{"id":"b4e8b1cc-bb80-4772-8bc6-cf6c59827b34","arxiv_id":"2506.20092","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Log Gromov-Witten invariants of Calabi-Yau 3-folds are canonically represented by holomorphic lagrangian cycles in a new Weinstein-style category, with a conjectural unitary bridge to Donaldson-Thomas invariants.","lead":"This paper builds a holomorphic version of Weinstein's symplectic category and proves that Gromov-Witten invariants of log Calabi-Yau 3-folds are encoded as holomorphic lagrangian correspondences. It also conjectures a natural unitary correspondence linking these invariants to Donaldson-Thomas invariants.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The exploded/log virtual-class comparison in §5.2 is asserted rather than proved, so Theorem 5.11 may hold only for exploded invariants.","rationale":"The reader identifies the weakest assumption as the comparison between the exploded-manifold virtual fundamental class and the log virtual fundamental class. My reading agrees: the proof of Theorem 5.11 and the surrounding text in §5.2 never establish that comparison. Proposition 5.7 is a statement about isomorphism classes of curves, and Lemma 5.10 is a statement about independence within the exploded construction. Theorem 5.11 therefore establishes a canonical lagrangian correspondence associated to exploded Gromov–Witten invariants. The additional step needed for Theorem 1.1, namely that this correspondence computes log Gromov–Witten invariants, is asserted in Remark 5.9 with the word 'morally' and a citation to birational invariance, not proved. This is a real obstruction to the advertised claim, but it does not invalidate the substantial construction in the exploded category. The paper already makes a strong, testable claim in the exploded setting; the appropriate verdict remains CONDITIONAL, pending either a full proof of the virtual-class comparison or a precise restatement of the main theorem as an exploded-manifold theorem. I would leave the reader's conditional verdict unchanged.","tokens_in":51006,"tokens_out":6375,"duration_ms":76278,"concrete_test":"Take X = (CP^3, toric boundary) and compute the genus-0, degree-1 invariant with one marked point of contact vector (1,1,1). Compute ev_*[M(X)] using the Gross–Siebert log virtual fundamental class [9] and compare it with the η obtained from the exploded virtual fundamental class [34] via Proposition 5.7. If the two agree for all closed forms θ in rH^2(X_{(1,1,1)}), the bridge is substantiated; if they differ, Theorem 5.11 must be read as a theorem about exploded-manifold invariants. The comparison should also be repeated for a base F0 → M(X) that is not representable by a scheme, since Proposition 5.7 uses a non-strict log base.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.1 claims log Gromov–Witten invariants of log Calabi–Yau 3-folds are encoded by holomorphic lagrangian correspondences. The construction of η in Theorem 5.11, however, uses only the exploded-manifold virtual fundamental class from [34, Definition 4.7] and the independence statement Lemma 5.10. Nothing in the proof integrates against the Gross–Siebert log virtual fundamental class [9, §4]. For Theorem 1.1 to be a theorem about log Gromov–Witten invariants, the exploded virtual class and the log virtual class must push forward to the same lagrangian cycle after the comparison of Proposition 5.7. Proposition 5.7 proves only that an explodable family contains every isomorphism class of stable exploded curve; it does not identify the virtual fundamental cycles. Remark 5.9 concedes this gap: it says 'morally speaking [M(X′)] is a logarithmic modification of [M(X)], [2]', but [2] is birational invariance in log Gromov–Witten theory, not an identification of the Gross–Siebert class with the exploded virtual class. Since evaluation integrals can be sensitive to virtual weights on boundary strata, a bijection on isomorphism classes of curves does not imply equality of the resulting η classes. Thus the central claim is load-bearing on an unproved comparison between two virtual fundamental classes.