Conformal Lifshitz gravity from holography

We show that holographic renormalization of relativistic gravity in asymptotically Lifshitz spacetimes naturally reproduces the structure of gravity with anisotropic scaling: the holographic counterterms induced near anisotropic infinity take the form of the action for gravity at a Lifshitz point, with the appropriate value of the dynamical critical exponent z. In the particular case of 3 + 1 bulk dimensions and z = 2 asymptotic scaling near infinity, we find a logarithmic counterterm, related to anisotropic Weyl anomaly of the dual CFT, and show that this counterterm reproduces precisely the action of conformal gravity at a z = 2 Lifshitz point in 2 + 1 dimensions, which enjoys anisotropic local Weyl invariance and satisfies the detailed balance condition. We explain how the detailed balance is a consequence of relations among holographic counterterms, and point out that a similar relation holds in the relativistic case of holography in AdS(5). Upon analytic continuation, analogous to the relativistic case studied recently by Maldacena, the action of conformal gravity at the z = 2 Lifshitz point features in the ground-state wavefunction of a gravitational system with an interesting type of spatial anisotropy.