# Homogeneous Symmetry Operators in Kerr–NUT–AdS Spacetimes

Finnian Gray<EMAIL_ADDRESS>University of Vienna, Faculty of
Physics, Boltzmanngasse 5, A 1090 Vienna, Austria  David Kubizňák
<EMAIL_ADDRESS>Institute of Theoretical Physics, Faculty of
Mathematics and Physics, Charles University, Prague, V Holešovičkách 2, 180 00
Prague 8, Czech Republic

(January 6, 2024)

###### Abstract

It is well known that the Kerr–NUT–AdS spacetimes possess hidden symmetries
encoded in the so-called principal Killing–Yano tensor. In this paper,
focusing on the four-dimensional case, we obtain a number of symmetry
operators for scalar, vector, and tensor perturbations, that are of degree two
(to be defined below) and homogeneous in the principal tensor. In particular,
by considering homogeneous operators that are linear, quadratic, and cubic in
the principal tensor, we recover a complete set of 4 mutually commuting
operators for scalar perturbations, underlying the separability of (massive)
scalar wave equation. Proceeding to vector and tensor perturbations of the
Kerr–NUT–AdS spacetimes, we find a set of 7 and 8 commuting operators,
respectively. It remains to be seen whether such operators can be used to
separate the corresponding spin 1 and spin 2 test field equations in these
spacetimes.

##  I Introduction

Hidden symmetries play an important role in black hole physics. As opposed to
explicit symmetries, that are ‘visible’ in the spacetime and are encoded in
Killing vector fields, hidden symmetries correspond to genuine symmetries of
the phase space, e.g. crampin1984hidden ; Cariglia:2014ysa , and are described
by higher rank tensors that obey a natural generalization of the Killing
vector equation. Among these, perhaps the most remarkable is the hidden
symmetry of the Principal Killing–Yano (PKY) tensor, see Frolov:2017kze  for a
review.

The PKY tensor is a non-degenerate closed conformal Killing–Yano 2-form
$k_{ab}$. This is an object whose covariant derivative is completely encoded
its divergence

$\xi_{a}=\frac{1}{d-1}\nabla^{b}k_{ba}\,,$ (1)

where $d$ stands for the number of spacetime dimensions. This divergence is a
Killing vector by construction and explicitly the PKY tensor satisfies the
following defining equation:

$\nabla_{c}k_{ab}=g_{ca}\xi_{b}-g_{cb}\xi_{a}\,.$ (2)

The principal tensor exists in the Kerr spacetime and its ‘square’ gives rise
to Carter’s famous constant for geodesics Carter:1968rr . More generally, the
PKY tensor can be found in a large class of the (off-shell) Kerr–NUT–AdS
spacetimes in any number of dimensions Kubiznak:2006kt ; Krtous:2008tb ;
Houri:2008ng , and underlies many of its remarkable properties, such as
complete integrability of the geodesic equations Page:2006ka , separability of
the Hamilton–Jacobi and Klein–Gordon equations Frolov:2006pe , separability of
the Dirac equation Oota:2007vx , or the special algebraic type of these
metrics Mason:2010zzc .

More recently Lunin:2017drx ; Frolov:2018ezx , it was also shown how, using
the PKY tensor, one can construct a ‘polarization tensor’
$B_{ab}=(g_{ab}+\lambda k_{ab})^{-1}$, which gives rise to a separable ansatz
for the (massive) vector perturbations:

$A^{a}=B^{ab}\partial_{b}Z\,,$ (3)

where $Z$ is the scalar function that takes a standard separable form, and
$\lambda$ is the additional separation constant, see Frolov:2018ezx  for
details.

Interestingly, in the scalar Carter:1977pq ; Sergyeyev:2007gf , conformal
scalar Gray:2020rtr ; Gray:2021wzf , and the Dirac Carter:1979fe ;
Cariglia:2011qb  cases, the separation can be linked to the existence of a
compete set of the mutually commuting operators (called symmetry operators)
whose common eigenfunction is the separable solution. Remarkably, such
operators take a simple form that is homogeneous (linear, quadratic, cubic,
and so on) in the principal tensor and contains a fixed number of derivatives
(some of which may act on the principal tensor while others are free act on
the field). We call this number the degree of the corresponding operator. In
this work, we study such homogeneous operators and restrict ourselves to
degree two. Such operators can be of the first-order or second-order in the
number of derivatives acting on the field.

For example, as we shall review in Sec. III the separability of the massive
scalar equation is characterized by the existence of 3 homogeneous degree two
operators that are linear, quadratic, and cubic in the PKY tensor. Together
with the scalar wave operator itself, these form a complete set of mutually
commuting operators that underlie the separability of the scalar field
equations. As discussed in App. A, similar result remains true also for the
Dirac case.

For vector and tensor perturbations, however, the situation is less
straightforward. While such perturbations can be tackled with the help of the
Teukolsky equation Teukolsky:1973ha ; Teukolsky:1972my , the relationship to
the principal tensor and whether there exists an underlying set of mutually
commuting operators is much less clear in this case. See Andersson:2014lca ;
Aksteiner:2016pjt ; Michishita:2019eeh ; Aksteiner:2016mol ; Jacobsson:2022mls
for related works on symmetry operators for various test field equations.
Moreover, recent work Houri:2019lnu  constructs the required symmetry
operators for the Maxwell case, using the equations resulting from the ansatz
(3). However, recently there has been a new breakthrough achieved in
Mei:2023pho , where some homogeneous operators were constructed directly for
the field equations of the vector and tensor perturbations of the vacuum Kerr
spacetime.

It is the aim of this paper, to systematically study such homogeneous symmetry
operators, picking up the threads of this recent work Mei:2023pho . In this
way we discover novel symmetry operators for the vector and tensor
perturbations of the Kerr–NUT–AdS spacetimes. Notably, contrasting with the
recent work, our basic building block is directly the principal tensor $k$,
instead of its dual Killing–Yano tensor $f=*k$. Since the PKY tensor comes to
the fore for Kerr–NUT–AdS spacetimes in arbitrary number of dimensions, our
results are readily generalizable to $d$ dimensions (with possibly
$d$-dependent) factors, thus providing symmetry operators for black hole
perturbations in higher dimensions myprep:2024 .

Our work is organized as follows. In the next section, focusing on four
dimensions, we overview the (off-shell) Kerr–NUT–AdS spacetimes and their
remarkable symmetries. In Sec. III we review the homogeneous symmetry
operators for the (massive) scalar field, their systematic derivation, and
application to separability of the scalar field equation. In Sec. IV we
systematically study symmetry operators for the (massive) vector
perturbations, while Sec. V is devoted to the massive tensor perturbations. We
conclude in Sec. VI. In App. A we review the symmetry operators for the Dirac
equation and its separability. App. B contains analogous study of constants of
motion for geodesic and classical spinning particle trajectories. App. C
explores the possibility of using the Hodge dual of the PKY tensor in the
study of linear vector symmetry operators.

##  II Kerr–NUT–AdS and its hidden symmetries

In this paper we shall study various perturbations of four-dimensional
Kerr–NUT–AdS spacetimes, which we write in a convenient Carter-like form
Carter:1968cmp :

$\displaystyle g$ $\displaystyle=$
$\displaystyle-\frac{\Delta_{r}}{\Sigma}(d\tau+y^{2}d\psi)^{2}+\frac{\Delta_{y}}{\Sigma}(d\tau-r^{2}d\psi)^{2}$
(4)
$\displaystyle+\frac{\Sigma}{\Delta_{r}}dr^{2}+\frac{\Sigma}{\Delta_{y}}dy^{2}\,,$

where

$\Sigma=\sqrt{-g}=r^{2}+y^{2}\,,$ (5)

and the metric functions $\Delta_{r}$ and $\Delta_{y}$ are functions of one
variable:

$\Delta_{y}=\Delta_{y}(y)\,,\quad\Delta_{r}=\Delta_{r}(r)\,.$ (6)

In what follows we shall consider three cases. i) The off-shell canonical
metric, for which the metric functions take the above general form (6). Many
of the geometric properties remain valid in this general case Frolov:2017kze .
ii) The Einstein space case, when the metric solve the vacuum Einstein
equations with the cosmological constant $\Lambda$, and the corresponding
metric functions take the following specific form:

$\begin{split}\Delta_{r}&=(r^{2}+a^{2})(1-\Lambda r^{2}/3)-2Mr\,,\\\
\Delta_{y}&=(a^{2}-y^{2})(1+\Lambda y^{2}/3)+2Ny\,.\end{split}$ (7)

Here, $M$ stands for the mass parameter, $a$ denotes the rotation, and $N$ is
the Newmann–Unti–Tamburino (NUT) NewmanEtal:1963  parameter. iii) Finally, we
shall consider the (pure) vacuum case, for which the metric functions take the
specific form (7), with

$\Lambda=0\,.$ (8)

The metric then becomes the vacuum solution of the Einstein equations – the
standard Kerr spacetime Kerr:1963ud  equipped with the NUT parameter $N$.

The remarkable property of the off-shell canonical metric (inherited by its
special on-shell subcases) is that it admits the powerful symmetry of the PKY
tensor $k$, obeying (2). Explicitly, it is given by

$k=ydy\wedge(d\tau-r^{2}d\psi)-rdr\wedge(d\tau+y^{2}d\psi)\,.$ (9)

This tensor generates all of the explicit and hidden symmetries, and
determines many remarkable properties of the geometry, many of which prevail
for higher-dimensional Kerr–NUT–AdS spacetimes as well, see Frolov:2017kze .

In particular, it is possible to show that $\xi$, from (1), is given by

$\xi=\partial_{\tau}\,.$ (10)

We call it the primary Killing vector. Since, due to the Killing vector
equation, its covariant derivative is completely antisymmetric, the complete
information about two derivatives of the PKY tensor $k$ is encoded in
$(d\xi)$. In other words, the following 3 objects encode information about
$k$, and its first, and second order derivatives:

$k_{ab}\,,\quad\xi^{a}\,,\quad(d\xi)_{ab}\,.$ (11)

In the following sections, such objects will be used to construct symmetry
operators for various test field equations.

The ‘square’ of the principal tensor gives rise to a Killing tensor

$K_{ab}=(k^{2})_{ab}+\frac{1}{2}k^{2}g_{ab}\,,$ (12)

obeying $\nabla_{(c}K_{ab)}=0$. Here, we have introduced

$\displaystyle(k^{2})_{ab}$ $\displaystyle=$ $\displaystyle
k_{ac}k^{c}{}_{b}\,,\quad k^{2}=k_{ab}k^{ab}\,,$

and denote for future convenience

$\displaystyle(k^{3})_{ab}$ $\displaystyle=$ $\displaystyle
k_{ac}(k^{2})^{c}{}_{b}\,,\quad k^{3}=(k^{3})^{a}{}_{a}=0\,.$ (14)

Moreover, the following vector, cubic in the PKY tensor:

$\zeta^{a}=K^{a}{}_{b}\xi^{b}=(\partial_{\psi})^{a}\,,$ (15)

is the secondary Killing vector of the spacetime.

Finally, the Hodge dual of the PKY tensor is a Killing–Yano tensor
Frolov:2017kze

$f=*k=rdy\wedge(d\tau-r^{2}d\psi)+ydr\wedge(d\tau+y^{2}d\psi)\,,$ (16)

a generalization of the Killing vector, obeying $\nabla_{(c}f_{a)b}=0$. It is
related to the above Killing tensor as follows: $K_{ab}=f_{ac}f^{c}{}_{b}$.
Since its covariant derivative is completely antisymmetric, it is encoded in
$(df)$. Moreover, due to the relation (1), the Hodge dual of $(df)$ is a
Killing vector. Thence, $df$ is a closed-conformal Killing–Yano 3-form. As
such, the derivative of this object is entirely captured by

$\nu^{ab}=\nabla_{c}(df)^{abc}\,.$ (17)

In other words, the following 3 objects encode information about $f$, and its
first, and second order derivatives:

$f_{ab}\,,\quad(df)_{abc}\,,\quad\nu_{ab}\,.$ (18)

They provide alternatives to (11), capturing the information about the Hodge
dual of $k$ and its derivatives; we shall return to this possibility in App C.

