Gromov - Witten invariants and integrable hierarchies

格罗莫夫 - 维滕不变量和可积层次结构

• 批准号: 0306316
• 负责人: Alexander Givental
• 金额: $ 12万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-08-01 至 2006-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0306316&HistoricalAwards=false
• 关键词: Gromov; Witten; invariants; integrable; hierarchies

AbstractAward: DMS-0306316Principal Investigator: Alexander GiventalGromov-Witten invariants characterize global properties of phasespaces in Hamiltonian mechanics. The project concerns withvarious problems of constructing and generalizing Gromov-Witteninvariants, developing methods of their computation, interpretingthe geometrical information they encode and understanding therules that govern relationships among them. In particular, therole of integrable hierarchies and their Virasoro symmetries inGromov-Witten theory and in its mirror partner - singularitytheory - is to be investigated. To this end, certain vertexoperator constuctions based on the theory of vanishing cycles insingularity theory will be explored as a promising tool fordescribing and manipulating with integrablehierarchies. Exploiting the twisted loop group of hidensymmetries discovered recently in the structure of Gromov-Witteninvariants and uncovering the origin of the symmetries are amongkey motives for this project.In a broader prospective, the projects continues the legacy ofthe past two centures in the development and application ofmodern mathematics. The methods of algebraic geometry, topology,group theory and quantum mechanics laid down in the classicalwork of Gauss, Riemann, Lie, Klein, Poincare and Weyl cometogether to address the problems posed lately by string theory inits search for the ultimate unity of fundamental interactions.

摘要奖：DMS-0306316主要研究者：Alexander GiventalGromov-Witten不变量刻画了哈密顿力学中相空间的整体性质。该项目涉及构造和推广Gromov-Wittenant不变量、开发它们的计算方法、解释它们所编码的几何信息以及理解它们之间关系的规则等各种问题。特别是，可积族及其Virasoro对称性在Gromov-Witten理论及其镜像伙伴奇点理论中的作用有待研究。为此，基于奇点理论中的零圈理论的某些顶点算子构造将被探索为描述和操纵可积层次的一个有前途的工具。利用最近在Gromov-Wittenant不变量结构中发现的隐藏对称性的扭曲环群并揭示对称性的起源是该项目的关键动机。从更广泛的角度来看，这些项目延续了过去两个世纪在现代数学发展和应用方面的遗产。在Gauss、Riemann、Lie、Klein、Poincare和Weyl Come的经典著作中，代数几何、拓扑学、群论和量子力学的方法共同解决了弦理论最近提出的问题，以寻求基本相互作用的终极统一。