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arxiv: 2604.27079 · v1 · submitted 2026-04-29 · 🧮 math.SG

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Holomorphic disks and tropical Lagrangians

Chris T. Woodward

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Pith reviewed 2026-05-07 08:49 UTC · model grok-4.3

classification 🧮 math.SG
keywords pseudoholomorphic diskstropical Lagrangiansalmost toric manifoldsquantum multiplicationfirst Chern classLagrangian isotopyholomorphic pants
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The pith

Tropical graphs provide a calculus for counting pseudoholomorphic disks bounded by Lagrangians in almost toric manifolds.

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

The paper develops a combinatorial method to count pseudoholomorphic disks with boundaries on tropical Lagrangians in almost toric manifolds by summing over interacting tropical graphs. This extends previous tropical enumeration techniques from spheres in toric varieties to the disk case in more general symplectic settings. Central to the method are explicit calculations of vertex multiplicities for disk configurations, including the holomorphic pant and univalent vertices, obtained through a specific Lagrangian isotopy in the del Pezzo surface of degree seven. The approach yields concrete results in dimension four under monotonicity assumptions. One application shows that all integer eigenvalues of non-maximal modulus for quantum multiplication by the first Chern class arise from holomorphic spheres constructed this way.

Core claim

We develop a calculus for counting pseudoholomorphic disks with boundary in tropical Lagrangians contained in almost toric manifolds, given as a sum over tropical graphs that interact with the tropical graph of the Lagrangian. The main contribution is the calculation of several multiplicities of vertices corresponding to disks, such as the holomorphic pant and various univalent vertices occurring at trivalent vertices of the graph of the Lagrangian, using a Lagrangian isotopy from the Lagrangian pair of pants in the del Pezzo of degree seven to the inverse image of a diagonal. We show that every integer eigenvalue of non-maximal modulus for quantum multiplication by the first Chern class is

What carries the argument

The sum over tropical graphs interacting with the Lagrangian's tropical graph, with computed multiplicities for disk vertices including the holomorphic pant.

If this is right

  • The calculus generalizes Mikhalkin-Nishinou-Siebert sphere counts and prior disk counts on moment fibers.
  • Explicit vertex multiplicities enable computation of disk invariants in almost toric four-manifolds.
  • All integer non-maximal eigenvalues of quantum multiplication by the first Chern class are realized by holomorphic spheres from the construction.
  • The technique extends in principle to higher dimensions and non-monotonic cases.

Where Pith is reading between the lines

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

  • This combinatorial approach could enable algorithmic calculations of quantum invariants for specific almost toric examples.
  • The isotopy between the Lagrangian pants and the diagonal inverse image might extend to other Lagrangian submanifolds for simplifying counts.
  • Connections to tropical geometry suggest potential links with mirror symmetry computations in these manifolds.

Load-bearing premise

The counting results rely on dimension four and monotonicity assumptions for the Lagrangians, although the underlying technique is asserted to work more generally.

What would settle it

A mismatch between the tropical graph sum prediction and an independent computation of the number of holomorphic disks bounding a tropical Lagrangian in a specific almost toric four-manifold, such as a del Pezzo surface.

Figures

Figures reproduced from arXiv: 2604.27079 by Chris T. Woodward.

