Scalar hairs and exact vortex solutions in 3D AdS gravity

We investigate three-dimensional (3D) Anti-de Sitter (AdS) gravity coupled to a complex scalar field with self-interaction potential V . We show that the mass of static, rotationally symmetric, AdS black hole with scalar hairs is determined algebraically by the scalar charges. We recast the field equations as a linear system of first order differential equations. Exact solutions, describing 3D AdS black holes with real spherical scalar hairs and vortex-black hole solutions are derived in closed form for the case of a scalar field saturating the Breitenlohner-Freedman (BF) bound and for a scalar field with asymptotic zero mass. The physical properties of these solutions are discussed. In particular, we show that the vortex solution interpolates between two different AdS3 vacua, corresponding respectively to a U(1)-symmetry-preserving maximum and to a symmetrybreaking minimum of the potential V .