On the Exact Quantum Integrability of the Membrane

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∞ ⊕ W̄∞ 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. PACS : 0465.+e; 02.40.+m