We shall now discuss symmetry operators for various test field equations in a
Kerr–NUT–AdS background spacetime. We seek these to be homogeneous in a number
of derivatives and number of PKY tensors. In particular, since we consider
wave-type equations that are second order in derivatives, we shall focus on
homogeneous symmetry operators that contain exactly two covariant derivatives
for each term, some possibly acting on the principal tensor. In the next
section, we start by reviewing the well known scalar case. We then proceed to
more complicated vector and tensor perturbations.

##  III Scalar field perturbations

###  III.1 Separability of scalar field equation

In a generic spacetime, there are only the following two objects available for
a construction of the scalar field operator:111In principle, one could also
add curvature terms. While such terms are important, for example for the
conformal wave equation (see e.g. Gray:2020rtr ; Gray:2021wzf ), in this paper
we do not consider this possibility, neither for the field operators, nor for
their symmetry operators. Whether the latter is a serious restriction remains
to be seen in future.

$g^{ab}\,,\quad\nabla_{a}\,.$ (19)

The only possibility for a 2nd-order in derivatives operator is thus

$\nabla^{2}=g^{ab}\nabla_{a}\nabla_{b}\,.$ (20)

The (massive) scalar equation can then be written as an eigenvalue equation:

$\nabla^{2}\Phi=\mu^{2}\Phi\,,$ (21)

where $\mu$ stands for the mass of the scalar field.

It turns out that the above scalar field equation admits separability in the
general off-shell canonical spacetimes Carter:1968cmp . Namely, a solution can
be found in the following separated form (e.g. Frolov:2017kze ):

$\Phi=R(r)Y(y)e^{-i\omega\tau+im\psi}\,,$ (22)

where functions $R(r)$ and $Y(y)$ obey the following 2nd-order ODE’s:

$\displaystyle\partial_{r}(\Delta_{r}\partial_{r}R)+\frac{{\cal
X}_{r}}{\Delta_{r}}R=0\,,$
$\displaystyle\partial_{y}(\Delta_{y}\partial_{y}Y)+\frac{{\cal
X}_{y}}{\Delta_{y}}Y=0\,,$ (23)

where

$\displaystyle{\cal X}_{r}$ $\displaystyle=$ $\displaystyle(\omega
r^{2}-m)^{2}-\Delta_{r}(\kappa+\mu^{2}r^{2})\,,$ $\displaystyle{\cal X}_{y}$
$\displaystyle=$ $\displaystyle-(\omega
y^{2}-m)^{2}+\Delta_{y}(\kappa-\mu^{2}y^{2})\,,$ (24)

and $\kappa$ is the separation constant.

Importantly, the above separability is possible due to the existence of a
complete set of four mutually commuting operators, one of which is scalar wave
operator $\nabla^{2}$, whose common eigenfunction is the separated solution.
Namely, one has the following set:

$\\{\nabla^{2},O^{(s)}_{l},O^{(s)}_{q},O^{(s)}_{c}\\}\,,$ (25)

where

$\displaystyle O^{(s)}_{l}$ $\displaystyle=$
$\displaystyle\xi^{a}\nabla_{a}\,,$ $\displaystyle O^{(s)}_{q}$
$\displaystyle=$ $\displaystyle\nabla_{a}(K^{ab}\nabla_{b})\,,$ $\displaystyle
O^{(s)}_{c}$ $\displaystyle=$ $\displaystyle\zeta^{a}\nabla_{a}\,,$ (26)

are homogeneous symmetry operators that are linear, quadratic, and cubic,
respectively in the PKY tensor and contain exactly two derivatives.222In
higher dimensions, the situation is even more remarkable, as in $d$
dimensions, one finds $(d-1)$ homogenous symmetry operators that contain
$1,2,\dots(d-1)$ powers of the PKY tensor $k$, see Frolov:2017kze  for more
details. The separated solution above is then the common eigenfunction of
these operators:

$\displaystyle\nabla^{2}\Phi$ $\displaystyle=$ $\displaystyle\mu^{2}\Phi\,,$
$\displaystyle O^{(s)}_{l}\Phi$ $\displaystyle=$
$\displaystyle-i\omega\Phi\,,$ $\displaystyle O^{(s)}_{q}\Phi$
$\displaystyle=$ $\displaystyle\kappa\Phi\,,$ $\displaystyle O^{(s)}_{c}\Phi$
$\displaystyle=$ $\displaystyle im\Phi\,.$ (27)

Obviously, the separation constant $\kappa$ is simply the eigenvalue of the
quadratic in $k$ operator $O^{(s)}_{q}$.

###  III.2 Homogeneous symmetry operators for scalar fields

Let us now show how the above symmetry operators can be systematically derived
by seeking the homogeneous operators that contain two derivatives and are
linear, quadratic, and cubic in the PKY tensor. To construct such operators,
we can use the following objects:

$g^{ab}\,,\ \nabla_{a}\,,\ k^{ab}\,,\ \xi^{a}\,,\ (d\xi)_{ab}\,,$ (28)

where the last three are uniquely constructed from $k$ by applying zero, one,
and two derivatives. To simplify our notation let us introduce the following
shorthands:

$\displaystyle\nabla_{\xi}$ $\displaystyle=$
$\displaystyle\xi^{a}\nabla_{a}\,,\quad\nabla_{\zeta}=\zeta^{a}\nabla_{a}\,,$
$\displaystyle\nabla^{2}_{k}$ $\displaystyle=$ $\displaystyle
k^{ab}\nabla_{a}\nabla_{b}\,,\quad\nabla^{2}_{k^{2}}=(k^{2})^{ab}\nabla_{a}\nabla_{b}\,,$
$\displaystyle\nabla^{2}_{k^{3}}$ $\displaystyle=$
$\displaystyle(k^{3})^{ab}\nabla_{a}\nabla_{b}\,.$ (29)

Of course, since there is no torsion, $\nabla^{2}_{k}$ and
$\nabla^{2}_{k^{3}}$ both vanish on scalars, but may be non-trivial when
applied to vectors and tensors.

####  III.2.1 Linear in $k$ operators

We start with linear in $k$ scalar operators, starting with no derivatives on
$k$, then one derivative, and finally two. As discussed above, since there is
no torsion, we have to discard $\nabla^{2}_{k}$, which is the only possibility
including $k$. The only operator containing 1 derivative on $k$ is

$O^{(s)}_{l}=\nabla_{\xi}={\cal L}_{\xi}=\partial_{\tau}\,.$ (30)

Moreover, there is no scalar operator with two derivatives on $k$, so
$O^{(s)}_{l}$ is the only linear operator at our disposal. One can easily show
that it commutes with the box operator:

$[O^{(s)}_{l},\nabla^{2}]=0\,.$ (31)

####  III.2.2 Quadratic in $k$ operators

Lets now proceed to operators with two $k$’s. Then we have the following
possibilities:

$\displaystyle O_{q}^{1}$ $\displaystyle=$
$\displaystyle\nabla^{2}_{k^{2}}\,,\quad O_{q}^{2}=k^{2}\nabla^{2}\,,$ (32)
$\displaystyle O_{q}^{3}$ $\displaystyle=$ $\displaystyle
k^{a}{}_{b}\xi^{b}\nabla_{a}\,,$ (33) $\displaystyle O_{q}^{4}$
$\displaystyle=$ $\displaystyle\xi^{2}\,,\quad O_{q}^{5}=k_{ab}(d\xi)^{ab}\,.$
(34)

The potential most general homogeneous operator quadratic in $k$ is thus

$O_{q}=\alpha_{1}O_{q}^{1}+\alpha_{2}O_{q}^{2}+\alpha_{3}O_{q}^{3}+\alpha_{4}O_{q}^{4}+\alpha_{5}O_{q}^{5}\,.$
(35)

We now require that

$[O_{q},\nabla^{2}]=0\,,$ (36)

for any off-shell functions $\Delta_{r}$ and $\Delta_{y}$. This implies the
following constraints:

$\alpha_{2}=\frac{\alpha_{1}}{2}\,,\quad\alpha_{3}=-2\alpha_{1}\,,\quad\alpha_{4}=0\,,\quad\alpha_{5}=0\,.$
(37)

So, we have a 1-parametric family of quadratic in $k$ operators. Setting
$\alpha_{1}$ w.l.o.g., the most general commuting operator thus reads

$O_{q}^{(s)}=\nabla^{2}_{k^{2}}+\frac{1}{2}k^{2}\nabla^{2}-2k^{a}{}_{b}\xi^{b}\nabla_{a}\,.$
(38)

In fact, this is nothing else than the known operator, constructed long time
ago by Carter Carter:1977pq , as we have

$\displaystyle O_{q}^{(s)}$
$\displaystyle=\nabla_{a}(k^{a}{}_{c}k^{cb}+\frac{1}{2}k^{2}g^{ab}\nabla_{b})$
$\displaystyle-\nabla_{a}(k^{a}{}_{c}k^{cb}+\frac{1}{2}k^{2}g^{ab})\nabla_{b}-2k^{a}{}_{b}\xi^{b}\nabla_{a}$
$\displaystyle=\nabla_{a}(K^{ab}\nabla_{b})\,,$ (39)

where in the final step we have made use of (2) and (12) to recover the usual
second order symmetry operator corresponding to Killing tensor $K^{ab}$.

####  III.2.3 Cubic in $k$ operators

Let us finally proceed to cubic in $k$ operators. Since on scalars
$\nabla^{2}_{k^{3}}$ vanishes, and we find

$(k^{2})^{ab}(d\xi)_{ab}=0\,,$ (40)

we have the following possibilities:

$\displaystyle O_{c}^{1}$ $\displaystyle=$
$\displaystyle(k^{2})_{ac}\xi^{c}\nabla^{a}\,,\quad
O_{c}^{2}=k^{2}\xi^{a}\nabla_{a}\,,$ (41)

The requirement that this commutes with $\nabla^{2}$ then yields

$\alpha_{2}=\frac{\alpha_{1}}{2}\,.$ (42)

Thus, the most general symmetry opertor of this order is given by

$O_{c}^{(s)}=(k^{2})_{ac}\xi^{c}\nabla^{a}+\frac{1}{2}k^{2}\xi^{a}\nabla_{a}\,.$
(43)

Of course, this operator is also well known, as we have

$O_{c}^{(s)}=K^{a}{}_{b}\xi^{b}\nabla_{a}=\zeta^{a}\nabla_{a}={\cal
L}_{\zeta}=\partial_{\phi}\,,$ (44)

on behalf of (15).

It is easy to show that all the above symmetry operators mutually commute. We
have thus systematically recovered the complete set (25) of mutually commuting
operators that intrinsically characterize the separability of scalar
perturbations in the off-shell Kerr–NUT–AdS spacetimes.333 This procedure also
applies to the construction of the symmetry operators of the conformal wave
equation Gray:2021wzf , where the additional terms present are also
homogeneous in $k$. For brevity, we do not repeat the corresponding
construction here, and refer the intersted reader to Gray:2021wzf .

##  IV Vector perturbations

###  IV.1 Vector operator

In this section we shall seek homogeneous symmetry operators for (massive)
vector perturbations. The corresponding field strength $F_{ab}$ is determined
from a vector potential $A^{a}$ by

$F=dA\,,$ (45)

i.e. $F_{ab}=\nabla_{a}A_{b}-\nabla_{b}A_{a}$. In what follows we directly
work with the vector potential $A^{a}$. Since such a field carries a vector
index, we seek ‘matrix’ operators $O^{a}{}_{b}$ that act on $A^{a}$ as
follows:

$(OA)^{a}\equiv O^{a}{}_{b}A^{b}\,.$ (46)

In a generic spacetime, where the only available objects are444In principle,
one can also consider a Hodge star operation which, however, changes rank of
the operator as the number of dimensions are varied. For example, in 4d one
could consider an operator  $\epsilon^{a}{}_{b}{}^{cd}\nabla_{c}\nabla_{d}\,,$
(47) where $\epsilon^{abcd}$ stands for the Levi-Civita tensor. In what
follows we do not consider this possibility, see however App. C.

