Exact moduli space metrics for hyperbolic vortices
Abstract
Exact metrics on some totally geodesic submanifolds of the moduli space of static hyperbolic N-vortices are derived. These submanifolds, denoted n,m, are spaces of Cn-invariant vortex configurations with n single vortices at the vertices of a regular polygon and m=N-n coincident vortices at the polygon's centre. The geometric properties of n,m are investigated, and it is found that n,n-1 is isometric to the hyperbolic plane of curvature -1/(3π n). Geodesic flow on n,m, and a geometrically natural variant of geodesic flow recently proposed by Collie and Tong, are analyzed in detail.
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