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

Research Project

Project/Area Number	13640017
Research Category	Grant-in-Aid for Scientific Research (C)
Allocation Type	Single-year Grants
Section	一般
Research Field	Algebra
Research Institution	KYOTO UNIVERSITY
Principal Investigator	KAWAI Toshiya Kyoto,University, RIMS, Associate Professor, 数理解析研究所, 助教授 (20293970)
Project Period (FY)	2001 – 2003
Project Status	Completed (Fiscal Year 2003)
Budget Amount *help	¥1,600,000 (Direct Cost: ¥1,600,000) Fiscal Year 2003: ¥800,000 (Direct Cost: ¥800,000) Fiscal Year 2002: ¥800,000 (Direct Cost: ¥800,000)
Keywords	String Duality / Calabi-Yau manifolds / Gromov-Witten invariants / Jacobi forms / Elliptic cohomology / ヤコビ形式 / 弦理論 / 双対性 / Calabi-Yau多様体 / Jacobi形式 / 不変式論 / Gromov-Witten不変量 / グロモフ-ウィッテン不変量
Research Abstract	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.