We construct and study a nonstandard t-structure on the derived category of equivariant coherent sheaves on the Braverman-Finkelberg-Nakajima space of triples ${\mathcal {R}}_{G,N}$ , where ${N}$ is a representation of a reductive group $G$ . Its heart ${\mathcal {KP}}_{G,N}$ is a finite-length, rigid, monoidal abelian category with renormalized $r$ -matrices. We refer to objects of ${\mathcal {KP}}_{G,N}$ as Koszul-perverse coherent sheaves. Simple objects of ${\mathcal {KP}}_{G,N}$ define a canonical basis in the quantized $K$ -theoretic Coulomb branch of the associated gauge theory. These simples possess various characteristic properties of Wilson-’t Hooft lines, and we interpret our construction as an algebro-geometric definition of the category of half-BPS line defects in a 4d $\mathcal{N}=2$ gauge theory of cotangent type.
