Gromov-Witten Theory of Calabi-Yau Varieties

弦理论是统一自然界所有力的一个主要理论候选者。弦论的宇宙模型是十维的，我们通常的四维时空加上一个非常小的六维空间，称为卡-丘流形。 如此小的内部空间将通过某些数学量（如Gromov-Witten不变量）影响我们的时空。这些不变量的计算一直是几何学和物理学的中心问题。 目前的项目设想发展一个理论框架以及技术工具来计算Gromov-Witten不变量。在过去的二十年里，数学和物理之间有很多的相互作用。物理学中的各种对应或对偶在数学中产生了巨大的影响。这些对应之一是朗道-金兹伯格/卡拉比-丘对应。物理学的一个相关发现是Gromov-Witten不变量的生成函数应该是一个准模形式，一个数论对象。该项目旨在开发一个全面的程序，以建立数学中的Landau-Ginzburg/Calabi-Yau对应和Gromov-Witten理论的模块性。一个重要的应用是计算紧致卡-丘流形的Gromov-Witten不变量，这是几何和物理中的一个中心问题，也是一个困难的问题。该项目是跨学科的性质;物理和数学思想发挥重要作用。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.