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).

我最近一直在探索仿射格拉斯曼上的相干层范畴。很自然地称之为凝聚的佐竹范畴，因为它的可构造类似物（仿射格拉斯曼上的可构造层范畴）通常被称为佐竹范畴。通常的（可构造的）佐竹范畴是数论和几何表示论中非常丰富的对象。特别是，它与朗兰兹纲领密切相关，并且通过卡普斯廷和维滕的相对较新的结果，与数学物理中的某些规范理论密切相关。在这种解释中，著名的朗兰兹对偶现象对应于电磁对偶性。同调的佐竹范畴同样与规范理论有关。然而，这个理论的表现不同，这个故事的数学对应物相对来说知之甚少。例如，可构造的Satake范畴是半单的，它的么半群结构是对称的。相反，凝聚的佐竹范畴既不是半单的，也不是对称的。相反，它的结构可以被描述为一个monoidal簇范畴。在最近的预印本与H。威廉姆斯，我们定义并研究了GL（n）的仿射Grassmannian的这种簇结构。这个仿射格拉斯曼的K理论是4d N=2规范场论的库仑分支的最简单的例子。这种团簇结构的出现在其他规范场理论中得到了更普遍的注意。我们的证明在很大程度上依赖于一个重正化的r-矩阵的建设，这是有意义的任何monoidal类别的产品是兼容的辅助手征类别。这表明，在对这种场论的其他库仑分支的研究和理解方面还有待于取得进展。这个建议的主要目的之一是开发新的工具（几何以及代表性理论），以扩展我们的结果。例如，一个明显的目标是为其他群的仿射格拉斯曼群定义一个聚类结构。相干佐竹范畴也与J. Kamnitzer在定义同调结不变量（例如Khovanov同调）和几何佐竹的量子K理论版本的背景下研究的卷积空间密切相关。在这些情况下，从几何和表示理论的各种工具的开发和应用。一个突出的工具是量子群的分类作用的想法。我们计划进一步发展和适应这些技术的连贯佐竹范畴（和其他类别产生的规范场理论）的背景下。