Figure 1
Figure 1. Figure 1: Graphs of Manin configurations in del Pezzo’s of excep￾tional type X Manin Root System ≈ Spec(c1⋆) P 2 3α, α3 = 1 P 1 × P 1 A1 4, 0 ⊕2 , −4 Bl1 P 2 −0.33, 3.8, −2.23 ± 1.94√ −1 Bl2 P 2 A1 (−1)⊕2 , 4.73, −2.86 ± 0.94√ −1 Bl3 P 2 A1 ⊕ A2 (−2)⊕3 ,(−3)⊕2 , 6 Bl4 P 2 A4 (−3)⊕5 , 8.09, −3.09 Bl5 P 2 D5 (−4)⊕7 , 12 Bl6 P 2 E6 (−6)⊕8 , 21 Bl7 P 2 E7 (−12)⊕9 , 52 Bl8 P 2 E8 (−60)⊕10 , 372 view at source ↗
Figure 2
Figure 2. Figure 2: ; in the depiction we have overlayed the dual polyhedral complex with the original polyhedral complex; otherwise the pictures become difficult to visualize. The left picture in view at source ↗
Figure 3
Figure 3. Figure 3: The holomorphic pant (right in blue) and its tropical graph (left in blue) The broken maps are required to satisfy matching conditions at the edges. In our previous work [35], we performed an additional degeneration so that the diagonal condition at the edges was deformed to split form. We show in Theorem 3.33 below that the moduli space MΓ(L) of broken maps with type Γ is a product MΓ(L) ∼= Y v∈Vert(Γ) MΓ… view at source ↗
Figure 4
Figure 4. Figure 4: Tropical graph for a holomorphic cylinder (b) (Holomorphic pairs of pants) As in Mikhalkin [25], the multiplicity m(v) = | det(T (e1)T (e2))| for trivalent vertices v ∈ Vert•(Γ) with adjacent edges e1, e2, e3 ∈ Edge(Γ) whose edge directions (exactly two of which are incoming) satisfying the balancing condition (2) T (e1) + T (e2) + T (e3) = 0; (c) (Disks meeting the toric boundary) As in Cho-Oh [5], the mu… view at source ↗
Figure 5
Figure 5. Figure 5: Tropical graph for a holomorphic pair of pants view at source ↗
Figure 6
Figure 6. Figure 6: Tropical graph for collisions with the boundary (d) (Multiple covers near focus-focus singularities) As in Bryan-Pandharipande [3], the multiplicity m(v) = (−1)ℓ(µ)−1 /ℓ(µ) 2 for the univalent closed vertices v ∈ Vert•(Γ) at the focus-focus singularities, where ℓ(µ) ∈ Z>0 is the lattice length of the direction µ = T (e) ∈ tP(v),Z of the adjacent edge e ∈ Edge(Γ); that is, µ = ℓ(µ)µ0 where µ0 is the primiti… view at source ↗
Figure 7
Figure 7. Figure 7: Tropical graph for collisions with the focus-focus values view at source ↗
Figure 8
Figure 8. Figure 8: Tropical graph for collisions with Lagrangian (f) (Holomorphic pant, Lagrangian with direction (±1, 1)) The multiplicity m(v) = (−1)| det(T (e◦)T (e•))|−1/2 | det(T (e◦)T (e•))| = 1 2 (−1)|2 det(T (e•)r(T (e•))|−1 | det(T (e•)r(T (e•))| for bivalent vertices v ∈ Vert◦(Γ) mapping to an edge ϵ of L with open edge e◦ = (1, 1) ⊂ ϵ and closed edges e•, e′ • with directions T (e•) = (x, y), r(T (e•)) = T (e ′ • … view at source ↗
Figure 9
Figure 9. Figure 9: Tropical graph for a holomorphic pant view at source ↗
Figure 10
Figure 10. Figure 10: Cartoon diagrams for disks with boundary in a La￾grangian pair of pants with a single strip-like end (h) (Two-ended strips in Lagrangian pants) The multiplicity m(v) = 1 for bivalent vertices v ∈ Vert◦(Γ) mapping to trivalent vertices ℓ ∈ Vert(Λ) with edge directions T (e1), T (e2) of adjacent edges e1 ∈ Edge◦ (Γ), e2 ∈ Edge◦ (Γ) with directions T (e1) = T (ϵ1), T (e2) = T (ϵ2) ∈ tP(v) equal to the direct… view at source ↗
Figure 11
Figure 11. Figure 11: Cartoon diagrams for disks with boundary in a La￾grangian pair of pants with two or three strip-like ends for trivalent vertices v ∈ Vert◦(Γ) with adjacent edges e1, e2, e3 having direc￾tions T (e1), T (e2), T (e3) ∈ tP(v) equal to the directions of the adjacent edges ϵ1, ϵ2, ϵ3 ∈ Edge(Λ) of L with a point constraint p ∈ L or a point constraint at a Reeb chord at one of the strip-like ends. (Recall that t… view at source ↗
Figure 12
Figure 12. Figure 12: Cartoon diagrams of Maslov-index-two disks in the de￾gree six del Pezzo Indeed, there are two rigid tropical graphs Γ1, Γ2 ⊂ Φ(X) representing Maslov￾index-two disks shown on the right-most two figures in blue in view at source ↗
Figure 13
Figure 13. Figure 13: Tropical graphs of Maslov-index-two disks in the degree seven del Pezzo Let p3 be a constraint with moment image Φ(p3) = (0, ϵ), ϵ > 0. There are two tropical graphs Γ1, Γ2 (shown in the upper part of view at source ↗
Figure 14
Figure 14. Figure 14: Cartoon diagrams of Maslov-index-two disks in the de￾gree six del Pezzo Consider first the case that the point constraint is on the lower-left leg with di￾rection (1, 1). In this case, there are two tropical graphs Γ1, Γ2 shown with two univalent vertices, each with contribution m(Γ1) = m(Γ2) = −1, and two tropical graphs Γ3, Γ4 shown with two univalent vertices and one bivalent vertex. The direc￾tion of … view at source ↗
Figure 15
Figure 15. Figure 15: 3 graphs each with multiplicity -1 for a total of -3 1 graphs multiplicity -1/2 for a total of 3/2 1 graphs multiplicity -1/2 for a total of 3/2 3 graphs each with multiplicity -1 for a total of -3 3 graphhs multiplicity -1/2 for a total of -3/2 3 graphhs multiplicity -1/2 for a total of -3/2 view at source ↗
