Quantized Coulomb Branches, Monopole Bubbling and Wall-Crossing Phenomena in 3d $\mathcal{N}=4$ Theories

TL;DR

This paper introduces a novel method combining supersymmetric localization and brane constructions to compute correlators in the quantized Coulomb branch of 3d $ ext{N}=4$ theories, revealing wall-crossing phenomena and non-commutative structures.

Contribution

It presents a new approach to calculate monopole and Casimir operator correlators using brane setups and matrix models, highlighting wall-crossing and non-commutative effects.

Findings

Explicit correlator calculations for non-bubbling monopoles

Identification of wall-crossing in monopole operator insertions

Realization of non-commutative product as a Moyal star product

Abstract

To study the quantized Coulomb branch of 3d $\mathcal{N}=4$ unitary SQCD theories, we propose a new method to compute correlators of monopole and Casimir operators that are inserted in the $\mathbb{R}\times\mathbb{R}^2_\epsilon$ Omega background. This method combines results from supersymmetric localization with inputs from the brane realisation of the correlators in type IIB string theory. The main challenge is the computation of the partition functions of certain Super-Matrix-Models (SMMs), which appear in the contribution of monopole bubbling sectors and are realised as the theory living on the D1 strings in the brane construction. We find that the non-commutativity arising in the monopole operator insertions is related to a wall-crossing phenomenon in the FI parameter space of the SMM. We illustrate our method in various examples and we provide explicit results for arbitrary correlators of non-bubbling bare monopole operators. We also discuss the realisation of the non-commutative product as a Moyal (star) product and use it to successfully test our results.