Geometric representation theory and crystals

几何表示理论和晶体

基本信息

批准号：
RGPIN-2018-04713
负责人：
Kamnitzer, Joel
金额：
$ 2.99万
依托单位：
University of Toronto
依托单位国家：
加拿大
项目类别：
Discovery Grants Program - Individual
财政年份：
2020
资助国家：
加拿大
起止时间：
2020-01-01 至 2021-12-31
项目状态：
已结题

来源：
https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=713192
关键词：
Geometric representation theory crystals

项目摘要

Symmetry is a fundamental area of mathematics. Mathematicians are particularly interested in continuous collections of symmetries, such as the group of rotations of a sphere. We call these objects Lie groups. An important problem is to understand how these Lie groups can arise as linear operators on vector spaces, these are called representations. One of the great achievements in mathematics in the late 19th and early 20th century was the classification of simple Lie groups and their representations. Much of this interest in Lie groups is motivated by theoretical physics. Lie groups and their representations have been used heavily since the 1930s in quantum mechanics to help describe and classify elementary particles. In recent years, there has been a resurgence of interaction between quantum field theory and representation theory, leading to important advances on both sides. In the past 20 years, mathematicians have developed geometric constructions in representation theory. In these constructions, we have a geometric object (an algebraic variety) whose topology encodes a representation of a Lie group. These geometric constructions are very beautiful and lead to deeper structures, such as categorification and canonical bases. There are two sources of these geometric constructions, which were invented roughly simultaneously. The first one comes from the theory of geometric Langlands duality, which was developed by Drinfeld as a geometric version of the famous Langlands conjectures, which have guided much of modern number theory. This first construction uses geometric objects called affine Grassmannians. The second construction was developed by Lusztig and Nakajima and involves geometric objects called quiver varieties. The first construction is more natural, but harder to work with; the second construction is more hands-on, but more ad-hoc. The existence of these two different, seemingly unrelated, geometric constructions was very mysterious and lead many mathematicians to the following question: What is the relationship between these two geometric constructions? In 2012, Webster, Weekes, Yacobi and I proposed an answer to this question through the mechanism of symplectic duality, a subtle relationship between certain classes of geometric objects, called conical symplectic resolutions. Symplectic duality has a beautiful origin in theoretical physics, more specifically from N = 4, 3-dimensional supersymmetric quantum field theories. To such a theory, we can consider all possible lowest energy states, which is called the moduli space of vacua. This moduli space has a few pieces, one of which is called the Higgs branch and another of which is called the Coulomb branch. These two branches of vacua form a symplectic dual pair. My current research focuses on exploring this symplectic duality and its consequences for categorification and special bases.

对称性是数学的一个基本领域。数学家对连续的对称性集合特别感兴趣，例如球体的旋转群。我们将这些对象称为李群。一个重要的问题是理解这些李群如何作为向量空间上的线性算子出现，这些称为表示。 19 世纪末和 20 世纪初数学的伟大成就之一是简单李群的分类及其表示。对李群的兴趣很大程度上是由理论物理学激发的。自 20 世纪 30 年代以来，李群及其表示在量子力学中被大量使用，以帮助描述和分类基本粒子。近年来，量子场论和表示论之间的相互作用重新兴起，双方都取得了重要进展。在过去的 20 年里，数学家发展了表示论中的几何构造。在这些构造中，我们有一个几何对象（代数簇），其拓扑编码了李群的表示。这些几何结构非常漂亮，并导致更深层次的结构，例如分类和规范基础。这些几何结构有两个来源，它们几乎是同时发明的。第一个来自几何朗兰兹对偶理论，该理论由德林菲尔德发展为著名朗兰兹猜想的几何版本，该猜想指导了许多现代数论。第一个构造使用称为仿射格拉斯曼函数的几何对象。第二种结构是由 Lusztig 和 Nakajima 开发的，涉及称为箭袋品种的几何对象。第一种结构更自然，但更难使用；第二种构建更需要实践，但也更临时。这两种不同的、看似无关的几何结构的存在非常神秘，并导致许多数学家提出以下问题：这两个几何结构之间有什么关系？ 2012 年，Webster、Weekes、Yacobi 和我通过辛对偶机制提出了这个问题的答案，辛对偶是某些类几何对象之间的微妙关系，称为圆锥辛分辨率。辛对偶性在理论物理学中有着美好的起源，更具体地说，源自 N = 4 的 3 维超对称量子场论。对于这样的理论，我们可以考虑所有可能的最低能态，称为真空模空间。这个模空间有几部分，其中一个称为希格斯分支，另一个称为库仑分支。这两个真空分支形成辛对偶对。我目前的研究重点是探索这种辛对偶性及其对分类和特殊基的影响。