We study the role of different terms in the N-body potential of glass
forming systems on the critical dynamics near the glass transition. Using a
simplified spin model with quenched disorder, where the different terms of the
real N-body potential are mapped into multi-spin interactions, we identified
three possible scenarios. For each scenario we introduce a ``minimal'' model
representative of the critical glassy dynamics near, both above and below, the
critical transition lin e. For each ``minimal'' model we discuss the low
temperature equilibrium dynamics.Comment: Completely revised version, 8 pages, 5 figures, typeset using
EURO-LaTeX, Europhysics Letters (in press
