Research Proposal in Algebraic Geometry and String Theory

该奖项是根据2009年《美国复苏和再投资法案》(Public Law 111-5)资助的。这项建议探索了代数几何与量子场论和弦理论相互作用的几个领域：数学方面的几何朗兰兹计划；物理方面的异质弦现象学和F理论；以及超弦测量，这是一个由物理推动的代数几何项目。最近在生产异类标准模型方面的突破是代数几何为物理学服务的力量的完美例证。使用构造非单连通的Calabi-Yau三重折叠及其上满足对其Chern类和上同调的各种约束的丛的技术，PI产生了唯一已知的杂交弦紧凑的例子，该例子恰好具有粒子和力的最小超对称标准模型(MSSM)谱，没有多余的外来物质。系统地研究了异质标准模型所在的弦状景观的高地地区。这包括对所有已知的非单连通Calabi-Yau三重数的研究，包括对它们上所有标准模型丛的分类，以及对它们的数学和唯象性质的分析。这些异质标准模型显然非常稀缺，这促使人们试图确定弦高国家的大致大小。最近基于F-理论的现象学突破强调了构建实现各种已知局部模型的全局几何模型的紧迫性。超弦测度的构造是弦理论中一个重要的基础性问题。最近的提案已经将这个问题转化为经典代数几何中的一个问题，与模形式、肖特基问题和theta恒等式密切相关。现有的建议并不十分奏效。幸运的是，一些代数几何成分的加入似乎可以克服这个障碍。几何朗兰兹猜想是代数几何和表示论中一个古老的和中心的公开问题。近年来，它也引起了物理学家的极大兴趣，他们将其嵌入到量子场论的背景中。他们的物理见解与非阿贝尔霍奇理论的最新突破和来自可积系统的旧思想相结合，提供了很快得到完整解决方案的真正可能性。F理论及其对杂弦的对偶性是代数几何能够对物理学做出强大贡献的另一个领域。国际和平研究所还建议继续开展广泛的教育活动，包括编制课程、编写教科书以及与本科生和研究生开展广泛的工作，目的是传播有关数学和高能物理相互作用的新知识。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).This proposal explores several areas where algebraic geometry interacts with quantum field theory and string theory: the Geometric Langlands Program, on the math side; heterotic string phenomenology and F theory, in physics; and the superstring measure, an algebraic geometry project motivated by physics. The recent breakthrough in producing a Heterotic Standard Model is a perfect illustration of the power of algebraic geometry at the service of physics. Using techniques for construction of non simply connected Calabi-Yau threefolds and of bundles on them satisfying various constraints on their chern classes and cohomology, the PI produced the only known example of a heterotic string compactification which has exactly the Minimal Supersymmetric Standard Model (MSSM) spectrum of particles and forces, with no unwanted exotic matter. A systematic study is proposed of the High Country region of the string Landscape, where the Heterotic Standard Models live. This includes investigation of all known non simply connected Calabi-Yau threefolds, incorporating a classification of all Standard Model bundles on them and analysis of their mathematical and phenomenological properties. The apparent great scarcity of these Heterotic Standard Models motivates attempts to determine the rough size of the string High Country. The recent phenomenological breakthroughs based on F-theory underlie the urgency of constructing global, geometric models realizing the various known local models. The construction, at all genera, of the superstring measure is an important foundational issue in string theory. Recent proposals have converted this to a question in classical algebraic geometry, closely related to modular forms, the Schottky problem, and theta identities. The existing proposals do not quite work. Fortunately, it seems likely that the addition of some algebro-geometric ingredients may overcome the obstruction.The Geometric Langlands Conjecture is an old and central open problem in algebraic geometry and representation theory. In recent years it has also been of great interest to physicists, who have embedded it into the context of quantum field theory. The combination of their physical insights with recent breakthroughs in non abelian Hodge theory and older ideas from integrable systems offers the real possibility of a complete solution soon.F-theory and its duality to the heterotic string are another area where algebraic geometry is able to make powerful contributions to the physics. The PI also proposes to continue a wide range of educational activities, including curriculum development, the writing of a textbook, and extensive work with undergraduate and graduate students, aimed at the dissemination of new knowledge concerning the interactions of mathematics and high energy physics.