  We focus on BPS solutions of the gauged O(3) Sigma model, due to Schroers,
and use these ideas to study the geometry of the moduli space. The model has an
asymmetry parameter $\tau$ breaking the symmetry of vortices and antivortices
on the field equations. It is shown that the moduli space is incomplete both on
the Euclidean plane and on a compact surface. On the Euclidean plane, the L2
metric on the moduli space is approximated for well separated cores and results
consistent with similar approximations for the Ginzburg-Landau functional are
found. The scattering angle of approaching vortex-antivortex pairs of different
effective mass is computed numerically and is shown to be different from the
well known scattering of approaching Ginzburg-Landau vortices. The volume of
the moduli space for general $\tau$ is computed for the case of the round
sphere and flat tori. The model on a compact surface is deformed introducing a
neutral field and a Chern-Simons term. A lower bound for the Chern-Simons
constant $\kappa$ such that the extended model admits a solution is shown to
exist, and if the total number of vortices and antivortices are different, the
existence of an upper bound is also shown. Existence of multiple solutions to
the governing elliptic problem is established on a compact surface as well as
the existence of two limiting behaviours as $\kappa \to 0$. A localization
formula for the deformation is found for both Ginzburg-Landau and the O(3)
Sigma model vortices and it is shown that it can be extended to the coalescense
set. This rules out the possibility that this is Kim-Lee's term in the case of
Ginzburg-Landau vortices, moreover, the deformation term is compared on the
plane with the Ricci form of the surface and it is shown they are different,
hence also discarding that this is the term proposed by Collie-Tong to model
vortex dynamics with Chern-Simons interaction.
