Periods and quasiperiods of modular forms and D-brane masses on the quintic

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 integral symplectic basis of $H_3(W,Z)$ at the point of maximal unipotent monodromy. It was discovered by Chad Schoen in 1986 that the singular fibre of the quintic at the conifold point gives rise to a Hecke eigen form of weight four $f_4$ on $\Gamma_0(25)$ whose Fourier coefficients $a_p$ are determined by counting solutions in that fibre over the finite field $\mathbb{F}_{p^k}$.  The D-brane masses at the conifold are given by the transition matrix $T_{mc}$  between the integral symplectic basis and a Frobenius basis at the conifold. We predict and verify to very high precision that the entries of $T_{mc}$ relevant for the D2 and D4 brane masses are given by the two periods (or L-values) of $f_4$. These values also determine 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 $f_4$ appear in $T_{mc}$. We extend the analysis to the other hypergeometric one parameter 3-folds and comment on simpler applications to local Calabi-Yau 3-folds and polarized K3 surfaces.

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