Lifshitz from AdS at finite temperature and top down models

We construct analytically an asymptotically Lifshitz black brane with dynamical exponent z = 1 + ? 2 in an Einstein-Proca model, where ? is a small parameter. In previous work we showed that the holographic dual QFT is a deformation of a CFT by the time component of a vector operator and the parameter ? is the corresponding deformation parameter. In the black brane background this operator additionally acquires a vacuum expectation value. We explain how the QFT Ward identity associated with Lifshitz invariance leads to a conserved mass and compute analytically the thermodynamic quantities showing that they indeed take the form implied by Lifshitz invariance. In the second part of the paper we consider top down Lifshitz models with dynamical exponent close to one and show that they can be understood in terms of vector deformations of conformal field theories. However, in all known cases, both the conformal field theory and its Lifshitz deformations have modes that violate the Breitenlohner-Freedman bound