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Hodge-theoretic Open/Closed Correspondence

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Pith's one-line read This paper proves that, after a Tate twist, the relative mixed Hodge structure of an open-string mirror is isomorphic to the closed-string mirror's cohomology, and that relative 3-cycles correspond to absolute 4-cycles with matching…

desk verdict The cycle-level correspondence is a genuine, well-argued theorem; the VMHS main theorem is honestly conditional on an unpublished companion paper. read the letter →

arxiv 2507.09941 v1 pith:FVJ65GPF submitted 2025-07-14 math.AG

Song Yu This is my paper · ORCID
classification math.AG MSC 14J3314D0714M2532G20
keywords open/closedcorrespondencetoricCalabi-YauorbifoldsHori-VafamirrorvariationsofmixedHodgestructuresPicard-FuchssystemsAganagic-VafabranesrelativeperiodsTatetwist
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

This paper establishes the B-model open/closed correspondence in Hodge-theoretic form: from a toric Calabi-Yau 3-orbifold with a framed Aganagic-Vafa brane, one constructs a toric Calabi-Yau 4-orbifold, and the paper proves the two Hori-Vafa mirrors carry the same period and mixed Hodge structure data once the open geometry is taken relative to a divisor family. The main theorems give an injective map from integral relative 3-cycles to integral 4-cycles that preserves periods up to a factor of $1/(2\pi\sqrt{-1})$, and an isomorphism of variations of mixed Hodge structures $H^3(X^\vee, Y; \mathbb{C}) \cong H^4(\tilde{X}^\vee; \mathbb{C}) \otimes T(1)$. The result matters because it turns a longstanding physics duality between open strings on a 3-fold and closed strings on a 4-fold into a precise statement about ordinary cohomology, periods, and Picard-Fuchs equations.

What carries the argument

The load-bearing structure is the Hori-Vafa mirror as a conic fibration $uv = H$ over an algebraic torus, with the mirror of the brane as a hypersurface divisor $Y$. Circle fibrations obtained from the $U(1)$ action on $(u, v)$ identify relative homology of the torus against its discriminant hypersurface with the middle homology of the mirror, reducing the cycle correspondence to a statement about affine hypersurfaces in tori. On the Hodge side, the machinery is the combinatorial description of the variations of mixed Hodge structures through graded rings $R_H$ built from Newton polytopes and their $E$- and $I$-filtrations; the key structural input is a short exact sequence $0 \to R_{H_0} \to R_{\tilde{H}} \to R_H \to 0$ compatible with the filtrations, which translates into the extension of VMHS underlying the desired isomorphism. Here $T(1)$ is the Tate twist by $1$, the one-dimensional Hodge structure of weight $-2$.

What would settle it

Compute the relative integral on the left and the 4-cycle integral on the right of Theorem 5.6 for the explicit example $X = \mathbb{C}^3$ with $f=1$ at a small nonzero value of $\tilde{q}$; if the numbers disagree, the cycle map is wrong, and the VMHS isomorphism would also collapse once the underlying combinatorial description is fixed. Alternatively, if the in-preparation combinatorial VMHS statement [4] used for Theorem 6.2 turns out to give different filtrations on a non-reflexive polytope, Theorem 6.7 would fail unless the extension sequence can be re-built.

Watch

Extended reading notes

Core claim

On the B-side, the paper's central claim is that the open-string data of $(X, L)$ and the closed-string data of $\tilde{X}$ are the same object after a Tate twist. Concretely, over a small parameter domain, there is an injective cycle map $\iota\colon H_3(X^\vee_q, Y_{q,x}; \mathbb{Z}) \to H_4(\tilde{X}^\vee_{\tilde{q}}; \mathbb{Z})$ that is an isomorphism over $\mathbb{Q}$ and satisfies $\int_\Gamma \Omega_q = \frac{1}{2\pi\sqrt{-1}} \int_{\iota(\Gamma)} \tilde{\Omega}_{\tilde{q}}$ for every relative 3-cycle $\Gamma$; dually, there is an isomorphism of variations of mixed Hodge structures $H^3(X^\vee_q, Y_{q,x}; \mathbb{C}) \cong H^4(\tilde{X}^\vee_{\tilde{q}}; \mathbb{C}) \otimes T(1)$ sending $[\Omega_q]$ to $[\tilde{\Omega}_{\tilde{q}}]$. The paper also shows that the Picard-Fuchs system of $\tilde{X}$ extends that of $X$, and characterizes the extra solutions as the open mirror map and B-model disk functions of all framed branes that give rise to the same $\tilde{X}$.