$\nabla_{a}\,,\quad g^{ab}\,,\quad\delta^{a}_{b}\,,\quad g_{ab}\,,$ (48)

we have the following three possibilities for operator $O$:

$\nabla^{2}\delta^{a}_{b}\,,\quad\nabla_{b}\nabla^{a}\,,\quad\nabla^{a}\nabla_{b}\,.$
(49)

(Note that the latter two coincide in vacuum spacetimes.) The vector operator
uses the first two and reads

$(O_{M})^{a}{}_{b}=\nabla^{2}\delta^{a}_{b}-\nabla_{b}\nabla^{a}\,,$ (50)

acting on the vector potential as

$(O_{M})^{a}{}_{b}A^{b}=\nabla_{b}F^{ba}=\nabla^{2}A^{a}-\nabla_{b}\nabla^{a}A^{b}\,.$
(51)

The massive vector equation can then be written as an eigenvalue equation:

$(O_{M})^{a}{}_{b}A^{b}=\mu^{2}A^{a}\,.$ (52)

Taking its divergence, one recovers the corresponding consistency condition:

$\nabla_{a}A^{a}=0\,.$ (53)

When $\mu=0$, which corresponds to the Maxwell case, the theory is a gauge
theory, and the latter equation is not a consequence of the equations of
motion, though it can be imposed as a Lorenz gauge fixing condition. Herein we
shall not explicitly impose (53).

To find the symmetry operators for the vector operator (50), that is to say
operators $O$ satisfying

$[O,O_{M}]^{a}{}_{b}=O^{a}{}_{c}(O_{M})^{c}{}_{b}-(O_{M})^{a}{}_{c}O^{c}{}_{b}=0\,,$
(54)

we proceed in a way similar to the scalar case, except we have now more
possibilities due to the fact that the field carries a non-trivial index. We
shall also distinguish two cases – the off-shell canonical case and the vacuum
(possibly with $\Lambda$) Kerr–NUT–AdS case. We start with linear in $k$
operators.

###  IV.2 Linear in $k$ operators

We have the following options for linear in $k$ matrix operators (for notation
simplicity not writing the matrix indices on operators):

$\displaystyle O_{1}$ $\displaystyle=$ $\displaystyle
k^{a}{}_{b}\nabla^{2}\,,\quad O_{2}=k_{cb}\nabla^{a}\nabla^{c}\,,\quad
O_{3}=k_{cb}\nabla^{c}\nabla^{a}\,,$ $\displaystyle O_{4}$ $\displaystyle=$
$\displaystyle\nabla^{2}_{k}\delta^{a}_{b}\,,\quad
O_{5}=k^{ac}\nabla_{c}\nabla_{b}\,,\quad O_{6}=k^{ac}\nabla_{b}\nabla_{c}$
$\displaystyle O_{7}$ $\displaystyle=$
$\displaystyle\nabla_{\xi}\delta^{a}_{b}\,,\quad
O_{8}=\xi_{b}\nabla^{a}\,,\quad O_{9}=\xi^{a}\nabla_{b}\,,$ $\displaystyle
O_{10}$ $\displaystyle=$ $\displaystyle(d\xi)^{a}{}_{b}\,,$ (55)

Let us first focus on the off-shell canonical spacetimes.555In the off-shell
case, we assume that the metric functions $\Delta_{r}(r)$ and $\Delta_{y}(y)$
are sufficiently general, that is, several of their lower derivatives are non-
zero and can be treated as ‘independent’. This will be true also on-shell for
dimensions higher than six, but does not continue to hold on-shell in four
dimension. Hence the need to distinguish the cases. By requiring commutativity
with the Maxwell operator, we find the following 6 constraints:

$\displaystyle\alpha_{3}$ $\displaystyle=$
$\displaystyle-\alpha_{1}-\alpha_{2}\,,\quad\alpha_{5}=-\alpha_{2}+\alpha_{4}-\alpha_{1}\,,$
$\displaystyle\alpha_{6}$ $\displaystyle=$
$\displaystyle\alpha_{2}-\alpha_{4}\,,\quad\alpha_{8}=-2\alpha_{1}\,,\quad\alpha_{9}=0\,,$
$\displaystyle\alpha_{10}$ $\displaystyle=$
$\displaystyle\alpha_{2}-\alpha_{4}+\frac{1}{2}\alpha_{7}\,.$ (56)

This means that in principle we have 4 operators of this kind that commute
with the Maxwell operator. However, there is a 2-parametric family of trivial
operators, characterized by an arbitrary choice of
$\\{\alpha_{2},\alpha_{4}\\}$. In what follows we eliminate this freedom by
setting

$\alpha_{2}=\alpha_{4}=0\,.$ (57)

We are just left with 2 independent non-trivial symmetry operators,
characterized by $\\{\alpha_{1},\alpha_{7}\\}$. Let us write their
representatives. First, setting $\alpha_{1}=1$ and $\alpha_{7}=0$, we have

$O_{k}^{(v)}=k^{a}{}_{b}\nabla^{2}-k_{cb}\nabla^{c}\nabla^{a}-k^{ac}\nabla_{c}\nabla_{b}-2\xi_{b}\nabla^{a}\,.$
(58)

This operator remains non-trivial upon imposing (53). Second, setting
$\alpha_{1}=0$ and $\alpha_{7}=1$ and the remaining ones to zero, we recover

$\displaystyle O^{(v)}_{\xi}$ $\displaystyle=$
$\displaystyle\xi^{c}\nabla_{c}\delta^{a}_{b}+\frac{1}{2}(d\xi)^{a}{}_{b}={\cal
L}_{\xi}\delta^{a}_{b}=\delta^{a}_{b}\partial_{t}\,.$ (59)

This is thus an operator corresponding to the Lie derivative along the primary
Killing vector $\xi$.

When the vacuum ($\Lambda=0$) on-shell Kerr–NUT–AdS spacetime is considered,
we find that the requirement for commutativity results only in 5 constraints:

$\displaystyle\alpha_{3}$ $\displaystyle=$
$\displaystyle-\alpha_{1}-\alpha_{2}\,,\quad\alpha_{6}=-\alpha_{1}-\alpha_{5}$
$\displaystyle\alpha_{8}$ $\displaystyle=$
$\displaystyle-2\alpha_{1}\,,\quad\alpha_{9}=0\,,$ $\displaystyle\alpha_{10}$
$\displaystyle=$ $\displaystyle\alpha_{2}-\alpha_{4}+\frac{1}{2}\alpha_{7}\,.$
(60)

However, in this case we find that there is a 3-parametric family of trivial
operators, characterized by arbitrary choice of
$\\{\alpha_{2},\alpha_{4},\alpha_{5}\\}$. To eliminate these, we could for
example set

$\alpha_{2}=\alpha_{4}=0\,,\quad\alpha_{5}=-\alpha_{1}\,.$ (61)

So again we are left with 2 independent symmetry operators, characterized by
$\\{\alpha_{1},\alpha_{7}\\}$ – proceeding as above, we then formally recover
‘the same’ symmetry operators as in the off-shell case.

###  IV.3 Quadratic in $k$ operators

Let us now turn to finding quadratic in PKY tensor homogeneous operators for
the vector perturbations. We have the following 22 possibilities:

$\displaystyle O_{1}$ $\displaystyle=$ $\displaystyle
k^{a}{}_{b}\nabla_{k}^{2}\,,\quad
O_{2}=k^{a}{}_{c}k_{bd}\nabla^{c}\nabla^{d}\,,$ $\displaystyle O_{3}$
$\displaystyle=$ $\displaystyle k^{a}{}_{c}k_{bd}\nabla^{d}\nabla^{c}\,,\quad
O_{4}=\nabla^{2}_{k^{2}}\delta^{a}_{b}\,,$ $\displaystyle O_{5}$
$\displaystyle=$ $\displaystyle(k^{2})^{a}{}_{e}\nabla^{e}\nabla_{b}\,,\quad
O_{6}=(k^{2})^{a}{}_{e}\nabla_{b}\nabla^{e}\,,$ $\displaystyle O_{7}$
$\displaystyle=$ $\displaystyle(k^{2})_{be}\nabla^{e}\nabla^{a}\,,\quad
O_{8}=(k^{2})_{be}\nabla^{a}\nabla^{e}\,,$ $\displaystyle O_{9}$
$\displaystyle=$ $\displaystyle k^{2}\nabla^{2}\delta^{a}_{b}\,,\quad
O_{10}=k^{2}\nabla^{a}\nabla_{b}\,,$ $\displaystyle O_{11}$ $\displaystyle=$
$\displaystyle k^{2}\nabla_{b}\nabla^{a}\,,\quad
O_{12}=(k^{2})^{a}{}_{b}\nabla^{2}\,,$ $\displaystyle O_{13}$ $\displaystyle=$
$\displaystyle k^{a}{}_{b}\nabla_{\xi}\,,\quad
O_{14}=k^{e}{}_{f}\xi^{f}\nabla_{e}\delta^{a}_{b}\,,$ (62) $\displaystyle
O_{15}$ $\displaystyle=$ $\displaystyle\xi^{a}k_{be}\nabla^{e}\,,\quad
O_{16}=\xi_{b}k^{a}{}_{e}\nabla^{e}\,,$ $\displaystyle O_{17}$
$\displaystyle=$ $\displaystyle k^{a}{}_{c}\xi^{c}\nabla_{b}\,,\quad
O_{18}=k_{eb}\xi^{e}\nabla^{a}\,,$ $\displaystyle O_{19}$ $\displaystyle=$
$\displaystyle(d\xi)^{ae}k_{be}\,,\quad O_{20}=(d\xi)_{be}k^{ae}\,,$
$\displaystyle O_{21}$ $\displaystyle=$
$\displaystyle\xi^{2}\delta^{a}_{b}\,,\quad O_{22}=\xi^{a}\xi_{b}\,.$

We note that not all of these are independent however for the purposes of
counting it is useful to list everything in this manner.

Focusing first on the off-shell Kerr–NUT–AdS spacetimes, we find that the
requirement for commutation (54) yields 16 constraints on the linear
combination of the above operators:

$\displaystyle\alpha_{1}$ $\displaystyle=$
$\displaystyle\alpha_{3}+2\alpha_{11}+\alpha_{16}+\alpha_{19}+\alpha_{20}\,,$
$\displaystyle\alpha_{2}$ $\displaystyle=$
$\displaystyle-\alpha_{3}-4\alpha_{11}-\alpha_{16}\,,\quad\alpha_{4}=-2\alpha_{11}\,,$
$\displaystyle\alpha_{5}$ $\displaystyle=$ $\displaystyle
2\alpha_{11}+\alpha_{16}+\alpha_{19}+\alpha_{20}\,,$ $\displaystyle\alpha_{6}$
$\displaystyle=$ $\displaystyle
2\alpha_{11}-\alpha_{19}-\alpha_{20}\,,\quad\alpha_{7}=2\alpha_{11}\,,$
$\displaystyle\alpha_{8}$ $\displaystyle=$ $\displaystyle
2\alpha_{11}+\alpha_{16}\,,\quad\alpha_{9}=-\alpha_{11}\,,\quad\alpha_{12}=-4\alpha_{11}-\alpha_{16}\,,$
$\displaystyle\alpha_{13}$ $\displaystyle=$
$\displaystyle-4\alpha_{11}-3\alpha_{16}\,,\quad\alpha_{14}=-\alpha_{16}\,,\quad\alpha_{15}=4\alpha_{11}\,,$
$\displaystyle\alpha_{17}$ $\displaystyle=$ $\displaystyle
4\alpha_{10}\,,\quad\alpha_{18}=8\alpha_{11}+3\alpha_{16}\,,$
$\displaystyle\alpha_{21}$ $\displaystyle=$
$\displaystyle-4\alpha_{11}-2\alpha_{16}\,,\quad\alpha_{22}=4\alpha_{11}+2\alpha_{16}\,.$
(63)

Moreover, it can be shown that we have a 3-parametric family of trivial
operators, encoded in arbitrary $\\{\alpha_{3},\alpha_{19},\alpha_{20}\\}$. We
can eliminate these by setting

$\alpha_{3}=\alpha_{19}=\alpha_{20}=0\,.$ (64)

This means that we are left with 3 independent quadratic operators,
characterized by specifying $\\{\alpha_{10},\alpha_{11},\alpha_{16}\\}$. We
write these as follows.

