Topological Field Theory and Integrable Systems

DMS-0072508Ezra Getzler紧致辛流形的Gromov-Witten不变量是具有背景V的二维拓扑重力的相关函数。这些不变量将代数几何的计数不变量推广为平面上亏格为g和d次的曲线的个数到3d+g-1点。对于每个亏格g，V的亏格g Gromov-Witten势是亏格g Gromov-Witten不变量的生成函数。本课题旨在研究这些位势所满足的微分方程及其几何意义。亏格零Gromov-Witten势满足Witten-Dijkgraaf-Verlinde-Verlinde方程，该方程在Dubrovin的Frobenius流形理论中找到了它的几何形式。PI以前的工作已经证明，亏格1的Gromov-Witten势满足定义在任何Frobenius流形上的一个微分方程。此外，Dubrovin和Zhang证明了该方程对任何半单Frobenius流形都有唯一解。PI建议研究亏格g大于1的类似情况以及它的多种结果。众所周知，平面上有一条线穿过两个点。同样，有一条二次曲线穿过平面上的三个点。Gromov-Witten不变量理论是对此的巨大推广：从直线推广到一般的代数曲线，从平面推广到更一般的空间。由此产生的计数问题与描述浅水中波的Kortweg-de Vrijs方程等可积系统的理论有关。在这个项目中，我们遵循杜布罗文和张的想法，试图更好地理解这种联系。因此，这个项目调查了最初由物理学推动的枚举几何领域的重大和令人惊讶的发展，并有可能阐明一些与现代物理学相关的问题。

DMS-0072508Ezra GetzlerThe Gromov-Witten invariants of a compact symplectic manifold are the correlation functions of a two-dimensional topological gravity with background V. These invariants generalize such enumerative invariants of algebraic geometry as the number of curves of genus g and degree d through 3d+g-1 points in the plane. For each genus g, the genus g Gromov-Witten potential of V is the generating function of the genus g Gromov-Witten invariants. This project intends to study the differential equationssatisfied by these potentials and their geometric significance. The genus zero Gromov-Witten potential satisfies theWitten-Dijkgraaf-Verlinde-Verlinde equation, which finds its geometric form in Dubrovin's theory of Frobenius manifolds. Previous work of the PI has shown that the genus one Gromov-Witten potential satisfies a differential equation defined on any Frobenius manifold. Moreover, Dubrovin andZhang have shown that this equation has a unique solution for any semisimple Frobenius manifold. The PI proposes to investigate analogous situation for genus g greater than one and its manifold consequences.It is well known that there is one line through two points in the plane. Similarly, there is one quadratic curve through three points in the plane. The theory of Gromov-Witten invariants is a tremendous generalization of this: one generalizes from lines to general algebraic curves, and from the plane to more general spaces. The resulting counting problems are related to the theory of such integrable systems as the Kortweg-de Vrijs equation describing waves in shallow water. In this project, following ideas of Dubrovin and Zhang, we attempt to understand this link better. Thus, thisproject investigates significant and surprising developments in enumerative geometry that were originally motivated physics, and potentially should cast light on a number of questions of relevance to modern physics.