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We consider the light-cone (LC) gauge and LC quantization of the dimensionalreduction of super Yang Mills theory from four to two dimensions. Afterintegrating out all unphysical degrees of freedom, the non-local LC Hamiltonianexhibits an explicit ${\cal N}=(2,2)$ supersymmetry. A further SUSY-preservingcompactification of LC-space on a torus of radius $R$, allows for a large-$N$numerical study where the smooth large-$R$ limit of physical quantities can bechecked. As a first step, we consider a simple, yet quite rich, "Coulombapproximation" that maintains an ${\cal N}=(1,1)$ subgroup of the originalsupersymmetry and leads to a non-trivial generalization of 't Hooft's modelwith an arbitrary --but conserved-- number of partons. We compute numericallythe eigenvalues and eigenvectors both in momentum and in position space. Ourresults, so far limited to the sectors with 2, 3 and 4 partons, directly andquantitatively confirm a simple physical picture in terms of a string-likeinteraction with the expected tension among pairs of nearest-neighbours alongthe single-trace characterizing the large-$N$ limit. Although broken by ourapproximation, traces of the full ${\cal N}=(2,2)$ supersymmetry are stillvisible in the low-lying spectrum.

Dimensionally reduced SYM$_4$ at large-$N$: an intriguing Coulomb approximation