, going beyond the truncated list does recover a missing letter class. So the truncation is not generally sufficient, and the six-point 'misses' may be artifacts. The authors acknowledge this in the conclusions, but it means the headline evidence is weaker than the presentation suggests.

2. The matching procedure is a search over multiplicative combinations of cluster coordinates, with arbitrary permutation completion. There is no baseline for how much matching would occur by chance. The 11/19 class rate is suggestive but uncalibrated. A random or permutation-generous basis of comparable size might do surprisingly well.

3. Everything is postdiction: the alphabets were already computed at two loops. There is no not-yet-computed observable predicted. The authors frame the paper as evidence, which is fair, but the reframing matters for how much weight the reader should give.

Is the central argument sound? As evidence, mostly yes; as a proof of relevance, no. The framework is plausible and the five-point part is solid. The six-point comparison is a useful data point but needs either a larger/complete truncation test, a chance baseline, or a prediction for a new observable.

Who this is for: amplitudes people working on symbol alphabets and cluster bootstrap, especially those interested in non-dual-conformal observables. It deserves a serious referee; the question is important and the claims are concrete and checkable. My recommendation: send to peer review, with a request that the authors either prove or carefully bound the truncation, provide a chance baseline, and avoid overclaiming the negatives.

Referee Report

3 major / 4 minor

Summary. The paper proposes that cluster algebras associated to partial flag varieties encode information about the symbol alphabets of massless scattering amplitudes without dual conformal symmetry. Two embeddings are studied: a momentum-twistor embedding into F(2,4;n), interpreting four-index Plücker coordinates as momentum twistors and two-index Plückers as angle brackets, and a spinor-helicity embedding into F(2,n-2;n), interpreting (n-2)-index Plückers as square brackets. For five points, both embeddings reproduce the known two-loop pentagon alphabet (up to permutation completion), and the momentum-twistor F(2,4;5) algebra is used to interpret certain triples of quadratic letters in existing two-loop data. For six points, the paper uses a tropical truncation of the Gr(4,8) cluster algebra to obtain 77 flag-compatible A-coordinates and reports that the momentum-twistor embedding captures 54 rational letters of the planar hexagon alphabet, rising to eleven of the nineteen permutation classes after permutation completion, while the spinor-helicity embedding captures a subset of the same classes. The paper also matches the five square-root letters and all algebraic letters to origin-cluster/limit-ray constructions in F(2,4;6), and analyzes adjacency relations in a two-loop Wilson-loop correlator. The central claim is presented as evidence, not a theorem.

Significance. If the evidence is sound, the paper extends cluster-algebra technology beyond dual-conformal planar MSYM to broader classes of massless observables, which is a potentially important step for symbol-bootstrap calculations in QCD-like settings. The strength of the paper is that it provides explicit formulas for the recovered letters, includes ancillary files documenting each identification and permutation, and uses no fitted numerical parameters; the partial flag cluster algebras are independently grounded in the geometry of momentum twistors and spinor helicity variables. The main limitations are that the six-point conclusions rely on an unproven tropical truncation, and the matching procedure is retrospective and searches over a very large space of permutations and multiplicative combinations of cluster variables. These issues do not invalidate the positive observations, but they do mean the negative statements about missing classes of letters cannot yet be read as genuine structural exclusions.

major comments (3)

