*mps*

# Ladder Symmetries of Black Holes and de Sitter Space:
Love Numbers and Quasinormal Modes

Roman Berens  Center for Theoretical Physics, Department of Physics, Columbia
University, New York, NY 10027 Lam Hui  Center for Theoretical Physics,
Department of Physics, Columbia University, New York, NY 10027 Zimo Sun
Princeton Gravity Initiative, Princeton University, Princeton, NJ 08544, USA

###### Abstract

In this note, we present a synopsis of geometric symmetries for (spin 0)
perturbations around (4D) black holes and de Sitter space. For black holes, we
focus on static perturbations, for which the (exact) geometric symmetries have
the group structure of SO(1,3). The generators consist of three spatial
rotations, and three conformal Killing vectors obeying a special melodic
condition. The static perturbation solutions form a unitary (principal series)
representation of the group. The recently uncovered ladder symmetries follow
from this representation structure; they explain the well-known vanishing of
the black hole Love numbers. For dynamical perturbations around de Sitter
space, the geometric symmetries are less surprising, following from the
SO(1,4) isometry. As is known, the quasinormal solutions form a non-unitary
representation of the isometry group. We provide explicit expressions for the
ladder operators associated with this representation. In both cases, the
ladder structures help connect the boundary condition at the horizon with that
at infinity (black hole) or origin (de Sitter space), and they manifest as
contiguous relations of the hypergeometric solutions.

###### Contents

  1. 1 Introduction
  2. 2 Black Hole Ladder
    1. 2.1 Geometric Symmetries
    2. 2.2 Ladder Structure and Love Numbers
  3. 3 de Sitter Ladders
    1. 3.1 Geometric Symmetries
    2. 3.2 Ladder Structures and Quasinormal Modes
    3. 3.3 Conformally Coupled Case
  4. 4 Discussion
  5. A (C)KVs of the 3D Effective Metric for Black Hole
  6. B (C)KVs of $\text{dS}_{4}$
  7. C Solutions to Equations of Motion
  8. D Hypergeometric Identities
  9. E Commutators and Lie Brackets
  10. F Melodic Conformal Killing Vectors
  11. G Ladder Identities in de Sitter space
  12. H Conditions for Horizontal Symmetries
  13. I Identities for General First Order Differential Operators
  14. J More Ways to Climb Ladders

##  1 Introduction

Geometric symmetries—isometries and conformal isometries—have always played an
important role in field theories. In this note, we wish to apply the geometric
viewpoint to two well known phenomena: the vanishing of the black hole Love
numbers, and the existence of quasinormal modes in de Sitter space. The reason
for focusing on them is that the underlying symmetries are exact, and they
share a number of common features as we shall see.

It was recently pointed out that static black hole perturbations enjoy certain
exact symmetries, termed ladder symmetries [1] (see also [2, 3]). They help
explain the well known vanishing of the Love numbers, which characterize black
holes’ response to static tidal fields [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
15, 16, 17, 18, 19, 20, 21].111In [22] a different set of symmetries, in a
near-zone approximation, were used to explain the vanishing of the Love
numbers. The representation-theoretic approach adopted in this paper is
similar to [22], but the symmetries are different. In particular, ours do not
involve time. It is possible to extend our exact ladder symmetries to
perturbations of small non-zero frequencies (for which the symmetries are only
approximate), but it requires using a different near-zone approximation from
that of [22]; the relevant near-zone approximation is the one proposed by
[23]. For further discussion, see [24]. For simplicity, in this note, we focus
on spin 0 perturbations. It can be shown that the spin 1 and 2 perturbations
are obtainable from the spin 0 ones by simple (derivative) operations;222The
spin raising (and lowering) operations are possible for uncharged black holes.
See [1]. the relevant spin 0 perturbations $\Phi$ obey:

$\displaystyle\Box_{\rm 4D}\Phi=0\quad;\quad ds_{\rm
4D}^{2}=-\frac{\Delta}{r^{2}}\,dt^{2}+\frac{r^{2}}{\Delta}\,dr^{2}+r^{2}(d\theta^{2}+{\,\rm
sin}^{2}\theta\,d\varphi^{2}),$ (1.1)

where $\Box_{\rm 4D}$ is defined with respect to the 4D metric given. Here, we
consider a general spherically symmetric (Reissner-Nordstöm) black hole, with
$\Delta\equiv r^{2}-r_{S}r+r_{Q}^{2}$, where $r_{S}\equiv 2GM$ and
$r_{Q}^{2}\equiv GQ^{2}$, with $M$ being the black hole mass and $Q$ being the
charge. For static $\Phi$, it can be shown that the equation of motion can be
equivalently expressed as:

$\displaystyle\Box\Phi=0\quad;\quad ds^{2}=dr^{2}+\Delta(d\theta^{2}+{\,\rm
sin}^{2}\theta\,d\varphi^{2}),$ (1.2)

where $\Box$ is now defined with respect to the 3D effective metric given. It
is worth stressing this 3D metric is not the same as the spatial part of the
4D starting point. The 3D effective metric has a vanishing Cotton tensor (but
non-vanishing and non-constant Ricci), and has 10 conformal Killing vectors
(CKVs).333That the 3D effective metric is conformally flat is not
surprising—this is true for $\Delta$ being any function of $r$. What is less
trivial is the condition on some of the CKVs (1.3). Three of these are Killing
vectors (KVs), generating the expected rotational symmetries. The rest at
first sight do not seem useful, since the scalar $\Phi$ is not conformally
coupled. Surprisingly, three of them turn out to generate symmetries for
$\Phi$. In [1], this was attributed to the fact that the 3D effective metric
is conformally related to an Euclidean AdS space. In this note, we wish to
point out a different way of viewing the surprise: it has to do with the fact
that each of these three conformal Killing vectors $X^{i}$ obey a special
condition which we term melodic:

$\displaystyle\Box\nabla_{i}X^{i}=0.$ (1.3)

where the derivatives are defined with respect to the 3D effective metric. A
main objective of this note is to explain the relevance of this condition for
generating symmetries. As we shall see, the three rotational KVs and the three
melodic CKVs form an $\mathfrak{so}(1,3)$ algebra. The solutions to Eq. (1.2),
organized by spherical harmonics, form a representation of the algebra. The
representation structure is what gives rise to the ladder symmetries pointed
out by [1].

This representation understanding of static black hole perturbations can be
usefully compared with the representation understanding of dynamical de Sitter
perturbations. It was first noticed in [25] that the de Sitter quasinormal
modes solved by [26] can be organized into representations of
$\text{SL}(2,\mathbb{R})$. Later in [27, 28], these quasinormal modes were
found to form a representation of the isometry algebra $\mathfrak{so}(1,4)$,
which was generalized to higher dimensions and higher spins by [29]. We shall
review that story, and provide expressions for the myriad of ladder operators,
analogous to those given for black holes.

A common theme in both stories is the connection of boundary conditions. In
both cases, the relevant radial equation (after separation of variables) is a
second order ordinary differential equation. Close to any boundary, there are
in general two asymptotic behaviors. For instance, for the black hole, the
static $\Phi$ approaches either a constant or a logarithm (or combinations
thereof) close to the horizon; it approaches either $r^{\ell}$ or
$1/r^{\ell+1}$ (or combinations thereof) at infinity, where $\ell$ is the
angular momentum quantum number. The Love number surprise, so to speak, has to
do with the fact that the regular solution, where $\Phi$ approaches a constant
at the horizon, has only the “growing” behavior $r^{\ell}$ at infinity, i.e.,
the “decaying” behavior $1/r^{\ell+1}$ is completely absent. Likewise, for
dynamical perturbations in the static patch of de Sitter space, $\Phi$
approaches either an ingoing or an outgoing wave (or combinations thereof) at
the horizon; it approaches either $r^{\ell}$ or $1/r^{\ell+1}$ (or
combinations thereof) at the origin. Quasinormal modes are those solutions at
special frequencies, such that $\Phi$ is purely outgoing at the horizon, and
purely $r^{\ell}$ at the origin.

As we shall see, the ladder structures help explain this single-asymptote
behavior, i.e., a single asymptotic behavior at one boundary is connected with
another single asymptotic behavior at a different boundary. At the level of
the radial solutions for $\Phi$, which are hypergeometric functions, this
phenomenon is associated with what are called connection formulas. The ladder
structures manifest as a generalized form of contiguous relations governing
hypergeometric functions, which we shall spell out.

It is worth stressing that much of this paper is a review of known results.
Our goal is thus a modest one: to view both the black hole Love numbers and
the de Sitter quasinormal modes through the same geometric/ representation-
theoretic lens. For black holes, the surprise, if there is one, is the
relevance of melodic CKVs. For de Sitter space, there is less of a surprise
since the symmetries arise straightforwardly from KVs. In both cases,
organizing the solutions in a representation of the symmetry algebra is a
helpful way of thinking about the ladder structures. For de Sitter space, we
provide expressions for operators (some of them new) for climbing up and down
the myriad ladders. In both cases, the ladder operators provide a simple way
to understand the single-asymptote behavior of the solutions.

Quasinormal modes exist also for black holes of course [30, 31, 32, 33]. The
reason we do not focus on them is that it is not known what exact symmetries
govern them, if any, beyond the usual rotational symmetries. There have been
interesting recent developments pointing out approximate symmetries [34, 35,
36, 37]. We shall discuss them in Section 4.

The paper is organized as follows. In Section 2, we highlight the importance
of the melodic CKVs in giving the static scalar perturbations around a
spherically symmetric black hole the symmetry algebra of $\mathfrak{so}(1,3)$.
The perturbation solutions form a unitary representation, from which the
ladder structures and symmetries of [1] follow. The arguments for the
vanishing of the Love numbers are reviewed here. In Section 3, we apply the
same geometric approach to a dynamical scalar of an arbitrary mass (or
coupling to Ricci) in de Sitter space. Multiple ladders exist, due to the
larger symmetry algebra. Explicit expressions for the ladder operators are
provided, and they are used to deduce the quasinormal spectrum, and the
single-asymptote nature of the quasinormal modes. If the scalar is conformally
coupled, more symmetries exist, the consequences of which are derived from a
representation perspective. In Section 4, we summarize our results and discuss
future directions. Appendices A and B give the (C)KVs of the relevant spaces
and their commutation relations. C contains the exact solutions for the
equations of motion. D shows the algebraic origin of the ladders from various
contiguous relations for hypergeometric functions. E shows the mutual
consistency among three different interpretations of the commutator. F
explores properties of the melodic CKVs. G collects various identities obeyed
by the ladder operators. H puts conditions on the existence of certain
“horizontal” symmetries useful for arguments in the main text. I presents the
general form of some useful identities for first order differential operators
of a certain type. Finally, J shows how to combine various ladder operators to
move flexibly throughout the space of solutions.

##  2 Black Hole Ladder

###  2.1 Geometric Symmetries

Our starting point is the 3D effective metric (1.2):

$\displaystyle ds^{2}=dr^{2}+\Delta(d\theta^{2}+{\,\rm
sin}^{2}\theta\,d\varphi^{2}),$ (2.1)

in which the static scalar $\Phi$ lives. It originates from thinking about a
massless, static scalar in the background of a general, spherically symmetric,
i.e., Reissner–Nordström, black hole (1.1).444The inner/outer horizons are at
$r=r_{\pm}\equiv(r_{S}\pm\sqrt{r_{S}^{2}-4r_{Q}^{2}})/2$. Note $\Delta\equiv
r^{2}-r_{S}r+r_{Q}^{2}$ can be rewritten as $(r-r_{+})(r-r_{-})$, and we have
$r_{+}+r_{-}=r_{S}$ and $r_{+}r_{-}=r_{Q}^{2}$. This 3D space has a vanishing
Cotton tensor, and thus is conformally flat. It has $10$ (C)KVs, forming an
$\mathfrak{so}(1,4)$ algebra.555A full account of the $10$ (C)KVs is given in
Appendix A. Three of these are the familiar KVs for rotation:

$\displaystyle J_{1}^{i}\partial_{i}$
$\displaystyle=-\sin\varphi\,\partial_{\theta}-\cos\varphi\cot\theta\,\partial_{\varphi}$
$\displaystyle J_{2}^{i}\partial_{i}$
$\displaystyle=\cos\varphi\,\partial_{\theta}-\cot\theta\sin\varphi\,\partial_{\varphi}$
$\displaystyle J_{3}^{i}\partial_{i}$ $\displaystyle=\partial_{\varphi}.$
(2.2)

The remaining seven—CKVs—do not at first sight seem useful, since $\Phi$ is
not conformally coupled. Surprisingly, this naïve expectation is false. It
turns out three CKVs are special:

$\displaystyle K_{1}^{i}\partial_{i}$
$\displaystyle=\Delta(r)\sin\theta\cos\varphi\,\partial_{r}+\frac{1}{2}\Delta^{\prime}(r)\left(-\cos\theta\cos\varphi\,\partial_{\theta}+\csc\theta\sin\varphi\,\partial_{\varphi}\right)$
$\displaystyle K_{2}^{i}\partial_{i}$
$\displaystyle=\Delta(r)\sin\theta\sin\varphi\,\partial_{r}-\frac{1}{2}\Delta^{\prime}(r)\left(\cos\theta\sin\varphi\,\partial_{\theta}+\csc\theta\cos\varphi\,\partial_{\varphi}\right)$
$\displaystyle K_{3}^{i}\partial_{i}$
$\displaystyle=\Delta(r)\cos\theta\,\partial_{r}+\frac{1}{2}\Delta^{\prime}(r)\sin\theta\,\partial_{\theta},$
(2.3)

where $\Delta^{\prime}\equiv\partial_{r}\Delta=2r-r_{S}$. They obey what we
call the melodic condition: a CKV $X$ is melodic if it satisfies

$\Box\nabla_{i}X^{i}=0.$ (2.4)

For the case at hand, $\nabla_{i}$ and $\Box$ are defined with respect to the
3D space (2.1). It can be shown that a melodic CKV generates a symmetry for a
massless scalar, even if the scalar is not conformally coupled. The symmetry
transformation it effects is:

$\delta_{X}\Phi=X^{i}\nabla_{i}\Phi+\frac{d-2}{2d}\nabla_{i}X^{i}\Phi,$ (2.5)

where $d=3$ in the case at hand. A proof is provided in the following inset.

Consider the following action for a scalar $\Phi$ in $d$ dimensions:

$S=\frac{1}{2}\int d^{d}x\sqrt{|g|}\,\left(\Phi\Box\Phi-\xi R\Phi^{2}\right),$
(2.6)

where the coupling $\xi$ need not take the conformal value
$\xi_{c}\equiv\frac{d-2}{4(d-1)}$. (We keep $\xi$ as a free parameter for the
sake of generality; for our black hole application, we are interested in
$\xi=0$.) It can be shown that under the transformation

$\delta_{X}\Phi=X^{i}\nabla_{i}\Phi+\frac{d-2}{2d}\nabla_{i}X^{i}\Phi,$ (2.7)

where $X$ is a CKV, the action transforms as

$\displaystyle\delta_{X}S$ $\displaystyle=\int
d^{d}x\sqrt{|g|}\left(\frac{1}{2}\nabla_{i}\left(X^{i}\Phi\Box\Phi-\xi
X^{i}R\Phi^{2}\right)-\frac{d-1}{d}\left(\xi-\xi_{c}\right)\left(\Box\nabla_{i}X^{i}\right)\Phi^{2}\right)$
(2.8) $\displaystyle=-\frac{d-1}{d}\left(\xi-\xi_{c}\right)\int
d^{d}x\sqrt{|g|}\,\left(\Box\nabla_{i}X^{i}\right)\Phi^{2},$ (2.9)

where we have dropped the boundary term in the last line. Thus, the CKV $X$
defines a symmetry of the action if $\xi=\xi_{c}$ (conformal coupling), or if
$\Box\nabla_{i}X^{i}=0$ ($X$ is a melodic CKV). A KV can be thought of a
trivial or degenerate case, since $\nabla_{i}X^{i}=0$.

The above discussion is at the level of the action. At the level of the
equation of motion, it can be shown that if $X$ is a CKV, then

$(\Box-\xi R)\delta_{X}\Phi=\overline{\delta}_{X}(\Box-\xi
R)\Phi-\frac{2(d-1)}{d}\left(\xi-\xi_{c}\right)\Box\nabla_{i}X^{i},$ (2.10)

where

$\overline{\delta}_{X}\Phi=X^{i}\nabla_{i}\Phi+\frac{d+2}{2d}\nabla_{i}X^{i}\Phi.$
(2.11)

In other words, if $\xi=\xi_{c}$ (conformal coupling), or
$\Box\nabla_{i}X^{i}=0$ ($X$ is a melodic CKV), $\delta_{X}$ passes through
the equation of motion operator $\Box-\xi R$ (and becomes
$\overline{\delta}_{X}$). This is how the symmetry manifests itself at the
level of the equation of motion. Incidentally, it can further be shown that
for the melodic CKVs $X_{1},X_{2},\ldots$ (or for $\xi=\xi_{c}$),

$(\Box-\xi
R)\delta_{X_{1}}\cdots\delta_{X_{n}}\Phi=\overline{\delta}_{X_{1}}\cdots\overline{\delta}_{X_{n}}(\Box-\xi
R)\Phi.$ (2.12)

These are related to higher spin symmetries [38], which can arise from
conformal Killing tensors (CKTs), and in this case the CKTs are decomposable,
i.e., they can be expressed as a symmetric product of CKVs. Further properties
of melodic CKVs are found in Appendix F.

