An Exact Membrane Quantization from W ∞ Symmetry

An exact quantization of the spherical membrane moving in flat target spacetime backgrounds is performed. Crucial ingredients are the exact integrabilty of the 3D SU (∞) continuous Toda equation and the quasi-finite highest weight irreducible representations of W ∞ algebras. Both continuous and discrete energy levels are found. The latter are found for periodic-like solutions. Membrane wavefunctionals solutions are found in terms of Bessel's functions and plausible relations to singleton field theory are outlined. Recently, [1] exact instanton solutions to D = 11 spherical membranes moving in flat target spacetime backgrounds were constructed. The starting point was dimensionally-reduced Super Yang-Mills theories based on the infinite dimensional SU (∞) algebra. The latter algebra is isomorphic to the area-preserving diffeomorphisms of the sphere. In this fashion the super Toda molecule equation was recovered preserving one supersymmetry out of the N = 16 expected. The expected critical target spacetime dimensions for the (super) membrane , D = 27(11) , was closely related to that of non-critical (super) W ∞ string theories. A BRST analysis revealed that the spectrum of the membrane must have a relationship to the first unitary minimal model of a W N algebra adjoined to a critical W N string in the N → ∞ limit [1]. It is the purpose of this work to push this connection forward. In II we briefly review the contents of [1] and show the crucial role that the continous Toda equation, a D = 3 integrable field theory, [2,3] has in the membrane quantization program. In the final section we quantize the continous Toda equation and establish the relationship between this quantization program and the construction of highest weight representations of W ∞ algebras [10,11]. This enables us to establish the connection between the membrane quantization and representations of W ∞ algebras via the continuous Toda theory. Brief comments about black holes, singleton-field theory and universal string theory are presented at the conclusion. II Based on the observation that the spherical membrane moving in D spacetime dimensions , in the light-cone gauge, is essentially equivalent to a D − 1 Yang-Mills theory, dimensionally reduced to one time dimension, of the SU (∞) group (see Duff [20] for a review) we look for solutions of the D = 10 Yang-Mills equations (dimensionally-reduced to one temporal dimension).