Gromov-Witten Theory of Calabi-Yau Varieties

弦理论是统一自然界中所有力的主要理论候选者。宇宙的弦论模型是十维的，我们通常的四维时空和一个非常小的六维空间一起被称为Calabi-Yau流形。如此小的内部空间将通过某些数学变量来影响我们的时空，例如Gromov-Witten不变量。这些不变量的计算一直是几何和物理学中的中心问题。目前的项目设想开发一个计算Gromov-Witten不变量的理论框架和技术工具。在过去的二十年里，数学和物理之间发生了大量的相互作用。物理学的各种对应关系或对偶关系对数学产生了重大影响。其中之一是兰道-金兹堡/卡拉比-尤的通信。物理学的一个相关发现是，Gromov-Witten不变量的母函数应该是准模形式，这是一个数论对象。该项目旨在开发一个全面的计划，以建立朗道-金兹堡/卡拉比-尤数学中的格罗莫夫-维腾理论的对应性和模块化。一个重要的应用是计算紧致Calabi-Yau流形的Gromov-Witten不变量，这是几何和物理中的一个中心和难点问题。该项目本质上是跨学科的；物理和数学思想都起着至关重要的作用。该项目还涉及博士后和学生的培训。

A leading theoretic candidate to unify all the forces in nature is string theory. The string-theoretic model of the universe is ten-dimensional, our usual four-dimensional space-time together with a very small six-dimensional space called a Calabi-Yau manifold. Such a small internal space will affect our space-time through certain mathematical quantities such as Gromov-Witten invariants. The computation of these invariants has been a central problem in geometry and physics. The current project envisages development of a theoretical framework as well as technical tools to compute Gromov-Witten invariants.During the past twenty years, there has been a great deal of interaction between mathematics and physics. Various correspondences or dualities from physics have had great impact in mathematics. One of these correspondences is the Landau-Ginzburg/Calabi-Yau correspondence. A related discovery from physics is that the generating function for Gromov-Witten invariants should be a quasi-modular form, a number theoretic object. This project aims to develop a comprehensive program to establish Landau-Ginzburg/Calabi-Yau correspondence and modularity of Gromov-Witten theory in mathematics. One important application is to compute Gromov-Witten invariants of compact Calabi-Yau manifolds, a central and still difficult problem in geometry and physics. The project is interdisciplinary in nature; both physical and mathematical ideas play essential roles. The project also involves training of postdocs and students.