[Section V, tropical truncation and negative claims] The saturation statement at the start of Section V ('generating more A-coordinates by performing many mutations does not yield anything further') is a computational observation, not a proof, and it is load-bearing for the paper's negative conclusions about the missing permutation classes S6, S7, S9, S10, S11, S12, S15, and S17. The paper's own 'Comment on unused flag variables' makes the dependence concrete: the variable p1378 is unused while its cyclic image p2478 is used, solely because a partner coordinate is absent from the truncated list. This shows that the recovery set depends on accidental availability of cofactors in the chosen truncation, so the failures to recover those classes are not yet established as genuine cluster-algebraic exclusions. Please either provide a systematic characterization of the relevant finite set of flag-compatible cluster variables (for example, by specifying the finite set of clusters and mutations explored and making the verification code available), or reformulate the six-point claims as 'not found within this truncation' rather than as failures of the cluster structure.
[Section V.A, recovery counts and permutation completion] The central six-point comparison is retrospective over a large search space: 54 rational letters are identified as multiplicative combinations of variables from a 77-element coordinate set, and all particle-label permutations are then used to complete the result to eleven of nineteen permutation classes. No baseline is given for how many of the 244 rational letters would be recovered by a similarly generous multiplicative closure over a random set of 77 coordinates, or over another subalgebra of comparable size. Without such a calibration, the 11/19 class rate and the 54/244 pre-permutation count are suggestive but do not by themselves establish that the cluster structure, rather than the size and flexibility of the coordinate set, is responsible for the matches. I would ask for a null-model estimate, for example the expected recovery rate under random choices of 77 coordinates with the same multiplicative and permutation closure, or at least a report of the number of distinct multiplicative combinations tested and the maximum degree of the products involved.
[Section IV.B and Section V.C, retrospective triple matching] The abstract states that the cluster structures 'correctly predict the appearance of certain triples of symbol letters related to cluster mutation relations,' but the triples in Eqs. (34)-(42) and in Section V.C are extracted from already computed two-loop data and then matched to a mutation pair in the flag algebra, with an additional permutation on particle labels allowed in the matching step (for example, the permutation {2,4,3,1,5,6} before the W23-W10 triple). This is a retrospective consistency check, not a prediction. I recommend softening the language to 'are consistent with' or 'are organized by,' and stating explicitly that the permutation step was not fixed in advance. This does not reduce the value of the observation, but it removes the appearance that the cluster algebra was used to anticipate the data.

minor comments (4)

[Table II caption] The caption of Table II contains 'presented in []'; the citation [30] should be inserted.
[Section VII, Conclusions] The text refers to 'F(2, n-4; n) with the spinor helicity embedding,' but the embedding was defined earlier as F(2, n-2; n) in Section III.B; this appears to be a typo and should be corrected.
[Section II.C, Example F(2,4;5)] In Eq. (10), the listed active coordinates are written as '{a1, a2, a4, a4}'; this should presumably be '{a1, a2, a3, a4}'.
[Section I, Introduction] The word 'Grasmmannian' in the third paragraph is a typo for 'Grassmannian'.

Circularity Check

2 steps flagged · score 4.0 of 10

Six-point evidence rests on a self-cited tropical truncation, and the 'predicted' triples are retrospective matches to known two-loop data; the cluster algebras themselves are independently defined, so the central claim is not fully circular.

ansatz smuggled in via citation [Section V, six-particle massless scattering, truncation paragraph]
"In particular, a truncated list of 272 active cluster coordinates was found to suffice to describe eight particle scattering at MHV and NMHV when supplemented with the eight frozen variables [17]. Of these 272 variables, 71 treat the points Z7, Z8 as a line and so have a realisation in the flag F(2, 4; 6); we supplement these with the six frozen coordinates of Gr(4, 8) which are compatible with the flag to give a list of 77 cluster variables."

The set of flag variables used for all six-point recovery claims is imported from the authors' own prior tropical-geometry papers [17,31]. Its sufficiency for the six-point alphabet is asserted only by the finite computational saturation statement 'generating more A-coordinates by performing many mutations does not yield anything further,' which is not a proof. Consequently the negative results (failure to recover permutation classes S6, S7, S9-S12, S15, S17) depend on this self-cited truncation. The paper's own 'Comment on unused flag variables' shows the recovered set is sensitive to accidental availability of cofactors (p1378 unused while p2478 used), so the missing classes are conditioned on the truncation rather than established by a complete cluster-algebraic exclusion.
fitted input called prediction [Abstract and Section IV.B.1 (five-point adjacency)]
"We also observe that the associated cluster structures correctly predict the appearance of certain triples of symbol letters related to cluster mutation relations. ... This cannot be realised in F(2 , 3; 5) using the spinor helicity cluster structure... However, in the momentum twistor cluster structure the letters V12 and V15 are related to the following F(2 , 4; 5) letters ..."

