Gkz Hypergeometric Systems and Applications to Mirror Symmetry

Mirror symmetry of Calabi-Yau manifolds is one of the most beautiful aspects of string theory. It has been applied with great success to do non-perturbative calculation of quantum cohomology rings1−10. More recently, new ideas have been developed to apply mirror symmetry to study the moduli space of the type II string vacua compactified on a Calabi-Yau manifold. Some of the recent work on verifying the so-called heterotic-type II string duality relies heavily on these new ideas 12 . One of the key ingredients for studying families of Calabi-Yau manifolds is the so-called Picard-Fuchs equations. They are differential equations which govern the period integrals of a Calabi-Yau manifold. In this report, we will review several aspects of the Picard-Fuchs equations which arise in mirror symmetry. We define the flat coordinates and use them to give a natural description of the quantum cohomology ring. We relate the flat coordinates to the general solutions of the Picard-Fuchs equations at the so-called point of maximally unipotent monodromy. The general solutions turn out to be in a subspace of the solutions to a Gel’fand-Kapranov-Zelevinski (GKZ) hypergeometric system. A GKZ system is therefore reducible in our case.