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 [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.

Periods of singular double octic Calabi-Yau threefolds and modular forms

“…After a quadratic or quartic base-change, totally ramified at such singularity, we can get an equation with local exponents (0, 1, 1, 2). As a consequence, the monodromy behaviour in this case is much better understood (see [2]). On the geometric side, we expect that a singular point of type 1 n C, i.e.…”

“…Coming back to the case of double octics, in [1] we observed (numerically) that for all one-parameter families of double octics the group of real periods Re L 0 P,t0 has rank one, while in nine cases the subgroup of imaginary periods Im L 0 P,t0 has rank 2. Thus in this case the inclusion Λ X ⊂ L 0 P,t0 is strict; note that due to results from [2] this cannot happen for a singularity of type Obviously the group L P,t0 depends not only on the smooth double octic X but also on the choice of a one-parameter smoothing. In fact birational models of rigid double octic can be realized as specializations of several one-parameter families.…”

“…We now consider the associated modular forms in order to check whether we can describe elements of L P,t0 in a way similar as we describe periods of a rigid Calabi-Yau threefold in terms of L(f, 1) and L(f, 2) 2πi . To this end we have to embed Λ f into some intrinsically defined group of greater rank, and therefore it is natural to try and write the special values generating Λ f as sums of some invariants of f .…”

“…In this section we shall propose a conjectural relation between the integrals M (f, 1) and M (f,3) 2π 2 and the period integrals of the degenerate element of a one-dimensional family. As in Section 3, we shall consider a one-parameter family X = (X t ) t∈∆ of projective varieties such that t ∈ {0, t 0 } the variety X t is a (smooth) Calabi-Yau threefold with h 1,2 (X t ) = 1.…”

Orphan Calabi-Yau threefold with arithmetic monodromy group

“…Hence the coefficients of the monodromy matrices M on B (M σ ) are a priori arbitrary complex numbers. If B m is the Frobenius basis at a point of maximal unipotent monodromy, it is conjectured that M on Bm (P) ⊂ GL(4, Q( ζ (3) (2πi) 3 )). For the Frobenius basis at a half-conifold point we have observed in [3] that coefficients of the monodromy matrices include special values L(f, 1) and L(f,2) 2πi of the L-function of a certain modular form but there also appears one yet unidentified constant.…”