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Gromov-Witten invariants of log Calabi-Yau 3-folds are holomorphic lagrangian correspondences

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Pith's one-line read Log Calabi–Yau 3-fold Gromov–Witten invariants are holomorphic lagrangian correspondences.

desk verdict A serious new framework that may reorganize log Gromov-Witten theory, but the headline claim about log Calabi-Yau 3-folds is proved only in the exploded category, with the bridge to the log virtual class asserted rather than demonstrated. read the letter →

arxiv 2506.20092 v1 pith:KZBRLAFM submitted 2025-06-25 math.AG math-phmath.MPmath.SG

Brett Parker This is my paper · ORCID
classification math.AGmath-phmath.MPmath.SG MSC 14N3553D1214J32
keywords Gromov-WitteninvariantslogCalabi-Yau3-foldslagrangiancorrespondencesholomorphicsymplecticcategorystarproductexplodedmanifoldsrefinedcohomologyDonaldson-Thomas
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

Gromov–Witten invariants of log Calabi–Yau 3-folds are usually infinite collections of numbers obtained by integrating closed forms over a virtual fundamental class. This paper claims that these numbers can be packaged into a single geometric object: a rational holomorphic lagrangian correspondence, that is, a half-dimensional holomorphic cycle in a product of holomorphic symplectic evaluation spaces. Theorem 5.11 states that the pushforward of the virtual fundamental class under the evaluation map is canonically represented by such a correspondence, so integrating any closed refined form over the correspondence reproduces the Gromov–Witten integral. If this is right, the invariants acquire a canonical chain-level virtual fundamental class, and the tropical gluing formula becomes composition of correspondences in a holomorphic symplectic category. The same language then gives a precise conjectural bridge to Donaldson–Thomas invariants through a unitary lagrangian correspondence.

What carries the argument

The central object is the holomorphic lagrangian correspondence and its composition law, the star product. A lagrangian correspondence from a holomorphic symplectic manifold $X$ to $Y$ is a half-dimensional holomorphic subvariety of $(X,-\omega_X)\times(Y,\omega_Y)$ with proper projection to $X$; the star product $r_1\star_X r_2=\pi_*\iota^!(r_1\times r_2)$ composes two such cycles through fibre product, pullback, and pushforward. In the logarithmic and exploded settings the product is implemented with log Chow rings or refined differential forms, and it is the associative, chain-level composition that will play the role of the tropical gluing formula. The complementary mechanism is the residue evaluation space: the holomorphic volume form on a log Calabi–Yau 3-fold induces, by residues at marked points, holomorphic symplectic forms $\omega_\nu$ on evaluation stacks, and the sum of the pullbacks of these forms vanishes on every holomorphic curve family. That vanishing is why the moduli image is isotropic and, in dimension three, half-dimensional.

What would settle it

Compute the lagrangian cycle from both sides in an explicit example with non-toric blowups, for instance the wall $z_3=0$ example with contact data $(\pm n,0)$, and compare the coefficient of $\{z_3=0\}$ and of the components $E_\pm$ with Equation (25). The theorem predicts the same coefficients from the exploded and logarithmic virtual fundamental classes for every $n$; one $n$ where the two coefficients differ would refute it.

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Extended reading notes

Core claim

For a compact log smooth Calabi–Yau 3-fold $(X,D,\Omega)$, fix genus, curve class, and contact data $\nu(1),\dots,\nu(m)$. The paper constructs evaluation spaces and stacks $X_{\nu(k)}$ carrying holomorphic symplectic forms obtained as residues of $\Omega$, and proves that the evaluation image $\operatorname{ev}(\mathcal M(X))$ of the moduli stack of stable log curves is algebraic, logarithmically proper, and isotropic. Because the virtual dimension equals $m$, half the dimension of the evaluation space, the pushforward of $[\mathcal M(X)]$ is a half-dimensional cycle. Theorem 5.11 asserts that there is a unique rational holomorphic lagrangian correspondence $$\eta \in \operatorname{Lag}\bigl(\prod_k X_{\nu(k)}\bigr)\otimes \mathbb Q$$ supported on $\operatorname{ev}(\mathcal M(X))$ such that for every closed refined form $\theta$ on any sufficiently small refinement neighbourhood, $\int_\eta \theta = \int_{[\mathcal M(X)]} \operatorname{ev}^*\theta$. This is what the paper means by saying that Gromov–Witten invariants of log Calabi–Yau 3-folds are canonically encoded by holomorphic lagrangian correspondences.

