Killing spinor initial data sets

A 3+1 decomposition of the twistor and valence-2 Killing spinor equation is made using the space-spinor formalism. Conditions on initial data sets for the Einstein vacuum equations are given so that their developments contain solutions to the twistor and/or Killing equations. These lead to the notions of twistor and Killing spinor initial data. These notions are used to obtain a characterisation of initial data sets whose developments are of Petrov type N or D.     

Symplectic twistor spaces

The paper describes the geometry of the bundle (M, ω) of the compatible complex structures of the tangent spaces of an (almost) symplectic manifold (M, ω), from the viewpoint of general twistor spaces [3], [9], [1]. It is shown that M has an either complex or almost Kaehler twistor space iff it has a flat symplectic connection. Applications of the twistor space to the study of the differential forms of M, and to the study of mappings : N → M, where N is a Kaehler manifold are indicated.     

Stability of twistor lifts for surfaces in four-dimensional manifolds as harmonic sections

We prove that the twistor lifts of certain twistor holomorphic surfaces in four-dimensional manifolds are weakly stable harmonic sections. As a corollary, if ambient spaces are self-dual Einstein manifolds with nonnegative scalar curvature, then the twistor lifts of twistor holomorphic surfaces are weakly stable. Moreover, for certain surfaces in four-dimensional hyperkähler manifolds, we show that the surfaces are twistor holomorphic if their twistor lifts are weakly stable harmonic sections. In particular, we characterize twistor holomorphic surfaces in four-dimensional Euclidean space by weak stability of the twistor lifts.     

On the Hyperkähler/Quaternion Kähler Correspondence

A hyperkähler manifold with a circle action fixing just one complex structure admits a natural hyperholomorphic line bundle. This observation forms the basis for the construction of a corresponding quaternionic Kähler manifold in the work of A.Haydys. In this paper the corresponding holomorphic line bundle on twistor space is described and many examples computed, including monopole and Higgs bundle moduli spaces. Finally a twistor version of the hyperkähler/quaternion Kähler correspondence is established.     

On Spin(7) holonomy metric based on SU(3)/U(1): II

We continue the investigation of Spin(7) holonomy metric of cohomogeneity one with the principal orbit SU(3)/U(1). A special choice of U(1) embedding in SU(3) allows more general metric ansatz with five metric functions. There are two possible singular orbits in the first-order system of Spin(7) instanton equation. One is the flag manifold SU(3)/T² also known as the twistor space of CP(2) and the other is CP(2) itself. Imposing a set of algebraic constraints, we find a two-parameter family of exact solutions which have SU(4) holonomy and are asymptotically conical. There are two types of asymptotically locally conical (ALC) metrics in our model, which are distinguished by the choice of S¹ circle whose radius stabilizes at infinity. We show that this choice of M theory circle selects one of the possible singular orbits mentioned above. Numerical analyses of solutions near the singular orbit and in the asymptotic region support the existence of two families of ALC Spin(7) metrics: one family consists of deformations of the Calabi hyper-Kähler metric, the other is a new family of metrics on a line bundle over the twistor space of CP(2).     

Projective geometry,Lagrangian subspaces,and twistor theory

The use of projective geometry for the characterization of Lagrangian subspaces and maps among them is of particular interest for the symplectic manifold that is twistor space. We raise some conjectures on how these should be interpreted on the space-time manifold by making use of the structure of projective twistor space.     

Coupling solutions of BGG-equations in conformal spin geometry

BGG-equations are geometric overdetermined systems of partial differential equations (PDEs) on parabolic geometries. Normal solutions of BGG-equations are particularly interesting, and we give a simple formula for the necessary and sufficient additional integrability conditions on a solution. We then discuss a procedure for coupling known solutions of BGG-equations to produce new ones. Employing a suitable calculus for conformal spin structures, this yields explicit coupling formulas and conditions between almost Einstein scales, conformal Killing forms, and twistor spinors. Finally, we discuss a class of generic twistor spinors that provides an invariant decomposition of conformal Killing fields.     

The Theory of H-space

The theory of $\text{H}$-space, the four-dimensional manifold of those complex null hypersurfaces of an asymptotically flat space-time which are asymptotically shear-free, is reviewed.In addition to a discussion of the origins of the theory, we present two independent formalisms for the derivation of the basic properties of $\text{H}$-space: that it is endowed with a natural holomorphic complex Riemannian metric which satisfies the vacuum Einstein equations and whose Weyl tensor is self-dual.We show the connection of our work on $\text{H}$-space to that of Plebanski and to the theory of deformed twistor spaces, due to Penrose.Finally, there is a discussion of equations of motion in $\text{H}$-space.     