The upshot is that, somewhat surprisingly, a static, massless scalar $\Phi$
around a spherically symmetric black hole enjoys a large amount of exact
symmetries: generated by three KVs (the $J_{i}$) and three CKVs (the $K_{i}$).
They obey an $\mathfrak{so}(1,3)$ algebra:

$\displaystyle\left[J_{i},J_{j}\right]=-\epsilon_{ijk}J_{k},\qquad\left[\bar{K}_{i},\bar{K}_{j}\right]=\epsilon_{ijk}J_{k},\qquad\left[J_{i},\bar{K}_{j}\right]=-\epsilon_{ijk}\bar{K}_{k},$
(2.13)

where $\bar{K}_{i}$ is simply a rescaled version of $K_{i}$:
$\bar{K}_{i}\equiv 2K_{i}/\sqrt{r_{S}^{2}-4r_{Q}^{2}}$.666 In the flat space
limit ($r_{S},r_{Q}\rightarrow 0$), it is better not to rescale. $K_{i}$
generates a special conformation transformation, and $[K_{i},K_{j}]=0$. The
extremal limit is similar, with $r$ replaced by $r-r_{S}/2$. In fact, the
exact solutions are simply $z^{\ell}$ and $z^{-(\ell+1)}$, with $z=r$ in the
flat space case and $z=r-r_{S}/2$ in the extremal case. In both cases the
algebra of the CKVs instead becomes the Lie algebra of the Euclidean group
$\text{ISO}(3)$. Here, we are abusing the notation somewhat: the brackets
$\left[\cdot,\cdot\right]$ can be interpreted as the Lie bracket of the
enclosed vectors or as the commutator of the symmetry transformations effected
by the enclosed vectors. (The symmetry transformation is not equal to the Lie
derivative in the case of the $K_{i}$, because $\nabla_{i}K^{i}\neq 0$. See
(2.5).) The interesting point is that the algebra is exactly the same
regardless of the interpretation. For a proof, see Appendix E.

The symmetry structure is identical to that of the Lorentz algebra, with the
$K_{i}$ playing the role of boosts. To see in detail how $J_{i}$ and $K_{i}$
act on the solutions of $\Box\Phi=0$, it is useful to decompose $\Phi$ in
spherical harmonics, with the solutions in harmonic space labeled by $\ell,m$:
$\Phi_{\ell m}(r,\theta,\varphi)\equiv\phi_{\ell}(r)Y_{\ell
m}(\theta,\varphi)$, where the radial function $\phi_{\ell}$ satisfies

$\partial_{r}\left(\Delta\partial_{r}\phi_{\ell}\right)-\ell(\ell+1)\phi_{\ell}=0.$
(2.14)

We have suppressed the $m$ label on the radial function because the equation
of motion does not depend on it, due to spherical symmetry of the background.

Imagine stacking the solutions $\phi_{\ell}Y_{\ell m}$, labeled by $\ell$ and
$m$, into a giant (infinite) column vector. We can form a representation of
the $\mathfrak{so}(1,3)$ algebra by asking how the generators act on this
column vector of solutions. It is clear how rotations act: for instance, under
the action of $J_{1}$ and $J_{2}$, spherical harmonics of different $m$ mix
while keeping $\ell$ the same. This gives the familiar ladder structure of the
$J_{i}$, where they are represented by anti-hermitian matrices.777In this work
we use the convention that a unitary representation of the Lie group means the
Lie algebra generators are represented as anti-hermitian matrices. From the
point of view of the $\mathfrak{so}(3)$ of rotations, the representation is
reducible to blocks of size $(2\ell+1)\times(2\ell+1)$, but the $K_{i}$ will
mix up solutions of different $\ell$ (and also $m$ for $K_{1}$ and $K_{2}$).

Let us illustrate with the action of $K_{3}$. Its effect on $\Phi$, according
to (2.5), is:

$\displaystyle\delta_{K_{3}}\Phi=\left[\Delta(r)\cos\theta\,\partial_{r}+\frac{1}{2}\Delta^{\prime}(r)\sin\theta\,\partial_{\theta}+\frac{1}{2}\Delta^{\prime}(r)\cos\theta\right]\Phi.$
(2.15)

Expanding in spherical harmonics, this implies (without assuming $\phi_{\ell}$
solves (2.14)):

$\delta_{K_{3}}\left(\phi_{\ell}Y_{\ell
m}\right)=-f(\ell)D_{\ell}^{+}\phi_{\ell}Y_{\ell+1,m}+f(\ell-1)D_{\ell}^{-}\phi_{\ell}Y_{\ell-1,m},$
(2.16)

where888Useful identities are:  $\displaystyle\cos\theta\,Y_{\ell m}$
$\displaystyle=f(\ell)Y_{\ell+1,m}+f(\ell-1)Y_{\ell-1,m}$
$\displaystyle\sin\theta\,\partial_{\theta}Y_{\ell m}$ $\displaystyle=\ell
f(\ell)Y_{\ell+1,m}-(\ell+1)f(\ell-1)Y_{\ell-1,m},$ (2.17)

$f(\ell)=\sqrt{\frac{(\ell-m+1)(\ell+m+1)}{(2\ell+1)(2\ell+3)}}.$ (2.18)

Here, the operators $D^{\pm}_{\ell}$ are defined by:

$\displaystyle
D_{\ell}^{+}\equiv-\Delta\partial_{r}-\frac{\ell+1}{2}\Delta^{\prime},\qquad
D_{\ell}^{-}\equiv\Delta\partial_{r}-\frac{\ell}{2}\Delta^{\prime}.$ (2.19)

Their significance can be deduced as follows. The transformation generated by
$K_{3}$, being a symmetry, maps solutions to solutions. Thus, if
$\phi_{\ell}Y_{\ell m}$ is a solution, $\delta_{K_{3}}(\phi_{\ell}Y_{\ell m})$
must be also. Looking at the right hand side of (2.16), the first term
multiplies $Y_{\ell+1,m}$; thus $D^{+}_{\ell}\phi_{\ell}$ must be the radial
solution at the $\ell+1$ level, assuming that all $\phi_{\ell}$ are regular at
$r_{+}$999The eq. (2.14) has two linearly independent solutions, one regular
at $r=r_{+}$, and the other one irregular at $r=r_{+}$. The operators
$D_{\ell}^{\pm}$ clearly preserve the (ir)regularity condition at $r=r_{+}$.
For the discussion of representation structure here, we always consider
regular solutions. See the next subsection for an extensive discussion on
(ir)regularity of the solutions to $\Box\Phi=0$.. Similarly, the second term
multiplies $Y_{\ell-1,m}$; thus $D^{-}_{\ell}\phi_{\ell}$ must be the radial
solution at the $\ell-1$ level. In other words, $D^{\pm}_{\ell}$ acting on the
radial solution $\phi_{\ell}$ at level $\ell$ maps it to $\phi_{\ell+1}$ or
$\phi_{\ell-1}$, i.e., they are raising and lowering operators.101010One can
abstract out from this reasoning a vertical ladder symmetry, following the
terminology of [1]: express the scalar action as sum in harmonic space, and
focus on two levels $\ell$ and $\ell-1$; it can be shown
$\delta\phi_{\ell}=D^{+}_{\ell-1}\phi_{\ell-1}$ and
$\delta\phi_{\ell-1}=-D^{-}_{\ell}\phi_{\ell}$ is a symmetry. This pairing of
levels is reminiscent of supersymmetry. See discussion in [1].

So far we have shown that the space of solutions to $\Box\Phi=0$, which is
spanned by all $\phi_{\ell}Y_{\ell m}$, furnishes an infinite dimensional
irreducible representation of $\rm{SO}(1,3)$. We can demonstrate the unitarity
of this representation for the regular solutions using the following inner
product:

$\displaystyle(\Phi_{1},\Phi_{2})=\int_{S^{2}}\,d\Omega\,\Phi^{*}_{1}(r_{+},\theta,\varphi)\Phi_{2}(r_{+},\theta,\varphi)~{},$
(2.20)

which is effectively the standard $L^{2}$ inner product on $S^{2}$. Choosing
the convenient normalization:111111This normalization is inappropriate in both
the flat space and extremal limits, as explained in footnote 6. However, the
existence of the ladder operators and the subsequent arguments about the
vanishing of the Love numbers remain valid.

$\displaystyle D^{+}_{\ell}\phi_{\ell}=N_{\ell+1}\phi_{\ell+1},\qquad
D^{-}_{\ell}\phi_{\ell}=N_{\ell}\phi_{\ell-1},$ (2.21)

where

$N_{\ell}\equiv\frac{\ell}{2}\sqrt{r_{S}^{2}-4r_{Q}^{2}}=\frac{\ell}{2}(r_{+}-r_{-}),$
(2.22)

we find the regular solutions are orthogonal and obey

$\displaystyle(\phi_{\ell}Y_{\ell m},\phi_{\ell}Y_{\ell
m})=\left|\phi_{\ell}(r_{+})\right|^{2}=\left|\phi_{0}\right|^{2}.$ (2.23)

Note that regularity at the horizon forces $\phi_{0}$ to be a constant. The
rotation generators are anti-hermitian automatically, and it can be easily
checked that all the $K_{i}$ are also anti-hermitian, for example

$\displaystyle(\Phi_{1},K_{3}\Phi_{2})=\frac{r_{+}-r_{-}}{2}\int_{S^{2}}\,d\Omega\,\Phi^{*}_{1}(r_{+},\theta,\varphi)\partial_{\theta}\left[\sin\theta\Phi_{2}(r_{+},\theta,\varphi)\right]=-(K_{3}\Phi_{1},\Phi_{2})~{}.$
(2.24)

Altogether, the regular solutions of $\Box\Phi=0$ equipped with the inner
product (2.20) carry a unitary irreducible representation of $\rm SO(1,3)$.

To identify this representation, we can use the SO$(1,3)$ Casimir. We have

$\displaystyle\delta^{ij}\delta_{J_{i}}\delta_{J_{j}}$
$\displaystyle=\nabla^{2}_{S^{2}}$ (2.25)
$\displaystyle\delta^{ij}\delta_{K_{i}}\delta_{K_{j}}$
$\displaystyle=\Delta\partial_{r}\left(\Delta\partial_{r}\right)+\frac{\left(\Delta^{\prime}\right)^{2}}{4}\nabla^{2}_{S^{2}}-\frac{r_{S}^{2}-4r_{Q}^{2}}{4},$
(2.26)

where $\nabla^{2}_{S^{2}}$ is the angular Laplacian.121212The operator $\Box$
according to the 3D effective metric is
$\Box=\Delta^{-2}\left(\delta^{ij}\delta_{K_{i}}\delta_{K_{j}}-\frac{r_{S}^{2}-4r_{Q}^{2}}{4}\delta^{ij}\delta_{J_{i}}\delta_{J_{j}}+\frac{r_{S}^{2}-4r_{Q}^{2}}{4}\right)$.
The 4D operator $\Box_{\rm 4D}$, restricted to static perturbations, is
$(\Delta/r^{2})\Box$. Acting on solutions in harmonic space, they give:

$\displaystyle\delta^{ij}\delta_{J_{i}}\delta_{J_{j}}\left(\phi_{\ell}Y_{\ell
m}\right)$ $\displaystyle=-\ell(\ell+1)\phi_{\ell}Y_{\ell m}$
$\displaystyle\delta^{ij}\delta_{K_{i}}\delta_{K_{j}}\left(\phi_{\ell}Y_{\ell
m}\right)$
$\displaystyle=-\left(\ell(\ell+1)+1\right)\frac{r_{S}^{2}-4r_{Q}^{2}}{4}\phi_{\ell}Y_{\ell
m}.$ (2.27)

Thus, the Casimir
$\delta^{ij}\delta_{J_{i}}\delta_{J_{j}}-\delta^{ij}\delta_{\bar{K}_{i}}\delta_{\bar{K}_{j}}=1$.
The set of solutions therefore form the principal series representation
$\mathcal{P}_{1,0}$ [39].131313We thank Alessandro Podo for discussions on
this point.

Before we close this subsection, let us introduce a generalization of (2.16),
following the treatment by [29]. Schematically, (2.16) takes the form
$\delta_{K_{3}}(\Phi_{\ell m})\sim\Phi_{\ell+1\,m}+\Phi_{\ell-1\,m}$, where
$\Phi_{\ell m}$ represents $\phi_{\ell}Y_{\ell m}$ and $\Phi_{\ell\pm 1,m}$
represents $D^{\pm}_{\ell}\phi_{\ell m}Y_{\ell\pm 1,m}$. Starting from
$\ell=0,m=0$, we see that $\delta_{K_{3}}(\Phi_{00})\sim\Phi_{10}$ (the term
$\Phi_{-10}=0$ due to the form of $f$ in (2.16)). Doing this again yields
$\delta_{K_{3}}\delta_{K_{3}}(\Phi_{00})\sim\Phi_{20}+\Phi_{00}$. Moving
$\Phi_{00}$ to the left, we see that there is a modification of
$\delta_{K_{3}}\delta_{K_{3}}$ that would raise the $\ell=0$, $m=0$ solution
to the $\ell=2$, $m=0$ solution. This is in fact general. There exists a
tensor $c^{\ell m}$ such that

$\sum_{i_{1},\ldots,i_{\ell}}c^{\ell m}_{i_{1}\cdots
i_{\ell}}\delta_{K_{i_{1}}}\cdots\delta_{K_{i_{\ell}}}\Phi_{00}\propto\Phi_{\ell
m},$ (2.28)

This procedure maps the $\Phi_{00}$ solution to a solution at an arbitrary
$\ell$ and $m$, $\Phi_{\ell m}=\phi_{\ell}Y_{\ell m}$, up to some
normalization constant (the constant depends on $\ell$ and $m$ if one adopts
the convention in (2.21)). The $\ell$-index tensor $c^{\ell m}$ is completely
symmetric and traceless, and turns out to be exactly the same as that used to
construct spherical harmonics:

$r^{\ell}Y_{\ell m}(\theta,\varphi)=\sum_{i_{1},\ldots,i_{\ell}}c^{\ell
m}_{i_{1}\cdots i_{\ell}}x^{i_{1}}\cdots x^{i_{\ell}},$ (2.29)

where the $x^{i}$ represent the Cartesian coordinates. A proof is given in
Appendix I. Equation (2.28) gives an economical way to build the entire
representation starting from the $\mathfrak{so}(3)$ singlet, a spherically
symmetric solution.

###  2.2 Ladder Structure and Love Numbers

To understand what the geometric symmetries have to say about the Love numbers
(one for each $\ell$), we need to work in harmonic space, and learn what the
ladder structure tells us about the nature of the solutions [1].

The equation of motion for the radial function $\phi_{\ell}$ (2.14) can be
rewritten as

$\displaystyle H_{\ell}\phi_{\ell}=0,\qquad
H_{\ell}\equiv-\Delta\left(\partial_{r}\left(\Delta\partial_{r}\right)-\ell(\ell+1)\right).$
(2.30)

Using $D^{+}_{\ell}$ and $D^{-}_{\ell}$ defined in (2.19), it can be shown
that

$H_{\ell+1}D_{\ell}^{+}=D_{\ell}^{+}H_{\ell},\qquad
H_{\ell-1}D_{\ell}^{-}=D_{\ell}^{-}H_{\ell}.$ (2.31)

Thus if $\phi_{\ell}$ is a solution to the equation of motion for level
$\ell$, $D^{\pm}_{\ell}\phi_{\ell}$ is a solution at level $\ell\pm 1$. This
reconfirms that $D^{\pm}_{\ell}$ indeed act as raising and lowering operators.
Furthermore, these operators satisfy the relations:

$\displaystyle
D_{\ell+1}^{-}D_{\ell}^{+}-D_{\ell-1}^{+}D_{\ell}^{-}=\frac{2\ell+1}{4}\left(r_{S}^{2}-4r_{Q}^{2}\right)$
(2.32) $\displaystyle
H_{\ell}=D_{\ell+1}^{-}D_{\ell}^{+}-\frac{(\ell+1)^{2}}{4}\left(r_{S}^{2}-4r_{Q}^{2}\right)=D_{\ell-1}^{+}D_{\ell}^{-}-\frac{\ell^{2}}{4}\left(r_{S}^{2}-4r_{Q}^{2}\right).$
(2.33)

The last expression implies that if $H_{\ell}\phi_{\ell}=0$, both
$D^{-}_{\ell+1}D^{+}_{\ell}\phi_{\ell}$ and
$D^{+}_{\ell-1}D^{-}_{\ell}\phi_{\ell}$ are proportional to $\phi_{\ell}$.
Note that the constant $r_{S}^{2}-4r_{Q}^{2}=(r_{+}-r_{-})^{2}$.

