Gromov - Witten invariants and integrable hierarchies

AbstractAward：DMS-0306316主要研究者：亚历山大GiventalGromov-Witten不变量描述了哈密顿力学中相空间的全局性质。该项目关注的是构建和推广Gromov-Witteninvariants的各种问题，开发其计算方法，解释它们编码的几何信息并理解管理它们之间关系的规则。特别是，therole的可积阶层和Virasoro对称在Gromov-Witten理论和它的镜像伙伴-奇异理论-是要调查。为此，基于奇点理论中的消失圈理论的某些顶点算子构造将被探索作为描述和操纵可积层次的有前途的工具。利用最近在Gromov-Witteninvariants结构中发现的隐藏对称性的扭环群，揭示对称性的起源，是该项目的重要动机，从更广阔的角度看，该项目延续了过去两个世纪现代数学发展和应用的遗产。 高斯、黎曼、李、克莱因、庞加莱和魏尔的经典著作中所提出的代数几何、拓扑学、群论和量子力学的方法，结合在一起，解决了最近弦理论在寻找基本相互作用的终极统一时所提出的问题。

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.