Representation theoretic methods in geometry and mathematical physics

几何和数学物理中的表示理论方法

• 批准号: RGPIN-2019-03961
• 负责人: Cautis, Sabin
• 金额: $ 1.89万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758491
• 关键词: Representation; theoretic; methods; geometry; mathematical

I have recently been exploring the category of coherent sheaves on the affine Grassmannian. It is natural to call this the coherent Satake category because its constructible analogue (the category of constructible sheaves on the affine Grassmannian) is usually called the Satake category. The usual (constructible) Satake category is a very rich object in both number theory and geometric representation theory. In particular, it is closely related to the Langlands program and, by relatively recent results of Kapustin and Witten, to certain gauge theories in mathematical physics. In this interpretation, the famous Langlands duality phenomenon corresponds to electric-magnetic duality. The coherent Satake category is likewise related to a gauge theory. However, this theory behaves differently and the mathematical counterpart of this story is relatively poorly understood. For example, the constructible Satake category is semisimple and its monoidal structure symmetric. In contrast, the coherent Satake category is neither semisimple nor symmetric. Instead, its structure can be described (conjecturally) as a monoidal cluster category. In a recent preprint with H. Williams, we define and study this cluster structure for the affine Grassmannian of GL(n). The K-theory of this affine Grassmannian is the simplest example of a Coulomb branch of a 4d N=2 gauge field theory. The appearance of such cluster structures has been noticed more generally for other gauge field theories. Our proof relies heavily on the construction of a renormalized r-matrix which makes sense in any monoidal category whose product is compatible with an auxiliary chiral category. This suggests a to make progress in the study and understanding of other Coulomb branches of such field theories. One of the main aims of this proposal is to develop new tools (geometric as well as representation-theoretic) in order to extend our results. For instance, an obvious goal is to define a cluster structure for affine Grassmannians of other groups. The coherent Satake category is also closely related to the convolution spaces studied earlier with J. Kamnitzer in the context of defining homological knot invariants (e.g. Khovanov homology) and a quantum K-theoretic version of geometric Satake. In those instances various tools from geometry and representation theory were developed and applied. One tool that stands out is the idea of categorical actions of quantum groups. We plan to develop further and adapt some of these techniques to the context of the coherent Satake category (and other categories arising from gauge field theories).

最近，我一直在探索Aggine Grassmannian上的连贯滑轮类别。自然而然地称其为连贯​​的萨克类别类别，因为它的可构造类似物（仿生草上的可构造束带的类别）通常称为Satake类别。在数字理论和几何表示理论中，通常（可构造的）萨克斯类别是一个非常丰富的对象。特别是，它与Langlands计划密切相关，并且通过Kapustin和Witten的最新结果与数学物理学中的某些规格理论相对较新。在这种解释中，著名的Langlands二元性现象对应于电磁双重性。连贯的Satake类别同样与仪表理论有关。但是，该理论的行为不同，这个故事的数学对应物相对较少理解。例如，可构造的萨克类别类别是半神经，其单体结构对称。相比之下，连贯的萨克类别既不是半圣事也不是对称的。取而代之的是，可以将其结构描述为（猜想）为单体群集类别。在最近与H. Williams的预印象中，我们为GL（N）的仿生司硕（Grassmannian）定义并研究了这种聚类结构。这种仿生的Grassmannian的K理论是4D N = 2量规场理论的库仑分支的最简单例子。对于其他量规场理论，已经更普遍地注意到了这种群集结构的外观。我们的证明在很大程度上依赖于构建重新归一化的R-matrix，该r-matrix在任何单型类别中都有意义，其产品与辅助手性类别兼容。这表明A在研究中取得进展并了解此类野外理论的其他库仑分支。该提案的主要目的之一是开发新工具（几何和代表理论），以扩大我们的结果。例如，一个明显的目标是为其他群体的植物植树群定义簇结构。在定义同源结（例如Khovanov同源性）的背景下，连贯的Satake类别也与前面与J. Kamnitzer一起研究的卷积空间密切相关。在这些情况下，开发和应用了几何学和表示理论的各种工具。一个脱颖而出的工具是量子组的分类作用的想法。我们计划进一步开发并将其中一些技术调整到连贯的Satake类别的背景下（以及量规场理论引起的其他类别）。