First, setting

$\alpha_{10}=1\,,\quad\alpha_{11}=0=\alpha_{16}\,,$ (65)

we recover the following commuting operator:

$O^{(v)}_{q_{1}}=k^{2}\nabla^{a}\nabla_{b}+4k^{a}{}_{c}\xi^{c}\nabla_{b}=\nabla^{a}(k^{2}\nabla_{b})\,.$
(66)

This operator will vanish for the Proca field, or upon imposing the Lorenz
gauge condition. In the vacuum Kerr case (with $\Lambda=0$ and no NUT
parameter), this operator agrees with the operator ${\cal K}_{2}$ in
Mei:2023pho .

Next, requiring that the absolute term is missing, we set

$\alpha_{10}=0\,,\quad\alpha_{11}=-\frac{1}{2}\,,\quad\alpha_{16}=1\,,$ (67)

to obtain the following operator:

$\displaystyle O_{q_{2}}^{(v)}$ $\displaystyle=$ $\displaystyle
k^{a}{}_{c}k_{bd}\nabla^{c}\nabla^{d}+\nabla^{2}_{k^{2}}\delta^{a}_{b}-(k^{2})^{a}{}_{e}\nabla_{b}\nabla^{e}$
(68)
$\displaystyle-(k^{2})_{be}\nabla^{e}\nabla^{a}+\frac{1}{2}k^{2}\nabla^{2}\delta^{a}_{b}-\frac{1}{2}k^{2}\nabla_{b}\nabla^{a}$
$\displaystyle+(k^{2})^{a}{}_{b}\nabla^{2}-k^{a}{}_{b}\nabla_{\xi}-k^{e}{}_{f}\xi^{f}\nabla_{e}\delta^{a}_{b}$
$\displaystyle-2\xi^{a}k_{be}\nabla^{e}+\xi_{b}k^{a}{}_{e}\nabla^{e}-k_{eb}\xi^{e}\nabla^{a}\,.$

This operator is gauge invariant and in the vacuum case (again with no NUT
parameter) identical to the operator ${\cal K}_{4}$ found recently in
Mei:2023pho .

Finally, setting

$\alpha_{10}=0=\alpha_{11}\,,\quad\alpha_{16}=1\,,$ (69)

we recover the following operator:

$\displaystyle O_{q_{3}}^{(v)}$ $\displaystyle=$ $\displaystyle
k^{a}{}_{b}\nabla_{k}^{2}-k^{a}{}_{c}k_{bd}\nabla^{c}\nabla^{d}+(k^{2})^{a}{}_{e}\nabla^{e}\nabla_{b}$
(70)
$\displaystyle+(k^{2})_{be}\nabla^{a}\nabla^{e}-(k^{2})^{a}{}_{b}\nabla^{2}-3k^{a}{}_{b}\nabla_{\xi}$
$\displaystyle-k^{e}{}_{f}\xi^{f}\nabla_{e}\delta^{a}_{b}+\xi_{b}k^{a}{}_{e}\nabla^{e}+3k_{eb}\xi^{e}\nabla^{a}$
$\displaystyle-2\xi^{2}\delta^{a}_{b}+2\xi^{a}\xi_{b}\,.$

Both these operators, $O_{q_{2}}^{(v)}$ and $O_{q_{3}}^{(v)}$ will not vanish
upon imposing $\nabla_{a}A^{a}=0$.

Let us next turn to the on-shell vacuum ($\Lambda=0$) Kerr–NUT spacetimes. In
this case, the operators $O_{10}$ and $O_{11}$ are no longer independent, and
the above construction only yields two commuting operators. However, we can
repeat the whole construction again, with the following results. The
requirement for commutation yields 13 constraints:

$\displaystyle\alpha_{1}$ $\displaystyle=$
$\displaystyle\alpha_{3}+2\alpha_{8}+\frac{1}{2}(\alpha_{12}-\alpha_{16})+\alpha_{19}+\alpha_{20}\,,$
$\displaystyle\alpha_{2}$ $\displaystyle=$
$\displaystyle-\alpha_{3}+\alpha_{12}\,,\quad\alpha_{4}=\frac{1}{2}(\alpha_{12}+\alpha_{16})\,,$
$\displaystyle\alpha_{5}$ $\displaystyle=$
$\displaystyle-\alpha_{6}-\alpha_{12}\,,\quad\alpha_{7}=-\alpha_{8}-\alpha_{12}\,,$
$\displaystyle\alpha_{9}$ $\displaystyle=$
$\displaystyle\frac{1}{4}(\alpha_{12}+\alpha_{16})\,,\quad\alpha_{13}=\alpha_{12}-2\alpha_{16}\,,$
$\displaystyle\alpha_{14}$ $\displaystyle=$
$\displaystyle-\alpha_{16}\,,\quad\alpha_{15}=-\alpha_{12}-\alpha_{16}\,,$
$\displaystyle\alpha_{17}$ $\displaystyle=$ $\displaystyle
4\alpha_{10}+4\alpha_{11}+\alpha_{12}+\alpha_{16}\,,\quad\alpha_{18}=\alpha_{16}-2\alpha_{12}\,,$
$\displaystyle\alpha_{21}$ $\displaystyle=$
$\displaystyle-\alpha_{8}+\frac{1}{2}(\alpha_{12}-\alpha_{16})\,,\quad\alpha_{22}=\alpha_{16}-\alpha_{12}\,.$
(71)

Moreover, one can show that there is a remaining 5-parametric family of
trivial (or identical) operators, encoded in
$\\{\alpha_{3},\alpha_{6},\alpha_{8},\alpha_{19},\alpha_{20}\\}$. To eliminate
these, we may set

$\alpha_{3}=\alpha_{6}=\alpha_{8}=\alpha_{19}=\alpha_{20}=0\,.$ (72)

At the same time, since $\alpha_{10}$ and $\alpha_{11}$ are now not
independent, we can also set

$\alpha_{11}=0\,.$ (73)

Thus, similar to Mei:2023pho , we are left with a 3-parametric family of (non-
trivial) commuting operators. In our case these are characterized by a choice
of

$\\{\alpha_{10}\,,\alpha_{12}\,,\alpha_{16}\\}\,.$ (74)

In particular, the operator ${\cal K}_{2}$ in Mei:2023pho  is reproduced upon
setting $\alpha_{10}=1\,\ \alpha_{12}=\alpha_{16}=0\,,$ and the operator
${\cal K}_{4}$ corresponds to $\alpha_{10}=-\frac{1}{2}\,,\
\alpha_{12}=\alpha_{16}=1\,.$ We note these remain generalizations of the
operators in Mei:2023pho  since they contain non-trivial NUT parameter.
Finally, setting

$\alpha_{10}=\alpha_{12}=0\,,\quad\alpha_{16}=-2\,,$ (75)

we recover

$\displaystyle\tilde{O}_{q_{3}}^{(v)}$ $\displaystyle=$ $\displaystyle
k^{a}{}_{b}\nabla_{k}^{2}-\nabla^{2}_{k^{2}}\delta^{a}_{b}-\frac{1}{2}k^{2}\nabla^{2}\delta^{a}_{b}$
(76)
$\displaystyle+4k^{a}{}_{b}\nabla_{\xi}+2k^{e}{}_{f}\xi^{f}\nabla_{e}\delta^{a}_{b}+2\xi^{a}k_{be}\nabla^{e}$
$\displaystyle-2\xi_{b}k^{a}{}_{e}\nabla^{e}-2k^{a}{}_{c}\xi^{c}\nabla_{b}-2k_{eb}\xi^{e}\nabla^{a}$
$\displaystyle+\xi^{2}\delta^{a}_{b}-2\xi^{a}\xi_{b}\,,$

completing the set of the on-shell commuting operators.

###  IV.4 Cubic in $k$ operators

We finally examine the cubic in $k$ vector operators of degree two. We
organize these according to how many derivatives act on the (three factors of
the) principal tensor. We start with operators with no derivatives, proceed to
operators with one derivative, and finalize with two derivatives. In this way
we find a total of 34 operators listed below. Note, however, that not all of
these operators are necessarily independent.

Let us start with operators where no derivatives act on the three copies of
the PKY tensor, that is operators of the symbolic “$k^{3}\nabla^{2}$” form.
These are given by:

$\displaystyle O_{1}$ $\displaystyle=$
$\displaystyle(k^{2})^{a}{}_{b}\nabla^{2}_{k}\,,\quad
O_{2}=k^{a}{}_{b}\nabla^{2}_{k^{2}}\,,$ $\displaystyle O_{3}$ $\displaystyle=$
$\displaystyle\nabla^{2}_{k^{3}}\delta^{a}_{b}\,,\quad
O_{4}=k^{a}{}_{c}(k^{2})_{be}\nabla^{c}\nabla^{e}\,,$ $\displaystyle O_{5}$
$\displaystyle=$ $\displaystyle
k^{a}{}_{c}(k^{2})_{be}\nabla^{e}\nabla^{c}\,,\quad
O_{6}=(k^{2})^{a}{}_{c}k_{be}\nabla^{c}\nabla^{e}\,,$ $\displaystyle O_{7}$
$\displaystyle=$
$\displaystyle(k^{2})^{a}{}_{c}k_{be}\nabla^{e}\nabla^{c}\,,\quad
O_{8}=k^{2}\nabla^{2}_{k}\delta^{a}_{b}\,,$ $\displaystyle O_{9}$
$\displaystyle=$ $\displaystyle(k^{3})^{a}{}_{e}\nabla^{e}\nabla_{b}\,,\quad
O_{10}=(k^{3})^{a}{}_{e}\nabla_{b}\nabla^{e}\,,$ $\displaystyle O_{11}$
$\displaystyle=$ $\displaystyle(k^{3})_{be}\nabla^{e}\nabla^{a}\,,\quad
O_{12}=(k^{3})_{be}\nabla^{a}\nabla^{e}\,,$ (77) $\displaystyle O_{13}$
$\displaystyle=$ $\displaystyle k^{2}k^{a}{}_{e}\nabla^{e}\nabla_{b}\,,\quad
O_{14}=k^{2}k^{a}{}_{e}\nabla_{b}\nabla^{e}\,,$ $\displaystyle O_{15}$
$\displaystyle=$ $\displaystyle k^{2}k_{be}\nabla^{e}\nabla^{a}\,,\quad
O_{16}=k^{2}k_{be}\nabla^{a}\nabla^{e}\,,$ $\displaystyle O_{17}$
$\displaystyle=$ $\displaystyle(k^{3})^{a}{}_{b}\nabla^{2}\,,\quad
O_{18}=k^{a}{}_{b}k^{2}\nabla^{2}\,.$

Note that the operators $\\{O_{1},\dots,O_{8}\\}$ have two contractions
between the $k$’s and the covariant derivative operators, the operators
$\\{O_{9},\dots,O_{16}\\}$ have one, and the operators $O_{17},O_{18}$ have
zero contractions.

Next, we consider operators with one derivative acting on $k$’s, that is, the
terms of the symbolic “$k^{2}\xi\nabla$” form. They read:

$\displaystyle O_{19}$ $\displaystyle=$
$\displaystyle(k^{2})^{a}{}_{b}\xi^{c}\nabla_{c}\,,\quad
O_{20}=k^{2}\xi^{c}\nabla_{c}\delta^{a}_{b}\,,$ $\displaystyle O_{21}$
$\displaystyle=$ $\displaystyle(k^{2})^{ca}\xi_{b}\nabla_{c}\,,\quad
O_{22}=(k^{2})_{cb}\xi^{a}\nabla^{c}\,,$ $\displaystyle O_{23}$
$\displaystyle=$
$\displaystyle(k^{2})^{cd}\xi_{d}\nabla_{c}\delta^{a}{}_{b}\,,$ $\displaystyle
O_{24}$ $\displaystyle=$ $\displaystyle k^{2}\xi^{a}\nabla_{b}\,,\quad
O_{25}=k^{2}\xi_{b}\nabla^{a}\,,$ $\displaystyle O_{26}$ $\displaystyle=$
$\displaystyle(k^{2})^{a}{}_{c}\xi^{c}\nabla_{b}\,,\quad
O_{27}=(k^{2})_{bc}\xi^{c}\nabla^{a}\,.$ (78)

Here, $\\{O_{19},\dots O_{23}\\}$ have one contraction between the $k$’s and
the covariant derivatives, and the operators $\\{O_{24},\dots,O_{27}\\}$ have
no such contraction.

