Statistical entropy of a BTZ black hole from loop quantum gravity

We compute the statistical entropy of a BTZ black hole in the context of three-dimensional Euclidean loop quantum gravity with a cosmological constant Lambda. As in the four-dimensional case, a quantum state of the black hole is characterized by a spin network state. Now however, the underlying colored graph Gamma lives in a two-dimensional spacelike surface Sigma, and some of its links cross the black hole horizon, which is viewed as a circular boundary of Sigma. Each link l crossing the horizon is colored by a spin j(l) (at the kinematical level), and the length L of the horizon is given by the sum L = Sigma(l) L-l of the fundamental length contributions L-l carried by the spins j(l) of the links l. We propose an estimation for the number N-Gamma(BTZ) (L, Lambda) of the Euclidean BTZ black hole microstates (defined on a fixed graph Gamma) based on an analytic continuation from the case Lambda > 0 to the case Lambda < 0. In our model, we show that N-Gamma(BTZ) (L, Lambda) reproduces the Bekenstein-Hawking entropy in the classical limit. This asymptotic behavior is independent of the choice of the graph Gamma provided that the condition L = Sigma(l) L-l is satisfied, as it should be in three-dimensional quantum gravity.