Gromov-Witten Invariants, Moduli Spaces of Curves, and Integrable Systems

Gromov-Witten 不变量、曲线模空间和可积系统

• 批准号: 0505835
• 负责人: Xiaobo Liu
• 金额: $ 12.19万
• 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: 2005
• 资助国家: 美国
• 起止时间: 2005-06-01 至 2009-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0505835&HistoricalAwards=false
• 关键词: Gromov; Witten; Invariants; Moduli; Spaces

AbstractAward: DMS-0505835Principal Investigator: Xiaobo LiuThese projects emphasize Gromov-Witten invariants of compactsymplectic manifolds, relations in the tautological rings ofstable curves, and interactions of these topics with integrablesystems. Roughly speaking, primary Gromov-Witten invariantscount numbers of pseudo-holomorphic curves in compact symplecticmanifolds. A larger class of invariants, called descendentGromov-Witten invariants, can be defined by taking intoconsideration powers of first Chern classes of some tautologicalline bundles over the moduli space of pointed pseudo-holomorphicmaps. In the case when the manifold is a point it wasconjectured by Witten and proved by Kontsevich that descendantGromov-Witten invariants are governed by the KdV hierarchy fromintegrable systems. The principal investigator is pursuinggeneralizations of the Witten conjecture and the Virasoroconjecture on by studying universal equations which come fromrelations in the tautological rings of the moduli spaces ofstable curves.Geometric descriptions of classical or quantum mechanical systemsare based in symplectic geometry, which takes as its basicmeasurement the area of two-dimensional surfaces. TheGromov-Witten invariants of a symplectic manifold can bedescribed as a family of number that count the number of surfacesof a nice sort that satisfy certain constraints, such asintersecting specified subsets of the manifold. Enumerativeinvariants of this kind are an ancient concern in algebraicgeometry, where we count the number of solutions to a system ofequations, and the subject was recently found to be connected tohigh energy physical theory, where the partition functions ofcertain quantum systems on a manifold have been found to berelated to generating functions that collect families ofGromov-Witten invariants into a manageable form.

摘要奖项：DMS-0505835首席研究员：刘晓波这些项目强调紧辛流形的 Gromov-Witten 不变量、稳定曲线同义反复环中的关系以及这些主题与可积系统的相互作用。 粗略地说，初级 Gromov-Witten 不变量计算紧辛流形中伪全纯曲线的数量。 更大的一类不变量，称为后代格罗莫夫-维滕不变量，可以通过考虑一些同义同义线丛的第一陈类在尖伪全纯映射的模空间上的考虑能力来定义。 在流形是点的情况下，由 Witten 猜想并由 Kontsevich 证明，后代 Gromov-Witten 不变量受可积系统的 KdV 层次结构控制。 主要研究者正在通过研究来自稳定曲线模空间同义反复环关系的通用方程来追求维滕猜想和维拉索罗猜想的推广。经典或量子力学系统的几何描述基于辛几何，辛几何将二维表面的面积作为其基本测量。 辛流形的格罗莫夫-维滕不变量可以描述为一个数字族，它计算满足某些约束（例如流形的指定子集）的良好排序表面的数量。 这种枚举不变量是代数几何中一个古老的问题，我们计算方程组的解的数量，最近发现这个主题与高能物理理论有关，其中流形上某些量子系统的配分函数被发现与将格罗莫夫-维滕不变量族收集成可管理形式的生成函数有关。