  This contribution is devoted to summarize the recent results obtained in the
construction of an "analytic continuation" of Loop Quantum Gravity (LQG). By
this, we mean that we construct analytic continuation of physical quantities in
LQG from real values of the Barbero-Immirzi parameter $\gamma$ to the purely
imaginary value $\gamma = \pm i$. This should allow us to define a quantization
of gravity with self-dual Ashtekar variables. We first realized in [1] that
this procedure, when applied to compute the entropy of a spherical black hole
in LQG for $\gamma=\pm i$, allows to reproduce exactly the Bekenstein-Hawking
area law at the semi-classical limit. The rigorous construction of the analytic
continuation of spherical black hole entropy has been done in [2]. Here, we
start with a review of the main steps of this construction: we recall that our
prescription turns out to be unique (under natural assumptions) and leads to
the right semi-classical limit with its logarithmic quantum corrections.
Furthermore, the discrete and $\gamma$-dependent area spectrum of the black
hole horizon becomes continuous and obviously $\gamma$-independent. Then, we
review how this analytic continuation could be interpreted in terms of an
analytic continuation from the compact gauge group $SU(2)$ to the non-compact
gauge group $SU(1,1)$ relying on an analysis of three dimensional quantum
gravity.
