Re: Re: Correspondence between TQFT and state sum models?

Re: Re: Correspondence between TQFT and state sum models?

[Urs Schreiber]

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
>
>
>