The triple V12 - V5V19/V2V4 - V15 is taken from already computed two-loop data [26-29]. The authors first report that the F(2,3;5) spinor-helicity embedding cannot realise it, then find a realisation in the momentum-twistor embedding of F(2,4;5), after which the abstract calls this a correct prediction. Because the embedding and the mutation pair are selected after inspecting the target data (and after permutation completion), the match is a postdiction rather than a derivation that would have singled out this triple beforehand. This is discrete model selection on the target data, although the cluster algebra itself is not constructed from the symbol letters.

full rationale

There is no equation-level circularity in the five-point alphabet recovery: the F(2,3;5) (D4) and F(2,4;5) (A4) cluster algebras are finite and independently defined, and the map from flag Plücker coordinates to spinor brackets is fixed by geometry, not by the alphabet. The six-point algebraic-letter construction from origin clusters and limit rays is also an independent check against the known square-root letters. The main weaknesses are evidentiary rather than definitional: the six-point rational-letter recovery is governed by a tropically truncated list of 272 Gr(4,8) A-coordinates imported from the authors' earlier work [17,31], with only a finite computational saturation check; and the 'predictions' of triples are retrospective matches to two-loop data in which the successful embedding was identified after the failure of the other embedding. These raise the circularity score to 4 because the central six-point claim leans on a self-cited, unproven truncation, but they do not make the derivation equivalent to its inputs, since the cluster structures have independent content and the matching computations are genuine consistency checks.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No fitted numerical parameters appear in the paper; the matching is algebraic. The main imported assumptions are the cluster structure on partial flags, the kinematic identifications, and the tropical truncation of Gr(4,8). No new physical entities are introduced; the square-root letters and limit rays are mathematical constructs from [17].

assumptions (6)

domain assumption Cluster algebra structure exists on partial flag varieties F(d;n) as used here.
Relied on from [2] and [23]; the paper does not prove the cluster structure but uses its A-coordinates.
domain assumption The kinematic identifications p_{ij} = angle bracket and p_{ij...} = momentum twistor or square bracket correctly encode massless scattering kinematics.
Section III defines the two embeddings; correctness is taken from prior work and checked only by examples.
domain assumption Pluecker relations on the flag varieties encode Schouten identities and momentum conservation.
Used in Section III.B for F(2,3;5); imported from [24].
ad hoc to paper The tropical truncation of Gr(4,8) to 272 A-coordinates, 77 of which are used for F(2,4;6), is sufficient.
Section V imports this finite list from [17,31]; saturation is asserted from computation, not proven.
domain assumption Symbol letters are expressible as multiplicative combinations of cluster A-coordinates and frozen variables.
Standard cluster bootstrap assumption from [15] and [5]; the paper searches for such combinations.
ad hoc to paper Completing recovered letters under all particle-label permutations is legitimate because different flag orientations are related by permutations.
Section IV.A uses permutation completion to reach the full alphabet; this enlarges the search space and weakens predictivity.

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Cite this review

Pith. "Pith review of Singularities of massless scattering and cluster algebras." pith.science (2026). https://pith.science/paper/AKNQN7DT

@misc{pith2026250701015,
  author       = {Pith},
  title        = {Pith review of: Singularities of massless scattering and cluster algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AKNQN7DT}},
  note         = {Machine review of arXiv:2507.01015}
}

read the original abstract

Partial flag varieties arise in the context of massless scattering kinematics. They can be associated to both spinor-helicity variables and momentum twistor variables in two separate yet natural ways. Here we report on evidence at five and six points that the cluster algebras associated to these partial flag varieties contain information relevant to non-dual conformal massless scattering amplitudes and related observables. At five points both spinor-helicity and momentum twistor cluster structures capture similar information about symbol alphabets. At six points we demonstrate that the momentum twistor structure captures a larger subset of the alphabet compared to the spinor-helicity one. We also observe that the associated cluster structures correctly predict the appearance of certain triples of symbol letters related to cluster mutation relations.

