1997/04/06 by Carlos Castro, Castro, Carlos
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Nonlinear Waves and Solitons #Quantum Mechanics and Non-Hermitian Physics #hep-th
paper · pdf · doi:10.48550/arxiv.hep-th/9704031
25 pages. Revised tex file. More references have been included and minor corrections
openalex publication_date 1997/04/06 · arxiv created 1997/04/08 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.