Self-dual Einstein Equations Research Articles

On a reduction of the generalized Darboux–Halphen system

The equations for the general Darboux–Halphen system obtained as a reduction of the self-dual Yang–Mills can be transformed to a third-order system which resembles the classical Darboux–Halphen system with a common additive terms. It is shown that the transformed system can be further reduced to a constrained non-autonomous, non-homogeneous dynamical system. This dynamical system becomes homogeneous for the classical Darboux–Halphen case, and was studied in the context of self-dual Einstein's equations for Bianchi IX metrics. A Lax pair and Hamiltonian for this reduced system is derived and the solutions for the system are prescribed in terms of hypergeometric functions.

Reductions of self-dual Einstein equations

Transformations between the Pleba´nski equation and other reductions of the selfdual Einstein equations are given. A new reduction is presented. It consists of two equations describing canonical transformations in a 2-dimensional phase space. Symmetries and examples of linearizations of these equations are given.

Self-dual metrics in Husain's approach

We show that Husain's reduction of the self-dual Einstein equations is equivalent to the Plebański equation. The Bäcklund transformation between these equations is found. Contact symmetries of the Husain–Park equation and corresponding conservation laws are derived.

Twistor bundles, Einstein equations and real structures

We consider S2 bundles \U0001d4ab and \U0001d4ab' of totally null planes of maximal dimension and opposite self-duality over a four-dimensional manifold equipped with a Weyl or Riemannian geometry. The fibre product \U0001d4ab\U0001d4ab' of \U0001d4ab and \U0001d4ab' is found to be appropriate for the encoding of both the self-dual and the Einstein - Weyl equations for the 4-metric. This encoding is realized in terms of the properties of certain well defined geometrical objects on \U0001d4ab\U0001d4ab'. The formulation is suitable for complex-valued metrics and unifies results for all three possible real signatures. In the purely Riemannian positive-definite case it implies the existence of a natural almost Hermitian structure on \U0001d4ab\U0001d4ab' whose integrability conditions correspond to the self-dual Einstein equations of the 4-metric. All Einstein equations for the 4-metric are also encoded in the properties of this almost Hermitian structure on \U0001d4ab\U0001d4ab'.

The dispersive self-dual Einstein equations and the Toda lattice

The Boyer-Finley equation, or $SU(\infty)$-Toda equation is both a reduction of the self-dual Einstein equations and the dispersionlesslimit of the $2d$-Toda lattice equation. This suggests that there should be a dispersive version of the self-dual Einstein equation which both contains the Toda lattice equation and whose dispersionless limit is the familiar self-dual Einstein equation. Such a system is studied in this paper. The results are achieved by using a deformation, based on an associative $\star$-product, of the algebra $sdiff(\Sigma^2)$ used in the study of the undeformed, or dispersionless, equations.

Self-dual gravity revisited

Reconsidering the harmonic space description of the self-dual Einstein equations, we streamline the proof that all self-dual pure gravitational fields allow a local description in terms of an unconstrained analytic prepotential in harmonic space. Our formulation yields a simple recipe for constructing self-dual metrics starting from any explicit choice of such prepotential; and we illustrate the procedure by producing a metric related to the Taub-NUT solution from the simplest monomial choice of prepotential.

INTEGRABLE FIELD THEORIES DERIVED FROM 4-D SELF-DUAL GRAVITY

We reformulate the self-dual Einstein equation as a trio of differential form equations for simple two-forms. Using them, we can quickly show the equivalence of the theory and 2-D sigma models valued in infinite-dimensional group, which was shown by Park and Husain earlier. We also derive other field theories including the 2-D Higgs bundle equation. This formulation elucidates the relation among these field theories.

Solutions of A∞ Toda equations based on noncompact group SU(1,1) and infinite-dimensional Grassmann manifolds

Special solutions of A∞ Toda equations are constructed based on noncompact group SU(1,1). These solutions are derived from the holomorphic construction of infinite-dimensional nonlinear Grassmann σ models. It is also shown that the solutions give a family of solutions of self-dual Einstein equations in a certain continuum limit.

