Three Dimensional Gauge Theories and Monopoles

TL;DR

The work proposes a deep geometric bridge between three‑dimensional $N=4$ $SU(N)$ gauge theories and the moduli spaces of $N$ BPS monopoles in $SU(2)$, identifying the Coulomb branch metric with the hyper‑Kähler metric on the centered monopole moduli space $M_N^0$. Using semiclassical analysis, circle compactification, and a rational‑map description, the authors connect the 3D Coulomb branch to the asymptotics and spectral data of monopole configurations, and show how the metric interpolates between the Taub‑NUT/Dirac‑monopole structure in 3D and the hyper‑elliptic curves of 4D $N=2$ theories in the $R o ext{large}$ limit. They develop explicit SU(2) and SU(3) examples, derive a three‑dimensional BPS mass formula, and discuss how spectral curves and ADE generalizations fit into this program, while highlighting instanton corrections and potential links to brane constructions. The results offer a novel geometric lens on 3D gauge dynamics and a pathway to unify monopole moduli spaces with low‑dimensional supersymmetric vacua, with implications for dualities and string‑theory embeddings.

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-Kähler 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.