Complexity and typical microstates

Typical black hole microstates in $\mathrm{AdS}/\mathrm{CFT}$ were recently conjectured to have a geometrical dual with a smooth horizon and a portion of a second asymptotic region. I consider the application of the holographic complexity conjectures to this geometry. The holographic calculation leads to divergent values for the complexity; I argue that this classical divergence is consistent with expectations for typical microstates.

Figure 1

The proposed geometry dual to a typical high energy state in a CFT. The CFT lives on the boundary on the right; the dotted line on the left indicates a limit of the geometrical description of the state.

Figure 2

In the CV calculation, the complexity is given by the divergent volume of the slice $B$, which approaches $B_{\mathrm{in}}$ in the left part of the diagram.

Figure 3

Two alternatives for the Wheeler-de Witt patch: on the left, the domain of development of $B$, and on the right, the domain of development of a Cauchy surface.