  Motivated by the recently established duality between elasticity of crystals
and a fracton tensor gauge theory, we combine it with boson-vortex duality, to
explicitly account for bosonic statistics of the underlying atoms. We thereby
derive a hybrid vector-tensor gauge dual of a supersolid, which features both
crystalline and superfluid order. The gauge dual describes a fracton state of
matter with full dipole mobility endowed by the superfluid order, as governed
by "mutual" axion electrodynamics between the fracton and vortex sectors of the
theory, with an associated generalized Witten effect. Vortex condensation
restores U(1) symmetry, confines dipoles to be subdimensional (recovering the
dislocation glide constraint of a commensurate quantum crystal), and drives a
phase transition between two distinct fracton phases. Meanwhile, condensation
of elementary fracton dipoles and charges, respectively, provide a gauge dual
description of the super-hexatic and ordinary superfluid. Consistent with
conventional wisdom, in the absence of crystalline order, U(1)-symmetric phases
are prohibited at zero temperature via a mechanism akin to deconfined quantum
criticality.
