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arXiv · 0803.3319

The Ponzano-Regge model

John W. Barrett·Ileana Naish-Guzman
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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 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.

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John W. Barrett, Ileana Naish-Guzman. 2009-03-31. The Ponzano-Regge model. https://doi.org/10.1088/0264-9381%2F26%2F15%2F155014

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