Rotating Hairy Black Holes in AdS$_5\times$S$^5$

TL;DR

This work numerically constructs fully non-linear, rotating, charged hairy black holes in AdS_5×S^5 within a consistent N=8 supergravity truncation, revealing solutions that extend the CLP family toward the BPS bound with finite entropy. Hairy black holes exist arbitrarily close to the BPS bound for all charges, saturating M = 2J + 3Q in certain limits, and can form a two-parameter supersymmetric extension of Gutowski–Reall on the BPS surface; in the planar limit, rotating hairy branes exhibit spontaneous currents and cannot be obtained by simple boosts. The study combines phase-diagram analysis, linear onset, and non-linear thermodynamics across microcanonical, canonical, and grand-canonical ensembles, highlighting intricate structures in the holographic thermodynamics of N=4 SYM. These findings illuminate the microstate structure and rich phase behavior of strongly coupled gauge theories via AdS/CFT and motivate further analytic and numerical exploration near extremality and in higher-dimensional truncations.

Abstract

We present a numerical study of fully non-linear, rotating and charged hairy black hole solutions in five-dimensional anti-de Sitter space, which originate from a consistent truncation of $\mathcal{N}=8$ supergravity, and can be consistently embedded in type IIB supergravity with AdS$_5\times$S$^5$ asymptotics. The hairy black holes have one scalar field charged under a $U(1)$ gauge field, and branch from the near-extremal Cvetič, Lü and Pope solutions. We give numerical evidence that the hairy solutions exist arbitrarily close to the BPS bound for all charges, and saturate it in the $T\rightarrow 0$ and $T\rightarrow\infty$ limits. We give further evidence for the conjecture of Markevičiūtė and Santos, that on the BPS bound, the rotating hairy black holes form a two-parameter family of solutions with finite entropy, and can be regarded as a one-parameter extension of the supersymmetric Gutowski and Reall black hole. We analyse the approach to the supersymmetric limit and explore the full phase diagram. In the planar horizon limit we find a two parameter family of rotating hairy black brane solutions which cannot be obtained via a Lorentz boost. The field theory dual exhibits a spontaneously generated current. The results of this paper suggest rich and intricate structure of hairy black hole solutions in AdS$_5\times$S$^5$, and highlight their importance in understanding the thermodynamics of $\mathcal{N}=4$ SYM.

Figures (21)

• Figure 1: Left: Microcanonical phase diagram of charged, non-rotating hairy black holes Bhattacharyya:2010ygMarkeviciute:2016ivy. The Reissner-Nordström black holes exist above their extremal limit (black dashed line), and the hairy black holes exist below the bold orange line, which shows the onset of the superradiant instability. These solutions extend down to the BPS bound, where they reduce to the smooth soliton (the purple wavy line) in the limit $T\rightarrow 0$, while keeping the value of the scalar field at the horizon $\varepsilon_H$ fixed. Right: Microcanonical phase diagram of charged, rotating hairy black holes at a constant angular momentum $J$. The hairy black holes exist between the merger line (orange) and the BPS bound (dashed red), in the region shaded in light red. The black disk is the supersymmetric Gutowski-Reall black hole Gutowski:2004ez, which lies at the intersection of the extremal and the BPS planes. The purple wavy line shows a line of the conjectured supersymmetric hairy black holes, which terminates at a finite charge $Q_\mathrm{max}(J)$, where the black holes become singular.
• Figure 2: Left: Phase diagram for the regular CLP black hole solutions, showing possible $Q$, $M$ and $J$ values. The grey plane shows the BPS limit $M=2 J+3 Q$, the blue plane is the extremal limit with $T=0$. The red line shows Real-Gutowski holes, which are extremal BPS solutions, and lie on the intersection of the extremal and BPS planes. The orange plane shows the $\Omega_H=1$ limit for regular black holes with $T>0$. Such black holes exist above the extremal plane, and regular solutions with $\Omega_H>1$ exist only between the orange and blue planes. For $J=0$, we recover RNAdS solutions, and for $Q=0$ we recover MPAdS holes. Right: Allowed parameter space for regular CLP black holes. It is bounded by the extremal surface, which also guarantees a non-negative entropy. The grey region on top shows the moduli space with $\Omega_H \geq 1$. The Gutowski-Reall black holes are the curve on the front left face, where the gray region meets the orange region.
• Figure 3: Convergence of fractional error in black hole energy $\Delta M_n=\left|1-M_{n+1}/M_n\right|$ against the grid size $n$. Left: Hairy black hole solution in DeTurck gauge. The scale is log-log, and the convergence is a power law. Right: Convergence in radial gauge, for the same hairy black hole. The scale now is log-linear, exhibiting an exponential decay.
• Figure 4: Thermodynamic quantities against the temperature $T$, for black holes with fixed angular momentum $J=0.05$ and horizon scalar field $\varepsilon_H=10^{-4}$ (black data points). The dashed gridlines show the values for the supersymmetric black hole with the same $J$.
• Figure 5: Left: The line of solutions showing the onset of the superradiance for the CLP black holes at constant charge $Q=0.1315$ (black points). These solutions extend arbitrarily close to the supersymmetric black holes (red square), with the temperature asymptotically approaching $T=0$. The grey dashed line shows the extremality, and the red bold line shows the BPS bound $M=3Q+2J$. Right: The difference $M-M_\mathrm{ext}$ against the charge $Q$, for black holes with fixed $J=0.05$ and $\varepsilon_H=10^{-4}$ (black data points), where $M_\mathrm{ext}$ is mass of the extremal CLP black hole with the same $Q$. Red solid line is the BPS bound, and the red star shows the Gutowski-Reall solution.