The Ponzano–Regge model

Abstract

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 regularization 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 the definition on the triangulation of the 3-manifold and on those arbitrary choices made in the regularization. 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 S 3 are computed explicitly. Alternative regularizations of the Ponzano–Regge model by the simple cut-off procedure and by the limit of the Turaev–Viro model are discussed, giving successes and limitations of these approaches.