Canonical Bases for Coulomb Branches

Following work of Kapustin-Saulina and Gaiotto-Moore-Neitzke, one expects half-BPS line defects in a 4d N=2 field theory to form a monoidal category with a rich structure (for example, a monoidal cluster structure in many cases). In this talk we explain a proposal for an algebro-geometric definition of this category in the case of gauge theories with polarizable matter. The proposed category is the heart of a nonstandard t-structure on the dg category of coherent sheaves on the derived Braverman-Finkelberg-Nakajima space of triples. We refer to its objects as Koszul-perverse coherent sheaves, as this t-structure interpolates between the perverse coherent t-structure and certain t-structures appearing in the theory of Koszul duality (specializing to these in the case of a pure gauge theory and an abelian gauge theory, respectively). As a byproduct, this defines a canonical basis in the associated quantized Coulomb branch by passing to classes of irreducible objects. This is joint work with Sabin Cautis.