Figures

Figures reproduced from arXiv: 2507.01015 by the authors.

**Figure 2.** FIG. 2. The initial cluster of Gr(3 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3. A choice of initial cluster for the flag F(2 [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: FIG. 4. The flag F(2 [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

**Figure 5.** Figure 5: FIG. 5. One of the 64 origin clusters associated with discrim [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

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Reference graph

Works this paper leans on

58 extracted references · 44 canonical work pages · cited by 2 Pith papers

[1]

Alphabet from F(2, 3; 5) As we previously noted, the partial flag variety F(2, n− 2; n) in the case n = 5 is F(2 , 3; 5) which is isomorphic to the Gr(3; 6) cluster algebra. This finite cluster al- gebra of D4 mutation type endows us with a total of 22 cluster variables: 20 minors of the form pijk with i, j, k∈ {1, 2, 3, 4, 5, 6} (and where we interpet pi...

work page
[2]

Alphabet from F(2, 4; 5) We may alternatively investigate the embedding of five particle scattering in the partial flag F(2 , 4; n), which in the present case is F(2 , 4; 5), a cluster algebra of finite A4 mutation type with 20 cluster coordinates distributed across 42 clusters. Interpreting two-index Pl¨ ucker coordinates as angle brackets and four-index...

work page
[3]

( Qi,j;k,l;m, Qi,k;j,m;l) does not appear in any two- loop data, but strikingly this is precisely the type of pair of quadratics mutually identifiable under some permutation inside F(2, 3; 5) using the spinor helic- ity embedding of the kinematics. Since we would expect such quadratics to be separated by at least two slots in the symbol, and we only have ...

work page
[4]

Johannes M. Henn. Multiloop integrals in dimen- sional regularization made simple. Physical Review Letters, 110(25), June 2013. ISSN 1079-7114. doi: 10.1103/physrevlett.110.251601. URL http://dx.doi. org/10.1103/PhysRevLett.110.251601

doi:10.1103/physrevlett.110.251601 2013
[5]

Written in terms of spinor brackets, letters V6 to V21 from the non-planar two-loop five-point alphabet are quadratic

Adjacency predictions from F(2, 4; 5) Given that the F(2, 3; 5) and F(2, 4; 5) cluster algebras are different, but enjoy identical success in generating the symbol alphabet for five point massless scattering, it is interesting to investigate to what extent either cluster algebra can make predictions about adjacency relations at five points. Written in ter...

work page
[6]

Each quadratic letter is of the form Qi,j;k,l;m = si,j − sk,l (37) for {i, j, k, l, m} = {1, 2, 3, 4, 5}

Classifying pairs of quadratic letters Let us consider in more detail the classes of quadratic letters which may appear separated in the same word. Each quadratic letter is of the form Qi,j;k,l;m = si,j − sk,l (37) for {i, j, k, l, m} = {1, 2, 3, 4, 5}. Note that we indeed have 15 quadratics under permutation completion since these enjoy the symmetry (up ...

work page
[7]

These are the pairs of quadratics which appear separated by one slot in presently computed two-loop planar data, e.g

( Qi,j;k,l;m, Qi,j;k,m;l). These are the pairs of quadratics which appear separated by one slot in presently computed two-loop planar data, e.g. the pentagonal Wilson loop with Lagrangian operator insertion [28], and planar QCD [29]. Such triples are also seen in non-planar data e.g. N = 8 su- pergravity [26], non-planar MSYM [27], including variations us...

work page
[8]

These appear separated by one slot in non-planar data, with a unique combi- nation sitting between

