GKZ Hypergeometric Series for the Hesse Pencil, Chain Integrals and Orbifold Singularities

TL;DR

This paper explores the extra solutions of the GKZ system for the Hesse pencil of elliptic curves, providing various interpretations and highlighting the role of orbifold singularities in the moduli space.

Contribution

It introduces new realizations of the additional GKZ solutions and connects them to geometric and algebraic structures like Calabi-Yau threefolds and orbifold singularities.

Findings

Extra solutions of the GKZ system are interpreted as oscillating, Eichler, and chain integrals.

Orbifold singularities influence the structure of the GKZ solutions.

Connections between elliptic curves, Calabi-Yau geometries, and moduli space are established.

Abstract

The GKZ system for the Hesse pencil of elliptic curves has more solutions than the period integrals. In this work we give different realizations and interpretations of the extra solution, in terms of oscillating integral, Eichler integral, chain integral on the elliptic curve, limit of a period of a certain compact Calabi-Yau threefold geometry, etc. We also highlight the role played by the orbifold singularity on the moduli space and its relation to the GKZ system.