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Platonic hyperbolic monopoles

Manton, N.S; Sutcliffe, P.M.

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N.S Manton


We construct a number of explicit examples of hyperbolic monopoles, with various charges and often with some platonic symmetry. The fields are obtained from instanton data in R 4 that are invariant under a circle action, and in most cases the monopole charge is equal to the instanton charge. A key ingredient is the identification of a new set of constraints on ADHM instanton data that are sufficient to ensure the circle invariance. Unlike for Euclidean monopoles, the formulae for the squared Higgs field magnitude in the examples we construct are rational functions of the coordinates. Using these formulae, we compute and illustrate the energy density of the monopoles. We also prove, for particular monopoles, that the number of zeros of the Higgs field is greater than the monopole charge, confirming numerical results established earlier for Euclidean monopoles. We also present some one-parameter families of monopoles analogous to known scattering events for Euclidean monopoles within the geodesic approximation.


Manton, N., & Sutcliffe, P. (2014). Platonic hyperbolic monopoles. Communications in Mathematical Physics, 325(3), 821-845.

Journal Article Type Article
Acceptance Date May 13, 2013
Online Publication Date Jan 4, 2014
Publication Date Feb 1, 2014
Deposit Date Jan 29, 2014
Publicly Available Date Feb 4, 2014
Journal Communications in Mathematical Physics
Print ISSN 0010-3616
Electronic ISSN 1432-0916
Publisher Springer
Peer Reviewed Peer Reviewed
Volume 325
Issue 3
Pages 821-845


Accepted Journal Article (1.7 Mb)

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