Torsionful Conformal Killing–Yano Forms

• Torsionful conformal Killing–Yano forms are antisymmetric tensors defined using a metric-compatible connection with torsion, extending classical hidden symmetries.
• They offer a refined framework where modified differential operators, duality properties, and integrability conditions yield generalized Dirac symmetry operators and anomaly insights.
• Their various formulations (e.g., CKYT, GCKY, GCCKY) underpin applications in black-hole metrics, supersymmetry, and G-structure classifications in differential geometry.

Searching arXiv for recent and foundational papers on torsionful conformal Killing–Yano forms. arXiv search query: "torsionful conformal Killing-Yano forms skew-symmetric torsion generalized conformal Killing-Yano" Torsionful conformal Killing–Yano forms are antisymmetric tensors defined by replacing the Levi–Civita derivative in the conformal Killing–Yano equation with a metric-compatible connection carrying torsion, most often a totally antisymmetric 3-form. In the literature this produces several closely related notions—CKYT, GCKY, GCCKY, CCKYT, and, for general metric-compatible torsion, CCCKY—whose common role is to extend hidden-symmetry geometry, Hodge duality, separability structures, and Dirac-type symmetry operators beyond the torsion-free setting. A persistent terminological subtlety is that some influential papers on Killing–Yano geometry with torsion study only the coclosed sector or the intrinsic torsion of $G$-structures, rather than a full torsionful CKY formalism (Houri et al., 2010, Chow, 2015, Batista, 2015, Santillan, 2011).

1. Definitions and terminology

For an ordinary $p$-form $f$, the torsion-free conformal Killing–Yano equation reviewed in the Killing–Yano survey is

$$\nabla_X f = \frac{1}{p+1} i_X df -\frac{1}{n-p+1} X^\flat\wedge d^*f,$$

and it reduces to the Killing–Yano equation when $d^*f=0$ (Santillan, 2011). The torsionful generalization keeps the same CKY decomposition but substitutes a torsionful covariant derivative and torsion-modified differential operators.

A systematic formulation on a pseudo-Riemannian spin manifold with totally antisymmetric torsion $T\in \Omega^3(M)$ defines a generalized conformal Killing–Yano $p$-form $\omega$ by

$$\nabla^T_X \omega = \frac{1}{p+1}\, X\lrcorner d^T\omega -\frac{1}{n-p+1}\, X^\flat\wedge \delta^T\omega,$$

with generalized Killing–Yano forms characterized by $\delta^T\omega=0$ and generalized closed conformal Killing–Yano forms by $p$0 (Houri et al., 2010). A closely related index formulation defines a conformal Killing–Yano form with torsion $p$1 by

$p$2

with $p$3; the coclosed and closed subclasses are called KYT and CCKYT, respectively (Chow, 2015). Batista, by contrast, allows a general metric-compatible torsion and defines a torsionful CKY $p$4-form $p$5 through

$p$6

with $p$7 determined by contraction; the covariantly closed subclass is termed CCCKY (Batista, 2015).

The nomenclature used across the literature is therefore not fully uniform.

Term	Defining restriction	Usage
CKYT / GCKY	Full torsionful CKY equation	Skew-torsion formulations (Chow, 2015, Houri et al., 2010)
KYT / GKY	$p$8 or $p$9	Coclosed torsionful sector (Chow, 2015, Houri et al., 2010)
CCKYT / GCCKY	$f$0 or $f$1	Closed torsionful sector (Chow, 2015, Houri et al., 2010)
CCCKY	$f$2	General metric-compatible torsion (Batista, 2015)

This suggests that the phrase “torsionful conformal Killing–Yano form” can refer either to the skew-torsion framework based on $f$3, or to the broader metric-compatible framework in which covariant closure replaces exterior closure.

2. Torsion-modified calculus, duality, and integrability

In the skew-torsion formalism of generalized CKY geometry, the natural operators are

$f$4

and, in general, $f$5 and $f$6 (Houri et al., 2010). A later algebraic treatment uses the notation $f$7 for the skew torsion and writes instead

$f$8

together with

$f$9

This suggests that sign conventions vary across the literature, while the structural replacement $$\nabla_X f = \frac{1}{p+1} i_X df -\frac{1}{n-p+1} X^\flat\wedge d^*f,$$0 remains the central idea (Ertem et al., 7 Aug 2025).

