Pure connection formalism and Plebanski's second heavenly equation

TL;DR

This paper introduces a new derivation of Plebanski's second heavenly equation for self-dual Einstein metrics using the pure connection formalism, offering insights into the structure of self-dual gravity and Yang-Mills theory.

Contribution

It provides a simplified derivation of Plebanski's equations via the pure connection approach and proposes a novel interpretation of the self-dual Yang-Mills kinematic algebra.

Findings

New derivation of Plebanski's second heavenly equation

Connection between self-dual gravity and complex structures on R4

Insight into the kinematic algebra of self-dual Yang-Mills

Abstract

Plebanski's second heavenly equation reduces the problem of finding a self-dual Einstein metric to solving a non-linear second-order PDE for a single function. Plebanski's original equation is for self-dual metrics obtained as perturbations of the flat metric. Recently, a version of this equation was discovered for self-dual metrics arising as perturbations around a constant curvature background. We provide a new simple derivation of both versions of the Plebanski second heavenly equation. Our derivation relies on the `pure connection' description of self-dual gravity. Our results also suggest a new interpretation to the kinematic algebra of self-dual Yang-Mills theory, as the Lie algebra of (0,1) vector fields on a R4 endowed with a complex structure.