1996/12/15 by Carlos Castro, Castro, Carlos
Physics and Astronomy · #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #hep-th
paper · pdf · doi:10.48550/arxiv.hep-th/9612160
15 pages, tex file. A review of work that appeared in the Journal of Chaos, Solitons and Fractals with some new results
arxiv created 1996/12/15 · arxiv updated 2009/11/30
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(∞) 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_∞ 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(∞) SDYM equations ) bears a direct connection to a critical W_∞ string spectrum adjoined to a q=N+1 unitary minimal model of the WN algebra in the N→ ∞ 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_∞ algebras from nonlinear integrable deformations of 4D self dual gravity was studied by the author.The full SU(∞) YM theory remains to be explored as well as the incipient role that noncritical nonlinear W_∞ strings might have in the full quantization program.