The Nonlinear Graviton as an Integrable System

The curved twistor theory is studied from the point of view of integrable systems. A twistor construction of the hierarchy associated with the anti-self-dual Einstein vacuum equations (ASDVE) is given. The recursion operator R is constructed and used to build an infinite-dimensional symmetry algebra of ASDVE. It is proven that R acts on twistor functions by multiplication. The recursion operator is used to construct Killing spinors. The method is illustrated on the example of the Sparling-Tod solution. An infinite number of commuting flows on extended space-time is constructed. It is proven that a moduli space of rational curves, with normal bundle O(n) ⊕ O(n) in twistor space, is canonically equipped with a Lax distribution for ASDVE hierarchies. It is demonstrated that the isomonodromy problem can, in the Fuchsian case, be understood in terms of curved twistor spaces. The solutions to the SL(2, C) Schlesinger equation are related to the flows of the heavenly hierarchy. The Lagrangian, Hamiltonian and bi-Hamiltonian formulations of heavenly equations are given. The symplectic form on the moduli space of solutions to heavenly equations is derived, and is proven to be compatible with the recursion operator. It is proven that a family of rational curves in the twistor space may be found by integrating the Hamiltonian system which has the second heavenly potential as its Hamiltonian. An alternative view of heavenly potentials as generating functions on the spin bundle is given. The potentials for linear fields on ASD vacuum backgrounds are constructed. It is shown that generalised zero–rest–mass field equations can be solved by means of functions on O(n) ⊕ O(n) twistor spaces. The moduli space of deformed O(n) ⊕ O(n) curves is shown to be foliated by four dimensional hyper-Kahler slices. The twistor theory of four-dimensional hyper-Hermitian manifolds is formulated as a combination of the Nonlinear Graviton Construction with the Ward transform for anti-self-dual Maxwell fields. The Lax formulation is found and used to derive a pair of potentials for a hyper-Hermitian metric. A class of examples of hyper-Hermitian metrics which depend on two arbitrary functions of two complex variables is given. The ASDV metrics with a conformal, non-triholomorphic Killing vector are considered. The symmetric solutions to the first heavenly equation are shown to give rise to a new integrable system in three dimensions, and to a new class of Einstein–Weyl geometries. The Lax representation, Lie point symmetries, hidden symmetries and the recursion operator associated with the reduced 3D system are found, and some group invariant solutions are considered. It is proven that if an Einstein–Weyl space admits a solution of a generalised monopole equation, which yields four dimensional ASD vacuum, or Einstein metrics, then the four-dimensional correspondence space is equipped with a closed and simple two-form. A class of Einstein–Weyl structures is given in terms of solutions to the dispersion-less Kadomtsev–Petviashvili equation. It is explained how to construct ASDVE metrics from solutions of various 2D integrable systems by exploiting the fact that the Lax formulations of both systems can be embedded in that of the anti-self-dual Yang–Mills equations. The explicit ASDVE metrics are constructed on R^2 × Σ, where Σ is a homogeneous space for a real subgroup of SL(2, C) associated with the two-dimensional system. The twistor interpretation of the construction is given.