UUITP-20/21

New currents with Killing-Yano tensors

Ulf Lindström a,b111email<EMAIL_ADDRESS>and Özgür Sarıoğlu
a222email<EMAIL_ADDRESS>
aDepartment of Physics, Faculty of Arts and Sciences,

Middle East Technical University, 06800, Ankara, Turkey

bDepartment of Physics and Astronomy, Theoretical Physics, Uppsala University

SE-751 20 Uppsala, Sweden

Abstract

New relations involving the Riemann, Ricci and Einstein tensors that have to
hold for a given geometry to admit Killing-Yano tensors are described. These
relations are then used to introduce novel conserved “currents” involving such
Killing-Yano tensors. For a particular current based on the Einstein tensor,
we discuss the issue of conserved charges and consider implications for matter
coupled to gravity. The condition on the background geometry to allow
asymptotic conserved charges for a current introduced by Kastor and Traschen
is found and a number of other new aspects of this current are commented on.
In particular we show that it vanishes for rank $(D-1)$ Killing-Yano tensors
in $D$ dimensions.

###### Contents

  1. 1 Introduction
  2. 2 Killing-Yano tensors
  3. 3 KYT identities
    1. 3.1 Generalisation of (3.3) for arbitrary rank $n$ KYT
    2. 3.2 A new identity
    3. 3.3 Generalisation of (3.9) for arbitrary rank $n$ KYT
  4. 4 New currents
  5. 5 Constraints on matter sources from (4.2) and (4.3)
    1. 5.1 The perfect fluid
    2. 5.2 The electromagnetic field
  6. 6 The KT-current
    1. 6.1 AD charges for the KT-current
    2. 6.2 KT-current in terms of the Weyl and Schouten tensors
  7. 7 Comments on the KT and related currents
  8. 8 Conclusions and comments
  9. A No AD charge for $K^{ab}$ in maximally symmetric spacetimes.
  10. B Conservation of $J_{E}$ in conformally flat geometries

##  1 Introduction

Killing-Yano tensors (KYTs) have long been studied in various settings. They
can be thought of as square roots of Killing tensors with which they share
some properties. In particular they are relevant to gravity, supergravity and
string theory for finding hidden symmetries for particles and backgrounds, for
separating variables in Hamilton-Jacobi equations and for finding the
symmetries of the Dirac equation and its super extensions. The literature is
vast and this is not a review, so we shall just mention some references that
we have found useful in our present endeavor.

A general background to Killing tensors and KYTs is the nice paper [1]. A
classical application to finding new supersymmetries is contained in “Susy in
the sky” [2]. Relevant for string theory are the more recent paper [3] and the
extensive treatise [4]. There are further applications in General Relativity
(GR) [5, 6] to $G$-structures [7, 8], to WZW models [9], to classical
mechanics [10] and to symmetries of the Dirac operator [11]. A comprehensive
survey of these topics, together with many more references, can be found in
[12]. Finally, supersymmetric conformal KYTs are discussed in [13], and partly
in [14].

We are interested in the effect of KYTs on the geometry. Part of our
motivation is purely mathematical, investigating the interplay between the
properties of a generic rank $n$ KYT and the rest of the geometry. As a
consequence, we are also able to construct conserved antisymmetric
contravariant tensors that we refer to as conserved currents. Not being
Noether currents, these tensors correspond to conserved integrals that are not
in general flux integrals. They can nevertheless in some cases lead to
conserved charges along the lines of the Abbott-Deser (AD) construction for a
Killing vector contracted with the energy momentum tensor. Our setting is GR
in $D$ dimensions coupled to matter. Assuming that this admits a KYT of rank
$n$, we derive two new identities for such KYTs and use them to find new
conserved currents. We apply our identities to several known solutions of GR
and discuss possible conserved charges for the new currents as well as other
constraints on the matter content.

Our discussion is inspired by a result of Kastor and Traschen [15, 16], who
constructed a conserved current for an arbitrary rank KYT. We show how this
KT-current111See section 6 for the definition of a KT-current. in general
splits into sums of conserved currents and how special gravitational
backgrounds allow particular such splittings. In [15, 16], it is stated that
any spacetime that allows asymptotic KYTs will give rise to conserved charges
using the KT-current. We find that in general there are obstructions to this,
and derive a relation that the perturbed background geometry has to satisfy
for these charges to exist. These obstructions can be traced back to the
linearized Bianchi identities needed for the conservation of the KT-current.

After the definition of KYTs in section 2, we describe the new identities in
section 3 and the currents in section 4. In section 5, we discuss some of the
consequences of the existence of a KYT on the matter fields coupled to GR.
Section 6 contains a reformulation of the KT-current in terms of the Weyl and
Schouten tensors. This rewrite allows us to show that the KT-current
identically vanishes in $D=3$ for all KYTs and for rank $n=D-1$ KYTs in $D\geq
4$. It also helps us to identify new constituent currents for special
dimensions and KYT ranks. Moreover, it contains the derivation of a condition
on the geometry for a general KT-current to give rise to asymptotic AD charges
[17]. Sections 5 and 6 also contain gratifying checks on our identities for
the FLRW geometry, the Kantowski-Sachs metric and the Kerr-Newman black hole.
Section 7 deals with various special cases of the $n=2$ KT-current. In
appendix A, we discuss and exclude AD charges for one of our new currents
based on the Einstein tensor. Appendix B contains the proof that another of
our currents is conserved for conformally flat geometries.

##  2 Killing-Yano tensors

The Killing-Yano tensors generalise Killing vectors and Killing tensors to
rank $n$ antisymmetric tensor fields with analogous properties. They can be
thought of as being the components of an $n$-form222So in $D$ dimensions, one
has $n\leq D$. $f_{a_{1}\dots a_{n}}=f_{[a_{1}\dots a_{n}]}$ satisfying

$\displaystyle\nabla_{(a}f_{b)a_{2}\dots a_{n}}=0~{},$ (2.1)

which implies the further properties

$\displaystyle\nabla_{a_{1}}f_{a_{2}\dots
a_{n+1}}=\nabla_{[a_{1}}f_{a_{2}\dots
a_{n+1}]}\quad\mbox{and}\quad\nabla_{a_{1}}f^{a_{1}\dots a_{n}}=0\,.$ (2.2)

These can be used to derive the nontrivial identity333We use “identity” in the
less strict sense where the properties of $f$ have to be taken into account.
[15]

$\displaystyle\nabla_{a}\nabla_{b}f_{c_{1}\dots
c_{n}}=(-1)^{n+1}\frac{(n+1)}{2}\,R^{d}\,_{a[bc_{1}}\,f_{c_{2}\dots
c_{n}]d}\,,$ (2.3)

which generalises the analogous formula for a Killing vector
$\nabla_{a}\nabla_{b}f_{c}=R^{d}\,_{abc}\,f_{d}$ when $n=1$.

##  3 KYT identities

Let us rewrite (2.3) explicitly for $n=2$:

$\nabla_{a}\nabla_{b}f_{cd}=-\frac{3}{2}\,R^{e}\,_{a[bc}\,f_{d]e}=\frac{1}{2}\,R^{e}\,_{abc}\,f_{ed}+\frac{1}{2}\,R^{e}\,_{acd}\,f_{eb}+\frac{1}{2}\,R^{e}\,_{adb}\,f_{ec}\,.$
(3.1)

We contract the $(a,c)$ indices in (3.1). Since $\nabla_{a}f^{ab}=0$, we find
that

$\displaystyle\nabla_{a}\nabla_{b}f^{ac}$ $\displaystyle=$
$\displaystyle[\nabla_{a},\nabla_{b}]f^{ac}=R_{ab}\,f^{ac}+R_{ab}\,^{c}\,{}_{d}\,f^{ad}$
(3.2) $\displaystyle=$
$\displaystyle\frac{1}{2}\,R_{ab}\,f^{ac}-\frac{1}{2}\,R^{ac}\,f_{ab}+\frac{1}{2}\,R_{da}\,^{c}\,{}_{b}\,f^{da}\,,$

where the first line follows from the definition of the commutator of
covariant derivatives and the second line from the contraction of indices on
the right hand side of (3.1). From the equality (3.2) we find

$\displaystyle\frac{1}{2}\left(R_{ab}\,f^{ac}+R^{ac}\,f_{ab}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\,R_{dab}\,^{c}\,f^{ad}+R_{abd}\,^{c}\,f^{ad}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\,R_{dab}\,^{c}\,f^{da}+R_{bda}\,^{c}\,f^{da}\,,$

using $R_{[abd]}\,^{c}=0$. We split the last term into two halves using the
dummy index pair $(a,d)$ and employ $R_{[abd]}\,^{c}=0$ again to arrive at

$\displaystyle R_{ab}\,f^{ac}+R^{ac}\,f_{ab}=0\,.$ (3.3)

To our knowledge the identity (3.3) was first reported in [18], but does not
seem to be widely known (see however [19, 20, 21]). It can alternatively be
derived by acting on the defining property (2.1) with a second covariant
derivative, considering various index combinations and applying the Ricci
identity. This also leads to an identity between the Weyl tensor and $f$ which
we omit. See [20, 21] for details.

