Abstract:
We consider one complex structure parameter mirror families W of Calabi-Yau 3-folds with Picard-Fuchs equations of hypergeometric type. By mirror symmetry the even D-brane masses of the orginal Calabi-Yau M can be identified with four periods w.r.t. to an inte- gral symplectic basis of H3(W, Z) at the point of maximal unipotent monodromy. It was discovered by Chad Schoen in 1986 that the sin- gular fibre of the quintic at the conifold point gives rise to a Hecke eigen form of weight four f4 on Γ0(25) whose Fourier coefficients ap are determined by counting solutions in that fibre over the finite field Fpk . The D-brane masses at the conifold are given by the transition matrix Tmc between the integral symplectic basis and a Frobenius basis at the conifold. We predict and verify to very high precision that the entries of Tmc relevant for the D2 and D4 brane masses are given by the two periods (or L-values) of f4. These values also deter- mine the behaviour of the Weil-Petersson metric and its curvature at the conifold. Moreover we describe a notion of quasi periods and find that the two quasi period of f4 appear in Tmc. We extend the analy- sis to the other hypergeometric one parameter 3-folds and comment on simpler applications to local Calabi-Yau 3-folds.