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The study of gapped quantum many-body systems in three spatial dimensions hasuncovered the existence of quantum states hosting quasiparticles that areconfined, not by energetics but by the structure of local operators, to movealong lower dimensional submanifolds. These so-called "fracton" phases arebeyond the usual topological quantum field theory description, and thus requirenew theoretical frameworks to describe them. Here we consider coupling fractonmodels to topological quantum field theories in (3+1) dimensions by startingwith 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 solvablelattice models that we study in detail for the case of the X-cube model andHaah's cubic code. The resulting phases host finite-energy non-Abelian immobilequasiparticles with robust degeneracies that depend on their relativepositions. The phases also host non-Abelian string excitations with robustdegeneracies that depend on the string geometry. Applying the construction toHaah's cubic code in particular provides an exactly solvable model with finiteenergy yet immobile non-Abelian quasiparticles that can only be created at thecorners of operators with fractal support.

Gauging fractons: immobile non-Abelian quasiparticles, fractals, and position-dependent degeneracies