Geometry of Langlands Duality

本研究项目旨在联系和统一数学和理论粒子物理学的最新发展。这些发展的数学起源是所谓的朗兰兹纲领，这是对数论这一经典数学学科的现代看法。朗兰兹纲领在今天的作用是数学的中心学科（如表示论、代数几何和数论）与物理学的中心学科（如规范理论和弦理论）之间的桥梁。该项目将在两个方向上使用这种联系，通过将物理学的思想应用于数学，通过局部性的新用途，并将数学结构应用于物理学，通过构建“理论X，“这六个国家--一维共形场论，它应该统一所有重要的低维量子场论。这个项目的中心主题是从维的对象扩展几何朗兰兹程序一个（在代数几何学中）到任意维度。该项目的目标是将概念上的突破发展成一个完整的数学机器。这些突破包括将几何朗兰兹纲领（所谓的几何类场论）的基本情况扩展到任意维。这一发展预计将在数学中产生影响，特别是通过引入相对动机上同调。一般的几何朗兰兹程序也将进行调查，根据最近的结果在建设和推广循环格拉斯曼从统计力学的角度来看。

This research project aims to relate and unify recent developments in mathematics and theoretical particle physics. The mathematical origin of these developments is the so-called Langlands program, which is the modern view on the classical mathematical discipline of number theory. The Langlands program functions today as a bridge between central disciplines of mathematics, such as representation theory, algebraic geometry, and number theory, and central disciplines in physics, such as gauge theory and string theory. This project will use this connection in both directions by applying ideas from physics to mathematics, through a new use of locality, and applying mathematical constructions to physics, by constructing the "theory X," the conjectural six-dimensional conformal field theory that should unify all important quantum field theories in lower dimensions.The central theme of this project is to extend the geometric Langlands program from objects of dimension one (in algebraic geometry) to arbitrary dimensions. The goal of the project is to develop conceptual breakthroughs into a full mathematical machinery. These breakthroughs include an extension of the basic case of the geometric Langlands program (the so called geometric Class Field Theory) to arbitrary dimension. This development is expected to be influential in mathematics, in particular through the introduction of the relative motivic cohomology. The general geometric Langlands program will also be investigated, based on recent results in constructing and generalizing the loop Grassmannian from the point of view of statistical mechanics.