Turaev-Viro amplitudes from 2+1 Loop Quantum Gravity

Abstract

The Turaev-Viro state sum model provides a covariant spin foam quantization of three-dimensional Riemannian gravity with a positive cosmological constant {\Lambda}. We complete the program to canonically quantize the theory in the BF formulation using the formalism of Loop Quantum Gravity. In particular, we show first how quantum group structures arise from the requirement of the constraint algebra to be anomaly free. This allows us to generalize the construction of the physical scalar product, from the {\Lambda} = 0 case, in presence of a positive {\Lambda}. We prove the equivalence between the covariant and canonical quantizations by recovering the spin foam amplitudes.