Figure 16
Figure 16. Figure 16: A focus-focus singularity and one of the Lagrangian van￾ishing thimbles holomorphic disk holomorphic disk Lagrangian disk Lagrangian disk view at source ↗
Figure 17
Figure 17. Figure 17: Holomorphic and Lagrangian disks near a focus-focus singularity The two-forms ωI = g(·, I·), ωJ = g(·, J·), ωK = g(·, K·) ∈ Ω 2 (X) are symplectic, and the hyper-K¨ahler rotation (X, g, I, J, K) 7→ (X, g, J, I, −K) interchanges special Lagrangians with respect to the form ωI with holomorphic sub￾manifolds with respect to I as in Strominger-Yau-Zaslow [33]. Explicitly, let X = C 2 = {(q1 + ip1, q2 + ip2)} … view at source ↗
Figure 18
Figure 18. Figure 18: Tropical graphs of tropical Lagrangian spheres in Bl2 P 2 Definition 2.7. For any tropical graph Λ with a trivalent vertex ν ∈ Vert(Λ), define the local self-intersection number δ(ν) = 1 2 (m(ν) − 1) where m(ν) is the determinant from Definition 1.5 (b). Choose a metric on B and for δ > 0 let Uδ(Vert(Λ)) ⊂ B denote an δ-neighborhood around the vertices Vert(Λ). 2We differ from Mikhalkin [27] in that we do… view at source ↗
Figure 19
Figure 19. Figure 19: Graphs of Manin configurations of tropical Lagrangians in del Pezzo surfaces Example 2.15. Let X be the blow-up of P 1 ×P 1 at a torus-fixed point corresponding to a vertex of the moment polytope Φ(X) = [0, 1]2 . There is a Lagrangian sphere L ⊂ X whose tropical graph Λ ⊂ R 2 is the trivalent graph shown on the right in view at source ↗
Figure 20
Figure 20. Figure 20: Behavior of tropical Lagrangians under nodal slides view at source ↗
Figure 21
Figure 21. Figure 21: Mutation of a Manin configuration in the del Pezzo of degree one Proposition 2.18. Let X = Bl2 P 2 = Bl1 (P 1 × P 1 ) equipped with a monotone sym￾plectic form. There is a unique Hamiltonian isotopy class of embedded Lagrangian spheres in X containing the inverse image L of the diagonal in P 1 × P 1 . In partic￾ular, the Lagrangian L ′ whose tropical graph Λ is the trivalent graph shown in the last diagram in view at source ↗
Figure 22
Figure 22. Figure 22: An isotopy of tropical Lagrangian spheres in Bl2 P 2 (a) each P is locally a simple rational convex polyhedra with respect to the affine structure on B − Bfoc (b) each face of P ∈ P is also in P, and (c) the intersection of any two elements of P is either empty or a face of each. Definition 3.2. A polyhedral decomposition P is admissible for X if each element P ∈ P is a simple polytope that is transverse … view at source ↗
Figure 23
Figure 23. Figure 23: Dual complex for the degree one del Pezzo The results in Venugopalan-Woodward [35] on the limits of holomorphic disks under neck-stretching extend to Lagrangian boundary conditions under translation￾invariance conditions on the Lagrangians near the facets of the polytope. Definition 3.3. (Partial translation actions on the necks) For each P ∈ P, identify a neighborhood of Φ−1 (P) × {0} in XP with a neighb… view at source ↗
Figure 24
Figure 24. Figure 24: A polyhedral decomposition of an almost toric diagram and cartoon diagrams of three Maslov-two broken disks bounding a moment fiber Definition 3.11. An open-closed map type Γ is an open-closed tropical graph Γ with vertices v ∈ Vert(Γ) labelled by homotopy classes [uv] of disks uv (for open vertices v) or spheres [uv] (for closed vertices v). Denote by MΓ(X, L) the moduli space of maps of type Γ. Definiti… view at source ↗
Figure 25
Figure 25. Figure 25: The moduli space as a product of moduli spaces for its pieces Definition 3.35. Let u = (uv)v∈Vert(Γ) be a broken map with tropical graph Γ. Denote the number of interior leaves d•(Γ) ∈ Z≥0 of the graph Γ. Let γ(u) ∈ {±1} denote the orientation sign defined using the relative spin structure. The unbroken holomorphic disk count with constraints Y is given as the sum over rigid maps u (20) m(L, Y ) = X Γ X u… view at source ↗
Figure 26
Figure 26. Figure 26: Curves with boundary in the anti-diagonal in P 1 × P 1 disks and spheres by [38, Proposition 5.4.6]. The constraints on the disk translate to constraints on the sphere v(z•) = p1, v(z•) = p2, v(z◦) = p Complex conjugation on P 1 is orientation reversing, and so maps the dual class [p1] to p1 to minus the dual class −[p1] of p1 in H2 (P 1 ). We claim that the count of holomorphic spheres v with these three… view at source ↗
Figure 27
Figure 27. Figure 27: Cartoon diagrams of Maslov-index-four disks bounding tropical Lagrangians Proof. An isotopy of the given tropical Lagrangian to the inverse image of the diag￾onal was described in Example 2.19. We first consider the possibilities for the inter￾section number with the exceptional divisor. Let Y = (p1, p2, p3) with p1, p2, p3 ∈ L generic. The moduli space M(L, Y ) consists of rigi Maslov-index-four disks u … view at source ↗
Figure 28
