# How can I understand the braiding terms introduced in the plaquette operator of the Walker-Wang TQFT

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

• Perhaps you can try rephrasing this so as to ask a very specific question. – Chris Ramsey Sep 25 '15 at 12:33

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.
• 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.