Universal construction of black hole microstates

We refine and extend a recent construction of sets of black hole microstates with semiclassical interiors that span a Hilbert space of dimension $e^S$, where $S$ is the black hole entropy. We elaborate on the definition and properties of microstates in statistical and black hole mechanics. The gravitational description of microstates employs matter shells in the interior of the black hole, and we argue that in the limit where the shells are very heavy, the construction acquires universal validity. To this end, we show it for very wide classes of black holes; we first extend the construction to rotating and charged black holes, including extremal and near-extremal solutions, with or without supersymmetry, and we sketch how the construction of microstates can be embedded in string theory. We then describe how the approach can include general quantum corrections, near or far from extremality. For supersymmetric black holes, the microstates we construct differ from other recent constructions in that the interior excitations are not confined within the near-extremal throat.

Figure 1

Euclidean saddle point bulk geometry preparing the semiclassical dual to the PEGS.

Figure 2

Different classes of shell microstates for near-extremal black holes. Above: Light shells enter the near-horizon ${\mathrm{AdS}}_2$ throat (light blue region). These microstates can be directly related to microstates in the JT theory that describes the throat and correspond to operator insertions in the Schwarzian theory (wiggly blue line) at the mouth of the throat. Below: In the heavy-shell limit, the shell does not probe the near-horizon region. The two-sided interior contains two ${\mathrm{AdS}}_2$ throats and a weakly gravitating region where the shell resides.

Figure 3

The two- and three-boundary wormhole contributions to the moments of the Gram matrix. Trajectories with arrows are identified. For general $n$, the solution consists of two Euclidean black holes glued along $n$ trajectories. The Euclidean solution contains $n$ disconnected asymptotic boundaries, each of which corresponds to an overlap. Each shell is taken to be different, so this solution is the only possible contraction of the operators.

Figure 4

Euclidean Pac-Man wormhole geometry computing $\overline{{\mathrm{\Psi }}_i(E){\mathrm{\Psi }}_i(E^{\prime }{)}^{*}}$. Note that the solution enforces $E=E^{\prime }$, as well as $Q_I=Q_I^{\prime }$. By construction, integrating the on shell action of the Pac-Man wormhole over $E$ and the charges $Q_I$ gives the normalization of the PEGS, $Z_{1,i}$. Higher moments such as $\overline{{\mathrm{\Psi }}_i(E){\mathrm{\Psi }}_j(E^{\prime }{)}^{*}{\mathrm{\Psi }}_j(E){\mathrm{\Psi }}_i(E^{\prime }{)}^{*}}$ semiclassically factorize into disconnected copies of the Pac-Man wormhole geometry. Gluing these copies together to form the semiclassical PEGS wormholes produces additional factors of $e^{-S(E,Q_I)}$ coming from Hayward corner terms.