  The study of gapped quantum many-body systems in three spatial dimensions has
uncovered the existence of quantum states hosting quasiparticles that are
confined, not by energetics but by the structure of local operators, to move
along lower dimensional submanifolds. These so-called "fracton" phases are
beyond the usual topological quantum field theory description, and thus require
new theoretical frameworks to describe them. Here we consider coupling fracton
models to topological quantum field theories in (3+1) dimensions by starting
with two copies of a known fracton model and gauging the $\mathbb{Z}_2$
symmetry that exchanges the two copies. This yields a class of exactly solvable
lattice models that we study in detail for the case of the X-cube model and
Haah's cubic code. The resulting phases host finite-energy non-Abelian immobile
quasiparticles with robust degeneracies that depend on their relative
positions. The phases also host non-Abelian string excitations with robust
degeneracies that depend on the string geometry. Applying the construction to
Haah's cubic code in particular provides an exactly solvable model with finite
energy yet immobile non-Abelian quasiparticles that can only be created at the
corners of operators with fractal support.