As explained above, the ladder structure originates from geometric symmetries.
There is another way to see that such a structure must be present: by
recalling contiguous relations of hypergeometric functions. Observe that
$\phi_{\ell}$ obeys a second order ordinary differential equation with three
regular singular points, at $r=r_{-}$, $r_{+}$, and $\infty$. Its solutions
are hypergeometric functions ${}_{2}F_{1}(a,b,c)$ whose arguments are related
to $\ell$. Contiguous relations relate hypergeometric functions with
neighboring arguments, i.e., they take the form of some operator acting on
${}_{2}F_{1}(a,b,c)$, giving another ${}_{2}F_{1}$ with $a,b$, and/or $c$
incremented by $1$, and by extension incrementing $\ell$ by $1$ also. Details
can be found in [1] and summarized in Appendix D. From this point of view, it
is not surprising that the ladder operators $D^{\pm}_{\ell}$ exist.

How is the ladder structure helpful for the Love number problem? Let us recall
what the problem is. Consider what the equation of motion (2.14) tells us
about the asymptotic behavior of $\phi_{\ell}$ at infinity and at the outer
horizon. As $r\rightarrow\infty$, it is straightforward to see that
$\phi_{\ell}$ goes as $r^{\ell}$ or $1/r^{\ell+1}$. As $r\rightarrow r_{+}$,
$\phi_{\ell}$ goes as a constant or as $\ln[(r-r_{+})/r_{+}]$. The phenomenon
of vanishing Love numbers has to do with the fact that the solution that is
regular at the horizon (i.e., going as a constant as $r\rightarrow r_{+}$) has
a purely $r^{\ell}$ asymptotic at large $r$. The naïve expectation, that it
has a mixture of “growing” $r^{\ell}$ and “decaying” $1/r^{\ell+1}$ at large
$r$, does not hold. This is the surprise of the vanishing Love number, defined
as the ratio of the decaying tail to the growing one. It is a problem of
connecting asymptotic behaviors at two different boundaries: how come
regularity at the horizon is connected with purely “growing” behavior at
infinity?

Let us look at (2.14) and (2.33). They tell us

$H_{\ell}\phi_{\ell}=\left[D^{+}_{\ell-1}D^{-}_{\ell}-\frac{\ell^{2}}{4}(r_{S}^{2}-4r_{Q}^{2})\right]\phi_{\ell}=0$
(2.34)

The constant term in $H_{\ell}$ vanishes for $\ell=0$, i.e.,
$H_{0}\phi_{0}=D^{+}_{-1}D^{-}_{0}\phi_{0}=0$. It suggests one possible
solution is given by a $\phi_{0}$ that is annihilated by $D^{-}_{0}$:

$D^{-}_{0}\phi_{0}=0.$ (2.35)

This is helpful, because we have turned a second order problem (i.e., $H_{0}$
involves two derivatives) into a first order one (i.e., $D^{-}_{0}$ involves
only one derivative). Thus, imposing regularity at the horizon is no longer
expected to yield generically an admixture of growing and decaying tail at
infinity. Indeed, $D^{-}_{0}$ is simply $\Delta\partial_{r}$, and so
$D^{-}_{0}\phi_{0}=0$ implies $\phi_{0}$ is a constant: this is clearly
regular at the horizon and purely “growing” at infinity (i.e., no $1/r$ tail).
Moreover, starting from any solution $\phi_{0}$, it is simple to construct a
solution at level $\ell$ by climbing the ladder:

$\phi_{\ell}\propto D^{+}_{\ell-1}\cdots D^{+}_{1}D^{+}_{0}\phi_{0}.$ (2.36)

Plugging in the $\phi_{0}=$ constant solution, and recalling the form of
$D^{+}_{\ell}=(r^{2}-r_{S}r+r_{Q}^{2})\partial_{r}-(\ell+1)(r-r_{S}/2)$, it is
straightforward to see that the resulting $\phi_{\ell}$ approaches a constant
at the horizon and is purely growing, i.e., it has the $r^{\ell}$ asymptotic
behavior with no $1/r^{\ell+1}$ tail (indeed, it cannot have any negative
power tail).

Strictly speaking, the above argument for the vanishing of the Love numbers
requires an extra comment. All we have shown is that there is a solution that
is regular at the horizon and purely growing at infinity. How do we know there
is not an independent solution that is also regular at the horizon but
contains a decaying tail at infinity? We know that cannot happen, because an
independent solution cannot also approach a constant at the horizon as its
leading behavior, i.e., it must approach the other possible asymptotic
behavior $\ln[(r-r_{+})/r_{+}]$ and is irregular at the horizon.141414On the
other hand, it is possible to have two independent solutions which both
approach the logarithm as their leading behavior at the horizon, because one
of them could have the constant asymptote hidden in its subleading terms. But
it is impossible to have two independent solutions both approaching the
constant as their leading behavior. 151515 Indeed, one can deduce further
statements about such irregular solutions. Going back to
$H_{0}\phi_{0}=D^{+}_{-1}D^{-}_{0}\phi_{0}=0$: instead of looking for a
solution that satisfies $D^{-}_{0}\phi_{0}=0$ (i.e., the constant solution),
we could look for a different solution such that $D^{-}_{0}\phi_{0}\neq 0$,
but $D^{+}_{-1}(D^{-}_{0}\phi_{0})=0$. Recalling the form of $D^{+}_{-1}$, we
see that this implies $D^{-}_{0}\phi_{0}$ equals a non-zero constant. Solving
this tells us $\phi_{0}\propto{\,\rm ln\,}[(r-r_{+})/(r-r_{-})]$. This
solution diverges logarithmically as $r\rightarrow r_{+}$, and goes as $1/r$
at large $r$. It can be shown that raising it by $D^{+}_{\ell}$ operators
yield solutions at higher $\ell$ that retain the logarithmic divergence at the
horizon. Their large $r$ behavior consists of a mixture of growing and
decaying terms.

There is another way to connect boundary conditions, by exploiting
conservation. The observation is that there is another kind of symmetry at
play, one at each level $\ell$, termed a horizontal symmetry by [1], following
the work of [40]. Note that:

$[Q_{0},H_{0}]=0,\quad\text{where}\quad Q_{0}\equiv
D_{0}^{-}=\Delta\partial_{r}.$ (2.37)

(This can be quickly understood by examining (2.31) for $\ell=0$ and noting
that $H_{-1}=H_{0}$.) It can be shown that $\delta\phi_{0}=Q_{0}\phi_{0}$ is a
symmetry at the level of the action. By climbing up and down the ladder, it
can be further shown that:

$[Q_{\ell},H_{\ell}]=0,\quad\text{where}\quad Q_{\ell}\equiv
D^{+}_{\ell-1}Q_{\ell-1}D^{-}_{\ell}.$ (2.38)

The transformation $\delta\phi_{\ell}=Q_{\ell}\phi_{\ell}$ is also a symmetry
at the level of the action (written as a sum over modes in harmonic space).
See Appendix H for a proof. From the horizontal symmetry at each level $\ell$,
a Noether current can be derived, which in our static context takes the form
$\partial_{r}J^{r}_{\ell}=0$, i.e., the $r$ component of the current is a
conserved quantity, in the sense of being $r$-independent. Carrying out this
procedure is straightforward though a bit cumbersome, and it is in fact
simpler to identify the conserved quantity by inspection. Observing that the
$\ell=0$ equation of motion is $\partial_{r}(\Delta\partial_{r}\phi_{0})=0$,
we see that

$P_{0}=\Delta\partial_{r}\phi_{0}$ (2.39)

is conserved on the equation of motion. For any higher $\ell$, it is natural
to define

$P_{\ell}\equiv\Delta\partial_{r}\left(D_{1}^{-}D_{2}^{-}\cdots
D_{\ell}^{-}\phi_{\ell}\right),$ (2.40)

which is also conserved on the equation of motion, i.e.,
$\partial_{r}P_{\ell}=0$, since the string of lowering operators merely serve
to lower $\phi_{\ell}$ to $\phi_{0}$ (up to a constant). It is worth noting
that actually carrying out the Noether procedure would yield
$J^{r}_{\ell}=P_{\ell}^{2}$. Obviously, its conservation is equivalent to the
conservation of $P_{\ell}$.161616It can be seen that $P_{0}$ is in fact the
Wronskian between $\phi_{0}$ and the constant solution. Normally, the
conservation of Wronskian follows from a trivial symmetry of a linear system:
one can always add to the field a solution of the equation of motion. What is
curious here is that a less trivial symmetry, i.e.
$\delta\phi_{0}=Q_{0}\phi_{0}$, gives rise to a conserved quantity whose
square root turns out to be $P_{0}$.  Henceforth, slightly abusing the
terminology, we shall refer to $P_{\ell}$ as a conserved charge.171717Suppose
$\phi_{\ell}$ in (2.40) is built by raising from some $\phi_{0}$, i.e.,
$A_{\ell}\phi_{\ell}=D^{+}_{\ell-1}\cdots D^{+}_{0}\phi_{0}$, where $A_{\ell}$
follows from a string of normalization factors (2.21):
$A_{\ell}=\prod_{i=1}^{\ell}N_{i}=\ell!\left(r_{S}^{2}-4r_{Q}^{2}\right)^{\ell/2}/2^{\ell}$.
It can be shown that
$P_{\ell}=\frac{\ell!}{2^{\ell}}\left(r_{S}^{2}-4r_{Q}^{2}\right)^{\ell/2}P_{0}.$
(2.41) This follows from noting that
$P_{\ell}=A_{\ell}^{-1}\Delta\partial_{r}(D^{-}_{1}\cdots(D^{-}_{\ell}D^{+}_{\ell-1})D^{+}_{\ell-2}\cdots
D^{+}_{0}\phi_{0})$. We highlight by parentheses $D^{-}_{\ell}D^{+}_{\ell-1}$,
and observe that to its right is essentially $\phi_{\ell-1}$ which solves
$H_{\ell-1}\phi_{\ell-1}=0$. This tells us
$[D^{-}_{\ell}D^{+}_{\ell-1}-\ell^{2}(r_{S}^{2}-4r_{Q}^{2})/4]\phi_{\ell-1}=0$,
so we can replace $D^{-}_{\ell}D^{+}_{\ell-1}$ by
$\ell^{2}(r_{S}^{2}-4r_{Q}^{2})/4$. Repeating the procedure to lower rungs of
the ladder yields the above expression.

How do we use the conservation of $P_{\ell}$ to tackle the problem of
connecting boundary conditions? Following [1]: first of all, observe that if
$\phi_{\ell}$ approaches a constant at the horizon, $D^{-}_{1}\cdots
D^{-}_{\ell}\phi_{\ell}$ would produce something regular at the horizon, and
so $P_{\ell}=0$ when evaluated at the horizon, due to the factor of $\Delta$
in (2.40). In other words, the regular solution has a vanishing $P_{\ell}$
charge. Conversely, if $\phi_{\ell}$ approaches the divergent logarithm
$\ln[(r-r_{+})/r_{+}]$ at the horizon, $P_{\ell}\neq 0$ when evaluated at the
horizon. Thirdly, if $\phi_{\ell}\sim 1/r^{\ell+1}$ at large $r$, the form of
$D^{-}_{\ell}$ tells us that $P_{\ell}\neq 0$ when evaluated at large $r$.
Conservation of $P_{\ell}$ thus tells us a solution with a purely decaying
tail $1/r^{\ell+1}$ at large $r$ must diverge at the horizon, consistent with
Bekenstein’s no-hair theorem [41]. Lastly, let’s deduce the large $r$ behavior
of the solution that is regular at the horizon, which we shall call
$\phi_{\ell}^{\rm reg.}$. As noted above, $P_{\ell}=0$ at the horizon for
$\phi_{\ell}^{\rm reg.}$, and thus by conservation, $P_{\ell}=0$ at large $r$
too. Looking at (2.40), this implies $D^{-}_{1}\cdots
D^{-}_{\ell}\phi_{\ell}^{\rm reg.}$ at large $r$, when written as a power
series, starts at most at $r^{0}$. Crucially, the subleading terms cannot have
a $1/r$ contribution; otherwise $P_{\ell}$ would be nonzero. It is useful at
this point to recall $D^{-}_{1}\cdots D^{-}_{\ell}\phi_{\ell}^{\rm reg.}$ is
by construction a solution at the $\ell=0$ level, for which we know the most
general solution is a superposition of a constant and a logarithm (see
footnote 15). Importantly, the logarithm has a $1/r$ contribution at large
$r$. This means the only way for $\phi_{\ell}^{\rm reg.}$ to yield a zero
$P_{\ell}$ at large $r$ is for $D^{-}_{1}\cdots D^{-}_{\ell}\phi_{\ell}^{\rm
reg.}$ to be just a constant. Inverting this, we can say $\phi_{\ell}^{\rm
reg.}$ has to be equal to $D^{+}_{\ell-1}\cdots D^{+}_{0}$ acting on a
constant. By inspecting the form of $D^{+}_{\ell}$, we conclude
$\phi_{\ell}^{\rm reg.}$ goes as $r^{\ell}$ at large $r$ and has no negative
power law tail, and in particular no $1/r^{\ell+1}$ contribution.

##  3 de Sitter Ladders

We now turn our attention to de Sitter space. We shall work in the static
patch, with the metric given by

$ds^{2}=-\frac{\Delta}{r^{2}}\,dt^{2}+\frac{r^{2}}{\Delta}\,dr^{2}+r^{2}d\Omega^{2},$
(3.1)

where now $\Delta\equiv r^{2}\left(1-r^{2}/L^{2}\right)$. We consider a scalar
field in de Sitter space with an arbitrary coupling $\xi$ to the Ricci scalar.
The equation of motion is

$\left(\Box-\xi R\right)\Phi=0.$ (3.2)

Since $R=\frac{d(d-1)}{L^{2}}$ is a constant (with $d=4$ in our case), one
could also interpret $\xi R$ as mass squared. The scalar is conformally
coupled if $\xi=\xi_{c}\equiv\frac{d-2}{4(d-1)}$. It turns out to be more
convenient to work with the parameter

$\alpha\equiv\frac{(d-1)^{2}}{4}-\xi\,d(d-1).$ (3.3)

Conformal coupling corresponds to $\alpha=\frac{1}{4}$, while masslessness
corresponds to $\alpha=\frac{(d-1)^{2}}{4}$.

Let us emphasize again many of the results below are not new. The quasinormal
spectrum was worked out in [26], and its relation to the geometric symmetries
was pointed out by [25, 27, 28, 29], though the expressions for some of the
ladder operators, and the conserved charges (for the conformally coupled
case), are new. The main motivation for including a discussion of de Sitter
space is to emphasize the many similar features shared with the black hole
problem. The way the ladder structures emerge out of representations is the
same. And the way asymptotic behaviors at separate boundaries can be
connected—relevant for Love numbers for black holes, and quasinormal modes for
de Sitter space—is also the same.

###  3.1 Geometric Symmetries

For de Sitter space, there does not appear to be the analog of melodic CKVs,
i.e., CKVs that obey condition (2.4) in a non-trivial way. Thus, we focus on
the KVs. We shall comment on the extra symmetries that a conformally coupled
scalar enjoys in Section 3.3.

