Picard-Fuchs equations and mirror maps for hypersurfaces

TL;DR

This paper presents a method using Picard-Fuchs equations to compute mirror maps and Yukawa couplings for hypersurfaces, providing predictions for rational curves that are confirmed by algebraic geometry.

Contribution

It introduces a variant of the Griffiths technique for computing Picard-Fuchs equations and applies it to specific hypersurfaces, extending previous methods for quintic hypersurfaces.

Findings

Computed Picard-Fuchs equations for four hypersurfaces

Predicted numbers of rational curves on hypersurfaces

Confirmed some predictions with classical algebraic geometry techniques

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.