Picard-Fuchs equations and mirror maps for hypersurfaces
David R. Morrison
TL;DR
This work develops a Picard–Fuchs based framework to compute Yukawa couplings and the mirror map for Calabi–Yau hypersurfaces with $h^{2,1}=1$, using Griffiths’ residue method to derive PF equations in weighted projective spaces. It performs explicit PF and mirror-map computations for four one-parameter families, establishing maximally unipotent monodromy and extracting integer-valued instanton numbers via carefully chosen constants tied to degree and integrality constraints. The main output is a set of predictions $n_j$ for the numbers of rational curves of degree $j$ on the mirror manifolds, with several checks against classical results for quintics and related geometries. The results reinforce the bridge between PF-period computations and enumerative predictions from mirror symmetry and extend the quintic paradigm to weighted-hypersurface Calabi–Yau threefolds.
Abstract
We describe a strategy for computing Yukawa couplings and the mirror map, based on the Picard-Fuchs equation. (Our strategy is a variant of the method used by Candelas, de la Ossa, Green, and Parkes in the case of quintic hypersurfaces.) We then explain a technique of Griffiths which can be used to compute the Picard-Fuchs equations of hypersurfaces. Finally, we carry out the computation for four specific examples (including quintic hypersurfaces, previously done by Candelas et al.). This yields predictions for the number of rational curves of various degrees on certain hypersurfaces in weighted projective spaces. Some of these predictions have been confirmed by classical techniques in algebraic geometry.