Towards the Turaev-Viro amplitudes from a Hamiltonian constraint

3D Loop Quantum Gravity with a vanishing cosmological constant can be related to the quantization of the $\textrm{SU}(2)$ BF theory discretized on a lattice. At the classical level, this discrete model characterizes discrete flat geometries and its phase space is built from $T^\ast \textrm{SU}(2)$. In a recent paper \cite{HyperbolicPhaseSpace}, this discrete model was deformed using the Poisson-Lie group formalism and was shown to characterize discrete hyperbolic geometries while being still topological. Hence, it is a good candidate to describe the discretization of $\textrm{SU}(2)$ BF theory with a (negative) cosmological constant. We proceed here to the quantization of this model. At the kinematical level, the Hilbert space is spanned by spin networks built on $\mathcal{U}_{q}(\mathfrak{su}(2))$ (with $q$ real). In particular, the quantization of the discretized Gauss constraint leads naturally to $\mathcal{U}_{q}(\mathfrak{su}(2))$ intertwiners. We also quantize the Hamiltonian constraint on a face of degree 3 and show that physical states are proportional to the quantum 6j-symbol. This suggests that the Turaev-Viro amplitude with $q$ real is a solution of the quantum Hamiltonian. This model is therefore a natural candidate to describe 3D loop quantum gravity with a (negative) cosmological constant.

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