**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 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.

Fri 15 Jun, 11:30 - 12:30, Aula Dini

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