D = 11 Supermembrane instantons, W∞ strings and the super Toda molecule

Exact instanton solutions to D = 11 spherical supermembranes moving in flat target space-time backgrounds are constructed. Our starting point is super Yang-Mills theories, based on the infinite-dimensional SU(∞) group, dimensionally reduced to one time dimension. In this fashion the super Toda molecule equation is recovered preserving only one supersymmetry out of the N = 16 that one would have obtained otherwise. It is conjectured that the expected critical target space-time dimensions for the (super)membrane, (D = 11) D = 27 is closely related to that of the noncritical (super) W∞ strings. A BRST analysis of these symmetries should yield information about the quantum consistency of the (D = 11) D = 27 dimensional (super)membrane. Comments on the role that skyrmions might play in the two types of spinning membrane actions constructed so far is presented in the conclusion. Finally, the importance that integrability on light lines in complex superspaces has in other types of solutions is emphasized.