HyperK\"{a}hler Quotient Construction of BPS Monopole Moduli Spaces

TL;DR

This paper constructs explicit hyperKähler metrics for monopole moduli spaces using quotient methods, explores their topology and symmetries, and introduces new metrics relevant to gauge theories and monopole dynamics.

Contribution

It provides a new hyperKähler quotient construction of monopole moduli spaces, including novel metrics and analysis of their geometric properties and physical relevance.

Findings

Explicit monopole moduli space metrics derived

Identification of conditions for metric completeness and singularities

Introduction of new hyperKähler metrics related to gauge theories

Abstract

We use the HyperK\"{a}hler 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.