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。该项目的目标是进一步开发技术工具和方法，用于帮助从弦理论中提取现实世界的物理，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言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“紧化”，其中额外的维度被卷起在一些小的紧化空间中。可以证明，这个小的紧致空间的几何和拓扑结构决定了低能四维物理。该项目将进一步开发工具和技术，以理解弦理论的紧化，并预测低能四维物理的各种紧化。从技术上讲，夏普教授将研究“测量线性西格玛模型”（GLSM），这是用于研究弦紧化的最强大工具之一。这些工具在过去几年中经历了一系列惊人的进步。其中一部分工作将围绕在二维理论中扩展超对称局域化技术展开。作为另一部分，Sharpe教授将研究二维非abel（2,2）和（0,2）超对称规范理论中的对偶性。Sharpe教授还将继续发展量子束上同调，这是普通量子上同调的数学推广，它决定了异质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abel GLSM构建的物体中的量子束上同调，作为计算紧凑Calabi-Yau流形中的量子束上同调的基石。

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.