Correspondence between TQFT and state sum models?
Hi all categorists Here are other questions i think about and need your help with. 2> Is there a correspondence in general between TQFTs and state sum models? If not what restrictions are necessary? I think there is a correspondence but i am not sure. If it is, what is the correspondence then called? I am als interested in who discovered or proved it and in which year? Also, do anyone have references about how to actually construct the correspondence both ways? Best regards Rafael Borowiecki
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
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
participants (3)
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Hasse Riemann -
John Baez -
Urs Schreiber