1996/05/06 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/9605029
35 pages; Revised Tex file; some minor details have been added
arxiv created 1996/05/17 · arxiv updated 2009/11/30
The exact quantum integrability problem of the membrane is investigated. It is found that the spherical membrane moving in flat target spacetime backgrounds is an exact quantum integrable system for a particular class of solutions of the light-cone gauge equations of motion : a dimensionally-reduced SU(∞) Yang-Mills theory to one temporal dimension. Crucial ingredients are the exact integrability property of the 3D~SU(∞) continuous Toda theory and its associated dimensionally-reduced SU(∞) Toda molecule equation whose symmetry algebra is the U_∞ algebra obtained from a dimensional-reducion of the W_∞ ⊕ W_∞ algebras that act naturally on the original 3D continuous Toda theory. The U_∞ algebra is explicitly constructed in terms of exact quantum solutions of the quantized continuous Toda equation. Highest weight irreducible representations of the W_∞ algebras are also studied in detail. Continuous and discrete energy levels are both found in the spectrum . Other relevant topics are discussed in the conclusion.