Hodge duality survives torsion in a strong form. In the skew-torsion setting, GKY and GCCKY forms are exchanged by the Hodge star, and the same statement holds for KYT and CCKYT in the lift framework (Houri et al., 2010, Chow, 2015). Batista proves an analogous statement for a general metric-compatible connection: a Killing–Yano $$\nabla_X f = \frac{1}{p+1} i_X df -\frac{1}{n-p+1} X^\flat\wedge d^*f,$$1-form with respect to the torsionful connection is Hodge dual to a covariantly closed conformal Killing–Yano $$\nabla_X f = \frac{1}{p+1} i_X df -\frac{1}{n-p+1} X^\flat\wedge d^*f,$$2-form with respect to the same connection, because the volume form remains covariantly constant under any metric-compatible connection (Batista, 2015).

A central torsion-specific issue is the distinction between exterior closure and covariant closure. For a $$\nabla_X f = \frac{1}{p+1} i_X df -\frac{1}{n-p+1} X^\flat\wedge d^*f,$$3-form $$\nabla_X f = \frac{1}{p+1} i_X df -\frac{1}{n-p+1} X^\flat\wedge d^*f,$$4,

$$\nabla_X f = \frac{1}{p+1} i_X df -\frac{1}{n-p+1} X^\flat\wedge d^*f,$$5

so $$\nabla_X f = \frac{1}{p+1} i_X df -\frac{1}{n-p+1} X^\flat\wedge d^*f,$$6 and $$\nabla_X f = \frac{1}{p+1} i_X df -\frac{1}{n-p+1} X^\flat\wedge d^*f,$$7 are no longer equivalent when torsion is present (Batista, 2015). This is why GCCKY and CCCKY are not interchangeable notions.

Integrability theory is correspondingly richer. For GCKY forms one has

$$\nabla_X f = \frac{1}{p+1} i_X df -\frac{1}{n-p+1} X^\flat\wedge d^*f,$$8

together with a torsionful Weitzenböck identity and curvature terms involving both $$\nabla_X f = \frac{1}{p+1} i_X df -\frac{1}{n-p+1} X^\flat\wedge d^*f,$$9 and $d^*f=0$0 (Houri et al., 2010). For CCCKY forms, Batista derives a coupled system involving curvature, torsion, and the derivative of the trace field $d^*f=0$1, and also isolates a purely torsional obstruction,

$d^*f=0$2

which has no torsion-free analogue (Batista, 2015).

3. Dirac operators, particle models, and anomalies

A major reason torsionful CKY geometry matters is that it survives into symmetry-operator theory for spinors. In the skew-torsion spin setting, the relevant Dirac operator is not the naive torsion Dirac operator $d^*f=0$3, but the modified Bismut operator

$d^*f=0$4

For a GCKY $d^*f=0$5-form $d^*f=0$6, one constructs a first-order operator

$d^*f=0$7

and the deviation from exact symmetry is an explicit anomaly $d^*f=0$8 built from $d^*f=0$9, $T\in \Omega^3(M)$0, $T\in \Omega^3(M)$1, and $T\in \Omega^3(M)$2. The anomaly splits into classical and quantum parts; when it vanishes, GKY and GCCKY forms generate graded symmetry operators of the massless Dirac equation (Houri et al., 2010).

The same paper identifies strong KT and strong HKT manifolds as particularly clean torsion backgrounds. On a KT manifold the Hermitian form $T\in \Omega^3(M)$3 satisfies

$T\in \Omega^3(M)$4

and the anomaly reduces to $T\in \Omega^3(M)$5, so it vanishes whenever $T\in \Omega^3(M)$6. On a strong HKT manifold the three Kähler forms $T\in \Omega^3(M)$7 satisfy the analogous relations and each generates a commuting symmetry of the modified Dirac operator (Houri et al., 2010).

The charged-particle and lift literature reveals a second mechanism. If $T\in \Omega^3(M)$8 is a KYT $T\in \Omega^3(M)$9-form and the Kaluza–Klein field strength satisfies

$p$0

then the quadratic quantity built from the associated Killing–Stäckel tensor is conserved for charged particle motion, and the same algebraic condition appears in first-order Dirac symmetry operators and in pseudo-classical spinning-particle transformations (Chow, 2015).

Papadopoulos derives the torsionful Killing–Yano equation directly from invariance of an $p$1 supersymmetric worldline action with torsion 3-form $p$2,

$p$3

supplemented by an extra torsion constraint and a magnetic-field condition. That paper does not formulate a full torsionful CKY equation, but it identifies the mechanical origin of the coclosed sector and shows that the torsion solving the KY invariance equations need not be unique (Papadopoulos, 2011).