###  3.1 Generalisation of (3.3) for arbitrary rank $n$ KYT

We repeat the steps above for the generic rank $n$ case starting from (2.3).
Contracting the $(a,c_{n})$ indices gives

$g^{ac_{n}}\nabla_{a}\nabla_{b}f_{c_{1}\dots
c_{n}}=\nabla^{c_{n}}\nabla_{b}f_{c_{1}\dots
c_{n}}=\left[\nabla^{c_{n}},\nabla_{b}\right]f_{c_{1}\dots c_{n}}\,,$ (3.4)

which yields

$\displaystyle R^{d}\,_{b}\,f_{[c_{1}\dots
c_{n-1}]d}+(-1)^{n}(n-1)\,R_{bda[c_{1}}\,f_{c_{2}\dots
c_{n-1}]}\,^{ad}\qquad\qquad\qquad\qquad\qquad\qquad$
$\displaystyle=R^{d}\,_{[b}\,f_{c_{1}\dots
c_{n-1}]d}+(-1)^{n}\frac{(n-1)}{2}\,R_{ad[bc_{1}}\,f_{c_{2}\dots
c_{n-1}]}\,^{ad}\,.$ (3.5)

Since

$\displaystyle R^{d}\,_{[b}\,f_{c_{1}\dots c_{n-1}]d}$ $\displaystyle=$
$\displaystyle\frac{1}{n}\left(R^{d}\,_{b}\,f_{[c_{1}\dots
c_{n-1}]d}+(-1)^{n-1}(n-1)\,R^{d}\,_{[c_{1}}\,f_{c_{2}\dots
c_{n-1}]bd}\right)\,,$ $\displaystyle R_{ad[bc_{1}}\,f_{c_{2}\dots
c_{n-1}]}\,^{ad}$ $\displaystyle=$
$\displaystyle\frac{1}{n}\left(4\,R_{bda[c_{1}}\,f_{c_{2}\dots
c_{n-1}]}\,^{ad}+(-1)^{n-1}(n-2)\,R_{ad[c_{1}c_{2}}\,f_{c_{3}\dots
c_{n-1}]b}\,^{ad}\right)\,,$

(3.5) can be recast as

$\displaystyle R^{d}\,_{b}\,f_{[c_{1}\dots
c_{n-1}]d}+(-1)^{n}\,R^{d}\,_{[c_{1}}\,f_{c_{2}\dots
c_{n-1}]bd}\qquad\qquad\qquad\qquad\qquad\qquad$
$\displaystyle+(n-2)\left((-1)^{n}R_{bda[c_{1}}\,f_{c_{2}\dots
c_{n-1}]}\,^{ad}+\frac{1}{2}\,R_{ad[c_{1}c_{2}}\,f_{c_{3}\dots
c_{n-1}]b}\,^{ad}\right)$ $\displaystyle=0\,.$ (3.6)

This is the generalisation of (3.3) for a rank $n$ KYT, and to our knowledge,
has not been reported elsewhere in the literature444It has been pointed out to
us by one of the referees that it might be related to the material in
subsection 3.4 of [22]. Indeed the integrability condition in [22] gives the
relation (3.3) for a $n=2$ KYT when traced over one set of indices. For a
$n=3$ KYT tracing and anti-symmetrising two different index pairs recovers our
(3.6)..

As a quick check, it identically reduces to (3.3) when $n=2$. Note that when
any pair of free indices are contracted in (3.6), one gets identically zero on
the left hand side and there is nothing to infer from such contractions.

###  3.2 A new identity

Let us go back to (3.1) for a rank $n=2$ KYT. This time we differentiate, i.e.
consider555The analogs of the steps taken here for the case of a Killing
vector $f$, i.e. $n=1$ case, gives the well-known result that the Lie
derivative of the Riemann tensor along the Killing vector vanishes, i.e.
${\cal L}_{f}R_{abcd}=0$, which leads to ${\cal L}_{f}R_{ab}=0\,,{\cal
L}_{f}R=0$ and hence to ${\cal L}_{f}G_{ab}=0$.

$\nabla_{a}\left(\nabla_{b}\nabla_{c}f_{de}\right)-\nabla_{b}\left(\nabla_{a}\nabla_{c}f_{de}\right)=[\nabla_{a},\nabla_{b}]\nabla_{c}f_{de}\,,$
(3.7)

and use (3.1) on the left hand side of (3.7). Using the Bianchi identity
$\nabla_{[a}\,R_{bc]d}\,^{e}=0$ and multiplying by an overall factor of 2, one
gets

$f_{i[c}\,\nabla^{i}R_{de]ab}=R^{i}\,_{b[cd}\,\nabla_{e]}f_{ia}+R^{i}\,_{a[cd}\,\nabla_{e]}f_{bi}+2R_{ab[c}\,^{i}\nabla_{d}f_{e]i}\,.$
(3.8)

Contracting the $(a,e)$ indices in the latter and multiplying by an overall
factor of 3 then gives

$\displaystyle 2f_{a[d}\nabla^{a}R_{c]b}+f^{ia}\nabla_{i}R_{abcd}$
$\displaystyle=$ $\displaystyle
2R_{iba[c}\nabla_{d]}f^{ia}+2R^{a}\,_{[d}\nabla_{c]}f_{ba}+R_{iacd}\nabla_{b}f^{ia}$
$\displaystyle+4R_{abi[d}\nabla_{c]}f^{ai}+2R^{a}\,_{b}\nabla_{c}f_{da}\,$
$\displaystyle=$ $\displaystyle
3R_{abi[c}\nabla_{d]}f^{ia}+2R_{iab[c}\nabla_{d]}f^{ia}+3R^{a}\,_{[d}\nabla_{c}f_{b]a}$
$\displaystyle+R^{a}\,_{b}\nabla_{c}f_{da}+R_{iacd}\nabla_{b}f^{ia}\,.$

Finally contracting the $(b,c)$ indices in the last equality gives

$f_{ad}\nabla^{a}R-f^{ab}\nabla_{a}R_{db}-f^{ba}\nabla_{b}R_{da}=0\,,$

which is equivalent to

$\displaystyle f^{ab}\nabla_{a}G_{bd}=0\,,$ (3.9)

where $G_{ab}$ denotes the Einstein tensor. As far as we know, this identity
has not been reported elsewhere.

###  3.3 Generalisation of (3.9) for arbitrary rank $n$ KYT

It is again worth repeating the steps taken from (3.7) to (3.9) for a generic
rank $n$ KYT. Starting from (2.3), we have, in analogy to (3.7),

$\nabla_{a}\left(\nabla_{b}\nabla_{c}f_{c_{1}\dots
c_{n}}\right)-\nabla_{b}\left(\nabla_{a}\nabla_{c}f_{c_{1}\dots
c_{n}}\right)=[\nabla_{a},\nabla_{b}]\nabla_{c}f_{c_{1}\dots c_{n}}\,.$ (3.10)

Using (2.3), the Bianchi identity $\nabla_{[a}\,R_{bc]d}\,^{e}=0$ and some
algebra, one finds

$\left(\nabla_{d}R_{ab[cc_{1}}\right)f_{c_{2}\dots
c_{n}]}\,^{d}=2R_{abd[c}\,\nabla_{c_{1}}\,f_{c_{2}\dots
c_{n}]}\,^{d}+R_{bd[cc_{1}}\,\nabla_{|a|}\,f_{c_{2}\dots
c_{n}]}\,^{d}+R_{da[cc_{1}}\,\nabla_{|b|}\,f_{c_{2}\dots c_{n}]}\,^{d}$ (3.11)

analogous to (3.8). On both sides of (3.11), if one contracts first the index
pair $(a,c_{n})$ and then the pair $(b,c)$, one finds that the right hand side
vanishes identically. However the left hand side yields

$\displaystyle(n-1)\left(\nabla^{b}\,R^{a}\,_{[c_{1}}\right)f_{c_{2}\dots
c_{n-1}]ab}+\frac{1}{2}\,\left(\nabla^{a}R\right)f_{a[c_{1}\dots
c_{n-1}]}=0\,.$ (3.12)

This is the generalisation of (3.9) for a rank $n$ KYT, and reduces to (3.9)
when $n=2$. To our knowledge, this identity is also new.

##  4 New currents

Let us return to the $n=2$ case, and the associated identities (3.3) and
(3.9). The antisymmetry of the KYT and (3.3) immediately give

$G_{ab}\,f^{ac}+G^{ac}\,f_{ab}=0\,,$ (4.1)

i.e. the analogous identity for the Einstein tensor. This suggests defining
the “current”666Apart from [23], the relation (4.1) appears neither to have
been considered nor used.

$\displaystyle K^{ab}\equiv
2\,G_{c}\,^{[a}\,f^{b]c}=G^{a}\,_{c}\,f^{bc}-G^{b}\,_{c}\,f^{ac}=2\,G^{a}\,_{c}\,f^{bc}\,,$
(4.2)

where the last equality follows from (4.1). It is easy to see that this
antisymmetric tensor is covariantly conserved

$\displaystyle\nabla_{a}K^{ab}=0\,.$ (4.3)

This can be shown in at least two separate ways. The easier one starts by
using the last equality in (4.2), and employing (2.2) and the property
$\nabla_{a}G^{ab}=0$. Alternatively, one can use the penultimate equality in
(4.2). This results in a total of four terms for $\nabla_{a}K^{ab}$, three of
which cancel out by (2.2) and $\nabla_{a}G^{ab}=0$ as before. The remaining
piece, $(\nabla_{a}\,G^{b}\,_{c})\,f^{ac}$, does vanish due to (3.9).

A question that comes to mind is whether the current $K^{ab}$ (4.2) can be
used for finding new conserved Killing charges, in the sense of e.g. [17, 15].
The stakes are high because of the presence of the Einstein tensor, which
through the Einstein field equations, naturally relates to the matter sources.
It seems unlikely, since the current $K^{ab}$ (4.2) does not have a Noether
origin, i.e. conservation is not modulo field equations and it is not derived
as a Noether current for a symmetry, but that fact does not exclude asymptotic
charges in the sense of [17], (AD charges)777See subsection 6.1 for a detailed
discussion. for the KT-current. In appendix A we explicitly show and explain
the absence of an asymptotic AD-charge for maximally symmetric spacetimes.

