The Generalized Moyal Nahm and Continuous Moyal Toda Equations

TL;DR

This paper explores solutions to 4D SU(∞) Moyal ASDYM equations, deriving new Moyal Toda equations, and establishing links between Moyal Nahm equations and Toda systems through reductions and ansatz methods.

Contribution

It introduces a generalized Moyal Nahm framework that encompasses continuous SU(∞) Moyal Toda equations and provides explicit solution mappings and embeddings.

Findings

Derived SU(2) and SU(∞) Moyal Toda equations.

Proposed implicit solutions using Lax-Brockett formalism.

Established a conjecture linking Moyal Nahm and Toda equations.

Abstract

We present in detail a class of solutions to the $4D SU(\infty)$ 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.