The phase diagram of the D1-D5 CFT and localized black holes

Abstract

In this paper we analyze the phases that dominate the microcanonical ensemble at various energies in the D1-D5 CFT, which is dual to type II string theory on $AdS_3 \times S^3\times T^4$. We focus on black hole solutions, and on the dependence of the phase structure on the ratio of the size of the torus to the AdS scale; as small localized black holes (with horizon topology $S^8$) grow, they can start to fill the $S^3$ or the $T^4$ or both, and we analyze the general aspects of the transitions between the various phases of uniform and non-uniform black holes, incorporating known solutions and discussing the properties of additional unknown solutions. Some features of the transitions between these phases are similar to higher dimensional AdS spaces, while other features are different. We provide evidence that when the torus is much larger than the AdS radius, there is a large range of energies where the typical states are a novel phase, described by a lattice (in the $T^4$ directions) of black holes with horizon topology $S^5\times S^3$. In this phase the entropy is linear in the energy, with a coefficient that is of order the AdS radius.

Figures (9)

• Figure 1: Schematic plots of the microcanonical entropy as a function of energy for string theory on $AdS_{5}\times S^{5}$ (at large $N$ and large 't Hooft coupling $\lambda$). Left: The phase that dominates in each energy range. Right: A magnified view of the two dominant black hole phases and the transition between them. The green dot marks the location of the highest Gregory--Laflamme zero mode of the black hole that is uniform on the compact $S^{5}$, while the red dot indicates the first-order phase transition. The cyan dot denotes the conjectured topology-changing transition between the $S^{8}$ black hole branch and the lumpy black holes.
• Figure 2: Entropy-Energy relation in the symmetric orbifold theories studied in Hartman:2014oaa. The low, medium and high regimes are denoted on the horizontal axis.
• Figure 3: The entropy as a function of energy for $AdS_3 \times S^3 \times T^4$ in the microcanonical ensemble for $R \gg R_T \sim l_s$. Note that if $Q_1$ is not large enough (compared to $Q_5$), $g_s$ is of order one (see \ref{['eq:g_s_as_RT']}) and there is no phase of free strings.
• Figure 4: The entropy as a function of energy for $AdS_3 \times S^3 \times T^4$ in the microcanonical ensemble for $R \gg R_T \gg l_s$. As in figure \ref{['fig:microcanonical_ads3_rt_sim_ls']}, if $Q_1$ is not large enough (compared to $Q_5$), there is no phase of free strings.
• Figure 5: A schematic drawing of the entropy as a function of the energy for the three types of black hole solutions that dominate the microcanonical ensemble for $R \gg R_T \gg l_s$.