Building on work of Kashiwara-Rouquier, Gan-Ginzburg, and others Braden-Licata-Proufoot-Webster argued that the natural context for these results was the deformation quantization of conical symplectic resolutions: The ring of differential operators on G/B is a deformation quantization of the symplectic manifold T^*G/B and the Beilinson-Bernstein theorem is equivalent to the fact that T^*G/B resolves the singularities of the cone of nilpotent elements in g^*. Moreover, they showed that one can define a version of category O for any conical symplectic resolution X and conjectured that each symplectic resolution X had a symplectic dual X^! such that O(X) is Koszul dual to O(X^!).
Soon, it became clear that BLPW's list of conjectural symplectic dual pairs coincided with the list of known Higgs and Coulomb branches of 3d N=4 theories but the physical interpretation of category O was unclear until Bullimore, Dimofte, and Gaiotto, and I identified it with the category of (2,2)-boundary conditions. I will show how to interpret this work terms of the construction of the Coulomb branch by Braverman-Finkelberg-Nakajima. Part of this work is joint with Kamnitzer-Weekes.