Load-bearing premise

The whole statement rests on the identification of the counting cycle built from exploded manifolds with the counting cycle used in logarithmic Gromov–Witten theory, made canonically and independent of logarithmic modifications; if that identification fails, the constructed correspondence would not describe the log invariants.

Editorial extensions

If this is right

  • Log Gromov–Witten invariants of Calabi–Yau 3-folds acquire a canonical chain-level virtual fundamental class, independent of the auxiliary perturbations used to construct it.
  • The tropical gluing formula for log Gromov–Witten invariants is expressed by composition, or star product, of lagrangian correspondences, making gluing a categorical operation.
  • Restricting the correspondences to interiors gives numerical invariants independent of logarithmic modifications.
  • The Gromov–Witten partition function is an exponential $\exp(\eta)$ in a ring of lagrangian cycles, so disconnected curve counts become products in that ring.
  • The conjectured relationship with sheaf counting says that, after the change of variables $q^{1/2}=ie^{i\hbar/2}$, the Gromov–Witten and Donaldson–Thomas cycles are related by the unitary lagrangian correspondence $L$.

Reading between the lines

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

  • The chain-level presentation suggests that the lagrangian weights of $\eta$ could be computed by intersecting with test submanifolds pointwise on moduli spaces, yielding an explicit algorithm in toric examples; the paper does not spell out such an algorithm.
  • Remark 5.1 on boundary conditions indicates the same residue construction could define real lagrangian correspondences for holomorphic curves with Lagrangian boundary, a setting outside log Gromov–Witten theory as usually formulated.
  • If the conjectured GW/DT identity holds, the unitary correspondence $L$ would turn a Donaldson–Thomas partition function into the Gromov–Witten partition function by a single move in the category, so integrality properties of curve counts would follow from the unitarity identity $L^\dagger\star L=\Delta$.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper proposes a holomorphic analogue of Weinstein's symplectic category and uses it to encode Gromov-Witten invariants of log Calabi-Yau 3-folds. Sections 2-3 develop lagrangian correspondences and a 'star product' composition law, first for smooth algebraic spaces and then for log schemes; Section 4 switches to exploded manifolds and refined cohomology, where the star product is associative and Poincare duals of lagrangian cycles exist. Section 5 constructs a holomorphic symplectic evaluation space from residues of the volume form, proves in Proposition 5.8 that the evaluation image of the moduli stack of stable exploded curves is logarithmically proper and isotropic, and states as Theorem 5.11 that for each choice of contact data there is a unique rational holomorphic lagrangian correspondence eta representing integration against the exploded-manifold virtual fundamental class. The paper closes with generating functions and conjectural GW/PT/DT correspondences via a unitary lagrangian correspondence to Hilbert schemes.

Significance. If valid, the main theorem is significant: it would upgrade numerical log Gromov-Witten invariants to a canonical chain-level object, make the tropical gluing formula a composition law in a holomorphic Weinstein category, and give a precise conjectural GW/DT correspondence. The proof has genuine substance: the residue computations in Lemmas 5.3-5.5, the isotropy statement, the refined-cohomology star product, and the uniqueness argument in Theorem 5.11 are worked out in detail. The Gromov-Witten/Donalson-Thomas conjectures are clearly separated from the theorems, and the worked integrality examples are informative. The main caveat is that the headline statement for log Calabi-Yau 3-folds depends on a comparison between the exploded virtual fundamental class and the Gross-Siebert log virtual fundamental class that is asserted rather than proved.