Perhaps naively but naturally, one is also tempted to generalise the
expression (4.2) for $K^{ab}$ and define

$\displaystyle
J_{E}^{c_{1}...c_{n}}=G^{d[c_{1}}\,f^{c_{2}...c_{n}]}_{~{}~{}~{}~{}~{}~{}~{}~{}d}\,.$
(4.4)

as a possible new current. It should be noted that the covariant conservation
of this expression requires

$\displaystyle\nabla_{a}J_{E}^{ac_{2}...c_{n}}=\frac{1}{n}\Big{(}\nabla_{a}G^{da}f^{c_{2}...c_{n}}_{~{}~{}~{}~{}~{}~{}~{}~{}d}+(-1)^{n+1}(n-1)\nabla_{a}G^{d[c_{2}}f^{c_{3}...c_{n}]a}_{~{}~{}~{}~{}~{}~{}~{}~{}~{}d}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+G^{da}\nabla_{a}f^{c_{2}...c_{n}}_{~{}~{}~{}~{}~{}~{}~{}~{}d}+(-1)^{n+1}(n-1)G^{d[c_{2}}\nabla_{a}f^{c_{3}...c_{n}]a}_{~{}~{}~{}~{}~{}~{}~{}~{}~{}d}\Big{)}=0~{}.$
(4.5)

Using (2.2) and $\nabla_{a}G^{ab}=0$, the latter becomes

$\displaystyle\nabla_{a}J_{E}^{ac_{2}...c_{n}}=(-1)^{n+1}\frac{(n-1)}{n}\,\nabla_{a}\,G^{d[c_{2}}\,f^{c_{3}...c_{n}]a}_{~{}~{}~{}~{}~{}~{}~{}~{}~{}d}=0~{}.$
(4.6)

We first observe that this expression vanishes for $n=1$. This reproduces the
well-known covariant conservation of the Killing vector current, e.g. in [15].
Secondly, we note that (4.6) vanishes _if_

$\displaystyle
G^{d[c_{2}}f^{c_{3}...c_{n}]a}_{~{}~{}~{}~{}~{}~{}~{}~{}~{}d}\sim
G^{da}f^{[c_{2}c_{3}...c_{n}]}_{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}d}\,,$ (4.7)

which is true for $n=2$ according to (3.9). Nevertheless, it does not vanish
for general $n$, which can be seen from the $n$-dependent coefficient in
(3.12). However, it does vanish for special cases, such as conformally flat
geometries (See appendix B).

Closer scrutiny of (3.12) reveals that one can in fact generalise (4.2) for a
generic rank $n$ KYT by defining

$\displaystyle K^{a_{1}\dots a_{n}}\equiv R_{c}\,^{[a_{1}}\,f^{a_{2}\dots
a_{n}]c}+\frac{(-1)^{n}}{n}\,R\,f^{a_{1}\dots a_{n}}\,,$ (4.8)

that is covariantly conserved, $\nabla_{a_{1}}K^{a_{1}\dots a_{n}}=0$, and
reduces to (4.2) for $n=2$.

##  5 Constraints on matter sources from (4.2) and (4.3)

In this section we restrict our attention to the consequences of (4.2) and
(4.3) on continuous matter distributions that are described by a stress-energy
tensor $T_{ab}$, which acts as a source in Einstein’s field equations. To keep
the discussion concise, we only consider the stress tensors of a perfect fluid
and of an electromagnetic field.

###  5.1 The perfect fluid

The stress tensor of a perfect fluid is given by

$T_{ab}=\rho u_{a}u_{b}+p\left(g_{ab}+u_{a}u_{b}\right)\,,$ (5.1)

where $u^{a}$ is a unit timelike 4-velocity of the fluid with $u^{a}u_{a}=-1$
and the functions $p$ and $\rho$, respectively, denote the pressure and the
mass-density of the fluid. The stress tensor satisfies the equations of motion

$\nabla^{a}T_{ab}=0\,,$ (5.2)

which yields

$\displaystyle u^{a}\nabla_{a}\rho+(\rho+p)\nabla^{a}u_{a}$ $\displaystyle=$
$\displaystyle 0\,,$ (5.3)
$\displaystyle(p+\rho)u^{a}\nabla_{a}u_{b}+\left(g_{ab}+u_{a}u_{b}\right)\nabla^{a}p$
$\displaystyle=$ $\displaystyle 0\,.$ (5.4)

If the spacetime of interest admits a KYT of rank $n=2$, then (4.3), or
equivalently (3.9) which becomes $f^{ab}\nabla_{a}T_{bc}=0$, imposes yet
another set of conditions in analogy to (5.3) and (5.4) above. These read

$\displaystyle f^{ab}u_{b}\nabla_{a}\rho+(\rho+p)f^{ab}\nabla_{a}u_{b}$
$\displaystyle=$ $\displaystyle 0\,,$ (5.5)
$\displaystyle(p+\rho)f^{ab}u_{b}\nabla_{a}u_{c}+\left(g_{bc}+u_{b}u_{c}\right)f^{ab}\nabla_{a}p$
$\displaystyle=$ $\displaystyle 0\,.$ (5.6)

The new identities (5.5) and (5.6) can be checked by using e.g. the Robertson-
Walker metric written as

$ds^{2}=-dt^{2}+a^{2}(t)\big{(}dr^{2}+b^{2}(r)(d\theta^{2}+\sin^{2}{\theta}\,d\varphi^{2})\big{)}\,,$
(5.7)

where $b(r)\equiv\sin{r},r,\sinh{r}$ corresponding to the three spatial –
spherical, Euclidean, hyperboloidal, respectively – geometries. This metric
admits four independent rank $n=2$ KYTs[21], the components of which read

$\displaystyle f_{(1)\,\theta r}=2a^{3}b\sin{\varphi}\,,\;f_{(1)\,\varphi
r}=a^{3}b\cos{\varphi}\sin{2\theta}\,,\;f_{(1)\,\theta\varphi}=2a^{3}b^{2}b^{\prime}\cos{\varphi}\sin^{2}{\theta}\,;$
$\displaystyle f_{(2)\,r\theta}=2a^{3}b\cos{\varphi}\,,\;f_{(2)\,\varphi
r}=a^{3}b\sin{\varphi}\sin{2\theta}\,,\;f_{(2)\,\theta\varphi}=2a^{3}b^{2}b^{\prime}\sin{\varphi}\sin^{2}{\theta}\,;$
$\displaystyle
f_{(3)\,r\varphi}=2a^{3}b\sin^{2}{\theta}\,,\;f_{(3)\,\theta\varphi}=a^{3}b^{2}b^{\prime}\sin{2\theta}\,;$
$\displaystyle f_{(4)\,\theta\varphi}=2a^{3}b^{3}\sin{\theta}\,.$ (5.8)

Here we have omitted the arguments of the metric functions $a$ and $b$, and
used a prime over $b$ to indicate differentiation with respect to $r$. One can
show separately for each KYT (5.8) that (5.5) and (5.6) (as well as (5.3) and
(5.4), of course) are satisfied for the Robertson-Walker metric.

As for another example, one can consider the Kantowski-Sachs metric in $D=4$:

$ds^{2}=-dt^{2}+X^{2}(t)dr^{2}+Y^{2}(t)(d\theta^{2}+\sin^{2}{\theta}\,d\varphi^{2})\,.$
(5.9)

This is a solution of the Einstein field equations for dust and admits the
rank $n=2$ KYT[21] with a single component

$f_{\theta\varphi}=2Y^{3}(t)\sin{\theta}\,.$ (5.10)

It follows easily that (5.5) and (5.6) (as well as (5.3) and (5.4), of course)
are satisfied for the Kantowski-Sachs metric.

###  5.2 The electromagnetic field

The electromagnetic stress tensor is given by

$T_{ab}=F_{ac}F_{b}\,^{c}-\frac{1}{4}g_{ab}F_{de}F^{de}\,.$ (5.11)

From the Einstein field equations, one must again have that (5.2) is
satisfied. Using $\nabla_{[a}F_{bc]}=0$ carefully, this yields

$\nabla^{a}T_{ab}=(\nabla^{a}F_{ac})F_{b}\,^{c}=0\,.$ (5.12)

If Maxwell’s equations admit a current, then they read

$\nabla^{a}F_{ab}=j_{b}\,,$ (5.13)

and (5.12) can be thought of as $F^{bc}j_{c}=0$, a non-trivial requirement to
be satisfied by the components of the current. For a nontrivial solution for
the current $j_{c}$, the “coefficients” $F^{bc}$ must be such that ${\rm
det}(F^{bc})=0$888In $D=4$, one has ${\rm
det}(F^{ab})\sim(F_{ab}\,^{*}F^{ab})^{2}$, of course.. Put in another way, one
must have $\nabla^{a}F_{ab}=0$ provided ${\rm det}(F^{bc})\neq 0$.

If the spacetime of interest admits a KYT of rank $n=2$, then (4.3), or
equivalently (3.9) which becomes $f^{ab}\nabla_{a}T_{bc}=0$, imposes

$(f^{ab}\nabla_{a}F_{bd})F_{c}\,^{d}+\frac{3}{2}F^{bd}\nabla_{a}\big{(}f^{a}\,_{[b}\,F_{cd]}\big{)}=0\,.$
(5.14)

The celebrated Kerr-Newman solution in $D=4$ is an example for which the new
identities put forward can be checked. The metric and the vector potential are
given by

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\left(\frac{\Delta-a^{2}\sin^{2}\theta}{\Sigma}\right)dt^{2}-\frac{2a\sin^{2}\theta\left(r^{2}+a^{2}-\Delta\right)}{\Sigma}\,dt\,d\phi$
(5.15)
$\displaystyle+\left(\frac{\left(r^{2}+a^{2}\right)^{2}-\Delta\,a^{2}\sin^{2}\theta}{\Sigma}\right)\sin^{2}\theta\,d\phi^{2}+\frac{\Sigma}{\Delta}\,dr^{2}+\Sigma\,d\theta^{2}\,,$
$\displaystyle A_{a}\,dx^{a}$ $\displaystyle=$
$\displaystyle-\frac{qr}{\Sigma}\left(dt-a\sin^{2}\theta\,d\phi\right)\,,$
(5.16)

where

$\Sigma=r^{2}+a^{2}\cos^{2}\theta\qquad\mbox{and}\qquad\Delta=r^{2}+a^{2}+q^{2}-2Mr\,.$
(5.17)

One has $G_{ab}=2T_{ab}$ and $\nabla^{a}F_{ab}=0$ here, with
$F_{ab}=2\partial_{[a}A_{b]}$ as usual. Kerr-Newman metric shares the same
rank $n=2$ KYT with the Kerr metric, i.e. (5.15) with $q=0$. Its components
explicitly read

$f_{rt}=a\,\cos{\theta}\,,\;\;f_{t\theta}=ar\sin{\theta}\,,\;\;f_{\phi
r}=a^{2}\cos{\theta}\,\sin^{2}\theta\,,\;\;f_{\theta\phi}=r(r^{2}+a^{2})\sin{\theta}\,.$
(5.18)

One can show explicitly that the identities (3.3), (3.9), (4.3) (together with
(4.2)) and (5.14) are all nontrivially satisfied for the Kerr-Newman metric.

