  We study the structure of the ground state wave functions of bosonic Symmetry
Protected Topological (SPT) insulators in 3 space dimensions. We demonstrate
that the differences with conventional insulators are captured simply in a dual
vortex description. As an example we show that a previously studied bosonic
topological insulator with both global U(1) and time-reversal symmetry can be
described by a rather simple wave function written in terms of dual "vortex
ribbons". The wave function is a superposition of all the vortex ribbon
configurations of the boson, and a factor (-1) is associated with each
self-linking of the vortex ribbons. This wave function can be conveniently
derived using an effective field theory of the SPT in the strong coupling
limit, and it naturally explains all the phenomena of this SPT phase discussed
previously. The ground state structure for other 3d bosonic SPT phases are also
discussed similarly in terms of vortex loop gas wave functions. We show that
our methods reproduce known results on the ground state structure of some 2d
SPT phases.