Load-bearing premise

The paper assumes the combinatorial description of the variations of mixed Hodge structures of Hori-Vafa mirrors for general polytopes (Theorem 6.2) from an in-preparation companion paper; the VMHS comparison Theorem 6.7 rests on it, even though the cycle correspondence in Theorem 5.6 is argued directly in this text.

Editorial extensions

If this is right

  • The full solution space of the extended Picard-Fuchs system is spanned by the closed solutions plus the open mirror map and the B-model disk functions of all branes sharing the same closed 4-fold, making the 4-fold's I-function a generating object for open Gromov-Witten data.
  • Every relative period of the open geometry is, up to the constant $1/(2\pi\sqrt{-1})$, a period of the closed geometry, so open mirror maps and disk instanton series can be computed from 4-cycle integrals.
  • The relative cohomology $H^3(X^\vee, Y; \mathbb{C})$ carries exactly the same VMHS as the closed cohomology after a Tate twist, so the open/closed duality is compatible with the Hodge filtration, the weight filtration, and their variations.
  • The same closed 4-fold arises from several open phases, and the correspondence identifies the different brane framings with different bases of the same solution space, giving a geometric picture of open-string wall-crossing.
  • On integral homology the rank identity $\operatorname{rank} H_4(\tilde{X}^\vee) = \operatorname{rank} H_3(X^\vee) + \operatorname{rank} H_2(Y)$ holds, with injectivity of the cycle map proven over $\mathbb{Z}$ and surjectivity conjectured over $\mathbb{Z}$.

Reading between the lines

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

  • If the VMHS correspondence extends to all orders in the quantum parameter, the quantum D-module of the open geometry should be a natural subquotient of the closed geometry's D-module, giving an open analogue of the Gelfand-Kapranov-Zelevinsky D-module formalism.
  • The divisor-boundary formulation suggests that Aganagic-Vafa B-brane boundary conditions are one representative of a more general class of divisor boundaries; a testable extension would construct the same correspondence for other hypersurface boundaries inside $X^\vee$.
  • Because the same closed 4-fold is shared by several framed branes, the correspondence predicts relations among their disk functions; these relations can be checked numerically from the explicit I-function of $\tilde{X}$.
  • The integrality question left open in Remark 1.4, namely surjectivity of $\iota$ over $\mathbb{Z}$, is decidable in examples such as $X = \mathbb{C}^3$, and a failure would mean the open/closed period correspondence holds rationally but not integrally.
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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 / 4 minor

Summary. The paper develops a B-model mathematical foundation for the open/closed correspondence of Mayr and Lerche-Mayr in the toric setting. For an open geometry consisting of a semi-projective toric Calabi-Yau 3-orbifold X with a framed Aganagic-Vafa outer brane (L,f), and the associated closed toric Calabi-Yau 4-orbifold X~, the paper studies the Hori-Vafa mirrors. It proves that the Picard-Fuchs system of X~ extends that of X and identifies the additional solutions with the open mirror map and B-model disk functions (Propositions 1.2, 4.5, 4.6). It constructs an injective integral cycle map from relative 3-cycles of (X^vee,Y) to 4-cycles of X~^vee that matches periods up to the standard factor (Theorem 5.6), and states an isomorphism of variations of mixed Hodge structures H^3(X^vee,Y;C) =~ H^4(X~^vee;C) tensor T(1) identifying [Omega] with [Omega~] (Theorem 6.7).

