A Moyal Quantization of the Continuous Toda Field

TL;DR

This paper develops a Moyal deformation framework for the continuous Toda field theory, linking it to the quantization of self-dual membranes and gravity via Weyl-Wigner-Moyal techniques.

Contribution

It introduces a novel Moyal quantization approach for the continuous Toda theory connected to self-dual membranes and 4D gravity.

Findings

Moyal deformations of the 3D Toda field equation are constructed.

The approach relates to the quantization of $SU( abla)$ Nahm equations.

Provides a new perspective on noncommutative geometry in gravity theories.

Abstract

Since the lightcone self dual spherical membrane, moving in flat target backgrounds, has a direct correspondence with the $SU(\infty)$ Nahm equations and the continuous Toda theory, we construct the Moyal deformations of the self dual membrane in terms of the Moyal deformations of the continuous Toda theory. This is performed by using the Weyl-Wigner-Moyal quantization technique 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 associated with 4D Self Dual Gravity.