Exact moduli space metrics for hyperbolic vortex polygons

Exact metrics on some totally geodesic submanifolds of the moduli space of static hyperbolic N-vortices are derived. These submanifolds, denoted as 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 center. 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 ?(3 pi n)^{?1}. The geodesic flow on Sigma_{n,m} and a geometrically natural variant of geodesic flow recently proposed by Collie and Tong [“The dynamics of Chern-Simons vortices,” Phys. Rev. D Part. Fields Gravit. Cosmol. 78, 065013 (2008);e-print arXiv:hep-th/0805.0602] are analyzed in detail.