This may depend on what exactly one understands under "state sum models". The Fukuma-Hosono-Kawai construction of 2d TQFTs from semisimple algebras has tradionally been called a state sum model description. Lauda and Pfeiffer have described it at great length in Lauda-Pfeiffer State sum construction of two-dimensional open-closed Topological Quantum Field Theories http://arxiv.org/abs/math.QA/0602047 When one internalizes these constructions from Vect into a modular tensor category, one obtains the state-sum-like construction of 2d CFT by Fuchs-Runkel-Schweigert, a review of which is for instance here I. Runkel, J. Fjelstad, J. Fuchs, Ch. Schweigert Topological and conformal field theory as Frobenius algebras math.CT/0512076. The Turaev-Viro model for 3d TQFT is also frequently called state sum model. I don't find the good review of Turaev-Viro that I wanted to link to right this moment, but googling shows up lots or useful links, it seems. Best, Urs On 5/13/09, John Baez <john.c.baez@gmail.com> wrote:
Rafael Borowiecki writes:
Is there a correspondence in general between TQFTs and state sum models?
There should be a correspondence between *extended* TQFTs and state sum models.
The theory of extended TQFTs is only beginning to be developed, so this expected correspondence has not yet been proved. I recommend taking a look at this paper:
Jacob Lurie On the Classification of Topological Field Theories http://arxiv.org/abs/0905.0465
Best,
jb