Abelian Higgs hair for an AdS-Schwarzschild black hole

We show that the Abelian Higgs field equations in the background of the four dimensional AdS-Schwarzschild black hole have a vortex line solution. This solution, which has axial symmetry, is a generalization of the AdS spacetime Nielsen-Olesen string. From a numerical study of the field equations, we show that black hole could support an Abelian Higgs field as its Abelian hair. Also, we consider the self-gravity of the Abelian Higgs field both in pure AdS spacetime and the AdS-Schwarzschild black hole background and show that the effect of a string as black hole hair is to induce a deficit angle in the AdS-Schwarzschild black hole.