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In the context of the geometrical interpretation of the spin network statesof Loop Quantum Gravity, we look at the holonomies of the Ashtekar-Barberoconnection on loops embedded in space-like hyperboloids. We use this simplesetting to illustrate two points. First, the Ashtekar-Barbero connection is nota space-time connection, its holonomies depend on the spacetime embedding ofthe canonical hypersurface. This fact is usually interpreted as aninconvenience, but we use it to extract the extrinsic curvature from theholonomy and separate it from the 3d intrinsic curvature. Second, we show thelimitations of this reconstruction procedure, due to a periodicity of theholonomy in the Immirzi parameter, which underlines the role of a real Immirziparameter as a cut-off for general relativity at the quantum level in contrastwith its role of a mere coupling constant at the classical level.

Ashtekar-Barbero holonomy on the hyperboloid: Immirzi parameter as a Cut-off for Quantum Gravity