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

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

“…In [4] we observed that L 0 P,s contains (some integer multiple of) Λ f . In many cases L 0 P,s ⊗ Q = Λ f ⊗ Q or, more precisely, both L 0 P,s and Λ f are lattices and they are commensurable.…”

“…In the next sections we present explicit results concerning the form of the transition matrix T B Fc between the Doran-Morgan basis B and the Frobenius basis F c at a singularity of type 1 n C. Here we begin with presenting our main motivation for considering it. It comes from results presented in [4], which could be vastly generalized if the form of the transition matrix T B Fc was known. In the rest of this paper we work in the following setup: P is a Picard-Fuchs operator of order 4 with a MUM point at 0 and a singularity of type 1 n C at s. M 0 and M s are some fixed local monodromies around the respective singularities, N 0 := M 0 − Id and N s := M s − Id.…”

“…On the other hand, this connection between L 0 P,s and the transition matrix T B Fc can also be used in a following way. From [4] we know that the special values L(f, 1) and L(f, 2) 2πi are elements of L 0 P,s ⊗ Q. Since M B 0 ∈ GL(4, Q) and special values are (expected to be) irrational, it indicates that they could perhaps be identified among the coefficients of the matrix αT B Fc .…”

“…Both a motivation and a way of approaching this problem is provided by our results presented in [4]. There we use Picard-Fuchs operator P for numerical identification of period integrals of certain rigid Calabi-Yau threefolds.…”

“…In section 5 we present basic facts concerning modular forms and the modularity theorem for Calabi-Yau threefolds. In section 6 we recall results form [4] and formulate the problems we consider. The last three sections contain main results of the paper.…”