Conformal Lifshitz Gravity from Holography

TL;DR

This paper demonstrates how holographic renormalization in Lifshitz spacetimes naturally yields gravity theories with anisotropic scaling, specifically connecting counterterms to Lifshitz conformal gravity and revealing implications for dual field theories.

Contribution

It establishes a direct link between holographic counterterms and Lifshitz gravity actions, including the derivation of anisotropic Weyl anomalies and detailed balance conditions in Lifshitz holography.

Findings

Counterterms reproduce Lifshitz gravity actions with specific z

Logarithmic counterterm related to anisotropic Weyl anomaly

Analytic continuation links Lifshitz conformal gravity to ground-state wavefunctions

Abstract

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.