The Lagrangian of a Self-Dual Gravitational Field as a Limit of the SDYM Lagrangian

TL;DR

This paper demonstrates how the action for su(N) self-dual Yang-Mills equations converges to a six-dimensional heavenly equation in the large N limit, exploring symmetry reductions and Moyal deformations relevant to self-dual gravity.

Contribution

It establishes a connection between SDYM actions and heavenly equations via a large N limit, including symmetry reductions and deformations.

Findings

SDYM action converges to six-dimensional heavenly equation as N approaches infinity

Symmetry reductions recover known self-dual gravity equations

Moyal deformation extends the framework to noncommutative geometries

Abstract

The action for the su(N) SDYM equations is shown to give in the limit $N \to \infty$ the action for the six-dimensional version of the second heavenly equation. The symmetry reductions of this latter equation to the well known equations of self-dual gravity are given. The Moyal deformation of the heavenly equations are also considered.