Conformal Killing–Yano Tensors

• Conformal Killing–Yano tensors are antisymmetric p-form fields that generalize Killing–Yano tensors by including a metric trace part and conformal invariance.
• They underpin hidden symmetries and separability in complex geometries, notably in higher-dimensional black-hole spacetimes and torsionful string theory backgrounds.
• Their rich structure leads to conserved currents, integrability conditions, and classifications in conformal geometry, offering practical tools in theoretical physics.

Conformal Killing–Yano tensors are antisymmetric tensor fields whose first covariant derivative decomposes into a totally antisymmetric part and a metric trace part. They generalize Killing–Yano tensors in the same way that conformal Killing vectors generalize Killing vectors, and they form a common framework for hidden symmetries, separability, curvature restrictions, conformal invariants, and conserved currents. In the recent literature they appear in conformal geometry, especially in the study of conformally Kähler metrics, in higher-dimensional black-hole spacetimes admitting principal tensors, in torsionful string-theoretic backgrounds, and in constructions of quasi-local and topological charges (Batista, 2014, Yasui et al., 2011, Dunajski et al., 2024).

1. Defining equations and basic variants

For a totally antisymmetric $p$-form $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$ on an $n$-dimensional manifold with Levi–Civita connection $\nabla_a$, the conformal Killing–Yano equation can be written as

$$\nabla_a Y_{b_1\cdots b_p}+\nabla_{b_1}Y_{a b_2\cdots b_p} = 2\,g_{a[b_1}\,h_{b_2\cdots b_p]} + 2\,g_{b_1[a}\,h_{b_2\cdots b_p]},$$

for a $(p-1)$-form $h$. Equivalently,

$$\nabla_aY_{b_1\cdots b_p} = \nabla_{[a}Y_{b_1\cdots b_p]} + 2\,g_{a[b_1}\,h_{b_2\cdots b_p]}, \qquad h_{b_2\cdots b_p} = \frac{p}{2(n+1-p)}\,\nabla^aY_{a b_2\cdots b_p}.$$

Thus the trace part is fixed by the divergence of $Y$ (Batista, 2014).

In differential-form language the same condition is

$$\nabla_X h = \frac{1}{p+1}\,X\lrcorner\, d h - \frac{1}{n-p+1}\,X^\flat\wedge \delta h,$$

which makes the decomposition into curl and divergence explicit. For rank $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$0, a closed conformal Killing–Yano tensor satisfies

$Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$1

and this closed rank-$Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$2 case is the standard starting point for principal tensors in higher-dimensional geometry (Frolov et al., 2017).

Object	Additional condition	Consequence
Killing–Yano tensor	$Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$3 or $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$4	Co-closed antisymmetric symmetry tensor
Closed CKY tensor	$Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$5 or $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$6	Hodge dual is a Killing–Yano tensor
Conformal Yano–Killing tensor (4D usage)	$Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$7 with $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$8	Divergence defines a conformal Killing vector

Hodge duality exchanges Killing–Yano and closed conformal Killing–Yano forms. In particular, if $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$9 is a Killing–Yano form of rank $n$0, then $n$1 is a closed conformal Killing–Yano form of rank $n$2, and conversely (Cariglia et al., 2011). In four dimensions one often encounters the notation “CYK tensor” or “conformal Yano–Killing tensor” for a CKY $n$3-form $n$4; its divergence $n$5 is then a conformal Killing vector, and the Killing–Yano case is recovered when $n$6 (Jezierski et al., 2014).

2. Conformal covariance, Weyl identities, and integrability

The CKY equation is conformally natural. For the $n$7-form equation used in conformal geometry, one checks that under a conformal rescaling $n$8, the equation is invariant provided the CKY $n$9-form has weight zero (Dunajski et al., 2024). For a rank-$\nabla_a$0 CKYT, the Weyl-rescaling law takes the corrected form

$\nabla_a$1

together with a nontrivial transformation for its trace $\nabla_a$2; the conserved currents built from these tensors remain covariantly conserved under general conformal rescaling (Lindström et al., 2022).

