Exact moduli space metrics for hyperbolic vortices

Exact metrics on some totally geodesic submanifolds of the moduli space of static hyperbolic N-vortices are derived. These submanifolds, denoted \Sigma_{n,m}, are spaces of C_n-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 \Sigma_{n,m} are investigated, and it is found that \Sigma_{n,n-1} is isometric to the hyperbolic plane of curvature -1/(3\pi n). Geodesic flow on \Sigma_{n,m}, and a geometrically natural variant of geodesic flow recently proposed by Collie and Tong, are analyzed in detail.