Coulomb Branches in 3d $\mathcal{N} = 4$ Revisited

Using ideas from the gauge theory approach to the geometric Langlands program, we revisit supersymmetric localization with monopole operators in 3d $\mathcal{N} = 4$ supersymmetric gauge theories subject to $\Omega$-deformation. The key novel feature of our setup is a pair of dual boundary conditions, which drastically simplify the dynamics of the theory and the nature of the localization loci. From a careful calculation with these boundary conditions, the mathematical definition of Coulomb branches proposed by Braverman, Finkelberg and Nakajima emerges naturally. It is straightforward to incorporate codimension two defects in the setup, and in this way we gain insight into Webster's construction of tilting bundles on Coulomb branches.