The isometry algebra of de Sitter space is $\mathfrak{so}(1,4)$. For the
purpose of understanding the ladder structures, it is helpful to see their
explicit form in static patch coordinates:

$\displaystyle J_{1}^{\mu}\partial_{\mu}$
$\displaystyle=-\sin\varphi\,\partial_{\theta}-\cos\varphi\cot\theta\,\partial_{\varphi}$
$\displaystyle J_{2}^{\mu}\partial_{\mu}$
$\displaystyle=\cos\varphi\,\partial_{\theta}-\cot\theta\sin\varphi\,\partial_{\varphi}$
$\displaystyle J_{3}^{\mu}\partial_{\mu}$ $\displaystyle=\partial_{\varphi}$
$\displaystyle D^{\mu}\partial_{\mu}$ $\displaystyle=-L\partial_{t}$
$\displaystyle P_{1}^{\mu}\partial_{\mu}$
$\displaystyle=e^{-t/L}\left(-\frac{r^{2}}{\sqrt{\Delta}}\sin\theta\cos\varphi\,\partial_{t}+L\frac{\sqrt{\Delta}}{r}\left(\sin\theta\cos\varphi\,\partial_{r}+\frac{1}{r}\cos\theta\cos\varphi\,\partial_{\theta}-\frac{1}{r}\csc\theta\sin\varphi\,\partial_{\varphi}\right)\right)$
$\displaystyle P_{2}^{\mu}\partial_{\mu}$
$\displaystyle=e^{-t/L}\left(-\frac{r^{2}}{\sqrt{\Delta}}\sin\theta\sin\varphi\,\partial_{t}+L\frac{\sqrt{\Delta}}{r}\left(\sin\theta\sin\varphi\,\partial_{r}+\frac{1}{r}\cos\theta\sin\varphi\,\partial_{\theta}+\frac{1}{r}\csc\theta\cos\varphi\,\partial_{\varphi}\right)\right)$
$\displaystyle P_{3}^{\mu}\partial_{\mu}$
$\displaystyle=e^{-t/L}\left(-\frac{r^{2}}{\sqrt{\Delta}}\cos\theta\,\partial_{t}+L\frac{\sqrt{\Delta}}{r}\left(\cos\theta\,\partial_{r}-\frac{1}{r}\sin\theta\,\partial_{\theta}\right)\right)$
$\displaystyle K_{1}^{\mu}\partial_{\mu}$
$\displaystyle=e^{t/L}\left(\frac{r^{2}}{\sqrt{\Delta}}\sin\theta\cos\varphi\,\partial_{t}+L\frac{\sqrt{\Delta}}{r}\left(\sin\theta\cos\varphi\,\partial_{r}+\frac{1}{r}\cos\theta\cos\varphi\,\partial_{\theta}-\frac{1}{r}\csc\theta\sin\varphi\,\partial_{\varphi}\right)\right)$
$\displaystyle K_{2}^{\mu}\partial_{\mu}$
$\displaystyle=e^{t/L}\left(\frac{r^{2}}{\sqrt{\Delta}}\sin\theta\sin\varphi\,\partial_{t}+L\frac{\sqrt{\Delta}}{r}\left(\sin\theta\sin\varphi\,\partial_{r}+\frac{1}{r}\cos\theta\sin\varphi\,\partial_{\theta}+\frac{1}{r}\csc\theta\cos\varphi\,\partial_{\varphi}\right)\right)$
$\displaystyle K_{3}^{\mu}\partial_{\mu}$
$\displaystyle=e^{t/L}\left(\frac{r^{2}}{\sqrt{\Delta}}\cos\theta\,\partial_{t}+L\frac{\sqrt{\Delta}}{r}\left(\cos\theta\,\partial_{r}-\frac{1}{r}\sin\theta\,\partial_{\theta}\right)\right).$
(3.4)

Defining:

$\begin{split}M_{ij}=\epsilon_{ijk}J_{k},\quad
M_{0i}=\frac{1}{2}\left(P_{i}-K_{i}\right),\quad
M_{i4}=\frac{1}{2}\left(P_{i}+K_{i}\right),\quad M_{04}=D,\end{split}$ (3.5)

the commutation relations are summarized by

$\left[M_{AB},M_{CD}\right]=\eta_{AD}M_{BC}+\eta_{BC}M_{AD}-\eta_{AC}M_{BD}-\eta_{BD}M_{AC},$
(3.6)

with $\eta\equiv{\rm diag\,}(-1,1,1,1,1)$.

In particular, the $J_{i}$ commute with $D$ which is simply time translation.
Thus, it is natural to label the solutions by $\omega,\ell,m$ where $i\omega
L$ is the eigenvalue of $D$ (i.e., the solution’s dependence on $t$, $\theta$,
and $\varphi$ takes the form $e^{-i\omega t}Y_{\ell m}(\theta,\varphi)$). As
we shall see, it turns out to be convenient to label the solutions by $\gamma$
in place of $\omega$:

$\gamma\equiv i\omega L-\ell.$ (3.7)

Thus, applying the separation of variables, solutions to (3.2) take the form
of $\Phi_{\gamma\ell m}(t,r,\theta,\varphi)\equiv\phi_{\gamma\ell}(r)Y_{\ell
m}(\theta,\varphi)e^{-i\omega_{\gamma\ell}t}$.181818Our notation, in which
$\omega$ carries $\gamma$ and $\ell$ labels, might seem unmotivated. It will
make more sense in a moment. For now, it is perfectly acceptable to think of
the solutions as labeled instead by $\omega$, $\ell$, and $m$, i.e.,
$\Phi_{\omega\ell m}=\phi_{\omega\ell}Y_{\ell m}e^{-i\omega t}$. They form a
representation of the $\mathfrak{so}(1,4)$ isometry algebra.

As in the case of the black hole, $J_{1}$ and $J_{2}$ mix solutions of
different $m$ without affecting $\ell$ and $\gamma$. The $K_{i}$ and $P_{i}$
have non-trivial commutators with $J_{i}$ and $D$, and will mix different
$\ell$ and $\gamma$ (and in general $m$ too).191919The complete set of
commutation relations is shown in Appendix B. For instance, $K_{3}$, $P_{3}$
and $D$ form an $\mathfrak{sl}(2,\mathbb{R})$ subalgebra:

$\left[D,P_{3}\right]=P_{3},\quad\left[D,K_{3}\right]=-K_{3},\quad\left[K_{3},P_{3}\right]=2D.$
(3.8)

(The same statement can be made with $K_{3},P_{3}$ replaced by $K_{1}$,
$P_{1}$ or $K_{2}$, $P_{2}$.) Thus $K_{3}$ and $P_{3}$ can be thought of as
raising and lowering operators, incrementing the eigenvalue of $D$, i.e.,
$i\omega L$. This can also be seen explicitly by noting that $K_{3}$ and
$P_{3}$ contain factors of $e^{\pm t/L}$. Since $K_{3}$ and $P_{3}$ do not
commute with $J_{1}$ and $J_{2}$, they at the same time mix up solutions with
different $\ell$ (without affecting $m$):

$\displaystyle\delta_{P_{3}}\left(\phi_{\gamma\ell}Y_{\ell
m}e^{-i\omega_{\gamma\ell}t}\right)$
$\displaystyle=Le^{-(i\omega_{\gamma\ell}L+1)t/L}\left(f(\ell)E^{+}_{\gamma\ell}\phi_{\gamma\ell}Y_{\ell+1,m}-f(\ell-1)F^{+}_{\gamma\ell}\phi_{\gamma\ell}Y_{\ell-1,m}\right)$
$\displaystyle\delta_{K_{3}}\left(\phi_{\gamma\ell}Y_{\ell
m}e^{-i\omega_{\gamma\ell}t}\right)$
$\displaystyle=Le^{-(i\omega_{\gamma\ell}L-1)t/L}\left(f(\ell)F^{-}_{\gamma\ell}\phi_{\gamma\ell}Y_{\ell+1,m}-f(\ell-1)E^{-}_{\gamma\ell}\phi_{\gamma\ell}Y_{\ell-1,m}\right),$
(3.9)

where $f(\ell)$ is again given by (2.18), and we have defined the operators

$\displaystyle E_{\gamma\ell}^{+}$
$\displaystyle\equiv\frac{\sqrt{\Delta}}{r}\partial_{r}-\ell\frac{\sqrt{\Delta}}{r^{2}}+i\omega_{\gamma\ell}\frac{r^{2}}{L\sqrt{\Delta}}$
$\displaystyle E_{\gamma\ell}^{-}$
$\displaystyle\equiv-\frac{\sqrt{\Delta}}{r}\partial_{r}-(\ell+1)\frac{\sqrt{\Delta}}{r^{2}}+i\omega_{\gamma\ell}\frac{r^{2}}{L\sqrt{\Delta}}$
$\displaystyle F^{+}_{\gamma\ell}$
$\displaystyle\equiv-\frac{\sqrt{\Delta}}{r}\partial_{r}-(\ell+1)\frac{\sqrt{\Delta}}{r^{2}}-i\omega_{\gamma\ell}\frac{r^{2}}{L\sqrt{\Delta}}$
$\displaystyle F^{-}_{\gamma\ell}$
$\displaystyle\equiv\frac{\sqrt{\Delta}}{r}\partial_{r}-\ell\frac{\sqrt{\Delta}}{r^{2}}-i\omega_{\gamma\ell}\frac{r^{2}}{L\sqrt{\Delta}}.$
(3.10)

By virtue of being symmetries, $K_{3}$ and $P_{3}$ map solutions to solutions.
Running an argument similar to the black hole case, we see that
$E^{\pm}_{\gamma\ell}$ effects $\ell\rightarrow\ell\pm 1$ and
$i\omega_{\gamma\ell}L\rightarrow i\omega_{\gamma\ell}L\pm 1$, and
$F^{\pm}_{\gamma\ell}$ effects $\ell\rightarrow\ell\mp 1$ and
$i\omega_{\gamma\ell}L\rightarrow i\omega_{\gamma\ell}L\pm 1$. Notice how the
action of $E^{\pm}_{\gamma\ell}$ can be alternatively described as effecting
$\ell\rightarrow\ell\pm 1$ while keeping $i\omega_{\gamma\ell}L-\ell$ fixed.
This is what motivates the introduction of the symbol $\gamma$ in (3.7), i.e.,
$E^{\pm}_{\gamma\ell}$ effects $\ell\rightarrow\ell\pm 1$ without changing
$\gamma$.

Is there a way to effect a shift in $i\omega_{\gamma\ell}L$, without changing
$\ell$ (or $m$)? The answer is yes, and unsurprisingly, it involves
spherically symmetric combinations of the $K_{i}$ and $P_{i}$:

$\displaystyle\delta^{ij}\delta_{P_{i}}\delta_{P_{j}}\left(\phi_{\gamma\ell}Y_{\ell
m}e^{-i\omega_{\gamma\ell}t}\right)$
$\displaystyle=-L^{2}G^{+}_{\gamma\ell}\phi_{\gamma\ell}Y_{\ell
m}e^{-(i\omega_{\gamma\ell}L+2)t/L}$
$\displaystyle\delta^{ij}\delta_{K_{i}}\delta_{K_{j}}\left(\phi_{\gamma\ell}Y_{\ell
m}e^{-i\omega_{\gamma\ell}t}\right)$
$\displaystyle=-L^{2}G^{-}_{\gamma\ell}\phi_{\gamma\ell}Y_{\ell
m}e^{-(i\omega_{\gamma\ell}L-2)t/L},$ (3.11)

where $G^{\pm}_{\gamma\ell}$ is given by:

$\displaystyle G^{+}_{\gamma\ell}\equiv
E^{+}_{\gamma+2,\ell-1}F^{+}_{\gamma\ell},\qquad G^{-}_{\gamma\ell}\equiv
E^{-}_{\gamma-2,\ell+1}F^{-}_{\gamma\ell}.$ (3.12)

In other words, $\delta^{ij}\delta_{P_{i}}\delta_{P_{j}}$ (or
$G^{+}_{\gamma\ell})$ effects $i\omega_{\gamma\ell}L\rightarrow
i\omega_{\gamma\ell}L+2$, while $\delta^{ij}\delta_{K_{i}}\delta_{K_{j}}$ (or
$G^{-}_{\gamma\ell})$ effects $i\omega_{\gamma\ell}L\rightarrow
i\omega_{\gamma\ell}L-2$. Equivalently, we can also say $G^{\pm}_{\gamma\ell}$
effects $\gamma\rightarrow\gamma\pm 2$ without changing $\ell$. As before, all
of these ladders can also be seen as arising from contiguous relations for the
hypergeometric solutions, which are shown in Appendix D.

With the above ladder operators defined, it is easy to see how one could
construct a representation of $\mathfrak{so}(1,4)$. Start from a solution
labeled by $i\omega_{\gamma\ell}L$ and $\ell$, or equivalently $\gamma$ and
$\ell$ (we suppress $m$ since incrementing it is straightforward using $J_{1}$
and $J_{2}$); use $E^{\pm}_{\gamma\ell}$ to increment $\ell$ by $\pm 1$
without changing $\gamma$, or use $G^{\pm}_{\gamma\ell}$ to increment $\gamma$
by $\pm 2$ without changing $\ell$. This way, we fill out a whole table of
solutions, which we will also refer to as states. So far, the only restriction
is that $\ell\geq 0$, i.e., the table of $(\gamma,\ell)$ has $\ell$ taking all
possible non-negative integer values, and $\gamma=\gamma_{0}+2n$ for any
integer $n$ and some base value $\gamma_{0}$ (with $\gamma_{0}$ arbitrary for
the moment). A further restriction on $\gamma$ will come later when we discuss
quasinormal modes. Incidentally, since $\gamma$ in general has a non-zero real
part, and therefore $i\omega_{\gamma\ell}L$ as well, we have a non-unitary
representation (e.g. the matrix representation for time-translation $D$ is not
anti-hermitian).

We close this subsection with a description of an economical method to build
the entire representation of $\mathfrak{so}(1,4)$, given by [39]. (This
mirrors what was given at the end of Section 2.1.) Taking $c^{\ell m}$ to be
the traceless symmetric tensor associated with $Y_{\ell m}$ (2.29), we have

$\displaystyle c^{\ell m}_{i_{1}\cdots
i_{\ell}}\delta_{P_{i_{1}}}\cdots\delta_{P_{i_{\ell}}}\Phi_{\gamma
00}(t,r,\theta,\varphi)$ $\displaystyle\propto\Phi_{\gamma\ell
m}(t,r,\theta,\varphi)$ $\displaystyle c^{\ell m}_{i_{1}\cdots
i_{\ell}}\delta_{K_{i_{1}}}\cdots\delta_{K_{i_{\ell}}}\Phi_{\gamma
00}(t,r,\theta,\varphi)$ $\displaystyle\propto\Phi_{\gamma-2\ell,\ell
m}(t,r,\theta,\varphi),$ (3.13)

up to some irrelevant normalization constant. Proofs are once again in
Appendix I. In this language, (3.11) can be recast as:

$\displaystyle\delta^{ij}\delta_{P_{i}}\delta_{P_{j}}\Phi_{\gamma\ell
m}(t,r,\theta,\varphi)$ $\displaystyle\propto\Phi_{\gamma+2,\ell
m}(t,r,\theta,\varphi)$
$\displaystyle\delta^{ij}\delta_{K_{i}}\delta_{K_{j}}\Phi_{\gamma\ell
m}(t,r,\theta,\varphi)$ $\displaystyle\propto\Phi_{\gamma-2,\ell
m}(t,r,\theta,\varphi).$ (3.14)

For further operators connecting different solutions, see Appendix J.

###  3.2 Ladder Structures and Quasinormal Modes

The upshot of the last subsection is that de Sitter space has multiple
ladders: besides a ladder in $\ell$, there is also a ladder in $\gamma$,
defined as $i\omega L-\ell$. (A ladder in $m$ is present as well, but is no
different from the familiar one of spherical harmonics.) Our next task is to
learn what the ladder structures tell us about the nature of the quasinormal
solutions.

After the separation of variables, the radial function $\phi_{\gamma\ell}(r)$
obeys

$\widetilde{H}_{\gamma\ell}\phi_{\gamma\ell}=0,\qquad\widetilde{H}_{\gamma\ell}\equiv-\frac{r^{2}}{L^{2}\Delta}\left(\partial_{r}\left(\Delta\partial_{r}\right)+\frac{\omega_{\gamma\ell}^{2}r^{4}}{\Delta}+\left(\alpha-\frac{9}{4}\right)\frac{r^{2}}{L^{2}}-\ell(\ell+1)\right).$
(3.15)

The asymptotic behaviors of $\phi_{\gamma\ell}$ are as follows:
$\phi_{\gamma\ell}$ goes as $r^{\ell}$ or $1/r^{\ell+1}$ as $r\rightarrow 0$;
$\phi_{\gamma\ell}$ goes as
$\left(1-\frac{r^{2}}{L^{2}}\right)^{-i\omega_{\gamma\ell}L/2}$ (outgoing at
horizon) or $\left(1-\frac{r^{2}}{L^{2}}\right)^{i\omega_{\gamma\ell}L/2}$
(incoming at horizon) as $r\rightarrow L$. Linear combinations thereof are of
course allowed. What is special about quasinormal modes is that they have such
special frequencies that only a single asymptotic behavior is realized at each
boundary. In particular, the behavior at the origin is purely regular
$r^{\ell}$ and the behavior at the horizon is purely outgoing
$\left(1-\frac{r^{2}}{L^{2}}\right)^{-i\omega_{\gamma\ell}L/2}$. This is
reminiscent of the Love number problem in the case of the black hole:
regularity at the horizon and purely growing behavior at infinity. Our task in
this subsection is to understand this single-asymptote behavior for the
quasinormal solutions.

Our strategy is similar to that in the black hole case (2.33): write the
equation of motion operator $\widetilde{H}_{\gamma\ell}$ as a product of
raising and lowering operators plus a constant. We shall use
$G^{\pm}_{\gamma\ell}$ which raises or lowers $\gamma$ by $2$ while keeping
$\ell$ constant. For other ways to rewrite the equation of motion, using
$E^{\pm}_{\gamma\ell}$ or $F^{\pm}_{\gamma\ell}$, see Appendix G.

