权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Problems in Enumerative Geometry Related to String Theory and Their Relations to Automorphic Forms

与弦理论相关的枚举几何问题及其与自守形式的关系

基本信息

批准号：
13640017
负责人：
KAWAI Toshiya
金额：
$ 1.02万
依托单位：
KYOTO UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2003
项目状态：
已结题

项目摘要

The first result of this research project is that, based on string duality conjecture, I uniformly constructed conjectural expressions of the Gromov-Witten potentials of elliptic Calabi-Yau threefolds with sections fibered over Hirzebruch surfaces in a certain limit by investigating appropriate conformal field theories on elliptic curves. I made various non-trivial consistency checks on this result. This research revealed various rich relations to elliptic cohomology, Borcherds products, Jacobi forms and hyperbolic reflection groups. I am currently preparing a paper for the results I obtained.The second result of this project has already been reported as an article ("String and Vortex"). It is about the properties of Gromov-Witten potentials of generic Calabi-Yau threefolds. In my previous work with Kota Yoshioka we proposed some connections between relative Hilbert schemes of points on curves and Gromov-Witten invariants. This was based on the experience on specific Calabi-Yau threefolds. In the present work I extended this proposal for more general cases. On the other hand there was another proposal by Gopakumar and Vafa in which the relative compactified Jacobians are relevant. I discussed the consistency of the two proposals in favorable situations. Along the way I found a formula for the Euler characteristic of Hilbert schemes of points on nodal curves which may be viewed as a non-trivial generalization of Macdonald's classical formula on the Euler characteristic of symmetric products of non-singular curves.

本研究项目的第一个成果是，基于弦对偶猜想，通过研究椭圆曲线上适当的共形场论，统一构造了在一定限度内在 Hirzebruch 曲面上纤维化的椭圆 Calabi-Yau 三重体的 Gromov-Witten 势的猜想表达式。我对此结果进行了各种重要的一致性检查。这项研究揭示了椭圆上同调、Borcherds 积、雅可比形式和双曲反射群的各种丰富关系。我目前正在为我获得的结果准备一篇论文。这个项目的第二个结果已经被报道为一篇文章（“弦和涡”）。它是关于一般 Calabi-Yau 三倍体的 Gromov-Witten 势的性质。在我之前与 Kota Yoshioka 的合作中，我们提出了曲线上点的相对希尔伯特方案与 Gromov-Witten 不变量之间的一些联系。这是基于特定 Calabi-Yau 三重的经验。在目前的工作中，我将此提案扩展到更一般的情况。另一方面，Gopakumar 和 Vafa 提出了另一个与相对紧凑的雅可比行列式相关的提案。我讨论了在有利的情况下这两个提案的一致性。一路上，我发现了节点曲线上点的希尔伯特方案的欧拉特征的公式，它可以被视为麦克唐纳关于非奇异曲线对称乘积的欧拉特征的经典公式的非平凡概括。