Walker-Wang models as introduced in (3+1)-TQFTS and Topological Insulators are an example of 3+1D lattice TQFTS. In analogy with the Levin-Wen models in 2+1D the authors define an exactly solvable Hamiltonian in terms of mutually commuting operators acting on vertices and plaquettes of a 3D trivalent-cubic lattice.

Reading the papers of Levin-Wen and the book by Zhenghan Wang I understand the form of the F-moves in the definition of the plaquette operator using the graphical manipulations of a fusion category/pre-modular catgegory. The step I'm struggling to understand is the argument about how to consistently introduce the R-matrices into the plaquette operator. I would like to understand this question with the motivation of defining the Walker-Wang model on other 3D trivalent lattices aside from the cubic one introduced in the paper.

Thanks in advance for any insight

  • $\begingroup$ Perhaps you can try rephrasing this so as to ask a very specific question. $\endgroup$ Sep 25, 2015 at 12:33

1 Answer 1


The role of the $R$-matrices in the plaquette term is described starting on page 10 of http://arxiv.org/pdf/1104.2632v2.pdf. It is important to keep in mind that the various graphs in the figures there actually depict ribbon graphs via the blackboard framing convention.

If you are interested in writing down analogous formulas for a different lattice (or for some arbitrary cellulation of a 3-manifold), you should keep the following in mind.

  • Within the 1-skeleton of your cellulation, certain cycles $\{C_i\}$ are the boundaries of 2-cells (plaquettes). When we choose a ribbon graph structure (thickening) of the 1-skeleton, it is convenient if the framing of $C_i$ coming from the ribbon graph structure agrees with the framing of $C_i$ coming from the associated 2-cell. If this is not the case then you will need to incorporate $T$-matrices (twist factors) into your formula.

  • Consider the edges {$e_j\}$ incident to some cycle $C_i$. Using the ribbon graph structure, each $e_j$ either points "out" (away from the plaquette) or "in" (toward the plaquette). Each outward-pointing edge gives rise to an $F$-matrix factor in the plaquette term, similar to Levin-Wen. The inward-pointing edges are more complicated. In the original WW paper, we push them outward (using an $R$-matrix), do the usual LW move ($F$-matrix), then push the edge back inward (inverse $R$-matrix), for a total of two $R$-matrices and one $F$-matrix. Alternatively, choose a difference sequence of ribbon graph manipulations, resulting in one $R$-matrix and two $F$-matrices. The two answers are, of course, related by the hexagon relation.

If you understand the LW model and you also understand how to manipulate labeled ribbon graphs using $F$, $R$ and $T$ matrices, then it is easy to write down the WW model.


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