Hyper-Kähler Hierarchies and their twistor theory

A twistor construction of the hierarchy associated with the hyper-Kahler 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 innite-dimensional symmetry algebra and in particular higher ows for the hyper-Kahler 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 N is constructed whose extra dimensions correspond to higher ows of the hierarchy. It is shown that N is a moduli space of rational curves with normal bundle O(n) O(n) in twistor space and is canonically equipped with a Lax distribution for ASDVE hierarchies. The space N is shown to be foliated by four dimensional hyper-Kahler 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.