Hyper-K{\"a}hler Hierarchies and their twistor theory

TL;DR

This paper develops a twistor-theoretic framework for hyper-K"ahler hierarchies, constructing recursion operators, symmetry algebras, and extended moduli spaces, with explicit examples and Hamiltonian formulations.

Contribution

It introduces a twistor construction for hyper-K"ahler hierarchies, including recursion operators, extended moduli spaces, and explicit links to heavenly equations.

Findings

Constructed a recursion operator R acting on twistor data.

Built an infinite-dimensional symmetry algebra for hyper-K"ahler equations.

Established a Lax distribution and symplectic structure on the moduli space.

Abstract

A twistor construction of the hierarchy associated with the hyper-K\"ahler 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-K\"ahler 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 ${\cal O}(n)\oplus{\cal O}(n)$ 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-K{\"a}hler 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.