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We construct and study a nonstandard t-structure on the derived category ofequivariant coherent sheaves on the Braverman-Finkelberg-Nakajima space oftriples $\mathcal{R}_{G,N}$, where $N$ is a representation of a reductive group$G$. Its heart $\mathcal{KP}_{G,N}$ is a finite-length, rigid, monoidal abeliancategory 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 $K$-theoreticCoulomb branch of the associated gauge theory. These simples possess variouscharacteristic properties of Wilson-'t Hooft lines, and we interpret ourconstruction as an algebro-geometric definition of the category of half-BPSline defects in a 4d $\mathcal{N}=2$ gauge theory of cotangent type.

Canonical bases for Coulomb branches of 4d $\mathcal{N}=2$ gauge theories