The Ponzano-Regge model

The definition of the Ponzano-Regge state-sum model of three-dimensional quantum gravity with a class of local observables is developed. The main definition of the Ponzano-Regge model in this paper is determined by its reformulation in terms of group variables. The regularisation is defined and a proof is given that the partition function is well-defined only when a certain cohomological criterion is satisfied. In that case, the partition function may be expressed in terms of a topological invariant, the Reidemeister torsion. This proves the independence of our definition on the triangulation of the 3-manifold and on those arbitrary choices made in the regularisation. A further corollary is that when the observable is a knot, the partition function (when it exists) can be written in terms of the Alexander polynomial of the knot. Various examples of observables in the three-sphere are computed explicitly. Alternative regularisations of the Ponzano-Regge model by the simple cutoff procedure and by the limit of the Turaev-Viro model are discussed, giving successes and limitations of these approaches.