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）的组成部分是某些称为卡拉比-丘流形的几何对象。 PI 将重点研究这些空间的几何特性，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，并涉及多个数学学科，包括几何、拓扑、代数、组合学和表示论。首席研究员将继续研究 GW 和 DT 理论之间奇怪的猜想对应关系，并尝试发现它们与数学和物理学这些分支的联系。研究人员计划利用这些学科提供的框架为高中生、本科生和研究生以及博士后研究人员创建和教授课程并指导他们。更具体地说，主要研究人员将研究二维滑轮的 DT 不变量卡拉比-丘的三重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 不变量的模块化，以及 2) 找到它们与根据弦理论对偶性预测的“D6-D2-D0 系统的 BPS 不变量”的关系。后者的 BPS 不变量表现为一维滑轮的 DT 不变量以及 GW 不变量。 PI 还计划开发一种算法来计算三重环面的 2 阶 DT 不变量，然后研究它们的属性。这被称为“拓扑顶点算法”，是计算三重环面的 GW 和 1 阶 DT 不变量的非常强大的工具。这些项目将使用的先进工具包括定位、变形、退化、分类、穿墙和镜像对称技术。