SU(∞) q-MOYAL-NAHM EQUATIONS AND QUANTUM DEFORMATIONS OF THE SELF DUAL MEMBRANE

ABSTRACT Since the lightcone self dual spherical membrane, moving in flat target backgrounds, has a direct correspondence with the SU(∞) Nahm equations and the continuous Toda theory, we construct the quantum/Moyal deformations of the self dual membrane in terms of the q-Moyal star product . The q deformations of the SU(∞) Nahm equations are studied and explicit solutions are given. The continuum limit of the q Toda chain equations are obtained furnishing q deformations of the self dual membrane. Finally, the continuum Moyal-Toda chain equation is embedded into the SU(∞) Moyal-Nahm equations, rendering the relation with the Moyal deformations of the self dual membrane. W∞ and q-W∞ algebras arise as the symmetry algebras and the role of ( the recently developed ) quantum Lie algebras associated with quantized universal enveloping algebras is pointed out pertaining the formulation of a q Toda theory. We review as well the Weyl-Wigner-Moyal quantization of the 3D continuous Toda field equation, and its associated 2D continuous Toda molecule, based on Moyal deformations of rotational Killing symmetry reductions of Plebanski first heavenly equation.