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H 1 The coherent Satake category

Auteurs : Williams, Harold (Auteur de la Conférence)
CIRM (Editeur )

Résumé : 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 $G_{rG}$ - 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 $G_{rG}$. 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 $GL_n$ 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 $\mathcal{N} = 2$ gauge theory, just as simple perverse sheaves correspond to ’t Hooft operators in $\mathcal{N} = 4$ 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 $E_1$-category, providing a new way of understanding the ubiquity of cluster algebras in $\mathcal{N} = 2$ 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).

Keywords : coherent Satake category; monoidal cluster categorification; renormalized r-matrices

Codes MSC :
14F05 - Sheaves, derived categories of sheaves and related constructions
14M15 - Grassmannians, Schubert varieties, flag manifolds
17B37 - Quantum groups and related deformations
18D10 - Monoidal, symmetric monoidal and braided categories
81T13 - Yang-Mills and other gauge theories
14D24 - Geometric Langlands program: algebro-geometric aspects
13F60 - Cluster algebras

 Informations sur la Vidéo Langue : Anglais Date de publication : 28/03/2018 Date de captation : 20/03/2018 Collection : Research talks ; Algebraic and Complex Geometry ; Mathematical Physics Format : MP4 Durée : 00:54:04 Domaine : Algebraic & Complex Geometry ; Mathematical Physics Audience : Chercheurs ; Doctorants , Post - Doctorants Download : https://videos.cirm-math.fr/2018-03-20_H_Williams.mp4 Informations sur la rencontre Nom de la rencontre : Cluster algebras: twenty years on / Vingt ans d'algèbres amasséesOrganisateurs de la rencontre : Amiot, Claire ; Baur, Karin ; Dupont, Grégoire ; Marsh, Robert J. ; Morier-Genoud, Sophie ; Palu, Yann ; Pilaud, Vincent ; Plamondon, Pierre-Guy ; Qin, FanDates : 19/03/2018 - 23/03/2018 Année de la rencontre : 2018 URL Congrès : https://conferences.cirm-math.fr/1777.htmlCitation Data DOI : 10.24350/CIRM.V.19382703 Cite this video as: Williams, Harold (2018). The coherent Satake category. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19382703 URI : http://dx.doi.org/10.24350/CIRM.V.19382703

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Bibliographie

1. Cautis, S., & Williams, H. (2018). Cluster theory of the coherent Satake category. - https://arxiv.org/abs/1801.08111

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