HyperKähler Quotient Construction of BPS Monopole Moduli Spaces

TL;DR

This paper develops explicit HyperKähler quotient constructions from flat space to produce metric models of monopole moduli spaces, enabling direct analysis of topology, completeness, and isometries while clarifying massless-monopole limits via moment-map level sets. By applying quotients with tri-holomorphic torus actions, it yields concrete metrics such as Taub-NUT, Lee-Weinberg-Yi, Calabi, Taubian-Calabi, and cyclic ALE/ALF spaces, and analyzes their behavior in zero- and infinite-mass limits. The work connects these metrics to physically relevant settings, including SU$(m{+}2) o$ SU$(m) imes U(1)^2$ and related symmetry-breaking patterns, where massless clouds and infinite inertia emerge as key features. The results offer explicit tools for studying monopole dynamics, S-duality checks in gauge theories, and the moduli of three-dimensional gauge theories, with broader implications for geodesic integrability and the global geometry of hyperkähler quotients.

Abstract

We use the HyperKähler quotient of flat space to obtain some monopole moduli space metrics in explicit form. Using this new description, we discuss their topology, completeness and isometries. We construct the moduli space metrics in the limit when some monopoles become massless, which corresponds to non-maximal symmetry breaking of the gauge group. We also introduce a new family of HyperK"{a}hler metrics which, depending on the ``mass parameter'' being positive or negative, give rise to either the asymptotic metric on the moduli space of many SU(2) monopoles, or to previously unknown metrics. These new metrics are complete if one carries out the quotient of a non-zero level set of the moment map, but develop singularities when the zero-set is considered. These latter metrics are of relevance to the moduli spaces of vacua of three dimensional gauge theories for higher rank gauge groups. Finally, we make a few comments concerning the existence of closed or bound orbits on some of these manifolds and the integrability of the geodesic flow.