## 学术活动

1. 中心公开课 短期课程 讨论班 学术报告 会议论坛

## 讨论班简介 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,\mathbb{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 connection 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 \$D_2\$ and \$D_4\$ 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.