Effective computation of quantum cohomology across conifold transitions: a blow-up formula along Weil divisors

Let $X \rightsquigarrow \bar{X} \leftarrow Y$ be a projective conifold transition of Calabi-Yau threefolds: $X$ is a smoothing of a nodal threefold $\bar{X}$ with finitely many ordinary double points, and $Y \to \bar{X}$ is a small resolution, so that $Y = \mathrm{Bl}_W \bar{X}$ is the blow-up of a non-Cartier, non-complete-intersection Weil divisor $W$. Lee-Lin-Wang's $A+B$ theory proves that the non-extremal GW theory of $Y$ is encoded in 'linked' invariants on $X$ refined by the vanishing spheres (the invariants of $Y$ in exceptional classes being extremal ray invariants, computable explicitly as local data independent of the transition), but leaves open how to compute the linked invariants: the degeneration formula expresses GW invariants of $X$ only as a finite sum over curve classes on $Y$, with no method to isolate individual terms. The open problem is an effective blow-up formula for GW theory whose center is the Weil divisor $W$ - computing $QH(Y)$, and more generally the individual linked invariants, from the A- and B-model data of $X$ together with intrinsic data of $W$ such as the linear system $|W|$, with determinantal transitions as the test class. This is the effective bottleneck of the transition step of the quantum minimal model program.

Background

Gromov-Witten (GW) invariants are enumerative counts of stable holomorphic maps into a smooth projective variety; the genus-0 three-point invariants serve as the structure constants of the quantum cohomology ring $QH(X)$, a deformation of the classical cohomology ring in which curve classes are tracked by Novikov variables $q^\beta$, and equivalently of the quantum D-module (the Dubrovin connection, a flat connection whose flatness encodes the WDVV relations). A basic functoriality question is how $QH$ behaves under the birational maps and transitions produced by the minimal model program (MMP). For $K$-equivalent maps (flops, i.e. birational maps that extract no divisor and are trivial outside codimension two), the graph correspondence induces ring isomorphisms after analytic continuation, proved for ordinary flops in Lee-Lin-Wang's series of works summarized in the source survey; for non-$K$-equivalent transformations such as flips, divisorial contractions, and blow-ups, the theory genuinely changes and one asks for a decomposition or comparison formula.

A conifold transition of Calabi-Yau threefolds proceeds in two steps: the smoothing $X$ degenerates to a nodal threefold $\bar{X}$ whose singularities are ordinary double points (nodes, locally $xy - zw = 0$ in $\mathbb{C}^4$), and a small resolution $Y \to \bar{X}$ (a proper birational map that is an isomorphism in codimension one and replaces each node by an exceptional curve $C_i \cong \mathbb{P}^1$ with normal bundle $\mathcal{O}(-1) \oplus \mathcal{O}(-1)$) restores smoothness at the cost of changing the topology. The degeneration formula of Jun Li and Li-Ruan then implies, in the Calabi-Yau case, that $\langle - \rangle^X_\beta = \sum_{\gamma \mapsto \beta} \langle - \rangle^Y_\gamma$, a finite sum over curve classes on $Y$ mapping to $\beta$, interpretable as setting the exceptional Novikov variable $q^\ell = 1$. Lee-Lin-Wang's $A+B$ theory refines this: they prove that the non-extremal GW theory (A model) of $Y$ is equivalent to 'linked' GW invariants on the open variety $X^\circ = X \setminus \bigcup_i S_i$, where the $S_i$ are the vanishing 3-spheres (Lagrangian spheres in the symplectic picture) shrinking to the nodes - the invariants are refined by how source curves link the $S_i$ - while the GW invariants of $Y$ supported on the exceptional curves are extremal ray invariants, computable explicitly as local data independent of the transition, and dually the variation of Hodge structure (B model) of $X$ is reconstructed from $Y$ linked along the exceptional curves. This gives a mutual, but so far only existential, determination of the combined A and B models of $X$ and $Y$.

