Topics in Algebraic Geometry: Gromov-Witten Theory and Donaldson-Thomas Theory

代数几何专题：格罗莫夫-维滕理论和唐纳森-托马斯理论

• 批准号: 1600997
• 负责人: Yunfeng Jiang
• 金额: $ 13.33万
• 依托单位: University of Kansas Center for Research Inc
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-09-01 至 2020-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1600997&HistoricalAwards=false
• 关键词: Topics; Algebraic; Geometry; Gromov; Witten

This project focuses on two related topics in algebraic geometry, the study of solutions of systems of polynomial equations, that are motivated by their connections with physical theories (string theory and gauge theory). The first topic is Gromov-Witten (GW) theory, which roughly speaking is about a systematic way of counting numbers of curves with particular constraints in a space defined by a set of polynomial equations. For example on a plane there are four conics passing three given points and two given lines in general positions. The second topic is Donaldson-Thomas (DT) theory, which studies properties of the space parametrizing sets of polynomials defining one dimensional objects with certain topological constraints in a space. In a so-called Calabi-Yau space, a series of remarkable conjectures say that counting using GW theory is equivalent to that using DT theory. Different branches of mathematics are linked together by these two theories and deep properties of geometric objects have been uncovered by calculating invariants they give rise to. In this project the PI will investigate a conjectured duality between these two theories and relate them to other branches of mathematics and physics. In more detail, the projects in this proposal are designed to study several aspects of enumerative properties of the moduli spaces of geometric objects in algebraic geometry. The first topic is Gromov-Witten theory. The PI will study birational transformations between GW invariants, calculate the quantum cohomology of symplectic Deligne-Mumford stacks, and relate the monodromy of the quantum connections to the monodromy coming from the equivalence between the derived categories. The second topic is Donaldson-Thomas theory. The PI will study local DT invariants via Berkovich analytic spaces, apply the cotangent invariants of the PI and R. Thomas to find more dualities for geometric spaces, and investigate the motivic Donaldson-Thomas invariants by the method of motivic integration.

该项目侧重于代数几何学的两个相关主题，即多项式方程系统解决方案的研究，它们是由它们与物理理论的联系（弦理论和仪表理论）的动机。第一个主题是Gromov-Witten（GW）理论，该理论大概是关于在由一组多项式方程定义的空间中计算曲线数量的系统方式。例如，在平面上有四个圆锥形通过三个给定点和两条给定线的一般位置。第二个主题是Donaldson-Thomas（DT）理论，该理论研究了定义一个尺寸对象的多项式空间参数化集的属性，并在空间中具有某些拓扑约束。在所谓的Calabi-yau空间中，一系列出色的猜想表明，使用GW理论进行计数等同于使用DT理论。这两种理论将数学的不同分支链接在一起，并且通过计算产生的不变式来发现几何对象的深度特性。在这个项目中，PI将研究这两种理论之间的二元性，并将其与数学和物理学的其他分支相关联。更详细地，该提案中的项目旨在研究代数几何形状中几何对象模量空间的枚举特性的几个方面。第一个主题是Gromov-Witten理论。 PI将研究GW不变性之间的异性转化，计算符号deLigne-Mumford堆栈的量子同谋，并将量子连接的单构型与来自派生类别之间的等效性产生的量子连接的单差。第二个主题是唐纳森 - 托马斯理论。 PI将通过Berkovich分析空间研究局部DT不变性，应用Pi和R. Thomas的无关紧要的几何二重性，并通过动机整合方法研究动机Donaldson-Thomas不变性。