Abstract
Yang–Mills–Higgs monopoles and vortices in hyperbolic space can be constructed from and invariant Yang–Mills instantons, respectively. We use this fact to describe a large class of hyperbolic monopoles directly in terms of hyperbolic vortices embedded into 3 dimensions, yielding a remarkably simple relation between their Higgs fields. The class of monopoles we obtain are fixed relative to a plane in hyperbolic space, in a way which will be made clear by a study of the monopole spectral curve. We will use the correspondence between vortices and monopoles to give new insight into the moduli space of hyperbolic monopoles. Finally, our technique allows an explicit construction of the fields of a hyperbolic monopole invariant under a action, which we compare to periodic monopoles in Euclidean space.
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