  We introduce a class of gapped three-dimensional models, dubbed "cage-net
fracton models," which host immobile fracton excitations in addition to
non-Abelian particles with restricted mobility. Starting from layers of
two-dimensional string-net models, whose spectrum includes non-Abelian anyons,
we condense extended one-dimensional "flux-strings" built out of point-like
excitations. Flux-string condensation generalizes the concept of anyon
condensation familiar from conventional topological order and allows us to
establish properties of the fracton ordered (equivalently, flux-string
condensed) phase, such as its ground state wave function and spectrum of
excitations. Through the examples of doubled Ising and SU(2)$_k$ cage-net
models, we demonstrate the existence of strictly immobile Abelian fractons and
of non-Abelian particles restricted to move only along one dimension. In the
doubled Ising cage-net model, we show that these restricted-mobility
non-Abelian excitations are a fundamentally three-dimensional phenomenon, as
they cannot be understood as bound states amongst two-dimensional non-Abelian
anyons and Abelian particles. We further show that the ground state wave
function of such phases can be understood as a fluctuating network of extended
objects -- cages -- and strings, which we dub a cage-net condensate. Besides
having implications for topological quantum computation in three dimensions,
our work may also point the way towards more general insights into quantum
phases of matter with fracton order.
