Hyperbolic monopole data

TL;DR

This paper reformulates hyperbolic monopole data using real matrices satisfying quartic equations, enabling new solutions and a novel family of charge 4 hyperbolic monopoles with square symmetry.

Contribution

It introduces a new matrix-based formulation of hyperbolic monopoles and constructs a new family of charge 4 solutions with square symmetry.

Findings

Reformulation of hyperbolic monopoles via real matrices and quartic equations.

Recovery of known monopole examples through this new formulation.

Construction of a new charge 4 hyperbolic monopole family with square symmetry.

Abstract

It is known that hyperbolic monopoles, with a particular value of the curvature, can be obtained from ADHM instanton data that satisfies additional constraints. Here this data is reformulated in terms of a triplet of real matrices that satisfy a set of quartic equations, with solutions associated with representations of su(2). Many of the known examples of hyperbolic monopoles can easily be recovered in this formulation by evaluating Nahm data for Euclidean monopoles at the centre of its domain. Toda reductions of Nahm's equation correspond to cyclic Euclidean monopoles, and this is adapted to the hyperbolic setting to obtain solutions, even when the corresponding Nahm data is not tractable. A new family of charge 4 hyperbolic monopoles with square symmetry is presented as an example.