Lifshitz black holes in string theory

TL;DR

This work embeds Lifshitz black holes with $z=2$ into string theory by a consistent truncation of type IIB on ${\rm E}_5\times{\rm S}^1$, retaining the axio-dilaton and a circle modulus that together with a flux-induced massive vector supports Lifshitz asymptotics. The authors obtain an infinite family of $d=4$, $z=2$ Lifshitz black holes, show that there is no extremal limit, and reveal thermodynamic instability for small black holes, hinting at a Hawking-Page–like transition between large Lifshitz black holes and thermal Lifshitz space. The extra scalars (dilaton and circle modulus) crucially influence the near-horizon and UV behaviour, leading to qualitative differences from bottom-up Lifshitz models. This construction provides a UV-complete setting for non-relativistic holography at finite temperature and motivates further exploration of charged solutions and the Lifshitz holographic dictionary in string theory.

Abstract

We provide the first black hole solutions with Lifshitz asymptotics found in string theory. These are expected to be dual to models enjoying anisotropic scale invariance with dynamical exponent z=2 at finite temperature. We employ a consistent truncation of type IIB supergravity to four dimensions with an arbitrary 5-dimensional Einstein manifold times a circle as internal geometry. New interesting features are found that significantly differ from previous results in phenomenological models. In particular, small black holes are shown to be thermodynamically unstable, analogously to the usual AdS-Schwarzschild black holes, and extremality is never reached. This signals a possible Hawking-Page like phase transition at low temperatures.

Figures (7)

• Figure 1: Metric functions $f(r)$ (solid) and $g(r)$ (dashdotted) for various different pairs of initial values. The red (inner) and the blue (outer) lines correspond to $\tau_0=2$ and $\tau_0=1$ for $r_{\rm H}=1/2$ (left), and to $\tau_0=2$ and $\tau_0=1/2$ for $r_{\rm H}=20$ (right), respectively.
• Figure 2: Form fields $h(r)$ (solid) and $j(r)$ (dashdotted) for various different pairs of initial values. The red (inner) and the blue (outer) lines correspond to $\tau_0=2$ and $\tau_0=1$ for $r_{\rm H}=1/2$ (left), and to $\tau_0=2$ and $\tau_0=1/2$ for $r_{\rm H}=20$ (right), respectively.
• Figure 3: Scalar fields $\gamma(r)$ (solid) and $\tau(r)$ (dashdotted) for various different pairs of initial values. The red (upper) and the blue (lower) lines correspond to $\tau_0=2$ and $\tau_0=1$ for $r_{\rm H}=1/2$ (left), and to $\tau_0=2$ and $\tau_0=1/2$ for $r_{\rm H}=20$ (right), respectively.
• Figure 4: Horizon 1-form flux $h_0$ as a function of the ratio of the scalar fields $\gamma_0/\tau_0$ for fixed $r_{\rm H}=1/2,\,1,\,3/2,\,2,\,5,\,20$ and $r_{\rm H}=\infty$, from left to right.
• Figure 5: (Left) Horizon dilaton $\gamma_0$ as a function of $r_{\rm H}$ for fixed $\tau_0=6,\,2,\,1,\,1/2$, from top to down. (Right) Horizon 1-form flux $h_0$ as a function of $r_{\rm H}$ for fixed $\tau_0=6,\,2,\,1,\,1/2$, from down to top, normalized by the flat solution value: $h_0^{\infty}=0.0549,\,0.459,\,1.57,\,4.79$, respectively. First branch (solid) is connected to the $k=0$ case, while the second branch (dashdotted) is unstable.