Small Hairy Black Holes in $AdS_5 \times S^5$

TL;DR

This work constructs and analyzes small, charged, hairy black holes in a consistent truncation of ${\cal N}=8$ gauged supergravity with a single charged scalar, showing near-extremal RNAdS black holes are unstable to superradiant condensation and end on hair-bearing solutions. The authors develop a perturbative expansion around RNAdS and soliton solutions, and numerically explore the supersymmetric soliton space, uncovering a singular fixed point $S$ that governs large-charge behavior and a rich spiraling structure of regular and singular SUSY solutions. They show that, at leading order, hairy black hole thermodynamics can be captured by a non-interacting mix of a small RNAdS black hole and a SUSY soliton, and they extend these ideas to rotating (Kerr RNAdS) configurations, predicting a two-parameter SUSY hairy family and BPS behavior across charges and angular momenta. Together, these results illuminate a qualitatively new near-BPS phase structure for ${\cal N}=4$ SYM at finite $N^2$ and suggest broad, testable predictions for rotating hairy black holes and their dual field theory interpretations.

Abstract

We study small hairy black holes in a consistent truncation of ${\cal N}=8$ gauged supergravity that consists of a single charged scalar field interacting with the metric and a U(1) gauge field. Small very near extremal RNAdS black holes in this system are unstable to decay by superradiant emission. The end point of this instability is a small hairy black hole that we construct analytically in a perturbative expansion in its charge. Unlike their RNAdS counterparts, hairy black hole solutions exist all the way down to the BPS bound, demonstrating that ${\cal N}=4$ Yang Mills theory has an ${\cal O}(N^2)$ entropy at all energies above supersymmetry. At the BPS bound these black holes reduce to previously discussed regular, supersymmetric horizon free solitons. We use numerical methods to continue the construction of these solitons to large charges and find that the line of soliton solutions terminates at a singular solution $S$ at a finite charge. We conjecture that a one parameter family of singular supersymmetric solutions, which emerges out of $S$, constitutes the BPS limit of hairy black holes at larger values of the charge. At leading order in their charge, the thermodynamics of the small hairy black holes constructed in this paper turns out to be correctly reproduced by modeling these objects as a non interacting mix of an RNAdS black hole and the supersymmetric soliton in thermal equilibrium. Assuming that a similar non interacting model continues to apply upon turning on angular momentum, we also predict a rich family of rotating hairy black holes, including new hairy supersymmetric black holes. This analysis suggests interesting structure for the space of (yet to be constructed) hairy charged rotating black holes in $AdS_5\times S^5$, particularly in the near BPS limit.

Figures (25)

• Figure 1: Phase diagram as a function of charge $q$ (x axis) and mass $m$ (y axis) at small $q$. The solid blue line at the bottom is the BPS bound along which the soliton lives. Hairy black holes exist - and are the dominant phase - in the shaded region. RNAdS black holes are the only known solutions (so in particular the dominant phase) in the unshaded region above the solid red curve at the top. RNAdS black holes also exist (but are dynamically unstable and thermodynamically sub dominant) between the solid red curve and the dashed curve. The solid red curve is described by $m=3 q + 3 q^2 +{\cal O}(q^4)$, while the blue curve by $m=3q$. The dashed curve corresponds to extremal RNAdS black holes and is given by $m=3q+3q^2 -6q^3+{\cal O}(q^4)$. The curves have not been drawn to scale to make the diagram more readable.
• Figure 2: Conjectured phase diagram as a function of charge $q$ (x axis) and mass $m$ (y axis) for all values of $q$. The blue line at the bottom is the BPS bound along which the regular soliton lives (straight part) and the singular supersymmetric solutions (wiggly part). The solid red curve at the top marks the phase transition between the regime of RNAdS black holes (above the line) and that of smooth hairy black holes (below). The black curve indicates a phase transition between different types of hairy black holes. This curve originates from the BPS line at the black dot which is close to the point $q=q_c$ and could end either in the bulk of the hairy black hole region or could extend all the way up to the red line.
• Figure 3: Convergence of the numerical solutions for the regular solitons to the special singular solution $S$ as we increase $h_0 = h(0)$. The black line corresponds to the solution $S$ with $\rho^{-2/3}$ behavior near $\rho=0$. The blue lines correspond to regular solitons of $h_0 = 2,3,4,6,8$, starting from the lowest blue curve and going up.
• Figure 4: The damped oscillations of $q$ around the critical value $q_c$ for large $h_0$.
• Figure 5: Convergence of the numerical solutions for singular solitons with an ${a\over \rho}$ singularity to the special singular solution $S$ as we decrease $a$. The black line corresponds to the solution $S$ with $\rho^{-2/3}$ behavior near $\rho=0$. The red lines correspond to singular solitons of $a = 0.35, 0.5, 1, 2.5,5$, starting from the lowest red curve and going up.