Geometry of Langlands Duality

这项研究项目旨在将数学和理论粒子物理的最新发展联系起来并统一起来。这些发展的数学起源是所谓的朗兰兹计划，这是对数论经典数学学科的现代观点。朗兰兹计划今天起到了数学中心学科(如表示论、代数几何和数论)和物理中心学科(如规范理论和弦理论)之间的桥梁作用。这个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到数学中，通过局域性的新使用，以及通过将数学结构应用到物理中，通过构建猜想的六维共形场理论，它应该统一所有低维的重要量子场论。这个项目的中心主题是将几何朗兰兹计划从一维对象(代数几何)扩展到任意维。该项目的目标是将概念上的突破发展成一台完整的数学机器。这些突破包括将几何朗兰兹程序的基本情形(所谓的几何类场理论)推广到任意维度。这一发展预计将对数学产生影响，特别是通过引入相对理据上同调。还将根据从统计力学的观点构造和推广格拉斯曼环的最新结果，对一般几何朗兰兹程序进行研究。

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.