Lifshitz as a deformation of Anti-de Sitter

We consider holography for Lifshitz spacetimes with dynamical exponent z = 1+ϵ2, where ϵ is small. We show that the holographically dual field theory is a specific deformation of the relativistic CFT, corresponding to the z = 1 theory. Treating ϵ as a small expansion parameter we set up the holographic dictionary for Einstein-Proca models up to order ϵ2 in three and four bulk dimensions. We explain how renormalization turns the relativistic conformal invariance into non-relativistic Lifshitz invariance with dynamical exponent z = 1 + ϵ2. We compute the two-point function of the conserved spin two current for the dual two-dimensional field theory and verify that it is Lifshitz invariant. Using only QFT arguments, we show that a particular class of deformations of CFTs generically leads to Lifshitz scaling invariance and we construct examples of such deformations.