Figure 28. Figure 28: Curves with boundary in the anti-diagonal in P 1 × P 1 Proof of Theorem 1.4 case (e). The case of primitive directions was proved in Lemma 4.11. To prove the claim for arbitrary directions, we consider the case of holomor￾phic disks u in X = Bl(P 1 × P 1 ) with a tangency at order d ∈ Z≥0 at a point z ∈ C mapping to the exceptional divisor E. These are naturally in bijection with disks u ′ with tangency o… view at source ↗
Figure 29
Figure 29. Figure 29: Cartoon diagrams for broken disks in the del Pezzo of degree two • The tropical graphs Γ1 corresponding to Maslov-index-two disks bounding L1 have initial direction (0, ±1) and interact with 2 out of the 4 possible focus-focus singularities, for a total of WL1 = 2(−1)6 = −12. • For the tropical Lagrangians L2 with vertical graph Λ2 the possible initial directions of the tropical graphs Γ′ 2 , Γ ′′ 2 corre… view at source ↗
Figure 30
Figure 30. Figure 30: The holomorphic pant and a capping disk Remark 4.15. A holomorphic disk with boundary in the Lagrangian pair of pants as in Theorem 1.4 case Definition 1.5 (f) can be described explicitly as follows. Consider a holomorphic pair of pants u : C = P 1 → P 1 × P 1 . Define u(z) = (z, 1 − z) of so that u has bidegree (1, 1). Let u −1 (∆) denote the inverse image of the anti￾diagonal L = {z2 = z1}. We have u −1… view at source ↗
Figure 31
Figure 31. Figure 31: Decomposing into rectangles view at source ↗
Figure 32
Figure 32. Figure 32: Polyhedral decomposition computing the disks bound￾ing a Lagrangian pair of pants Remark 4.19. For the standard complex structure, the Lagrangian pair of pants in Lemma 4.18 above can be described explicitly as follows. Consider the holomorphic pair of pants H = {1 + z1 + z2 = 0} in C 2 , and let L ⊂ R 4 be its Lagrangian hyper-K¨ahler rotation. The diagonal ∆ = {(z, z), z ∈ C} is divided into two pieces … view at source ↗
Figure 33
Figure 33. Figure 33: A disk bounding a Lagrangian pair of pants Proof. We recall the model problem in Lemma 4.10: For the trivalent graph shown in view at source ↗
Figure 34
Figure 34. Figure 34: Cartoon diagrams for disks bounding a Lagrangian sphere in Bl2 P 2 view at source ↗
Figure 35
Figure 35. Figure 35: A polyhedral decomposition P on Bl8 P 2 defined as the intersection of Pann and Pin. Lemma 4.22. For any ρ0 > 0, there exists a real number T > 0 so that the number of tropical graphs Γ for gluing data ρ > ρ0 of broken Maslov-index-two disks u = (uv) is at most T. Proof. By monotonicity, the set of broken Maslov-index-two map has energy, hence area, bounded by some number E0. On the other hand, suppose th… view at source ↗
Figure 36
Figure 36. Figure 36: Curves hitting the boundary and continuing in a pro￾jected direction Proof. The proof is a computation with intersection multiplicities. Let u : C → X be a map with such a graph Γ. Let X′ = P 2 equipped with a toric birational equivalence with X, and u ′ : C → X′ the map given by composing with the birational equivalence. The map u ′ intersects the boundary of X′ exactly twice. The image is therefore a li… view at source ↗
Figure 37
Figure 37. Figure 37: Removing collisions on the interior by increasing ρ edge direction resulting from such a collision has positive resp. negative projection onto the first component. Since the two types of edges never intersect, there is no connected graph which involves combinations of these edges, which is a contradic￾tion. Repeating the process for each graph Γ (without tropical structure) produces a constant ρ0 so that … view at source ↗
Figure 38
Figure 38. Figure 38: Cartoon diagrams of Maslov-index-two disks in the de￾gree seven del Pezzo vertices v ∈ Vert◦(Γ) mapping to an edge ϵ of L with open edges e◦, e′ ◦ ⊂ ϵ and closed edge e• with directions satisfying the balancing condition T (e ′ ◦ ) + T (e◦) + T (e•) + T (r(e•)) = 0 view at source ↗
Figure 39
Figure 39. Figure 39: The half-trident graph The tropical graph Γ is half of a tropical graph Γ with a four-valent vertex ˜ v˜ ∈ Vert( ˜Γ) that corresponds to a holomorphic sphere. As explained in [35], for holo￾morphic sphere counts, four-valent vertices v ∈ Vert(Γ), |v| = 4 can be be avoided by choosing the constraints generically, so after perturbation two three-valent vertices v1, v2 ∈ Vert(Γ) appear rather than a single f… view at source ↗
Figure 40
Figure 40. Figure 40: Cartoon diagrams of Maslov-index-two disks in the de￾gree six del Pezzo bounding (top) the monotone Lagrangian torus (middle) a Lagrangian sphere meeting two fixed points (bottom) the Lagrangian sphere meeting three fixed points Example 1.8. The del Pezzo of degree two was treated in Example 4.12. We show view at source ↗
Figure 41
Figure 41. Figure 41: Cartoon diagrams for disks bounding a Lagrangian sphere in Bl8 P 2 References [1] Paul Biran and Octav Cornea. Lagrangian cobordism. I. J. Amer. Math. Soc., 26(2):295–340, 2013. [2] Kenneth Blakey, Soham Chanda, Yuhan Sun, and Chris T. Woodward. Augmentation varieties and disk potentials i, 2024. [3] Jim Bryan and Rahul Pandharipande. The local Gromov-Witten theory of curves. J. Amer. Math. Soc., 21(1):10… view at source ↗
read the original abstract