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":51337,"tokens_out":6534,"duration_ms":72930,"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":[{"comment":"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.","section":"§5.2, Theorem 5.11 and Remark 5.9"},{"comment":"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.","section":"§5.2, Proposition 5.8"}],"minor_comments":[{"comment":"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'.","section":"§1, after Theorem 1.1"},{"comment":"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.","section":"§3.1, Corollary 3.20"},{"comment":"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.","section":"§5.1, Eq. (19)"},{"comment":"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.","section":"§5.3, restriction to tropical part 0"},{"comment":"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.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper's central claim rests on a chain of the author's own exploded-manifold foundations [25]-[34]. I did not independently verify all of these foundations, and the main risk is the missing comparison between the exploded virtual fundamental class and the Gross-Siebert log virtual fundamental class. If the author can supply a proof that ev_*[M_log] equals the class eta constructed in Theorem 5.11, the result would be very strong; without that comparison, the paper proves a theorem about exploded manifolds rather than the stated theorem about log Calabi-Yau 3-folds."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper is worth your attention. It builds a holomorphic Weinstein category with lagrangian correspondences, defines a workable star product, extends it to log schemes and exploded manifolds, and proves a definite theorem: the exploded-manifold virtual fundamental class for log Gromov-Witten invariants pushes forward to a canonical rational lagrangian cycle eta in a holomorphic symplectic evaluation space (Theorem 5.11). That is a real result, new as far as I know, and it supplies a chain-level replacement for the usual numerical invariants. The residue construction of the evaluation form and the isotropy argument are carefully done. The unitary correspondence L to Hilbert schemes of points is a nice addition, and the conjectural GW/DT statement is coherent.\n\nThe soft spot is exactly where the stress-test put it. Theorem 1.1 claims log Calabi-Yau 3-folds, but the proof of Theorem 5.11 integrates only against the exploded virtual fundamental class from [34]. Proposition 5.7 shows the exploded family contains every isomorphism class of stable exploded curve; it does not identify the Gross-Siebert log virtual cycle with the exploded one. Remark 5.9 says the log class is 'morally' a logarithmic modification and cites birational invariance, which is not the same as equality of virtual cycles. So the advertised statement about log Gromov-Witten invariants rests on an unproved comparison. I don't see a sign that the comparison is false; it may well be true. But it is load-bearing, and the paper should prove it or restrict the main theorem to the exploded category and label the log statement a conjecture.\n\nThe reliance on the author's own exploded-manifold machinery is heavy but legitimate; the machinery is published and the citations are to actual papers, not placeholders.\n\nFor whom is this? Anyone working in logarithmic Gromov-Witten theory, enumerative geometry, or the symplectic category will get value from the framework even if the gap bothers them. The paper deserves a serious referee; I would send it out. But I would ask the author to fix the scope mismatch before publication.","headline":"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.","tokens_in":51792,"tokens_out":1736,"would_cite":true,"duration_ms":18043,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14N35","53D12","14J32"],"pacs":[],"model":"deepseek-v4-flash","headline":"Log Calabi–Yau 3-fold Gromov–Witten invariants are holomorphic lagrangian correspondences.","keywords":["Gromov-Witten invariants","log Calabi-Yau 3-folds","lagrangian correspondences","holomorphic symplectic category","star product","exploded manifolds","refined cohomology","Donaldson-Thomas invariants"],"falsifier":"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.","tokens_in":50720,"feed_emoji":"📐","tokens_out":12330,"duration_ms":123294,"temperature":0.7,"pith_summary":"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.","feed_headline":"Calabi–Yau 3-fold counts become lagrangian correspondences","feed_subtitle":"If the proof holds, every log Gromov–Witten invariant is a canonical chain-level cycle and gluing is composition.