Multi-Monopole Moduli Spaces for SU(n) Gauge Group

TL;DR

This paper characterizes the moduli space of multi-monopoles in SU(n) gauge theories, proving conjectures and proposing new hyperk"ahler metrics that model complex monopole interactions.

Contribution

It proves the conjectured form of the multi-monopole moduli space in SU(n) theories and introduces new hyperk"ahler metrics using the generalized Legendre transform.

Findings

Proved the form of the moduli space for n-1 monopoles.

Proposed hyperk"ahler metrics with features of Atiyah-Hitchin metrics.

Conjectured solutions for arbitrary charge configurations.

Abstract

The moduli space describing the low-energy dynamics of BPS multi-monopoles for several charge configurations is presented. We first prove the conjectured form of the moduli space of $n-1$ distinct monopoles in a spontaneously broken SU(n) gauge theory. We further propose the solution where one of the charge components has two units, hence asymptotically corresponds to embeddings of two monopoles of one charge type and the rest different. The latter hyperk\"ahler metrics possess features of the two-monopole Atiyah-Hitchin metric. We also conjecture classes of solutions to multi-monopole moduli spaces with arbitrary charge and no more than two units in each component, which models the gluing together of Atiyah-Hitchin metrics. Our approach here uses the generalized Legendre transform technique to find the new hyperk\"ahler manifolds and rederive previously conjectured ones.