We develop a calculus for counting pseudoholomorphic disks with boundary in tropical Lagrangians contained in almost toric manifolds, using our previous work with Venugopalan. The results are mostly in dimension four under monotonicity assumptions although in principle the same technique works in any dimension and without monotonicity. The calculus is given as a sum over tropical graphs that interact with the tropical graph of the Lagrangian, generalizing results of Mikhalkin and Nishinou-Siebert for holomorphic spheres in toric varieties, and our previous result with Venugopalan which dealt with disks bounding almost toric moment fibers. The main contribution of this paper is the calculation of several multiplicities of vertices corresponding to disks, such as the holomorphic pant (half of the holomorphic pair of pants) and various univalent vertices occuring at trivalent vertices of the graph of the Lagrangian; a key tool is a Lagrangian isotopy from the Lagrangian pair of pants in the del Pezzo of degree seven to the inverse image of a diagonal, which is a special case of a results of Hind and Evans. We show that every integer eigenvalue of non-maximal modulus for quantum multiplication by the first Chern class is realized by such a sphere.

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 manuscript develops a tropical calculus for counting pseudoholomorphic disks with boundary on tropical Lagrangians in almost toric manifolds, expressed as a sum over tropical graphs interacting with the Lagrangian graph. This generalizes Mikhalkin/Nishinou-Siebert sphere counts and the authors' prior work with Venugopalan on disks bounding almost toric moment fibers. The central technical contribution is the explicit computation of vertex multiplicities (including the holomorphic pant and univalent vertices at trivalent Lagrangian points) via a Lagrangian isotopy in the degree-7 del Pezzo surface from the pair-of-pants to the inverse image of a diagonal, invoking a special case of Hind-Evans results. The main theorem asserts that every integer eigenvalue of non-maximal modulus for quantum multiplication by the first Chern class is realized by a sphere obtained from this counting procedure. The results are presented primarily in dimension four under monotonicity assumptions.

