Gromov-Witten and Donaldson-Thomas theories in dimensions two and three

第二维和第三维的 Gromov-Witten 和 Donaldson-Thomas 理论

• 批准号: 1406788
• 负责人: Amin Gholampour
• 金额: $ 15.25万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-15 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1406788&HistoricalAwards=false
• 关键词: Gromov; Witten; Donaldson; Thomas; theories

String theory, a branch of physics, predicts that the building blocks of spacetime (which is the fabric of the universe) are certain geometric objects called Calabi-Yau manifolds. The PI will focus on the investigation of the geometric properties of these spaces, mainly by computing important invariants associated to them, called Gromov-Witten and Donaldson-Thomas invariants. Both Gromov-Witten (GW) and Donaldson-Thomas (DT) theories are inspired by theoretical physics and involve several mathematical subjects, including geometry, topology, algebra, combinatorics, and representation theory. The principal investigator will continue his research on the curious conjectural correspondences between GW and DT theories and will try to discover their links to these branches of mathematics and physics. The investigator plans to use the framework provided by these disciplines to create and teach courses for and mentor high school, undergraduate, and graduate students, as well as postdoctoral researchers.More specifically, the principal investigator will investigate the DT invariants of 2-dimensional sheaves in Calabi-Yau threefolds inspired by "M5-brane elliptic genera" and "BPS invariants of D4-D2-D0 systems" studied by string theorists. The PI plans to prove 1) the modularity of these DT invariants and 2) find their relation to the "BPS invariants of D6-D2-D0 systems" predicted from dualities in string theory. The latter BPS invariants are manifested in DT invariants of 1-dimensional sheaves as well as in GW invariants. The PI also plans to develop an algorithm for computing the rank 2 DT invariants of toric threefolds and then study their properties. This is known as the "Topological Vertex Algorithm" and has been a very powerful tool for computing GW and rank 1 DT invariants of the toric threefolds. The advanced tools that will be employed for these projects are localization, deformation, degeneration, categorification, wall-crossing, and mirror symmetry techniques.

弦理论是物理学的一个分支，它预测时空（宇宙的结构）的组成部分是某些被称为卡拉比-丘流形的几何物体。PI将重点研究这些空间的几何性质，主要是通过计算与它们相关的重要不变量，称为Gromov-Witten和Donaldson-Thomas不变量。Gromov-Witten （GW）和Donaldson-Thomas （DT）理论都受到理论物理学的启发，并涉及几个数学学科，包括几何、拓扑、代数、组合学和表示理论。首席研究员将继续研究GW和DT理论之间奇怪的推测对应关系，并试图发现它们与这些数学和物理分支的联系。研究者计划使用这些学科提供的框架来为高中生、本科生、研究生以及博士后研究人员创建和教授课程。更具体地说，首席研究员将研究Calabi-Yau三倍二维轴的DT不变量，其灵感来自弦理论家研究的“m5膜椭圆属”和“D4-D2-D0系统的BPS不变量”。PI计划证明1)这些DT不变量的模性，2)找到它们与从弦理论对偶中预测的“D6-D2-D0系统的BPS不变量”的关系。后一种BPS不变量既表现在一维轴的DT不变量中，也表现在GW不变量中。PI还计划开发一种计算环面三倍的二阶DT不变量的算法，然后研究它们的性质。这被称为“拓扑顶点算法”，已经成为计算环面三倍体的GW和1阶DT不变量的一个非常强大的工具。这些项目将采用的先进工具是本地化、变形、退化、分类、墙壁穿越和镜像对称技术。