Скачать или смотреть Harold Williams: The coherent Satake category

Abstract: The geometric Satake equivalence identifies the Satake category of a reductive group G - that is, the category of equivariant perverse sheaves on the affine Grassmannian GrG - with the representation category of its Langlands dual group G∨. While the Satake category is topological in nature, it has a poorly understood algebro-geometric cousin: the category of perverse coherent sheaves on GrG. This category is not semi-simple and its monoidal product is not symmetric. We show however that it is rigid and admits renormalized r-matrices similar to those appearing in the theory of quantum loop or KLR algebras. Applying the framework developed by Kang-Kashiwara-Kim-Oh in their proof of the dual canonical basis conjecture, we use these results to show that the coherent Satake category of GLn is a monoidal cluster categorification in the sense of Hernandez-Leclerc. This clarifies the physical meaning of the coherent Satake category: simple perverse coherent sheaves correspond to Wilson-’t Hooft operators in N=2 gauge theory, just as simple perverse sheaves correspond to ’t Hooft operators in N=4 gauge theory following the work of Kapustin-Witten. Our results also explain the appearance of identical quivers in the work of Kedem-Di Francesco on Q-systems and in the context of BPS quivers. More generally, our construction of renormalized r-matrices works in any chiral E1-category, providing a new way of understanding the ubiquity of cluster algebras in N=2 field theory: the existence of renormalized r-matrices, hence of iterated cluster mutation, is a formal feature of such theories after passing to their holomorphic-topological twists. This is joint work with Sabin Cautis (arXiv:1801.08111).

Recording during the thematic meeting "Cluster algebras: twenty years on" the March 20, 2018 at the Centre International de Rencontres Mathématiques (Marseille, France)

Filmmaker: Guillaume Hennenfent

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