Advertisement

Communications in Mathematical Physics

, Volume 213, Issue 3, pp 641–672 | Cite as

Hyper-Kähler Hierarchies and Their Twistor Theory

  • Maciej Dunajski
  • Lionel J. Mason

Abstract:

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 ? is constructed whose extra dimensions correspond to higher flows of the hierarchy. It is shown that ? is a moduli space of rational curves with normal bundle ?(n)⊕?(n) in twistor space and is canonically equipped with a Lax distribution for ASDVE hierarchies. The space ? 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.

Keywords

Rational Function Modulus Space High Flow Extra Dimension Symplectic Form 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Maciej Dunajski
    • 1
  • Lionel J. Mason
    • 1
  1. 1.The Mathematical Institute, 24-29 St Giles, Oxford OX1 3LB, UK. E-mail: dunajski@maths.ox.ac.ukGB

Personalised recommendations