  We study $1+1$-dimensional $SU(N)$ gauge theories with adjoint scalar matter
representations, based on a dimensional truncation of $2+1$ and
$3+1$-dimensional pure QCD, which approximate the dynamics of transversely
polarized gluons. The glueballs are investigated non-perturbatively using
light-front quantisation, detailed spectra and wavefunctions being obtained for
the large-$N$ limit. In general there is some qualitative agreement of the
spectra with lattice Monte Carlo data from the higher dimensional QCD. From the
light-front wavefunctions we calculate (polarized) structure functions and
interpret the gluon and spin content of glueballs. We discuss the phase
structure of the reduced theories in relation to matrix models for relativistic
non-critical strings.
