Conformal Lifshitz Gravity from Holography

TL;DR

This work shows that holographic renormalization in asymptotically Lifshitz spacetimes naturally yields gravity with anisotropic scaling, where near anisotropic infinity the counterterms reproduce a Lifshitz-conformal gravitational action with the correct dynamical exponent $z$. In the key case of $3+1$ bulk dimensions and $z=2$, a logarithmic counterterm emerges that matches the action of $z=2$ conformal gravity in $2+1$ dimensions and enforces detailed balance, a relation that persists when bulk scalars are included. The authors develop a Hamiltonian holographic renormalization framework, compute the divergences for bulk gravity with a massive vector (and with scalars), and show that detailed balance is a structural consequence of the holographic counterterm relations, with an analytic continuation linking the result to a de Sitter-like regime and a peculiar, spatially anisotropic ground-state wavefunction. Overall, the paper provides a string-theory-inspired holographic route to multicritical gravity and clarifies the role of anisotropic Weyl invariance and detailed balance in the Lifshitz holographic context, offering paths to generalize to higher $z$ and richer bulk embeddings.

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.