##  6 The KT-current

In this section we discuss under what condition conservation of a general rank
$n$ KT-current gives rise to asymptotically conserved charges, rewrite the KT-
current in terms of the Weyl and Schouten tensors and show that this current
vanishes for rank $n=D-1$ KYTs in $D$ dimensions.

In [15], a covariantly conserved current999We shall refer to (6.1) as the _KT-
current_ henceforth. was constructed

$\displaystyle j^{a_{1}\dots
a_{n}}=-\frac{(n-1)}{4}\,R^{[a_{1}a_{2}}\,_{bc}\,f^{a_{3}\dots
a_{n}]bc}+(-1)^{n+1}\,R_{c}\,^{[a_{1}}\,f^{a_{2}\dots
a_{n}]c}-\frac{1}{2n}\,R\,f^{a_{1}\dots a_{n}}\,,$ (6.1)

with $\nabla_{a_{1}}j^{a_{1}\dots a_{n}}=0$, for a spacetime that admits a
rank $n$ KYT. To show the conservation of $j^{a_{1}\dots a_{n}}$ the following
Bianchi identities are needed:

$\displaystyle\nabla_{[a}R_{bc]}{}^{de}=0\,,\quad\nabla_{a}R_{bc}{}^{da}+2\nabla_{[b}R_{c]}^{~{}d}=0\,,\quad\nabla_{a}R^{a}{}_{b}-{\textstyle{\frac{1}{2}}}\nabla_{b}R=0\,.$
(6.2)

A look at the newly found current (4.8) shows that one can in fact split the
KT-current into two separately covariantly conserved pieces. To see this,
introduce

$\displaystyle\tilde{K}^{a_{1}\dots
a_{n}}=-\frac{(n-1)}{4}\,R^{[a_{1}a_{2}}\,_{bc}\,f^{a_{3}\dots
a_{n}]bc}+\frac{1}{2n}\,R\,f^{a_{1}\dots a_{n}}\,,$ (6.3)

with $\nabla_{a_{1}}\tilde{K}^{a_{1}\dots a_{n}}=0$ and write the KT-current
as [24]

$\definecolor{tcbcol@origin}{rgb}{0,0,0}\definecolor{.}{rgb}{0,0,0}\definecolor{.}{rgb}{0,0,0}\leavevmode\hbox
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to0.0pt{{}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{}{}{}{}{}\definecolor[named]{pgffillcolor}{rgb}{0.25,0.25,0.25}\pgfsys@color@gray@fill{0.25}\pgfsys@invoke{
}\pgfsys@fill@opacity{1.0}\pgfsys@invoke{
}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{0.0pt}{4.26788pt}\pgfsys@lineto{0.0pt}{26.88518pt}\pgfsys@curveto{0.0pt}{29.2423pt}{1.91077pt}{31.15306pt}{4.26788pt}{31.15306pt}\pgfsys@lineto{121.65031pt}{31.15306pt}\pgfsys@curveto{124.00743pt}{31.15306pt}{125.9182pt}{29.2423pt}{125.9182pt}{26.88518pt}\pgfsys@lineto{125.9182pt}{4.26788pt}\pgfsys@curveto{125.9182pt}{1.91077pt}{124.00743pt}{0.0pt}{121.65031pt}{0.0pt}\pgfsys@lineto{4.26788pt}{0.0pt}\pgfsys@curveto{1.91077pt}{0.0pt}{0.0pt}{1.91077pt}{0.0pt}{4.26788pt}\pgfsys@closepath\pgfsys@fill\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{}{}{}{}{}\definecolor[named]{pgffillcolor}{rgb}{0.95,0.95,0.95}\pgfsys@color@gray@fill{0.95}\pgfsys@invoke{
}\pgfsys@fill@opacity{1.0}\pgfsys@invoke{
}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}\pgfsys@moveto{1.42262pt}{4.26788pt}\pgfsys@lineto{1.42262pt}{26.88518pt}\pgfsys@curveto{1.42262pt}{28.45659pt}{2.69647pt}{29.73044pt}{4.26788pt}{29.73044pt}\pgfsys@lineto{121.65031pt}{29.73044pt}\pgfsys@curveto{123.22173pt}{29.73044pt}{124.49557pt}{28.45659pt}{124.49557pt}{26.88518pt}\pgfsys@lineto{124.49557pt}{4.26788pt}\pgfsys@curveto{124.49557pt}{2.69647pt}{123.22173pt}{1.42262pt}{121.65031pt}{1.42262pt}\pgfsys@lineto{4.26788pt}{1.42262pt}\pgfsys@curveto{2.69647pt}{1.42262pt}{1.42262pt}{2.69647pt}{1.42262pt}{4.26788pt}\pgfsys@closepath\pgfsys@fill\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@fill@opacity{1.0}\pgfsys@invoke{
}{{{}}{{}}{{}}{{}}{{}}{{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.64897pt}{12.20842pt}\pgfsys@invoke{
}\hbox{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\hbox{\set@color{$\displaystyle
j^{a_{1}\dots a_{n}}=\tilde{K}^{a_{1}\dots a_{n}}+(-1)^{n-1}K^{a_{1}\dots
a_{n}}$}}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,.$ (6.4)

###  6.1 AD charges for the KT-current

A covariantly conserved antisymmetric rank $n$ tensor field is equivalent to a
co-closed $n$-form. By the extension of the Poincaré lemma to the exterior co-
derivative this means that it is equal to the co-derivative of an $(n+1)$-form
in an open set, under quite general conditions on this set. In what follows we
apply this fact to the background geometry to construct conserved charges for
linearized currents.

In[15, 25], the existence of asymptotic charges based on the KT-current was
shown for asymptotically flat and asymptotically AdS geometries. The method is
a generalisation of the idea of employing asymptotic Killing vectors [17] to
define the corresponding conserved charges.

We first treat the current based on a rank-2 KYT. So consider a
$D$-dimensional spacetime $\bar{g}_{ab}$, which is often referred to as “the
background spacetime” with a completely antisymmetric rank-2 KYT
$\bar{f}_{ab}$ satisfying

${\bar{\nabla}}_{a}\,\bar{f}_{bc}+{\bar{\nabla}}_{b}\,\bar{f}_{ac}=0\,.$ (6.5)

Now the spacetime $g_{ab}$ whose _new_ Killing-Yano charge(s) we are after
does not necessarily have to admit exact KYTs. We assume that the metric
$g_{ab}$ can be asymptotically split into a background plus a deviation as

$g_{ab}\equiv\bar{g}_{ab}+h_{ab}\quad\mbox{so that}\quad
g^{ab}=\bar{g}^{ab}-h^{ab}+{\cal O}(h^{2})\,,$ (6.6)

where $h^{ab}=\bar{g}^{ac}h_{cd}~{}\\!\bar{g}^{db}$. In what follows, all
indices are raised and lowered with the generic background metric
$\bar{g}_{ab}$, e.g. $h\equiv\bar{g}^{ab}h_{bc}$ and $\bar{{\vrule
height=7.22743pt,depth=0.72229pt,width=0.72229pt\vrule
height=0.0pt,depth=0.72229pt,width=7.22743pt\hbox to0.0pt{\hss\vrule
height=7.22743pt,depth=-6.50514pt,width=7.94972pt}\hskip-0.72229pt\vrule
height=7.22743pt,depth=0.72229pt,width=0.72229pt}}\equiv{\bar{\nabla}}^{c}\,{\bar{\nabla}}_{c}$.
To ${\cal O}(h)$ his leads to the following linearized curvature, Ricci tensor
and curvature scalar:

$\displaystyle(R_{ab}{}^{cd})_{L}$ $\displaystyle=$
$\displaystyle\bar{R}_{abe}{}^{[c}h^{d]e}+2\,\bar{\nabla}_{[a}\bar{\nabla}^{[d}h_{b]}{}^{c]}\,,$
(6.7) $\displaystyle(R^{a}{}_{b})_{L}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\bar{\nabla}^{c}\bar{\nabla}^{a}h_{bc}+\bar{\nabla}_{c}\bar{\nabla}_{b}h^{ac}-\bar{\nabla}^{a}\bar{\nabla}_{b}h-\bar{{\vrule
height=7.22743pt,depth=0.72229pt,width=0.72229pt\vrule
height=0.0pt,depth=0.72229pt,width=7.22743pt\hbox to0.0pt{\hss\vrule
height=7.22743pt,depth=-6.50514pt,width=7.94972pt}\hskip-0.72229pt\vrule
height=7.22743pt,depth=0.72229pt,width=0.72229pt}}h^{a}{}_{b}\right)-h^{ac}\bar{R}_{bc}\,,$
(6.8) $\displaystyle R_{L}$ $\displaystyle=$
$\displaystyle{\bar{\nabla}}_{a}{\bar{\nabla}}_{b}h^{ab}-\bar{{\vrule
height=7.22743pt,depth=0.72229pt,width=0.72229pt\vrule
height=0.0pt,depth=0.72229pt,width=7.22743pt\hbox to0.0pt{\hss\vrule
height=7.22743pt,depth=-6.50514pt,width=7.94972pt}\hskip-0.72229pt\vrule
height=7.22743pt,depth=0.72229pt,width=0.72229pt}}h-h^{ab}\bar{R}_{ab}\,.$
(6.9)

To see if the linearised KT-current is conserved, we shall need the following
versions of the identities (6.2) that hold modulo terms of ${\cal O}(h^{2})$
and higher:

$\displaystyle\bar{\nabla}_{[a}(R_{bc]}{}^{de})_{L}+(\Gamma_{[a})_{L}\cdot\big{(}\bar{R}_{bc]}{}^{de}\big{)}=0\,,$
$\displaystyle\bar{\nabla}_{a}(R_{bc}{}^{da})_{L}+2\bar{\nabla}_{[b}(R_{c]}^{~{}d})_{L}+(\Gamma_{a})_{L}\cdot\big{(}\bar{R}_{bc}{}^{da}\big{)}+2(\Gamma_{[b})_{L}\cdot\big{(}\bar{R}_{c]}^{~{}d}\big{)}=0\,,$
(6.10)
$\displaystyle\bar{\nabla}_{a}(R^{a}{}_{b})_{L}-{\textstyle{\frac{1}{2}}}\bar{\nabla}_{b}R_{L}+(\Gamma_{a})_{L}\cdot\big{(}\bar{R}^{a}{}_{b}\big{)}=0\,.$

Here $(\Gamma_{a})\cdot$ denotes the usual action of a connection on a tensor
as exemplified by

$\displaystyle(\Gamma_{a})\cdot(T^{b}{}_{c})=\Gamma^{b}{}_{ae}T^{e}{}_{c}-\Gamma^{e}{}_{ac}T^{b}{}_{e}\,.$
(6.11)

The linearized connection is

$\displaystyle(\Gamma_{~{}ab}^{c})_{L}={\textstyle{\frac{1}{2}}}\bar{g}^{ce}\big{(}\bar{\nabla}_{a}h_{be}+\bar{\nabla}_{b}h_{ae}-\bar{\nabla}_{e}h_{ab}\big{)}\,.$
(6.12)

Note that for flat or maximally symmetric backgrounds, the relations (6.10)
become the same as (6.2) with all curvature objects replaced by their
linearized counterparts. It is this form that is needed for background
conservation of the linearised current. We shall also need the assumption that
$g_{ab}$ asymptotically admits KYTs due to this splitting and that $h_{ab}$
vanishes sufficiently fast at the hypersurface of interest $\Sigma$ (see
(6.15) below) which is used for defining the charges. When the linearized
connection terms in (6.10) vanish, the current $j^{ab}$ is background
covariantly conserved, i.e. $\bar{\nabla}_{a}(j^{ac})_{L}=0$. Since the
current is antisymmetric, the covariant conservation is expected to give rise
to an ordinary conservation law via

$\bar{\nabla}_{a}(j^{ac})_{L}=\frac{1}{\sqrt{|\bar{g}|}}\,\partial_{a}\big{(}\sqrt{|\bar{g}|}\,(j^{ac})_{L}\big{)}=0\,.$
(6.13)

From this we infer as usual that the integral

$\displaystyle\int d^{D-1}x\,{\sqrt{|\bar{g}|}\,(j^{0b}})_{L}$ (6.14)

is constant. In [15, 25] the latter is turned into a flux integral over a
$(D-3)$-dimensional hypersurface101010For a rank $n$ KYT, the analogous step
involves an integral over a hypersurface of dimension $D-1-n$. by further
invoking the Stokes’ theorem: The crucial step is the determination of the
potential for the current, as described in the beginning of this section. We
thus need to express $(j^{ac})_{L}$ as the divergence of a completely
antisymmetric rank-3 tensor
$(j^{ac})_{L}={\bar{\nabla}}_{d}\,\bar{\ell}^{acd}\,,$ where
$\bar{\ell}^{acd}=\bar{\ell}^{[acd]}$. Then, up to a trivial normalization,
the conserved “charge” can be obtained by

$Q^{ac}\sim\int_{\Sigma}\,dS_{i}\,\sqrt{|\bar{\gamma}|}\,\bar{\ell}^{aci}\,,$
(6.15)

where $i$ ranges over the $(D-3)$-dimensional hypersurface $\Sigma$ and
$\gamma$ is the induced metric on $\Sigma$.

The asymptotic charges for the KT-current were given in [15] for an arbitrary
rank $n$ KYT in an asymptotically flat background and in [25] for an arbitrary
rank $n$ KYT in a maximally symmetric background. Their existence again rests
on the KT-current being expressible as the covariant divergence of an
$(n+1)$-form. Since the construction of such a potential is nontrivial, here
we complement this discussion by deriving a condition that the background has
to satisfy for such an $(n+1)$-form to exist.

Following [15], the general rank KT-current can be written as

$\displaystyle j^{a_{1}\dots a_{n}}=N_{n}\,\delta^{a_{1}\dots
a_{n}d_{1}d_{2}}_{b_{1}\dots b_{n}c_{1}c_{2}}\,f^{b_{1}\dots
b_{n}}\,R_{d_{1}d_{2}}{}^{c_{1}c_{2}}\,,$ (6.16)

where $\delta^{a_{1}\dots a_{m}}_{b_{1}\dots
b_{m}}=\delta^{[a_{1}}_{b_{1}}\cdots\delta^{a_{m}]}_{b_{m}}$ is totally
antisymmetric in all up and down indices, and

$\displaystyle N_{n}=-\frac{(n+1)(n+2)}{4n}\,.$ (6.17)

As explained above, we are only interested in the linearized part of (6.16)
and find

$\displaystyle(j^{a_{1}\dots a_{n}})_{L}=N_{n}\,\delta^{a_{1}\dots
a_{n}d_{1}d_{2}}_{b_{1}\dots b_{n}c_{1}c_{2}}\,\bar{f}^{b_{1}\dots
b_{n}}\,(R_{d_{1}d_{2}}{}^{c_{1}c_{2}})_{L}\,.$ (6.18)

In terms of the linearized Riemann tensor in (6.7), the current may be written

$\displaystyle(j^{a_{1}\dots a_{n}})_{L}=N_{n}\,\delta^{a_{1}\dots
a_{n}d_{1}d_{2}}_{b_{1}\dots b_{n}c_{1}c_{2}}\,\bar{f}^{b_{1}\dots
b_{n}}\,\big{(}\bar{R}_{d_{1}d_{2}e}{}^{[c_{1}}\,h^{c_{2}]e}+2\,\bar{\nabla}_{d_{1}}\bar{\nabla}^{c_{2}}h_{d_{2}}{}^{c_{1}}\big{)}\,.$
(6.19)

In [15] it is shown that, for a flat background, this may be written as

$\displaystyle(j^{a_{1}\dots
a_{n}})_{L}=\bar{\nabla}_{e}\,\bar{\ell}^{ea_{1}\dots a_{n}}$ (6.20)

where the $(n+1)$-form $\bar{\ell}^{ea_{1}\dots
a_{n}}=\bar{\ell}^{[ea_{1}\dots a_{n}]}$ is

$\displaystyle\bar{\ell}^{ea_{1}\dots a_{n}}=2N_{n}\,\delta^{a_{1}\dots
a_{n}ed_{2}}_{b_{1}\dots b_{n}c_{1}c_{2}}\,\bar{f}^{b_{1}\dots
b_{n}}\,\bar{\nabla}^{c_{2}}\,h_{d_{2}}{}^{c_{1}}-\frac{1}{2n}\Big{(}h\,\bar{\nabla}^{e}\,\bar{f}^{a_{1}\dots
a_{n}}-(n+1)\,h^{d_{2}[e}\,\bar{\nabla}_{d_{2}}\,\bar{f}^{a_{1}\dots
a_{n}]}\Big{)}\,.$ (6.21)

Similar manipulations as in [15] give the following result for the general
case111111Note that there are no additional curvature terms generated in the
process.

$\displaystyle(j^{a_{1}\dots
a_{n}})_{L}=\bar{\nabla}_{e}\,\bar{\ell}^{ea_{1}\dots
a_{n}}+N_{n}\Big{(}\bar{f}^{[a_{1}\dots
a_{n}}\,\bar{R}_{c_{1}c_{2}e}{}^{c_{1}}\,h^{c_{2}]e}+2\,h_{c_{2}}{}^{[c_{1}}\,\bar{\nabla}^{c_{2}}\,\bar{\nabla}_{c_{1}}\,\bar{f}^{a_{1}\dots
a_{n}]}\Big{)}$ (6.22)

with $\bar{\ell}$ as in (6.21).

Using (2.3) and the explicit antisymmetrisation, vanishing of the terms in
parenthesis (6.22) can be expressed in terms of the background curvature
as121212When $n=1$, (6.23) simply reads
$h\bar{R}^{ab}\bar{f}_{b}-h^{bc}\bar{R}_{bc}\bar{f}^{a}=0$.

(6.23)

For the KT construction of asymptotic charges, the condition (6.23) has to
hold. It is clearly fulfilled for the flat case which leads to the results in
[15]. For a maximally symmetric background

$\bar{R}_{abcd}=\frac{2\Lambda}{(D-1)(D-2)}\,(\bar{g}_{ac}\,\bar{g}_{bd}-\bar{g}_{ad}\,\bar{g}_{bc})\,,\;\;\bar{R}_{ab}=\frac{2\Lambda}{(D-2)}\,\bar{g}_{ab}\,,\;\;\bar{R}=\frac{2\Lambda
D}{(D-2)}\,,\;\;\bar{G}_{ab}=-\Lambda\,\bar{g}_{ab}\,.$

(6.23) is also fulfilled and leads to the results in [25]. This agrees with
the known cases where the linearized Bianchi identities (6.10) ensure
conservation of the KT-current.

Using the expansion of the Riemann tensor in terms of the Weyl and Schouten
tensors, given in (6.25) below, (6.23) may alternatively be written as

$\displaystyle\bar{f}^{[a_{1}\dots
a_{n}}\,\bar{C}_{c_{1}c_{2}e}{}^{c_{1}}\,h^{c_{2}]e}+2(-1)^{n}h_{c_{2}}{}^{[c_{2}}\bar{C}_{e}{}^{c_{1}}{}_{c_{1}}{}^{a_{1}}\,\bar{f}^{a_{2}\dots
a_{n}]e}\qquad\qquad\qquad$
$\displaystyle-\textstyle{\frac{2(D-(n+1))}{n+2}}\,\left(\bar{f}^{[a_{1}\dots
a_{n}}\,h^{d_{1}]d_{2}}\,\bar{S}_{d_{1}d_{2}}+h_{d_{1}}{}^{[a_{1}}\,\bar{S}_{d_{2}}{}^{d_{1}}\,\bar{f}^{|d_{2}|a_{2}\dots
a_{n}]}\right)$ $\displaystyle=0\,.$ (6.24)

This expression may be further simplified using the traceless property of the
Weyl tensor.