Existence of a CKY tensor imposes algebraic curvature restrictions. For a CKY $\nabla_a$3-form $\nabla_a$4, Batista derived a purely algebraic integrability condition expressed entirely in terms of the Weyl tensor and $\nabla_a$5, with an auxiliary fully antisymmetric tensor

$\nabla_a$6

The resulting condition is conformally invariant because it involves only $\nabla_a$7 and $\nabla_a$8 (Batista, 2014).

Several important specializations follow. In an Einstein space, if $\nabla_a$9 is a CKY $$\nabla_a Y_{b_1\cdots b_p}+\nabla_{b_1}Y_{a b_2\cdots b_p} = 2\,g_{a[b_1}\,h_{b_2\cdots b_p]} + 2\,g_{b_1[a}\,h_{b_2\cdots b_p]},$$0-form then its divergence

$$\nabla_a Y_{b_1\cdots b_p}+\nabla_{b_1}Y_{a b_2\cdots b_p} = 2\,g_{a[b_1}\,h_{b_2\cdots b_p]} + 2\,g_{b_1[a}\,h_{b_2\cdots b_p]},$$1

is a Killing–Yano $$\nabla_a Y_{b_1\cdots b_p}+\nabla_{b_1}Y_{a b_2\cdots b_p} = 2\,g_{a[b_1}\,h_{b_2\cdots b_p]} + 2\,g_{b_1[a}\,h_{b_2\cdots b_p]},$$2-form precisely when an algebraic Weyl constraint holds; for $$\nabla_a Y_{b_1\cdots b_p}+\nabla_{b_1}Y_{a b_2\cdots b_p} = 2\,g_{a[b_1}\,h_{b_2\cdots b_p]} + 2\,g_{b_1[a}\,h_{b_2\cdots b_p]},$$3 this always happens, so every rank-$$\nabla_a Y_{b_1\cdots b_p}+\nabla_{b_1}Y_{a b_2\cdots b_p} = 2\,g_{a[b_1}\,h_{b_2\cdots b_p]} + 2\,g_{b_1[a}\,h_{b_2\cdots b_p]},$$4 CKY tensor in an Einstein space has Killing-vector divergence (Batista, 2014). In maximally symmetric spaces, the covariant derivative of any Killing–Yano $$\nabla_a Y_{b_1\cdots b_p}+\nabla_{b_1}Y_{a b_2\cdots b_p} = 2\,g_{a[b_1}\,h_{b_2\cdots b_p]} + 2\,g_{b_1[a}\,h_{b_2\cdots b_p]},$$5-form is a closed CKY $$\nabla_a Y_{b_1\cdots b_p}+\nabla_{b_1}Y_{a b_2\cdots b_p} = 2\,g_{a[b_1}\,h_{b_2\cdots b_p]} + 2\,g_{b_1[a}\,h_{b_2\cdots b_p]},$$6-form, and every CKY $$\nabla_a Y_{b_1\cdots b_p}+\nabla_{b_1}Y_{a b_2\cdots b_p} = 2\,g_{a[b_1}\,h_{b_2\cdots b_p]} + 2\,g_{b_1[a}\,h_{b_2\cdots b_p]},$$7-form decomposes uniquely as the sum of a Killing–Yano form and a closed CKY form (Batista, 2014).

These integrability statements rule out a common oversimplification: a CKY tensor is not merely a formal generalization of a Killing–Yano tensor. Its existence can force strong restrictions on the Weyl tensor, on Ricci alignment, and on the allowed conformal class.

3. Prolongation, tractor geometry, and conformally Kähler structures

In even-dimensional Riemannian conformal geometry, Dunajski and Gover recast the CKY $$\nabla_a Y_{b_1\cdots b_p}+\nabla_{b_1}Y_{a b_2\cdots b_p} = 2\,g_{a[b_1}\,h_{b_2\cdots b_p]} + 2\,g_{b_1[a}\,h_{b_2\cdots b_p]},$$8-form equation as a prolongation problem on a tractor-type bundle. On an even-dimensional Riemannian manifold $$\nabla_a Y_{b_1\cdots b_p}+\nabla_{b_1}Y_{a b_2\cdots b_p} = 2\,g_{a[b_1}\,h_{b_2\cdots b_p]} + 2\,g_{b_1[a}\,h_{b_2\cdots b_p]},$$9, $(p-1)$0, a skew $(p-1)$1-form $(p-1)$2 is a CKY $(p-1)$3-form if

$(p-1)$4

together with the extra conditions that $(p-1)$5 is non-zero and $(p-1)$6 (Dunajski et al., 2024).

