  It has recently been realized that zero modes with projective non-Abelian
statistics, generalizing the notion of Majorana bound states, may exist at the
interface between a superconductor and a ferromagnet along the edge of a
fractional topological insulator (FTI). Here we study two-dimensional
architectures of these non-Abelian zero modes, whose interactions are generated
by the charging and Josephson energies of the superconductors. We derive
low-energy Hamiltonians for two different arrays of FTIs on the plane,
revealing an interesting interplay between the real-space geometry of the
system and its topological properties. On the one hand, in a geometry where the
length of the FTI edges is independent on the system size, the array has a
topologically ordered phase, giving rise to a qudit toric code Hamiltonian in
perturbation theory. On the other hand, in a geometry where the length of the
edges scales with system size, we find an exact duality to an Abelian lattice
gauge theory and no topological order.
