Turaev-Viro amplitudes from 2+1 Loop Quantum Gravity
Daniele Pranzetti
TL;DR
The paper completes the canonical quantization of 2+1 dimensional Riemannian gravity with a positive cosmological constant by showing that the quantum-curvature constraint requires a quantum-group (U_q(su(2))) structure. This leads to a modified projector for the physical inner product, introducing quantum dimensions and q-deformed recoupling, and allows the recovery of the Turaev-Viro spin foam amplitudes from the canonical loop quantization. The work demonstrates the exact equivalence between covariant TV quantization and the canonical LQG approach even for Λ>0, providing a nontrivial consistency check and guiding insights for higher-dimensional spin foams. It also clarifies how quantum-group ideas emerge dynamically from anomaly considerations rather than being imposed a priori.
Abstract
The Turaev-Viro state sum model provides a covariant spin foam quantization of three-dimensional Riemannian gravity with a positive cosmological constant Λ. 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 Λ = 0 case, in presence of a positive Λ. We prove the equivalence between the covariant and canonical quantizations by recovering the spin foam amplitudes.