Classical and Quantum Geometric Langlands Correspondence

经典与量子几何朗兰兹对应

• 批准号: 1063470
• 负责人: Dennis Gaitsgory
• 金额: $ 70.06万
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1063470&HistoricalAwards=false
• 关键词: Classical; Quantum; Geometric; Langlands; Correspondence

For a smooth complete algebraic curve X over an algebraically closed field k, and a reductive group G, there is a fundamental object called the moduli space of G-bundles on X, and denoted Bun(G). The theory of automorphic sheaves, which is a geometrization of the classical theory of automorphic functions, studies various categories of sheaves on Bun(G). Most fundamentally, we study the (derived) category of quasi-coherent sheaves QCoh(Bun(G)), as well as various categories of twisted D-modules. The goal of Geometric Langlands Correspondence is to study the remarkable relation of this category to the similarly defined category for another reductive group, called the Langlands dual of G. As was discovered in the middle of the 20th century, one can give a list of all fundamental laws of symmetry that occur in nature--these are classified by Dynkin diagrams, also known as root systems. Our fundamental object of study, the theory of automorphic sheaves, couples these symmetry laws with another fundamental object in modern mathematics, namely algebraic curves (also known in their incarnation as Riemann surfaces). The discovery of Langlands correspondence indicates that the theory of automorphic sheaves admits additional symmetries of its own. The current project aims to study these new symmetries at their most fundamental.

对于代数闭域k上的光滑完全代数曲线X和约化群G，存在一个基本对象，称为X上G束的模空间，记作Bun(G)。自同构轴理论是经典自同构函数理论的几何化，研究了Bun(G)上的各种类型的轴。最根本的是，我们研究了准相干束QCoh（Bun(G)）的（派生）范畴，以及扭曲d模的各种范畴。几何朗兰兹对偶的目标是研究这个范畴与另一个约化群的相似定义范畴之间的显著关系，这个约化群被称为g的朗兰兹对偶。正如20世纪中叶发现的那样，人们可以列出自然界中出现的所有基本对称定律——这些定律被Dynkin图（也称为根系统）分类。我们研究的基本对象，自同构轴理论，将这些对称定律与现代数学中的另一个基本对象，即代数曲线（也称为黎曼曲面）相结合。朗兰兹对应的发现表明自同构轴理论承认其自身的额外对称性。目前的项目旨在从最基本的角度研究这些新的对称性。