  A number of exactly solvable spin models, including the Kitaev toric code in
two and three dimensions and the X-cube model in three dimensions, can be
related to their respective parent lattice gauge theories (LGT) through the
mathematical process of 'Higgsing'. Field theories of the low-energy
excitations of these spin models can be developed subsequently, building upon
the symmetry of the parent LGTs. Recently, two of the present authors proposed
a variant of the three-dimensional toric code which we now call the F3 model,
whose elementary excitations consist of freeon and fluxon excitations of the
three-dimensional toric code and fracton excitations of the X-cube model. In
this work, we identify the parent LGT of the F3 model as the hybrid rank-1 and
rank-2 U(1) LGT, and develop the corresponding field theory. The resulting
Lagrangian of the F3 model is that of a three-dimensional toric code with an
extra term, which ties the dynamics of fractons to that of fluxons. The matter
part of the effective action for the F3 model can be derived as well, by
carefully keeping track of the gauge invariance of the F3 model. Hydrodynamic
equations of motion of the quasiparticles are derived, which properly reflect
their constrained dynamics. Finally, we present a tight-binding model for the
quasiparticle motion in the F3 model.
