Conformal Killing-Yano Research Articles

Charges and topology in linearised gravity

Covariant conserved 2-form currents for linearised gravity are constructed by contracting the linearised curvature with conformal Killing-Yano tensors. The corresponding conserved charges were originally introduced by Penrose and have recently been interpreted as the generators of generalised symmetries of the graviton. We introduce an off-shell refinement of these charges and find the relation between these improved Penrose charges and the linearised version of the ADM momentum and angular momentum. If the graviton field is globally well-defined on a background Minkowski space then some of the Penrose charges give the momentum and angular momentum while the remainder vanish. We consider the generalisation in which the graviton has Dirac string singularities or is defined locally in patches, in which case the conventional ADM expressions are not invariant under the graviton gauge symmetry in general. We modify them to render them gauge-invariant and show that the Penrose charges give these modified charges plus certain magnetic gravitational charges. We discuss properties of the Penrose charges, generalise to toroidal Kaluza-Klein compactifications and check our results in a number of examples.

Covariant generalized conserved charges of General Relativity

Motivated by the current research of generalized symmetries and the construction of conserved charges in pure Einstein gravity linearized over Minkowski spacetime in Cartesian coordinates, we investigate, from a purely classical point of view, the construction of these charges in a coordinate- and frame-independent language in order to generalize them further. We show that all the charges constructed in that context are associated to the conformal Killing-Yano 2-forms of Minkowski spacetime. Furthermore, we prove that those associated to closed conformal Killing-Yano 2-forms are identical to the charges constructed by Kastor and Traschen for their dual Killing-Yano (d − 2)-forms. We discuss the number of independent and non-trivial gravitational charges that can be constructed in this way.

TCFHs and hidden symmetries of type IIA AdS backgrounds

We present the twisted covariant form hierarchies (TCFHs) of warped (massive) IIA AdS backgrounds. As a consequence we demonstrate that all Killing spinor form bilinears satisfy a generalisation of the conformal Killing-Yano equation with respect to the TCFH connections. We also explore some of the properties of TCFHs which include the reduced holonomy of the minimal TCFH connections for generic backgrounds. Furthermore, we investigate the interplay between TCFHs and hidden symmetries of probes propagating on IIA AdS backgrounds. We find that some of the Killing spinor form bilinears of near horizon geometries of a class of IIA intersecting brane configurations are Killing-Yano forms and so generate hidden symmetries for spinning particle probes.

The TCFHs of D = 11 AdS backgrounds and hidden symmetries

We present the TCFHs of all AdS backgrounds of 11-dimensional supergravity and explore some of the properties of the associated connections. Therefore, we demonstrate that all Killing spinor bilinears satisfy a generalisation of the conformal Killing-Yano equations with respect to the TCFH connection. In addition, we explore the hidden symmetries of spinning particle probes propagating on these backgrounds. We give several examples of hidden symmetries for probes on the maximal supersymmetric AdS backgrounds as well as on some AdS backgrounds that arise as near horizon geometries of intersecting M-branes.

Geometry, conformal Killing-Yano tensors and conserved “currents”

In this paper we discuss the construction of conserved tensors (currents) involving conformal Killing-Yano tensors (CKYTs), generalising the corresponding constructions for Killing-Yano tensors (KYTs). As a useful preparation for this, but also of intrinsic interest, we derive identities relating CKYTs and geometric quantities. The behaviour of CKYTs under conformal transformations is also given, correcting formulae in the literature. We then use the identities derived to construct covariantly conserved “currents”. We find several new CKYT currents and also include a known one by Penrose which shows that “trivial” currents are also useful. We further find that rank-n currents based on rank-n CKYTs k must have a simple form in terms of dk. By construction, these currents are covariant under a general conformal rescaling of the metric. How currents lead to conserved charges is then illustrated using the Kerr-Newman and the C-metric in four dimensions. Separately, we study a rank-1 current, construct its charge and discuss its relation to the recently constructed Cotton current for the Kerr-Newman black hole.

TCFHs, IIB warped AdS backgrounds and hidden symmetries

We present the twisted covariant form hierarchies (TCFHs) on the internal spaces of all type IIB warped AdS backgrounds. As a result we demonstrate that the form bilinears on the internal spaces satisfy a generalisation of the conformal Killing-Yano equation. We also explore some of the properties of the TCFHs, like for example the holonomy of the TCFH connections. In addition, we present examples where the form bilinears generate hidden symmetries for particle probes propagating on the internal spaces of some AdS backgrounds. These include the maximally supersymmetric AdS5 solution as well as some of the near horizon geometries of intersecting IIB branes.

