# Coulomb Branch Quantization and Abelianized Monopole Bubbling

## TL;DR

This paper develops a new algebraic framework for understanding Coulomb branch operators in 3D $
abla=4$ gauge theories, extending abelianization techniques to non-abelian cases through monopole bubbling analysis.

## Contribution

It introduces a star product algebra structure on Coulomb branches and extends abelianization to non-abelian theories via monopole bubbling quantification.

## Key findings

- Constructed a noncommutative star product algebra for Coulomb branches.
- Provided a trace map for good and ugly theories to compute correlation functions.
- Validated the abelianization approach through algebraic consistency and monopole bubbling analysis.

## Abstract

We develop an approach to the study of Coulomb branch operators in 3D $\mathcal{N}=4$ 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 and noncommutative star product algebra on the Coulomb branch. For `good' and `ugly' theories (according to the Gaiotto-Witten classification), we also exhibit a trace map on this algebra, which allows for the computation of 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, 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.