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?


2 Answers 2


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.

  • $\begingroup$ 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? $\endgroup$
    – Meng Cheng
    Jan 23, 2015 at 5:44
  • 1
    $\begingroup$ 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. $\endgroup$ Jan 23, 2015 at 14:32
  • $\begingroup$ @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. $\endgroup$ Jan 23, 2015 at 14:40
  • $\begingroup$ (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. $\endgroup$ 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.

  • $\begingroup$ 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. $\endgroup$ Jan 22, 2015 at 16:50

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.