Abstract
The geodesic approximation is a powerful method for studying the dynamics of BPS solitons. However, there are systems, such as BPS monopoles in three-dimensional hyperbolic space, where this approach is not applicable because the moduli space metric defined by the kinetic energy is not finite. In the case of hyperbolic monopoles, an alternative metric has been defined using the abelian connection on the sphere at infinity, but its relation to the dynamics of hyperbolic monopoles is unclear. Here this metric is placed in a more general context of boundary metrics on soliton moduli spaces. Examples are studied in systems in one and two space dimensions, where it is much easier to compare the results with simulations of the full nonlinear field theory dynamics. It is found that geodesics of the boundary metric provide a reasonable description of soliton dynamics.
Highlights
JHEP01(2022)118 in a more general context of boundary metrics on BPS soliton moduli spaces
It is found that geodesics of the boundary metric provide a reasonable description of soliton dynamics
The section provides the details of the construction of the boundary metric for hyperbolic monopoles, and further sections consider a radial lump in the CP1 sigma model in the plane, and kink dynamics on a line
Summary
This section concerns SU(2) magnetic monopoles in three-dimensional hyperbolic space of curvature −1, with metric. The moduli space, MN , of charge N monopole solutions of the Bogomolny equation (2.2), up to gauge transformations, is a (4N − 1)-dimensional manifold [5]. The condition of orthogonality to gauge orbits yields a finite limit for Φ (1 − r2)−2 as r → 1, so there is no divergence of the kinetic energy as b → 1 from this first term. On this boundary sphere Gauss’ law (2.6) projected onto the abelian component reduces to ar = 0 and the requirement that the angular components must satisfy With this condition, the renormalized kinetic energy on the monopole moduli space becomes that of an abelian gauge theory on the sphere. For any initial condition of this form, causality considerations imply that for a sufficiently large m, the evolution within any given spatial region will be insensitive to the value of m for evolution up to some time limit that can be increased by increasing m
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