Figure 1

$F$ versus $\Phi$ in three dimensions ($p=1$), at $4\pi T=0$, 1, 2, 3 (top to bottom). The minimum of each curve is a BTZ black hole (at $T=0$ the minimum is at $\Phi =0$ and represents the zero-mass black hole). The third curve is at the Hawking-Page temperature.Reuse & Permissions

Figure 2

$\kappa$ and $F$ versus $\Phi$ in five dimensions ($p=3$), at $(2\pi T{)}^2=5$, 6, 7, 8, 9, 10 (top to bottom). From the bottom, the second curve is at $T_{\mathrm{HP}}$, the third at $T_{\mathrm{crit}}$, and the fifth at $T_{\mathrm{crit}}^{\prime }$ (where the curves’ inflection points merge and disappear).Reuse & Permissions

Figure 3

Phase diagram in the $\kappa \mathrm{\text{−}}T$ plane for the five-dimensional system ($p=3$), whose free energy diagram is shown in Fig. 2. The dashed line represents a second-order transition at an infinitesimal value of $\kappa$ separating thermal AdS from a tiny black hole. The solid curve represents a first-order transition separating the tiny and large black hole phases. This is an extension of the Hawking-Page transition and ends at a critical point.Reuse & Permissions

Figure 4

$\kappa$ versus $\Phi$ for the same set of temperatures as shown in Fig. 2, namely, (from top to bottom) $(2\pi T{)}^2=5$, 6, 7, 8, 9, 10. $2\pi T=8^{1/2}$ is the critical temperature, while $2\pi T=3$ is the Hawking-Page temperature. (The curves for some higher temperatures are shown in Fig. 9.) The solutions corresponding to the points labeled $a$, $b$, $c$, $d$ are plotted in Fig. 5.Reuse & Permissions

Figure 5

A selection of supergravity hedgehog black hole solutions, corresponding to the labeled points in Fig. 4. (a) $(2\pi T,\Phi ,\kappa /N^2,F/N^2)=({10}^{1/2},0.37,0.15,0.04)$; (b)  $({10}^{1/2},0.83,-0.30,-0.04)$; (c) $({10}^{1/2},1.14,0,-0.10)$ (this is the conventional large black hole at this temperature); (d) $(5^{1/2},0.55,0.33,0.09)$. The horizontal axis is proper radial distance from the horizon. The curves marked $S^{1,3,5}$ represent the proper radii of the respective spheres, and the one marked $e^{\Phi }$ represents the local string coupling (relative to its asymptotic value $g_{\mathrm{s}}$). In case (c) (the conventional black hole) both the dilaton and $S^5$ radius are constant.Reuse & Permissions

Figure 6

Free energy diagram for the same set of temperatures as in previous figures. Only the lower branch of solutions is plotted, since they obviously have lower free energy at a given value of $\Phi$ than the solutions on the upper branch.Reuse & Permissions

Figure 7

Conjectural sketch of the complete free energy diagram, at the representative temperature $T_{\mathrm{HP}}$, when the possibility of spontaneous $SO(6)$ breaking is included. The upper curve is the same one shown in the previous figure, and represents $SO(6)$-preserving hedgehogs. The points marked $a$ and $b$ are the five-dimensional small and large black holes, respectively. The lower curve is a conjectural sketch of a new branch of $SO(6)$-breaking hedgehogs, which range from $\Phi =0$ (thermal AdS) to $\Phi ={\Phi }_{*}$, where they connect continuously onto the upper curve. The maximum of this new branch ($c$) is a conventional black hole (since it has $\kappa =0$), and is presumably the ten-dimensional black hole, which is thermodynamically unstable but otherwise stable. The value of ${\Phi }_{*}$ could be determined by a linearized perturbation analysis of the $SO(6)$-preserving hedgehogs; the rest of the curve would require a difficult computation (a warm-up problem would be to find the conventional ten-dimensional black hole solution $c$). Somewhere between the point $c$ and ${\Phi }_{*}$, the new branch of hedgehogs would undergo a topology-changing transition, from ten-dimensional black holes localized on the $S^5$ to five-dimensional black holes that are nonuniform over the $S^5$.Reuse & Permissions

Figure 8

Conjectural sketch of the phase diagram in the presence of a chemical potential for the PML.Reuse & Permissions

Figure 9

Left: $\kappa$ versus $\Phi$ for (top to bottom) $(2\pi T{)}^2=10,20,40,80,\infty$. The bottom curve is the result for the gauge theory on ${\mathbf{R}}^3$. Note that each curve—including the one for ${\mathbf{R}}^3$—goes to a minimum value of $\Phi$. Here $\kappa$ is the string tension per unit volume on the boundary; ${\kappa }_{\mathrm{here}}={\kappa }_{\mathrm{above}}/{\Omega }_3$. Right: The left figure blown up near the origin, showing the curves for $(2\pi T{)}^2=\infty$ (left) and 80 (right). The curve for $T=\infty$ presumably ends at the origin, representing AdS, although this region is difficult to access computationally. The solutions corresponding to the points $a$, $b$, $c$ are plotted in Fig. 11.Reuse & Permissions

Figure 10

Left: Free energy diagram for the gauge theory on ${\mathbf{R}}^3$. Here $F$ is the free energy per unit volume on the boundary. Right: left figure blown up near the origin.Reuse & Permissions

Figure 11

A selection of supergravity hedgehog black hole solutions, corresponding to the labeled points in Fig. 9. (a) (shown twice for clarity): $(\Phi /T,\kappa /N^2{T}^3,F/N^2{T}^4)=(-0.14,-0.0007,-0.0001)$; (b) $(1.9,-0.20,-0.22)$; (c)  $(\pi ,0,-0.39)$ (this is the conventional black brane). The horizontal axis is proper radial distance from the horizon. The curves marked $S^1$ and $S^5$ represent the proper radii of the respective spheres, the one marked $R^3$ represents (the square root of) the warp factor for the ${\mathbf{R}}^3$ part of the metric, and the one marked $e^{\Phi }$ represents the local string coupling (relative to its asymptotic value $g_{\mathrm{s}}$). In case (c) both the dilaton and $S^5$ radius are constant.Reuse & Permissions