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

本研究项目的第一个成果是，在弦对偶猜想的基础上，通过研究椭圆曲线上合适的共形场论，在一定极限下，统一构造了截面在Hirzebruch曲面上纤维化的椭圆型Calabi-Yau三折的Gromov-Witten势的猜想表达式。我对这个结果进行了各种重要的一致性检查。研究揭示了椭圆上同、Borcherds积、Jacobi形和双曲反射群之间的丰富关系。我目前正在为我得到的结果准备一篇论文。该项目的第二个结果已经作为一篇文章（“String and Vortex”）进行了报道。讨论了泛型Calabi-Yau三倍体的Gromov-Witten势的性质。在我之前与Kota Yoshioka的工作中，我们提出了曲线上点的相对希尔伯特格式与Gromov-Witten不变量之间的一些联系。这是根据具体的卡拉比-丘三倍的经验得出的。在目前的工作中，我将这一建议扩展到更一般的情况。另一方面，Gopakumar和Vafa提出了另一种与相对紧化雅可比矩阵相关的建议。我在有利的情况下讨论了两个建议的一致性。在这个过程中，我发现了一个关于节点曲线上点的希尔伯特方案的欧拉特性的公式，它可以被视为麦克唐纳关于非奇异曲线对称积的欧拉特性的经典公式的非平凡推广。

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.