The overdetermined system can be prolonged to a first-order parallel-section problem on a rank-$(p-1)$7 vector bundle $(p-1)$8, equivalently $(p-1)$9, where $h$0 is the standard tractor bundle. The normal tractor connection does not suffice; one must introduce a modified tractor connection $h$1 on $h$2, and the system $h$3 is equivalent to the CKY equation together with its first integrability conditions. This realizes the PDE as a conformally invariant parallel-transport problem (Dunajski et al., 2024).

Parallel-section integrability yields explicit algebraic obstructions from the Weyl tensor. The top slot of the curvature gives the necessary condition

$h$4

Equivalently, one constructs an endomorphism $h$5 of $h$6, and in higher dimensions a necessary condition for any non-degenerate CKY two-form is

$h$7

The vanishing of the associated scalar invariants of the Weyl tensor is the first obstruction to admitting any CKY $h$8-form (Dunajski et al., 2024).

The same framework isolates the conformally Kähler problem. A general parallel tractor corresponds to a CKY form, but not every CKY $h$9-form yields a Kähler metric in its conformal class. Imposing that $$\nabla_aY_{b_1\cdots b_p} = \nabla_{[a}Y_{b_1\cdots b_p]} + 2\,g_{a[b_1}\,h_{b_2\cdots b_p]}, \qquad h_{b_2\cdots b_p} = \frac{p}{2(n+1-p)}\,\nabla^aY_{a b_2\cdots b_p}.$$0 squares, up to scale, to an almost complex structure $$\nabla_aY_{b_1\cdots b_p} = \nabla_{[a}Y_{b_1\cdots b_p]} + 2\,g_{a[b_1}\,h_{b_2\cdots b_p]}, \qquad h_{b_2\cdots b_p} = \frac{p}{2(n+1-p)}\,\nabla^aY_{a b_2\cdots b_p}.$$1 requires the nonlinear algebraic constraint

$$\nabla_aY_{b_1\cdots b_p} = \nabla_{[a}Y_{b_1\cdots b_p]} + 2\,g_{a[b_1}\,h_{b_2\cdots b_p]}, \qquad h_{b_2\cdots b_p} = \frac{p}{2(n+1-p)}\,\nabla^aY_{a b_2\cdots b_p}.$$2

together with additional equations labelled $$\nabla_aY_{b_1\cdots b_p} = \nabla_{[a}Y_{b_1\cdots b_p]} + 2\,g_{a[b_1}\,h_{b_2\cdots b_p]}, \qquad h_{b_2\cdots b_p} = \frac{p}{2(n+1-p)}\,\nabla^aY_{a b_2\cdots b_p}.$$3 and $$\nabla_aY_{b_1\cdots b_p} = \nabla_{[a}Y_{b_1\cdots b_p]} + 2\,g_{a[b_1}\,h_{b_2\cdots b_p]}, \qquad h_{b_2\cdots b_p} = \frac{p}{2(n+1-p)}\,\nabla^aY_{a b_2\cdots b_p}.$$4 in the paper. These conditions define an algebraic subvariety $$\nabla_aY_{b_1\cdots b_p} = \nabla_{[a}Y_{b_1\cdots b_p]} + 2\,g_{a[b_1}\,h_{b_2\cdots b_p]}, \qquad h_{b_2\cdots b_p} = \frac{p}{2(n+1-p)}\,\nabla^aY_{a b_2\cdots b_p}.$$5, and a Riemannian metric $$\nabla_aY_{b_1\cdots b_p} = \nabla_{[a}Y_{b_1\cdots b_p]} + 2\,g_{a[b_1}\,h_{b_2\cdots b_p]}, \qquad h_{b_2\cdots b_p} = \frac{p}{2(n+1-p)}\,\nabla^aY_{a b_2\cdots b_p}.$$6 is locally conformal to a Kähler metric if and only if there is a non-zero $$\nabla_aY_{b_1\cdots b_p} = \nabla_{[a}Y_{b_1\cdots b_p]} + 2\,g_{a[b_1}\,h_{b_2\cdots b_p]}, \qquad h_{b_2\cdots b_p} = \frac{p}{2(n+1-p)}\,\nabla^aY_{a b_2\cdots b_p}.$$7-parallel section $$\nabla_aY_{b_1\cdots b_p} = \nabla_{[a}Y_{b_1\cdots b_p]} + 2\,g_{a[b_1}\,h_{b_2\cdots b_p]}, \qquad h_{b_2\cdots b_p} = \frac{p}{2(n+1-p)}\,\nabla^aY_{a b_2\cdots b_p}.$$8 taking values in $$\nabla_aY_{b_1\cdots b_p} = \nabla_{[a}Y_{b_1\cdots b_p]} + 2\,g_{a[b_1}\,h_{b_2\cdots b_p]}, \qquad h_{b_2\cdots b_p} = \frac{p}{2(n+1-p)}\,\nabla^aY_{a b_2\cdots b_p}.$$9 (Dunajski et al., 2024). This directly corrects another common misconception: the existence of a CKY $Y$0-form is necessary but not sufficient for conformal Kählerity.

