  We derive a bulk-boundary correspondence for three-dimensional (3D)
symmetry-protected topological (SPT) phases with unitary symmetries. The
correspondence consists of three equations that relate bulk properties of these
phases to properties of their gapped, symmetry-preserving surfaces. Both the
bulk and surface data appearing in our correspondence are defined via a
procedure in which we gauge the symmetries of the system of interest and then
study the braiding statistics of excitations of the resulting gauge theory. The
bulk data is defined in terms of the statistics of bulk excitations, while the
surface data is defined in terms of the statistics of surface excitations. An
appealing property of this data is that it is plausibly complete in the sense
that the bulk data uniquely distinguishes each 3D SPT phase, while the surface
data uniquely distinguishes each gapped, symmetric surface. Our correspondence
applies to any 3D bosonic SPT phase with finite Abelian unitary symmetry group.
It applies to any surface that (1) supports only Abelian anyons and (2) has the
property that the anyons are not permuted by the symmetries.
