No abstract

Doran and Morgan introduced in [10] 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.

By the modularity theorem, every rigid Calabi–Yau threefold X has associated modular form f such that the equality of L‐functions holds. In this case, period integrals of X are expected to be expressible in terms of the special values and . We propose a similar interpretation of period integrals of a nodal model of X. It is given in terms of certain variants of a Mellin transform of f. We provide numerical evidence toward this interpretation based on a case of double octics.

We study monodromy groups of Picard-Fuchs operators of one-parameter families of Calabi-Yau threefolds without a point of Maximal Unipotent Monodromy (orphan operators). We construct rational symplectic bases for the monodromy action for all orphan double octic Picard-Fuchs operators of order 4. As a consequence we show that monodromy groups of all double octic orphan operators are dense in Sp(4, Z) and identify maximally unipotent elements in all of them, except one. Finally, we prove that the monodromy group of one of these orphan operators is arithmetic.

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 the considered Calabi-Yau threefold.