Testing the role of the Barbero-Immirzi parameter and the choice of connection in loop quantum gravity

We study the role of the Barbero-Immirzi parameter $\gamma$ and the choice of connection in the construction of (a symmetry-reduced version of) loop quantum gravity. We start with the four-dimensional Lorentzian Holst action that we reduce to three dimensions in a way that preserves the presence of $\gamma$. In the time gauge, the phase space of the resulting three-dimensional theory mimics exactly that of the four-dimensional one. Its quantization can be performed, and on the kinematical Hilbert space spanned by SU(2) spin network states 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 flatness constraint of an $\mathfrak{s}\mathfrak{l}(2,\mathbb{C})$ connection, and we show that this latter has to satisfy a linear simplicitylike condition analogous to the one used in the construction of spin foam models. The physically relevant solution to this constraint singles out the noncompact subgroup SU(1, 1), which in turn leads to the disappearance of the Barbero-Immirzi parameter and to a continuous length spectrum, in agreement with what is expected from Lorentzian three-dimensional gravity.