Significance. The cycle-level correspondence is a concrete and largely self-contained contribution: Lemmas 5.13-5.15 give explicit period identities, Proposition 5.8 gives a splitting of the relevant homology sequence, and the running example X=C3 is worked out in detail in Examples 4.7, 5.16, and 6.8. If Theorem 6.7 is fully established, the result would elevate the open/closed correspondence from an equality of periods to an isomorphism of variations of mixed Hodge structures, tying together the B-model disk functions, extended Picard-Fuchs systems, and Hodge-theoretic mirror symmetry. The main caveat is that Theorem 6.7 relies on Theorem 6.2, whose proof for general polytopes is delegated to the in-preparation paper [4]; thus the VMHS part is conditional, while the cycle correspondence itself is not affected by this dependency.

major comments (2)
  1. [Section 6.3-6.4 (Theorem 6.2, Corollary 6.5, Theorem 6.7)] The main Hodge-theoretic result is not self-contained. Theorem 6.2 is stated for general polytopes with attribution to [45,4], but the text notes that Konishi-Minabe proved the reflexive case and that the general case is delegated to the in-preparation paper [4] by Aleshkin and Yu. Corollary 6.5 and Theorem 6.7 use Theorem 6.2 essentially, so the VMHS open/closed correspondence is conditional on [4] appearing with compatible hypotheses covering arbitrary semi-projective toric Calabi-Yau orbifolds and rational framings with a>1. The cycle-level Theorem 5.6 does not invoke Theorem 6.2 and is not affected by this gap.
  2. [Section 6.5 (Lemma 6.10, Proposition 6.4)] The proof of Proposition 6.4 depends on Lemma 6.10's assertion that the image of the map iota is exactly the span of monomials in which the Z-power is at least 1. The construction of iota and the verification that it descends to the quotient are detailed, but the proof of the image statement is not. In particular, the c=1 base case of the induction is dismissed with 'follows from the argument above' after the descent computation; no explicit argument is given that a general monomial X_0^k X^a Y^b Z is congruent modulo the D-ideal to an iota-monomial. Since exactness at the middle of (29) and hence Corollary 6.5 use this identification, the proof needs to be completed.
minor comments (4)
  1. [Section 3.2] The phrase 'in terms fo the Q-basis' should read 'in terms of the Q-basis'.
  2. [Section 5.4] Several displayed isomorphisms contain corrupted glyphs such as '/leftrggtlne'; these appear to be source-rendering artifacts and should be cleaned up before publication.
  3. [Section 6.4 (Lemma 6.6)] The sentence 'This form also represents a non-trivial class in the previous term H^2(X^vee,Y)' is confusing; the needed input for the short exact sequence (32) is the vanishing of the restriction map H^2(X^vee;C) -> H^2(Y;C), and the sentence should be rephrased accordingly.
  4. [Section 5.5-5.6] The proof of Theorem 5.6 uses rank equality (19) to conclude surjectivity over Q; it would be helpful to state explicitly the finite-generation and torsion-freeness facts about the homology groups that justify comparing ranks with Betti numbers.

Circularity Check

1 steps flagged · score 4.0 of 10

The VMHS half of the paper rests on the unpublished self-cited companion [4] for the general-polytope case of Theorem 6.2, while the integral-cycle correspondence is independently argued.

  1. self citation load bearing [Section 6.3, paragraph before Theorem 6.2; Theorem 6.2; used in Corollary 6.5 and Theorem 6.7]
    "Konishi-Minabe [45] showed that if the 2-dimensional polytope ∆ is reflexive, so that the toric Calabi-Yau 3-orbifold X is a local surface, RH also describes the VMHS on the middle-dimensional cohomology H3(X∨q; C) of the Hori-Vafa mirror X∨q. In upcoming joint work [4] with Aleshkin, we extend this result to general polytopes of any dimensions based on the methods of [6, 45]."

