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. (C) 2010 American Institute of Physics. [doi: 10.1063/1.3277189]