Research Proposal in Algebraic Geometry and String Theory

代数几何和弦理论的研究计划

• 批准号: 0908487
• 负责人: Ron Donagi
• 金额: $ 48.08万
• 依托单位: University of Pennsylvania
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-07-01 至 2013-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0908487&HistoricalAwards=false
• 关键词: Research; Proposal; Algebraic; Geometry; String

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.

该奖项由 2009 年美国复苏和再投资法案（公法 111-5）资助。该提案探讨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；物理学中的异质弦现象学和F理论；以及超弦测度，这是一个由物理学推动的代数几何项目。最近在生成异质标准模型方面取得的突破完美地说明了代数几何为物理学服务的力量。使用构造非简单连接的 Calabi-Yau 三重及其束的技术，满足其陈类和上同调的各种约束，PI 生成了唯一已知的杂质弦紧致化示例，该示例具有精确的最小超对称标准模型 (MSSM)粒子和力的光谱，没有不需要的外来物质。提出了对杂种标准模型居住的弦景观高地地区进行系统研究。这包括对所有已知的非简单连接的 Calabi-Yau 三重的调查，纳入对它们的所有标准模型束的分类以及对其数学和现象学属性的分析。这些杂种标准模型的明显稀缺性促使我们尝试确定高地国家串的粗略规模。最近基于 F 理论的现象学突破凸显了构建实现各种已知局部模型的全局几何模型的紧迫性。总而言之，超弦测度的构造是弦理论中的一个重要基础问题。最近的提案已将其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。现有的建议不太有效。幸运的是，添加一些代数几何成分似乎可以克服障碍。几何朗兰兹猜想是代数几何和表示论中一个古老而核心的开放问题。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以及可积系统的旧思想相结合，为很快提供完整解决方案提供了真正的可能性。F 理论及其与杂异弦的对偶性是代数几何能够实现的另一个领域对物理学的巨大贡献。 PI还建议继续开展广泛的教育活动，包括课程开发、教材编写以及与本科生和研究生的广泛合作，旨在传播有关数学和高能物理相互作用的新知识。