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.

该项目重点关注代数几何中的两个相关主题，即多项式方程组解的研究，其动机是它们与物理理论（弦理论和规范理论）的联系。第一个主题是 Gromov-Witten (GW) 理论，粗略地说，该理论是关于在由一组多项式方程定义的空间中计算具有特定约束的曲线数量的系统方法。例如，在平面上，在一般位置上有四个二次曲线通过三个给定点和两条给定线。第二个主题是唐纳森-托马斯 (DT) 理论，该理论研究空间参数化多项式集的属性，这些多项式定义了空间中具有某些拓扑约束的一维对象。在所谓的 Calabi-Yau 空间中，一系列引人注目的猜想表明，使用 GW 理论进行计数与使用 DT 理论进行计数是等价的。这两种理论将数学的不同分支联系在一起，并且通过计算它们产生的不变量来揭示几何对象的深层属性。在这个项目中，PI 将研究这两种理论之间推测的二元性，并将它们与数学和物理学的其他分支联系起来。更详细地说，该提案中的项目旨在研究代数几何中几何对象模空间的枚举性质的几个方面。第一个主题是格罗莫夫-维滕理论。 PI 将研究 GW 不变量之间的双有理变换，计算辛 Deligne-Mumford 堆栈的量子上同调，并将量子连接的单向性与来自派生类别之间的等价性的单向性联系起来。第二个主题是唐纳森-托马斯理论。 PI将通过Berkovich解析空间研究局部DT不变量，应用PI和R.Thomas的余切不变量来寻找几何空间的更多对偶性，并通过动机积分的方法研究动机Donaldson-Thomas不变量。