Yano Tensors Research Articles

Symmetries of the Dirac operator with skew-symmetric torsion

In this paper, we consider the symmetries of the Dirac operator derived from a connection with skew-symmetric torsion, ∇T. We find that the generalized conformal Killing–Yano tensors give rise to symmetry operators of the massless Dirac equation, provided an explicitly given anomaly vanishes. We show that this gives rise to symmetries of the Dirac operator in the case of strong Kähler with torsion (KT) and strong hyper-Kähler with torsion (HKT) manifolds.

SEPARABILITY OF GRAVITATIONAL PERTURBATION IN GENERALIZED KERR–NUT–DE SITTER SPACE–TIME

Generalized Kerr–NUT–de Sitter space–time is the most general space–time which admits a rank-2 closed conformal Killing–Yano tensor. It contains the higher-dimensional Kerr–de Sitter black holes with partially equal angular momenta. We study the separability of gravitational perturbations in the generalized Kerr–NUT–de Sitter space–time. We show that a certain type of tensor perturbations admits the separation of variables. The linearized perturbation equations for the Einstein condition are transformed into the ordinary differential equations of Fuchs type.

Killing–Yano tensors and multi-Hermitian structures

We show that the Euclidean Kerr–NUT-(A)dS metric in 2 m dimensions locally admits 2 m Hermitian complex structures. These are derived from the existence of a non-degenerate closed conformal Killing–Yano tensor with distinct eigenvalues. More generally, a conformal Killing–Yano tensor, provided its exterior derivative satisfies a certain condition, algebraically determines 2 m almost complex structures that turn out to be integrable as a consequence of the conformal Killing–Yano equations. In the complexification, these lead to 2 m maximal isotropic foliations of the manifold and, in Lorentz signature, these lead to two congruences of null geodesics. These are not shear-free, but satisfy a weaker condition that also generalises the shear-free condition from four dimensions to higher dimensions. In odd dimensions, a conformal Killing–Yano tensor leads to similar integrable distributions in the complexification. We show that the recently discovered five-dimensional solution of Lü, Mei and Pope also admits such integrable distributions, although this does not quite fit into the story as the obvious associated two-form is not conformal Killing–Yano. We give conditions on the Weyl curvature tensor imposed by the existence of a non-degenerate conformal Killing–Yano tensor; these give an appropriate generalisation of the type D condition on a Weyl tensor from four dimensions.

HIGHER ORDER FIRST INTEGRALS OF MOTION IN A GAUGE COVARIANT HAMILTONIAN FRAMEWORK

The higher order symmetries are investigated in a covariant Hamiltonian formulation. The covariant phase-space approach is extended to include the presence of external gauge fields and scalar potentials. The special role of the Killing–Yano tensors is pointed out. Some nontrivial examples involving Runge–Lenz type conserved quantities are explicitly worked out.

Generalized Killing–Yano equations in [formula omitted] gauged supergravity

We propose a generalization of the (conformal) Killing–Yano equations relevant to D=5 minimal gauged supergravity. The generalization stems from the fact that the dual of the Maxwell flux, the 3-form ∗F, couples naturally to particles in the background as a ‘torsion’. Killing–Yano tensors in the presence of torsion preserve most of the properties of the standard Killing–Yano tensors — exploited recently for the higher-dimensional rotating black holes of vacuum gravity with cosmological constant. In particular, the generalized closed conformal Killing–Yano 2-form gives rise to the tower of generalized closed conformal Killing–Yano tensors of increasing rank which in turn generate the tower of Killing tensors. An example of a generalized Killing–Yano tensor is found for the Chong–Cvetič–Lü–Pope black hole spacetime [Z.W. Chong, M. Cvetic, H. Lu, C.N. Pope, hep-th/0506029]. Such a tensor stands behind the separability of the Hamilton–Jacobi, Klein–Gordon, and Dirac equations in this background.

Nonlinear symmetries on spaces admitting Killing tensors

Nonlinear symmetries corresponding to Killing tensors are investigated. The intimate relation between Killing–Yano tensors and non-standard supersymmetries is pointed out. The gravitational anomalies are absent if the hidden symmetry is associated with a Killing–Yano tensor. In the case of the nonlinear symmetries the dynamical algebras of the Dirac-type operators is more involved and could be organized as infinite dimensional algebras or superalgebras. The general results are applied to some concrete spaces involved in theories of modern physics. As a first example it is considered the 4-dimensional Euclidean Taub-NUT space and its generalizations introduced by Iwai and Katayama. One presents the infinite dimensional superalgebra of Dirac type operators on Taub-NUT space that could be seen as a graded loop superalgebra of the Kac-Moody type. The axial anomaly, interpreted as the index of the Dirac operator, is computed for the generalized Taub-NUT metrics. Finally the existence of the conformal Killing–Yano tensors is investigated for some spaces with mixed Sasakian structures.

