Representation theoretic methods in geometry and mathematical physics

我最近一直在探索仿射格拉斯曼的相干滑轮范畴。很自然地将其称为连贯佐竹范畴，因为它的可构造类似物（仿射格拉斯曼上的可构造滑轮范畴）通常称为佐竹范畴。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，并且根据卡普斯汀和维滕相对较新的结果，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二象性。相干佐竹范畴同样与规范场理论相关。然而，这个理论的表现有所不同，而且这个故事的数学对应物相对来说还不太被理解。例如，可构造的佐竹范畴是半简单的，其幺半群结构是对称的。相比之下，连贯的 Satake 范畴既不是半简单的，也不是对称的。相反，它的结构可以（推测地）描述为幺半群簇类别。在最近与 H. Williams 合作的预印本中，我们定义并研究了 GL(n) 仿射格拉斯曼函数的簇结构。该仿射格拉斯曼的 K 理论是 4d N=2 规范场理论库仑分支的最简单示例。这种簇结构的出现在其他规范场理论中已被更普遍地注意到。我们的证明在很大程度上依赖于重正化 r 矩阵的构造，该矩阵在任何其乘积与辅助手征范畴兼容的幺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取得进展。该提案的主要目标之一是开发新工具（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格拉斯曼人定义一个集群结构。相干 Satake 范畴也与 J. Kamnitzer 早期在定义同调结不变量（例如 Khovanov 同源）和几何 Satake 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，几何和表示理论的各种工具被开发和应用。一个突出的工具是量子群的分类行为的想法。我们计划进一步开发并使其中一些技术适应相干佐竹范畴（以及规范场理论产生的其他范畴）的背景。

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).