Consider an unoriented bipartite quad-graph $\Gamma$ embedded in a a closed surface $\Sigma$ and a state space $\mathcal{H} = \bigotimes_{\text{edges}} \, \Bbb{C}_e^2$ consisting of spins situated on the edges $e$ of $\Gamma$. Equip $\mathcal{H}$ with a family local operators $A_v = \bigotimes_{\text{edges}} a_e$ and $B_p = \bigotimes_{\text{edges}} b_e$ respectively attached to vertices $v$ and plaquettes $p$ (i.e. faces) of the graph defined in the following manner: Each tensor factor $a_e$ (respectively $b_e$) act trivially unless $v \in \partial e$ (respectively $e \in \partial p$) in which case it acts by Pauli-matrices $a_e = \sigma_x$ (respectively $b_e = \sigma_z$). Define a hamiltonian $H$ by summing these local operators, specifically

\begin{equation} H \ = \ \sum_{v} \, A_v \, + \, \sum_{p} \, B_p \end{equation}

Each orientation of $\omega$ of $\Gamma$ can be encoded as spin-configuration $\vec{u}_\omega = \bigotimes_{\text{edges}} \vec{u}_e$ in $\mathcal{H}$ where $\vec{u}_e$ is the standard basis vector $\vec{e}_1$ or $\vec{e}_2$ depending on whether $\omega$ agrees or disagrees with the canonical bipartite orientation (i.e. black points to white). To each Kasteleyn orientation $K$ an eigenstate $[ \vec{u}_K ]$ can be constructed by averaging over all spin-configurations $\vec{u}_\omega$ which are gauge-equivalent to $K$; and such an eigenstate is topological in the sense that its eigenvalue depends only on the Euler characteristic $\chi$ of the surface $\Sigma$.

For each edge $e$ consider replacing each spin-factor $\Bbb{C}^2_e$ by a finite dimensional irreducible representation $V_e$ of $\text{SU}(2)$ and forming the more general state space $\mathcal{H} = \bigotimes_{\text{edges}} \, V_e$.

**Question:** How should the definition of vertex and plaquette operators $A_v$ and $B_p$ be modified so that the
corresponding hamiltonian $H$ possesses topological eigenstates ?

regards, A. Leverkühn

*Note*: The Kasteleyn condition together with the fact that all the faces of $\Gamma$ are quadrilateral implies that $B_p \, \big[ \vec{u}_K \big] \, = \, - \big[ \vec{u}_K \big]$ and consequently

\begin{equation} \begin{array}{ll} H \, \big[ \vec{u}_K \big] &= \ \Bigg( \# \, \big\{ \text{vertices of $\Gamma$} \big\} - \ \# \, \big\{ \text{faces of $\Gamma$} \big\} \Bigg) \, \big[ \vec{u}_K \big] \\ \\ &= \ \chi\big( \Sigma \big) \, \big[ \vec{u}_K \big] \end{array} \end{equation}