Vortex hair on AdS black holes
  We analyse vortex hair for charged rotating asymptotically AdS black holes in
the abelian Higgs model. We give analytical and numerical arguments to show how
the vortex interacts with the horizon of the black hole, and how the solution
extends to the boundary. The solution is very close to the corresponding
asymptotically flat vortex, once one transforms to a frame that is non-rotating
at the boundary. We show that there is a Meissner effect for extremal black
holes, with the vortex flux being expelled from sufficiently small black holes.
The phase transition is shown to be first order in the presence of rotation,
but second order without rotation. We comment on applications to holography.
