Stability analysis of holographic RG flows in 3d supergravity

Abstract

We study holographic RG flows in a 3d supergravity model from the side of the dynamical system theory. The gravity equations of motion are reduced to an autonomous dynamical system. Then we find equilibrium points of the system and analyze them for stability. We also restore asymptotic solutions near the critical points. We find two types of solutions: with asymptotically AdS metrics and hyperscaling violating metrics. We write down possible RG flows between an unstable (saddle) UV fixed point and a stable (stable node) IR fixed point. We also analyze bifurcations in the model.

Geometric characteristics of the metric and stress-energy tensor

The metric in the domain wall coordinates is described by the following expression:

$$\begin{aligned} ds^2=e^{2A(w)}\left( -dt^2+dx^2\right) +dw^2. \end{aligned}$$

(A.1)

Then non-zero Ricci tensor components take the form:

$$\begin{aligned} R_{tt}= e^{2 A} ( {\ddot{A}}+2{\dot{A}}^2),\quad R_{xx}=-e^{2 A} ({\ddot{A}}+2 {\dot{A}}^2),\quad R_{ww}=-2 ({\ddot{A}}+{\dot{A}}^2), \end{aligned}$$

(A.2)

and the Ricci scalar is given by:

$$\begin{aligned} R=-2 (2 {\ddot{A}}+3 {\dot{A}}^2). \end{aligned}$$

(A.3)

The components of the Einstein tensor are:

$$\begin{aligned} G_{tt}=-e^{2A}({\ddot{A}}+{\dot{A}}^2),\quad G_{xx}=e^{2A}({\ddot{A}}+{\dot{A}}^2),\quad G_{ww}={\dot{A}}^2. \end{aligned}$$

(A.4)

Stress-energy-momentum tensor, defined as

$$\begin{aligned} T_{\mu \nu }=\frac{1}{a^2}\left( \partial _{\mu }\phi \partial _{\nu }\phi -\frac{1}{2}g_{\mu \nu }\partial _{\sigma }\phi \partial ^{\sigma }\phi \right) -\frac{1}{2}g_{\mu \nu }V, \end{aligned}$$

(A.5)

will have the following non-zero components:

$$\begin{aligned} T_{tt}= & {} \frac{1}{a^2}\left( -g_{tt}{\dot{\phi }}^2g^{ww}\right) -\frac{1}{2}g_{tt}V=\frac{e^{2 A}}{2}\left( \frac{{\dot{\phi }}^2}{a^2}+V\right) , \end{aligned}$$

(A.6)

$$\begin{aligned} T_{xx}= & {} \frac{1}{a^2}\left( -\frac{1}{2}g_{xx}{\dot{\phi }}^2g^{ww}\right) -\frac{1}{2}g_{xx}V=-\frac{e^{2 A}}{2}\left( \frac{{\dot{\phi }}^2}{a^2}+V\right) , \end{aligned}$$

(A.7)

$$\begin{aligned} T_{ww}= & {} \frac{1}{a^2}\left( {\dot{\phi }}^2-\frac{1}{2}{\dot{\phi }}^2\right) -\frac{1}{2}g_{ww}V=\frac{1}{2}\left( \frac{{\dot{\phi }}^2}{a^2}-V\right) . \end{aligned}$$

(A.8)