The Noncritical W(infinity) String Sector of the Membrane

The exact quantum integrability aspects of a sector of the membrane is investigated. It is found that spherical membranes ( in the lightcone gauge) moving in flat target spacetime backgrounds admit a class of integrable solutions linked to $SU(\infty)$ SDYM equations ( dimensionally reduced to one temporal dimension) which, in turn, are related to Plebanski 4D SD Gravitational equations. A further rotational Killing-symmetry reduction yields the 3D continuous Toda theory. It is precisely the latter which bears a direct relationship to non critical $W_\infty$ string theory. The expected critical dimensions for the ( super) membrane , (D=11) and D=27, are easily obtained. This suggests that this particular sector of the membrane's spectrum (connected to the $SU(\infty)$ SDYM equations ) bears a direct connection to a critical $W_\infty$ string spectrum adjoined to a q=N+1 unitary minimal model of the W_N algebra in the $N\rightarrow \infty$ limit. Final comments are made about the connection to Jevicki's observation that the 4D quantum membrane is linked to dilatonic-self dual gravity plus matter . 2D dilatonic ( super) gravity was studied by Ikeda and its relation to nonlinear $W_\infty$ algebras from nonlinear integrable deformations of 4D self dual gravity was studied by the author.The full $SU(\infty)$ YM theory remains to be explored as well as the incipient role that noncritical nonlinear $W_\infty$ strings might have in the full quantization program.