Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 030, 32 pages      arXiv:1606.08352      https://doi.org/10.3842/SIGMA.2017.030
Contribution to the Special Issue on Modular Forms and String Theory in honor of Noriko Yui

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

Jie Zhou
Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada

Received October 01, 2016, in final form May 14, 2017; Published online May 20, 2017

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.

Key words: GKZ system; chain integral; orbifold singularity; Hesse pencil.

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