Flows involving Lifshitz solutions

TL;DR

The authors construct holographic RG flows between relativistic AdS vacua and non-relativistic Lifshitz geometries in two settings: a simple massive-vector theory and a six-dimensional $F(4)$ gauged supergravity that can be embedded in string theory. They analyze linearized perturbations to determine operator dimensions, and numerically realize domain-wall solutions that interpolate among AdS and Lifshitz fixed points, including flows from 6D AdS to 4D AdS/Lifshitz, AdS to Lifshitz, and Lifshitz to Lifshitz. Their results illuminate how Lifshitz holography can be understood within a controlled UV completion and establish a constructive description of the Lifshitz dual via twisted compactifications of higher-dimensional theories, while revealing dynamical instabilities in some Lifshitz solutions. The work points to new avenues for connecting AdS/CFT techniques with Lifshitz holography and for exploring the field-theory duals of non-relativistic fixed points.

Abstract

We construct gravity solutions describing renormalization group flows relating relativistic and non-relativistic conformal theories. We work both in a simple phenomenological theory with a massive vector field, and in an N=4, d=6 gauged supergravity theory, which can be consistently embedded in string theory. These flows offer some further insight into holography for Lifshitz geometries: in particular, they enable us to give a description of the field theory dual to the Lifshitz solutions in the latter theory. We also note that some of the AdS and Lifshitz solutions in the N=4, d=6 gauged supergravity theory are dynamically unstable.

Figures (17)

• Figure 1: The vacua of the massive vector model in $d+1$ dimensions, labelled according to whether or not they possess an irrelevant perturbation within our ansatz. This plot was made using $d=3$, but is qualitatively the same at higher $d$. The arrows denote the holographic RG flows.
• Figure 2: Holographic RG flow in $d=3$ from a Lifshitz spacetime with $z=1.6$ in the UV to an AdS$_4$ spacetime in the IR. In our numerical analysis we use a radial coordinate $\rho=\ln r$. Note that $\partial_\rho F \to z$ as we approach one of the AdS or Lifshitz solutions.
• Figure 3: Holographic RG flow in $d=3$ from a Lifshitz spacetime with $z=1.333$ in the UV to one with $z=3$ in the IR. In our numerical analysis we use a radial coordinate $\rho=\ln r$. Note that $\partial_\rho F \to z$ as we approach one of the AdS or Lifshitz solutions.
• Figure 4: Holographic RG flow in $d=3$ from an AdS$_4$ spacetime in the UV to a Lifshitz spacetime with $z=6$ in the IR. In our numerical analysis we use a radial coordinate $\rho=\ln r$. Note that $\partial_\rho F \to z$ as we approach one of the AdS or Lifshitz solutions.
• Figure 5: Plots of $D_0$ and $H_0$ for the AdS and Lifshitz solutions as a function of $\varphi_0^2$. The AdS solutions are shown in grey, the smaller Lifshitz in blue, and the larger $z$ Lifshitz in dashed red.