Gromov-Witten and Donaldson-Thomas theories in dimensions two and three

弦理论是物理学的一个分支，它预言时空（宇宙的结构）的组成部分是某些称为卡-丘流形的几何对象。PI将专注于研究这些空间的几何性质，主要是通过计算与它们相关的重要不变量，称为Gromov-Witten和Donaldson-Thomas不变量。Gromov-Witten（GW）和Donaldson-Thomas（DT）理论都受到理论物理的启发，并涉及几个数学学科，包括几何，拓扑，代数，组合学和表示论。首席研究员将继续他的研究之间的GW和DT理论的好奇artifural对应关系，并将试图发现他们的联系，这些分支的数学和物理。研究人员计划使用这些学科提供的框架来创建和教授课程，并指导高中，本科和研究生以及博士后研究人员。更具体地说，主要研究者将研究Calabi-Yau三重中二维层的DT不变量，其灵感来自于“M5-膜椭圆属”和“D4-D2-D 0系统的BPS不变量”弦理论家们研究的东西PI计划证明1）这些DT不变量的模块性，2）找到它们与“D 6-D2-D 0系统的BPS不变量”的关系，这些不变量是从弦理论的对偶中预测出来的。后者的BPS不变量表现在1维层的DT不变量以及GW不变量中。PI还计划开发一种算法来计算复曲面三重的秩2 DT不变量，然后研究它们的性质。这被称为“拓扑顶点算法”，并且已经成为计算复曲面三重的GW和秩1 DT不变量的非常强大的工具。将用于这些项目的先进工具是本地化，变形，退化，分类，跨壁和镜像对称技术。

String theory, a branch of physics, predicts that the building blocks of spacetime (which is the fabric of the universe) are certain geometric objects called Calabi-Yau manifolds. The PI will focus on the investigation of the geometric properties of these spaces, mainly by computing important invariants associated to them, called Gromov-Witten and Donaldson-Thomas invariants. Both Gromov-Witten (GW) and Donaldson-Thomas (DT) theories are inspired by theoretical physics and involve several mathematical subjects, including geometry, topology, algebra, combinatorics, and representation theory. The principal investigator will continue his research on the curious conjectural correspondences between GW and DT theories and will try to discover their links to these branches of mathematics and physics. The investigator plans to use the framework provided by these disciplines to create and teach courses for and mentor high school, undergraduate, and graduate students, as well as postdoctoral researchers.More specifically, the principal investigator will investigate the DT invariants of 2-dimensional sheaves in Calabi-Yau threefolds inspired by "M5-brane elliptic genera" and "BPS invariants of D4-D2-D0 systems" studied by string theorists. The PI plans to prove 1) the modularity of these DT invariants and 2) find their relation to the "BPS invariants of D6-D2-D0 systems" predicted from dualities in string theory. The latter BPS invariants are manifested in DT invariants of 1-dimensional sheaves as well as in GW invariants. The PI also plans to develop an algorithm for computing the rank 2 DT invariants of toric threefolds and then study their properties. This is known as the "Topological Vertex Algorithm" and has been a very powerful tool for computing GW and rank 1 DT invariants of the toric threefolds. The advanced tools that will be employed for these projects are localization, deformation, degeneration, categorification, wall-crossing, and mirror symmetry techniques.