Representation theoretic methods in geometry and mathematical physics

最近，我一直在研究仿射Grassmanian上的凝聚层范畴。将其称为凝聚的Satake范畴是很自然的，因为它的可构造性类似物(仿射Grassmanian上的可构造层的范畴)通常被称为Satake范畴。通常的(可构造的)Satake范畴在数论和几何表示理论中都是一个非常丰富的对象。特别是，它与朗兰兹计划密切相关，根据卡普斯汀和维腾最近的结果，它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。相干性的萨塔克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关系相对较少被理解。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，连贯的Satake范畴既不是半单的，也不是对称的。相反，它的结构可以(推测)描述为一元簇范畴。在最近与H.Williams的预印本中，我们定义并研究了GL(N)的仿射Grassman的这种簇结构。这个仿射格拉斯曼的K理论是4dN=2规范场理论中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广泛的关注。我们的证明在很大程度上依赖于重整化r-矩阵的构造，它在任何乘积与辅助手征范畴相容的么半群范畴中都是有意义的。这表明，在研究和理解此类场论的其他库仑分支方面，有必要取得进展。这项提议的主要目的之一是开发新的工具(几何和表示理论)，以扩展我们的结果。例如，一个明显的目标是为其他群体的仿射Grassmannians定义一个集群结构。在定义同调纽结不变量(例如Khovanov同调)和几何Satake的量子K理论版本的背景下，凝聚Satake范畴也与J.Kamnitzer早先研究的卷积空间密切相关。在这些情况下，几何学和表象理论的各种工具被开发和应用。其中一个突出的工具是量子群的绝对作用的想法。我们计划进一步发展这些技术，并使其中一些技术适用于相干的Satake范畴(以及从规范场理论产生的其他范畴)。

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).