The missing ingredient is effectiveness. The linked invariants are defined as components of a virtual-class decomposition; no generating technique computes them. Concretely, the small resolution is $Y = \mathrm{Bl}_W \bar{X}$, the blow-up of a Weil divisor $W \subset \bar{X}$ (a codimension-one cycle that is not locally principal, hence not a complete-intersection center), so computing $QH(Y)$ from $QH(X)$ - identifying the GW theory of the singular $\bar{X}$, itself undefined in the literature, with that of $X$ via deformation invariance of logarithmic GW theory - amounts to a blow-up formula whose center is a non-complete-intersection Weil divisor. For blow-ups along smooth or complete-intersection centers, quantum Lefschetz-type formulas provide exactly such blow-up formulas - here the I-function, the hypergeometric generating series of genus-0 GW invariants delivered by mirror theorems, of the blow-up is obtained by an explicit hypergeometric factor; for Weil-divisor centers nothing is available. The survey's test class is that of determinantal transitions: $Y \subset S \times \mathbb{P}^n$ is cut out by sections of line bundles $L_i \boxtimes \mathcal{O}(1)$ and contracts onto $\bar{X} = \{\det s_{ij} = 0\}$; there quantum Lefschetz on $Y$ yields the extrinsic hypergeometric expression $I_{Y/X} = \prod_i (L_i + h) / ((h)^{n+1} \sum_i L_i)$ with $h = c_1(\mathcal{O}_{\mathbb{P}^n}(1))$, and the challenge is to reinterpret it intrinsically through the linear system $|W|$, where $\psi: Y \to \bar{X}$ is the contraction and $W = \psi_*(h)$ is the Weil divisor making $Y = \mathrm{Bl}_W \bar{X}$. The same authors' broader goal is a quantum minimal model program (QMMP) - tracking quantum cohomology through all MMP steps, ultimately for singular varieties - of which this conifold-transition step is the effectiveness bottleneck.

Problem Statement

For a projective conifold transition of Calabi-Yau threefolds $X \rightsquigarrow \bar{X} \leftarrow Y$, where $X$ is a smoothing of the nodal threefold $\bar{X}$, $Y \to \bar{X}$ is a small resolution, and $Y = \mathrm{Bl}_W \bar{X}$ for the non-Cartier Weil divisor $W \subset \bar{X}$, give an effective blow-up formula for Gromov-Witten theory with the Weil divisor $W$ as center. Concretely: (1) a proven formula or algorithm expressing the genus-0 quantum cohomology $QH(Y)$ - equivalently every individual linked GW invariant $\langle \cdots \rangle_{X, \gamma}$ for each non-extremal curve class $\gamma$ in the Mori cone $\mathrm{NE}(Y)$ of effective curve classes on $Y$, together with the invariants of $Y$ in exceptional classes, which the $A+B$ theory treats as extremal ray invariants - in terms of the GW theory of $X$ linked along the vanishing spheres (with the GW theory of the singular $\bar{X}$ identified, as in the sources, with that of $X$ via deformation invariance of logarithmic GW theory), the variation-of-Hodge-structure data of $X$, and intrinsic data of $W$ such as the linear system $|W|$; (2) the intrinsic formulation for determinantal transitions, in which the extrinsic ambient data $(L_0, \ldots, L_n)$ entering the hypergeometric factor $I_{Y/X} = \prod_i (L_i + h) / ((h)^{n+1} \sum_i L_i)$ is replaced by data of $|W|$ alone; and (3) validation of the formula on the computable classes, such as determinantal (complete-intersection Calabi-Yau) transitions, where the $Y$-side is independently accessible via the quantum Lefschetz theorem. Part (1) must cover arbitrary non-Cartier Weil-divisor centers rather than only the determinantal test class, with parts (2) and (3) as complementary refinements and consistency checks. The question asks for a result at least in genus 0 with proof, the all-genera effective determination of the linked invariants of the A model of $Y$ being the broader form. This is precisely the effectiveness gap recorded in the source survey and left open by the $A+B$ theory; it supplies the conifold-transition step of the quantum minimal model program.

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