Self-dual gravity as a two-dimensional theory and conservation laws

Starting from the Ashtekar Hamiltonian variables for general relativity, the self-dual Einstein equations (SDE) may be rewritten as evolution equations for three divergence-free vector fields given on a three-dimensional surface with a fixed volume element. From this general form of the SDE, it is shown how they may be interpreted as the field equations for a two-dimensional field theory. It is further shown that these equations imply an infinite number of non-local conserved currents. A specific way of writing the vector fields allows an identification of the full SDE with those of the two-dimensional chiral model, with the gauge group being the group of area-preserving diffeomorphisms of a two-dimensional surface. This gives a natural Hamiltonian formulation of the SDE in terms of that of the chiral model.

Self-dual gravity and the chiral model.

The self-dual Einstein equation (SDE) is shown to be equivalent to the two dimensional chiral model, with gauge group chosen as the group of area preserving diffeomorphisms of a two dimensional surface. The approach given here leads to an analog of the Plebanski equations for general self-dual metrics, and to a natural Hamiltonian formulation of the SDE, namely that of the chiral model.

Nonlinear Grassmann?-models, Toda equations, and self-dual Einstein equations: supplements to previous papers

We review recent results on the relations between classical solutions of nonlinearσ-models and Toda equations, and point out some relations among Toda equations (two-dimensions), the continuum Toda equation (three-dimensions), and self-dual Einstein equations (four-dimensions).

The Moyal algebra and integrable deformations of the self-dual Einstein equations

A two-dimensional chiral model is studied whose gauge potentials are valued in the Moyal algebra. This algebra has recently been shown to be the most general two-index Lie algebra, and contains (either as special cases or as subalgebras) the finite classical algebras as well as the algebras of volume preserving diffeomorphisms of a two-surface. The equations so obtained are a deformation of the self-dual Einstein equations, and solutions may be constructed (using a number of standard constructions) as a power series in the deformation parameter.

SDiff(2) Toda equation — Hierarchy, Tau function, and symmetries

A continuum limit of the Toda lattice field theory, called the SDiff(2) Toda equation, is shown to have a Lax formalism and an infinite hierarchy of higher flows. The Lax formalism is very similar to the case of the self-dual vacuum Einstein equation and its hyper-K\"ahler version, however now based upon a symplectic structure and the group SDiff(2) of area preserving diffeomorphisms on a cylinder $S^1 \times \R$. An analogue of the Toda lattice tau function is introduced. The existence of hidden SDiff(2) symmetries are derived from a Riemann-Hilbert problem in the SDiff(2) group. Symmetries of the tau function turn out to have commutator anomalies, hence give a representation of a central extension of the SDiff(2) algebra.

An action principle for self-dual Yang-Mills and Einstein equations

Abstract We propose actions from which the self-dual Yang-Mills and Einstein equations in four dimensions follow by straightforward variation. This off-shell formulation essentially involves harmonic variables. After reformulation in harmonic space, the self-duality constraints can be derived from actions with the help of Lagrange multipliers. The latter turn out not to describe additional degrees of freedom. The off-shell theories obtained can be quantized in a standard way.

Self-dual gravity as a large- N limit of the 2D non-linear sigma model

The 4D self-dual gravity is shown to be equivalent to a large- N limit of the 2D non-linear sigma model with Wess-Zumino terms only. Infinite dimensional symmetries of the self-dual Einstein equations are derived and shown to form a closed algebra which includes W ∞—a large- N limit of Zamolodchikov's W N algebra. These are interpreted in terms of Penrose's twistor theory.

Some solutions of the self-dual Einstein equation

Some solutions of the self-dual Einstein equation

ℋ -Space with a cosmological constant

It is demonstrated that a real-analytic 3-manifold with Riemannian conformal metric is naturally the conformal infinity of a germ-unique real-analytic 4-manifold with real-analytic Riemannian metric satisfying the self-dual Einstein equations with cosmological constant — 1. Moreover, this result holds if ‘Riemannian’ is replaced in the first case by ‘Lorentzian’ (i. e. signature + - - ) and in the second case by ‘pseudo-Riemannian with signature + + - - ’, or if ‘real-analytic’ is replaced by ‘complex-analytic’ and 'Riemannian’ is replaced by ‘holomorphic’. This provides a cosmological-constant analogue of Newman’s ℋ-space construction (Newman 1976, 1977).