Finally, we consider terms with two derivatives acting on $k$’s, that is terms
of the symbolic “$\xi\xi k$” or “$(d\xi)kk$” form. These are:

$\displaystyle O_{28}$ $\displaystyle=$ $\displaystyle
k^{a}{}_{c}\xi^{c}\xi_{b}\,,\quad O_{29}=k_{bc}\xi^{c}\xi^{a}\,,\quad
O_{30}=\xi^{2}k^{a}{}_{b}\,,$ $\displaystyle O_{31}$ $\displaystyle=$
$\displaystyle(d\xi)^{a}{}_{c}(k^{2})^{c}{}_{b}\,,\quad
O_{32}=(d\xi)_{bc}(k^{2})^{ca}\,,$ $\displaystyle O_{33}$ $\displaystyle=$
$\displaystyle k^{a}{}_{e}(d\xi)^{ef}k_{fb}\,,\quad
O_{34}=k^{2}d\xi^{a}{}_{b}\,.$ (79)

For the off-shell Kerr–NUT–AdS spacetimes one can check, through tedious
computer algebra calculations, that the only non trivial cubic operator that
commutes with the vector operator is the Lie derivative along the secondary
Killing vector $\zeta$, explicitly given by

$O^{(v)}_{\zeta}=O_{23}+\frac{1}{2}O_{20}+\frac{1}{4}(O_{31}-O_{32})\\\
+\frac{1}{4}O_{34}+(O_{28}-O_{29})-O_{30}\,,$ (80)

or, upon using the definitions of $\zeta$ (15), $K$ (12), and $k$ (2):

$\displaystyle O^{(v)}_{\zeta}$ $\displaystyle=$
$\displaystyle\zeta^{c}\nabla_{c}\delta^{a}_{b}+\frac{1}{2}(d\zeta)^{a}{}_{b}={\cal
L}_{\zeta}\delta^{a}_{b}=\delta^{a}_{b}\partial_{\psi}\,.$ (81)

A similar procedure shows that there are no additional non-trivial symmetry
operators when we are on-shell, neither with nor without $\Lambda$.

###  IV.5 Vector symmetry operators: summary

To summarize, for the off-shell Kerr–NUT–AdS spacetimes we have obtained the
following 6 symmetry operators:

$\\{O^{(v)}_{\xi},O^{(v)}_{k},O_{q_{1}}^{(v)},O_{q_{2}}^{(v)},O_{q_{3}}^{(v)},O_{\zeta}^{(v)}\\}\,.$
(82)

Together with the vector operator $O_{M}$, these form a set of 7 mutually
commuting operators. For the on-shell vacuum case, one has an analogous set of
7 symmetry operators.

It remains to be seen whether by considering the corresponding eigenvalue
problem, one could obtain the associated R-separated solution for vector
potential perturbations, as attempted in Mei:2023pho .

##  V Tensor perturbations

###  V.1 Lichnerowicz operator

Let us finally turn to tensor perturbations,

$g_{ab}\to g_{ab}+h_{ab}\,.$ (83)

Then, to the linear order in the perturbation theory, we find the Lichnerowicz
operator:

$\displaystyle(Lh)_{ab}$ $\displaystyle=$ $\displaystyle\delta R_{ab}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\nabla^{c}(\nabla_{a}h_{bc}+\nabla_{b}h_{ac}-\nabla_{c}h_{ab})-\frac{1}{2}\nabla_{a}\nabla_{b}h\,,$

where $h=h_{ab}g^{ab}$. That means that we can write the Lichnerowicz operator
as

$(Lh)_{ab}=L_{ab}{}^{cd}h_{cd}\,,$ (85)

where

$L_{ab}{}^{cd}=\nabla^{(d}\nabla_{(a}\delta^{c)}{}_{b)}-\frac{1}{2}\nabla^{2}\delta^{c}_{a}\delta^{d}_{b}-\frac{1}{2}\nabla_{(a}\nabla_{b)}g^{cd}\,,$
(86)

highlighting manifestly that $L$ is symmetric in both $(ab)$ and $(cd)$
indices.

In what follows, we seek symmetry operators for $L$, of the form

$(Oh)_{ab}\equiv O_{ab}{}^{cd}h_{cd}\,$ (87)

that are manifestly symmetric in $(ab)$ and $(cd)$ indices, and commute with
$L$:

$[O,L]_{ab}{}^{cd}=O_{ab}{}^{ef}L_{ef}{}^{cd}-L_{ab}{}^{ef}O_{ef}{}^{cd}=0\,.$
(88)

To simplify our notation for the $O$’s, we assume but do not explicitly write
symmetrization in $(ab)$ and $(cd)$ indices, and only restate it for the final
result.

###  V.2 Linear operators

Let us first construct linear in $k$ tensor symmetry operators. We have the
following 20 possibilities:

$\displaystyle O_{1}$ $\displaystyle=$
$\displaystyle\nabla^{2}_{k}\delta_{a}^{c}\delta_{b}^{d}\,,\quad
O_{2}=\nabla_{k}^{2}g^{cd}g_{ab}\,,$ $\displaystyle O_{3}$ $\displaystyle=$
$\displaystyle k^{ec}\nabla_{e}\nabla^{d}g_{ab}\,,\quad
O_{4}=k^{ec}\nabla^{d}\nabla_{e}g_{ab}\,,$ $\displaystyle O_{5}$
$\displaystyle=$ $\displaystyle
k^{ec}\nabla_{e}\nabla_{a}\delta^{d}_{b}\,,\quad
O_{6}=k^{ec}\nabla_{a}\nabla_{e}\delta^{d}_{b}\,,$ $\displaystyle O_{7}$
$\displaystyle=$ $\displaystyle
k^{e}{}_{a}\nabla_{e}\nabla^{c}\delta^{d}_{b}\,,\quad
O_{8}=k^{e}{}_{a}\nabla^{c}\nabla_{e}\delta^{d}_{b}\,,$ $\displaystyle O_{9}$
$\displaystyle=$ $\displaystyle k^{e}{}_{a}\nabla_{e}\nabla_{b}g^{cd}\,,\quad
O_{10}=k^{e}{}_{a}\nabla_{b}\nabla_{e}g^{cd}\,,$ $\displaystyle O_{11}$
$\displaystyle=$ $\displaystyle k^{c}{}_{a}\nabla^{2}\delta^{d}_{b}\,,\quad
O_{12}=k^{c}{}_{a}\nabla^{d}\nabla_{b}\,,$ $\displaystyle O_{13}$
$\displaystyle=$ $\displaystyle k^{c}{}_{a}\nabla_{b}\nabla^{d}\,,\quad
O_{14}=\nabla_{\xi}g^{cd}g_{ab}\,,$ (89) $\displaystyle O_{15}$
$\displaystyle=$
$\displaystyle\nabla_{\xi}\delta^{c}_{a}\delta^{d}_{b}\,,\quad
O_{16}=\xi^{c}\nabla^{d}g_{ab}\,,$ $\displaystyle O_{17}$ $\displaystyle=$
$\displaystyle\xi^{c}\nabla_{a}\delta^{d}_{b}\,,\quad
O_{18}=\xi_{a}\nabla^{c}\delta^{d}_{b}\,,$ $\displaystyle O_{19}$
$\displaystyle=$ $\displaystyle\xi_{a}\nabla_{b}g^{cd}\,,\quad
O_{20}=(d\xi)^{c}{}_{a}\delta^{d}_{b}\,.$

As always, not all of these operators are independent; in fact, the operator
$O_{2}$ is trivial since it is the anti-symmetrized second derivative of a
scalar.

Let us first focus on the off-shell Kerr–NUT–AdS spacetimes. Then, by
requiring the linear combination of these operators commutes with the
Lichnerowicz operator, we find 15 constraints. However, 4 of the remaining
linear combinations are trivial. We are thus left with only one commuting
operator, the primary Killing vector operator:

$O^{(t)}_{\xi}=\nabla_{\xi}\delta^{c}_{a}\delta^{d}_{b}-(d\xi)^{(c}{}_{(a}\delta^{d)}{}_{b)}=\delta^{c}_{a}\delta^{d}_{b}{\cal
L}_{\xi}=\delta^{c}_{a}\delta^{d}_{b}\partial_{t}\,.$ (90)

As we shall see, the situation is much more interesting in the on-shell case.

Namely, let us consider on-shell Kerr-NUT-AdS spacetimes with non-trivial
$\Lambda$. Then, by requiring that the linear combination of the above
operators commutes with the Lichnerowicz operator, we find the following 12
constraints:

$\displaystyle\alpha_{4}$ $\displaystyle=$
$\displaystyle-\alpha_{3}\,,\quad\alpha_{6}=-\alpha_{1}-\alpha_{5}\,,\quad\alpha_{8}=\alpha_{1}-\alpha_{7}\,,$
$\displaystyle\alpha_{9}$ $\displaystyle=$
$\displaystyle-\alpha_{1}-\alpha_{10}-\alpha_{14}\,,\quad\alpha_{11}=-\alpha_{1}\,,$
$\displaystyle\alpha_{12}$ $\displaystyle=$
$\displaystyle\alpha_{1}+\alpha_{5}+\alpha_{7}\,,\quad\alpha_{13}=\alpha_{1}-\alpha_{5}-\alpha_{7}+2\alpha_{14}\,,$
$\displaystyle\alpha_{16}$ $\displaystyle=$
$\displaystyle-\alpha_{1}-2\alpha_{14}\,,\quad\alpha_{17}=-3\alpha_{1}\,,\quad\alpha_{18}=\alpha_{1}+2\alpha_{14}\,,$
$\displaystyle\alpha_{19}$ $\displaystyle=$
$\displaystyle-\alpha_{14}\,,\quad\alpha_{20}=4\alpha_{1}+\alpha_{5}-\alpha_{15}\,.$
(91)

Moreover, there is a 5-parametric family of trivial operators, characterized
by $\\{\alpha_{2},\alpha_{3},\alpha_{5},\alpha_{7},\alpha_{10}\\}$; we
eliminate these by setting

$\alpha_{2}=\alpha_{3}=\alpha_{5}=\alpha_{7}=\alpha_{10}=0\,.$ (92)

We are just left with 3 non-trivial symmetry operators, characterized by
$\\{\alpha_{1},\alpha_{14},\alpha_{15}\\}$. In particular, by setting

$\alpha_{15}=1\,,\quad\alpha_{1}=\alpha_{14}=0\,,$ (93)

we recover the above Killing vector operator $O_{\xi}^{(t)}$, (90). Next,
setting

$\alpha_{1}=1\,,\quad\alpha_{14}=\alpha_{15}=0\,,$ (94)

we recover the following operator:

$\displaystyle O^{(t)}_{l_{1}}$ $\displaystyle=$
$\displaystyle\nabla^{2}_{k}\delta_{a}^{c}\delta_{b}^{d}-k^{e(c}\nabla_{(a}\nabla_{|e|}\delta^{d)}_{b)}+k^{e}{}_{(a}\nabla^{(c}\nabla_{|e|}\delta^{d)}_{b)}$
(95)
$\displaystyle-k^{e}{}_{(a}\nabla_{|e|}\nabla_{b)}g^{cd}-k^{(c}{}_{(a}\nabla^{2}\delta^{d)}_{b)}+k^{(c}{}_{(a}\nabla^{d)}\nabla_{b)}$
$\displaystyle+k^{(c}{}_{(a}\nabla_{b)}\nabla^{d)}-\xi^{(c}\nabla^{d)}g_{ab}-3\xi^{(c}\nabla_{(a}\delta^{d)}{}_{b)}$
$\displaystyle+\xi_{(a}\nabla^{(c}\delta^{d)}{}_{b)}+4(d\xi)^{(c}{}_{(a}\delta^{d)}{}_{b)}\,.$

Finally, setting

$\alpha_{14}=1\,,\quad\alpha_{1}=\alpha_{15}=0\,,$ (96)

we have the following symmetry operator:

$\displaystyle O^{(t)}_{l_{2}}$ $\displaystyle=$
$\displaystyle-k^{e}{}_{(a}\nabla_{|e|}\nabla_{b)}g^{cd}+2k^{(c}{}_{(a}\nabla_{b)}\nabla^{d)}+\nabla_{\xi}g^{cd}g_{ab}$
(97)
$\displaystyle-2\xi^{(c}\nabla^{d)}g_{ab}+2\xi_{(a}\nabla^{(c}\delta^{d)}{}_{b)}-\xi_{(a}\nabla_{b)}g^{cd}\,.$

Of course, both these symmetry operators remain symmetry operators also for
the case of $\Lambda=0$, while no new independent operators are obtained in
that case.

