# 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?

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

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.

• I was expecting Kevin to answer this question. I'll just add that in the last example, Kitaev's original finite group model corresponds to the trivial class in $\pi_{\leq n} BG$, so some generalization is needed (although it is fairly clear how to generalize). Also, to define a state sum is it necessary to have a fully extended TQFT? Jan 23, 2015 at 5:44
• Yes, one of the details that I suppressed was the distinction between twisted DW theory and untwisted DW theory. If you have an appropriate cocycle in $C^{n+1}(BG; U(1))$, then you can define a twisted version of $\pi_{\le n}(BG)$ and apply the machinery to that $n$-category. Jan 23, 2015 at 14:32
• @Meng, one can define a state sum without ever thinking about a TQFT. That's how it was done in the old days; I think the term "state sum" predates "TQFT" by a few years. But I think it would be difficult to construct a state sum from a TQFT that does not extend all the down to points. For example, no one has constructed a state sum (for 3-manifolds) that computes the WRT/Chern-Simons path integral, and that TQFT is only one level shy of being fully extended. Jan 23, 2015 at 14:40
• (continuing) There is, of course, a state sum on 4-manifolds (modified Crane-Yetter sum) that computes the WRT invariant of the boundary of the 4-manifold. This corresponds to the fact that WRT-theory can be thought of as living on the boundary of the CYK TQFT, and the CYK TQFT is fully extended. Jan 23, 2015 at 14:43

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.

• Right, in Crane-Yetter, triangle faces and tetrahedra (dual edges) are labelled. I thought that labelling the triangles is reminiscent of the procedure of replacing all (higher-valent) vertices in Walker-Wang by trivalent vertices. Jan 22, 2015 at 16:50