Open Gromov-Witten Theory, Mirror Symmetry, and Toric Geometry

数学和物理在传统上是密切相关的。近年来，由于试图研究亚原子粒子性质的影响，许多新的数学学科被创建。这种新数学的一个重要例子是迅速发展的Gromov-Witten理论。Gromov-Witten理论的重要性的一个原因是它与许多其他学科的深刻联系，如弦理论，偏微分方程的可积系统，几何对象的计数和几何对象的参数空间。另一个例子是镜像对称，这是一种来自物理学的对偶性，在数学中被解释为两个看似完全无关的几何对象的等价性。这种高度非平凡的对应关系本身就非常有趣，同时对理论物理也很重要。本研究项目将探讨Gromov-Witten理论与镜像对称性之间的深层关系中的几个关键问题。这项研究将进一步推进镜像对称、全纯圆盘计数和曲面参数空间几何的知识，也将进一步促进代数几何、辛几何、组合数学、数学物理和弦理论之间的相互作用。本研究项目旨在研究开Gromov-Witten理论和镜像对称性的几个方面：紧致Calabi-Yau流形的开Gromov-Witten不变量的计算;理解用周期积分定义的镜像映射与开Gromov-Witten不变量之间的关系;证明辛复曲面轨道折叠的开Gromov-Witten不变量的镜像对称性陈述;研究了开Gromov-Witten不变量在双有理变换下的行为。PI计划通过将开放的Gromov-Witten不变量与其他量（如封闭的Gromov-Witten不变量）联系起来来研究开放的Gromov-Witten不变量，并推导出开放的Gromov-Witten不变量之间的关系。技术，如虚拟本地化，镜像定理，分析Kuranishi空间，热带几何，赛德尔表示，和量子微分方程将被用于研究。

Mathematics and physics are traditionally closely related. In recent years many new mathematical subjects have been created due to the influence of attempts to study the nature of subatomic particles. One important example of such new mathematics is the rapidly-developing subject of Gromov-Witten theory. One reason for the importance of Gromov-Witten theory is its deep connections with many other subjects, such as string theory, integrable systems of partial differential equations, counting of geometric objects, and parameter spaces for geometric objects. Another example is mirror symmetry, which is a duality arising from physics and is interpreted in mathematics as the equivalence of two seemingly completely unrelated geometric objects. Such a highly nontrivial correspondence is extremely interesting in its own right, while also important for theoretical physics. This research project will explore several key questions in the deep relations between Gromov-Witten theory and mirror symmetry. The research will further advance knowledge about mirror symmetry, enumeration of holomorphic disks, and the geometry of parameter spaces of surfaces, and it will also further promote the existing interactions between algebraic geometry, symplectic geometry, combinatorics, mathematical physics, and string theory. This research project aims to study several aspects of open Gromov-Witten theory and mirror symmetry: calculations of open Gromov-Witten invariants of compact Calabi-Yau manifolds; understanding the relations between mirror maps defined using period integrals and open Gromov-Witten invariants; proving mirror symmetry statements for open Gromow-Witten invariants of symplectic toric orbifolds; and studying the behaviors of open Gromov-Witten invariants under birational transformations. The PI plans to study open Gromov-Witten invariants by relating them to other quantities, such as closed Gromov-Witten invariants, and to derive relations among open Gromov-Witten invariants. Techniques such as virtual localization, mirror theorems, analysis of Kuranishi spaces, tropical geometry, Seidel representations, and quantum differential equations will be used in the research.