Dunajski and Gover also show that existence of a non-degenerate CKY $Y$1-form forces the Weyl tensor to be of type $Y$2 in the higher-dimensional classification of Coley–Milson–Pravda–Pravdová, since the CKY $Y$3-form commutes with the Weyl tensor viewed as an endomorphism of $Y$4 (Dunajski et al., 2024).

4. Principal closed CKY tensors and hidden symmetries

A rank-$Y$5 closed CKY tensor is called principal when it is non-degenerate. For such a tensor $Y$6,

$Y$7

and non-degeneracy means that $Y$8, viewed as a matrix, has maximal rank and its independent eigenvalues have linearly independent gradients (Frolov et al., 2017).

A single rank-$Y$9 closed CKY tensor generates a hierarchy of hidden symmetries. Writing $$\nabla_X h = \frac{1}{p+1}\,X\lrcorner\, d h - \frac{1}{n-p+1}\,X^\flat\wedge \delta h,$$0, one forms

$$\nabla_X h = \frac{1}{p+1}\,X\lrcorner\, d h - \frac{1}{n-p+1}\,X^\flat\wedge \delta h,$$1

Each $$\nabla_X h = \frac{1}{p+1}\,X\lrcorner\, d h - \frac{1}{n-p+1}\,X^\flat\wedge \delta h,$$2 is a closed CKY form, each $$\nabla_X h = \frac{1}{p+1}\,X\lrcorner\, d h - \frac{1}{n-p+1}\,X^\flat\wedge \delta h,$$3 is a Killing–Yano form, and from each $$\nabla_X h = \frac{1}{p+1}\,X\lrcorner\, d h - \frac{1}{n-p+1}\,X^\flat\wedge \delta h,$$4 one obtains a rank-$$\nabla_X h = \frac{1}{p+1}\,X\lrcorner\, d h - \frac{1}{n-p+1}\,X^\flat\wedge \delta h,$$5 Killing tensor $$\nabla_X h = \frac{1}{p+1}\,X\lrcorner\, d h - \frac{1}{n-p+1}\,X^\flat\wedge \delta h,$$6. Together with the primary Killing vector obtained from $$\nabla_X h = \frac{1}{p+1}\,X\lrcorner\, d h - \frac{1}{n-p+1}\,X^\flat\wedge \delta h,$$7, these structures produce mutually commuting Killing tensors and Killing vectors, yielding Liouville integrability of the geodesic flow and separability of the Hamilton–Jacobi, Klein–Gordon, and Dirac equations in the canonical higher-dimensional Kerr–NUT–(A)dS family (Yasui et al., 2011).

