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

开放格罗莫夫-维滕理论、镜像对称和环面几何

• 批准号: 1506551
• 负责人: Hsian-Hua Tseng
• 金额: $ 19.2万
• 依托单位: Ohio State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-09-15 至 2019-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1506551&HistoricalAwards=false
• 关键词: Open; Gromov; Witten; Theory; Mirror

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.

传统上，数学和物理学密切相关。近年来，由于研究亚原子颗粒的性质的影响，已经创建了许多新的数学主题。这种新数学的一个重要例子是Gromov-Witten理论的快速发展的主题。 Gromov-Witten理论重要性的原因之一是它与许多其他主题的深入联系，例如字符串理论，部分微分方程的集成系统，几何对象的计数以及几何对象的参数空间。另一个例子是镜像对称性，它是由物理学引起的二元性，并在数学中解释为两个看似完全无关的几何对象的等效性。如此高度不平凡的对应本身非常有趣，同时对理论物理学也很重要。该研究项目将探讨格罗莫夫（Gromov-Witten）理论与镜像对称性之间的深厚关系中的几个关键问题。这项研究将进一步提高有关镜像对称性，全态磁盘的枚举以及表面参数空间几何形状的知识，并且还将进一步促进代数几何形状，符号几何，组合物，数学物理学和弦理论之间的现有相互作用。该研究项目的目的是研究开放式Gromov-Witten理论和镜像对称性的几个方面：紧凑的Calabi-yau歧管的开放式Gromov-witten不变性的计算；了解使用时期积分定义的镜像图和开放式gromov-witten不变性的关系；证明镜像对称性的陈述，用于符合旋转曲线的开放式gromow-witten不变性；并研究在异性转变下的开放性格罗莫夫 - 理通不变性的行为。 PI计划通过将其与其他数量（例如Gromov-Witten Invantiants）联系起来，并在开放的Gromov-witten不变式之间获得关系来研究开放的Gromov-Witten不变性。研究将在研究中使用诸如虚拟定位，镜像定理，kuranishi空间的分析，热带几何形状，SEIDEL表示和量子微分方程等技术。