On relating multiple M2 and D2-branes

Due to the difficulties of finding superconformal Lagrangian theories for multiple M2-branes, we will in this paper instead focus on the field equations. By relaxing the requirement of a Lagrangian formulation we can explore the possibility of having structure constants f^ABC_D satisfying the fundamental identity but which are not totally antisymmetric. We exemplify this discussion by making use of an explicit choice of a non-antisymmetric f^ABC_D constructed from the Lie algebra structure constants f^ab_c of an arbitrary gauge group. Although this choice of f^ABC_D does not admit an obvious Lagrangian description, it does reproduce the correct SYM theory for a stack of N D2-branes to leading order in g_YM^-1 upon reduction and, moreover, it sheds new light on the centre of mass coordinates for multiple M2-branes.