Lifshitz as a deformation of Anti-de Sitter

TL;DR

This work shows that Lifshitz holography with z = 1 + ε^2 can be understood as a controlled deformation of a relativistic CFT by a vector primary of dimension d. Using a perturbative ε-expansion in Einstein–Proca gravity, the authors derive a holographic dictionary, compute renormalized one-point functions, and establish Ward identities that encode Lifshitz invariance at order ε^2. They also demonstrate, via conformal perturbation theory and detailed OPE analyses, that a broad class of vector deformations generically flow to Lifshitz-invariant fixed points, with the holographic and field-theory pictures in agreement. In 2D, Lifshitz-invariant two-point functions of the conserved spin-two current are fully determined by Ward identities and exhibit the expected anisotropic scaling, while in higher dimensions the analysis clarifies how the metric beta-function arises from universal JJ OPE data. These results illuminate how non-relativistic Lifshitz symmetry emerges from relativistic CFTs both holographically and in general QFT terms, providing a framework for exploring near-Lifshitz fixed points in strongly and weakly coupled theories.

Abstract

We consider holography for Lifshitz spacetimes with dynamical exponent z=1+epsilon^2, where epsilon 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 epsilon as a small expansion parameter we set up the holographic dictionary for Einstein-Proca models up to order epsilon^2 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+epsilon^2. 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.