Gromov-Witten invariants and integrable systems

Gromov-Witten 不变量和可积系统

• 批准号: 0905227
• 负责人: Xiaobo Liu
• 金额: $ 18.68万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-07-15 至 2013-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0905227&HistoricalAwards=false
• 关键词: Gromov; Witten; invariants; integrable; systems

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.

该奖项根据 2009 年美国复苏和再投资法案（公法 111-5）提供资金。在该提案中，提出者打算研究通用方程和紧辛流形的 Gromov-Witten 不变量的其他性质，以及它们与可积系统的相互作用。 Gromov-Witten 不变量由从黎曼曲面到紧辛流形的稳定伪全纯映射模空间的交集理论定义。他们对紧辛流形中的伪全纯曲线计数有直观的解释。通用方程是偏微分方程，它控制所有紧辛流形的 Gromov-Witten 不变量的基本行为，以及较高自旋曲线模空间上的交集数。它们为计算这些不变量提供了强大的机制，并且在 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 KdV 猜想（由 Kontsevich 证明）的推广，是 Gromov-Witten 理论和可积系统之间的桥梁。普适方程的研究也有助于理解曲线模空间同义反复环的结构。格罗莫夫-维滕不变量理论在辛几何、代数几何、规范论和可积系统中有着重要的应用。拟议的工作涉及该理论的基本结构。除了在数学中的应用之外，格罗莫夫-维滕理论还对应于弦理论中的拓扑西格玛模型。因此，这个提议的项目也与理论物理学有很强的联系。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。这里提出的问题的解决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西格玛模型。