The operators $G^{\pm}_{\gamma\ell}$ defined in (3.12) are a bit cumbersome to
use, because they involve second derivatives. They can be improved by
subtracting off $\widetilde{H}_{\gamma\ell}$ (or something proportional to it,
keeping in mind that the objective is to act on solutions, which are
annihilated by $\widetilde{H}_{\gamma\ell}$):

$\displaystyle\mathcal{G}^{+}_{\gamma\ell}$
$\displaystyle\equiv\left(\gamma+\ell+1\right)^{-1}\left(G^{+}_{\gamma\ell}-\frac{L^{2}\Delta}{r^{4}}\widetilde{H}_{\gamma\ell}\right)$
$\displaystyle=-\frac{2r}{L^{2}}\partial_{r}+\frac{1}{L^{2}}\left(-\frac{2(\gamma+\ell)}{\Delta}r^{2}+\frac{(\gamma-1)(\gamma+2\ell)+\alpha-\frac{9}{4}}{(\gamma+\ell+1)}\right)$
(3.16) $\displaystyle\mathcal{G}^{-}_{\gamma\ell}$
$\displaystyle\equiv\left(\gamma+\ell-1\right)^{-1}\left(G^{-}_{\gamma\ell}-\frac{L^{2}\Delta}{r^{4}}\widetilde{H}_{\gamma\ell}\right)$
$\displaystyle=\frac{2r}{L^{2}}\partial_{r}+\frac{1}{L^{2}}\left(-\frac{2(\gamma+\ell)}{\Delta}r^{2}+\frac{\gamma(\gamma+2\ell+1)+\alpha-\frac{9}{4}}{(\gamma+\ell-1)}\right).$
(3.17)

It can be shown that

$\widetilde{H}_{\gamma+2,\ell}\,\mathcal{G}^{+}_{\gamma\ell}=\mathcal{G}^{+}_{\gamma\ell}\,\widetilde{H}_{\gamma\ell},\qquad\widetilde{H}_{\gamma-2,\ell}\,\mathcal{G}^{-}_{\gamma\ell}=\mathcal{G}^{-}_{\gamma\ell}\,\widetilde{H}_{\gamma\ell}.$
(3.18)

This reconfirms what we already know from geometric arguments, that given a
solution $\phi_{\gamma\ell}$,
$\mathcal{G}^{\pm}_{\gamma\ell}\phi_{\gamma\ell}$ is a solution at level
$\gamma\pm 2$ and the same $\ell$.202020 It is also useful to note that
$\mathcal{G}^{-}_{\gamma+2,\ell}\mathcal{G}^{+}_{\gamma\ell}-\mathcal{G}^{+}_{\gamma-2,\ell}\mathcal{G}^{-}_{\gamma\ell}=\frac{4}{L^{4}}\frac{(\gamma+\ell)\left(A_{\gamma\ell}+B_{\gamma\ell}\left(\alpha-\frac{9}{4}\right)-\left(\alpha-\frac{9}{4}\right)^{2}\right)}{(\gamma+\ell+1)^{2}(\gamma+\ell-1)^{2}},\\\
$ $\displaystyle A_{\gamma\ell}$
$\displaystyle\equiv\left(\gamma(\gamma+2\ell)-\ell+1\right)\left(\gamma(\gamma+2\ell)+(2\ell+3)(\ell-1)\right),$
$\displaystyle B_{\gamma\ell}$ $\displaystyle\equiv 2(\ell-1)(\ell+2).$ (3.19)
The raising and lowering operators above are the same as those in Section 3.2
of [25].

With this setup, we are ready to rewrite $\widetilde{H}_{\gamma\ell}$ in terms
of $\mathcal{G}^{\pm}_{\gamma\ell}$:

$\widetilde{H}_{\gamma\ell}=\frac{L^{2}}{4}\left(\mathcal{G}^{+}_{\gamma-2,\ell}\mathcal{G}^{-}_{\gamma\ell}-\frac{1}{L^{4}}\frac{a_{\gamma\ell}b_{\gamma\ell}}{\left(\gamma+\ell-1\right)^{2}}\right)=\frac{L^{2}}{4}\left(\mathcal{G}^{-}_{\gamma+2,\ell}\mathcal{G}^{+}_{\gamma\ell}-\frac{1}{L^{4}}\frac{a_{\gamma+2,\ell}b_{\gamma+2,\ell}}{\left(\gamma+\ell+1\right)^{2}}\right),$
(3.20)

where

$\displaystyle
a_{\gamma\ell}\equiv\left(\gamma-\gamma_{+}+2\ell+1\right)\left(\gamma-\gamma_{-}+2\ell+1\right),\qquad
b_{\gamma\ell}\equiv\left(\gamma-\gamma_{+}\right)\left(\gamma-\gamma_{-}\right),$
(3.21)

with

$\gamma_{\pm}\equiv\frac{3}{2}\pm\sqrt{\alpha}.$ (3.22)

Our strategy is similar to the one adopted for the black hole: observe that
the equation of motion $\widetilde{H}_{\gamma\ell}\phi_{\gamma\ell}=0$
simplifies if $\gamma=\gamma_{\pm}$ or $\gamma=\gamma_{\pm}-2\ell-1$, such
that the constant term in $\widetilde{H}_{\gamma\ell}$ (the one $\propto
a_{\gamma\ell}b_{\gamma\ell}$) vanishes. With these choices for $\gamma$, the
equation of motion becomes

$\mathcal{G}^{+}_{\gamma-2,\ell}\,\mathcal{G}^{-}_{\gamma\ell}\,\phi_{\gamma\ell}=0,$
(3.23)

for which a possible solution is

$\mathcal{G}^{-}_{\gamma\ell}\,\phi_{\gamma\ell}=0.$ (3.24)

As before, this reduction of a second order differential equation to a first
order one is what lies behind the single-asymptote behavior. Analysis of
(3.24), for the choice $\gamma=\gamma_{\pm}$, shows that the solution
$\phi_{\gamma_{\pm},\ell}$ approaches $r^{\ell}$ at the origin, and
$\left(1-\frac{r^{2}}{L^{2}}\right)^{-(\gamma_{\pm}+\ell)/2}$ at the horizon.
These match precisely the desired boundary conditions for a quasinormal mode,
i.e. regularity at the origin and outgoing at the horizon.212121 We do not
need it, but the exact quasinormal solution is
$\phi_{\gamma_{\pm},\ell}(r)=\left(\frac{r}{L}\right)^{\ell}\left(1-\frac{r^{2}}{L^{2}}\right)^{-(\gamma_{\pm}+\ell)/2}.$
(3.25) Recalling (3.7), we see that the quasinormal frequency is given by

$i\omega_{\gamma_{\pm}\ell}L=\gamma_{\pm}+\ell.$ (3.26)

Thus, we have identified two quasinormal frequencies (for a given $\ell$), one
for $\gamma_{+}$ and one for $\gamma_{-}$. Each corresponding quasinormal mode
serves as some sort of “ground state”. The idea is to build the “excited
states” by successively applying $\mathcal{G}^{+}$. This is reminiscent of how
the simple harmonic oscillator problem is solved algebraically.

Before doing so, let us remark upon the other possibility
$\gamma=\gamma_{\pm}-2\ell-1$. In that case, it can be shown that (3.24)
implies the irregular asymptotic behavior $r^{-{(\ell+1)}}$ at the origin,
which does not match the desired quasinormal boundary conditions.222222We do
not need this one either, but the exact irregular solution is
$\phi_{\gamma_{\pm}-2\ell-1,\ell}(r)=\left(\frac{L}{r}\right)^{\ell+1}\left(1-\frac{r^{2}}{L^{2}}\right)^{-(\gamma_{\pm}-\ell-1)/2}.$
(3.27) 232323To round out the discussion, one could go back to (3.23)
$\mathcal{G}^{+}_{\gamma-2,\ell}\mathcal{G}^{-}_{\gamma\ell}\phi_{\gamma\ell}=0$,
and ask what happens if instead of (3.24), one has
$\mathcal{G}^{-}_{\gamma\ell}\phi_{\gamma\ell}\neq 0$. In that case, by
studying what $\mathcal{G}^{+}_{\gamma-2,\ell}$ annihilates, it can be shown
that the choice $\gamma=\gamma_{\pm}$ yields $\phi_{\gamma\ell}$ solutions
irregular at the origin, whereas the choice $\gamma=\gamma_{\pm}-2\ell-1$
yields solutions that are incoming at the horizon. Neither satisfies the
desired quasinormal boundary conditions.

Returning thus to the choice $\gamma=\gamma_{\pm}$, having found the “ground
state” $\phi_{\gamma_{\pm},\ell}$ (for given $\ell$), we can raise to a higher
$\gamma$ by successively applying the raising operator (recalling that each
time, $\gamma$ increments by $2$):

$\phi_{\gamma_{\pm}+2n,\ell}\propto\mathcal{G}^{+}_{\gamma_{\pm}+2n-2,\ell}\mathcal{G}^{+}_{\gamma_{\pm}+2n-4,\ell}\cdots\mathcal{G}^{+}_{\gamma_{\pm}+2,\ell}\mathcal{G}^{+}_{\gamma_{\pm},\ell}\phi_{\gamma_{\pm},\ell},$
(3.28)

where $n$ an integer. It can be shown by inspecting the form of
$\mathcal{G}^{\pm}_{\gamma\ell}$ that the resulting
$\phi_{\gamma_{\pm}+2n,\ell}$ has the correct quasinormal boundary conditions,
i.e. $\phi_{\gamma_{\pm}+2n,\ell}$ approaches $r^{\ell}$ at the origin, and
approaches $\left(1-\frac{r^{2}}{L^{2}}\right)^{-(\gamma_{\pm}+2n+\ell)/2}$ at
the horizon. This way, we can construct all the quasinormal modes, covering
the whole spectrum:

$i\omega_{\gamma\ell}L=\gamma+\ell=\gamma_{\pm}+2n+\ell,$ (3.29)

where $n$ is a non-negative integer. One can further show that the
$E^{+}_{\gamma\ell}$ operators (which raises $\ell$) also preserve the
quasinormal boundary conditions. So we can even combine
$\mathcal{G}^{+}_{\gamma\ell}$ and $E^{+}_{\gamma\ell}$ to construct the
entire spectrum from the quasinormal solution $\phi_{\gamma_{\pm},0}$, which
can be regarded as the ground state.242424There are some exceptional cases
where the procedure—constructing the quasinormal modes by raising $\gamma$
starting from $\phi_{\gamma_{\pm},\ell}$—requires a modification. This
subtlety happens only for the $\gamma_{-}$ branch: it turns out if
$\gamma_{-}=-\ell$ (or equivalently, $2\sqrt{\alpha}=2\ell+3$),
$\mathcal{G}^{+}_{\gamma_{-},\ell}$ is proportional to
$\mathcal{G}^{-}_{\gamma_{-},\ell}$ which means the former cannot be used to
raise $\gamma$ from $\phi_{\gamma_{-},\ell}$ which is annihilated by the
latter. To construct $\phi_{\gamma_{-}+2,\ell}$, one could alternatively use
$E^{+}$ instead. For instance, if $\alpha=9/4$ (massless scalar) such that
$\gamma_{-}=0$, the $\phi_{00}$ solution is annihilated by both
$\mathcal{G}^{+}_{00}$ and $\mathcal{G}^{-}_{00}$. One could instead use
$\phi_{20}$ as the ground state from which to build out the quasinormal
spectrum at $\ell=0$, by acting successively with $\mathcal{G}^{+}$. To find
$\phi_{20}$, use the fact that $E^{-}_{20}\phi_{20}=0$. In other words, forget
about the state $\phi_{00}$ (its corresponding frequency is zero anyway);
instead, start from $\phi_{20}$ which can be obtained by solving
$E^{-}_{20}\phi_{20}=0$. Or more precisely, there is a ground state for
$\gamma_{+}$, and a ground state for $\gamma_{-}$. From each, a whole table of
states labeled by $\gamma=\gamma_{\pm}+2n$ and $\ell$ can be constructed. In
other words, they form two separate (non-unitary) representations of
$\mathfrak{so}(1,4)$.

###  3.3 Conformally Coupled Case

A conformally coupled scalar ($\xi=\xi_{c}$ or $\alpha=1/4$ in (3.2)) enjoys
additional symmetries effected by the CKVs. The five CKVs of de Sitter space
are

$\displaystyle\widetilde{K}_{1}^{\mu}\partial_{\mu}$
$\displaystyle=\frac{\Delta}{r^{2}}\sin\theta\cos\varphi\,\partial_{r}+\frac{1}{r}\cos\theta\cos\varphi\,\partial_{\theta}-\frac{1}{r}\csc\theta\sin\varphi\,\partial_{\varphi}$
$\displaystyle\widetilde{K}_{2}^{\mu}\partial_{\mu}$
$\displaystyle=\frac{\Delta}{r^{2}}\sin\theta\sin\varphi\,\partial_{r}+\frac{1}{r}\cos\theta\sin\varphi\,\partial_{\theta}+\frac{1}{r}\csc\theta\cos\varphi\,\partial_{\varphi}$
$\displaystyle\widetilde{K}_{3}^{\mu}\partial_{\mu}$
$\displaystyle=\frac{\Delta}{r^{2}}\cos\theta\,\partial_{r}-\frac{1}{r}\sin\theta\,\partial_{\theta}$
$\displaystyle\widetilde{K}_{+}^{\mu}\partial_{\mu}$
$\displaystyle=e^{-t/L}\left(\frac{r}{\sqrt{\Delta}}\partial_{t}-\frac{\sqrt{\Delta}}{L}\partial_{r}\right)$
$\displaystyle\widetilde{K}_{-}^{\mu}\partial_{\mu}$
$\displaystyle=e^{t/L}\left(\frac{r}{\sqrt{\Delta}}\partial_{t}+\frac{\sqrt{\Delta}}{L}\partial_{r}\right).$
(3.30)

The commutation relations among them and the KVs are given in Appendix B.
Together, they form an $\mathfrak{so}(2,4)$ algebra. From it, one can deduce
how they act on solutions, labeled by $\gamma,\ell,m$ as in the previous
subsection. We can also roughly guess their effects based on their explicit
form given in (3.30). For instance, $\widetilde{K}_{3}$ does not involve $t$
or $\varphi$, but involves $r$ and $\theta$. Thus we expect
$\delta_{\widetilde{K}_{3}}$ acting on $\phi_{\gamma\ell}Y_{\ell
m}e^{-i\omega_{\gamma\ell}t}$ to effect $\ell\rightarrow\ell\pm 1$ without
changing $m$ or the frequency. But because the frequency is proportional to
$\gamma+\ell$ (3.7), keeping the frequency unchanged while incrementing $\ell$
means $\gamma\rightarrow\gamma\mp 1$ at the same time. Explicit expressions
for the effect of $\delta_{\widetilde{K}_{i}}$, and the corresponding raising
and lowering operators, are given in Appendix B. Suffice to say the
$\widetilde{K}_{i}$ in general mix up solutions with different $\ell$ and $m$
(while keeping frequency fixed) in a way reminiscent of how the $K_{i}$ act in
the case of the black hole.

The two CKVs $\widetilde{K}_{\pm}$ introduce something more novel. They
involve a factor of $e^{\pm t/L}$ and hence increment $i\omega L$ by $1$. They
do not involve $\theta,\varphi$ and thus do not affect $\ell,m$. They are
therefore the analogs of $\delta^{ij}\delta_{P_{i}}\delta_{P_{j}}$ and
$\delta^{ij}\delta_{K_{i}}\delta_{K_{j}}$ (3.11) which raise and lower
$\gamma$ (by $2$) without changing $\ell,m$. A crucial difference is that here
there is no need to form quadratic combinations of generators, and so
$\widetilde{K}_{\pm}$ can increment $\gamma$ by $1$:

$\displaystyle\delta_{\widetilde{K}_{\pm}}\left(\phi_{\gamma\ell}Y_{\ell
m}e^{-i\omega_{\gamma\ell}t}\right)$
$\displaystyle=\frac{1}{L}\widetilde{G}^{\pm}_{\gamma\ell}\phi_{\gamma\ell}Y_{\ell
m}e^{-(i\omega_{\gamma\ell}L\pm 1)t/L},$ (3.31)

where

$\displaystyle\widetilde{G}^{+}_{\gamma\ell}$
$\displaystyle\equiv-\sqrt{\Delta}\partial_{r}-\frac{\sqrt{\Delta}}{r}-i\omega_{\gamma\ell}L\frac{r}{\sqrt{\Delta}}$
(3.32) $\displaystyle\widetilde{G}^{-}_{\gamma\ell}$
$\displaystyle\equiv\sqrt{\Delta}\partial_{r}+\frac{\sqrt{\Delta}}{r}-i\omega_{\gamma\ell}L\frac{r}{\sqrt{\Delta}},$
(3.33)

with $\widetilde{G}^{\pm}_{\gamma\ell}$ effecting $\gamma\pm 1$. These ladder
operators are equivalent to those in Section 3.1 of [25]. The various
identities obeyed by $\widetilde{G}_{\gamma\ell}^{\pm}$, showing explicitly
how they effect changes to the equation of motion operator, are in Appendix G.
As before, the corresponding contiguous relations for hypergeometric functions
are shown in Appendix D. And as before, an economical way to build the entire
(non-unitary) representation of the $\mathfrak{so}(2,4)$ is to use the
traceless symmetric tensor from (2.29) to recast one of our ladders:

$\displaystyle c^{\ell m}_{i_{1}\cdots
i_{\ell}}\delta_{\widetilde{K}_{i_{1}}}\cdots\delta_{\widetilde{K}_{i_{\ell}}}\Phi_{\gamma
00}(t,r,\theta,\varphi)$ $\displaystyle\propto\Phi_{\gamma-\ell,\ell
m}(t,r,\theta,\varphi).$ (3.34)

The proof is given in Appendix I. Recasting (3.31) in the same language, we
have

$\delta_{\widetilde{K}_{\pm}}\Phi_{\gamma\ell m}\propto\Phi_{\gamma\pm 1,\ell
m}.$ (3.35)

The derivation of the quasinormal spectrum given in the last subsection is
general, and makes no assumption about the value of the coupling to Ricci.
Thus (3.29), namely $i\omega_{\gamma\ell}L=\gamma+\ell=\gamma_{\pm}+2n+\ell$,
should hold true for conformal coupling as well. What might seem puzzling is
the fact that this spectrum has $\gamma$ (or $i\omega_{\gamma\ell}L$)
incrementing by $2$, while as argued earlier, $\widetilde{K}_{\pm}$ increments
$\gamma$ by $1$. The two statements are in fact consistent, since for
conformal coupling, $\gamma_{-}=1$ and $\gamma_{+}=2$. In other words, one
spectrum goes as $\gamma=1,3,5,\ldots$, and the other spectrum goes as
$\gamma=2,4,6,\ldots$. From the point of view of the isometry algebra
$\mathfrak{so}(1,4)$, these form two separate representations. But viewed
through the lens of the larger algebra $\mathfrak{so}(2,4)$, the two together
form a single irreducible representation, thanks to $\widetilde{K}_{\pm}$
which effects $\gamma\rightarrow\gamma\pm 1$.

