On String Theory Duals of Lifshitz-like Fixed Points

TL;DR

The authors construct and analyze Type IIB supergravity backgrounds dual to Lifshitz-like fixed points arising from D3-D7 intersections, achieving an IR anisotropic scaling with $z=3/2$ and isotropic AdS$_5$ behavior in the UV. They demonstrate interpolating RG flows between AdS$_5$ and the Lifshitz-like geometry, interpret the flows as anisotropic perturbations (notably a linearly $w$-dependent $ heta$-angle) triggering the IR fixed point, and extend the solutions to finite temperature via black branes. The work further quantifies physical properties of the dual theories, including entropies, shear and bulk viscosities, and entanglement entropies along different directions, and analyzes perturbative stability with hints that suitable Einstein manifolds for $X_5$ may stabilize the background. Additionally, a D4-D6 analogue is discussed, broadening the scope of anisotropic holography beyond D3-D7, with implications for RG flows and potential condensed-matter realizations.

Abstract

We present type IIB supergravity solutions which are expected to be dual to certain Lifshitz-like fixed points with anisotropic scale invariance. They are expected to describe a class of D3-D7 systems and their finite temperature generalizations are straightforward. We show that there exist solutions that interpolate between these anisotropic solutions in the IR and the standard AdS5 solutions in the UV. This predicts anisotropic RG flows from familiar isotropic fixed points to anisotropic ones. In our case, these RG flows are triggered by a non-zero theta-angle in Yang-Mills theories that linearly depends on one of the spatial coordinates. We study the perturbations around these backgrounds and discuss the possibility of instability. We also holographically compute their thermal entropies, viscosities, and entanglement entropies.

Figures (2)

• Figure 1: The $(B,C)$ flow diagram. The horizontal and vertical axes are $B\equiv\dot{b}$ and $C\equiv\dot{c}$, respectively. The arrows point in the direction from the UV ($r=\infty$) to the IR ($r=0$). In the left figure, the blue dot at $(\frac{7}{\sqrt{33}},\frac{1}{\sqrt{33}}-1)\simeq (1.22,-0.83)$ is the unstable fixed point corresponding to the scaling solution; the green dot at $(1,-1)$ is the stable UV fixed point corresponding to the $AdS_5$ solution. The green line running through the scaling fixed point corresponds to the negative eigenvalue $\lambda_1$ while the red line corresponds to the positive one $\lambda_2$. The black curve is given by $9B-8C-8= \sqrt{33B^2+48}$ and corresponds to a pure D3 solution; the allowed D3-D7 solutions are above this curve.
• Figure 2: An interpolating solution with fluxes satisfying $\frac{\beta^2}{64(\frac{\pi^4}{\hbox{Vol}(X_5)})^2N^2}=1$. In the left figure, $(B,C)\equiv (\dot{b},\dot{c})$ flow from $(\frac{7}{\sqrt{33}},\frac{1}{\sqrt{33}}-1)\simeq (1.22,-0.83)$ in the IR to $(1,-1)$ in the UV. The right figure shows that, in the Einstein frame, the scalings of the $(t,x,y)$-directions, the $w$-direction, and $e^{\phi}$ flow from $(\frac{8}{\sqrt{33}},\frac{4}{\sqrt{33}},\frac{4}{\sqrt{33}}) \simeq (1.04,0.70,0.70)$ in the IR to $(1,1,0)$ in the UV.