Turaev-Viro, Levin-Wen, 3D gravity

In summary, the relationship between the TV and 3D gravity side is that they are related to each other through a topological quantum field theory. However, the relationship between the Turaev-Viro model and the Levin-Wen models is not clear yet.
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atyy said:
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 [tex]G[/tex], one can construct the representation category of [tex]G[/tex] 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.
 
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element4 said:
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 [tex]G[/tex], one can construct the representation category of [tex]G[/tex] 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 :smile:
 
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FAQ: Turaev-Viro, Levin-Wen, 3D gravity

What is Turaev-Viro, Levin-Wen, 3D gravity?

Turaev-Viro, Levin-Wen, 3D gravity is a mathematical theory that describes the dynamics of three-dimensional spacetime using topological quantum field theory. It combines concepts from topology, quantum mechanics, and general relativity to provide a framework for studying the quantum properties of gravity in three dimensions.

What are the key principles behind Turaev-Viro, Levin-Wen, 3D gravity?

The key principles behind Turaev-Viro, Levin-Wen, 3D gravity are the use of topological invariants to describe the dynamics of spacetime, the concept of quantum entanglement between different regions of space, and the idea of a discrete lattice structure that governs the interactions between particles.

What are the applications of Turaev-Viro, Levin-Wen, 3D gravity?

Turaev-Viro, Levin-Wen, 3D gravity has applications in theoretical physics, specifically in the study of quantum gravity and the unification of fundamental forces. It also has potential applications in other areas such as condensed matter physics, quantum information theory, and computer science.

What are the main challenges in studying Turaev-Viro, Levin-Wen, 3D gravity?

One of the main challenges in studying Turaev-Viro, Levin-Wen, 3D gravity is the complexity of the mathematical framework involved. This theory requires a deep understanding of topology, quantum field theory, and general relativity, making it a challenging subject for researchers. Additionally, the lack of experimental data in the three-dimensional quantum gravity regime poses a challenge for testing and validating the theory.

What current research is being done on Turaev-Viro, Levin-Wen, 3D gravity?

Current research on Turaev-Viro, Levin-Wen, 3D gravity focuses on developing a deeper understanding of the theory and its implications for quantum gravity. This includes investigating the dualities between different formulations of the theory and exploring its connections to other areas of physics. Researchers are also working on developing new mathematical techniques and computational methods to better study and test the theory.

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