    Corollary 6.5 is introduced as 'In view of Theorems 6.1, 6.2, it directly translates into the following result,' and Theorem 6.7 is then obtained by 'standard arguments' comparing the relative-cohomology sequence (32) with the extension (30) from Corollary 6.5. The general-polytope part of Theorem 6.2—the only part covering the arbitrary semi-projective toric Calabi-Yau orbifolds and rational framings used in this paper—is not proved here; it is explicitly deferred to the in-preparation paper [4] by Aleshkin and Yu. Hence the VMHS open/closed isomorphism is not derived within this paper for the non-reflexive cases: it inherits its content from an unverified self-cited companion.

full rationale

The paper's main derivation for the cycle correspondence (Theorem 5.6) is self-contained: it is built from the long exact sequence of the pair, Lemmas 5.10–5.12, explicit sections, and standard period-matching results [24, 13], with no fitted parameters and no dependence on the unpublished companion. The Picard-Fuchs extension Proposition 4.2 is a direct verification from the charge matrices, and the solution-space description in Propositions 4.5 and 4.6 imports the hypergeometric open/closed correspondence from [53]; this is a self-citation to prior work by the same authors, but it is not a claim whose content is established by the present paper's own assumptions. The VMHS theorem, in contrast, depends on Theorem 6.2, whose reflexive case is due to Konishi-Minabe and whose general-polytope case is explicitly assigned to [4], an in-preparation paper by the present author and Aleshkin. Because [4] is not independently verified and is not machine-checked, code-reproduced, or externally falsified within the paper, the central Hodge-theoretic open/closed correspondence is conditional on a load-bearing self-citation; this prevents a score of 0–2. However, there is no definitional or fitted-input circularity, and the integral-cycle half is independently argued, so the score is moderate rather than severe.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard background results (Batyrev, Givental, Iritani, Konishi-Minabe) plus one unpublished input: the VMHS description of Hori-Vafa mirrors for general polytopes (Theorem 6.2) is partly attributed to [4], an in-preparation paper. Assumption 5.3 (regularity of Laurent polynomials on a small parameter domain) is explicitly stated and justified by a cited lemma. No free parameters or invented entities are introduced; the framing f and the choice of bases are part of the geometric input, not fitted constants.

assumptions (5)
  • standard math Batyrev's combinatorial description of VMHS of affine hypersurfaces in algebraic tori (Theorem 6.1)
    Invoked in Section 6.2 to identify the E- and I-filtrations on R_H, R_{H~}, R_{H0} with Hodge and weight filtrations; cited to [6, 58, 45].
  • domain assumption Regularity of Laurent polynomials (Assumption 5.3)
    Fixes a small ϵ so H, H~, H0 are ∆-regular, ensuring smoothness of the mirror families and the validity of the Batyrev VMHS description; the existence is justified by [39, Lemma 3.8].
  • ad hoc to paper VMHS description of Hori-Vafa mirrors for general polytopes (Theorem 6.2)
    Attributed to [45] and the in-preparation [4]; the part for general polytopes is not proven in this manuscript and is load-bearing for the proof of Theorem 6.7.
  • standard math Givental and Iritani mirror theorems identifying solutions of Picard-Fuchs systems with coefficients of I-functions
    Used in Section 4.2 (Theorems 4.3 and 4.4) to count and generate solutions of the Picard-Fuchs systems; cited to [33, 39, 29].
  • standard math Period integrals over middle-dimensional cycles generate Picard-Fuchs solution spaces (Theorem 5.4)
    Used in Sections 5 and 6 to link period integrals to Picard-Fuchs equations; cited to [38, 45, 13].

