Picard-Fuchs equations and mirror maps for hypersurfaces

TL;DR

This paper presents a method for computing Yukawa couplings and mirror maps for hypersurfaces using Picard-Fuchs equations, with explicit examples including quintic hypersurfaces, leading to predictions about rational curves.

Contribution

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

Findings

Predictions for the number of rational curves on hypersurfaces

Verification of some predictions by classical algebraic geometry

Explicit computation of mirror maps and Yukawa couplings for examples

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.