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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 $L^\\dagger\\star L=\\Delta$."],"forward_implications":["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 $L$."],"supporting_citations":[{"why":"Defines the moduli stack of basic stable log curves and proves its properness, the algebraic input Proposition 5.7 compares with exploded curves.","marker":"[9]"},{"why":"Establishes the correspondence between log schemes and exploded manifolds used to explode basic log curves into universal tropical families.","marker":"[26]"},{"why":"Provides universal tropical structures for stable curves in exploded manifolds, used to cover every exploded curve class by a basic log curve.","marker":"[28]"},{"why":"Constructs the Kuranishi structure and weighted branched sections on moduli spaces of exploded curves that underlie the virtual fundamental class.","marker":"[27]"},{"why":"Develops refined differential forms and refined cohomology, giving the chain-level pullbacks, pushforwards, and Poincare duals used in the star product.","marker":"[32]"},{"why":"Builds the exploded-manifold virtual fundamental class and its pushforward, the cycle Theorem 5.11 represents as a lagrangian correspondence.","marker":"[34]"},{"why":"Defines the evaluation stacks and the tropical gluing formula, interpreted here as composition of correspondences.","marker":"[31]"},{"why":"Supplies the dimension and regularity statements fixing the virtual dimension as half the evaluation space dimension.","marker":"[33]"},{"why":"Gives a proper surjective cover of a Deligne-Mumford stack, used in Proposition 5.7 to produce an explodable family containing every curve class.","marker":"[22]"}],"fun_headline_variants":["Lagrangian correspondences encode log Calabi-Yau 3-fold counts","Holomorphic symplectic category encodes Gromov-Witten invariants","GW invariants of log CY3 as Lagrangian correspondences","Log CY3 counts are holomorphic Lagrangian correspondences"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Lagrangian correspondences encode log Calabi-Yau 3-fold counts","Holomorphic symplectic category encodes Gromov-Witten invariants","GW invariants of log CY3 as Lagrangian correspondences","Log CY3 counts are holomorphic Lagrangian correspondences"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001292,"raw_usage":{"total_tokens":5256,"prompt_tokens":904,"completion_tokens":4352,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":520,"completion_tokens_details":{"reasoning_tokens":4280}},"tokens_in":520,"tokens_out":4352,"duration_ms":32223,"temperature":1.0,"reasoning_tokens":4280,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T18:21:15.487423+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Logarithmic Gromov-Witten invariants","cited_arxiv_id":null,"evidence_quote":"Defines the moduli stack of basic stable log curves and proves its properness, the algebraic input Proposition 5.7 compares with exploded curves."},{"cited_title":"Log geometry and exploded manifolds","cited_arxiv_id":null,"evidence_quote":"Establishes the correspondence between log schemes and exploded manifolds used to explode basic log curves into universal tropical families."},{"cited_title":"Universal tropical structures for curves in exploded manifolds","cited_arxiv_id":"1301.4745","evidence_quote":"Provides universal tropical structures for stable curves in exploded manifolds, used to cover every exploded curve class by a basic log curve."},{"cited_title":"Holomorphic curves in exploded manifolds: Kuranishi structure","cited_arxiv_id":"1301.4748","evidence_quote":"Constructs the Kuranishi structure and weighted branched sections on moduli spaces of exploded curves that underlie the virtual fundamental class."},{"cited_title":"De Rham theory of exploded manifolds","cited_arxiv_id":null,"evidence_quote":"Develops refined differential forms and refined cohomology, giving the chain-level pullbacks, pushforwards, and Poincare duals used in the star product."},{"cited_title":"Holomorphic curves in exploded manifolds: virtual fundamental class","cited_arxiv_id":null,"evidence_quote":"Builds the exploded-manifold virtual fundamental class and its pushforward, the cycle Theorem 5.11 represents as a lagrangian correspondence."},{"cited_title":"Holomorphic curves in exploded manifolds: regularity","cited_arxiv_id":null,"evidence_quote":"Supplies the dimension and regularity statements fixing the virtual dimension as half the evaluation space dimension."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives a proper surjective cover of a Deligne-Mumford stack, used in Proposition 5.7 to produce an explodable family containing every curve class."}],"review_version":2}