  The dislocation-mediated quantum melting of solids into quantum liquid
crystals is extended from two to three spatial dimensions, using a
generalization of boson-vortex or Abelian-Higgs duality. Dislocations are now
Burgers-vector-valued strings that trace out worldsheets in spacetime while the
phonons of the solid dualize into two-form (Kalb-Ramond) gauge fields. We
propose an effective dual Higgs potential that allows for restoring
translational symmetry in either one, two or three directions, leading to the
quantum analogues of columnar, smectic or nematic liquid crystals. In these
phases, transverse phonons turn into gapped, propagating modes while
compressional stress remains massless. Rotational Goldstone modes emerge
whenever translational symmetry is restored. We also consider electrically
charged matter, and find amongst others that as a hard principle only two out
of the possible three rotational Goldstone modes are observable using
electromagnetic means.
