2000/01/03 by Maciej Dunajski, Lionel Mason, Lionel J. Mason · 2 citations
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Black Holes and Theoretical Physics #Bundle #Hamiltonian (control theory) #Mathematical analysis #Mathematical physics #Mathematics #Moduli space #Nonlinear Waves and Solitons #Pure mathematics #Space (punctuation) #Symplectic geometry #Twistor space #Twistor theory #gr-qc #math.DG #msc:35Q58 #msc:53B35 #nlin.SI
paper · pdf · doi:10.1007/pl00005532
published as Commun.Math.Phys. 213 (2000) 641-672 · 23 pages, 1 figure
arxiv created 2000/01/03 · openalex publication_date 2000/10/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
A twistor construction of the hierarchy associated with the hyper-Kähler equations on a metric (the anti-self-dual Einstein vacuum equations, ASDVE, in four dimensions) is given. The recursion operator R is constructed and used to build an infinite-dimensional symmetry algebra and in particular higher flows for the hyper-Kähler equations. It is shown that R acts on the twistor data by multiplication with a rational function. The structures are illustrated by the example of the Sparling-Tod (Eguchi-Hansen) solution. An extended space-time \cal N is constructed whose extra dimensions correspond to higher flows of the hierarchy. It is shown that \cal N is a moduli space of rational curves with normal bundle \cal O(n)⊕\cal O(n) in twistor space and is canonically equipped with a Lax distribution for ASDVE hierarchies. The space \cal N is shown to be foliated by four dimensional hyper-Kähler slices. The Lagrangian, Hamiltonian and bi-Hamiltonian formulations of the ASDVE in the form of the heavenly equations are given. The symplectic form on the moduli space of solutions to heavenly equations is derived, and is shown to be compatible with the recursion operator.