Research in Geometry, String Compactifications, and Mathematical String Theory

该奖项资助弗吉尼亚理工学院和州立大学埃里克·夏普教授的研究活动。这个项目的目标是进一步开发用于帮助从弦理论中提取现实世界物理学的技术工具和方法，弦理论是统一广义相对论和量子场论的主要竞争者，这是20世纪物理学最重要的两个发展。人们为这种统一付出的代价是预测世界有四个以上的时空维度。解决这种差异的一个标准建议是通过弦理论的“紧凑化”，即在一些小的紧凑空间上卷起额外的维度。可以证明，这个小而紧凑的空间的几何和拓扑决定了低能四维物理。这个项目将进一步开发工具和技术来理解弦理论的紧凑化和对低能四维物理的各种紧凑化的预测。在技术上，夏普教授将研究用于研究弦紧密化的最强大的工具之一--规范线性西格玛模型(GLSM‘s)。在过去的几年里，这些工具经历了一系列惊人的进步。这项努力的一部分将围绕着扩展二维理论中的超对称局部化技术。作为另一部分，夏普教授将研究二维非阿贝尔(2，2)和(0，2)超对称规范理论中的对偶。夏普教授还将继续开发量子束上同调，这是普通量子上同调的数学推广，它确定了对杂性弦紧化中带电物质耦合的弦非微扰修正，特别关注理解Grassmannians和其他通过非交换GLSM建立的物体中的量子束上同调，作为计算紧致Calabi-Yau流形中量子束上同调的垫脚石。

This award funds the research activities of Professor Eric Sharpe at Virginia Polytechnic Institute and State University. The goal of this project is to further develop technical tools and methods used to help extract real-world physics from string theory, the leading contender to unify general relativity with quantum field theory, two of the most significant developments of twentieth-century physics. The price one pays for that unification is a prediction that the world has more than four spacetime dimensions. One standard proposal to resolve that discrepancy is via a `compactification' of string theory, in which the extra dimensions are rolled up on some small compact space. It can be shown that the geometry and topology of that small compact space determine low-energy four-dimensional physics. This project will further develop tools and techniques for understanding compactifications of string theory and the predictions of various compactifications for low-energy four-dimensional physics.Technically, Professor Sharpe will study `gauged linear sigma models' (GLSM's), one of the most powerful tools used to study string compactifications. These tools have undergone a striking series of advances within the last few years. One part of that effort will revolve around extending the technology of supersymmetric localization in two-dimensional theories. As another part, Professor Sharpe will study dualities in nonabelian (2,2) and (0,2) supersymmetric gauge theories in two dimensions. Professor Sharpe will also continue developing quantum sheaf cohomology, a mathematical generalization of ordinary quantum cohomology that determines stringy nonperturbative corrections to charged matter couplings in heterotic string compactifications, with a special focus on understanding quantum sheaf cohomology in Grassmannians and other objects built via nonabelian GLSM's, as a stepping-stone to computing quantum sheaf cohomology in compact Calabi-Yau manifolds.