Computing period integrals of rigid double octic Calabi-Yau threefolds with Picard-Fuchs operator

TL;DR

This paper introduces a numerical method to compute period integrals of rigid double octic Calabi-Yau threefolds using Picard-Fuchs operators, enabling calculations without detailed geometric data.

Contribution

The authors develop a novel approach leveraging Fuchsian equations and MAPLE computations to evaluate period integrals of rigid Calabi-Yau threefolds without explicit geometric information.

Findings

Successfully computed period integrals for rigid double octic Calabi-Yau threefolds.

Provided approximations for integrals related to singular models of the threefolds.

Demonstrated the method's effectiveness using only differential equation theory and computational tools.

Abstract

We present a method for numerical computation of period integrals of a rigid Calabi-Yau threefold using Picard-Fuchs operator of a one-parameter smoothing. Our method gives a possibility of computing the lattice of period integrals of a rigid double octic without any explicit knowledge of its geometric properties, employing only simple facts from the theory of Fuchsian equations and computations in MAPLE with a library for differential equations. As a surprising consequence we also get approximations of additional integrals related to a singular (nodal) model of considered Calabi-Yau threefold.