  This thesis deals with planar gauge theories in which some gauge group G is
spontaneously broken to a finite subgroup H. The spectrum consists of magnetic
vortices, global H charges and dyonic combinations exhibiting topological
Aharonov-Bohm interactions. Among other things, we review the Hopf algebra D(H)
related to this residual discrete H gauge theory, which provides an unified
description of the spin, braid and fusion properties of the aforementioned
particles. The implications of adding a Chern-Simons (CS) term to these models
are also addressed. We recall that the CS actions for a compact gauge group G
are classified by the cohomology group H^4(BG,Z). For finite groups H this
classification boils down to the cohomology group H^3(H,U(1)). Thus the
different CS actions for a finite group H are given by the inequivalent
3-cocycles of H. It is argued that adding a CS action for the broken gauge
group G leads to additional topological interactions for the vortices governed
by a 3-cocycle for the residual finite gauge group H determined by a natural
homomorphism from H^4(BG,Z) to H^3(H,U(1)). Accordingly, the related Hopf
algebra D(H) is deformed into a quasi-Hopf algebra. These general
considerations are illustrated by CS theories in which the direct product of
some U(1) gauge groups is broken to a finite subgroup H. It turns out that not
all conceivable 3-cocycles for finite abelian gauge groups H can be obtained in
this way. Those that are not reached are the most interesting. A Z_2 x Z_2 x
Z_2 CS theory given by such a 3-cocycle, for instance, is dual to an ordinary
gauge theory with nonabelian gauge group the dihedral group of order eight.
Finally, the CS theories with nonabelian finite gauge group a dihedral or
double dihedral group are also discussed in full detail.