###  6.2 KT-current in terms of the Weyl and Schouten tensors

It is also instructive to rewrite the KT-current (6.1) using the decomposition
of the Riemann tensor in terms of the Weyl tensor $C$ and the Schouten tensor
$S$

$\displaystyle S_{ab}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{(D-2)}\left(R_{ab}-\frac{1}{2(D-1)}R\,g_{ab}\right)\,,$
$\displaystyle R_{ab}$ $\displaystyle=$
$\displaystyle(D-2)S_{ab}+S\,g_{ab}\quad\mbox{with}\quad S\equiv
g^{ab}S_{ab}\;\;\mbox{so that}\;\;R=2(D-1)S\,,$ $\displaystyle R^{ab}{}_{cd}$
$\displaystyle=$ $\displaystyle
C^{ab}{}_{cd}+4\delta^{[a}{}_{[c}\,S^{b]}{}_{d]}\quad\mbox{and}\quad
G_{ab}=(D-2)\left(S_{ab}-S\,g_{ab}\right)\,.$ (6.25)

These let one express (6.1) alternatively as

$\displaystyle j^{a_{1}\dots a_{n}}$ $\displaystyle=$
$\displaystyle-\frac{(n-1)}{4}\,C^{[a_{1}a_{2}}\,_{bc}\,f^{a_{3}\dots
a_{n}]bc}+(-1)^{n-1}\,\left(\frac{D-(n+1)}{D-2}\right)\,R_{c}\,^{[a_{1}}\,f^{a_{2}\dots
a_{n}]c}$ (6.26)
$\displaystyle+\left(\frac{n-1}{2(D-1)(D-2)}-\frac{1}{2n}\right)\,R\,f^{a_{1}\dots
a_{n}}\,,$ $\displaystyle=$
$\displaystyle-\frac{(n-1)}{4}\,C^{[a_{1}a_{2}}\,_{bc}\,f^{a_{3}\dots
a_{n}]bc}$
$\displaystyle+(D-(n+1))\,\left((-1)^{n-1}\,S_{c}\,^{[a_{1}}\,f^{a_{2}\dots
a_{n}]c}-\frac{1}{n}\,S\,f^{a_{1}\dots a_{n}}\right)\,.$

The latter equality shows that when the rank $n=D-1$, the KT-current (6.1)
reduces to

$j^{a_{1}\dots
a_{n}}\Big{|}_{n=D-1}=-\frac{(D-2)}{4}\,C^{[a_{1}a_{2}}\,_{bc}\,f^{a_{3}\dots
a_{n}]bc}\,.$ (6.27)

Note also that since the Weyl tensor vanishes identically in $D=3$, so does
the whole KT-current $j^{ab}$ for $n=2$. Moreover, when $D=4$ one has a
special current for a rank $n=3$ KYT from (6.27)

$j^{a_{1}a_{2}a_{3}}=-\frac{1}{2}\,C^{[a_{1}a_{2}}\,_{bc}\,f^{a_{3}]bc}\,.$
(6.28)

In fact one can show that this also vanishes and thus the KT-current does not
exist in this case either. The Hodge dual of a KYT is a closed conformal
Killing tensor (KT) [26]. In particular this means that a rank $n=D-1$ KYT is
dual to a closed conformal Killing vector, (defined in (7.9) below), as
discussed in [27]. We thus first dualize the $n=3$ KYT to a closed conformal
Killing vector $\tilde{f}_{a}$ (defined in (7.9) below) to write

$\tilde{f}_{a}=\frac{\sqrt{|g|}}{3!}\,\epsilon_{a}{}^{bcd}\,{f}_{bcd}\quad\Rightarrow\quad{f}_{bcd}=\tilde{f}_{a}\,\epsilon^{a}{}_{bcd}$
(6.29)

satisfying

$\nabla_{a}\tilde{f}_{b}=\frac{1}{4}(\nabla_{c}\tilde{f}^{c})\,g_{ab}\,.$
(6.30)

Dualizing also $j^{a_{1}a_{2}a_{3}}$, we may then write the relation (6.28) up
to some signs and factors as

$\displaystyle\epsilon_{da_{1}a_{2}a_{3}}j^{a_{1}a_{2}a_{3}}\sim\epsilon_{da_{1}a_{2}a_{3}}\,C^{a_{1}a_{2}}{}_{bc}\,\epsilon^{a_{3}bce}\,\tilde{f}_{e}\,.$
(6.31)

Using the formula for contracting one index on the Levi-Civita symbol and the
traceless property of the Weyl tensor then shows that the right hand side
vanishes, and thus that $j^{a_{1}a_{2}a_{3}}=0$. This can also be seen,
perhaps more directly, from the fact that

$\displaystyle C_{abcd}\,\tilde{f}^{d}=0$ (6.32)

in $D=4$ when $\tilde{f}^{d}$ satisfies (6.29), see e.g. [20]. We have

$\displaystyle
C^{a_{1}a_{2}}{}_{bc}\,\epsilon^{a_{3}bce}\,\tilde{f}_{e}=2\,\tilde{f}_{e}\,C^{a_{1}a_{2}a_{3}e}\,\star=2\,\star
C^{a_{1}a_{2}a_{3}e}\,\tilde{f}_{e}=0\,,$ (6.33)

where we used a relation between the right and left duals of the Weyl tensor
(see, e.g. [28]) and the last equality follows by (6.32).

The condition (6.32) implies that either $\tilde{f}_{e}$ is a null vector or
the space is conformally flat. It is gratifying to see that at least for the
conformally flat case the existence of the charge condition (6.24) also
vanishes.

Clearly the argument leading from (6.28) to (6.33) holds equally well for a
KT-current based on a rank $n=D-1$ KYT in $D$ dimensions, so that such a KT-
current also has to vanish.

##  7 Comments on the KT and related currents

The KT current has many interesting special cases for particular geometries.
We also found that a number of conserved “currents” related to the KT current
can be defined. In this section we summarize these cases for completeness and
collect their interrelations in a table.

We start by reproducing the $n=2$ KT-current (6.1), for convenience:

$\displaystyle-4j^{ab}$ $\displaystyle=$ $\displaystyle
R^{abcd}\,f_{cd}+4R_{c}\,^{[a}\,f^{b]c}+R\,f^{ab}\,.$ (7.1) $\displaystyle=$
$\displaystyle
C^{abcd}\,f_{cd}+2(D-3)\left(2S^{c[a}\,f^{b]}\,_{c}+Sf^{ab}\right)\,,$ (7.2)

It is interesting to note that the expression multiplying $(D-3)$ in (7.2)

$\displaystyle J_{(1)}^{ab}\equiv 2f^{c[a}\,S_{c}{}^{b]}+f^{ab}S$ (7.3)

is conserved for certain geometries. We have

$\displaystyle\nabla_{a}J_{(1)}^{ab}=f^{ca}\,\nabla_{a}S^{b}{}_{c}$ (7.4)

which vanishes when $S_{ab}$ is a Codazzi tensor, i.e. a symmetric 2-tensor
whose covariant derivative is also symmetric

$\displaystyle\nabla_{a}S_{c}{}^{b}=\nabla_{c}S_{a}{}^{b}\,.$ (7.5)

The Weyl-Schouten theorem [29, 30] states that:

> A Riemannian manifold of dimension $D$ with $D\geq 3$ is conformally flat if
> and only if the Schouten tensor is a Codazzi tensor for $D=3$, or the Weyl
> tensor vanishes for $D>3$.

Hence we need a conformally flat metric in $D=3$ for $J_{(1)}^{ab}$ to be
conserved. For higher dimensions, we note that a metric $g$ has a harmonic
Weyl tensor

$\displaystyle\nabla_{a}C_{cd}^{~{}~{}ab}=0~{},$ (7.6)

if and only if its Schouten tensor is a Codazzi tensor. In this case we see
from (7.2) that $j^{ab}$ is the sum of two independently conserved currents,
one proportional to $J_{(1)}^{ab}$ and a new current

$\displaystyle J_{(2)}^{ab}\equiv f^{cd}\,C_{cd}^{~{}~{}ab}~{},$ (7.7)

according to ($D>3$)

$\displaystyle-4j^{ab}=J_{(2)}^{ab}+2(D-3)J_{(1)}^{ab}~{}.$ (7.8)

A rank $n$ conformal Killing-Yano tensor ($CKYT$) $\hat{f}$ obeys

$\displaystyle\nabla_{b}\hat{f}_{a_{1}...a_{n}}=\nabla_{[b}\hat{f}_{a_{1}...a_{n}]}+ng_{b[a_{1}}\bar{f}_{a_{2}...a_{n}]}$
(7.9)

with

$\displaystyle\bar{f}_{a_{2}\dots
a_{n}}\equiv\frac{1}{D-n+1}\nabla_{b}f^{b}_{~{}a_{2}...a_{n}}\,.$ (7.10)

When the first term in (7.9) vanishes, the tensor is called a closed conformal
Killing-Yano tensor (CCKYT). A differential form is a KYT if, and only if, its
Hodge dual is a CCKYT.

The current $J_{(2)}^{ab}$ in (7.7) can be extended to involve a conformal
Yano 2-form $\hat{f}$. When acting on by the covariant derivative

$\displaystyle\nabla_{a}\big{(}\hat{f}_{bc}\,C^{cdaf}\big{)}=\big{(}\nabla_{[a}\hat{f}_{bc]}+2g_{a[b}\bar{f}_{c]}\big{)}C^{cdaf}+\hat{f}_{bc}\nabla_{a}C^{cdaf}\,,$
(7.11)

the first term vanishes due to the anti-symmetrization of $\nabla\hat{f}$
which imposes the first Bianchi identity on $C$, the second vanishes since $C$
is trace-free and the third since the Weyl tensor is harmonic.

