Geometry of Pseudoholomorphic Curves and Gromov-Witten Invariants

伪全纯曲线的几何和 Gromov-Witten 不变量

• 批准号: 0604874
• 负责人: Aleksey Zinger
• 金额: $ 11.2万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0604874&HistoricalAwards=false
• 关键词: Geometry; Pseudoholomorphic; Curves; Gromov; Witten

AbstractAward: DMS-0604874Principal Investigator: Aleksey ZingerThe theory of Gromov-Witten invariants plays a prominent role insymplectic topology, enumerative algebraic geometry, and stringtheory. Via Gromov-Witten theory, string theorists have madecompletely unexpected predictions concerning counts of complex(holomorphic) curves in algebraic manifolds. Some of thesepredictions have been verified mathematically, but most havenot. Many connections between Gromov-Witten theory andenumerative geometry have been discovered independently of stringtheory as well. However, many others remain to be found. The mostfundamental object in the Gromov-Witten theory is the modulispace of (pseudo-) holomorphic maps. The PI has developed anapproach for studying its local structure using analytictechniques of symplectic topology and a separate topologicalapproach for recovering global information from the local data.Combined together, the two approaches have led to a variety ofresults, in enumerative geometry and in symplectic topology, forcounts of curves of low genus, primarily zero and one. Among themis a geometric relation between genus-one invariants of acomplete intersection and those of the ambient projectivespace. One objective of this project is to verify thelong-standing mirror symmetry conjecture for counts of genus-onecurves in the quintic threefold, using this geometricrelation. Another objective is to apply the results to countinggenus-one curves in projective varieties. However, the primaryobjective is to extend the detailed description of the modulispace of genus-one maps already obtained to higher-genus cases,especially genus two.The broader impact of this project is potentially farranging. Its aim is to advance the fundamental understanding ofthe Gromov-Witten theory, which in turn should lead to newapplications in symplectic topology and enumerativegeometry. Furthermore, it should open a way for testing a numberof mathematical predictions of string theory. If some of thesepredictions were shown to fail, string theory would require atleast some modification, perhaps with implications forunderstanding physical phenomena.

AbstractAward：DMS-0604874首席研究员：Aleksey Zinger Gromov-Witten不变量理论在辛拓扑学、枚举代数几何和弦理论中起着重要作用。 通过Gromov-Witten理论，弦理论家对代数流形中复（全纯）曲线的计数有了完全出乎意料的预言。其中一些预测已经得到了数学上的验证，但大多数还没有。Gromov-Witten理论和计数几何之间的许多联系也独立于弦理论而被发现。Gromov-Witten理论中最基本的对象是（伪）全纯映射的模空间。 PI已经开发了一种使用辛拓扑的分析技术来研究其局部结构的方法，以及一种从局部数据中恢复全局信息的单独的拓扑方法。结合在一起，这两种方法在枚举几何和辛拓扑中产生了各种结果，用于低亏格曲线的计数，主要是零和一。其中包括完全交的亏格一不变量与周围射影空间的亏格一不变量之间的几何关系。本项目的一个目标是利用这个几何关系来验证长期以来关于五次三重曲线中亏格为一的曲线数的镜像对称猜想。另一个目的是将结果应用于射影簇中的countinggenus-one曲线。 然而，我们的主要目标是将已经获得的亏格1映射的模空间的详细描述扩展到更高亏格的情况，特别是亏格2。其目的是推进Gromov-Witten理论的基本理解，这反过来又会导致辛拓扑和枚举几何的新应用。此外，它应该为检验弦理论的一些数学预测开辟一条道路。如果这些预言中的一些被证明是失败的，那么弦理论至少需要一些修改，也许会对理解物理现象产生影响。