4. Principal torsionful CKY structures and local metrics

The most developed local geometry arises from the principal Killing–Yano tensor with torsion. This is defined as a non-degenerate rank-2 $p$4-closed GCKY tensor $p$5 satisfying

$p$6

It is the torsionful analogue of the usual principal closed conformal Killing–Yano tensor (Houri et al., 2012).

In dimension $p$7, the principal tensor admits a Darboux frame in which

$p$8

The torsion is then strongly constrained: in even dimensions only $p$9-components may survive, while in odd dimensions $\omega$0-components may also be nonzero. Because $\omega$1 is only $\omega$2-closed, not necessarily $\omega$3-closed, torsion creates genuinely new local possibilities unavailable in torsion-free principal CKY geometry (Houri et al., 2012).

The hidden-symmetry tower persists. The wedge powers

$\omega$4

are $\omega$5-closed GCKY forms, their Hodge duals are generalized KY forms, and they generate explicit rank-2 Killing tensors

$\omega$6

that mutually commute under the Schouten–Nijenhuis bracket (Houri et al., 2012). The local classification reduces to a nonlinear first-order PDE system and splits into types A, B, and C; Type A generalizes the torsionless Kerr–NUT–(A)dS pattern, while Types B and C occur only when torsion is present (Houri et al., 2012).

In five dimensions, Houri, Oota, and Yasui analyze the dual picture of a rank-2 GKY tensor $\omega$7 and its rank-3 GCCKY dual $\omega$8. Under the additional assumption that the zero-eigenvalue eigenvector is Killing, the local metrics again fall into types A, B, and C. Type A contains charged rotating Kaluza–Klein black holes and black strings; the physical torsion is

$\omega$9

in minimal supergravity and

$$\nabla^T_X \omega = \frac{1}{p+1}\, X\lrcorner d^T\omega -\frac{1}{n-p+1}\, X^\flat\wedge \delta^T\omega,$$0

in heterotic supergravity. The associated Killing–Stäckel tensor yields additive separability of the Hamilton–Jacobi equation, and in the heterotic branch the relevant scalar equation is a deformed Klein–Gordon equation rather than the ordinary one (Houri et al., 2012).

These local models intersect supergravity black-hole geometry in several ways. The principal CCKYT 2-form appearing in charged Kerr–NUT solutions of string theory and gauged supergravity has Darboux form

$$\nabla^T_X \omega = \frac{1}{p+1}\, X\lrcorner d^T\omega -\frac{1}{n-p+1}\, X^\flat\wedge \delta^T\omega,$$1

and its powers lift under a Kaluza–Klein ansatz whenever the gauge field strength is aligned with the same Darboux 2-planes (Chow, 2015).

5. $$\nabla^T_X \omega = \frac{1}{p+1}\, X\lrcorner d^T\omega -\frac{1}{n-p+1}\, X^\flat\wedge \delta^T\omega,$$2-structures, intrinsic torsion, and adjacent classifications

Not all torsion-related Killing–Yano literature develops torsionful CKY forms in the modern sense. The review of Killing–Yano tensors and applications reproduces Papadopoulos’ list for $$\nabla^T_X \omega = \frac{1}{p+1}\, X\lrcorner d^T\omega -\frac{1}{n-p+1}\, X^\flat\wedge \delta^T\omega,$$3-structures by analyzing intrinsic torsion classes and testing whether $$\nabla^T_X \omega = \frac{1}{p+1}\, X\lrcorner d^T\omega -\frac{1}{n-p+1}\, X^\flat\wedge \delta^T\omega,$$4-invariant forms satisfy the ordinary Levi–Civita Killing–Yano equation

$$\nabla^T_X \omega = \frac{1}{p+1}\, X\lrcorner d^T\omega -\frac{1}{n-p+1}\, X^\flat\wedge \delta^T\omega,$$5

