Periods of fibre products of elliptic surfaces and the Gamma conjecture

TL;DR

This paper develops an algorithm to compute homology bases and period matrices of fibre products of elliptic surfaces, using this data to explore the Gamma conjecture for associated Calabi-Yau threefolds, supported by a SageMath implementation.

Contribution

It introduces a new algorithm for homology and period computations of elliptic surface fibre products and applies it to test the Gamma conjecture in specific Calabi-Yau cases.

Findings

A universal formula for a list of 105 fibre product operators

Validation of the Gamma conjecture for certain Calabi-Yau threefolds

Implementation of the algorithm in SageMath

Abstract

We provide an algorithm for computing a basis of homology of fibre products of elliptic surfaces over $\mathbb P^1$, along with the corresponding intersection product and period matrices. We use this data to investigate the Gamma conjecture for Calabi-Yau threefolds obtained in this manner. We find a formula that works for all operators of a list of 105 fibre products, as well as for fourth order operators of the Calabi-Yau database. This algorithm comes with a SageMath implementation.