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.
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