Algebraic Geometry in String Theory

本文探讨了代数几何与量子场论和弦论相互作用的一些主要研究领域，包括：超黎曼曲面的模以及超弦测量和超弦摄动理论的相关问题；几何朗兰兹程序；异质弦现象学；F理论;和量子束上同调。该提案还考虑了曲线和阿贝尔变体的模空间中的一些问题，以及一些更具探索性的数学/物理联系，如振幅，格拉斯曼和扭转；以及关于六维共形场理论及其数学结果的一些猜想。这个项目的意义在于发展数学（主要是代数几何）和高能物理（主要是弦理论）之间的联系。这些领域中的每一个都涉及数学和物理学的混合问题，并且大多数将在涉及数学家和物理学家的团队中进行探索。特别是第一个研究领域预计将对微扰超弦理论的（数学）基础产生重大影响，而第二个研究领域则使用新工具解决数学中的一个主要开放问题，至少部分是由高能物理思想启发的。PI还建议继续开展范围广泛的影响广泛的社区和教育活动，包括：发展和指导一系列强调物理和数学相互作用的会议，以及传播有关数学和高能物理相互作用的新知识的其他渠道；研究生和本科生的课程发展；为数学家编写字符串文本；与研究生和本科生的广泛合作以及对宾夕法尼亚大学数学专业的评估；是美国医学辅助学会专业委员会等国家机构的成员，并担任几本期刊和丛书的编辑。该奖项由DMS和PHY共同资助。

This proposal explores some of the main research areas where algebraic geometry interacts with quantum field theory and string theory including: the moduli of super Riemann surfaces and the related issues of superstring measure and superstring perturbation theory; the geometric Langlands program; heterotic string phenomenology; F theory; and quantum sheaf cohomology. The proposal also considers some issues in moduli spaces of curves and abelian varieties, and some more exploratory math/physics connections such as amplitudes, Grassmannians, and twistors; and some conjectures regarding the 6-dimensional conformal field theory and its mathematical consequences.The significance of this project is in developing the connections between mathematics (mostly algebraic geometry) and high energy physics (mostly string theory). Each of these areas involves a mixture of issues from math and from physics, and most will be explored in teams involving both mathematicians and physicists. The first research area in particular is expected to have a major impact on the (mathematical) foundations of perturbative superstring theory, while the second addresses one of the major open problems in mathematics using new tools inspired, at least in part, by high energy physics ideas. The PI proposes also to continue a wide range of broad impact community and educational activities, including: the development and guidance of a series of conferences emphasizing the interactions of physics and mathematics, and of other channels for the dissemination of new knowledge concerning interactions of mathematics and high energy physics; curriculum development at the graduate and undergraduate level; writing a strings-for-mathematicians text; extensive work with graduate and undergraduate students and evaluation of the math major at Penn; membership of national bodies such as the AMS Committee on the Profession, and editorship of several journals and book series.This award is co-funded by DMS and PHY.