How are the Walker-Wang TQFT and the Crane-Yetter TQFT related? Mathematical physicists in solid state physics and topological insulators talk a lot about Walker-Wang models, 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, 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 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?
 A: They are indeed the same, in the sense that Crane-Yetter TQFT is a state sum (i.e. partition function) while Walker-Wang model is the Hamiltonian formulation. There are some technical differences, e.g. in Walker-Wang model edges of a 3D trivalent lattice are labeled, while in Crane-Yetter (I believe) faces of the triangulation are labeled, so they are "dual" to each other in this sense.
A: 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
In general, given an $n$-category with the right sort of duality, there is a standard procedure for constructing


*

*A fully extended $n{+}1$-dimensional TQFT

*A state sum computing the path integral for an $n{+}1$-manifold equipped with a cell decomposition

*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

*

*A (generalized) Turaev-Viro type TQFT (full extended)

*Generalized Turaev-Viro state sum

*Levin-Wen model (again generalized)


*Input: premodular category (i.e. a 3-category with trivial 0- and 1-morphisms)

*

*Premodular (also known as Crane-Yetter-Kauffman) TQFT

*Crane-Yetter state sum

*Walker-Wang model


*Input: $\pi_{\le n} BG$, for $G$ and finite group

*

*$n{+}1$-dimensional Dijkgraaf-Witten TQFT

*DW state sum

*Kitaev finite group model



I've suppressed a few details above.
