Turaev-Viro, Levin-Wen, 3D gravity

[atyy]

What is the relationship between these?

On the TV and LW side, these seem relevant
http://golem.ph.utexas.edu/category/2011/03/liang_kong_on_levinwen_models.html
http://arxiv.org/abs/1004.1533
http://arxiv.org/abs/1004.1737v2

On the TV and 3D gravity side, this seems relevant
http://arxiv.org/abs/gr-qc/0012035

 

[atyy]

So it looks like the Turaev-Viro model is "related to" 3D Riemannian pure gravity with positive cosmological constant.
http://arxiv.org/abs/hep-th/9110057
http://arxiv.org/abs/gr-qc/9905087
http://arxiv.org/abs/gr-qc/0311066
http://arxiv.org/abs/1012.4784

However, when the cosmological constant is positive, there are no known observables for someone inside the spacetime.
http://arxiv.org/abs/0706.3359

Now, what's the relationship between the Turaev-Viro and Levin-Wen models?

 

[element4]

Now, what's the relationship between the Turaev-Viro and Levin-Wen models?

Mathematically, a topological quantum field theory (TQFT) is a certain (modular) functor from a category of cobordisms to the category of vector spaces. For example, one can construct such a TQFT for each modular tensor category. I think that Turaev-Viro TQFT is a specific construction of certain (doubled/non-chiral) TQFT's, which takes a unitary (spherical?) tensor category (UTC) as input (for example for any finite group $$G$$, one can construct the representation category of $$G$$ with simple objects as irreducible representations).

I think that Levin-Wen models are essentially explicit Hamiltonian formulation of Turaev-Viro TQFT. This can give rise to models, where continuum limits are given by doubled Chern-Simons theories.

I am in no way an expert in this subject, and what I just wrote contains many inaccuracies. But this response might provoke more knowledgeable people to intervene.

 

[atyy]

Mathematically, a topological quantum field theory (TQFT) is a certain (modular) functor from a category of cobordisms to the category of vector spaces. For example, one can construct such a TQFT for each modular tensor category. I think that Turaev-Viro TQFT is a specific construction of certain (doubled/non-chiral) TQFT's, which takes a unitary (spherical?) tensor category (UTC) as input (for example for any finite group $$G$$, one can construct the representation category of $$G$$ with simple objects as irreducible representations).

I think that Levin-Wen models are essentially explicit Hamiltonian formulation of Turaev-Viro TQFT. This can give rise to models, where continuum limits are given by doubled Chern-Simons theories.

I am in no way an expert in this subject, and what I just wrote contains many inaccuracies. But this response might provoke more knowledgeable people to intervene.

Thanks! Now reading the Kirillov and Balsam paper I linked above, they seem to think the Levin-Wen models are microscopic Hamiltonian models of Barrett-Westbury TQFTs, which are a generalization from Turaev-Viro. Kirillov and Balsam use your language, which I'm not familiar with. I found another paper http://arxiv.org/abs/1103.6264 which uses language that is easier for me. Bahr et al say that BF theory with finite gauge group describe string net models. Do you know whether Kirillov and Balsam are saying the same thing as Bahr et al?

It seems the link between these models and condensed matter are much tighter and solid than those for quantum gravity.

So strings are good for condensed matter, and so is LQG