Pure connection formalism and Plebanski’s second heavenly equation

Krasnov, Kirill; Lipstein, Arthur

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.

Citation

Krasnov, K., & Lipstein, A. (2025). Pure connection formalism and Plebanski’s second heavenly equation. Journal of High Energy Physics, 2025(3), Article 152. https://doi.org/10.1007/jhep03%282025%29152

• Journal Article Type: Article
• Acceptance Date: Feb 11, 2025
• Online Publication Date: Mar 20, 2025
• Publication Date: Mar 20, 2025
• Deposit Date: Mar 24, 2025
• Publicly Available Date: Mar 24, 2025
• Journal: Journal of High Energy Physics
• Electronic ISSN: 1029-8479
• Publisher: Springer Verlag
• Peer Reviewed: Peer Reviewed
• Volume: 2025
• Issue: 3
• Article Number: 152
• DOI: https://doi.org/10.1007/jhep03%282025%29152
• Public URL: https://nottingham-repository.worktribe.com/output/46994784
• Publisher URL: https://link.springer.com/article/10.1007/JHEP03(2025)152