Algebraic Geometry in String Theory

这个建议探讨了一些主要的研究领域，其中代数几何相互作用与量子场论和弦理论，包括：超黎曼曲面的模量和超弦测量和超弦微扰理论的相关问题;几何朗兰兹计划;杂合弦现象学; F理论;和量子层上同调。该计划还考虑了曲线和阿贝尔簇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，格拉斯曼和扭曲;以及一些关于6维共形场论及其数学后果的说明。该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到数学和物理学的问题，大多数将在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，预计将有一个重大影响的（数学）基础微扰超弦理论，而第二个地址的一个主要开放的问题，在数学中使用新的工具，启发，至少部分地，由高能物理的想法。PI还建议继续广泛的影响社区和教育活动，包括：开发和指导一系列会议，强调物理和数学的相互作用，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;研究生和本科生课程开发;编写数学家字符串文本;与研究生和本科生的广泛工作和宾夕法尼亚大学数学专业的评估;国家机构的成员，如AMS委员会的专业，和几个期刊和丛书的编辑。这个奖项是由DMS和物理共同资助。

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.