It may also be of interest to consider a metric $g$ with a harmonic Riemann
tensor

$\displaystyle\nabla_{a}R_{cd}^{~{}~{}ab}=0~{}.$ (7.12)

This requires the Ricci tensor to be a Codazzi tensor, instead of the Schouten
tensor:

$\displaystyle\nabla_{a}R_{bc}=\nabla_{b}R_{ac}~{}.$ (7.13)

Returning to the form (7.1) for the current $J^{ab}$ we note that

$\displaystyle J_{(3)}^{ab}\equiv f^{cd}\,R_{cd}^{~{}~{}ab}$ (7.14)

satisfies

$\displaystyle\nabla_{a}{J}^{ab}_{(3)}=g_{ae}\nabla^{[e}\,f^{cd]}\,R_{cd}^{~{}~{}ab}+f^{cd}\nabla_{a}R_{cd}^{~{}~{}ab}=0\,,$
(7.15)

where the first term vanishes due to the first Bianchi identity and the second
due to (7.12). Since the full current $j^{ab}$ is conserved, we realize that
writing

$\displaystyle J_{(4)}^{ab}\equiv j^{ab}-J^{ab}_{(3)}$ (7.16)

yields, in analogy to the harmonic Weyl tensor case, a third current, which
must be conserved,

$\displaystyle\nabla_{a}J_{(4)}^{ab}=0~{},$ (7.17)

due to (7.13), which may also be explicitly verified.

Current | Cons. Conditions |  Relation to the KT-current $j^{ab}$
---|---|---
$J^{ab}_{(1)}=2f^{c[a}\,S^{~{}b]}_{c}+f^{ab}S$ |  $S_{ab}$ Codazzi | $-4j^{ab}=C^{abcd}\,f_{cd}+2(D-3)J_{(1)}^{ab}$
$J_{(2)}^{ab}\equiv f^{cd}\,C_{cd}^{~{}~{}ab}$ | Weyl harmonic | $-4j^{ab}=J_{(2)}^{ab}+2(D-3)J_{(1)}^{ab}$
$J_{(3)}^{ab}\equiv f^{cd}\,R_{cd}^{~{}~{}ab}$ | Riemann harmonic | $-4j^{ab}=J_{(3)}^{ab}+4R_{c}\,^{[a}\,f^{b]c}+R\,f^{ab}$
$J_{(4)}^{ab}=4R_{c}\,^{[a}\,f^{b]c}+R\,f^{ab}$ |  $R_{ab}$ Codazzi | $-4j^{ab}=J_{(3)}^{ab}+{J_{(4)}}^{ab}$
Table 1: Relations between various currents in section 7.

##  8 Conclusions and comments

In this paper we have presented new identities for KYTs and shown how they may
be used to find new conserved currents. These currents are all of the Kastor-
Traschen type, i.e. not Noether currents. As shown in [15, 25], such currents
may nevertheless lead to asymptotically conserved charges of AD type. We found
a condition for such conserved charges to exist for the KT-current. We also
displayed the linearized form of the Bianchi identities and pointed out that
only for certain backgrounds do they directly lead to background conserved
linearized KT-currents. An interesting question is if there are other
backgrounds and/or modifications of the current that allow such conservation
using these linearized identities.

For our current $K^{ab}$, based on the Einstein tensor, we investigated this
possibility too and showed that it does not give an AD charge for a maximally
symmetric space time (see appendix A). There are however many more cases, both
currents and backgrounds, that should be studied.

It is particularly interesting to note that we were able to find new conserved
currents for $n>2$ KY forms. These should be relevant for higher dimensional
solutions to Einstein’s equation.

There are several directions into which the present efforts may be extended:
Treating conformal KYTs as we touched upon in the text. Extending the geometry
to allow for torsion which will introduce modified Killing-Yano equations as
in e.g. [8]. This opens up for supersymmetric extensions, such as discussed in
[13].

Acknowledgments
Ö.S. would like to thank D.O. Devecioğlu for help with xAct in the early
stages of this work. The research of U.L. is supported in part by the 2236 Co-
Funded Scheme2 (CoCirculation2) of TÜBİTAK (Project No:120C067)131313However
the entire responsibility for the publication is ours. The financial support
received from TÜBİTAK does not mean that the content of the publication is
approved in a scientific sense by TÜBİTAK.. We are grateful to Dr. Vojtech
Witzany for asking us a question that led to the clarification around (6.10)
and to an anonymous referee for helping us to further refine the related
arguments.

APPENDICES

##  Appendix A No AD charge for $K^{ab}$ in maximally symmetric spacetimes.

This appendix serves as an illustration for the method of deriving AD charges
and its limitations. We adapt and apply the arguments given in subsection
(6.1) to the currents $K^{a}=G^{a}_{~{}c}f^{c}$ and
$K^{ab}=G^{[a}_{~{}c}f^{b]c}$ (4.2) for the maximally symmetric and flat
backgrounds. We show explicitly that only the first can be used in defining
new conserved quantities as done in [16, 25] for the KT-current.

So one starts with a $D$-dimensional background $\bar{g}_{ab}$ admitting a
rank-2 KYT $\bar{f}_{ab}$ satisfying (6.5). For such a maximally symmetric
spacetime, one has

$\bar{R}_{abcd}=\frac{2\Lambda}{(D-1)(D-2)}\,(\bar{g}_{ac}\,\bar{g}_{bd}-\bar{g}_{ad}\,\bar{g}_{bc})\,,\;\;\bar{R}_{ab}=\frac{2\Lambda}{(D-2)}\,\bar{g}_{ab}\,,\;\;\bar{R}=\frac{2\Lambda
D}{(D-2)}\,,\;\;\bar{G}_{ab}=-\Lambda\,\bar{g}_{ab}\,.$

Then one finds the following which are frequently used in the ensuing
calculations:

$\displaystyle{\bar{\nabla}}_{a}\,\bar{f}^{ab}$ $\displaystyle=$
$\displaystyle
0\,,\qquad{\bar{\nabla}}_{a}\,\bar{f}_{bc}={\bar{\nabla}}_{b}\,\bar{f}_{ca}={\bar{\nabla}}_{c}\,\bar{f}_{ab}\,,$
(A.1) $\displaystyle{\bar{\nabla}}_{a}\,{\bar{\nabla}}_{b}\,\bar{f}_{cd}$
$\displaystyle=$
$\displaystyle\frac{2\Lambda}{(D-1)(D-2)}(\bar{g}_{ab}\,\bar{f}_{dc}+\bar{g}_{ac}\,\bar{f}_{bd}+\bar{g}_{ad}\,\bar{f}_{cb})\,,$
(A.2) $\displaystyle\bar{{\vrule
height=7.22743pt,depth=0.72229pt,width=0.72229pt\vrule
height=0.0pt,depth=0.72229pt,width=7.22743pt\hbox to0.0pt{\hss\vrule
height=7.22743pt,depth=-6.50514pt,width=7.94972pt}\hskip-0.72229pt\vrule
height=7.22743pt,depth=0.72229pt,width=0.72229pt}}\,\bar{f}_{ab}$
$\displaystyle=$
$\displaystyle\frac{2\Lambda}{(D-1)}\,\bar{f}_{ba}\,,\qquad{\bar{\nabla}}^{a}\,{\bar{\nabla}}_{b}\,\bar{f}_{ac}=\frac{2\Lambda}{(D-1)}\,\bar{f}_{bc}\,.$
(A.3)

The “linearized” version of $K^{a}=G^{a}_{~{}c}f^{c}$ is background
covariantly conserved, i.e. $\bar{\nabla}_{a}(K^{a})_{L}=0$. It should
therefore have a potential $(K^{a})_{L}=\bar{\nabla}_{d}\bar{\ell}^{[da]}$
according to the general argument.

Keeping in mind that all indices are raised and lowered with the maximally
symmetric background metric $\bar{g}_{ab}$, one finds that the linearized
Ricci tensor and the Ricci scalar read141414These easily follow by adapting
(6.9) accordingly to a maximally symmetric background.

$\displaystyle(R_{ab})_{L}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left({\bar{\nabla}}^{c}\,{\bar{\nabla}}_{b}\,h_{ac}+{\bar{\nabla}}^{c}\,{\bar{\nabla}}_{a}\,h_{bc}-\bar{{\vrule
height=7.22743pt,depth=0.72229pt,width=0.72229pt\vrule
height=0.0pt,depth=0.72229pt,width=7.22743pt\hbox to0.0pt{\hss\vrule
height=7.22743pt,depth=-6.50514pt,width=7.94972pt}\hskip-0.72229pt\vrule
height=7.22743pt,depth=0.72229pt,width=0.72229pt}}\,h_{ab}-{\bar{\nabla}}_{a}\,{\bar{\nabla}}_{b}\,h\right)\,,$
(A.4) $\displaystyle R_{L}$ $\displaystyle=$
$\displaystyle{\bar{\nabla}}^{c}\,{\bar{\nabla}}^{d}\,h_{cd}-\bar{{\vrule
height=7.22743pt,depth=0.72229pt,width=0.72229pt\vrule
height=0.0pt,depth=0.72229pt,width=7.22743pt\hbox to0.0pt{\hss\vrule
height=7.22743pt,depth=-6.50514pt,width=7.94972pt}\hskip-0.72229pt\vrule
height=7.22743pt,depth=0.72229pt,width=0.72229pt}}\,h-\frac{2\Lambda}{(D-2)}h\,.$
(A.5)