( Qi,j;k,l;m, Qi,k;j,l;m). These appear separated by one slot in non-planar data, with a unique combi- nation sitting between. For instance, V6 ⊗ V3V19 V4V17 ⊗ V12 (41) is a triple observed in both non-planar MSYM [27] and N = 8 supergravity [26]. This does not have an interpretation inside F(2, 4; 5), where there is not a permutation under which V6 and V...

work page

Show all 58 references

[9]

Cluster algebras i: Foundations, 2001

Sergey Fomin and Andrei Zelevinsky. Cluster algebras i: Foundations, 2001. URL https://arxiv.org/abs/math/ 15 0104151

2001
[10]

Adjacency relations in integrable triples and quadruples and beyond Since the available two-loop data is only of weight four, with stringent restrictions on the first entry and (conjec- turally) the second entry, it is instructive to probe the structure of integrable triples (...
[11]

Five of the algebraic letters are these naked discriminants

Algebraic letters in the planar hexagon alphabet The six point planar alphabet contains 45 algebraic letters containing five distinct square roots r1 to r5. Five of the algebraic letters are these naked discriminants. The square roots are given by r1+i ≡ C iq s2 12 + s2 34 + s...
[12]

An example of an origin cluster is shown in Figure 5

A review of Gr(4, 8) origin clusters and limit rays Although Gr(4, 8) is an infinite cluster algebra, it is of finite mutation type, with infinitely many A-coordinates specifically arising from origin clusters . An example of an origin cluster is shown in Figure 5. 10 In [17],...
[13]

(18) The map in the other direction, from four-index Pl¨ ucker coordinates to square and angle brackets, is sim- ilarly straightforward

= 1 p31 p6134 p61 p34 − p6123 p61 p23 − p1234 p12 p34 . (18) The map in the other direction, from four-index Pl¨ ucker coordinates to square and angle brackets, is sim- ilarly straightforward. Since we concern ourselves here 5 with the scattering of at most six particles, we c...
[14]

The only difference is that there are now only 16 different origin clusters, each of which asymptotes to the same ray

Algebraic letters from origin clusters in F(2, 4; 6) The case of F(2, 4; 6) is very similar to that of Gr(4, 8) given that the former can be embedded in the latter as a cluster subalgebra. The only difference is that there are now only 16 different origin clusters, each of whi...
[15]

This class seems to have a realisation in the flag, specifi- cally using the momentum twistor embedding

W23 ⊗ W1W7 W2W28 ⊗ W10 and cyclic permutations. This class seems to have a realisation in the flag, specifi- cally using the momentum twistor embedding. For example, if one interchanges the particle labels un- der the permutation {2, 4, 3, 1, 5, 6}, W23 has a fac- tor which mu...
[16]

Since W61 is one of the letters which we fail to reproduce in either embedding of the kinematics, we cannot hope to realise this inside F(2 , 4; 6)

W26 ⊗ W1W31 W61 ⊗ W22 and cyclic permutations. Since W61 is one of the letters which we fail to reproduce in either embedding of the kinematics, we cannot hope to realise this inside F(2 , 4; 6). However, it is interesting to note that these letters do obey the exchange-type r...
[17]

Express these in terms of the letters of the planar pentagon al- phabet

Take homogeneous combinations of the cluster vari- ables associated to the partial flag variety F(2, 4; 5), and construct all weight-four integrable words which obey A4 cluster adjacency. Express these in terms of the letters of the planar pentagon al- phabet
[18]

Complete the weight-four integrable words under arbitrary permutations on the particle momenta, to obtain a complete basis of cluster-adjacent weight four integrable words coming from any orientation of F(2, 4; 5)
[19]

Test whether the symbols of the coefficients of the Wilson loop correlator may be expressed as a linear combination of these basis elements. Doing so, we were able to express the symbols of the coefficients of the leading singularities for the pentago- nal Wilson loop with Lag...
[20]

Summer School in Total Positivity and Quantum Field Theory

to reproduce the two-loop planar pentagon alphabet using cyclic permutations only, it would be interesting 13 to investigate whether any different truncations of the Gr(4, 8) alphabet could yield candidates for the missing three-loop pentagon letters, and whether the square ro...

