# 6j symbols with Majorana indices

The Levin-Wen model is a Hamiltonian formulation of Turaev-Viro (2 + 1)d TQFTs. It can be constructed from a unitary fusion category $\mathcal{C}$, which can be equivalently defined using $6j$ symbols: An equivalence class of solutions of the pentagon equation satisfying certain normalizations.

Gu, Wen, Wang proposed a generalization of the above construction in http://arxiv.org/abs/1010.1517. The mathematical framework for such a generalization is the theory of enriched categories. By considering special enriched categories, the so-called projective super fusion categories, which are enriched categories over the category of super Hilbert spaces up to projective even unitary transformations, one can get a Hamiltonian formulation of the fermionic Turaev-Viro TQFTs. The projective super fusion category can be equivalently defined in terms of the fermionic $6j$ symbols $F^{ijm,\alpha\beta}_{knt,\eta\varphi}$, which are solutions of the fermionic pentagon equation below:

\begin{align} & \sum_{t} \sum_{\eta=1}^{N^{jk}_{t}} \sum_{\varphi=1}^{N^{it}_{n}} \sum_{\kappa=1}^{N^{tl}_{s}} F^{ijm,\alpha\beta}_{knt,\eta\varphi} F^{itn,\varphi\chi}_{lps,\kappa\gamma} F^{jkt,\eta\kappa}_{lsq,\delta\varphi} = (-)^{s^{ij}_{m}(\alpha) s^{kl}_{q}(\delta)} \sum_{\epsilon=1}^{N^{mq}_{p}} F^{mkn,\beta\chi}_{lpq,\delta\epsilon} F^{ijm,\alpha\epsilon}_{qps,\varphi\gamma}, \end{align} where we use English letters (bosonic indices) to label states on the edges, Greek letters (fermionic indices) to label states on the vertex. $s^{ij}_{m}(\alpha)$ is used to denote the fermion parity of the state $\alpha$.

I'd like to know if there are generalizations of the Levin-Wen model in which the relevant enriched category is defined in terms of the "Majorana $6j$ symbols $F^{ijm,\alpha\beta}_{knt,\eta\varphi}$", where there could be a single Majorana fermion sitting on the vertices. In this case, one should allow the fermion number on each vertex to fluctuate, and the Greek indices $\alpha,\beta,\eta,\varphi$ labeling the vertex states should be Majorana variables. Naturally, these "Majorana $6j$ symbols" should be solutions of a "Majorana pentagon equation".

• I'm not completely certain, but it seems that a recent paper of Aasen-Lake-Walker answers this question. I think their analogue of Majorana fermions at the vertices they call $q$-type particles, and they discuss the generalization of the Levin-Wen model in §9. Their versions of the Majorana $6j$ symbols and Majorana pentagon equation are discussed in §§8.7 and 8.8. – Arun Debray Jul 18 '18 at 19:28