Ponzano-Regge model revisited II: Equivalence with Chern-Simons

TL;DR

This paper establishes a mathematical framework connecting the Ponzano-Regge spin foam model with Chern-Simons theory, demonstrating their equivalence in 3D quantum gravity and extending to observable computations.

Contribution

It provides a rigorous definition of the gauge-fixed Ponzano-Regge model and proves its equivalence to Chern-Simons quantization using quantum group techniques.

Findings

Ponzano-Regge measure on flat connections is well-defined.

Equivalence between spin foam and Chern-Simons quantizations is established.

Extension to physical observables and particle insertions is demonstrated.

Abstract

We provide a mathematical definition of the gauge fixed Ponzano-Regge model showing that it gives a measure on the space of flat connections whose volume is well defined. We then show that the Ponzano-Regge model can be equivalently expressed as Reshetikhin-Turaev evaluation of a colored chain mail link based on D(SU(2)): a non compact quantum group being the Drinfeld double of SU(2) and a deformation of the Poincare algebra. This proves the equivalence between spin foam quantization and Chern-Simons quantization of three dimensional gravity without cosmological constant. We extend this correspondence to the computation of expectation value of physical observables and insertion of particles.