# 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".

• There is a construction by Kevin Walker, see canyon23.net/math/talks/IPAM%20201501b%20compressed.pdf However the way he derived it is somewhat different from a naive generalization of Gu-Wen pentagon equation. In fact, for the simplest kind of "Majorana pentagon equation", one can consider just one type of nontrivial label on edge, call it $\sigma$, and allow the vertex $\sigma\times\sigma\rightarrow 1$ to have $0$ or $1$ fermion. A straightforward generalization of the pentagon equation does not yield a unitary solution. – Meng Cheng Sep 4 '15 at 1:24
• Thanks for the reference, Meng. I also noticed that the fermionic pentagon equation is obtained in page 27 of the paper by Gaiotto and Kapustin arxiv.org/abs/1505.05856 (in a somewhat different language). They made an assumption in the fusion rules of simple objects (Eq. 6.2). They neglect the possibility of some simple objects being Majorana and assumed that all simple objects are bosonic, so that all $H^k_{ij}$ have a well-defined grading, which leads to the fermionic pentagon equation. I think loosen this constraint might give us a "Majorana pentagon equation". – Zitao Wang Sep 4 '15 at 7:09
• Well in a way it does not make sense to have a single Majorana mode at a site, since one can not define the Hilbert space for a single Majorana mode. So unless one is willing to give up the (fermionic) tensor product structure of the lattice model, it is not clear what "Majorana pentagon equation" means. Of course it does not mean this is impossible, but one probably needs to define the Hilbert space structure first. – Meng Cheng Sep 4 '15 at 7:25
• In my opinion, what Kevin had is very close: he considered that the endomorphism of a simple object can be fermionic, which leads to all kinds of subtle issues in the diagrammatic calculus, and remarkably resolved them. In the end he had an exactly solvable model of an Ising phase (not the doubled one). – Meng Cheng Sep 4 '15 at 7:26
• Doesn't Kevin have a paper with Scott that covers some of the content in that presentation? – Matthew Titsworth Sep 4 '15 at 18:56