Twistor Spinors with Zero on Lorentzian 5-Space

We present in this paper a C ¹-metric of Lorentzian signature (1,4) on an open neighbourhood of the origin in \(\mathbb{R}^{5}\), which admits a solution to the twistor equation for spinors with a unique isolated zero at the origin. The metric is not conformally flat in any neighbourhood of the origin and the associated conformal Killing vector to the twistor generates a one-parameter group of essential conformal transformations. The construction is based on the Eguchi-Hanson metric in dimension 4.     

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.     

The twistor equation on Riemannian manifolds

It is shown that a twistor spinor on a Riemannian manifold defines a conformal deformation to an Einstein manifold. Twistor spinors on 4-manifolds are considered. A characterization of the hyperbolic space is given. Moreover the solutions of the twistor equation on warped products Mⁿ × , where Mⁿ is an Einstein manifold, are described.     

Anti-self-dual Conformal Structures with Null Killing Vectors from Projective Structures

Using twistor methods, we explicitly construct all local forms of four–dimensional real analytic neutral signature anti–self–dual conformal structures (M, [g]) with a null conformal Killing vector. We show that M is foliated by anti-self-dual null surfaces, and the two-dimensional leaf space inherits a natural projective structure. The twistor space of this projective structure is the quotient of the twistor space of (M, [g]) by the group action induced by the conformal Killing vector. We obtain a local classification which branches according to whether or not the conformal Killing vector is hyper-surface orthogonal in (M, [g]). We give examples of conformal classes which contain Ricci–flat metrics on compact complex surfaces and discuss other conformal classes with no Ricci–flat metrics. Dedicated to the memory of Jerzy Plebański     

Supersymmetry in Lorentzian Curved Spaces and Holography

We consider superconformal and supersymmetric field theories on four-dimensional Lorentzian curved space-times, and their five-dimensional holographic duals. As in the Euclidean signature case, preserved supersymmetry for a superconformal theory is equivalent to the existence of a charged conformal Killing spinor. Differently from the Euclidean case, we show that the existence of such spinors is equivalent to the existence of a null conformal Killing vector. For a supersymmetric field theory with an R-symmetry, this vector field is further restricted to be Killing. We demonstrate how these results agree with the existing classification of supersymmetric solutions of minimal gauged supergravity in five dimensions.     

Killing spinors for the bosonic string and Kaluza–Klein theory with scalar potentials

The paper consists mainly of two parts. In the first part, we obtain well-defined Killing spinor equations for the low-energy effective action of the bosonic string with the conformal anomaly term. We show that the conformal anomaly term is the only scalar potential that one can add into the action that is consistent with the Killing spinor equations. In the second part, we demonstrate that Kaluza–Klein theory can be gauged so that the Killing spinors are charged under the Kaluza–Klein vector. This gauging process generates a scalar potential with a maximum that gives rise to an AdS spacetime. We also construct solutions of these theories.     

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.     

Twistor forms on Riemannian products

We study twistor forms on products of compact Riemannian manifolds and show that they are defined by Killing forms on the factors. The main result of this note is a necessary step in the classification of compact Riemannian manifolds with non-generic holonomy carrying twistor forms.     

Hidden supersymmetry of domain walls and cosmologies

We show that all domain-wall solutions of gravity coupled to scalar fields for which the world-volume geometry is Minkowski or anti-de Sitter admit Killing spinors, and satisfy corresponding first-order equations involving a superpotential determined by the solution. By analytic continuation, all flat or closed Friedmann-Lema?tre-Robertson-Walker cosmologies are shown to satisfy similar first-order equations arising from the existence of "pseudo Killing" spinors.     

Spinor methods in conformal Killing transport

The equations of conformal Killing transport are discussed using tensor and spinor methods. It is shown that, in Minkowski space-time, the equations for a null conformal Killing vector ξ ^a are completely determined by the corresponding spinor ω ^A and its covariant derivative, which defines a spinor π _A′ . In conformally flat space-time, the covariant derivative of π _A′ is also involved. Some applications to twistor theory are briefly mentioned.     

Twistor Geometry of a Pair of Second Order ODEs

We discuss the twistor correspondence between path geometries in three dimensions with vanishing Wilczynski invariants and anti-self-dual conformal structures of signature (2, 2). We show how to reconstruct a system of ODEs with vanishing invariants for a given conformal structure, highlighting the Ricci-flat case in particular. Using this framework, we give a new derivation of the Wilczynski invariants for a system of ODEs whose solution space is endowed with a conformal structure. We explain how to reconstruct the conformal structure directly from the integral curves, and present new examples of systems of ODEs with point symmetry algebra of dimension four and greater which give rise to anti–self–dual structures with conformal symmetry algebra of the same dimension. Some of these examples are (2, 2) analogues of plane wave space–times in General Relativity. Finally we discuss a variational principle for twistor curves arising from the Finsler structures with scalar flag curvature.     

