Hyperbolic monopole data

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.

Figures (2)

• Figure 1: Energy density isosurfaces for the family $G_4^2$, with the parameter value from left to right given by $q_0=-0.45,-0.30,-0.25,-0.20,-0.05.$
• Figure 2: Energy density isosurfaces for the family $G_4^1$, with the parameter value from left to right given by $p_1=0.05,0.20,0.25,0.28,0.32.$