Hyperkahler Metrics from Periodic Monopoles

TL;DR

This paper studies the geometry of hyperkahler manifolds arising from periodic monopoles, revealing their asymptotic metrics, topology, and complex structure, and connects them to Hitchin moduli spaces.

Contribution

It introduces new hyperkahler metrics from periodic monopoles and relates them to gravitational instantons via moduli spaces of Hitchin equations.

Findings

Asymptotic behavior of metrics for well-separated monopoles

Construction of hyperkahler metrics with self-dual curvature

Topological and complex geometric analysis of these manifolds

Abstract

Relative moduli spaces of periodic monopoles provide novel examples of Asymptotically Locally Flat hyperkahler manifolds. By considering the interactions between well-separated periodic monopoles, we infer the asymptotic behavior of their metrics. When the monopole moduli space is four-dimensional, this construction yields interesting examples of metrics with self-dual curvature (gravitational instantons). We discuss their topology and complex geometry. An alternative construction of these gravitational instantons using moduli spaces of Hitchin equations is also described.