7-Dimensional compact Riemannian manifolds with Killing spinors

Using a link between Einstein-Sasakian structures and Killing spinors we prove a general construction principle of odd-dimensional Riemannian manifolds with real Killing spinors. In dimensionn=7 we classify all compact Riemannian manifolds with two or three Killing spinors. Finally we classify nonflat 7-dimensional Riemannian manifolds with parallel spinor fields.     

Twistor theory: An approach to the quantisation of fields and space-time

Twistor theory offers a new approach, starting with conformally-invariant concepts, to the synthesis of quantum theory and relativity. Twistors for flat space-time are the SU(2,2) spinors of the twofold covering group O(2,4) of the conformal group. They describe the momentum and angular momentum structre of zero-rest-mass particles. Space-time points arise as secondary concepts corresponding to linear sets in twistor space. They, rather than the null cones, should become “smeared out” on passage to a quantised gravitational theory. Twistors are represented here in two-component spinor terms. Zero-rest-mass fields are described by holomorphic functions on twistor space, on which there is a natural canonical structure leading to a natural choice of canonical quantum operators. The generalisation to curved space can be accomplished in three ways; i) local twistors, a conformally invariant calculus, ii) global twistors, and iii) asymptotic twistors which provide the framework for an S-matrix approach in asymptotically flat space-times. A Hamiltonian scattering theory of global twistors is used to calculate scattering cross-sections. This leads to twistor analogues of Feynman graphs for the treatment of massless quantum electrodynamics. The recent development of methods for dealing with massive (conformal symmetry breaking) sources and fields is briefly reviewed.     

Conformal angular momentum and lie derivative of spinor field in Riemann space

It is shown that in Riemann space—time the conformal angular momentum and Lie derivative of spinors differ only by a part involving multiplication by a factor from the conformal Killing equations.Yakutsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 99–101, November, 1994.     

Homothetic and conformal symmetries of solutions to Einstein's equations

We present several results about the nonexistence of solutions of Einstein's equations with homothetic or conformal symmetry. We show that the only spatially compact, globally hyperbolic spacetimes admitting a hypersurface of constant mean extrinsic curvature, and also admitting an infinitesimal proper homothetic symmetry, are everywhere locally flat; this assumes that the matter fields either obey certain energy conditions, or are the Yang-Mills or massless Klein-Gordon fields. We find that the only vacuum solutions admitting an infinitesimal proper conformal symmetry are everywhere locally flat spacetimes and certain plane wave solutions. We show that if the dominant energy condition is assumed, then Minkowski spacetime is the only asymptotically flat solution which has an infinitesimal conformal symmetry that is asymptotic to a dilation. In other words, with the exceptions cited, homothetic or conformal Killing fields are in fact Killing in spatially compact or asymptotically flat spactimes. In the conformal procedure for solving the initial value problem, we show that data with infinitesimal conformal symmetry evolves to a spacetime with full isometry.     

Complete Classification of Conformal Killing Symmetries of Plane-Symmetric Static Spacetimes in Teleparallel Theory

We have studied the conformal, homothetic and Killing vectors in the context of teleparallel theory of gravitation for plane-symmetric static spacetimes. We have solved completely the non-linear coupled teleparallel conformal Killing equations. This yields the general form of teleparallel conformal vectors along with the conformal factor for all possible cases of metric functions. We have found four solutions which are divided into one Killing symmetries and three conformal Killing symmetries. One of these teleparalel conformal vectors depends on x only and other is a function of all spacetime coordinates. The three conformal Killing symmetries contain three proper homothetic symmetries where the conformal factor is an arbitrary non-zero constant.     

Arbitrary dimensional cosmological multi-black holes

We find cosmological black hole solutions for spacetimes of arbitrary dimension (greater than three) that are asymptotically de Sitter, and we show that these solutions can be extended to give multi-black hole solutions. We investigate the motion of a charged massive test particle in a five-dimensional extreme Reissner-Nordström de Sitter background. Furthermore we obtain Killing spinors for Reissner-Nordström de Sitter spacetimes. We also find five-dimensional cosmological black hole solutions in an asymptotically anti de Sitter spacetime and we show that these solutions are supersymmetric in the sense that they admit a supercovariantly constant spinor.