Three Dimensional Gauge Theories and Monopoles

TL;DR

This paper explores the relationship between three-dimensional supersymmetric gauge theories and monopole moduli spaces, proposing a hyper-Kähler metric connection and comparing it to four-dimensional theories via compactification.

Contribution

It introduces a direct link between 3D $N=4$ supersymmetric Yang-Mills theories and monopole moduli spaces, and compares these theories to 4D $N=2$ theories through circle compactification.

Findings

Hyper-Kähler metric of $N=4$ $SU(N)$ Yang-Mills is given by the charge $N$ monopole moduli space.

Rational maps are effective tools for comparing 3D and 4D theories.

A BPS mass formula for particles and strings in these theories is derived.

Abstract

The coulomb branch of $N=4$ supersymmetric Yang-Mills gauge theories in $d=2+1$ is studied. A direct connection between gauge theories and monopole moduli spaces is presented. It is proposed that the hyper-K\"ahler metric of supersymmetric $N=4$ $SU(N)$ Yang-Mills theory is given by the charge $N$ centered moduli space of BPS monopoles in $SU(2)$. The theory is compared to $N=2$ supersymmetric Yang-Mills theory in four dimensions through compactification on a circle of the latter. It is found that rational maps are appropriate to this comparison. A BPS mass formula is also written for particles in three dimensions and strings in four dimensions.