Geometry of Langlands Duality

朗兰兹对偶的几何

• 批准号: 0901768
• 负责人: Ivan Mirkovic
• 金额: $ 17.97万
• 依托单位: University of Massachusetts Amherst
• 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 2013-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901768&HistoricalAwards=false
• 关键词: Geometry; Langlands; Duality

This project is directed towards search for geometry that is expected to underlie the Langlands conjectures. One aspect is the interpretation of Langlands duality of reductive groups in terms of homotopy theory (in the framework of Derived Algebraic Geometry). Another aspect is a continuation of the work on developing a particular geometric Koszul duality (``Linear Koszul Duality'') and applying it to representation theory. In relation to physics, the project envisions a mathematical formalism of t'Hooft operators and a construction of a certain extended topological field theory in four dimensions, with applications to knot theory. The project also contains a speculation on the relation of number theory and physics. Here, a geometrization of number theory is expected to arise through a connection to quantum field theory. The project aims towards relating and unifying developments in mathematics and theoretical particle physics. The origin of these developments in mathematics is the so called Langlands program, which is the modern view on a classical discipline of number theory. In time this program incorporated a number of central disciplines of mathematics, starting with representation theory, then algebraic geometry and currently the homotopy theory. The relation to physics is a part of the current melting of barriers between mathematics and physics which arose in a period when the two subjects developed separately. The central impact on mathematics in the last quarter century was the import of the ideas from quantum field theory (in particular string theory), which is the part of physics that studies elementary particles. Recently, Witten, Kapustin and Gukov established a long sought bridge between Langlands program and quantum field theory (in particular gauge theory and string theory). The proposed work attempts to work in both directions by applying ideas from physics to mathematics and mathematical constructions to physics. It also aims towards deeper understanding of the relation between quantum field theory and number theory. The proposal also contains more standard topics within representation theory and Langlands program.

这个项目的目的是寻找有望成为朗兰兹猜想基础的几何图形。一方面是用同伦理论(在导代数几何的框架下)解释约化群的朗兰兹对偶。另一个方面是继续发展一种特殊的几何Koszul对偶(‘线性Koszul对偶’)并将其应用于表示理论的工作。在物理学方面，该项目设想了t‘Hooft算符的数学形式，以及在四维空间中构建特定的扩展拓扑场论，并将其应用于纽结理论。该项目还包含了对数论和物理学关系的推测。在这里，数论的几何化有望通过与量子场论的联系而出现。该项目旨在将数学和理论粒子物理学的发展联系起来并统一起来。数学上这些发展的起源是所谓的朗兰兹程序，这是对一个经典数论学科的现代观点。随着时间的推移，这个程序包含了许多数学的中心学科，从表示论开始，然后是代数几何，现在是同伦理论。与物理的关系是目前数学和物理之间障碍融化的一部分，这种障碍是在这两个学科单独发展的时期产生的。在过去的四分之一个世纪里，对数学的主要影响是从量子场论(特别是弦理论)中引进了思想，量子场论是物理学中研究基本粒子的部分。最近，Witten、Kapustin和Gukov在朗兰兹计划和量子场论(特别是规范理论和弦理论)之间建立了一座寻求已久的桥梁。这项拟议的工作试图通过将物理学的思想应用于数学，将数学构造应用于物理学，来进行双向的工作。旨在加深对量子场论和数论之间关系的理解。该提案还包含了表征理论和朗兰兹计划中更标准的主题。