In the principal case there is a strong classification theorem: locally, any Einstein manifold admitting a principal CKY $$\nabla_X h = \frac{1}{p+1}\,X\lrcorner\, d h - \frac{1}{n-p+1}\,X^\flat\wedge \delta h,$$8-form is exactly the higher-dimensional Kerr–NUT–(A)dS metric, while the more general non-principal case leads to generalized Kerr–NUT–(A)dS metrics fibred over Kähler–Einstein bases (Yasui et al., 2011). At the operator level, first-order symmetries of the Dirac operator are in one-to-one correspondence with CKY forms; closed CKY forms produce Clifford-odd symmetries and Killing–Yano forms produce Clifford-even ones. For the principal tensor background, the tower $$\nabla_X h = \frac{1}{p+1}\,X\lrcorner\, d h - \frac{1}{n-p+1}\,X^\flat\wedge \delta h,$$9 yields a complete subset of mutually commuting first-order symmetry operators underlying Dirac separability (Cariglia et al., 2011).

The earlier belief that off-shell Kerr–NUT–(A)dS metrics exhaust geometries admitting a principal tensor depends on signature assumptions. Frolov, Krtouš, and Kubizňák showed that in Lorentzian and other indefinite signatures one may allow null eigenvalues of the principal tensor, and then new off-shell canonical metrics arise while retaining the same closed CKY $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$00-form $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$01 (Frolov et al., 2017). This shows that the Euclidean-signature uniqueness result does not extend unchanged to indefinite signatures.

5. Generalizations with skew-symmetric torsion

CKY theory extends naturally to metric connections with totally antisymmetric torsion. If

$Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$02

then the torsion-twisted differential operators are

$Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$03

and a $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$04-form $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$05 is a generalized conformal Killing–Yano tensor if

$Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$06

The specializations are generalized Killing–Yano tensors when $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$07 and generalized closed conformal Killing–Yano tensors when $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$08 (Houri et al., 2010).

Many structural properties survive. The Hodge dual of a GCCKY $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$09-form is a GKY $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$10-form, wedge products of GCCKY forms remain GCCKY, and a GKY $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$11-form gives a rank-$Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$12 Killing tensor. In the Kerr–Sen black hole and in Chow’s higher-dimensional charged Kerr–NUT metrics, the natural torsion is the $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$13-form field strength $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$14, and the spacetimes admit non-degenerate GCCKY $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$15-forms. These forms generate commuting Killing tensors, establish Liouville integrability of geodesic motion, and ensure separability of the scalar and Dirac equations (Houri et al., 2010).

For the Dirac operator with skew-symmetric torsion,

$Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$16

a GCKY $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$17-form $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$18 defines a first-order operator $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$19, but a torsion anomaly $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$20 must vanish for this operator to be an on-shell symmetry. The anomaly splits into classical and quantum parts, and in strong KT and strong HKT geometries the Kähler forms are GCKY and generate genuine symmetries of $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$21 (Houri et al., 2010).

In Riemann–Cartan geometry more generally, CKY and covariantly closed CKY tensors satisfy torsion-dependent integrability conditions involving the curvature $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$22, the torsion $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$23, and the divergence field $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$24. For rank $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$25 Killing–Yano tensors, Batista obtained a complete local classification in adapted coordinates. In the totally skew torsion case, a maximally symmetric space with torsion still has metric curvature of the usual constant-curvature form, and no further restriction on the skew torsion arises (Batista, 2015).

6. Local classifications, low-dimensional reductions, and Lie-group examples

In four-dimensional Lorentzian geometry, Ferrando and Sáez showed that the usual Jebsen–Birkhoff hypothesis can be weakened substantially. If a spacetime admits a CKY $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$26-form $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$27 and the Ricci tensor commutes with $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$28, then the complex vector

$Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$29

is a Killing vector, or vanishes identically. In the non-null case this leads to the class of $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$30-metrics, including vacuum and charged Kerr–NUT, A-metrics, and B-metrics, and the additional symmetry appears as the invariant Killing vector $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$31 rather than being assumed a priori (Ferrando et al., 2015). The result shows that “there is a CKY tensor” can replace “there is a 3-parameter isometry group” in the Birkhoff analysis.

The null case has a different algebraic structure. For a null CKY $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$32-form, Ferrando and Sáez showed that the Weyl tensor is of Petrov type $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$33 or $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$34, and if it is type $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$35 then the self-dual CKY bivector is a repeated principal bivector. When the divergence is Killing, the non-flat possibilities are exhausted by three classes: conformally flat pure-radiation plus $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$36 metrics, generalized pp-waves, and Siklos A-pp-waves (Ferrando et al., 2015).

