The Coherent Satake Category, Clusters, and Wilson-'t Hooft Operators

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We discuss recent work showing that in type A_n the category of equivariant perverse coherent sheaves on the affine Grassmannian categorifies the cluster algebra associated to the BPS quiver of pure N=2 gauge theory. Physically, this can be understood as a statement about line operators in this theory, following ideas of Gaiotto-Moore-Neitzke, Costello, and Kapustin-Saulina -- in short, coherent IC sheaves are the precise algebro-geometric counterparts of Wilson-'t Hooft line operators. The proof relies on techniques developed by Kang-Kashiwara-Kim-Oh in the setting of KLR algebras. A key moral is that the appearance of cluster structures is in large part forced by the compatibility between chiral and tensor structures on the category in question (i.e. by formal features of holomorphic-topological field theory). This is joint work with Sabin Cautis.