Enumerative Geometry of Calabi-Yau Manifolds and String Theory

我们研究了Calabi-Yau三重的Borcherd产品的类似物。从几何角度来看，这类产品应该计入歧管上的滑轮或D-膜。也期望它们通过一定的渐近展开式与Gromov-Witten势相联系。在弦理论的背景下，他们应该计算紧凑在Calabi-Yau流形上的IIA型弦理论中的BPS态。在这项研究中，我们试图开发一种系统的方法来构造这种产物或部分产物，当Calabi-Yaus是椭圆光纤或K3光纤时。在一定程度上，我们成功地完成了这项任务，并取得了具体的表现。已经进行了一些一致性检查，并正在调查功能特性。我目前正在准备报告这些结果的手稿。作为一个相关的主题，我们已经研究了节点曲线和尖角曲线上的涡旋特性。

We investigated analogs of Borcherds products for Calabi-Yau threefolds. From a geometrical viewpoint, such products should count sheaves or D-branes on the manifolds. It is also expected that they are related to the Gromov-Witten potentials through certain asymptotic expansions. In the context of string theory they should count BPS states in type IIA string theory compactified on the Calabi-Yau manifolds. In this research we tried to develop a systematic method to construct such products or partial products when the Calabi-Yaus are elliptically fibered or K3 fibered. To certain extent, we have succeeded in this mission and obtained concrete expressions. Some consistency checks have been made and functional properties are being investigated. I am currently preparing the manuscript reporting these results.As a related subject we have studied the properties of vortices on nodal and cuspidal curves.