2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/149344The 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 $SU(\infty)$ Yang-Mills theory to one temporal dimension. Crucial ingredients are the exact integrability property of the $3D~SU(\infty)$ continuous Toda theory and its associated dimensionally-reduced $SU(\infty)$ Toda $molecule$ equation whose symmetry algebra is the $U_\infty$ algebra obtained from a dimensional-reducion of the $W_\infty \oplus {\bar W}_\infty$ algebras that act naturally on the original $3D$ continuous Toda theory. The $U_\infty$ algebra is explicitly constructed in terms of exact quantum solutions of the quantized continuous Toda equation. Highest weight irreducible representations of the $W_\infty$ 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.35 pages; Revised Tex file; some minor details have been addedHigh Energy Physics - Theorytext