Significance. If the multiplicity calculations hold, the work supplies an explicit geometric mechanism for realizing eigenvalues of quantum multiplication by c1 via holomorphic spheres, extending tropical methods to Lagrangian settings in almost toric geometry. The isotopy-based computation of specific vertex multiplicities (holomorphic pant and univalent vertices) is a concrete strength that could facilitate further applications to quantum cohomology invariants.

major comments (2)
  1. [the section on Lagrangian isotopy and vertex multiplicity calculation] The transfer of disk multiplicities through the Lagrangian isotopy in the degree-7 del Pezzo (invoked to compute the holomorphic pant and univalent vertex contributions) requires explicit verification that the isotopy induces a bijection on the relevant pseudoholomorphic disks and that the almost-toric almost-complex structures introduce no extraneous contributions not captured by the tropical graphs. Without this, the multiplicity values feeding the sum-over-tropical-graphs formula are not guaranteed to be correct, undermining the eigenvalue realization claim.
  2. [the statement of the main theorem and the preceding multiplicity computations] The headline statement that every integer eigenvalue of non-maximal modulus is realized by such a sphere rests on the computed multiplicities entering the generalized Mikhalkin/Nishinou-Siebert formula; the manuscript must confirm that the tropical graph enumeration exhausts all contributions under the monotonicity and dimension-four hypotheses, with no missing or overcounted terms from the almost toric structure.
minor comments (2)
  1. The abstract and introduction could more clearly distinguish the new multiplicity calculations from the results that follow directly from prior work with Venugopalan.
  2. Notation for the tropical graphs and their interaction with the Lagrangian graph would benefit from an additional illustrative figure or explicit low-degree example.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying points where additional clarification would strengthen the presentation. We address each major comment below and have revised the manuscript to incorporate explicit verifications of the isotopy and the completeness of the tropical enumeration under the stated hypotheses.

read point-by-point responses

  1. Referee: [the section on Lagrangian isotopy and vertex multiplicity calculation] The transfer of disk multiplicities through the Lagrangian isotopy in the degree-7 del Pezzo (invoked to compute the holomorphic pant and univalent vertex contributions) requires explicit verification that the isotopy induces a bijection on the relevant pseudoholomorphic disks and that the almost-toric almost-complex structures introduce no extraneous contributions not captured by the tropical graphs. Without this, the multiplicity values feeding the sum-over-tropical-graphs formula are not guaranteed to be correct, undermining the eigenvalue realization claim.

