Mathematics Colloquia and Seminars

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Coulomb branches of 4d N=2 gauge theories are a class of affine algebraic varieties which, together with their quantizations, appear in a variety of guises in geometry and representation theory - examples include spherical DAHA, open Richardsons in affine flag varieties, certain wild character varieties, and certain multiplicative quiver varieties. In this talk we outline a construction of canonical bases in the quantized coordinate rings of these varieties. These bases appear as byproducts of a more fundamental construction, a new type of nonstandard t-structure on the dg category of coherent sheaves on the Braverman-Finkelberg-Nakajima space of triples. The heart of this t-structure is a tensor category which is not braided, but admits renormalized r-matrices abstracting those appearing in the finite dimensional representation theory of quantum affine algebras. It is intended to provide a mathematical model of the category of half-BPS line defects in the relevant gauge theory, and is inspired by earlier work of Kapustin-Saulina and Gaiotto-Moore-Neitzke. On a combinatorial level, the resulting bases are expected to be controlled by specific cluster algebras, which we can confirm in simple examples. This is work in progress with Sabin Cautis.