Quantum geometry of orbifolds and their resolutions

在弦理论的物理学中，宇宙是一个十维空间。虽然其中四个维度包括通常的空间和时间概念，但其余六个维度卷曲成微小的复杂几何空间，称为Calabi-Yau三重空间。理解Calabi-Yau三折线的几何对于代数几何和物理学都是至关重要的。在弦理论中，Calabi-Yau三重的几何性质决定了通过它的粒子(即弦)的物理性质。在代数几何中，Calabi-Yau三重空间在格罗莫夫-威腾理论、Donaldson-Thomas理论和双曲几何等诸多研究领域中都有着特殊的作用。例如，弦理论中的Calabi-Yau三重空间通常被认为是光滑空间，但也可以是温和奇异的空间，称为orbiolds。在代数几何中，通过所谓的奇点分解，可以用光滑空间代替奇异空间。与Calabi-Yau条件相容的奇点分解称为Crepant分解。物理学中的一条指导原则认为，弦理论在任何分辨率下都应该等价于弦理论。此外，与Calabi-Yau三重不变量相关的不变量有多种-它们的Gromov-Witten不变量，它们的Donaldson-Thomas不变量，以及它们的Gopakumar-Vafa不变量。这些不变量在弦理论中都有物理解释。这个项目的长期目标是在数学上理解这样一个原理，即在任何一种确定的解上，弦理论与弦理论是等价的。具体地说，我想在一些重要的情况下公式化并证明各种Crepant归结猜想，即与Calabi-Yau orbiold相关的各种不变量与它的Crepant归结(S)之间的等价性。

In the physics of string theory, the universe is a ten dimensional space. While four of the dimensions comprise the usual notions of space and time, the remaining six are curled up into tiny complicated geometric spaces called Calabi-Yau threefolds. Understanding the geometry of Calabi-Yau threefolds is central to both algebraic geometry and physics. In string theory, geometric properties of a Calabi-Yau threefold determine the physical properties of the particles (i.e. strings) that move through it. In algebraic geometry, Calabi-Yau threefolds play a special role in a variety of fields of study, for example Gromov-Witten theory, Donaldson-Thomas theory, and birational geometry.      The Calabi-Yau threefolds of string theory are usually taken to be smooth spaces, but can also be mildly singular spaces called orbifolds. In algebraic geometry, a singular space can be replaced by a smooth space by a so-called resolution of singularities. A resolution of singularities that is compatible with the Calabi-Yau condition is called a crepant resolution. A guiding principle in physics asserts that string theory on an orbifold should be equivalent to string theory on any crepant resolution.     There are a variety of invariants associated to Calabi-Yau threefolds --- their Gromov-Witten invariants, their Donaldson-Thomas invariants, and their Gopakumar-Vafa invariants. These invariants all have physical interpretations in string theory. The long term objective of this project is to make mathematical sense of the principle that string theory on an orbifold is equivalent to string theory on any of its crepant resolutions. In particular, I want to formulate, and prove in some important cases, the various Crepant Resolution Conjectures, namely the equivalence between the various invariants associated to a Calabi-Yau orbifold and its crepant resolution(s).