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

该项目主要关注代数几何中的两个相关主题，即多项式方程系统的解的研究，这是由它们与物理理论（弦理论和规范理论）的联系所激发的。第一个主题是Gromov-Witten （GW）理论，粗略地说，它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，在平面上有四条经三个给定点和两条给定直线的曲线。第二个主题是Donaldson-Thomas （DT）理论，研究在空间中定义具有一定拓扑约束的一维对象的多项式的空间参数化集的性质。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，通过计算它们产生的不变量，揭示了几何物体的深层性质。在这个项目中，PI将调查这两种理论之间的推测对偶性，并将它们与数学和物理的其他分支联系起来。更详细地说，本提案中的项目旨在研究代数几何中几何对象的模空间的枚举性质的几个方面。第一个主题是Gromov-Witten理论。PI将研究GW不变量之间的双分变换，计算辛Deligne-Mumford堆栈的量子上同调，并将量子连接的单调性与派生范畴之间的等价性产生的单调性联系起来。第二个主题是Donaldson-Thomas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。

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.