major comments (2)
  1. [§5.2, Theorem 5.11 and Remark 5.9] The theorem as stated does not establish Theorem 1.1 for log Gromov-Witten invariants in the sense of Gross-Siebert. The construction of eta in Theorem 5.11 integrates against the exploded virtual fundamental class from [34, Definition 4.7], while the Gross-Siebert log virtual fundamental class from [9] is never shown to push forward to the same cycle. Proposition 5.7 proves only that the exploded family contains every isomorphism class of stable exploded curve; a bijection on isomorphism classes does not determine virtual weights on boundary strata, and evaluation integrals can be sensitive to those weights. Remark 5.9 concedes this gap: it says 'morally speaking [M(X')] is a logarithmic modification of [M(X)], [2]', but [2] proves birational invariance of log Gromov-Witten invariants, not equality of the exploded and log virtual cycles, nor independence of the choice of logarithmic modification at the chain level. The abstract and Theorem 1.1 should either be restricted to exploded log Calabi-Yau 3-folds or the proof must supply the missing comparison.
  2. [§5.2, Proposition 5.8] The reduction to contact data with a single divisor also relies on the same unproved comparison. The proof says 'We take X' to be a refinement...' and later 'Assume that each vector nu(k)... This can be achieved by taking a further log modification', but the virtual fundamental class of the original log moduli stack is not shown to be the image of the virtual fundamental class of the modified stack. Isotropy and logarithmic properness for the original evaluation image can be recovered by Lemma 5.6 and Corollary 4.7, but the statement of Theorem 5.11 concerns the pushforward of the virtual fundamental class, and that pushforward is only computed after passing to a modification. Since Proposition 5.7 does not identify virtual fundamental classes, this step is load-bearing for the claimed lagrangian-cycle theorem for general contact data.
minor comments (5)
  1. [§1, after Theorem 1.1] The phrase 'canonical chain level virtual fundamental class' is stronger than what is constructed in §5.2, where eta is a cycle representing integration against the virtual fundamental class rather than a virtual fundamental chain on the moduli space; please clarify the intended meaning of 'chain level'.
  2. [§3.1, Corollary 3.20] The statement of the star product via the log Chow ring assumes that a single logarithmic modification M makes both pi_1^* ell_1 and pi_2^* ell_2 intersect the divisor nicely; the proof in Lemma 3.19 indicates how this is achieved, but adding a one-sentence justification in the corollary would help the reader.
  3. [§5.1, Eq. (19)] The vector field partial_theta_nu used to characterise omega_nu is not explicitly normalized; specifying its normalization with respect to the C*t[0,infinity) action would make Lemma 5.3 and equation (19) easier to check.
  4. [§5.3, restriction to tropical part 0] The passage from exploded stacks to schemes by taking points with tropical part 0 is described in one sentence; since this restriction is used to state Conjecture 5.14, a brief explanation of why the lagrangian property is preserved under this operation would be useful.
  5. [Throughout] There are numerous small typographical issues: 'lagrangian' is used with inconsistent capitalization, 'Zarski' should be 'Zariski', 'Laurant' should be 'Laurent', and the reference [3] contains garbled author names; these should be cleaned up in revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: eta is a representation of the virtual fundamental class, not a fitted input; the exploded/log comparison is an unproved bridge, not a circular reduction.

