Horizon entropy with loop quantum gravity methods

We show that the spherically symmetric isolated horizon can be described in terms of an SU(2)SU(2) connection and an su(2)su(2)-valued one-form, obeying certain constraints. The horizon symplectic structure is precisely the one of 3d gravity in a first order formulation. We quantize the horizon degrees of freedom in the framework of loop quantum gravity, with methods recently developed for 3d gravity with non-vanishing cosmological constant. Bulk excitations ending on the horizon act very similarly to particles in 3d gravity. The Bekenstein–Hawking law is recovered in the limit of imaginary Barbero–Immirzi parameter. Alternative methods of quantization are also discussed.