First BGG operators on homogeneous conformal geometries

We study first BGG operators and their solutions on homogeneous conformal geometries. We focus on conformal Killing tensors, conformal Killing–Yano forms and twistor spinors in particular. We develop an invariant calculus that allows us to find solutions explicitly using only algebraic computations. We also discuss applications to holonomy reductions and conserved quantities of conformal circles. We demonstrate our result on examples of homogeneous conformal geometries coming mostly from general relativity.

TCFHs, hidden symmetries and M-theory backgrounds

We present the twisted covariant form hierarchy (TCFH) of 11-dimensional supergravity and so demonstrate that the form bilinears of supersymmetric solutions satisfy a generalisation of the conformal Killing-Yano equation with resepct to the TCFH connection. We also compute the Killing-Stäckel, KY and closed conformal Killing-Yano tensors of all spherically symmetric M-branes that include the M2-brane, M5-brane, KK-monopole and pp-wave and demonstrate that their geodesic flows are completely integrable by giving all independent conserved charges in involution. We then find that all form bilinears of pp-wave and KK-monopole solutions generate (hidden) symmetries for spinning particle probes propagating on these backgrounds. Moreover, there are Killing spinors such that some of the 1-, 2- and 3-form bilinears of the M2-brane solution generate symmetries for spinning particle probes. We also explore the question on whether the form bilinears are sufficient to prove the integrability of particle probe dynamics on 11-dimensional supersymmetric backgrounds.

TCFHs, hidden symmetries and type II theories

We present the twisted covariant form hierarchies (TCFH) of type IIA and IIB 10-dimensional supergravities and show that all form bilinears of supersymmetric backgrounds satisfy the conformal Killing-Yano equation with respect to a TCFH connection. We also compute the Killing-Stäckel, Killing-Yano and closed conformal Killing-Yano tensors of all spherically symmetric type II brane backgrounds and demonstrate that the geodesic flow on these solutions is completely integrable by giving all independent charges in involution. We then identify all form bilinears of common sector and D-brane backgrounds which generate hidden symmetries for particle and string probe actions. We also explore the question on whether charges constructed from form bilinears are sufficient to prove the integrability of probes on supersymmetric backgrounds.

Invariant Conformal Killing–Yano 2-Forms on Five-Dimensional Lie Groups

We study left invariant conformal Killing–Yano (CKY) 2-forms on Lie groups endowed with a left invariant metric. We classify all 5-dimensional metric Lie algebras carrying CKY tensors that are obtained as a one-dimensional central extension of 4-dimensional metric Lie algebras endowed with an invertible parallel skew-symmetric tensor. On the other hand, we also classify 5-dimensional metric Lie algebras with center of dimension greater than one admitting strict CKY tensors. In addition, we determine all possible CKY tensors on these metric Lie algebras. In particular, we exhibit the first examples of CKY 2-forms on metric Lie algebras which do not admit any Sasakian structure.

Closed conformal Killing–Yano initial data

Through an exhaustive search, we produce a five-parameter family of propagation identities for the closed conformal Killing–Yano (cCYK) equation on two-forms, which hold on an Einstein cosmological vacuum spacetime in any dimension n > 4. It is well-known that spacetimes admitting a non-degenerate two-form of this type are exhausted by the Kerr-NUT-(A)dS family of exact higher dimensional black hole solutions. As a consequence, we identify a set of necessary and sufficient conditions ensuring that the cosmological vacuum development of an initial data set for Einstein’s field equations admits a cCYK two-form. We refer to these conditions as closed conformal Killing–Yano initial data equations. The four-dimensional case is special and is treated separately, where we can also handle the conformal Killing–Yano equation without the closed condition.

Killing-Yano Cotton currents

We discuss conserved currents constructed from the Cotton tensor and (conformal) Killing-Yano tensors (KYTs). We consider the corresponding charges generally and then exemplify with the four-dimensional Plebański-Demiański metric where they are proportional to the sum of the squares of the electric and the magnetic charges. As part of the derivation, we also find the two conformal Killing-Yano tensors of the Plebański-Demiański metric in the recently introduced coordinates of Podolsky and Vratny. The construction of asymptotic charges for the Cotton current is elucidated and compared to the three-dimensional construction in Topologically Massive Gravity. For the three-dimensional case, we also give a conformal superspace multiplet that contains the Cotton current in the bosonic sector. In a mathematical section, we derive potentials for the currents, find identities for conformal KYTs and for KYTs in torsionful backgrounds.

