Singular Monopoles and Supersymmetric Gauge Theories in Three Dimensions

TL;DR

This paper extends the Hanany-Witten proposal linking Coulomb branches of 3D N=4 gauge theories to monopole moduli spaces, providing new geometric constructions and comparisons with field theory predictions.

Contribution

It generalizes the monopole correspondence to theories with matter and constructs explicit hyperkähler metrics for their Coulomb branches.

Findings

Constructed metrics on Coulomb branches using hyperkähler quotients.

Identified Coulomb branches with known gravitational instantons for specific theories.

Compared complex structures with field theory results, confirming the correspondence.

Abstract

According to the proposal of Hanany and Witten, Coulomb branches of N=4 SU(n) gauge theories in three dimensions are isometric to moduli spaces of BPS monopoles. We generalize this proposal to gauge theories with matter, which allows us to describe the metrics on their spaces of vacua by means of the hyperk\"ahler quotient construction. To check the identification of moduli spaces a comparison is made with field theory predictions. For SU(2) theory with k fundamental hypermultiplets the Coulomb branch is expected to be the D_k ALF gravitational instanton, so our results lead to a construction of such spaces. In the special case of SU(2) theory with four or fewer fundamental hypermultiplets we calculate the complex structures on the moduli spaces and compare them with field-theoretical results. We also discuss some puzzles with brane realizations of three-dimensional N=4 theories.