Enumerative Geometry of Calabi-Yau Manifolds and String Theory

我们研究了Calabi-Yau三重Borcherds乘积的类似物。从几何的观点来看，这样的乘积应该计算流形上的层或D-膜。也可以预期，它们通过某些渐近展开与Gromov-Witten势相关。在弦理论的上下文中，他们应该在卡-丘流形上紧化的IIA型弦理论中计算BPS态。在本研究中，我们试图发展一个系统的方法来构造这样的产品或部分产品时，Calabi-Yaus椭圆纤维或K3纤维。在一定程度上，我们成功地完成了这一使命，并得到了具体的体现。已经进行了一些一致性检查，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，我们研究了节曲线和尖点曲线上的旋涡性质。

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.