full rationale

The construction of eta in Theorem 5.11 is a representation theorem: eta is defined as the unique lagrangian cycle satisfying ∫_eta θ = ∫_[M] ev*θ for all closed forms θ in a refined neighbourhood of ev(M(X)). The coefficients of eta are not fitted parameters; they are determined by integrating the virtual fundamental class against test forms (Poincare duals of transverse submanifolds), and Lemma 5.10 establishes independence of the choice of virtual perturbation. The isotropy of the support is proved in Proposition 5.8 from Lemma 5.5 (sum of residues vanishes), where the symplectic form ω_ν is defined from the Calabi-Yau volume form by a residue pushforward; this is a mathematical identity, not an assumption of the conclusion. The paper relies on the author's earlier exploded-manifold machinery ([27], [34]) and on Gross-Siebert's log moduli stack ([9]) as external results with stated hypotheses; citing them is not circular. The only serious gap is stated by the paper itself in Remark 5.9: 'Morally speaking [M(X′)] is a logarithmic modification of [M(X)], [2]' — here [2] is birational invariance in log Gromov-Witten theory, not an identification of the Gross-Siebert virtual class with the exploded virtual class. That is an unproved comparison on which the log-Calabi-Yau wording of Theorem 1.1 depends, but it is a correctness/completeness concern, not a circular reduction: no equation in the paper forces the exploded VFC to equal the log VFC by definition. Hence no circular step is present.

Assumptions & free parameters 0 free parameters · 4 assumptions · 2 invented entities

No numerical parameters are fitted to data anywhere in the paper. The axioms are standard results in algebraic geometry plus the author's prior exploded-manifold technology, on which the main theorem depends. The invented entities are mathematical structures introduced by the paper; the unitary correspondence is conjectural and its role in GW/DT is unverified.

assumptions (4)
  • domain assumption The exploded-manifold machinery of [24]-[34] is correct: refined differential forms, Kuranishi structures, virtual fundamental class, compactness, and the comparison between exploded and log curves.
    Theorem 5.11 is proved in the exploded setting; the paper's central claim about log Calabi-Yau 3-folds depends on this machinery via Proposition 5.7, Lemma 5.10, and the references in Section 5.2.
  • domain assumption The moduli stack of basic stable log curves in a proper log smooth scheme is a proper Deligne-Mumford stack with a virtual fundamental class, per Gross-Siebert [9].
    Used in Proposition 5.7 and Remark 5.9 to identify the exploded curve moduli with the log curve moduli.
  • standard math Resolution of singularities holds (Hironaka [11]; log resolution [1, Theorem 1.2.4]).
    Used in Proposition 4.4 to obtain explodable resolutions and in Remark 3.5 to represent subvarieties as images of log smooth domains.
  • standard math Fulton's intersection theory and the refined intersection product satisfy the compatibilities (8)-(10) used for the associativity of the star product.
    Section 2.1 and Lemma 2.20 depend on these formal properties of Chow groups and Borel-Moore homology.
invented entities (2)
  • Holomorphic Weinstein category
    purpose: Categorical framework whose morphisms are holomorphic lagrangian correspondences; used to encode Gromov-Witten invariants.
    New mathematical framework introduced in Sections 2-3. No external falsifiable handle; its validity is exactly the content of the paper's theorem.
  • Unitary lagrangian correspondence L between the disjoint union of Y_mu and Y[n] (and its relative versions)
    purpose: Conjectural bridge between Gromov-Witten and Donaldson-Thomas invariants in Conjecture 5.14.
    Defined in Example 2.29 using known Hilbert scheme geometry; the conjectural GW/DT relation is not proven, so the entity has no independent confirmation.

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Pith. "Pith review of Gromov-Witten invariants of log Calabi-Yau 3-folds are holomorphic lagrangian correspondences." pith.science (2026). https://pith.science/paper/KZBRLAFM

@misc{pith2026250620092,
  author       = {Pith},
  title        = {Pith review of: Gromov-Witten invariants of log Calabi-Yau 3-folds are holomorphic lagrangian correspondences},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KZBRLAFM}},
  note         = {Machine review of arXiv:2506.20092}
}
read the original abstract

We introduce a holomorphic version of Weinstein's symplectic category, in which objects are holomorphic symplectic manifolds, and morphisms are holomorphic lagrangian correspondences. We then extend this category to log schemes, and prove that Gromov-Witten invariants of log Calabi-Yau 3-folds are naturally encoded as holomorphic lagrangian correspondences. Gromov-Witten invariants and Donaldson-Thomas invariants are then conjecturally related by a natural unitary lagrangian correspondence.

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