###  V.3 quadratic operators

Let us next proceed to quadratic tensor operators. Denoting

$(d\xi\cdot k)=(d\xi)^{ab}k_{ab}\,,$ (98)

we find 52 possible operators which we list below.

First, we have the operators of the type “$k^{2}\nabla^{2}$”. These are given
by:

$\displaystyle O_{1}$ $\displaystyle=$
$\displaystyle\nabla^{2}_{k^{2}}g_{ab}g^{cd}\,,\quad
O_{2}=\nabla^{2}_{k^{2}}\delta^{c}_{a}\delta^{d}_{b}\,,\quad
O_{3}=k^{c}{}_{a}\nabla^{2}_{k}\delta^{d}_{b}\,,$ $\displaystyle O_{4}$
$\displaystyle=$ $\displaystyle k_{ae}k_{bf}\nabla^{e}\nabla^{f}g^{cd}\,,\quad
O_{5}=k^{ce}k^{df}\nabla_{e}\nabla_{f}g_{ab}\,,$ $\displaystyle O_{6}$
$\displaystyle=$ $\displaystyle
k^{ce}k_{af}\nabla_{e}\nabla^{f}\delta^{d}_{b}\,,\quad
O_{7}=k^{ce}k_{af}\nabla^{f}\nabla_{e}\delta^{d}_{b}\,,$ $\displaystyle O_{8}$
$\displaystyle=$ $\displaystyle(k^{2})^{ce}\nabla_{e}\nabla^{d}g_{ab}\,,\quad
O_{9}=(k^{2})^{ce}\nabla^{d}\nabla_{e}g_{ab}\,,$ $\displaystyle O_{10}$
$\displaystyle=$
$\displaystyle(k^{2})^{ce}\nabla_{e}\nabla_{a}\delta^{d}_{b}\,,\quad
O_{11}=(k^{2})^{ce}\nabla_{a}\nabla_{e}\delta^{d}_{b}\,,$ $\displaystyle
O_{12}$ $\displaystyle=$
$\displaystyle(k^{2})_{ae}\nabla^{e}\nabla_{b}g^{cd}\,,\quad
O_{13}=(k^{2})_{ae}\nabla_{b}\nabla^{e}g^{cd}\,,$ $\displaystyle O_{14}$
$\displaystyle=$
$\displaystyle(k^{2})_{ae}\nabla^{e}\nabla^{c}\delta^{d}_{b}\,,\quad
O_{15}=(k^{2})_{ae}\nabla^{c}\nabla^{e}\delta^{d}_{b}\,,$ $\displaystyle
O_{16}$ $\displaystyle=$ $\displaystyle
k^{c}{}_{a}k^{d}{}_{e}\nabla^{e}\nabla_{b}\,,\quad
O_{17}=k^{c}{}_{a}k^{d}{}_{e}\nabla_{b}\nabla^{e}\,,$ $\displaystyle O_{18}$
$\displaystyle=$ $\displaystyle k^{2}\nabla^{2}g^{cd}g_{ab}\,,\quad
O_{19}=k^{2}\nabla^{2}\delta^{c}_{a}\delta^{d}_{b}\,,$ $\displaystyle O_{20}$
$\displaystyle=$ $\displaystyle k^{c}{}_{a}k^{d}{}_{b}\nabla^{2}\,,\quad
O_{21}=(k^{2})^{c}{}_{a}\nabla^{d}\nabla_{b}\,,$ (99) $\displaystyle O_{22}$
$\displaystyle=$ $\displaystyle(k^{2})^{c}{}_{a}\nabla_{b}\nabla^{d}\,,\quad
O_{23}=(k^{2})_{a}{}^{c}\nabla^{2}\delta_{b}{}^{d}\,,$ $\displaystyle O_{24}$
$\displaystyle=$ $\displaystyle
k^{2}\nabla_{a}\nabla^{c}\delta_{b}^{d}\,,\quad
O_{25}=k^{2}\nabla^{c}\nabla_{a}\delta_{b}^{d}\,,$ $\displaystyle O_{26}$
$\displaystyle=$ $\displaystyle k^{2}\nabla_{a}\nabla_{b}g^{cd}\,,\quad
O_{27}=k^{2}g_{ab}\nabla^{c}\nabla^{d}\,.$

Here, the operators $\\{O_{1},\dots O_{7}\\}$ have two contractions between
$k$’s and $\nabla$’s, the operators $\\{O_{8},\dots O_{17}\\}$ have one, and
operators $\\{O_{18},\dots O_{27}\\}$ have no such contractions.

We then move to operators of the type “$k\xi\nabla$”

$\displaystyle O_{28}$ $\displaystyle=$ $\displaystyle
k^{c}{}_{a}\nabla_{\xi}\delta^{d}_{b}\,,\quad
O_{29}=\xi^{c}k^{d}{}_{e}\nabla^{e}g_{ab}\,,$ $\displaystyle O_{30}$
$\displaystyle=$ $\displaystyle\xi^{c}k_{ae}\nabla^{e}\delta^{d}_{b}\,,\quad
O_{31}=\xi_{a}k^{c}{}_{e}\nabla^{e}\delta^{d}_{b}\,,$ $\displaystyle O_{32}$
$\displaystyle=$ $\displaystyle\xi_{a}k_{be}\nabla^{e}g^{cd}\,,\quad
O_{33}=k^{f}{}_{e}\xi^{e}\nabla_{f}g^{cd}g_{ab}\,,$ $\displaystyle O_{34}$
$\displaystyle=$ $\displaystyle
k^{f}{}_{e}\xi^{e}\nabla_{f}\delta^{c}_{a}\delta^{d}_{b}\,,$ $\displaystyle
O_{35}$ $\displaystyle=$ $\displaystyle\xi^{c}k^{d}{}_{a}\nabla_{b}\,,\quad
O_{36}=\xi_{a}k^{c}{}_{b}\nabla^{d}\,,$ $\displaystyle O_{37}$
$\displaystyle=$ $\displaystyle k^{c}{}_{e}\xi^{e}\nabla^{d}g_{ab}\,,\quad
O_{38}=k^{c}{}_{e}\xi^{e}\nabla_{a}\delta^{d}_{b}\,,$ $\displaystyle O_{39}$
$\displaystyle=$ $\displaystyle k_{ae}\xi^{e}\nabla_{b}g^{cd}\,,\quad
O_{40}=k_{ae}\xi^{e}\nabla^{c}\delta^{d}_{b}\,.$ (100)

Here, the operators $\\{O_{28},\dots,O_{34}\\}$ have 1 contraction between
$k$’s and $\nabla$, and operators $\\{O_{35},\dots,O_{40}\\}$ have no such
contractions.

Finally, we have operators of the type “$\xi\xi$” and “$k(d\xi)$”. These read:

$\displaystyle O_{41}$ $\displaystyle=$ $\displaystyle(d\xi\cdot
k)g^{cd}g_{ab}\,,\quad O_{42}=(d\xi\cdot k)\delta^{c}_{a}\delta^{d}_{b}\,,$
$\displaystyle O_{43}$ $\displaystyle=$
$\displaystyle(d\xi)^{c}{}_{e}k^{ed}g_{ab}\,,\quad
O_{44}=(d\xi)^{c}{}_{e}k^{e}{}_{a}\delta^{d}_{b}\,,$ $\displaystyle O_{45}$
$\displaystyle=$ $\displaystyle(d\xi)_{ae}k^{e}{}_{b}g^{cd}\,,\quad
O_{46}=(d\xi)_{ae}k^{ec}\delta^{d}_{b}\,,$ $\displaystyle O_{47}$
$\displaystyle=$ $\displaystyle(d\xi)^{c}{}_{a}k^{d}{}_{b}\,,\quad
O_{48}=\xi^{c}\xi^{d}g_{ab}\,,$ $\displaystyle O_{49}$ $\displaystyle=$
$\displaystyle\xi^{c}\xi_{a}\delta^{d}_{b}\,,\quad
O_{50}=\xi_{a}\xi_{b}g^{cd}\,,$ $\displaystyle O_{51}$ $\displaystyle=$
$\displaystyle\xi^{2}g^{cd}g_{ab}\,,\quad
O_{52}=\xi^{2}\delta^{c}_{a}\delta^{d}_{b}\,.$ (101)

Note $(d\xi\cdot k)_{ab}=(d\xi\cdot k)_{ba}$ but $(d\xi\cdot
k)_{a}{}^{c}=-(d\xi\cdot k)_{c}{}^{a}$

Interestingly, we find that there are no non-trivial symmetry operators for
the off-shell Kerr-NUT-AdS spacetimes of this type.

Moving on to the on-shell spacetimes with non-trivial $\Lambda$, we find 42
constraints and 7-parametric family of trivial operators. We are thus left
with 3 non-trivial symmetry operators, in our case characterized by a choice
of $\\{\alpha_{2},\alpha_{26},\alpha_{37}\\}$. First, setting

$\alpha_{26}=1\,,\quad\alpha_{2}=0=\alpha_{37}\,,$ (102)

we find the following operator:

$\displaystyle O^{(t)}_{q_{1}}$ $\displaystyle=$
$\displaystyle-2k^{2}\nabla_{(a}\nabla^{(c}\delta^{d)}{}_{b)}+k^{2}\nabla_{(a}\nabla_{b)}g^{cd}$
(103)
$\displaystyle+4k_{(a|e|}\xi^{e}\nabla_{b)}g^{cd}-8k_{(a|e|}\xi^{e}\nabla^{(c}\delta^{d)}{}_{b)}\,.$

Next, setting

$\alpha_{37}=2\,,\quad\alpha_{2}=\alpha_{26}=0\,,$ (104)

we find

$\displaystyle O^{(t)}_{q_{2}}$ $\displaystyle=$
$\displaystyle(k^{2})_{(a|e|}\nabla_{b)}\nabla^{e}g^{cd}-2(k^{2})^{(c}{}_{(a}\nabla_{b)}\nabla^{d)}$
(105)
$\displaystyle-\xi_{(a}k_{b)e}\nabla^{e}g^{cd}-k^{f}{}_{e}\xi^{e}\nabla_{f}g^{cd}g_{ab}$
$\displaystyle-2\xi_{(a}k^{(c}{}_{b)}\nabla^{d)}+2k^{(c}{}_{e}\xi^{|e|}\nabla^{d)}g_{ab}$
$\displaystyle-
k_{(a|e|}\xi^{e}\nabla_{b)}g^{cd}+2k_{(a|e|}\xi^{e}\nabla^{(c}\delta^{d)}{}_{b)}\,.$

Finally, setting

$\alpha_{2}=1\,,\quad\alpha_{26}=\alpha_{37}=0\,,$ (106)

we have

$\displaystyle O^{(t)}_{q_{3}}$ $\displaystyle=$
$\displaystyle\nabla^{2}_{k^{2}}\delta^{c}_{a}\delta^{d}_{b}-2(k^{2})^{(c|e|}\nabla_{e}\nabla_{(a}\delta^{d)}{}_{b)}$
(107)
$\displaystyle+2(k^{2})^{(c|e|}\nabla_{(a}\nabla_{|e|}\delta^{d)}{}_{b)}+\frac{1}{2}k^{2}\nabla^{2}\delta^{c}_{a}\delta^{d}_{b}$
$\displaystyle+k^{2}\nabla_{(a}\nabla^{(c}\delta^{d)}{}_{b)}-k^{2}\nabla^{(c}\nabla_{(a}\delta^{d)}{}_{b)}$
$\displaystyle+8k^{(c}{}_{(a}\nabla_{\xi}\delta^{d)}{}_{b)}+4\xi^{(c}k_{(a|e|}\nabla^{e}\delta^{d)}_{b)}$
$\displaystyle-4\xi_{(a}k^{(c}{}_{|e|}\nabla^{|e|}\delta^{d)}{}_{b)}-2k^{f}{}_{e}\xi^{e}\nabla_{f}\delta^{c}_{a}\delta^{d}_{b}$
$\displaystyle-4k^{(c}{}_{e}\xi^{|e|}\nabla_{(a}\delta^{d)}{}_{b)}+4k_{(a|e|}\xi^{e}\nabla^{(c}\delta^{d)}{}_{b)}$
$\displaystyle-(d\xi)_{(a|e|}k^{e}{}_{b)}g^{cd}+3(d\xi)_{(a|e|}k^{e(c}\delta^{d)}{}_{b)}$
$\displaystyle-(d\xi)^{(c}{}_{(a}k^{d)}{}_{b)}-2\xi^{(c}\xi^{d)}g_{ab}+8\xi^{(c}\xi_{(a}\delta^{d)}{}_{b)}$
$\displaystyle-2\xi_{a}\xi_{b}g^{cd}+2\xi^{2}g^{cd}g_{ab}-6\xi^{2}\delta^{c}_{a}\delta^{d}_{b}\,.$

Of course, all three of these operators remain also valid in the vacuum,
$\Lambda=0$, case. Then, and further in the absence of the NUT charge, the
operators $O^{(t)}_{q_{1}}$ and $O^{(t)}_{q_{2}}$ coincide with ${\cal K}_{3}$
and ${\cal K}_{2}$ respectively in Mei:2023pho , while the operator
$O^{(t)}_{q_{3}}$ is related to ${\cal K}_{4}$ therein. We have not discovered
any new independent non-trivial symmetry operators of this type in the vacuum
case.

