Black Hole Entropy from complex Ashtekar variables

In loop quantum gravity, black holes can be described in terms of an SU(2) Chern-Simons theory on a punctured 2-sphere. The level $k$ of the Chern-Simons theory depends on both the Barbero-Immirzi parameter $\gamma$ and the horizon area $a_H$. In this framework, the number of microstates of the black hole is a function which is expressed in terms of the dimension of the SU(2) Chern-Simons theory Hilbert space. We propose an analytic continuation of this number of microstates to a purely imaginary value of $\gamma$, and we give an interpretation based on the analytic continuation of SU(2) Chern-Simons theory to a complex gauge group. We show that the number of microstates behaves as exp$(a_H/(4lp^2))$ for large area $a_H$ if $\gamma=\pm i$, and finally discuss the relation between this striking result and quantum gravity in terms of the original complex Ashtekar variables.