Higgs Branches in the Omega-background via the Category of Line Operators

The vacuum manifold $\mathcal{M}$ of a topological twist of a 3d $\mathcal{N}=4$ gauge theory is a hyper-K\"ahler variety; deformations and quantizations of $\mathcal{M}$ can be constructed in the framework of 3 dimensional topological quantum field theories. In particular, based on physics arguments, turning on an omega-background results in the quantization of $\mathbb{C}[\mathcal{M}]$ as a Poisson algebra. In this paper, we implement this idea mathematically, in the context of the B-twist of 3d $\mathcal{N}=4$ gauge theories, namely for Higgs branches. Our strategy is to implement omega-background in the category of line operators via the choice of a ribbon twist, and obtain the quantum Hamiltonian reduction as derived endomorphisms in the equivariant category, the category where the ribbon twist acts trivially. We apply quadratic Koszul duality to perform this computation.