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We study the role of the Barbero-Immirzi parameter $\gamma$ and the choice ofconnection in the construction of (a symmetry-reduced version of) loop quantumgravity. We start with the four-dimensional Lorentzian Holst action that wereduce to three dimensions in a way that preserves the presence of $\gamma$. Inthe time gauge, the phase space of the resulting three-dimensional theorymimics exactly that of the four-dimensional one. Its quantization can beperformed, and on the kinematical Hilbert space spanned by SU(2) spin networkstates the spectra of geometric operators are discrete and $\gamma$-dependent.However, because of the three-dimensional nature of the theory, its SU(2)Ashtekar-Barbero Hamiltonian constraint can be traded for the flatnessconstraint of an sl(2,C) connection, and we show that this latter has tosatisfy a linear simplicity-like condition analogous to the one used in theconstruction of spin foam models. The physically relevant solution to thisconstraint singles out the non-compact subgroup SU(1,1), which in turn leads tothe disappearance of the Barbero-Immirzi parameter and to a continuous lengthspectrum, in agreement with what is expected from Lorentzian three-dimensionalgravity.

Testing the role of the Barbero-Immirzi parameter and the choice of connection in Loop Quantum Gravity