Coulomb Branch Quantization and Abelianized Monopole Bubbling

We develop an approach to study Coulomb branch operators in 3D N=4 gauge theories and the associated quantization structure of their Coulomb branches. This structure is encoded in a one-dimensional TQFT subsector of the full 3D theory, which we describe by combining several techniques and ideas. The answer takes the form of an associative non-commutative star-product algebra on the Coulomb branch. For `good' and `ugly' theories (according to Gaiotto-Witten classification), we also have a trace map on this algebra, which allows to compute correlation functions and, in particular, guarantees that the star-product satisfies a truncation condition. This work extends previous work on Abelian theories to the non-Abelian case by quantifying the monopole bubbling that describes screening of GNO boundary conditions. In our approach, the monopole bubbling is determined from the algebraic consistency of the OPE. This also yields a physical proof of the Bullimore-Dimofte-Gaiotto abelianization description of the Coulomb branch.