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 eq...
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SpringerOpen
2025-03-01
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| Series: | Journal of High Energy Physics |
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| Online Access: | https://doi.org/10.1007/JHEP03(2025)152 |
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| author | Kirill Krasnov Arthur Lipstein |
| author_facet | Kirill Krasnov Arthur Lipstein |
| author_sort | Kirill Krasnov |
| collection | DOAJ |
| description | 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 ℝ 4 endowed with a complex structure. |
| format | Article |
| id | doaj-art-ced0b53d6dab4d48a549687436161fcb |
| institution | OA Journals |
| issn | 1029-8479 |
| language | English |
| publishDate | 2025-03-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of High Energy Physics |
| spelling | doaj-art-ced0b53d6dab4d48a549687436161fcb2025-08-20T02:11:55ZengSpringerOpenJournal of High Energy Physics1029-84792025-03-012025311710.1007/JHEP03(2025)152Pure connection formalism and Plebanski’s second heavenly equationKirill Krasnov0Arthur Lipstein1School of Mathematical Sciences, University of NottinghamDepartment of Mathematical Sciences, Durham UniversityAbstract 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 ℝ 4 endowed with a complex structure.https://doi.org/10.1007/JHEP03(2025)152de Sitter spaceClassical Theories of GravityIntegrable Field Theories |
| spellingShingle | Kirill Krasnov Arthur Lipstein Pure connection formalism and Plebanski’s second heavenly equation Journal of High Energy Physics de Sitter space Classical Theories of Gravity Integrable Field Theories |
| title | Pure connection formalism and Plebanski’s second heavenly equation |
| title_full | Pure connection formalism and Plebanski’s second heavenly equation |
| title_fullStr | Pure connection formalism and Plebanski’s second heavenly equation |
| title_full_unstemmed | Pure connection formalism and Plebanski’s second heavenly equation |
| title_short | Pure connection formalism and Plebanski’s second heavenly equation |
| title_sort | pure connection formalism and plebanski s second heavenly equation |
| topic | de Sitter space Classical Theories of Gravity Integrable Field Theories |
| url | https://doi.org/10.1007/JHEP03(2025)152 |
| work_keys_str_mv | AT kirillkrasnov pureconnectionformalismandplebanskissecondheavenlyequation AT arthurlipstein pureconnectionformalismandplebanskissecondheavenlyequation |