###  V.4 Tensor symmetry operators: summary

In addition to the above operators, we also have the cubic in $k$ secondary
Killing vector operator:

$O^{(t)}_{\zeta}=\zeta^{e}\nabla_{e}\delta^{c}_{a}\delta^{d}_{b}-(d\zeta)^{c}{}_{a}\delta^{d}_{b}=\delta^{c}_{a}\delta^{d}_{b}{\cal
L}_{\zeta}=\delta^{c}_{a}\delta^{d}_{b}\partial_{\varphi}\,.$ (108)

Of course, such an operator is valid both on-shell and off-shell. While we
have not checked this explicitly, we expect that, similar to the vector case,
this is the only cubic in $k$ symmetry operator.

If so, this means that, we have found the following 7 operators:

$\\{O^{(t)}_{\xi}\,,\ O^{(t)}_{l_{1}}\,,\ O^{(t)}_{l_{2}}\,,\
O^{(t)}_{q_{1}}\,,\ O^{(t)}_{q_{2}}\,,O^{(t)}_{q_{2}},O^{(t)}_{\zeta}\\}\,,$
(109)

that commute with the Lichnerowicz operator for the on-shell Kerr–NUT-AdS
spacetimes with arbitrary cosmological constant $\Lambda$. Together with $L$,
we thus have a set of 8 operators. It remains to be seen, if these are enough
to directly separate gravitational perturbations in these spacetimes.
Interestingly, for the off-shell Kerr–NUT–AdS spacetimes, only the 2 Killing
vector operators commute with the Lichnerowicz operator. Perhaps this could be
amended by adding curvature terms into our collection of potential operators –
we leave this for future study.

##  VI Summary

In this work, we have systematically studied homogeneous symmetry operators
for scalar, vector, and tensor perturbations of (general) four-dimensional
Kerr–NUT–AdS spacetimes, which are known to admit a fundamental hidden
symmetry of the principal Killing–Yano tensor $k$. In particular, we have
concentrated on operators of degree two in the sum of the number of
derivatives acting on $k$ and acting on the corresponding perturbation. In
this way we have obtained linear, quadratic, and cubic in $k$ operators that
commute with the corresponding field operator for both off-shell and on-shell
Kerr–NUT–AdS metrics.

Our study has developed upon the recent work Mei:2023pho  – we have
generalized it in a number ways. First, the work in Mei:2023pho  fully
concentrated on the vacuum Kerr case, whereas ours deals with more general
(possibly off-shell) Kerr–NUT–AdS spacetimes. Second, we have found a number
of new (predominantly linear) symmetry operators. Third, we have directly
worked with the principal tensor $k$ (rather than its Hodge dual $*k$) which
is the fundamental symmetry of the Kerr–NUT–AdS spacetimes that survives in
any number of dimensions.

For this reason, our results readily generalize to higher dimensions – the
operators will take the same form as those in 4d, apart from dimension-
dependent factors. For example, the new linear in $k$ 4d operator (58)
generalizes to 5d off-shell Kerr–NUT–AdS spacetimes as follows:

$O^{(v)}_{k}=k^{a}{}_{b}\nabla^{2}-k_{cb}\nabla^{c}\nabla^{a}-k^{ac}\nabla_{c}\nabla_{b}-\frac{9}{4}\xi_{b}\nabla^{a}\,,$
(110)

that is, the last factor simply changed from $(-2)$ to $(-9/4)$.

In higher dimensions, however, apart from generalizing the obtained operators
mentioned above that are linear, quadratic, and cubic in $k$, one will also
have to consider operators that are higher-order in $k$. For example, for the
scalar case in $d$ dimensions, the complete set of commuting operators
involves operators that are homogeneous in all possibles powers of $k$,
ranging from zero (the box operator) to $(d-1)$. We shall return to this issue
in the follow up paper myprep:2024 .

The main purpose of studying symmetry operators is to probe whether one has
enough ‘symmetries’ that would allow for separability of the corresponding
test field equations. Contrary to for example Teukolsky:1972my ;
Teukolsky:1973ha , in our approach we directly work with perturbations
themselves (with the vector potential or the metric perturbation). In this
way, we have found a set of 7 mutually commuting operators for the off-shell
vector case and a set of 8 operators for the vacuum tensor case.
Interestingly, for the scalar and Dirac perturbations one only needs 4 such
operators – the separated solution is simply the common eigenfunction of such
operators. It remains to be seen, if the obtained symmetry operators ‘are
enough’ to directly separate the (massive) vector and tensor perturbations in
these spacetimes.

### Acknowledgements

D.K. is grateful for support from GAČR 23-07457S grant of the Czech Science
Foundation.

##  Appendix A Dirac symmetry operators

In this appendix we review, following Carter:1979fe ; Oota:2007vx ;
Cariglia:2011qb , the symmetry operators responsible for separability of the
massive Dirac equation in Kerr–NUT–AdS spacetimes. As it turns out, these are
also given by the homogeneous first-order and 2nd-order operators. For
simplicity, we restrict to the four-dimensional case Carter:1979fe , and refer
to Oota:2007vx ; Cariglia:2011qb  for a generalization to higher dimensions.

The Dirac operator reads

$D=\gamma^{A}\nabla_{A}\,,$ (111)

where $A,B,\dots$ refer to the orthonormal frame indices, and the
$\gamma$-matrices obey

$\gamma^{A}\gamma^{B}+\gamma^{B}\gamma^{A}=2g^{{A}{B}}\,.$ (112)

Then, in 4d off-shell canonical Kerr–NUT–AdS spacetimes, we find the following
linear in PKY and linear in derivatives operator:

$M=\gamma^{ABC}k_{{A}{B}}\nabla_{{C}}+2\xi_{A}\gamma^{A}\,,$ (113)

where we use $\gamma^{AB\dots C}=\gamma^{[A}\gamma^{B}\dots\gamma^{C]}$ to
denote antisymmetrization of the $\gamma$-matrices. Of course, we also have
the symmetry operators associated with the Killing directions. Namely, we have
a linear in PKY and quadratic in derivatives operator

$K_{\xi}=\xi^{A}\nabla_{A}+\frac{1}{8}\gamma^{AB}(d\xi)_{{A}{B}}\,,$ (114)

and a cubic in PKY and quadratic in derivatives operator

$K_{\zeta}=\zeta^{A}\nabla_{A}+\frac{1}{8}\gamma^{AB}(d\zeta)_{{A}{B}}\,.$
(115)

These operators form a set of 4 mutually commuting operators:

$\\{D,M,K_{\xi},K_{\zeta}\\}\,.$ (116)

Interestingly, such a set is enough to characterize the separability of the
Dirac equation, whose separated solution can be found as a common
eigenfunction of these operators.

In particular, introducing the following natural orthonormal basis:

$g=\eta_{{A}{B}}e^{A}e^{B}\,,$ (117)

where

$\displaystyle e^{0}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\Delta_{r}}{\Sigma}}(d\tau+y^{2}d\psi)\,,\quad
e^{1}=\sqrt{\frac{\Sigma}{\Delta_{r}}}dr\,,$ $\displaystyle e^{\hat{2}}$
$\displaystyle=$
$\displaystyle\sqrt{\frac{\Delta_{y}}{\Sigma}}(d\tau-r^{2}d\psi)\,,\quad
e^{3}=\sqrt{\frac{\Sigma}{\Delta_{y}}}dy\,,$ (118)

and choosing the following representation for the $\gamma$ matrices:

$\displaystyle\gamma^{0}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}0&-I\\\
I&0\end{array}\right)\,,\quad\gamma^{1}=\left(\begin{array}[]{cc}\ 0&\ I\\\ \
I&\ 0\end{array}\right)\,,$ (123) $\displaystyle\gamma^{2}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\sigma^{2}&0\\\
0&-\sigma^{2}\end{array}\right)\,,\quad\gamma^{3}=\left(\begin{array}[]{cc}\sigma^{1}&0\\\
0&-\sigma^{1}\end{array}\right)\,,$ (128)

where $I$ is a unit $2\times 2$ matrix and $\sigma^{i}$ are Pauli matrices,
the R-separated 666R-separability is a generalized notion of multiplicative
separability, where each component of the field takes a multiplicative
separable form, up to a ‘known’ non-separable factor (the so called
$R$-factor), which is in principle a function of all coordinates, and has to
be ‘guessed’, see (131) above.  solution $\Psi$ of the Dirac massive equation:

$(D+\mu)\Psi=0\,,$ (129)

is the eigenfunction of the following eigenvalue problem:

$\displaystyle D\Psi$ $\displaystyle=$ $\displaystyle-\mu\Psi\,,$
$\displaystyle M\Psi$ $\displaystyle=$ $\displaystyle\kappa\Psi\,,$
$\displaystyle K_{\xi}\Psi$ $\displaystyle=$ $\displaystyle-i\omega\Psi\,,$
$\displaystyle K_{\zeta}\Psi$ $\displaystyle=$ $\displaystyle im\Psi\,,$ (130)

where $\omega,m$ stands for the frequency, azimuthal number, respectively, and
$\kappa$ is the separation constant. Namely, it reads

$\psi=\left(\begin{array}[]{c}(r-iy)^{-1/2}R_{+}Y_{+}\\\
(r+iy)^{-1/2}R_{+}Y_{-}\\\ (r+iy)^{-1/2}R_{-}Y_{+}\\\
(r-iy)^{-1/2}R_{-}Y_{-}\end{array}\right)\,e^{i(m\psi-\omega t)}\,,$ (131)

with functions $R_{\pm}=R_{\pm}(r)$ and $Y_{\pm}=Y_{\pm}(y)$ obey the
following ODEs, e.g. Frolov:2017kze :

$\displaystyle\frac{dR_{\pm}}{dr}+R_{\pm}\frac{\Delta_{r}^{\prime}\pm
V_{r}}{4\Delta_{r}}+R_{\mp}\frac{\mu r\mp\kappa}{\sqrt{\Delta_{r}}}$
$\displaystyle=$ $\displaystyle 0\,,$
$\displaystyle\frac{dY_{\pm}}{dy}+Y_{\pm}\frac{\Delta_{y}^{\prime}\pm
V_{y}}{4\Delta_{y}}-Y_{\mp}\frac{\kappa\pm i\mu y}{\sqrt{\Delta_{y}}}$
$\displaystyle=$ $\displaystyle 0\,$ (132)

where $V_{r}$ and $V_{y}$ are given by

$V_{r}=4i(m-\omega r^{2})\,,\quad V_{y}=4(m+\omega y^{2})\,.$ (133)

Let us stress that in principle the ansatz (131) leads to 8 equations with
four different separation constants. However, the consistency of these
equations implies that only 1 of these separation constants is independent,
given by the eigenvalue of the operator $M$. In other words, with the special
separability ansatz above, the 4 operators (116) form a complete set of
commuting operators responsible for separation of the Dirac equation. A
similar situation happens in higher dimensions, where a set of $D$ mutually
commuting operators underlies separability of the Dirac equation in
$D$-dimensional Kerr–NUT–AdS spacetimes Oota:2007vx ; Cariglia:2011qb . This
raises a hope that perhaps similar statement is, with a proper separation
ansatz, also true for other (vector or tensor) fields.

