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Hyperbolic monopoles from hyperbolic vortices

Certain hyperbolic monopoles and all hyperbolic vortices can be constructed from SO(2) and SO(3) invariant Euclidean instantons, respectively. This observation allows us to describe a large class of hyperbolic monopoles as hyperbolic vortices embedded into $\mathbb{H}^3$ and yields a remarkably simple relation between the two Higgs fields. This correspondence between vortices and monopoles gives new insight into the geometry of the spectral curve and the moduli space of hyperbolic monopoles. It also allows an explicit construction of the fields of a hyperbolic monopole invariant under a $\mathbb{Z}$ action, which we compare to periodic monopoles in Euclidean space.