Stationary spinning strings and symmetries of classical spacetimes

We explore the symmetries of classical stationary spacetimes in terms of the dynamics of a spinning string described by a worldsheet supersymmetric action. We show that for stationary configurations of the string, the action reduces to that for a pseudo-classical spinning point particle in an effective space, which is a conformally scaled quotient space of the original spacetime. As an example, we consider the stationary spinning string in the Kerr–Newman spacetime, whose motion is equivalent to that of the spinning point particle in the three-dimensional effective space. We present the Killing tensor as well as the spin-valued Killing vector of this space. However, the nongeneric supersymmetry corresponding to the Killing–Yano tensor of the Kerr–Newman spacetime is lost in the effective space.

Geodesics and symmetries of doubly spinning black rings

This paper studies various properties of the Pomeransky–Sen'kov doubly spinning black ring spacetime. I discuss the structure of the ergoregion, and then go on to demonstrate the separability of the Hamilton–Jacobi equation for null, zero energy geodesics, which exist in the ergoregion. These geodesics are used to construct geometrically motivated coordinates that cover the black hole horizon. Finally, I relate this weak form of separability to the existence of a conformal Killing tensor in a particular four-dimensional spacetime obtained by Kaluza–Klein reduction, and show that a related conformal Killing–Yano tensor only exists in the singly spinning case.

On the supersymmetric limit of Kerr-NUT-AdS metrics

Generalizing the scaling limit of Martelli and Sparks [D. Martelli, J. Sparks, Phys. Lett. B 621 (2005) 208, hep-th/0505027] into an arbitrary number of spacetime dimensions we re-obtain the (most general explicitly known) Einstein–Sasaki spaces constructed by Chen et al. [W. Chen, H. Lü, C.N. Pope, Class. Quantum Grav. 23 (2006) 5323, hep-th/0604125]. We demonstrate that this limit has a well-defined geometrical meaning which links together the principal conformal Killing–Yano tensor of the original Kerr-NUT-(A)dS spacetime, the Kähler 2-form of the resulting Einstein–Kähler base, and the Sasakian 1-form of the final Einstein–Sasaki space. The obtained Einstein–Sasaki space possesses the tower of Killing–Yano tensors of increasing rank—underlined by the existence of Killing spinors. A similar tower of hidden symmetries is observed in the original (odd-dimensional) Kerr-NUT-(A)dS spacetime. This rises an interesting question whether also these symmetries can be related to the existence of some ‘generalized’ Killing spinor.

Symmetry operators and the separability of massive Klein–Gordon and Dirac equations in the general five-dimensional Kerr–(anti-)de Sitter black hole background

It is shown that the Dirac equation is separable by variables in a five-dimensional rotating Kerr–(anti-)de Sitter black hole with two independent angular momenta. A first-order symmetry operator that commutes with the Dirac operator is constructed in terms of a rank-3 Killing–Yano tensor whose square is a second-order symmetric Stäckel–Killing tensor admitted by the five-dimensional Kerr–(anti-)de Sitter spacetime. We highlight the construction procedure of such a symmetry operator. In addition, the first law of black hole thermodynamics has been extended to the case that the cosmological constant can be viewed as a thermodynamical variable.

Closed conformal Killing–Yano tensor and the uniqueness of generalized Kerr–NUT–de Sitter spacetime

The higher-dimensional Kerr–NUT–de Sitter spacetime describes the general rotating asymptotically de Sitter black hole with NUT parameters. It is known that such a spacetime possesses a rank-2 closed conformal Killing–Yano (CKY) tensor as a ‘hidden’ symmetry which provides the separation of variables for the geodesic equations and Klein–Gordon equations. We present a classification of higher-dimensional spacetimes admitting a rank-2 closed CKY tensor. This provides a generalization of the Kerr–NUT–de Sitter spacetime. In particular, we show that the Kerr–NUT–de Sitter spacetime is the only spacetime with a non-degenerate CKY tensor.