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Pith. "Pith review of Hodge-theoretic Open/Closed Correspondence." pith.science (2026). https://pith.science/paper/FVJ65GPF

@misc{pith2026250709941,
  author       = {Pith},
  title        = {Pith review of: Hodge-theoretic Open/Closed Correspondence},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FVJ65GPF}},
  note         = {Machine review of arXiv:2507.09941}
}
abstract

We continue the B-model development of the open/closed correspondence proposed by Mayr and Lerche-Mayr, complementing the A-model study in the preceding joint works with Liu and providing a Hodge-theoretic perspective. Given a corresponding pair of open geometry on a toric Calabi-Yau 3-orbifold $\mathcal{X}$ relative to a framed Aganagic-Vafa brane $\mathcal{L}$ and closed geometry on a toric Calabi-Yau 4-orbifold $\widetilde{\mathcal{X}}$, we consider the Hori-Vafa mirrors $\mathcal{X}^\vee$ and $\widetilde{\mathcal{X}}^\vee$, where the mirror of $\mathcal{L}$ can be given by a family of hypersurfaces $\mathcal{Y} \subset \mathcal{X}^\vee$. We show that the Picard-Fuchs system associated to $\widetilde{\mathcal{X}}$ extends that associated to $\mathcal{X}$ and characterize the full solution space in terms of the open string data. Furthermore, we construct a correspondence between integral 4-cycles in $\widetilde{\mathcal{X}}^\vee$ and relative 3-cycles in $(\mathcal{X}^\vee, \mathcal{Y})$ under which the periods of the former match the relative periods of the latter. On the dual side, we identify the variations of mixed Hodge structures on the middle-dimensional cohomology of $\widetilde{\mathcal{X}}^\vee$ with that on the middle-dimensional relative cohomology of $(\mathcal{X}^\vee, \mathcal{Y})$ up to a Tate twist.

Figures

Figures reproduced from arXiv: 2507.09941 by the authors.

Figure 1
Figure 1. The projection relating ∆ and ∆0. that is, (9) Vol(∆̃) = Vol(∆) + Vol(∆0). 2.5. A 2-dimensional geometry. The data of the additional 4-cones in Σ̃(4) ∖ ι(Σ(3)) and specifically the interval ∆0 with the induced subdivision can be packaged into a toric Calabi￾Yau 2-orbifold X0 with semi-projective coarse moduli space. In more detail, the stacky fan of X0 is a triple Σ′ 0 = (N0, Σ0, α′ 0 ) where N0 ≅ Z 2 . The fan Σ0 c… view at source ↗
Figure 2
Figure 2. Triangulated polytopes in the example of X = C 3 and f = 1. τ s and write I ′ σs = I ′ τ s ⊔ {i s 1}. We then consider the Z-basis {v s 1 , vs 2 , vs 3} of N under which we have the coordinates: bi s 1 = (r(τ s , σ s ),−s(τ s , σ s ), 1), bi s 2 = (0,m(τ s , σ s ), 1), bi s 3 = (0, 0, 1) where s(τ s , σs ) ∈ {0, . . . ,r(τ s , σs ) − 1}. Under the Z-basis {v s 1 , vs 2 , vs 3 , v4} of Ñ, the vector ̃bR+1 now has co… view at source ↗
Figure 3
Figure 3. Construction of the section s in Lemma 5.12. Proof of Proposition 5.8. Lemma 5.9 implies that (23) is exact at H3(X ∨ q ;Z). Lemma 5.12 implies that (23) is exact at H2(Yq,x;Z) and is split. □ 5.6. Constructing 4-cycles and matching periods. In this subsection, we prove Theorem 5.6. In view of Proposition 5.8, we will construct the desired map ι ∶ H3(X ∨ q ,Yq,x;Z) → H4(X̃∨ q̃ ;Z) on the direct summands H3(X ∨ q ;Z)… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Construction of the 3-chain ̃γ1 in Lemma 5.15. Note that when t ∈ (0, 1), (X(t, θ), Y (t)) is perturbed off the hypersurface {XaY b = −q̃ a R−2 } and the denominator above is non-zero. To further validate this definition, we first note that the limit limt→0,1 Z1(t, θ) …
Figure 5
Figure 5. Figure 5: Aganagic-Vafa B-branes C∗ and Cp. Fix a branch of log Y that contains {p∗}∪Pq,x and assume that the path γp is chosen within this branch. The superpotential of the brane Cp is an integral of a 2-form over all of Cp and is reduced, in the radial direction of v, to the l…

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