Bekenstein-Hawking Entropy as Topological Entanglement Entropy

Abstract

Black holes in 2+1 dimensions enjoy long range topological interactions similar to those of non-abelian anyon excitations in a topologically ordered medium. Using this observation, we compute the topological entanglement entropy of BTZ black holes, via the established formula S_top = log(S^a_0), with S_b^a the modular S-matrix of the Virasoro characters chi_a(tau). We find a precise match with the Bekenstein-Hawking entropy. This result adds a new twist to the relationship between quantum entanglement and the interior geometry of black holes. We generalize our result to higher spin black holes, and again find a detailed match. We comment on a possible alternative interpretation of our result in terms of boundary entropy.

Figures (3)

• Figure 1: The black hole horizon forms a geodesic $\Gamma$. The entanglement entropy between the inside and outside regions $A$ and $B$ is equal to Length$(\Gamma)/4$.
• Figure 2: The classical space-time geometry is specified by the holonomies around the paths $\gamma_i$. In the quantum theory, states are identified with conformal blocks of 2D Liouville CFT.
• Figure 3: Liouville vertex operators fall into two classes. Those with $\Delta< \frac{1}{4} Q^2$ create elliptic solutions (punctures), those with $\Delta > \frac{1}{4} Q^2$ create hyperbolic solutions (macroscopic holes) Nati.