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

摘要奖：DMS-0505835主要研究员：刘晓波这些项目强调紧辛流形的Gromov-Witten不变量，稳定曲线重言环中的关系，以及这些主题与可积系统的相互作用。粗略地说，主Gromov-Witten不变量计算紧致辛流形中伪全纯曲线的个数。一类更大的不变量，称为后代Gromov-Witten不变量，可以通过考虑某些重言丛的第一类Chern在点伪全纯映射模空间上的幂来定义。当流形是点时，Witten猜想，Kontsevich证明了子孙Gromov-Witten不变量是由可积系统的KdV谱支配的。主要研究人员通过研究由稳定曲线的模空间的重言环中的关系产生的普适方程来推广Witten猜想和Virasoro猜想。经典或量子力学系统的几何描述基于辛几何，辛几何是其基本测量二维曲面的面积。辛流形的Gromov-Witten不变量可以描述为一个数族，它计算满足某些约束条件的曲面的数量，例如流形的指定子集相交。这类枚举不变量在代数几何中是一个古老的问题，在代数几何中，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，在高能物理理论中，流形上某些量子系统的配分函数被发现与生成函数有关，这些生成函数将Gromov-Witten不变量族收集成可管理的形式。

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.