It explicitly does not introduce a torsion-modified CKY equation. Within that Levi–Civita framework, the almost Kähler form $$\nabla^T_X \omega = \frac{1}{p+1}\, X\lrcorner d^T\omega -\frac{1}{n-p+1}\, X^\flat\wedge \delta^T\omega,$$6 is KY in the nearly Kähler case, the $$\nabla^T_X \omega = \frac{1}{p+1}\, X\lrcorner d^T\omega -\frac{1}{n-p+1}\, X^\flat\wedge \delta^T\omega,$$7 3-form $$\nabla^T_X \omega = \frac{1}{p+1}\, X\lrcorner d^T\omega -\frac{1}{n-p+1}\, X^\flat\wedge \delta^T\omega,$$8 is KY for nearly parallel $$\nabla^T_X \omega = \frac{1}{p+1}\, X\lrcorner d^T\omega -\frac{1}{n-p+1}\, X^\flat\wedge \delta^T\omega,$$9, and on Sasakian manifolds one has

$\delta^T\omega=0$0

so $\delta^T\omega=0$1 is of conformal Killing–Yano type, but still with respect to the Levi–Civita derivative (Santillan, 2011).

Papadopoulos’ worldline paper moves closer to the torsionful setting. For $\delta^T\omega=0$2, it studies the torsionful KY equation generated by the supersymmetric particle action and determines when compatible skew-torsion connections exist and whether they are unique. A central outcome is non-uniqueness for $\delta^T\omega=0$3, $\delta^T\omega=0$4, and $\delta^T\omega=0$5 in certain sectors, contrasted with uniqueness and actual parallelism for $\delta^T\omega=0$6, $\delta^T\omega=0$7, and $\delta^T\omega=0$8. That paper remains in the coclosed sector: it contains no torsionful codifferential term and no full conformal trace term (Papadopoulos, 2011).

This neighboring $\delta^T\omega=0$9-structure literature is therefore best interpreted as background for torsionful CKY theory rather than as a complete theory of torsionful conformal Killing–Yano forms. It shows how intrinsic torsion, compatible skew-torsion connections, and hidden symmetry interact, but it also clarifies that the full conformal equation entered the literature only in more specialized treatments (Santillan, 2011, Papadopoulos, 2011).

6. Lifts, graded brackets, and generalized extensions

A substantial structural development concerns higher-dimensional lifts. Under the Kaluza–Klein ansatz

$p$00

a KYT $p$01-form $p$02 lifts directly if

$p$03

while a CCKYT $p$04-form lifts as

$p$05

Because wedge products of liftable CCKYT forms remain liftable, principal torsionful conformal forms generate towers of higher-degree liftable hidden symmetries in supergravity black-hole backgrounds (Chow, 2015).

Recent work has also addressed the algebra of torsionful CKY forms themselves. For a pseudo-Riemannian manifold with skew torsion $p$06, one introduces

$p$07

and derives torsionful CKY integrability conditions in terms of $p$08, $p$09, and $p$10. A graded bracket

$p$11

is obtained from the ordinary CKY bracket by replacing $p$12 with $p$13. The bracket closes on a special subset of torsionful CKY forms when $p$14 is closed and $p$15-parallel, and when the forms satisfy

$p$16

Under these assumptions, one gets a graded Lie algebra on constant-curvature manifolds, and on Einstein manifolds for the corresponding normal subset (Ertem et al., 7 Aug 2025).

An even broader generalization arises from supergravity Killing spinor equations. Papadopoulos shows that bilinears of Killing spinors satisfy twisted covariant form hierarchies

$p$17

which imply a generalized CKY equation for a collection of forms on $p$18. In heterotic and ungauged $p$19, $p$20 supergravity, this hierarchy reduces to the standard skew-torsion connection $p$21. In minimal $p$22 supergravity, the 2-form bilinear is a CKY form with torsion $p$23. In $p$24 and $p$25, however, the hierarchy generally mixes form degree and goes beyond the standard torsionful CKY setting (Papadopoulos, 2020).

A useful boundary marker is provided by the Cotton-current paper, which writes the natural torsionful CKY definition for completely skew torsion,

$p$26

but then develops only a torsionful second-derivative identity for KY 2-forms, not a full torsionful CKY integrability theory. Its detailed CKY identities remain torsion-free (Lindström et al., 2021).

Taken together, these results identify torsionful conformal Killing–Yano forms as a hierarchy of hidden-symmetry objects ranging from coclosed KY-type forms to closed conformal subclasses, principal tensors, lifted black-hole symmetries, graded Lie brackets, and supergravity flux-twisted generalizations. The subject is unified by the replacement of Levi–Civita data with torsionful or flux-twisted operators, but internally differentiated by the choice of torsion class, the notion of closure, and the extent to which conformal, spinorial, and algebraic structures remain intact (Houri et al., 2010, Chow, 2015, Batista, 2015, Ertem et al., 7 Aug 2025)