Towards the Turaev-Viro amplitudes from a Hamiltonian constraint

Understand

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.