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Collateral reading:
Turaev Viro paper (Topology, 1992)
here
arXiv:1106.6033
The full equivalence proof between string nets and the Turaev Viro model by Kirillov Jr.
arXiv:cond-mat/0506438, Appendix E, often with physical motivations for some of the axioms.
arXiv:hep-th/0401076
Freidel and Louapre
,
Ponzano-Regge model of 3D quantum gravity
arXiv:gr-qc/0410141
The same model as above. Amplitudes are proved to be given by the Reshetikhin-Turaev evaluation of a coloured chain mail links based on the D(SU(2)) quantum group.
Roberts (Topology, 1995) The proof that (square of absolute values of) amplitudes of SU(2) Chern-Simons theory are given by Turave-Viro invariants (based on SU_q(2).
Karowski and Schrader (Comm. Math. Phys. 1992)
http://link.springer.com/article/10.1007%2FBF02096773?LI=true#page-1
Here observables are also in the game.
Ambjorn, Durhuus, Jonsoon: Quantum geometry: a statistical field theory approach
Karowski et al ;
and see ...
More:
Coecke's "Categories for the practising physicist"! .