2024
[21]

Tropicalised braid automorphisms of grassmannian cluster algebras

James Drummond. Tropicalised braid automorphisms of grassmannian cluster algebras. Amplitudes 2023, 2023. URL https://cds.cern.ch/record/2867520

2023
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Cluster structures on spinor helicity and momentum twistor varieties

Lara Bossinger and Jian-Rong Li. Cluster structures on spinor helicity and momentum twistor varieties. 8 2024

2024
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Symbol alphabets in qcd and flag cluster algebras, 2025

Andrzej Pokraka, Marcus Spradlin, Anastasia Volovich, and He-Chen Weng. Symbol alphabets in qcd and flag cluster algebras, 2025. URL https://arxiv.org/abs/ 2506.11895

2025
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Bossinger, J

L. Bossinger, J. M. Drummond, and R. Glew. Adjacency for scattering amplitudes from the Gr¨ obner fan, 2023. URL https://arxiv.org/abs/2212.08931

2023 arXiv
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Dixon, James M

Lance J. Dixon, James M. Drummond, and Johannes M. Henn. Bootstrapping the three-loop hexagon. Journal of High Energy Physics , 2011(11), November 2011. ISSN 1029-8479. doi:10.1007/jhep11(2011)023. URL http:// dx.doi.org/10.1007/JHEP11(2011)023

2011 doi
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Dixon, James M

Lance J. Dixon, James M. Drummond, and Johannes M. Henn. Analytic result for the two-loop six-point nmhv amplitude in N = 4 super yang-mills theory. Journal of High Energy Physics , 2012(1), January 2012. ISSN 1029-8479. doi:10.1007/jhep01(2012)024. URL http:// dx.doi.org/10.1...

2012 doi
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Here we review the recovery of the symbol letters using both F(2 , n− 4; n) and F(2, 4; n) in the case n = 5

[28] [29]. Here we review the recovery of the symbol letters using both F(2 , n− 4; n) and F(2, 4; n) in the case n = 5
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Dixon, James M

Lance J. Dixon, James M. Drummond, Claude Duhr, and Jeffrey Pennington. The four-loop remainder function and multi-regge behavior at nnlla in planar N = 4 super- yang-mills theory. Journal of High Energy Physics , 2014 (6), 2014. ISSN 1029-8479. doi:10.1007/jhep06(2014)116. UR...

2014 doi
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Cluster adjacency and the four-loop nmhv heptagon

James Drummond, Jack Foster, ¨Omer G¨ urdo˘ gn, and Georgios Papathanasiou. Cluster adjacency and the four-loop nmhv heptagon. Journal of High Energy Physics, 2019(3), March 2019. ISSN 1029-8479. doi: 10.1007/jhep03(2019)087. URL http://dx.doi.org/10. 1007/JHEP03(2019)087

2019 doi
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Joshua S. Scott. Grassmannians and cluster algebras,
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Tropical grassmannians, clus- ter algebras and scattering amplitudes

James Drummond, Jack Foster, ¨Omer G¨ urdo˘ gan, and Chrysostomos Kalousios. Tropical grassmannians, clus- ter algebras and scattering amplitudes. Journal of High Energy Physics , 2020(4), April 2020. ISSN 1029-8479. doi:10.1007/jhep04(2020)146. URL http://dx.doi.org/ 10.1007/...

