Speaker
Albrecht Klemm
Description
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.