New cohomogeneity one metrics with Spin(7) holonomy

We construct new explicit non-singular metrics that are complete on non-compact Riemannian 8-manifolds with holonomy Spin(7). One such metric, which we denote by , is complete and non-singular on . The other complete metrics are defined on manifolds with the topology of the bundle of chiral spinors over S⁴, and are denoted by , and . The metrics on and occur in families with a non-trivial parameter. The metric on arises for a limiting value of this parameter, and locally this metric is the same as the one for . The new Spin(7) metrics are asymptotically locally conical (ALC): near infinity they approach a circle bundle with fibres of constant length over a cone whose base is the squashed Einstein metric on . We construct the covariantly constant spinor and calibrating 4-form. We also obtain an L²-normalisable harmonic 4-form for the manifold, and two such 4-forms (of opposite dualities) for the manifold.     

Twistor spinors on Lorentzian symmetric spaces

An indecomposable Riemannian symmetric space which admits non-trivial twistor spinors has constant sectional curvature. Furthermore, each homogeneous Riemannian manifold with parallel spinors is flat. In the present paper we solve the twistor equation on all indecomposable Lorentzian symmetric spaces explicitly. In particular, we show that there are — in contrast to the Riemannian case — indecomposable Lorentzian symmetric spaces with twistor spinors, which have non-constant sectional curvature and non-flat and non-Ricci flat homogeneous Lorentzian manifolds with parallel spinors.     

On the twistor space of a quaternionic contact manifold

In this note, we prove that the CR manifold induced from the canonical parabolic geometry of a quaternionic contact (qc) manifold via a Fefferman-type construction is equivalent to the CR twistor space of the qc manifold defined by O. Biquard.     

A class of Einstein–Weyl spaces associated to an integrable system of hydrodynamic type

HyperCR Einstein–Weyl equations in 2+1 dimensions reduce to a pair of quasi-linear PDEs of hydrodynamic type. All solutions to this hydrodynamic system can in principle be constructed from a twistor correspondence, thus establishing the integrability. Simple examples of solutions including the hydrodynamic reductions yield new Einstein–Weyl structures.     

Eigenvalues of the transversal Dirac operator on Kähler foliations

In this paper, we prove Kirchberg-type inequalities for any Kähler spin foliation. Their limiting-cases are then characterized as being transversal minimal Einstein foliations. The key point is to introduce the transversal Kählerian twistor operators.     

Geometrical aspects of Schlesinger''s equation

The equation (Schlesinger's equation) for the isomonodromic deformations of an (SL (2, C) connection with four simple poles on the projective line is shown to describe a holomorphic projective structure on a surface. The space of geodesics of this structure is, by a primitive version of twistor theory, a two-dimensional complex Poisson manifold containing complete rational curves. The Poisson structure degenerates on a divisor and it is shown that the complement of the divisor is a symplectic manifold which can be identified with the quotient of the moduli space of representations of a free group on three generators in SL (2, ) by the action of a braid group.     

Hyper-Kähler Hierarchies and Their Twistor Theory

A twistor construction of the hierarchy associated with the hyper-K?hler 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?hler 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-K?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. Received: 27 January 2000 / Accepted: 20 March 2000     

Moduli Space of IIB Superstring and SUYM in AdS5 × S5

This paper consists of two parts. In part I, we interpret the hidden symmetry of the moduli space of IIB superstring on AdS₅×S⁵ in terms of the chiral embedding in AdS₅, which turns out to be the CP³ conformal affine Toda model. We review how the position μ of poles in the Riemann-Hilbert formulation of dressing transformation and the value of loop parameter μ in the vertex operator of affine algebra determine the moduli space of the soliton solutions, which describes the moduli space of the Green-Schwarz superstring. We show also how this affine SU(4) symmetry affinizes the conformal symmetry in the twistor space, and how a soliton string corresponds to a Robinson congruence with twist and dilation spin coefficients μ of twistor. In part II, by extending the dressing symmetric action of IIB string in AdS₅×S⁵ to the D₃ brane, we find a gauged WZW action of Higgs Yang-Mills field including the 2-cocycle of axially anomaly. The left and right twistor structures of left and right α-planes glue into an ambitwistor. The symmetry group of Nahm equations is centrally extended to an affine group, thus we explain why the spectral curve is given by affine Toda.     

Eingenvalues of the Dirac operator on compact Kähler manifolds

Kählerian twistor operators are introduced to get lower bounds for the eigenvalues of the Dirac operator on compact spin Kähler manifolds. In odd complex dimensions, manifolds with the smallest eigenvalues are characterized by an over determined system of differential equations similar to the Riemannian case. In these dimensions, we show the existence of a unique natural Kählerian twistor operator. It is also proved that, on a Kähler manifold with nonzero scalar curvature, the space of Riemannian twistor-spinors is trivial.This work has been partially supported by the EEC programme GADGET Contract Nr. SC1-0105