Let us close this section with the remark that there are analogs of horizontal
symmetries as in the black hole case (2.37) (2.38) (one for each
($\gamma,\ell$)), from which conserved charges can be deduced. They can also
be used to derive the quasinormal spectrum, and to explain the single-
asymptote behavior of quasinormal modes. Explicit expressions for the
symmetries and charges are given in Appendix G.

##  4 Discussion

In this note we have adopted a geometric point of view that is applicable to
spin $0$ perturbations around both the black hole and de Sitter space. For the
black hole, static, scalar perturbations enjoy an exact SO(1,3) symmetry,
whose generators consist of 3 rotational KVs and 3 “boost” CKVs. The latter
might come as a surprise since the scalar in question is not conformally
coupled. But a special condition we call melodic (2.4) is responsible for the
relevance of the 3 “boost” CKVs.252525It is not surprising that the effective
3D metric (2.1) seen by the static scalar is conformally flat, and thus has
CKVs. What is non-trivial is that some of its CKVs are melodic. For de Sitter
space, the dynamical scalar perturbations enjoy an exact SO(1,4) symmetry
following from the isometry, which is extended to SO(2,4) if the scalar is
conformally coupled.

In both the case of the black hole, and that of de Sitter space, the
perturbation solutions form representations of the corresponding algebra. The
non-trivial algebra means the matrix representations of some symmetry
generators must have non-vanishing off-diagonal terms. This is the origin of
ladder structures. The most familiar example is the algebra of rotation
generators $J_{1},J_{2},J_{3}$: in a basis in which $J_{3}$ is diagonal (i.e.
with the solutions labeled by $m$), $J_{1}$ and $J_{2}$ are represented by
non-diagonal matrices which means they give rise to ladder operators that can
be used to raise or lower $m$. Static, scalar perturbations around the black
hole can be labeled by $\ell,m$ (the quantum numbers of rotations), forming a
unitary (principal series) representation of SO(1,3): the boost CKVs, which
have a non-trivial algebra with rotations, effect transformations that raise
or lower $\ell$. Dynamical scalar perturbations around de Sitter space are
labeled by $\ell,m$ and frequency, forming a non-unitary representation of
SO(1,4) (or SO(2,4) in the case of conformal coupling). Those symmetry
generators that do not commute with the time-translation KV, effect
transformations that raise or lower the frequency.

For both the black hole and de Sitter space, the nature of the perturbation
solution in the “ground state”, from which the whole representation can be
generated by applying ladder operators, is what lies behind the remarkable
single-asymptote property: i.e., the black hole perturbation solution that is
regular at the horizon has purely “growing” tidal behavior but no “decaying”
(Love number) tail at infinity; the quasinormal solution around de Sitter
space is regular at the origin and purely outgoing at the horizon.

There are a number of interesting follow-up questions. How could this
geometric/ representation-theoretic understanding be extended to spin 1 and
spin 2 perturbations? For the black hole, hints of this can be gleaned from
the spin ladder discussed in [1], where the spin ladder was used to explain
the vanishing of spin 1 and 2 Love numbers for Kerr black holes. How could the
geometric/representation-theoretic understanding be extended to non-static
perturbations around a black hole? Near-zone approximations exist, which are
applicable to low frequency black hole perturbations. One of them has a larger
symmetry group than SO(1,3), in fact SO(2,4) [24]. This might be useful for
understanding the dissipative tidal response of the black hole. More
ambitiously, to understand black hole quasinormal modes, one must go beyond
the low frequencies associated with near-zone approximations. An interesting
recent development is the identification of symmetries associated with a near-
light-ring approximation [34, 35, 36]. Could there be further hidden exact
symmetries governing dynamical black hole perturbations? We hope to address
some of these questions in the future.

We are grateful to Frederik Denef, Manvir Grewal, Klaas Parmentier, Alessandro
Podo, and Luca Santoni for very helpful discussions and comments, and to
collaborators Austin Joyce, Riccardo Penco, Luca Santoni, and Adam Solomon for
many useful insights. Research for this work was supported in part by a Simons
Fellowship in Theoretical Physics and the Department of Energy DE-SC011941. ZS
is supported by the US National Science Foundation under Grant No. PHY-2209997
and the Gravity Initiative at Princeton University.

##  Appendix A (C)KVs of the 3D Effective Metric for Black Hole

The three KVs and three melodic CKVs of the 3D effective metric for the black
hole are listed in (2.2) and (2.3). The four non-melodic CKVs are

$\displaystyle\widetilde{K}_{1}^{i}\partial_{i}$
$\displaystyle=\frac{1}{2}\Delta^{\prime}(r)\sqrt{\Delta(r)}\sin\theta\cos\varphi\,\partial_{r}+\sqrt{\Delta(r)}\left(-\cos\theta\cos\varphi\,\partial_{\theta}+\csc\theta\sin\varphi\,\partial_{\varphi}\right)$
$\displaystyle\widetilde{K}_{2}^{i}\partial_{i}$
$\displaystyle=\frac{1}{2}\Delta^{\prime}(r)\sqrt{\Delta(r)}\sin\theta\sin\varphi\,\partial_{r}-\sqrt{\Delta(r)}\left(\cos\theta\sin\varphi\,\partial_{\theta}+\csc\theta\cos\varphi\,\partial_{\varphi}\right)$
$\displaystyle\widetilde{K}_{3}^{i}\partial_{i}$
$\displaystyle=\frac{1}{2}\Delta^{\prime}(r)\sqrt{\Delta(r)}\cos\theta\,\partial_{r}+\sqrt{\Delta(r)}\sin\theta\,\partial_{\theta}$
$\displaystyle\widetilde{D}^{i}\partial_{i}$
$\displaystyle=\sqrt{\Delta(r)}\partial_{r}.$ (A.1)

The commutation relations obeyed by the (C)KVs are

$\displaystyle\left[J_{i},J_{j}\right]$ $\displaystyle=-\epsilon_{ijk}J_{k}$
$\displaystyle\left[K_{i},K_{j}\right]$
$\displaystyle=\beta^{2}\epsilon_{ijk}J_{k}$
$\displaystyle\left[\widetilde{K}_{i},\widetilde{K}_{j}\right]$
$\displaystyle=-\beta^{2}\epsilon_{ijk}J_{k}$
$\displaystyle\left[J_{i},K_{j}\right]$ $\displaystyle=-\epsilon_{ijk}K_{k}$
$\displaystyle\left[J_{i},\widetilde{K}_{j}\right]$
$\displaystyle=-\epsilon_{ijk}\widetilde{K}_{k}$
$\displaystyle\left[K_{i},\widetilde{K}_{j}\right]$
$\displaystyle=-\beta^{2}\delta_{ij}\widetilde{D}$
$\displaystyle\left[\widetilde{D},J_{i}\right]$ $\displaystyle=0$
$\displaystyle\left[\widetilde{D},K_{i}\right]$
$\displaystyle=\widetilde{K}_{i}$
$\displaystyle\left[\widetilde{D},\widetilde{K}_{i}\right]$
$\displaystyle=K_{i},$ (A.2)

where $\beta=\frac{1}{2}\left(r_{+}-r_{-}\right).$ By taking

$M_{ij}=\epsilon_{ijk}J_{k},\quad M_{0i}=\frac{1}{\beta}K_{i},\quad
M_{i4}=-\frac{1}{\beta}\widetilde{K}_{i},\quad M_{04}=\widetilde{D},$ (A.3)

we can show that these CKVs form an $\mathfrak{so}(1,4)$ algebra. There are
several interesting subalgebras. The $J_{i}$ form the isometry algebra
$\mathfrak{so}(3)$. The $K_{i}$ are melodic CKVs; together with the $J_{i}$,
they form the melodic conformal algebra $\mathfrak{so}(1,3)$.

##  Appendix B (C)KVs of $\text{dS}_{4}$

The 10 KVs and the five CKVs of de Sitter space are listed in (3.4) and
(3.30). The commutation relations obeyed by the (C)KVs are

$\displaystyle\left[J_{i},J_{j}\right]$ $\displaystyle=-\epsilon_{ijk}J_{k}$
$\displaystyle\left[P_{i},P_{j}\right]$ $\displaystyle=0$
$\displaystyle\left[K_{i},K_{j}\right]$ $\displaystyle=0$
$\displaystyle\left[\widetilde{K}_{i},\widetilde{K}_{j}\right]$
$\displaystyle=\frac{1}{L^{2}}\epsilon_{ijk}J_{k}$
$\displaystyle\left[J_{i},P_{j}\right]$ $\displaystyle=-\epsilon_{ijk}P_{k}$
$\displaystyle\left[J_{i},K_{j}\right]$ $\displaystyle=-\epsilon_{ijk}K_{k}$
$\displaystyle\left[\widetilde{K}_{i},P_{j}\right]$
$\displaystyle=-\delta_{ij}\widetilde{K}_{+}$
$\displaystyle\left[\widetilde{K}_{i},K_{j}\right]$
$\displaystyle=\delta_{ij}\widetilde{K}_{-}$
$\displaystyle\left[J_{i},\widetilde{K}_{j}\right]$
$\displaystyle=\epsilon_{ijk}\widetilde{K}_{k}$
$\displaystyle\left[K_{i},P_{j}\right]$
$\displaystyle=2\delta_{ij}D-2\epsilon_{ijk}J_{k}$
$\displaystyle\left[\widetilde{K}_{+},P_{j}\right]$ $\displaystyle=0$
$\displaystyle\left[\widetilde{K}_{-},P_{j}\right]$
$\displaystyle=-2L\widetilde{K}_{i}$
$\displaystyle\left[\widetilde{K}_{+},K_{i}\right]$
$\displaystyle=2\widetilde{K}_{i}$
$\displaystyle\left[\widetilde{K}_{-},K_{i}\right]$ $\displaystyle=0$
$\displaystyle\left[\widetilde{K}_{+},\widetilde{K}_{i}\right]$
$\displaystyle=\frac{1}{L^{2}}P_{i}$
$\displaystyle\left[\widetilde{K}_{-},\widetilde{K}_{i}\right]$
$\displaystyle=-\frac{1}{L^{2}}K_{i}$
$\displaystyle\left[\widetilde{K}_{\pm},J_{i}\right]$ $\displaystyle=0$
$\displaystyle\left[\widetilde{K}_{+},\widetilde{K}_{-}\right]$
$\displaystyle=-\frac{2}{L^{2}}D$ $\displaystyle\left[D,J_{i}\right]$
$\displaystyle=0$ $\displaystyle\left[D,P_{i}\right]$ $\displaystyle=P_{i}$
$\displaystyle\left[D,K_{i}\right]$ $\displaystyle=-K_{i}$
$\displaystyle\left[D,\widetilde{K}_{i}\right]$ $\displaystyle=0$
$\displaystyle\left[D,\widetilde{K}_{\pm}\right]$
$\displaystyle=\pm\widetilde{K}_{\pm}.$ (B.1)

We saw in (3.5) the change of basis that turns the KVs into the standard basis
of $\mathfrak{so}(1,4)$. To this we add

$M_{05}=\frac{L}{2}\left(\widetilde{K}_{+}+\widetilde{K}_{-}\right),\quad
M_{i5}=L\widetilde{K}_{i},\quad
M_{45}=-\frac{L}{2}\left(\widetilde{K}_{+}-\widetilde{K}_{-}\right),$ (B.2)

which shows that the (C)KVs form the conformal algebra $\mathfrak{so}(2,4)$,
where we take $\eta\equiv\text{diag}(-1,1,1,1,1,-1)$.

Once again, there are several interesting subalgebras. The $J_{i}$ again form
an $\mathfrak{so}(3)$ algebra. The $J_{i}$ and $\widetilde{K}_{i}$ together
form an $\mathfrak{so}(1,3)$ algebra. The four sets
$\\{D,\widetilde{K}_{+},\widetilde{K}_{-}\\}$ and $\\{D,K_{i},P_{i}\\}$ for
any $i$ form $\mathfrak{sl}\left(2,\mathbb{R}\right)\cong\mathfrak{so}(1,2)$
algebras.

The fact that $J_{i}$ and $\widetilde{K}_{i}$ form an $\mathfrak{so}(1,3)$
algebra is similar to the black hole case. For instance, the analog of (2.16)
is

$\displaystyle\delta_{\widetilde{K}_{3}}\left(\phi_{\gamma\ell}Y_{\ell
m}e^{-i\omega_{\gamma\ell}t}\right)$
$\displaystyle=e^{-i\omega_{\gamma\ell}t}\left(f(\ell)\widetilde{D}^{+}_{\ell}\phi_{\gamma\ell}Y_{\ell+1,m}-f(\ell-1)\widetilde{D}^{-}_{\ell}\phi_{\gamma\ell}Y_{\ell-1,m}\right),$
(B.3)

where

$\displaystyle\widetilde{D}^{+}_{\ell}$
$\displaystyle\equiv\left(1-\frac{r^{2}}{L^{2}}\right)\partial_{r}-\frac{r}{L^{2}}-\frac{\ell}{r}$
$\displaystyle\widetilde{D}^{-}_{\ell}$
$\displaystyle\equiv-\left(1-\frac{r^{2}}{L^{2}}\right)\partial_{r}+\frac{r}{L^{2}}-\frac{\ell+1}{r}.$
(B.4)

We see that $\widetilde{D}^{+}_{\ell}$ effect $\ell\pm 1$ while keeping
$\omega$ fixed. However, when indexing the solutions with $\gamma$, holding
$\omega$ fixed while raising/lowering $\ell$ means lowering/raising $\gamma$
to compensate. Thus the $\omega_{\gamma\ell}$ factor in the exponent in (B.3)
should really be considered $\omega_{\gamma-1,\ell+1}$ when multiplying the
first term in parentheses and $\omega_{\gamma+1,\ell-1}$ when multiplying the
second term, though of course these both equal $\omega_{\gamma\ell}$.

##  Appendix C Solutions to Equations of Motion

For the Reissner–Nordström black hole, the two independent solutions to (2.14)
are

$\displaystyle\phi_{\ell}^{(1)}(r)$
$\displaystyle={}_{2}F_{1}\left(-\ell,\ell+1,1,x\right)$ (C.1)
$\displaystyle\begin{split}\phi_{\ell}^{(2)}(r)&={}_{2}F_{1}\left(-\ell,\ell+1,1,x\right)\log\left(\frac{x}{x-1}\right)\\\
&\quad+\sum_{k=0}^{\ell-1}\left[(-1)^{k}\binom{\ell+k}{k}\binom{\ell}{k}\left(\psi(\ell-k+1)+\psi(\ell+k+1)-2\psi(k+1)\right)x^{k}\right.\\\
&\quad\left.-\left(\sum_{j=0}^{k-1}(-1)^{j}\binom{\ell+j}{j}\binom{\ell}{j}\frac{1}{k-j}\right)x^{k}\right],\end{split}$
(C.2)

where $x=\frac{r-r_{-}}{r_{+}-r_{-}}$. At large $r$, the first goes at
$r^{\ell}$, and the second goes as $1/r^{\ell+1}$.