A useful constructive method exists when the spacetime admits a hypersurface-orthogonal Killing vector. Garfinkle and Glass decompose a CKY $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$37-form $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$38 as

$Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$39

with $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$40 and $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$41 orthogonal to the Killing vector $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$42. In four dimensions $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$43 may be dualized to a spatial vector $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$44, and the full CKY system reduces to conformal Killing vector equations for $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$45 and $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$46 on the three-dimensional orbit space together with two first-order Lie-derivative constraints. This replaces the original four-dimensional CKY PDE system by two simpler three-dimensional CKV problems (Garfinkle et al., 2013).

Invariant CKY forms also admit explicit Lie-algebraic classification in dimension five. Herrera and Origlia classified all $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$47-dimensional metric Lie algebras admitting strict left-invariant CKY $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$48-forms. Such algebras fall into two types: one-dimensional central extensions of $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$49-dimensional metric Lie algebras carrying an invertible parallel skew endomorphism, and a center-two family $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$50. The center-two family provides the first explicit examples of strict CKY $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$51-forms on metric Lie algebras that do not admit any Sasakian structure (Herrera et al., 2020).

7. Conserved currents, charges, and explicit ambient constructions

CKY tensors support several distinct constructions of conserved quantities. For a four-dimensional spin-$Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$52 field $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$53 and a CYK tensor $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$54, Jezierski and Migacz defined

$Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$55

and showed $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$56. The corresponding surface flux

$Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$57

is independent of the choice of $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$58. In Minkowski space the CYK solution space is $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$59-dimensional, while in the exact Schwarzschild metric only two global CYK solutions remain; on conformally flat slices with vanishing extrinsic curvature one may nevertheless define “momentary” charges from the ten spatial conformal Killing vectors (Jezierski et al., 2014).

In $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$60, Czajka and Jezierski gave an ambient-space construction of the full CYK space. The hyperboloid embedding in $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$61 yields $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$62 CYK tensors from pullbacks of constant ambient $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$63-forms and $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$64 more from Hodge duals of pullbacks of constant ambient $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$65-forms, producing the full $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$66-dimensional solution space. The same framework gives conserved charges for asymptotically $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$67 spacetimes by contracting the Weyl tensor with a CYK $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$68-form (Czajka et al., 2017).

At the level of curvature currents, CKY tensors generate several conserved objects. For a rank-$Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$69 CKYT $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$70, one has the divergence-free “trivial current”

$Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$71

For rank $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$72, the Einstein current

$Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$73

is conserved, and for a rank-$Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$74 CKYT $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$75 the Cotton current

$Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$76

is likewise conserved. These constructions are conformally covariant, and in Kerr–Newman the Einstein and Cotton currents reproduce, up to normalization, the Komar mass, whereas in the C-metric the trivial current yields a divergent charge reflecting non-asymptotic flatness (Lindström et al., 2022).

Lindström and Sarıoğlu further studied Cotton currents built from CKY and Killing–Yano tensors. In the four-dimensional Plebański–Demiański metric they identified two rank-$Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$77 CKY tensors in Podolský–Vrátný coordinates and found that the corresponding Cotton charges are proportional to $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$78, the sum of the squares of the electric and magnetic charges (Lindström et al., 2021).

A more recent higher-form formulation defines, for any CKY $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$79-form $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$80,

$Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$81

with divergence relation

$Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$82

The primary current $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$83 is strictly conserved, and the corresponding Penrose charge can be written as a boundary integral of the fully covariant $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$84. In the Killing–Yano case $Y_{a_1\cdots a_p}=Y_{[a_1\cdots a_p]}$85, the current is conserved off shell and defines a topological charge. Applications include Kerr–Newman, AdS–Kerr, and D-brane solutions in type II supergravity (Hull et al., 22 Aug 2025).

Taken together, these developments place CKY tensors at the intersection of conformal geometry, algebraic specialness, hidden symmetry algebras, separability theory, torsionful generalized geometry, and conserved-current constructions. Their rôle ranges from local curvature obstructions and tractor prolongations to explicit integrable black-hole metrics and higher-form charges.