Phases of Supersymmetric Ground States in $AdS_4$

Abstract

We construct the phase diagram of supersymmetric ground states in AdS$_4\times S^7$ supergravity. BPS black holes exist only when the conserved charges satisfy a certain non-linear constraint. For other charge sectors, we propose two component configurations comprised of a core black hole that carries macroscopic entropy, and a gas that carries a macroscopic fraction of the charge. The superconformal index counts ground states only modulo a linear constraint on the charges. Each index line includes a pure BPS black hole but we find that, when differences between R-charge exceed a certain threshold, the index is dominated by a two component configuration. We illuminate this result by studying the gravitational path integral in Euclidean signature, as function of supersymmetric boundary conditions. We find that a pure BPS black hole saddle dominates the index when the R-charges are comparable, but not when they differ sufficiently. The region where the Euclidean path integral becomes unstable is precisely where two component configurations dominate the index. We show that the KSW criteria are insensitive to this instability.

Figures (16)

• Figure 1: A plot of the "black hole sheet". It has boundaries at the bottom of the plot where it intersects the $j=0$ plane at the two lines $q=|\tilde{q}|$. As $j$ increases, the surface shrinks inwards from these lines. The lighter shade of green is in the foreground. The darker shade is where the sheet has bent over and there is a second branch hidden behind the translucent foreground. See Figure \ref{['fig:constj']} for a constant $j$ slice of the black hole sheet.
• Figure 2: Constant $j$ cross section of the black hole sheet, represented by the green curve. The curve moves to the right for larger $j$. Conversely, it moves left for smaller $j$, approaching the dashed lines as $j\to 0$. The surface formed by the dashed lines at all values of $j$ form the boundary of the allowed region. The black hole sheet splits this region into two components -- region $I$ and region $II$. The norm of the slope of the green curve is $>1$ at every point and asymptotically approaches unity at large charges.
• Figure 3: A projection of the charge space to a slice with constant $j$. The black hole sheet $j = j_c(q,\tilde{q})$ given in \ref{['sheetdef']} is defined inside the wedge created by the two black dashed lines $q=\pm \tilde{q}$. The green curve is the cross section of the black hole sheet at the same value of $j$ as the point $P$. The part of the black hole sheet with smaller values of $j$ lies to the left of the green curve, and the part with larger values is to the right. The grey shaded region is on the black hole sheet and contains all the core BPS black holes that can form a two component configuration with total charges at point $P$.
• Figure 4: A projection of the charge space to a slice with constant $j=j_P$ that includes the point $P$ in region $I$. The green curve is the cross section of the black hole sheet at $j=j_P$. The grey shaded region is on the black hole sheet and contains all the core black holes that can form a two component configuration at point $P$. This region of the black hole sheet is bounded by the boundaries of the allowed region (black dashed lines) and the boundaries of the gas constraints (grey dashed lines). Since the grey shaded region lies to the left of the green curve, it only contains points whose $j$ value is less than that of the point $P$. The point on the black hole sheet that is directly below the point $P$ has the maximum entropy in the shaded region.
• Figure 5: A projection of the charge space to a constant $j$ slice with a point $P$ in region $IIa$. The grey shaded surface is on the black hole sheet and contains the core black holes that can combine with a gas to form a two component configuration with total charges $P$. This region of the black hole sheet is bounded by the boundaries of the allowed region (black dashed lines), the boundaries of the gas constraints (grey dashed lines) and the black hole sheet (green curve). The two component configuration at point $P$ with the maximum entropy contains a core black hole at Point $T$.