Geons and the quantum information metric

We investigate the proposed duality between a quantum information metric in a ${\mathrm{CFT}}_{d+1}$ and the volume of a maximum time slice in the dual ${\mathrm{AdS}}_{d+2}$ for topological geons. Examining the specific cases of Banados-Teitelboim-Zannelli (BTZ) black holes and planar Schwarzschild–anti-de Sitter black holes, along with their geon counterparts, we find that the proposed duality relation for geons is the same apart from a factor of 4. The information metric therefore provides a probe of the topology of the bulk spacetime.

Figure 1

(a) Penrose diagram of the AdS (BTZ) black hole. The diagram shows two $2d$ CFTs each at one boundary of the AdS (BTZ) black hole as the AdS/CFT correspondence preconizes. (b) Penrose diagram of the AdS (BTZ) geon. This diagram appears to be the half of the AdS (BTZ) black hole one. In fact, the geon space is obtained via the identification (4) and consequently “splits” the original space into two pieces which are equivalent by a mirror symmetry. As the resulting space happens to be a one-sided black hole, an important and remarkable feature of this diagram is that the two CFTs are now identified and lie on the only remaining boundary.

Figure 2

(a) The contour chosen on the complex plane $\rho$ to compute $\mathrm{Vol}({\mathrm{\Sigma }}_{\max })$ for the BTZ black hole. ${\kappa }_{*}$ is the point on the imaginary axis. (b) Represents the contour chosen on the complex plane $z$ for the computation of $\mathrm{Vol}({\mathrm{\Sigma }}_{\max })$ for the AdS Schwarzschild planar black hole. $z_{*}$ on the imaginary axis is at $\sqrt{2}{z}_0$ and at $z_0$ on the real axis.