Black Hole Entropy from complex Ashtekar variables

In loop quantum gravity, the number \(N_\Gamma(A,\gamma)\) of microstates of a black hole for a given discrete geometry \(\Gamma\) depends on the so-called Barbero-Immirzi parameter \(\gamma\). Using a suitable analytic continuation of \(\gamma\) to complex values, we show that the number \(N_\Gamma(A,\pm i)\) of microstates behaves as \(\exp(A/(4\ell_\text{Pl}^2))\) for large area \(A\) in the large spin semiclassical limit. Such a correspondence with the semiclassical Bekenstein-Hawking entropy law points towards an unanticipated and remarkable feature of the original complex Ashtekar variables for quantum gravity.