Abstract
Magnetic monopoles in hyperbolic space are in correspondence with certain algebraic curves in mini-twistor space, known as spectral curves, which are in turn in correspondence with rational maps between Riemann spheres. Hyperbolic monopoles correspond to circle-invariant Yang–Mills instantons, with an identification of the monopole and instanton numbers, providing the curvature of hyperbolic space is tuned to a value specified by the asymptotic magnitude of the Higgs field. In previous work, constraints on ADHM instanton data have been identified that provide a non-canonical realization of the circle symmetry that preserves the standard action of rotations in the ball model of hyperbolic space. Here formulae are presented for the spectral curve and the rational map of a hyperbolic monopole in terms of its constrained ADHM matrix. This extends earlier results that apply only to the subclass of instantons of JNR type. The formulae are applied to obtain new explicit examples of spectral curves that are beyond the JNR class.
Highlights
Magnetic monopoles in three-dimensional hyperbolic space share many of the features of their counterparts in Euclidean space [1]
A hyperbolic monopole of charge N is equivalent to a circle-invariant charge N Yang–Mills instanton in four-dimensional Euclidean space, with the Higgs field of the monopole given by the component of the instanton gauge field associated with the circle action [1]
A simple formula has been presented for the spectral curves of hyperbolic monopoles in terms of ADHM matrices satisfying conditions that imply circle invariance of the associated instantons and commuting rotations that act canonically on the ball model of hyperbolic space
Summary
Magnetic monopoles in three-dimensional hyperbolic space share many of the features of their counterparts in Euclidean space [1]. The difficulty with implementing this approach lies in finding complex ADHM matrices, which is further complicated by the fact that the natural circle action used by Braam and Austin yields the upper half space model of hyperbolic space, where spatial rotations act in a non-canonical manner This has prevented the use of symmetry methods, despite the fact that symmetry reductions have been successfully applied in the past to find quaternionic ADHM matrices. Braam and Austin [3] provided a formula for the rational map from its complex ADHM matrix and within the JNR class this provides a simple expression for the rational map in terms of the free data of weighted points on a sphere [5]. A formula is presented for the rational map from its constrained quaternionic ADHM matrix
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