These further give

$\displaystyle(G^{a}\,_{b})_{L}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left({\bar{\nabla}}^{c}\,{\bar{\nabla}}^{a}\,h_{bc}+{\bar{\nabla}}_{c}\,{\bar{\nabla}}_{b}\,h^{ac}-\bar{{\vrule
height=7.22743pt,depth=0.72229pt,width=0.72229pt\vrule
height=0.0pt,depth=0.72229pt,width=7.22743pt\hbox to0.0pt{\hss\vrule
height=7.22743pt,depth=-6.50514pt,width=7.94972pt}\hskip-0.72229pt\vrule
height=7.22743pt,depth=0.72229pt,width=0.72229pt}}\,h^{a}\,_{b}-{\bar{\nabla}}^{a}\,{\bar{\nabla}}_{b}\,h\right)$
(A.6)
$\displaystyle-\frac{1}{2}\,\delta^{a}\,_{b}\,\left({\bar{\nabla}}_{c}\,{\bar{\nabla}}_{d}\,h^{cd}-\bar{{\vrule
height=7.22743pt,depth=0.72229pt,width=0.72229pt\vrule
height=0.0pt,depth=0.72229pt,width=7.22743pt\hbox to0.0pt{\hss\vrule
height=7.22743pt,depth=-6.50514pt,width=7.94972pt}\hskip-0.72229pt\vrule
height=7.22743pt,depth=0.72229pt,width=0.72229pt}}\,h-\frac{2\Lambda}{(D-2)}h\right)-\frac{2\Lambda}{(D-2)}\,h^{a}\,_{b}\,.$

With this linearized Einstein tensor the current can be rearranged to

$\displaystyle(K^{a})_{L}=(G^{a}\,_{b})_{L}\bar{f}^{b}=\bar{\nabla}_{d}\bar{\ell}^{[da]}~{}.$
(A.7)

Arguments analogous to those given in the discussion surrounding (6.13) can
now be repeated, replacing $(j^{ac})_{L}$ with $(K^{a})_{L}$, and lead to a
conserved charge as in (6.15).

The “linearized” version of the antisymmetric “current”151515As shown in
section 4, $\nabla_{a}K^{ac}=0$ if the spacetime $g_{ab}$ admits a KYT
$f_{ab}$ itself. $K^{ac}=2\,G_{b}\,^{[a}\,f^{c]b}$,

$(K^{ac})_{L}=-(G^{a}\,_{b})_{L}\,\bar{f}^{bc}+(G^{c}\,_{b})_{L}\,\bar{f}^{ba}$
(A.8)

can be similarly treated leading to

$\displaystyle(K^{ac})_{L}$ $\displaystyle=$ $\displaystyle
3\bar{\nabla}_{d}\left\\{\bar{f}^{b[a}\bar{\nabla}^{c}h^{d]}\,_{b}+h_{b}\,^{[d}\bar{\nabla}^{c}\bar{f}^{a]b}+\frac{1}{2}\bar{f}^{[dc}\bar{\nabla}^{a]}h\right\\}$
(A.9)
$\displaystyle+\bar{\nabla}_{d}\left\\{\bar{f}^{bd}\bar{\nabla}^{[c}h^{a]}\,_{b}+h_{b}\,^{d}\bar{\nabla}^{c}\bar{f}^{ba}+\frac{1}{2}\bar{f}^{ca}\bar{\nabla}^{d}h\right.$
$\displaystyle\qquad\quad\left.+\bar{f}^{b[a}\bar{\nabla}_{b}h^{c]d}+\bar{f}^{ac}\bar{\nabla}_{b}h^{bd}-h^{bd}\bar{\nabla}_{b}\bar{f}^{ac}+h\bar{\nabla}^{d}\bar{f}^{ac}\right\\}$
$\displaystyle+\frac{4\Lambda}{(D-1)(D-2)}\left(h\bar{f}^{ca}+2h_{b}\,^{[c}\bar{f}^{a]b}\right)\,.$

The first line is in the desired structure but the remaining parts of (A.9) do
not fulfill the requirements of a proper $\bar{\ell}$. This is so even when
one takes $\Lambda\to 0$, the same choice as in [15], to work in an
asymptotically flat background. This shows that the current $K^{ab}$ (4.2)
does not admit the construction of an $AD$-charge.

In retrospect the reason for this is clear. When defining
$K^{ac}=2\,G_{b}\,^{[a}\,f^{c]b}$, we needed to use (3.3):

${R_{ab}\,f^{ac}+R^{ac}\,f_{ab}=0\,.}$ (A.10)

When $f^{ac}=\bar{f}^{ac}$ is a background KYT, as in (A.8), this holds with
the background Ricci tensor $\bar{R}_{ab}$, and the background current
$\bar{K}^{ac}=2\,\bar{G}_{b}\,^{[a}\,\bar{f}^{c]b}$ is background conserved.
However this will not in general be the case for the linearized current
$(K^{ac})_{L}=2\,(G_{b}\,^{[a})_{L}\,\bar{f}^{c]b}$ since

$(R_{ab})_{L}\,\bar{f}^{ac}+(R^{ac})_{L}\,\bar{f}_{ab}\neq 0\,.$ (A.11)

Indeed the background covariant divergence of (A.9) is easily seen to be
nonvanishing.

##  Appendix B Conservation of $J_{E}$ in conformally flat geometries

In this section, we show that conformal flatness in fact guarantees the
conservation of the current $J_{E}$ in (4.4) for an arbitrary rank $n$ KYT.
Using

$\displaystyle R_{abc_{2}d}\,f_{c_{3}\dots
c_{n}}\,^{ad}=-\frac{1}{2}\,R_{adbc_{2}}\,f_{c_{3}\dots c_{n}}\,^{ad}\,,$
(B.1)

we rewrite (3.5) as

$\displaystyle\frac{(-1)^{n+1}}{2}\Big{(}2R^{d}_{~{}[b}\,f_{c_{2}\dots
c_{n}]d}+(-1)^{n+1}(n-1)\,R^{da}\,_{[bc_{2}}f_{c_{3}\dots c_{n}]ad}\Big{)}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad=(-1)^{n+1}\,R^{d}_{~{}b}\,f_{c_{2}\dots
c_{n}d}-\frac{(n-1)}{2}\,R_{adb[c_{2}}\,f_{c_{3}\dots c_{n}]}\,^{ad}\,.$ (B.2)

Using (6.25) gives

$\displaystyle\frac{(-1)^{n+1}}{2}\Big{(}2R^{d}_{~{}[b}\,f_{c_{2}\dots
c_{n}]d}+(-1)^{n+1}(n-1)\,C^{da}_{~{}~{}[bc_{2}}\,f_{c_{3}\dots
c_{n}]ad}-4(-1)^{n+1}(n-1)S^{d}_{~{}[c_{2}}\,f_{c_{3}\dots c_{n}b]d}\Big{)}$
$\displaystyle=(-1)^{n+1}R^{d}_{~{}b}\,f_{c_{2}\dots
c_{n}d}-\frac{(n-1)}{2}C^{ad}_{~{}~{}b[c_{2}}\,f_{c_{3}\dots
c_{n}]ad}-(n-1)\big{(}S^{d}_{~{}[c_{2}}\,f_{c_{3}\dots
c_{n}]bd}-S^{d}_{~{}b}\,f_{c_{3}\dots c_{n}c_{2}d}\big{)}\,.$ (B.3)

For vanishing Weyl tensor and ignoring the metric terms in $S_{ab}$ (B.3)
becomes

$\displaystyle AR^{d}_{~{}[b}\,f_{c_{2}\dots
c_{n}]d}=BR^{d}_{~{}b}\,f_{c_{2}\dots
c_{n}d}-CR^{d}_{~{}[c_{2}}\,f_{c_{3}\dots c_{n}]bd}\,,$ (B.4)

where

$A\equiv(-1)^{n+1}\big{(}1+2\alpha(1-n)\big{)}\,,\quad
B\equiv(-1)^{n+1}\big{(}1+\alpha(1-n)\big{)}\,,\quad C\equiv-\alpha(1-n)\,,$

with $\alpha=1/(D-2)$ depending on the dimension $D$ of the spacetime
according to (6.25). Observing that

$\displaystyle nR^{d}_{~{}[b}\,f_{c_{2}\dots
c_{n}]d}=R^{d}_{~{}b}\,f_{c_{2}\dots
c_{n}d}-(-1)^{n+1}(n-1)R^{d}_{~{}[c_{2}}\,f_{c_{3}\dots c_{n}]bd}\,,$ (B.5)

(B.4) can be rewritten as

$\displaystyle(Bn-A)R^{d}_{~{}b}\,f_{c_{2}...c_{n}d}=(Cn+(-1)^{n+1}A)R^{d}_{~{}[c_{2}}\,f_{c_{3}...c_{n}]bd}~{},$
(B.6)
$\displaystyle\iff(-1)^{n}\big{(}\alpha(n-1)n-n+1+2\alpha(1-n)\big{)}R^{d}_{~{}b}\,f_{~{}c_{2}...c_{n}d}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}=(n-1)\big{(}-\alpha
n+2\alpha+1\big{)}R^{d}_{~{}[c_{2}}\,f_{c_{3}...c_{n}]bd}$
$\displaystyle\iff(-1)^{n+1}R^{d}_{~{}b}\,f_{c_{2}...c_{n}d}=R^{d}_{~{}[c_{2}}\,f_{c_{3}...c_{n}]bd}$
$\displaystyle\iff
R^{d}_{~{}b}\,f_{c_{2}...c_{n}d}=R^{d}_{~{}[c_{2}}\,f_{bc_{3}...c_{n}]d}~{}.$
(B.7)

This leads to (4.7) which guarantees the conservation of $J_{E}$, provided
that the metric terms in the Schouten tensor also work out. However, from
(B.3) this requires

$\displaystyle-2\delta^{d}_{~{}[c_{2}}\,f_{c_{3}...c_{n}b]d}=-\delta^{d}_{~{}[c_{2}}\,f_{c_{3}...c_{n}]bd}+\delta^{d}_{~{}b}\,f_{c_{3}...c_{n}c_{2}d}~{},$

which indeed holds. So this proves that conformal flatness guarantees the
conservation of the current $J_{E}$ (4.4) for an arbitrary rank $n$ KYT.

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