The Generalized Moyal Nahm and Continuous Moyal Toda Equations

Abstract

We present in detail a class of solutions to the $4DSU(\infty )$ Moyal Anti Self Dual Yang Mills equations that are related to $reductions$ of the generalized Moyal Nahm quations using the Ivanova-Popov ansatz. The former yields solutions to the ASDYM/SDYM equations for arbitary gauge groups. A further dimensional reduction yields solutions to the Moyal Anti Self Dual Gravitational equations. The Self Dual Yang Mills /Self Dual Gravity case requires a separate study. SU(2) and $SU(\infty )$ (continuous) Moyal Toda equations are derived and solutions to the latter equations in $implicit$ form are proposed via the Lax-Brockett double commutator formalism . An explicit map taking the Moyal heavenly form (after a rotational Killing symmetry reduction) into the SU(2) Moyal Toda field is found. Finally, the generalized Moyal Nahm equations are conjectured that contain the continuous $SU(\infty )$ Moyal Toda equation after a suitable reduction. Three different embeddings of the three different types of Moyal Toda equations into the Moyal Nahm equations are discussed.

Revised TEX file. 31 pages. The Legendre transform between the Moyal heavenly form and the Moyal Toda field is solved

References in corpus (6)

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• A Moyal Quantization of the Continuous Toda Field
• Dispersionless Hierarchies, Hamilton-Jacobi Theory and Twistor Correspondences
• Matrix Membranes and Integrability
• Deformations of classical geometries and integrable systems

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• p-Branes from Generalized Yang-Mills Theory
• On the Moyal deformation of Nahm Equations in seven dimensions
• Self-dual gravity via Hitchin's equations