State sum invariants of 3-manifolds and quantum 6j-symbols

Turaev and Viro discovered an improved regularization based on quantum groups.

“The “topological” feature of the Ponzano-Regge model was made more precise by Turaev and Viro [262], who discovered an improved regularization, based on the technology of quantum groups.”

The quantum-deformed version of the three-dimensional amplitudes leads to the Turaev-Viro model, expected to correspond to three-dimensional Riemannian gravity with a positive cosmological constant.

“The quantum deformed version of the above amplitudes lead to the Turaev-Viro model [364], which is expected to correspond to the quantization of three-dimensional Riemannian gravity in the presence of a non-vanishing positive cosmological constant.”

The Turaev-Viro model is expected to correspond to the quantization of three-dimensional Riemannian gravity with a non-vanishing positive cosmological constant.

“The quantum deformed version of the above amplitudes lead to the Turaev-Viro model [364], which is expected to correspond to the quantization of three-dimensional Riemannian gravity in the presence of a non-vanishing positive cosmological constant.”

Equation (248) has been proven for simplicial decompositions, and its generalization to arbitrary cellular decomposition was obtained.

“Eq. (248) has been proven in this case for the case of simplicial decompositions in [364]; see also [363, 238]. The generalization for arbitrary cellular decomposition was obtained in [206].”

With a cosmological constant, the state-sum generalizes to the Turaev-Viro model defined in terms of SUq(2) with qn = 1, where the representations are finitely many.

“In the case of gravity with cosmological constant the state-sum generalizes to the Turaev-Viro model [364] defined in terms of SUq(2) with qn = 1, where the representations are finitely many.”

When the cosmological constant is nonzero, the Turaev-Viro model replaces the gauge group SU(2) with the quantum group SU_q(2), with q a function of the cosmological constant.

“When the cosmological constant is nonzero, one obtains instead the Turaev-Viro model [62], where the quantum group SU q (2) takes the place of the gauge group SU (2), with the deformation parameter q”

In three dimensions, the Ponzano-Regge model is based on SU(2) and the Turaev-Viro model is based on the quantum group SUq(2).

“In three dimensions one has the Ponzano-Regge [46] model based on SU (2) and the Turaev-Viro [62] model based on the quantum group SU q (2).”

In the covariant path integral approach, topology can change, and the path integral can include a sum over topologies, producing a foam-like structure of spacetime.

“Topology itself is also allowed to change, and the path integral can be completed by a sum over different topologies, giving rise to a foam-like structure of spacetime [39], with quantum fluctuation from one 4-metric to another for a given topology, but also from one topology to another.”

The PRTV model can be seen as a quantization of 3d GR, or a quantization of 3d Chern-Simon theory.

“The OCY model is a 4d generalization of the Ponzano-Regge-Turaev-Viro (PRTV) model [12,13], which, in turn, can be seen as a quantization of 3d GR, or a quantization of 3d Chern-Simon theory.”

The OCY model is a 4d generalization of the Ponzano-Regge-Turaev-Viro (PRTV) model.

“The OCY model is a 4d generalization of the Ponzano-Regge-Turaev-Viro (PRTV) model [12,13], which, in turn, can be seen as a quantization of 3d GR, or a quantization of 3d Chern-Simon theory.”