2020 doi
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Henn, and Georgios Pap- athanasiou

Dmitry Chicherin, Johannes M. Henn, and Georgios Pap- athanasiou. Cluster algebras for feynman integrals. Phys- ical Review Letters , 126(9), March 2021. ISSN 1079-

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Dixon, ¨Omer G¨ urdo˘ gan, Yu-Ting Liu, An- drew J

Lance J. Dixon, ¨Omer G¨ urdo˘ gan, Yu-Ting Liu, An- drew J. McLeod, and Matthias Wilhelm. An- tipodal Self-Duality for a Four-Particle Form Fac- tor. Phys. Rev. Lett. , 130(11):111601, 2023. doi: 10.1103/PhysRevLett.130.111601

2023 doi
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The other rational letters missed in in the momentum twistor embedding are all permutations of this letter, and so the entire rational alphabet is recovered

based on all Pl¨ ucker coordinates and their parity im- ages which include quadratic letters. The other rational letters missed in in the momentum twistor embedding are all permutations of this letter, and so the entire rational alphabet is recovered. Note that it is not entir...
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An excep- tional cluster algebra for Higgs plus jet production, 2025

Rigers Aliaj and Georgios Papathanasiou. An excep- tional cluster algebra for Higgs plus jet production, 2025. URL https://arxiv.org/abs/2408.14544

2025 arXiv
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For six particles, both partial flag varieties turn out to be F(2 , 4; 6)

Note1. For six particles, both partial flag varieties turn out to be F(2 , 4; 6). However the maps between cluster coordinates and kinematics are very different
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J. S. Scott. Grassmannians and Cluster Algebras. Proc. Lond. Math. Soc. III Ser. , 92:345–380, 2006. doi: 10.1112/S0024611505015571

2006 doi
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Paulos, Marcus Spradlin, and Anastasia Volovich

John Golden, Miguel F. Paulos, Marcus Spradlin, and Anastasia Volovich. Cluster polylogarithms for scattering amplitudes, 2014. URL https://arxiv.org/abs/1401. 6446

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Dixon, James Drummond, Thomas Har- rington, Andrew J

Lance J. Dixon, James Drummond, Thomas Har- rington, Andrew J. McLeod, Georgios Papathanasiou, and Marcus Spradlin. Heptagons from the stein- mann cluster bootstrap. Journal of High Energy Physics, 2017(2), February 2017. ISSN 1029-8479. doi: 10.1007/jhep02(2017)137. URL http:...

2017 doi
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Algebraic singularities of scat- tering amplitudes from tropical geometry, 2021

James Drummond, Jack Foster, ¨Omer G¨ urdogan, and Chrysostomos Kalousios. Algebraic singularities of scat- tering amplitudes from tropical geometry, 2021

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How tropical are seven- and eight-particle amplitudes? Journal of High Energy Physics , 2020(8), August 2020

Niklas Henke and Georgios Papathanasiou. How tropical are seven- and eight-particle amplitudes? Journal of High Energy Physics , 2020(8), August 2020. ISSN 1029-

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Singulari- ties of eight- and nine-particle amplitudes from cluster algebras and tropical geometry

Niklas Henke and Georgios Papathanasiou. Singulari- ties of eight- and nine-particle amplitudes from cluster algebras and tropical geometry. Journal of High Energy Physics, 2021(10), October 2021. ISSN 1029-8479. doi: 10.1007/jhep10(2021)007. URL http://dx.doi.org/10. 1007/JHE...

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Bourjaily, Andrew J

Jacob L. Bourjaily, Andrew J. McLeod, Marcus Spradlin, Matt von Hippel, and Matthias Wilhelm. Ellip- tic double-box integrals: Massless scattering ampli- tudes beyond polylogarithms. Physical Review Let- ters, 120(12), March 2018. ISSN 1079-7114. doi: 10.1103/physrevlett.120.1...

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Introduction to cluster algebras

Sergey Fomin, Lauren Williams, and Andrei Zelevinsky. Introduction to cluster algebras. chapters 1-3, 2024. URL https://arxiv.org/abs/1608.05735

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Cluster Adjacency Properties of Scattering Ampli- tudes in N = 4 Supersymmetric Yang-Mills The- ory

James Drummond, Jack Foster, and ¨Omer G¨ urdo˘ gan. Cluster Adjacency Properties of Scattering Ampli- tudes in N = 4 Supersymmetric Yang-Mills The- ory. Phys. Rev. Lett. , 120(16):161601, 2018. doi: 10.1103/PhysRevLett.120.161601

