Gromov-Witten invariants and integrable systems

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。本文拟研究紧辛流形的Gromov-Witten不变量的普适方程和其他性质，以及它们与可积系统的相互作用。利用从Riemann曲面到紧辛流形的稳定伪全纯映射的模空间上的交点理论定义了Gromov-Witten不变量。他们对紧辛流形中伪全纯曲线的计数有直观的解释。通称方程是控制所有紧辛流形的Gromov-Witten不变量的基本行为的偏微分方程，以及高自旋曲线模空间上的交数。它们为这些不变量的计算提供了强大的机制，并在Eguchi-Hori-Xiong和S. Katz的Virasoro猜想的研究中发挥了重要作用。Virasoro猜想是Witten的KdV猜想（由Kontsevich证明）的推广，是连接Gromov-Witten理论和可积系统的桥梁。通称方程的研究也有助于理解曲线模空间的重言环的结构。Gromov-Witten不变量理论在辛几何、代数几何、规范理论和可积系统中有着重要的应用。所提议的工作涉及这一理论的基本结构。除了在数学上的应用外，Gromov-Witten理论还对应于弦理论中的拓扑sigma模型。因此，所提出的项目也与理论物理有很强的联系。这个领域的一些最深刻的猜想，包括这个提议中的一些猜想，实际上来自物理学家。本文提出的问题的解决方案将为物理学家的直觉提供严格的支持，并有助于理解弦理论中的拓扑sigma模型。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). In this proposal, the proposer intends to study universal equations and other properties of Gromov-Witten invariants of compact symplectic manifolds, as well as their interaction with integrable systems. Gromov-Witten invariants are defined by the intersection theory on moduli spaces of stable pseudo-holomorphic maps from Riemann surfaces to compact symplectic manifolds. They have intuitive interpretation of counting pseudo-holomorphic curves in compact symplectic manifolds. Universal equations are partial differential equations which govern fundamental behavior of Gromov-Witten invariants of all compact symplectic manifolds, as well as for intersection numbers on moduli spaces of higher spin curves. They provide powerful machineries in the computation of these invariants and also play important roles in the study of the Virasoro conjecture of Eguchi-Hori-Xiong and S. Katz. The Virasoro conjecture is a generalization of Witten's KdV conjecture (proved by Kontsevich) and serves as a bridge between Gromov-Witten theory and integrable systems. The study of universal equations also helps the understanding of structures of tautological rings of moduli spaces of curves.The theory of Gromov-Witten invariants has important applications in symplectic geometry, algebraic geometry, gauge theory, and integrable systems. The proposed work concerns the fundamental structures of this theory. Besides its applications in mathematics, the Gromov-Witten theory corresponds to topological sigma models in string theory. Therefore the proposed project also has strong connections with theoretic physics.Some of the deepest conjectures in this field, including some conjectures in this proposal, actually came from physicists. The resolution of problems proposed here will give a rigorous support to intuitions of physicists and helps the understanding of the topological sigma model in string theory.