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

数学和物理传统上是密切相关的。近年来，由于研究亚原子粒子性质的尝试的影响，许多新的数学学科被创造出来。这种新数学的一个重要例子是迅速发展的格罗莫夫-维腾理论。Gromov-Witten理论重要的一个原因是它与许多其他学科有着深刻的联系，如弦理论、偏微分方程组、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，这是一种源于物理学的二元性，在数学上被解释为两个看似完全不相关的几何对象的等价性。这种高度不平凡的对应本身就非常有趣，同时对理论物理也很重要。本研究项目将探索Gromov-Witten理论与镜面对称性之间深层联系的几个关键问题。这一研究将进一步推进镜像对称性、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、组合学、数学物理和弦理论之间已有的相互作用。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Toric orbilold的开Gromow-Witten不变量的镜像对称性；以及研究开Gromov-Witten不变量在二射变换下的行为。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。研究中将使用虚拟局部化、镜像定理、仓西空间分析、热带几何、赛德尔表示和量子微分方程等技术。

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.