    Authors: We agree that the transfer of multiplicities via the isotopy requires a more explicit argument. In the revised manuscript we have added a dedicated paragraph in the relevant section that invokes the precise statement from Hind-Evans (the special case for the inverse image of a diagonal) to establish a bijection between the moduli spaces of pseudoholomorphic disks before and after the isotopy. Because the isotopy is through almost-toric Lagrangians and the almost-complex structure is chosen to be compatible with the toric fibration throughout the family, the maximum principle precludes disks that escape the almost-toric neighborhood; hence no extraneous contributions arise outside the tropical graphs. The vertex multiplicities for the holomorphic pant and the univalent vertices are therefore correctly transferred and may be used in the sum-over-graphs formula. revision: yes

  2. Referee: [the statement of the main theorem and the preceding multiplicity computations] The headline statement that every integer eigenvalue of non-maximal modulus is realized by such a sphere rests on the computed multiplicities entering the generalized Mikhalkin/Nishinou-Siebert formula; the manuscript must confirm that the tropical graph enumeration exhausts all contributions under the monotonicity and dimension-four hypotheses, with no missing or overcounted terms from the almost toric structure.

    Authors: We accept the need to make the exhaustion argument fully explicit. Under the monotonicity and dimension-four hypotheses the Maslov index is strictly positive for non-constant disks, and the almost-toric almost-complex structure forces every pseudoholomorphic disk (or sphere) to tropicalize to a graph in the base whose vertices lie on the Lagrangian graph. The revised proof of the main theorem now contains a short lemma showing that the tropicalization map is surjective onto the set of graphs appearing in the sum and that each such graph arises from a unique stable map in the Gromov compactification; multiplicities match by the vertex computations already performed. Consequently there are neither missing nor overcounted terms, and the eigenvalue realization follows directly from the generalized Mikhalkin/Nishinou-Siebert count. revision: yes

Circularity Check

1 steps flagged

Minor self-citation to prior tropical counting framework; central multiplicity calculations and eigenvalue realizations remain independent

specific steps
  1. self citation load bearing [Abstract]
    "We develop a calculus for counting pseudoholomorphic disks with boundary in tropical Lagrangians contained in almost toric manifolds, using our previous work with Venugopalan. ... The calculus is given as a sum over tropical graphs that interact with the tropical graph of the Lagrangian, generalizing results of Mikhalkin and Nishinou-Siebert for holomorphic spheres in toric varieties, and our previous result with Venugopalan which dealt with disks bounding almost toric moment fibers."

    The general counting formula and its application to eigenvalue realization are adopted from the author's prior self-cited work; while new multiplicities are computed separately, the framework itself is self-referential at the level of the sum-over-graphs method used to produce the spheres realizing the eigenvalues.

full rationale

The derivation generalizes a sum-over-tropical-graphs formula from Mikhalkin/Nishinou-Siebert and the author's prior work with Venugopalan to count disks bounding tropical Lagrangians. The load-bearing new step is the explicit computation of holomorphic pant and univalent vertex multiplicities via a Lagrangian isotopy in the degree-7 del Pezzo to the inverse image of a diagonal (special case of Hind-Evans). These multiplicities are fed into the formula to realize the integer eigenvalues of quantum multiplication by c1. The self-citation supports only the general counting method and is not load-bearing for the specific geometric computations or the final eigenvalue statement, which rest on independent isotopy-based input rather than reducing to fitted parameters or prior outputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated beyond the monotonicity assumption and reliance on prior isotopy results.

axioms (1)
  • domain assumption monotonicity assumptions
    Invoked to obtain results in dimension four; no further justification supplied in abstract.

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Reference graph

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