  We introduce a Hamiltonian coupling Majorana fermion degrees of freedom to a
quantum dimer model. We argue that, in three dimensions, this model has
deconfined quasiparticles supporting Majorana zero modes obeying nontrivial
statistics. We introduce two effective field theory descriptions of this
deconfined phase, in which the excitations have Coulomb interactions. A key
feature of this system is the existence of topologically non-trivial fermionic
excitations, called "Hopfions" because, in a suitable continuum limit of the
dimer model, such excitations correspond to the Hopf map and are related to
excitations identified in arXiv:1003.1964. We identify corresponding
topological invariants of the quantum dimer model (with or without fermions)
which are present even on lattices with trivial topology. The Hopfion energy
gap depends upon the phase of the model. We briefly comment on the possibility
of a phase with a gapped, deconfined $\mathbb{Z}_2$ gauge field, as may arise
on the stacked triangular lattice.
