Loop Quantum Gravity is a quantum background independent theory of gravity.
The starting point for quantization is the reformulation of General Relativity in terms of the socalled real Ashtekar's variables.
Those variables leads to nice polynomial constraints except for the Hamiltonian constraint. Therefore, it is possible from this formulation to build rigourously a kinematical Hilbert space (the loop quantization), but the question of the quantum dynamics remains open.
However, it is also possible to work with complex Ashtekar's variables which leads to pure polynomial constraint, even for the hamiltonian one. The difflculty enters now in the reality conditions linked to the complex nature of the variables.
Their resolution is still an open question, and so far, no one knows how to deal with complex Ashtekar gravity.
However, new insights have been proposed this last couple of years.
Motivated by results in three dimensional gravity and especially in black hole thermodynamics, it was possible to proposed a concrete analytical continuation of the results obtain in the real kinematical approach of Loop Quantum Gravity.
We present here the application of this analytical continuation to Loop Quantum Cosmology. We will see how the symmetric bouncing universe described by real LQC is trade for a non symmetric bouncing universe. This exercise could give new insights in the question of the quantum states involved in complex quantum Ashtekar' s gravity.
