Black Holes with Scalar Hair and Asymptotics in N=8 Supergravity

TL;DR

This work shows that in ${\cal N}=8$ gauged supergravity in $D=4$ and $D=5$, scalars with masses near the Breitenlohner–Freedman bound admit a one-parameter family of AdS-invariant boundary conditions that weaken metric falloffs while preserving AdS symmetry. They construct and analyze hairy black hole solutions under these boundary conditions, finding one-parameter families of static hairy black holes that exist above a critical mass and are disconnected from Schwarzschild–AdS; the hairy solutions generally have higher total energy than their hairless counterparts for the same horizon size, while Schwarzschild–AdS has larger entropy at fixed mass. The hairy configurations lift to black branes in higher dimensions and can be understood in the AdS/CFT framework as dual to multi-trace deformations of the boundary CFT, yielding a line of conformal fixed points. The results illuminate how generalized boundary conditions affect conserved charges, stability, and holographic interpretations in AdS spacetime with nonlocalized matter.

Abstract

We consider N=8 gauged supergravity in D=4 and D=5. We show one can weaken the boundary conditions on the metric and on all scalars with $m^2 <-{(D-1)^2 \over 4}+1$, while preserving the asymptotic anti-de Sitter (AdS) symmetries. Each scalar admits a one-parameter family of AdS-invariant boundary conditions for which the metric falls off slower than usual. The generators of the asymptotic symmetries are finite, but generically acquire a contribution from the scalars. For a large class of boundary conditions we numerically find a one-parameter family of black holes with scalar hair. These solutions exist above a certain critical mass and are disconnected from the Schwarschild-AdS black hole, which is a solution for all boundary conditions. We show the Schwarschild-AdS black hole has larger entropy than a hairy black hole of the same mass. The hairy black holes lift to inhomogeneous black brane solutions in ten or eleven dimensions. We briefly discuss how generalized AdS-invariant boundary conditions can be incorporated in the AdS/CFT correspondence.

Figures (10)

• Figure 1: The scalar field $\phi_e$ at the horizon as a function of horizon size $R_e$ in hairy black hole solutions of $D=5$${\cal N}=8$ supergravity. The two curves correspond to solutions with two different AdS-invariant boundary conditions, namely $f=0$ (bottom) and $f=1$ (top).
• Figure 2: The hair $\phi(r)$ (left) and $r^2\phi/\ln r$ (right) outside a black hole of size $R_e=.2$, with boundary conditions specified by $f=1$.
• Figure 3: The coefficient $\alpha$ that characterizes the asymptotic profile of the hair $\phi(r)$ as a function of horizon size $R_e$ in hairy black hole solutions of $D=5$${\cal N}=8$ supergravity. The two curves correspond to solutions with two different AdS-invariant boundary conditions, namely $f=0$ (bottom) and $f=1$ (top).
• Figure 4: The integration constant $M_0$ as a function of horizon size $R_e$ in hairy black hole solutions of $D=5$${\cal N}=8$ supergravity. The two curves correspond to solutions with two different AdS-invariant boundary conditions $f=0$ and $f=1$ (dotted line).
• Figure 5: left: The total mass $E_{h}/3\pi^2$ of hairy black holes as a function of horizon size $R_e$, in $D=5$${\cal N}=8$ supergravity with two different AdS-invariant boundary conditions $f=1$ (top) and $f=0$ (bottom). right: The ratio $E_{h}/E_{s}$ as a function of horizon size $R_e$, where $E_s$ is the mass of a Schwarschild-AdS black hole of the same size $R_e$.