  The vortex-boson (or Abelian-Higgs, XY) duality in 2+1 dimensions
demonstrates that the quantum disordered superfluid is equivalent to an ordered
superconductor and the other way around. Such a duality structure should be
ubiquitous but in 3+1 (and higher) dimensions a precise formulation of the
duality is lacking. The problem is that the topological defects become extended
objects, strings in 3+1D. We argue how the condensate of such vortex strings
must behave from the known physics of the disordered superfluid, namely the
Bose-Mott insulator. A flaw in earlier proposals is repaired, and a more direct
viewpoint, avoiding gauge fields, in terms of the physical supercurrent is laid
out, that also easily generalizes to higher-dimensional and more complicated
systems. Furthermore topological defects are readily identified; we demonstrate
that the Bose-Mott insulator supports line defects, which may be seen in cold
atom experiments.
