  In this paper, we apply the method of Fourier transform and basis rewriting
developed in arXiv:1910.13441 for the two-dimensional quantum double model of
topological orders to the three-dimensional gauge theory model (with a gauge
group $G$) of three-dimensional topological orders. We find that the gapped
boundary condition of the gauge theory model is characterized by a Frobenius
algebra in the representation category $\mathcal Rep(G)$ of $G$, which also
describes the charge splitting and condensation on the boundary. We also show
that our Fourier transform maps the three-dimensional gauge theory model with
input data $G$ to the Walker-Wang model with input data $\mathcal Rep(G)$ on a
trivalent lattice with dangling edges, after truncating the Hilbert space by
projecting all dangling edges to the trivial representation of $G$. This
Fourier transform also provides a systematic construction of the gapped
boundary theory of the Walker-Wang model. This establishes a correspondence
between two types of topological field theories: the extended Dijkgraaf-Witten
and extended Crane-Yetter theories.