##  Appendix B Motion of particles

The pattern seen for operators of various fields also translates to quantities
associated with motion of particles: homogeneous operators translate to
corresponding integrals of motion. Let us review this for geodesics and for
the motion of classical spinning particle.

###  B.1 Geodesics

The geodesic motion is described by the following Hamiltonian:

$H=\frac{1}{2}g^{ab}p_{a}p_{b}\,,$ (134)

where $p_{a}(\tau)$ are canonical momenta, conjugate to particle’s coordinates
$x^{a}(\tau)$. The Hamilton’s equations of motion are equivalent to the
geodesic equation.

To find an integral of motion $Q$, we seek a quantity that obeys

$\\{Q,H\\}=0\,,$ (135)

where the Poisson brackets are defined as

$\\{F,G\\}=\frac{\partial F}{\partial x^{a}}\frac{\partial G}{\partial
p_{a}}-\frac{\partial G}{\partial x^{a}}\frac{\partial F}{\partial p_{a}}\,.$
(136)

We may proceed in a way similar to what happens in the main text, seeking a
scalar quantity that is homogeneous in the number of PKY tensors and
homogeneous in the ‘sum of’ derivatives acting on $k$ and the number of
momenta $p_{a}$.

It is easy to see that there are no quantities that are linear in $k$ and of
degree one. There is one quantity that is linear in $k$ and of degree two:

$Q_{\xi}=\xi^{a}p_{a}\,,$ (137)

which corresponds to a constant of motion associated with the Killing vector
$\xi$. Moving to quadratic in $k$, there are no degree one quantities, but
there are the following degree two quantities:

$(k^{2})^{ab}p_{a}p_{b}\,,\quad k^{2}p^{2}\,,\quad
k^{a}{}_{b}\xi^{b}p_{a}\,,\quad\xi^{2}\,,\quad(d\xi)^{ab}k_{ab}\,,$ (138)

By requiring (135), the last 3 have to be eliminated, and we recover the
famous Carter’s constant Carter:1968rr

$Q_{K}=\Bigl{(}(k^{2})^{ab}+\frac{1}{2}k^{2}g^{ab}\Bigr{)}p_{a}p_{b}=K^{ab}p_{a}p_{b}\,.$
(139)

Considering finally constants that are cubic in $k$ and of degree two, we have
the following non-trivial possibilities:

$(k^{2})^{a}{}_{b}\xi^{b}p_{a}\,,\quad k^{2}\xi^{a}p_{a}\,.$ (140)

These combine to the following constant of motion:

$Q_{\zeta}=\Bigl{(}(k^{2})^{a}{}_{b}\xi^{b}+\frac{1}{2}k^{2}\xi^{a}\Bigr{)}p_{a}=\zeta^{a}p_{a}\,,$
(141)

associated with the Killing vector $\zeta$, which finalizes the complete set
of mutually commuting integrals of motion:

$\\{H,Q_{\xi},Q_{\zeta},Q_{K}\\}\,,$ (142)

which underlie the complete integrability of geodesic motion in Kerr–NUT–AdS
spacetimes.

###  B.2 Classical spinning particle

The above construction for geodesics can be generalized to classical spinning
particles, e.g. Rietdijk:1989qa ; Gibbons:1993ap . Such a motion can be
derived from the following Lagrangian:

${\cal
L}=\frac{1}{2}g_{ab}\dot{x}^{a}\dot{x}^{b}+\frac{i}{2}\eta_{AB}\theta^{A}\frac{D\theta^{B}}{d\tau}\,,$
(143)

where $x^{a}(\tau)$ ($a=1,\dots,d$) denote the particle’s worldline
coordinates, and the spin is described by a Lorentz vector of Grassmann-odd
coordinates $\theta^{A}(\tau)$ ($A=1,\dots,d$). In this subsection, $d$ stands
for the spacetime dimension, indices $A,B,\dots$ denote vielbein indices,
indices $a,b,\dots$ denote the curved space indices, and $e^{a}_{A}$ are the
veilbein components (that can be used to convert spacetime to vielbein indices
and vice versa). The above Lagrangian yields the following equations of
motion:

$\displaystyle\frac{D^{2}x^{a}}{d\tau^{2}}$ $\displaystyle=$
$\displaystyle\ddot{x}^{a}+\Gamma^{a}_{bc}\dot{x}^{b}\dot{x}^{c}=\frac{i}{2}R^{a}_{\
bAB}\theta^{A}\theta^{B}\dot{x}^{b}\,,$ (144)
$\displaystyle\frac{D\theta^{A}}{D\tau}$ $\displaystyle=$
$\displaystyle\dot{\theta}^{A}+\omega_{b}{}^{A}{}_{B}\dot{x}^{b}\theta^{B}=0\,,$
(145)

where $\Gamma^{a}_{bc}$ and $\omega_{b}{}_{AB}$ are the Levi-Civita and spin
connections, respectively, and $R_{abcd}$ is the Riemann tensor.

The theory possesses a generic supercharge $Q$,

$Q=\theta^{A}e_{A}{}^{a}\Pi_{a}\,,$ (146)

which obeys

$\\{H,Q\\}=0\,,\quad\\{Q,Q\\}=-2iH\,.$ (147)

Here, $H$ is the Hamiltonian,

$H=\frac{1}{2}\Pi_{a}\Pi_{b}g^{ab}\,,$ (148)

where $\Pi_{a}$ is related to $p_{a}$, the momentum canonically conjugate to
$x^{a}$, as follows:

$\Pi_{a}=p_{a}-\frac{i}{2}\theta^{A}\theta^{B}\omega_{aAB}=g_{ab}\dot{x}^{b}\,,$
(149)

and the Poisson brackets are defined as

$\\{F,G\\}=\frac{\partial F}{\partial x^{a}}\frac{\partial G}{\partial
p_{a}}-\frac{\partial F}{\partial p_{a}}\frac{\partial G}{\partial
x^{a}}+i(-1)^{a_{F}}\frac{\partial F}{\partial\theta^{A}}\frac{\partial
G}{\partial\theta_{A}}\,,$ (150)

where $a_{F}$ is the Grassmann parity of $F$, and the equations of motion are
accompanied by two physical (gauge) conditions

$2H=-1\,,\quad Q=0\,,$ (151)

which state that $\tau$ is the proper time and the particle’s spin is
spacelike.

A non-generic superinvariant $S$ is a quantity that Poisson commutes with the
generic supercharge:

$\\{Q,S\\}=0\,.$ (152)

Due to the Jacobi identity, any superinvariant is automatically a constant of
motion, $\\{H,S\\}=0$, and so is a new superinvariant $\\{S,S\\}$ (which may,
or may not be equal to $H$). Such superinvariants correspond to an enhanced
worldline (super)symmetry.

As shown in Gibbons:1993ap ; Kubiznak:2011ay , for Kerr–NUT–AdS spacetimes,
the spinning particle motion admits a number of homogeneous superinvariants.
In particular, focusing on 4d, we have the following two superinvariants
linear in the PKY tensor:

$\displaystyle Q_{\xi}$ $\displaystyle=$
$\displaystyle\xi^{a}\Pi_{a}-\frac{i}{4}\theta^{A}\theta^{B}(d\xi)_{AB}\,,$
(153) $\displaystyle Q_{h}$ $\displaystyle=$
$\displaystyle\theta^{A}(*k)^{a}{}_{A}\Pi_{a}-\frac{i}{9}\theta^{A}\theta^{B}\theta^{C}(d*k)_{ABC}\,.$
(154)

We also have a quadratic in $k$ superinvariant

$Q_{k^{2}}=K^{ab}\Pi_{a}\Pi_{b}+\theta^{A}\theta^{B}L_{AB}{}^{a}\Pi_{a}+\theta^{A}\theta^{B}\theta^{C}\theta^{D}M_{ABCD}\,,$
(155)

where

$\displaystyle K^{ab}$ $\displaystyle=$
$\displaystyle(*k)^{ae}(*k)^{b}{}_{e}=k_{ac}k_{b}^{\
c}+\frac{1}{2}k^{2}g_{ab}\,,$ $\displaystyle L_{abc}$ $\displaystyle=$
$\displaystyle-\frac{2i}{3}(*k)_{[a|e}(d*k)_{b]c}{}^{e}-\frac{2i}{3}(d*k)_{abe}(*k)_{c}{}^{e}\,,$
$\displaystyle=$ $\displaystyle
2i\big{(}\xi_{[a}k_{b]}{}_{c}+k_{ab}\xi_{c}+(\xi\cdot k)_{[a}g_{b]c}\big{)}$
$\displaystyle M_{abcd}$ $\displaystyle=$
$\displaystyle-\frac{i}{4}\nabla_{[a}L_{bcd]}=-\frac{i}{2}d\xi_{[ab}k_{cd]}\,.$
(156)

It also admits a cubic in $k$ superinvariant associated with the Killing
vector $\zeta$:

$Q_{\zeta}=\zeta^{a}\Pi_{a}-\frac{i}{4}\theta^{A}\theta^{B}(d\zeta)_{AB}\,.$
(157)

Note that $\\{Q_{\xi},Q_{k^{2}},Q_{\zeta}\\}$ are Grassmann even (bosonic) and
generalize the above constants of geodesic motion. One can show that in higher
dimensions, there will always be $d$ such bosonic constants for the
$d$-dimensional spinning particle motion, see Kubiznak:2011ay . On the other
hand superinvariants $\\{Q,Q_{h}\\}$ are Grassmann odd (fermionic), and are
reminiscent of the Dirac operator $D$ and the linear in $k$ operator $M$,
(113).

##  Appendix C Working with the Hodge dual?

In the main text, we have fully focused on working with the principal tensor
$k$. This is primarily motivated by the fact that $k$ is the fundamental
symmetry of the Kerr–NUT–AdS spacetimes that has the same rank (of a 2-form)
in any number of dimensions, whereas this is not true for its (dimension-
dependent) Hodge dual. However, one might wonder whether working with $(*k)$
would enlarge the number of commuting operators.777Of course, this may not be
the case for quadratic (or even power) in $(*k)$ operators, as in that case,
one can always eliminate the Hodge star and work directly with $k$’s.  In this
appendix, we gather a simple calculation for the linear in $(*k)$ vector
operators in 4d, showing that no new symmetry operators arise in this case.

When working with the Hodge dual of $k$,

$f=*k$ (158)

we have the following objects at our disposal:

$f_{ab}\,,\quad(df)_{abc}\,,\quad\nu^{ab}=\nabla_{e}(df)^{abe}\,,$ (159)

capturing the full information about the zeroth-order, 1st-order, and 2nd-
order derivatives of $f$. For the linear in $f$ degree two operators we thus
have the following 8 possibilities:

$\displaystyle O_{1}$ $\displaystyle=$ $\displaystyle
f^{a}{}_{b}\nabla^{2}\,,\quad O_{2}=f_{cb}\nabla^{a}\nabla^{c}\,,\quad
O_{3}=f_{cb}\nabla^{c}\nabla^{a}\,,$ $\displaystyle O_{4}$ $\displaystyle=$
$\displaystyle f_{cd}\nabla^{c}\nabla^{d}\delta^{a}_{b}\,,\quad
O_{5}=f^{ac}\nabla_{c}\nabla_{b}\,,\quad O_{6}=f^{ac}\nabla_{b}\nabla_{c}\,,$
$\displaystyle O_{7}$ $\displaystyle=$
$\displaystyle(df)^{a}{}_{be}\nabla^{e}\,,\quad O_{8}=\mu^{a}{}_{b}\,.$ (160)

One can then easily show that requiring the commutativity of the vector
operator with the linear combination of these operators, (54), does not yield
any non-trivial commuting operators in the off-shell Kerr–NUT–AdS spacetimes.
Thus, using $f$ in addition to $k$ does not bring any new operators in the
linear vector case. We suspect (but have not checked explicitly), that the
same conclusion remains true also in other (for example tensor) cases.

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