  We propose a systematic approach to constructing microscopic models with
fractional excitations in three-dimensional (3D) space. Building blocks are
quantum wires described by the (1+1)-dimensional conformal field theory (CFT)
associated with a current algebra $\mathfrak{g}$. The wires are coupled with
each other to form a 3D network through the current-current interactions of
$\mathfrak{g}_1$ and $\mathfrak{g}_2$ CFTs that are related to the
$\mathfrak{g}$ CFT by a nontrivial conformal embedding $\mathfrak{g} \supset
\mathfrak{g}_1 \times \mathfrak{g}_2$. The resulting model can be viewed as a
layer construction of a 3D topologically ordered state, in which the conformal
embedding in each wire implements the anyon condensation between adjacent
layers. Local operators acting on the ground state create point-like or
loop-like deconfined excitations depending on the branching rule. We
demonstrate our construction for a simple solvable model based on the conformal
embedding $SU(2)_1 \times SU(2)_1 \supset U(1)_4 \times U(1)_4$. We show that
the model possesses extensively degenerate ground states on a torus with
deconfined quasiparticles, and that appropriate local perturbations lift the
degeneracy and yield a 3D $Z_2$ gauge theory with a fermionic $Z_2$ charge.
