Abstract
The Atiyah-Hitchin manifold is the moduli space of parity inversion symmetric charge two SU(2) monopoles in Euclidean space. Here a hyperbolic analogue is presented, by calculating the boundary metric on the moduli space of parity inversion symmetric charge two SU(2) monopoles in hyperbolic space. The calculation of the metric is performed using a twistor description of the moduli space and the result is presented in terms of standard elliptic integrals.
Highlights
Spectral curves of hyperbolic monopolesThe mini-twistor space of three-dimensional hyperbolic space is the space of its oriented geodesics
JHEP01(2022)090 with the metric induced from the field theory kinetic energy, because the induced metric diverges
The qualitative similarity between the Atiyah-Hitchin manifold and its hyperbolic analogue indicates that hyperbolic monopole dynamics mirrors the Euclidean picture, within a geodesic description
Summary
The mini-twistor space of three-dimensional hyperbolic space is the space of its oriented geodesics. There is a bijective correspondence between hyperbolic monopoles and spectral curves, which are algebraic curves in mini-twistor space satisfying certain reality and non-singularity conditions [4]. The spectral curve describes all geodesics along which a certain linear operator, constructed from the monopole fields, has a normalizable solution This is equivalent to imposing non-singularity conditions on the algebraic curve that can be written in terms of relations between integrals of holomorphic differentials around particular cycles. Where X = (X1, X2, X3) is the point inside the unit ball at which the monopole is located (given by the vanishing of the Higgs field) This spectral curve gives all the geodesics that pass through the point X. The first example relevant for hyperbolic monopole scattering is constructed, by calculating the boundary metric on the hyperbolic analogue of the 4-dimensional Atiyah-Hitchin manifold for charge two monopoles with parity inversion symmetry
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