Generalized Hypergeometric Functions and Rational Curves on Calabi-Yau Complete Intersections in Toric Varieties
Victor V. Batyrev, Duco van Straten
TL;DR
The paper develops a unified framework linking the A-model variation of Hodge structure on Calabi–Yau complete intersections in toric varieties to generalized hypergeometric functions through the central series $\Phi_0(z)$. It shows that $\Phi_0$ satisfies a Picard–Fuchs (MU) system and yields canonical $q$-coordinates via logarithmic solutions, enabling explicit mirror constructions and Yukawa couplings. Through detailed computations across quintics, complete intersections in products of projective spaces, and toric cases, the work derives $q$-expansions and predicts Gromov–Witten invariants $n_d$ and line counts, linking intersection theory with hypergeometric data. The framework provides a principled method for constructing mirrors and intrinsic coordinates, with broad implications for computing Yukawa couplings and understanding mirror symmetry in high dimensions. Overall, it offers a versatile, hypergeometric-driven pathway to compute quantum cohomology data and to illuminate the geometry of Calabi–Yau mirrors.
Abstract
We formulate general conjectures about the relationship between the A-model connection on the cohomology of a $d$-dimensional Calabi-Yau complete intersection $V$ of $r$ hypersurfaces $V_1, \ldots, V_r$ in a toric variety ${\bf P}_Σ$ and the system of differential operators annihilating the special hypergeometric function $Φ_0$ depending on the fan $Σ$. In this context, the Mirror Symmetry phenomenon can be interpreted as the following twofold characterization of the series $Φ_0$. First, $Φ_0$ is defined by intersection numbers of rational curves in ${\bf P}_Σ$ with the hypersurfaces $V_i$ and their toric degenerations. Second, $Φ_0$ is the power expansion near a boundary point of the moduli space of the monodromy invariant period of the holomorphic differential $d$-form on an another Calabi-Yau $d$-fold $V'$ which is called Mirror of $V$. Using the generalized hypergeometric series, we propose a general construction for Mirrors $V'$ of $V$ and canonical $q$-coordinates on the moduli spaces of Calabi-Yau manifolds.