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

Abstract

Doran and Morgan introduced in [12] a 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.