Generalized Kerr–NUT–de Sitter metrics in all dimensions

We classify all spacetimes with a closed rank-2 conformal Killing–Yano tensor. They give a generalization of Kerr–NUT–de Sitter spacetimes. The Einstein condition is explicitly solved and written as an indefinite integral. It is characterized by a polynomial in the integrand. We briefly discuss the smoothness conditions of the Einstein metrics over compact Riemannian manifolds.

Higher-dimensional black holes: hidden symmetries and separation of variables

In this paper, we discuss hidden symmetries in rotating black hole spacetimes. We start with an extended introduction which mainly summarizes results on hidden symmetries in four dimensions and introduces Killing and Killing–Yano tensors, objects responsible for hidden symmetries. We also demonstrate how starting with a principal CKY tensor (that is a closed non-degenerate conformal Killing–Yano 2-form) in 4D flat spacetime one can ‘generate’ the 4D Kerr–NUT–(A)dS solution and its hidden symmetries. After this we consider higher-dimensional Kerr–NUT–(A)dS metrics and demonstrate that they possess a principal CKY tensor which allows one to generate the whole tower of Killing–Yano and Killing tensors. These symmetries imply complete integrability of geodesic equations and complete separation of variables for the Hamilton–Jacobi, Klein–Gordon and Dirac equations in the general Kerr–NUT–(A)dS metrics.

Non-standard supersymmetries on spaces admitting Killing–Yano tensors

We attempt to point out the intimate relation between Killing–Yano tensors and non-standard supersymmetries, along the way reviewing some recent results. The Euclidean Taub–NUT space is taken to illustrate some of the aspects of the general discussion about the connection between spacetime higher order geometrical symmetries and supersymmetries. The dynamical symmetry of this space leads in a natural way to an infinite dimensional twisted loop superalgebra of the conserved operators.

Conformal Yano–Killing tensors in anti-de Sitter spacetime

Conformal rescaling of conformal Yano–Killing tensors and relations between Yano and CYK tensors are discussed. The pullback of these objects to a submanifold is used to construct all solutions of a CYK equation in anti-de Sitter and de Sitter spacetimes. Properties of asymptotic conformal Yano–Killing tensors are examined for asymptotic anti-de Sitter spacetimes. Explicit asymptotic forms of them are derived. The results are used to construct asymptotic charges in asymptotic AdS spacetime.

Closed conformal Killing–Yano tensor and geodesic integrability

Assuming the existence of a single rank-2 closed conformal Killing–Yano tensor with a certain symmetry we show that there exists mutually commuting rank-2 Killing tensors and Killing vectors. We also discuss the condition of separation of variables for the geodesic Hamilton–Jacobi equations.

Closed conformal Killing–Yano tensor and Kerr–NUT–de Sitter space–time uniqueness

We study space–times with a closed conformal Killing–Yano tensor. It is shown that the D-dimensional Kerr–NUT–de Sitter space–time constructed by Chen–Lü–Pope is the only space–time admitting a rank-2 closed conformal Killing–Yano tensor with a certain symmetry.

Do Killing–Yano tensors form a Lie algebra?

Killing–Yano tensors are natural generalizations of Killing vectors. We investigate whether Killing–Yano tensors form a graded Lie algebra with respect to the Schouten–Nijenhuis bracket. We find that this proposition does not hold in general, but that it does hold for constant curvature spacetimes. We also show that Minkowski and (anti)-deSitter spacetimes have the maximal number of Killing–Yano tensors of each rank and that the algebras of these tensors under the SN bracket are relatively simple extensions of the Poincaré and (A)dS symmetry algebras.

IRREDUCIBLE KILLING TENSORS FROM THIRD RANK KILLING–YANO TENSORS

We investigate higher rank Killing–Yano tensors showing that third rank Killing–Yano tensors are not always trivial objects being possible to construct irreducible Killing tensors from them. We give as an example the Kimura IIC metric from two-rank Killing–Yano tensors to obtain a reducible Killing tensor and from third-rank Killing–Yano tensors, we obtain three Killing tensors, one reducible and two irreducible.

Conformal Yano–Killing tensors for the Taub–NUT metric

Symmetric conformal Killing tensors and (skew-symmetric) conformal Yano–Killing tensors for Euclidean Taub–NUT metric are given in an explicit form. Relations between Yano and CYK tensors in terms of conformal rescaling are discussed.