Coefficients of the monodromy matrices of one-parameter families of double octic Calabi-Yau threefolds at a half-conifold point

TL;DR

This paper computes the transition matrix of monodromy bases for a family of double octic Calabi-Yau threefolds at a half-conifold point, revealing entries as rational functions involving special L-values of modular forms.

Contribution

It provides a numerical computation of the monodromy transition matrix at a half-conifold point and links its entries to special values of modular form L-functions, extending understanding of monodromy in Calabi-Yau families.

Findings

Transition matrix entries are rational functions of L(f,1) and L(f,2)

Identifies the relation between monodromy and modular form L-values

Analyzes the rank of the period group generated by monodromy actions

Abstract

Doran and Morgan have introduced certain rational basis for the monodromy group of the Picard-Fuchs operator of a hypergeometric family of Calabi-Yau threefolds. In this paper we compute numerically the transition matrix between a generalization of the Doran-Morgan basis and the Frobenius basis at a half-conifold point of a one-parameter family of double octic Calabi-Yau threefolds. We identify the entries of this matrix as rational functions in the special values $L(f,1)$ and $L(f,2)$ of the corresponding modular form $f$ and one constant. We also present related results concerning the rank of the group of period integrals generated by the action of the monodromy group on the conifold period.