Non-abelian vortices on compact Riemann surfaces

Abstract

We consider the vortex equations for a U(n) gauge field A coupled to a Higgs field φ with values on the n × n matrices. It is known that when these equations are defined on a compact Riemann surface Σ, their moduli space of solutions is closely related to a moduli space of τ-stable holomorphic n-pairs on that surface. Using this fact and a local factorization result for the matrix φ, we show that the vortex solutions are entirely characterized by the location in Σ of the zeros of det φ 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. © The Author(s) 2009.