论文标题
卡拉比（Calabi-yau
D-brane masses at special fibres of hypergeometric families of Calabi-Yau threefolds, modular forms, and periods
论文作者
论文摘要
我们考虑了14个家族的$ W $ Calabi-yau三倍，其中一个复杂的结构参数和Picard-fuchs tore of Myperdodric类型，例如$ \ Mathbb {p}^4 $中的Quintic镜子。镜子对称性标识了镜子dism的质量 - 镜子的dism branes calabi-yau $ m $ yau $ m $，以及四个时期的holomorphic $（3,0）$ - 以$ h_3（w，w，\ mathbb {z}}）的符号基础上的形式。乍得·肖恩（Chad Schoen）发现，五五重柱的奇异纤维产生了重量的hecke特征形式，其中四个低于$γ_0（25）$，其hecke eigenvalues由Hasse-Weil Zeta功能确定，这可以由该纤维的计数而不是有限场上的计数来获得。类似的功能以其他13个情况而闻名。在两种情况下，我们进一步找到了特殊的常规点，即所谓的等级两个吸引子点，其中Hasse-Weil Zeta函数产生了重量四和两个的模块化形式。我们从数值上识别这些特殊纤维处的时期矩阵的条目是相关模块化形式的周期和准列。在一种情况下，我们通过构建对应关系和kuga-sato品种之间的对应关系来证明这一点。我们还对当地Calabi-yau三倍的应用程序进行了评论。
We consider the fourteen families $W$ of Calabi-Yau threefolds with one complex structure parameter and Picard-Fuchs equation of hypergeometric type, like the mirror of the quintic in $\mathbb{P}^4$. Mirror symmetry identifies the masses of even--dimensional D--branes of the mirror Calabi-Yau $M$ with four periods of the holomorphic $(3,0)$-form over a symplectic basis of $H_3(W,\mathbb{Z})$. It was discovered by Chad Schoen that the singular fiber at the conifold of the quintic gives rise to a Hecke eigenform of weight four under $Γ_0(25)$, whose Hecke eigenvalues are determined by the Hasse-Weil zeta function which can be obtained by counting points of that fiber over finite fields. Similar features are known for the thirteen other cases. In two cases we further find special regular points, so called rank two attractor points, where the Hasse-Weil zeta function gives rise to modular forms of weight four and two. We numerically identify entries of the period matrix at these special fibers as periods and quasiperiods of the associated modular forms. In one case we prove this by constructing a correspondence between the conifold fiber and a Kuga-Sato variety. We also comment on simpler applications to local Calabi-Yau threefolds.
