Isolated horizons and black hole entropy in Loop Quantum Gravity

We review the black hole entropy calculation in the framework of Loop Quantum Gravity based on the quasi-local definition of a black hole encoded in the isolated horizon formalism. We show, by means of the covariant phase space framework, the appearance in the conserved symplectic structure of a boundary term corresponding to a Chern–Simons theory on the horizon and present its quantization both in the $U(1)$ gauge fixed version and in the fully $SU(2)$ invariant one. We then describe the boundary degrees of freedom counting techniques developed for an infinite value of the Chern–Simons level case and, less rigorously, for the case of a finite value. This allows us to perform a comparison between the $U(1)$ and $SU(2)$ approaches and provide a state of the art analysis of their common features and different implications for the entropy calculations. In particular, we comment on different points of view regarding the nature of the horizon degrees of freedom and the role played by the Barbero–Immirzi parameter. We conclude by presenting some of the most recent results concerning possible observational tests for theory.