Abelian reductions of deformed $\mathcal{N}=4$ SYM

Following the work in \cite{Mohammed:2012gi}, where the massive ABJM model in 2+1 dimensions was shown to have an abelian reduction to the relativistic Landau-Ginzburg, and motivated by the implications for condensed matter through AdS/CFT, we show that a FI deformation of $\mathcal{N}=4$ SYM in 3+1 dimensions with a mass term can also be reduced to a relativistic Landau-Ginzburg model, with the possibility of coupling it to a real scalar, whereas the simply mass deformed $\mathcal{N}=4$ SYM reduces only to a massive ${\varphi }^4$ model (scalar QED) coupled to a real scalar. We study the classical solutions of the model, in particular vortex solutions.