Pure connection formalism and Plebanski’s second heavenly equation

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.