Wall-crossing and quantum invariants of Calabi-Yau threefolds

背景资料： 卡-丘三重性是代数几何数学和弦论物理的中心研究对象。在过去的几十年里，出现了卡-丘三重的微妙不变量，通常在数学和物理学中有平行的描述。在数学中，不变量来自于与卡-丘三重性相关联的各种模空间的几何;在物理学中，不变量来自于与卡-丘三重性相关联的各种量子场论。 这些不变量已经被证明具有惊人的丰富结构和惊人的许多连接到其他数学分支。因此，在过去15年中，它们已成为深入研究的对象。不变量的原始实例来自Gromov-Witten理论和Donaldson-Thomas理论。这些理论产生于卡拉比-丘几何的根本不同的方面：格罗莫夫-威滕理论来自于三重映射的模空间，唐纳森-托马斯理论来自于三重层上层的模。然而，这两个理论以一种非凡的方式被证明是等价的。这种对应是物理学中弦理论和规范理论之间深层对应的数学版本。 该项目的目标是完善基本的Gromov-Witten/Donaldson-Thomas对应关系，以包括Donaldson-Thomas理论中最近发现的许多结构。具体来说，Donaldson-Thomas不变量的动机改进和相关的跨壁现象应该在Gromov-Witten理论中有所体现。这种穿墙是物理学中边际稳定性思想的数学表现：随着理论参数的变化，粒子可以变得不稳定或稳定，从而从粒子谱中消失或出现。

Background: Calabi-Yau threefolds are central objects of research in both the mathematics of algebraic geometry and the physics of string theory. In the last few decades, subtle invariants of Calabi-Yau threefolds have arisen, often having parallel descriptions in math and physics. In math, the invariants are born from the geometry of various moduli spaces associated to the Calabi-Yau threefold; in physics, the arise out of various quantum field theories associated to the Calabi-Yau threefold. These invariants have turned out to have amazingly rich structure and surprisingly many connections to other branches of mathematics. They have consequently become objects of intense study in the last 15 years. The primal instances of the invariants are those from Gromov-Witten theory and from Donaldson-Thomas theory. These theories arise from fundamentally different aspects of the geometry of the Calabi-Yau: Gromov-Witten theory comes from the moduli space of maps to the threefold, Donaldson-Thomas theory comes from the moduli of sheaves on the threefold. However, these two theories are famously conjectured to be equivalent in a remarkable way. This correspondence is a mathematical version of a deep correspondence between string theory and gauge theory in physics. The goal of this project is to refine the basic Gromov-Witten/Donaldson-Thomas correspondence to include many of the structures recently discovered in Donaldson-Thomas theory. Specifically, the motivic refinements of Donaldson-Thomas invariants and the associated wall-crossing phenomena should have manifestations in Gromov-Witten theory. This wall-crossing is a mathematical manifestation of the idea of marginal stability in physics: as the parameters of a theory change, particles can become unstable or stable and consequently will disappear or appear from the particle spectrum.