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.

该奖项是根据2009年的《美国复苏与再投资法》（公法111-5）资助的。该提案探讨了几个领域，在数学方面，代数几何与量子场理论与弦理论相互作用：几何兰兰兹计划，数学方面；物理学中的杂弦现象学和F理论；以及SuperString Measure，这是一个由物理学动机的代数几何项目。在生产异性标准模型中，最近的突破是代数几何形状在物理服务服务方面的力量的完美说明。 PI使用技术来构建非简单连接的Calabi-yau三倍，并在它们上满足其Chern类和共同体的各种限制，PI产生了唯一已知的弦线紧凑型唯一已知的例子，它具有最小的Supersym-Supersym-Suppersym-Mepersymmetrymmetric Standard Model（MSSM）粒子和力量的粒子和无需不受欢迎的东西。提出了一项系统的研究，该研究对弦乐景观的高乡村地区进行了研究。这包括研究所有已知的非简单连接的calabi-yau三倍，其中包含了所有标准模型束的分类以及对其数学和现象学特性的分析。这些异性标准模型的明显稀缺性促使试图确定弦高国家的粗糙尺寸。基于F理论的最新现象学突破是构建全局的紧迫性，几何模型实现了各种已知的局部模型。超弦措施的所有属构造是字符串理论中的重要基础问题。最近的建议将其转换为经典代数几何形状，与模块化形式，肖特基问题和theta身份密切相关的问题。现有的建议不太有效。幸运的是，一些代数几何成分的添加似乎很可能会克服障碍物。几何兰兰斯猜想是代数几何学和代表理论中的一个古老而核心的开放问题。近年来，物理学家也非常感兴趣，物理学家将其嵌入了量子场理论的背景中。他们的物理见解与非亚伯利亚杂货理论的最新突破与综合系统的较旧思想的结合提供了很快的完整解决方案。F理论及其对杂种弦的双重性是代数几何学的另一个领域，可以对物理学做出强大的贡献。 PI还建议继续进行广泛的教育活动，包括课程发展，教科书的写作以及与本科生和研究生的广泛工作，旨在传播有关数学和高能量物理学相互作用的新知识。