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Dmitry Chicherin, Johannes Henn, and Vladimir Mitev. Bootstrapping pentagon functions. Journal of High En- ergy Physics , 2018(5), May 2018. ISSN 1029-8479. doi: 10.1007/jhep05(2018)164. URL http://dx.doi.org/10. 1007/JHEP05(2018)164

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Dixon, Enrico Herrmann, Ben Page, and Mao Zeng

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Pith tools

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

[1] [1]

Alphabet from F(2, 3; 5) As we previously noted, the partial flag variety F(2, n− 2; n) in the case n = 5 is F(2 , 3; 5) which is isomorphic to the Gr(3; 6) cluster algebra. This finite cluster al- gebra of D4 mutation type endows us with a total of 22 cluster variables: 20 minors of the form pijk with i, j, k∈ {1, 2, 3, 4, 5, 6} (and where we interpet pi...

work page

[2] [2]

Alphabet from F(2, 4; 5) We may alternatively investigate the embedding of five particle scattering in the partial flag F(2 , 4; n), which in the present case is F(2 , 4; 5), a cluster algebra of finite A4 mutation type with 20 cluster coordinates distributed across 42 clusters. Interpreting two-index Pl¨ ucker coordinates as angle brackets and four-index...

work page

[3] [3]

( Qi,j;k,l;m, Qi,k;j,m;l) does not appear in any two- loop data, but strikingly this is precisely the type of pair of quadratics mutually identifiable under some permutation inside F(2, 3; 5) using the spinor helic- ity embedding of the kinematics. Since we would expect such quadratics to be separated by at least two slots in the symbol, and we only have ...

work page

[4] [4]

Johannes M. Henn. Multiloop integrals in dimen- sional regularization made simple. Physical Review Letters, 110(25), June 2013. ISSN 1079-7114. doi: 10.1103/physrevlett.110.251601. URL http://dx.doi. org/10.1103/PhysRevLett.110.251601

doi:10.1103/physrevlett.110.251601 2013

[5] [5]

Written in terms of spinor brackets, letters V6 to V21 from the non-planar two-loop five-point alphabet are quadratic

Adjacency predictions from F(2, 4; 5) Given that the F(2, 3; 5) and F(2, 4; 5) cluster algebras are different, but enjoy identical success in generating the symbol alphabet for five point massless scattering, it is interesting to investigate to what extent either cluster algebra can make predictions about adjacency relations at five points. Written in ter...

work page

[6] [6]

Each quadratic letter is of the form Qi,j;k,l;m = si,j − sk,l (37) for {i, j, k, l, m} = {1, 2, 3, 4, 5}

Classifying pairs of quadratic letters Let us consider in more detail the classes of quadratic letters which may appear separated in the same word. Each quadratic letter is of the form Qi,j;k,l;m = si,j − sk,l (37) for {i, j, k, l, m} = {1, 2, 3, 4, 5}. Note that we indeed have 15 quadratics under permutation completion since these enjoy the symmetry (up ...

work page

[7] [7]

These are the pairs of quadratics which appear separated by one slot in presently computed two-loop planar data, e.g

( Qi,j;k,l;m, Qi,j;k,m;l). These are the pairs of quadratics which appear separated by one slot in presently computed two-loop planar data, e.g. the pentagonal Wilson loop with Lagrangian operator insertion [28], and planar QCD [29]. Such triples are also seen in non-planar data e.g. N = 8 su- pergravity [26], non-planar MSYM [27], including variations us...

work page

[8] [8]

These appear separated by one slot in non-planar data, with a unique combi- nation sitting between

( Qi,j;k,l;m, Qi,k;j,l;m). These appear separated by one slot in non-planar data, with a unique combi- nation sitting between. For instance, V6 ⊗ V3V19 V4V17 ⊗ V12 (41) is a triple observed in both non-planar MSYM [27] and N = 8 supergravity [26]. This does not have an interpretation inside F(2, 4; 5), where there is not a permutation under which V6 and V...

work page