Non-abelian vortices on compact Riemann surfaces

We consider the vortex equations for a U(n) gauge field A coupled to a Higgs field phi with values on the n x n matrices. It is known that when these equations are defined on a compact Riemann surface Sigma, their moduli space of solutions is closely related to a moduli space of tau-stable holomorphic n-pairs on that surface. Using this fact and a local factorization result for the matrix phi, we show that the vortex solutions are entirely characterized by the location in Sigma of the zeros of det phi and by the choice of a vortex internal structure at each of these zeros. We describe explicitly the vortex internal spaces and show that they are compact and connected spaces.