For four-dimensional de Sitter space, the two independent solutions to (3.2)
are

$\displaystyle\phi^{(1)}_{\gamma\ell}(r)$
$\displaystyle=\left(\frac{r^{2}}{L^{2}}\right)^{\ell/2}\left(1-\frac{r^{2}}{L^{2}}\right)^{\frac{i\omega_{\gamma\ell}L}{2}}\,{}_{2}F_{1}\left(a,b,c;\frac{r^{2}}{L^{2}}\right)$
(C.3) $\displaystyle\phi^{(2)}_{\gamma\ell}(r)$
$\displaystyle=\left(\frac{r^{2}}{L^{2}}\right)^{-(\ell+1)/2}\left(1-\frac{r^{2}}{L^{2}}\right)^{\frac{i\omega_{\gamma\ell}L}{2}}\,{}_{2}F_{1}\left(a-c+1,b-c+1,2-c;\frac{r^{2}}{L^{2}}\right),$
(C.4)

where

$a=\frac{1}{2}\left(\frac{3}{2}+\ell+i\omega_{\gamma\ell}L+\sqrt{\alpha}\right),\quad
b=\frac{1}{2}\left(\frac{3}{2}+\ell+i\omega_{\gamma\ell}L-\sqrt{\alpha}\right),\quad
c=\ell+\frac{3}{2},$ (C.5)

with $\alpha=\frac{9}{4}-12\xi$.

Near the origin, the first goes at $r^{\ell}$, and the second goes as
$1/r^{\ell+1}$.

##  Appendix D Hypergeometric Identities

The hypergeometric differential equation is

$z(1-z)\frac{d^{2}}{dz^{2}}F(z)+(c-(a+b+1)z)\frac{d}{dz}F(z)-abF(z)=0,$ (D.1)

and is solved by the two functions

${}_{2}F_{1}(a,b,c;z),\quad z^{1-c}{}_{2}F_{1}(a-c+1,b-c+1,2-c;z),$

where

${}_{2}F_{1}(a,b,c;z)=\sum_{n=0}^{\infty}\frac{(a)_{n}(b)_{n}}{(c)_{n}}\frac{z^{n}}{n!}.$
(D.2)

Note the symmetry in $a$ and $b$, which will be useful to us. When $c$ is a
non-positive integer, the second solution is instead given by the first times
$\log(z)$, plus a different power series. This is relevant for the
Reissner–Nordström case, as we can see in (C.2) above.

Hypergeometric functions with the parameters $a$ $b$, or $c$ offset by 1 can
be related to the original function via the 15 contiguous relations, which
relate ${}_{2}F_{1}(a,b,c;z)$ to any two of the six functions

${}_{2}F_{1}(a\pm 1,b,c;z),\quad{}_{2}F_{1}(a,b\pm
1,c;z),\quad{}_{2}F_{1}(a,b,c\pm 1;z).$ (D.3)

Repeated applications of these relations can yield a relation between any
${}_{2}F_{1}(a,b,c;z)$ and any two hypergeometric functions with parameters
offset by any three integers. We shall discuss the identities used to uncover
the various ladders here. For an extensive discussion of hypergeometric
functions and their various identities, see [42, 43, 44].

The $D^{\pm}_{\ell}$ operators raise and lower $\ell$ for the static solutions
in the Reissner–Nordström background. We see from (C.1) that raising $\ell$
corresponds to lowering $a$ and raising $b$, and vice versa for lowering
$\ell$. Thus the ladder arises directly from these two identities:

$\displaystyle\left((b-a-1)z(1-z)\frac{d}{dz}-a(c-b+(b-a-1)z)\right){}_{2}F_{1}(a,b,c;z)$
$\displaystyle=a(c-b)\,{}_{2}F_{1}(a+1,b-1,c;z)$ (D.4)
$\displaystyle\left((a-b-1)z(1-z)\frac{d}{dz}-b(c-a-(a-b-1)z)\right){}_{2}F_{1}(a,b,c;z)$
$\displaystyle=b(c-a)\,{}_{2}F_{1}(a-1,b+1,c;z).$ (D.5)

The $E^{\pm}_{\gamma\ell}$ operators raise and lower $\ell$ while keeping
$\gamma$ fixed for the solutions in de Sitter. Examining (C.3) and (C.5), we
see that this entails raising and lowering $a$, $b$, and $c$ all at once. Thus
we need the identities

$\displaystyle\frac{d}{dz}\,{}_{2}F_{1}(a,b,c;z)$
$\displaystyle=\frac{ab}{c}\,{}_{2}F_{1}(a+1,b+1,c+1;z)$ (D.6)
$\displaystyle\frac{d}{dz}\left(z^{c-1}(1-z)^{a+b-c}\,{}_{2}F_{1}(a,b,c;z)\right)$
$\displaystyle=(c-1)z^{c-2}(1-z)^{a+b-c-1}\,{}_{2}F_{1}(a-1,b-1,c-1;z).$ (D.7)

Of course, because the de Sitter equation of motion requires a change of
variables to $z=r^{2}/L^{2}$ and a field redefinition to turn bring it into
hypergeometric form, the exact form of the ladder operators is not immediately
apparent from these identities.

The $F^{\pm}_{\gamma\ell}$ operators raise and lower $\gamma$ while
simultaneously lowering and raising $\ell$ and thus require identities which
keep $a$ and $b$ fixed while raising and lowering $c$. These are easily
supplied:

$\displaystyle\frac{d}{dz}\left((1-z)^{a+b-c}\,{}_{2}F_{1}(a,b,c;z)\right)$
$\displaystyle=\frac{(c-a)(c-b)}{c}(1-z)^{a+b-c-1}\,{}_{2}F_{1}(a,b,c+1;z)$
(D.8) $\displaystyle\frac{d}{dz}\left(z^{c-1}\,{}_{2}F_{1}(a,b,c;z)\right)$
$\displaystyle=(c-1)z^{c-2}\,{}_{2}F_{1}(a,b,c-1;z)$ (D.9)

The $G^{\pm}_{\gamma\ell}$ operators, which raise and lower $\gamma$ by 2
while keeping $\ell$ fixed, can be written in terms of previous two, but also
arise from identities which raise and lower both $a$ and $b$ while keeping $c$
fixed:

$\displaystyle\left((A+1)z\frac{d}{dz}+ab\right)\,{}_{2}F_{1}(a,b,c;z)$
$\displaystyle=ab(1-z)\,{}_{2}F_{1}(a+1,b+1,c;z)$ (D.10)
$\displaystyle\left((A-1)z(1-z)\frac{d}{dz}-\left(B-C(1-z)\right)\right){}_{2}F_{1}(a,b,c;z)$
$\displaystyle=-(c-a)(c-b)\,{}_{2}F_{1}(a-1,b-1,c;z),$ (D.11)

where

$A=a+b-c,\quad B=A(A-1),\quad C=a(A-1)+(b-1)(b-c).$

Moving to the conformally coupled case, the $\widetilde{D}^{\pm}_{\ell}$
operators raise and lower $\ell$ while simultaneously lowering and raising
$\gamma$ to keep $\omega$ fixed. This would seem to indicate that we need an
identity that raises and lowers $a$ and $b$ by $1/2$ and $c$ by 1. To our
knowledge, no such identity exists. However, since the hypergeometric function
is symmetric in $a$ and $b$ and in the conformally coupled case $a=b+1/2$,
raising and lowering both $a$ and $b$ by $1/2$ is equivalent to keeping $a$
fixed and raising $b$ by 1 and lowering $a$ by 1 and keeping $b$ fixed,
respectively. Thus the identities we require are

$\displaystyle\frac{d}{dz}\left((1-z)^{b}\,{}_{2}F_{1}(a,b,c;z)\right)$
$\displaystyle=\frac{b(a-c)}{c}(1-z)^{b-1}\,{}_{2}F_{1}(a,b+1,c+1;z)$ (D.12)
$\displaystyle\frac{d}{dz}\left(z^{c-1}(1-z)^{b-c+1}\,{}_{2}F_{1}(a,b,c;z)\right)$
$\displaystyle=(c-1)z^{c-2}(1-z)^{b-c}\,{}_{2}F_{1}(a-1,b,c-1;z).$ (D.13)

Lastly, the $\widetilde{G}^{\pm}_{\gamma\ell}$ operators raise and lower
$\gamma$ while keeping $\ell$ fixed. This seems to requires raising and
lowering both $a$ and $b$ by 1/2, but by the same logic as before, we can
instead use the following identities:

$\displaystyle\frac{d}{dz}\left(z^{b}\,{}_{2}F_{1}(a,b,c;z)\right)$
$\displaystyle=bz^{b-1}\,{}_{2}F_{1}(a,b+1,c;z)$ (D.14)
$\displaystyle\frac{d}{dz}\left(z^{c-a}(1-z)^{a+b-c}\,{}_{2}F_{1}(a,b,c;z)\right)$
$\displaystyle=(c-a)z^{c-a-1}(1-z)^{a+b-c-1}\,{}_{2}F_{1}(a-1,b,c;z).$ (D.15)

##  Appendix E Commutators and Lie Brackets

In this paper, when we write something like $[X,Y]=Z$ where $X,Y,Z$ are
vectors, we implicitly mean three different, mutually consistent statements.
The first is interpreting $[X,Y]$ as a Lie bracket (E.2), i.e.
$[X,Y]^{\mu}_{\rm LB}=Z^{\mu}$. The second is interpreting it as a statement
about Lie derivatives: $[\mathcal{L}_{X},\mathcal{L}_{Y}]=\mathcal{L}_{Z}$
(E.3). The third is interpreting it as a a statement about symmetry
transformation (E.1): $[\delta_{X},\delta_{Y}]=\delta_{Z}$ (E.5). The goal of
this Appendix is to demonstrate their mutual consistency.

In this paper, we are interested in symmetry formation of the form

$\delta_{X}=\mathcal{L}_{X}+c\left(\nabla_{\mu}X^{\mu}\right),$ (E.1)

where $X$ any vector, $c$ is any constant, and $\mathcal{L}_{X}$ denotes the
Lie derivative along $X$, obey the same algebra as the Lie derivatives along
the vectors themselves when acting on a general tensor. Here (and only here)
we shall carefully distinguish between commutators of differential operators
and Lie brackets of vectors, denoting the latter by

$\left[X,Y\right]_{\text{LB}}^{\mu}\equiv
X^{\mu}\nabla_{\mu}Y^{\nu}-Y^{\mu}\nabla_{\mu}X^{\nu}.$ (E.2)

We then cite the well-known property that for any tensor $T$ of rank $(r,s)$,

$\left[\mathcal{L}_{X},\mathcal{L}_{Y}\right]{T^{\mu_{1}\cdots\mu_{r}}}_{\nu_{1}\cdots\nu_{s}}=\mathcal{L}_{\left[X,Y\right]_{\text{LB}}}{T^{\mu_{1}\cdots\mu_{r}}}_{\nu_{1}\cdots\nu_{s}},$
(E.3)

and compute

$\displaystyle\left[\delta_{X},\delta_{Y}\right]{T^{\mu_{1}\cdots\mu_{r}}}_{\nu_{1}\cdots\nu_{s}}$
$\displaystyle=\big{(}\left[\mathcal{L}_{X},\mathcal{L}_{Y}\right]+c\left(\mathcal{L}_{X}\left(\nabla_{\mu}Y^{\mu}\big{)}-\mathcal{L}_{Y}\left(\nabla_{\mu}X^{\mu}\right)\right)\right){T^{\mu_{1}\cdots\mu_{r}}}_{\nu_{1}\cdots\nu_{s}}$
$\displaystyle=\left(\mathcal{L}_{\left[X,Y\right]_{\text{LB}}}+c\left(X^{\nu}\nabla_{\nu}\nabla_{\mu}Y^{\mu}-Y^{\nu}\nabla_{\nu}\nabla_{\mu}X^{\mu}\right)\right){T^{\mu_{1}\cdots\mu_{r}}}_{\nu_{1}\cdots\nu_{s}}~{}.$
(E.4)

In the second line of (E), replacing the double covariant derivative
$\nabla_{\nu}\nabla_{\mu}$ by $\nabla_{\mu}\nabla_{\nu}-R_{\mu\nu}$ allows us
to express
$X^{\nu}\nabla_{\nu}\nabla_{\mu}Y^{\mu}-Y^{\nu}\nabla_{\nu}\nabla_{\mu}X^{\mu}$
as the divergence of $[X,Y]_{\rm LB}$, and hence we find

$\displaystyle\left[\delta_{X},\delta_{Y}\right]{T^{\mu_{1}\cdots\mu_{r}}}_{\nu_{1}\cdots\nu_{s}}$
$\displaystyle=\delta_{\left[X,Y\right]_{\text{LB}}}{T^{\mu_{1}\cdots\mu_{r}}}_{\nu_{1}\cdots\nu_{s}}.$
(E.5)

Thus we see that any symmetry transformations of the form above obey the same
algebra as the corresponding vectors.

##  Appendix F Melodic Conformal Killing Vectors

We consider a $d$-dimensional pseudo-Riemannian manifold $\mathcal{M}_{d}$
with metric $g_{\mu\nu}$. A CKV of this manifold is a vector $X$ which
satisfies

$\nabla^{\mu}X^{\nu}+\nabla^{\nu}X^{\mu}=\frac{2}{d}\nabla_{\rho}X^{\rho}g^{\mu\nu}.$
(F.1)

An ordinary KV satisfies the above equation with $\nabla_{\rho}X^{\rho}=0$.
Taking derivatives of the conformal Killing equation yields the following
identities, which shall prove useful later:

$\displaystyle\Box X^{\mu}$
$\displaystyle=-\frac{d-2}{d}\nabla^{\mu}\nabla_{\nu}X^{\nu}-{R^{\mu}}_{\nu}X^{\nu}$
(F.2) $\displaystyle\Box\nabla_{\mu}X^{\mu}$
$\displaystyle=-\frac{1}{d-1}\left(R\nabla_{\mu}X^{\mu}+\frac{d}{2}X^{\mu}\nabla_{\mu}R\right)$
$\displaystyle=d\,\nabla_{\mu}\Box X^{\mu}.$ (F.3)

CKVs naturally arise in the context of conformal field theories. As shown in
the inset in Section 2.1, a CKV has a naturally action on scalars that is a
symmetry of the action for conformal coupling, but more surprisingly, for a
generic Ricci coupling, a melodic CKV, that is, one obeying

$\Box\nabla_{\mu}X^{\mu}=0,$ (F.4)

also yields a symmetry of the action.262626If we include a mass term as well,
the condition instead becomes
$(d-1)\left(\xi-\xi_{c}\right)\Box\nabla_{\mu}X^{\mu}-m^{2}\nabla_{\mu}X^{\mu}=0$.

A few properties of melodic CKVs are quickly apparent from (F). If the
manifold has $R=0$, all CKVs are melodic. If the manifold is maximally
symmetric but not flat, so $\nabla_{\mu}R=0$ but $R\neq 0$, then a CKV is
melodic if and only if $\nabla_{\mu}X^{\mu}=0$, which means $X$ is just an
ordinary KV. It can be shown that if $R\neq 0$ and $X$ is melodic CKV, a
rescaling of the metric $g_{\mu\nu}\to\Omega^{2}g_{\mu\nu}$ will make $X$ a KV
of the new metric if $\Omega^{2}=RL_{0}^{2}$, where $L_{0}$ is an arbitrary
length scale.

An important property is that the set of melodic CKVs is closed under Lie
brackets, which we can verify with the following identity that holds for any
two CKVs $X$ and $Y$:

$\begin{split}\Box\nabla_{\nu}\left(X^{\mu}\nabla_{\mu}Y^{\nu}-Y^{\mu}\nabla_{\mu}X^{\nu}\right)&=X^{\mu}\nabla_{\mu}\Box\nabla_{\nu}Y^{\nu}-Y^{\mu}\nabla_{\mu}\Box\nabla_{\nu}X^{\nu}\\\
&\quad+\frac{2}{d}\nabla_{\mu}X^{\mu}\Box\nabla_{\nu}Y^{\nu}-\frac{2}{d}\nabla_{\mu}Y^{\mu}\Box\nabla_{\nu}X^{\nu}.\end{split}$
(F.5)

Thus if both $X$ and $Y$ are melodic CKVs, so is $[X,Y]$.

