  Topological phases of matter in (2+1) dimensions are commonly equipped with
global symmetries, such as electric-magnetic duality in gauge theories and
bilayer symmetry in fractional quantum Hall states. Gauging these symmetries
into local dynamical ones is one way of obtaining exotic phases from
conventional systems. We study this using the bulk-boundary correspondence and
applying the orbifold construction to the (1+1) dimensional edge described by a
conformal field theory (CFT). Our procedure puts twisted boundary conditions
into the partition function, and predicts the fusion, spin and braiding
behavior of anyonic excitations after gauging. We demonstrate this for the
electric-magnetic self-dual $\mathbb{Z}_N$ gauge theory, the twofold symmetric
$SU(3)_1$, and the $S_3$-symmetric $SO(8)_1$ Wess-Zumino-Witten theories.
