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The vortex-boson (or Abelian-Higgs, XY) duality in 2+1 dimensionsdemonstrates that the quantum disordered superfluid is equivalent to an orderedsuperconductor and the other way around. Such a duality structure should beubiquitous but in 3+1 (and higher) dimensions a precise formulation of theduality is lacking. The problem is that the topological defects become extendedobjects, strings in 3+1D. We argue how the condensate of such vortex stringsmust behave from the known physics of the disordered superfluid, namely theBose-Mott insulator. A flaw in earlier proposals is repaired, and a more directviewpoint, avoiding gauge fields, in terms of the physical supercurrent is laidout, that also easily generalizes to higher-dimensional and more complicatedsystems. Furthermore topological defects are readily identified; we demonstratethat the Bose-Mott insulator supports line defects, which may be seen in coldatom experiments.

Condensing Nielsen-Olesen strings and the vortex-boson duality in 3+1 and higher dimensions