Conformal geodesics on gravitational instantons

AbstractWe study the integrability of the conformal geodesic flow (also known as the conformal circle flow) on the SO(3)–invariant gravitational instantons. On a hyper–Kähler four–manifold the conformal geodesic equations reduce to geodesic equations of a charged particle moving in a constant self–dual magnetic field. In the case of the anti–self–dual Taub NUT instanton we integrate these equations completely by separating the Hamilton–Jacobi equations, and finding a commuting set of first integrals. This gives the first example of an integrable conformal geodesic flow on a four–manifold which is not a symmetric space. In the case of the Eguchi–Hanson we find all conformal geodesics which lie on the three–dimensional orbits of the isometry group. In the non–hyper–Kähler case of the Fubini–Study metric on $\mathbb{CP}^2$ we use the first integrals arising from the conformal Killing–Yano tensors to recover the known complete integrability of conformal geodesics.

Generalized symmetry superalgebras

We generalize the symmetry superalgebras of isometries and geometric Killing spinors on a manifold to include all the hidden symmetries of the manifold generated by Killing spinors in all dimensions. We show that bilinears of geometric Killing spinors produce special Killing–Yano forms and special conformal Killing–Yano forms. After defining the Lie algebra structure of hidden symmetries generated by Killing spinors, we construct symmetry operators as the generalizations of the Lie derivative on spinor fields. All these constructions together constitute the structure of generalized symmetry superalgebras. We exemplify the construction on weak G2 and nearly Kähler manifolds.

Eigenvalue estimates for multi-form modified Dirac operators

We give estimates for the eigenvalues of multi-form modified Dirac operators which are constructed from a standard Dirac operator with the addition of a Clifford algebra element associated to a multi-degree form. In particular such estimates are presented for modified Dirac operators with a k-degree form 0≤k≤4, those modified with multi-degree (0,k)-form 0≤k≤3 and the horizon Dirac operators which are modified with a multi-degree (1,2,4)-form. In particular, we give the necessary geometric conditions for such operators to admit zero modes as well as those for the zero modes to be parallel with a respect to a suitable connection. We also demonstrate that manifolds which admit such parallel spinors are associated with twisted covariant form hierarchies which generalize the conformal Killing–Yano forms.

Twistor spinors and extended conformal superalgebras

We show that the first-order symmetry operators of twistor spinors can be constructed from conformal Killing–Yano forms in conformally-flat backgrounds. We express the conditions on conformal Killing–Yano forms to obtain mutually commuting symmetry operators of twistor spinors. Conformal superalgebras which consist of conformal Killing vectors and twistor spinors and play important roles in supersymmetric field theories in conformal backgrounds are extended to more general superalgebras by using the graded Lie algebra structure of conformal Killing–Yano forms and the symmetry operators of twistor spinors. The even part of the extended conformal superalgebra corresponds to conformal Killing–Yano forms and the odd part consists of twistor spinors.

Non-compact manifolds with Killing spinors

We present here a quick result for non-compact manifolds with invertible Dirac operator, where we link the presence of a massless Killing spinor, with a harmonic, closed conformal Killing–Yano tensor, if one exists for the specific manifold. A couple of examples are introduced.

Hidden symmetry and the separability of the Maxwell equation on the Wahlquist spacetime

We examine hidden symmetry and its relation to the separability of the Maxwell equation on the Wahlquist spacetime. After seeing that the Wahlquist spacetime is a type-D spacetime whose repeated principal null directions are shear-free and geodesic, we show that the spacetime admits three gauged conformal Killing–Yano (GCKY) tensors which are in a relation with torsional conformal Killing–Yano tensors. As a by-product, we obtain an ordinary CKY tensor. We also show that thanks to the GCKY tensors, the Maxwell equation reduces to three Debye equations, which are scalar-type equations, and two of them can be solved by separation of variables.

On quantum numbers for Rarita–Schwinger fields

We consider first order linear operators commuting with the operator appearing in the linearized equation of motion of Rarita–Schwinger fields which comes directly from the action. First we consider a simplified operator giving an equation equivalent to the original equation, and classify first order operators commuting with it in four dimensions. In general such operators are symmetry operators of the original operator, but we find that some of them commute with it. We extend this result in four dimensions to arbitrary dimensions and give first order commuting operators constructed of odd rank Killing–Yano and even rank closed conformal Killing–Yano tensors with additional conditions.

The Minkowski Formula and the Quasi-Local Mass

In this article, we estimate the quasi-local energy with reference to the Minkowski spacetime [16,17], the anti-de Sitter spacetime [4], or the Schwarzschild spacetime [3]. In each case, the reference spacetime admits a conformal Killing-Yano 2-form which facilitates the application of the Minkowski formula in [15] to estimate the quasi-local energy. As a consequence of the positive mass theorems in [9,13] and the above estimate, we obtain rigidity theorems which characterize the Minkowski spacetime and the hyperbolic space.