  We present an embedding of the 3-dimensional relativistic Landau-Ginzburg
model for condensed matter systems in an $\mathcal{N}=6$, $U(N)\times U(N)$
Chern-Simons-matter theory (the ABJM model) by consistently truncating the
latter to an abelian effective field theory encoding the collective dynamics of
${\cal O}(N)$ of the ${\cal O}(N^2)$ modes. In fact, depending on the VEV on
one of the ABJM scalars, a mass deformation parameter $\mu$ and the
Chern-Simons level number $k$, our abelianization prescription allows us to
interpolate between the abelian Higgs model with its usual multi-vortex
solutions and a $\phi^4$ theory. We sketch a simple condensed matter model that
reproduces all the salient features of the abelianization. In this context, the
abelianization can be interpreted as giving a dimensional reduction from four
dimensions.
