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

本研究的第一个成果是，在弦对偶猜想的基础上，通过研究椭圆曲线上适当的共形场理论，一致地构造了三重椭圆的Gromov-Witten势的猜想表达式，其中三重椭圆Calabi-Yau的截面位于Hirzebruch面上。我对这个结果进行了各种不平凡的一致性检查。这项研究揭示了与椭圆上同调、Borcherds积、Jacobi形式和双曲反射群之间的各种丰富的关系。我目前正在为我所取得的成果准备一篇论文。这个项目的第二个成果已经作为一篇文章(《弦与漩涡》)报道了。它是关于一般的三重Calabi-Yau的Gromov-Witten势的性质。在我以前和Kota Yoshioka的工作中，我们提出了曲线上点的相对Hilbert格式和Gromov-Witten不变量之间的一些联系。这是基于对特定的Calabi-Yau三折的经验。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提出的另一项建议与相对紧凑的雅各比人有关。我讨论了在有利的情况下这两项建议的一致性。一路上，我找到了节点曲线上点的希尔伯特格式的欧拉特征的一个公式，它可以看作是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.