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The dislocation-mediated quantum melting of solids into quantum liquidcrystals is extended from two to three spatial dimensions, using ageneralization of boson-vortex or Abelian-Higgs duality. Dislocations are nowBurgers-vector-valued strings that trace out worldsheets in spacetime while thephonons of the solid dualize into two-form (Kalb-Ramond) gauge fields. Wepropose an effective dual Higgs potential that allows for restoringtranslational symmetry in either one, two or three directions, leading to thequantum analogues of columnar, smectic or nematic liquid crystals. In thesephases, transverse phonons turn into gapped, propagating modes whilecompressional stress remains massless. Rotational Goldstone modes emergewhenever translational symmetry is restored. We also consider electricallycharged matter, and find amongst others that as a hard principle only two outof the possible three rotational Goldstone modes are observable usingelectromagnetic means.

Dual gauge field theory of quantum liquid crystals in three dimensions