Mathematical physicists in solid state physics and topological insulators talk a lot about [Walker-Wang models](http://arxiv.org/pdf/1104.2632.pdf), which are a family of Hamiltonians defined on a 3d lattice. Unfortunately, the original paper is lacking a lot of mathematical details which were promised to appear in a later article, but (to my knowledge) never did.


The model looks a lot like the 20 years older [Crane-Yetter model](http://arxiv.org/abs/hep-th/9409167), in that it needs a ribbon fusion category (also called "premodular, in that it doesn't need to be modular) as labelling data. [An article](http://arxiv.org/abs/1208.5128) treating (amongst many other things) the ground state degeneracy of Walker-Wang models suggests that for modular categories, the ground state is in fact nondegenerate for all spatial topologies, that is, the topological state space is 1-dimensional. This is the same behaviour as in Crane-Yetter for modular categories, where $CY(N) = n^{\sigma(N)} \implies CY(S^1 \times M) = 1$ also suggests 1-dimensional state spaces (the general case is, I think, unknown).


The article briefly mentions the Crane-Yetter model for modular categories, but I fail to find a definitive statement like "Walker-Wang and Crane-Yetter are the same TQFT" or "they're different" in the article or elsewhere.


Are they the same (as Turaev-Viro and Levin-Wen seem to be related as well) and Walker-Wang is just the hamiltonian formulation?



How are the Walker-Wang TQFT and the Crane-Yetter TQFT related?
Yes, the Walker-Wang model is related to the Crane-Yetter-Kauffman TQFT in the same way the Levin-Wen model is related to the Turaev-Viro TQFT. See, for example, the table on page 14 of the notes from the talk "Premodular TQFTs" found on [this page](http://canyon23.net/math/talks/)


In general, given an $n$-category with the right sort of duality, there is a standard procedure for constructing


1. A fully extended $n{+}1$-dimensional TQFT
2. A state sum computing the path integral for an $n{+}1$-manifold equipped with a cell decomposition
3. A commuting projection hamiltonian whose ground state is isomorphic to the Hilbert space for an $n$-manifold equipped with a cell decomposition.


Here are three main examples of this procedure.


* Input: a fairly general pivotal 2-category
	1. A (generalized) Turaev-Viro type TQFT (full extended)
	2. Generalized Turaev-Viro state sum
	3. Levin-Wen model (again generalized)
* Input: premodular category (i.e. a 3-category with trivial 0- and 1-morphisms)
	1. Premodular (also known as Crane-Yetter-Kauffman) TQFT
	2. Crane-Yetter state sum
	3. Walker-Wang model
* Input: $\pi\_{\le n} BG$, for $G$ and finite group
	1. $n{+}1$-dimensional Dijkgraaf-Witten TQFT
	2. DW state sum
	3. Kitaev finite group model


I've suppressed a few details above.