It follows that the melodic CKVs form a Lie algebra. Since the KVs form the
isometry algebra of the manifold, and the CKVs form the conformal algebra, the
algebra formed by the melodic CKVs must lie “in between” these two algebras.
Denoting this algebra with $\mathfrak{mconf}\left(\mathcal{M}_{d}\right)$ and
letting $\mathfrak{isom}\left(\mathcal{M}_{d}\right)$ and
$\mathfrak{conf}(\mathcal{M}_{d})$ be the isometry and conformal algebras,
respectively, we have the following hierarchy:

$\mathfrak{isom}\left(\mathcal{M}_{d}\right)\subset\mathfrak{mconf}\left(\mathcal{M}_{d}\right)\subset\mathfrak{conf}\left(\mathcal{M}_{d}\right).$
(F.6)

(Here we consider KVs to be trivially melodic CKVs.) We see from above that
when $\mathcal{M}_{d}$ is Ricci-scalar-flat ($R=0$),
$\mathfrak{mconf}\left(\mathcal{M}_{d}\right)=\mathfrak{conf}\left(\mathcal{M}_{d}\right)$
because all CKVs are melodic, and when $\mathcal{M}_{d}$ is maximally
symmetric but not flat,
$\mathfrak{isom}\left(\mathcal{M}_{d}\right)=\mathfrak{mconf}\left(\mathcal{M}_{d}\right)$,
as the only melodic CKVs are ordinary KVs. Outside of these extreme cases, the
algebra $\mathfrak{mconf}\left(\mathcal{M}_{d}\right)$ is somewhat mysterious.
The three-dimensional manifold that appeared in Section 2.1 as the effective
space for static scalars in a Reissner–Nordström background is the only space
we have found with a non-trivial algebra of melodic CKVs. Letting that space
be $\Sigma_{3}$, we have

$\mathfrak{isom}\left(\Sigma_{3}\right)=\mathfrak{so}(3)\subset\mathfrak{mconf}\left(\Sigma_{3}\right)=\mathfrak{so}(1,3)\subset\mathfrak{conf}\left(\Sigma_{3}\right)=\mathfrak{so}(1,4).$
(F.7)

Returning to a scalar field theory, we are naturally led to consider the
conserved currents of the symmetries associated with the melodic CKVs. The
Noether procedure yields

$J_{\mu}=T_{\mu\nu}X^{\nu}+Z_{\mu},$ (F.8)

where

$Z_{\mu}\equiv\frac{2(d-1)}{d}\left(\xi-\xi_{c}\right)\left(\frac{1}{2}\Phi^{2}\nabla_{\mu}\nabla_{\nu}X^{\nu}-\Phi\nabla_{\mu}\Phi\nabla_{\nu}X^{\nu}\right),$
(F.9)

and the conserved stress-energy tensor is

$T_{\mu\nu}=\nabla_{\mu}\Phi\nabla_{\nu}\Phi-\frac{1}{2}g_{\mu\nu}\nabla_{\rho}\Phi\nabla^{\rho}\Phi+\xi\Phi^{2}G_{\mu\nu}+\xi\left(g_{\mu\nu}\nabla_{\rho}\nabla^{\rho}-\nabla_{\mu}\nabla_{\nu}\right)\left(\Phi^{2}\right).$
(F.10)

The trace of the stress energy tensor is

${T^{\mu}}_{\mu}=(d-1)\left(\xi-\xi_{c}\right)\nabla_{\mu}\left(\Phi^{2}\right)-\frac{d-2}{2}\Phi\left(\Box-\xi
R\right)\Phi.$ (F.11)

Thus we see that for $\xi=\xi_{c}$, the stress-energy tensor is traceless on
the equation of motion, which is a well-known property of conformal field
theories. It can be shown that

$\nabla_{\mu}Z^{\mu}=-\frac{1}{d}{T^{\mu}}_{\mu}\nabla_{\nu}X^{\nu}+\frac{d-1}{d}\Phi^{2}\Box\nabla_{\mu}X^{\mu}+\frac{d-2}{2d}\nabla_{\mu}X^{\mu}\Phi\left(\Box-\xi
R\right)\Phi.$ (F.12)

The second term vanishes for a melodic CKV, and the third vanishes on the
equation of motion. We can use (F.12), combined with the conservation of
$T_{\mu\nu}$ on the equation of motion, the conformal Killing equation, and
the melodic property of the CKV to show that the current we have defined is
conserved on the equation of motion:

$\displaystyle\nabla^{\mu}J_{\mu}$
$\displaystyle=\nabla^{\mu}T_{\mu\nu}X^{\nu}+T_{\mu\nu}\nabla^{\mu}X^{\nu}+\nabla^{\mu}Z_{\mu}$
(F.13)
$\displaystyle=\frac{1}{d}{T^{\mu}}_{\mu}\nabla_{\nu}X^{\nu}+\nabla^{\mu}Z_{\mu}+\text{terms
that vanish on-shell}$ (F.14) $\displaystyle=0+\text{terms that vanish on-
shell}.$ (F.15)

##  Appendix G Ladder Identities in de Sitter space

In this appendix we collect several identities for the ladder operators in de
Sitter space. The identities show in an explicit way how the ladder operators
map a solution at one level to that at another level. We divide the discussion
based on the coupling to Ricci. We also provide the conserved horizontal
charges for the conformally coupled case.

Generic coupling. The equation of motion can be expressed as

$H_{\gamma\ell}\phi_{\gamma\ell}=0\quad,\quad
H_{\gamma\ell}\equiv-\frac{1}{r^{2}}\left(\partial_{r}\left(\Delta\partial_{r}\right)+\frac{\omega_{\gamma\ell}^{2}r^{4}}{\Delta}+\left(\alpha-\frac{9}{4}\right)\frac{r^{2}}{L^{2}}-\ell(\ell+1)\right).$
(G.1)

For the $E_{\gamma\ell}^{\pm}$ and $F_{\gamma\ell}^{\pm}$ operators defined in
(3.1), we have the following ladder relations:

$H_{\gamma,\ell+1}E_{\gamma\ell}^{+}=E_{\gamma\ell}^{+}H_{\gamma\ell},\quad
H_{\gamma,\ell-1}E_{\gamma\ell}^{-}=E_{\gamma\ell}^{-}H_{\gamma\ell}.$ (G.2)
$H_{\gamma+2,\ell-1}F_{\gamma\ell}^{+}=F_{\gamma\ell}^{+}H_{\gamma\ell},\quad
H_{\gamma-2,\ell+1}F_{\gamma\ell}^{-}=F_{\gamma\ell}^{-}H_{\gamma\ell},$ (G.3)

confirming what we saw from the geometric perspective in Section 3.1.

We can then write

$\displaystyle
H_{\gamma\ell}=E_{\gamma,\ell-1}^{+}E_{\gamma\ell}^{-}+\frac{1}{L^{2}}a_{\gamma\ell}=E_{\gamma,\ell+1}^{-}E_{\gamma\ell}^{+}+\frac{1}{L^{2}}a_{\gamma+2,\ell}$
(G.4) $\displaystyle
E^{-}_{\gamma,\ell+1}E^{+}_{\gamma\ell}-E^{+}_{\gamma,\ell-1}E^{-}_{\gamma\ell}=-\frac{2\left(2\gamma+4\ell+1\right)}{L^{2}}$
(G.5) $\displaystyle
H_{\gamma\ell}=F_{\gamma-2,\ell+1}^{+}F_{\gamma\ell}^{-}+\frac{1}{L^{2}}b_{\gamma\ell}=F_{\gamma+2,\ell-1}^{-}F_{\gamma\ell}^{+}+\frac{1}{L^{2}}b_{\gamma+2,\ell}$
(G.6) $\displaystyle
F^{-}_{\gamma+2,\ell-1}F^{+}_{\gamma\ell}-F^{+}_{\gamma-2,\ell+1}F^{-}_{\gamma\ell}=-\frac{2\left(2\gamma-1\right)}{L^{2}},$
(G.7)

where $a_{\gamma\ell}$ and $b_{\gamma\ell}$ are defined in (3.21).

Conformal coupling. For the conformally coupled scalar, to exhibit the ladder
structure due to the CKV $\widetilde{K}_{3}$, it turns out to be convenient to
write the equation of motion as

$H_{\gamma\ell}\phi_{\gamma\ell}=0\quad,\quad
H_{\gamma\ell}\equiv-\frac{\Delta}{r^{4}}\left(\partial_{r}\left(\Delta\partial_{r}\right)+\frac{\omega_{\gamma\ell}^{2}r^{4}}{\Delta}-2\frac{r^{2}}{L^{2}}-\ell(\ell+1)\right).$
(G.8)

This $H_{\gamma\ell}$ is proportional to the one in (G.1), with $\alpha=1/4$.
For the ladder operators $\widetilde{D}^{\pm}_{\ell}$ defined in (B), we have:

$H_{\gamma-1,\ell+1}\widetilde{D}_{\ell}^{+}=\widetilde{D}_{\ell}^{+}H_{\gamma\ell},\quad
H_{\gamma+1,\ell-1}\widetilde{D}_{\ell}^{-}=\widetilde{D}_{\ell}^{-}H_{\gamma\ell},$
(G.9)

as well as

$\displaystyle
H_{\gamma\ell}=\widetilde{D}_{\ell-1}^{+}\widetilde{D}_{\ell}^{-}-\omega_{\gamma\ell}^{2}-\frac{\ell^{2}}{L^{2}}=\widetilde{D}_{\ell+1}^{-}\widetilde{D}_{\ell}^{+}-\omega_{\gamma\ell}^{2}-\frac{(\ell+1)^{2}}{L^{2}}$
(G.10)
$\displaystyle\widetilde{D}^{-}_{\ell+1}\widetilde{D}^{+}_{\ell}-\widetilde{D}^{+}_{\ell-1}\widetilde{D}^{-}_{\ell}=\frac{2\ell+1}{L^{2}}.$
(G.11)

Observe that since $H_{\gamma+1,-1}=H_{\gamma 0}$ (recall that
$\omega_{\gamma\ell}$ is proportional to $\gamma+\ell$), we have

$\left[Q_{0},H_{\gamma 0}\right]=0\quad{\rm where}\quad
Q_{0}\equiv\widetilde{D}^{-}_{0}=-\frac{\Delta}{r^{3}}\partial_{r}r$ (G.12)

and thus $\delta\phi_{0}=Q_{0}\phi_{0}$ is a symmetry. This can be verified at
the level of the action as well. This is the horizontal symmetry at $\ell=0$,
just as in the black hole case. Climbing the ladder, it can be shown

$\left[Q_{\ell},H_{\gamma\ell}\right]=0\quad{\rm where}\quad
Q_{\ell}\equiv\widetilde{D}^{+}_{\ell-1}Q_{\ell-1}\widetilde{D}^{-}_{\ell},$
(G.13)

and again $\delta\phi_{\ell}=Q_{\ell}\phi_{\ell}$ can be shown to be a
symmetry of the action. See Appendix H for a proof. For $\ell=0$, the Noether
procedure yields the conserved charge

$P_{\gamma
0}\equiv\frac{\Delta^{2}}{r^{4}}\left(\partial_{r}\left(r\phi_{\gamma
0}\right)\right)^{2}+\omega_{\gamma 0}^{2}r^{2}\phi_{\gamma 0}^{2},$ (G.14)

in the sense that $\partial_{r}P_{\gamma 0}=0.$ Note that the charge depends
on $\gamma$, because of $\gamma$ dependence from both $\omega_{\gamma 0}$ and
$\phi_{\gamma 0}$/ We can again climb the ladder to define the charges at non-
zero $\ell$:

$P_{\gamma\ell}\equiv\frac{\Delta^{2}}{r^{4}}\left(\partial_{r}\left(r\widetilde{D}^{-}_{1}\widetilde{D}^{-}_{2}\cdots\widetilde{D}^{-}_{\ell}\phi_{\gamma\ell}\right)\right)^{2}+\omega_{\gamma\ell}^{2}r^{2}\left(\widetilde{D}^{-}_{1}\widetilde{D}^{-}_{2}\cdots\widetilde{D}^{-}_{\ell}\phi_{\gamma\ell}\right)^{2}.$
(G.15)

It can be shown these are precisely the Noether charges of the symmetries
effected by the $Q_{\ell}$ operators.272727Since the $D^{\pm}_{\ell}$
operators change both $\ell$ and $\gamma$, we must take care with the
subscripts here. The series of $\widetilde{D}^{-}_{\ell}$ operators acting on
$\phi_{\gamma\ell}$ produce $\phi_{\gamma+\ell,0}$, and we can then use
$\omega_{\gamma\ell}=\omega_{\gamma+\ell,0}$ and invoke conservation of
$P_{\gamma+\ell,0}$ to show that $P_{\gamma\ell}$ is conserved. Note that
unlike in the black hole case, the conserved charges are not complete squares
i.e. one cannot take square root to reduce them to be linear in the field. If
one constructs $\phi_{\gamma\ell}$ by
$\phi_{\gamma\ell}\propto\widetilde{D}^{+}_{\ell-1}...\widetilde{D}^{+}_{0}\phi_{\gamma+\ell,0}$,
it can be shown that

$P_{\gamma\ell}\propto\frac{1}{L^{4\ell}}\left(\frac{\left(\gamma\right)_{2\ell+1}}{\gamma+\ell}\right)^{2}P_{\gamma+\ell,0},$
(G.16)

where $(x)_{n}=\Gamma(x+n)/\Gamma(n)$ is the Pochhammer symbol. Note that for
$\gamma=-\ell$, the expression should be evaluated as a limit. Just as the
conserved horizontal charges for black hole allowed us to show the vanishing
of the Love numbers, the conservation of these charges gives us an alternative
way to derive the spectrum of quasinormal mode frequencies in the conformally
coupled case, which we shall not discuss here.

For the $\widetilde{G}_{\gamma\ell}^{\pm}$ operators associated with the CKVs
$\widetilde{K}_{\pm}$ (3.31), yet another choice of the equation of motion
operator turns out to be useful:

$\widetilde{H}_{\gamma\ell}\equiv\frac{r^{4}}{\Delta}H_{\gamma\ell},$ (G.17)

for which we find the ladder relations:

$\widetilde{H}_{\gamma+1,\ell}\widetilde{G}_{\gamma\ell}^{+}=\widetilde{G}_{\gamma\ell}^{+}\widetilde{H}_{\gamma\ell},\quad\widetilde{H}_{\gamma-1,\ell}\widetilde{G}_{\gamma\ell}^{-}=\widetilde{G}_{\gamma\ell}^{-}\widetilde{H}_{\gamma\ell},$
(G.18)

as well as

$\displaystyle\widetilde{H}_{\gamma\ell}=\widetilde{G}^{-}_{\gamma+1,\ell}\widetilde{G}_{\gamma\ell}^{+}-\gamma\left(\gamma+2\ell+1\right)=\widetilde{G}_{\gamma-1,\ell}^{+}\widetilde{G}_{\gamma,\ell}^{-}-\left(\gamma-1\right)\left(\gamma+2\ell\right)$
(G.19)
$\displaystyle\widetilde{G}^{-}_{\gamma+1,\ell}\widetilde{G}^{+}_{\gamma\ell}-\widetilde{G}^{+}_{\gamma-1,\ell}\widetilde{G}^{-}_{\gamma\ell}=2(\gamma+\ell).$
(G.20)

##  Appendix H Conditions for Horizontal Symmetries

In this appendix we shall establish the conditions for the existence of the
so-called horizontal symmetries used throughout the main text. We consider an
action of the form

$S=\sum_{n}S_{n}=\frac{1}{2}\sum_{n}\int
dr\,\phi_{n}^{*}\mathcal{H}_{n}\phi_{n},$ (H.1)

where

$\mathcal{H}_{n}\equiv\partial_{r}\Delta\partial_{r}+V_{n}(r).$ (H.2)

Here $\Delta(r)$ and $V_{n}(r)$ are arbitrary functions, and $n$ is an
abstract index that can represent multiple indices. The equation of motion is
naturally

$\mathcal{H}_{n}\phi_{n}=0.$ (H.3)

We now assume that we have some ladder operators, i.e., some
$\mathcal{D}^{\pm}_{n}$ that raise and lower $n$. We must have

$\mathcal{D}^{\pm}_{n}H_{n}\phi_{n}=H_{n\pm k}\mathcal{D}^{\pm}_{n}\phi_{n}.$
(H.4)

where the equation of motion operator is given by

$H_{n}\equiv-f(r)\mathcal{H}_{n}$ (H.5)

for some function $f(r)$, and we have introduced a minus sign for later
convenience. Like $n$, $k$ is an abstract index that could represent the
shifting of many different indices by different amounts. We may assume that
the ladder operators are first order, since we may always remove higher
derivatives by subtracting $H_{n}$ or its derivatives.

Lastly, for any differential operator $\mathcal{O}$, we let
$\widehat{\mathcal{O}}$ be the operator that arises from integrating by parts,
i.e., if

$\mathcal{O}=\sum_{i}a_{i}(r)\partial_{r}^{i},$ (H.6)

then

$\widehat{\mathcal{O}}=\sum_{i}(-1)^{i}\partial_{r}^{i}a_{i}(r),$ (H.7)

where derivatives act on everything to the right. We can then write

$\int dr\,\phi\mathcal{O}\psi=\int dr\,\psi\widehat{\mathcal{O}}\phi$ (H.8)

by dropping a boundary term.

We now take $n=0$ (or any specific value, but 0 is convenient) and consider
some first order differential operator $Q_{0}$ which obeys $Q_{0}^{*}=Q_{0}$.
We define the transformation $\delta\phi_{0}=Q_{0}\phi_{0}$ and find that the
action transforms as

$\displaystyle\delta S_{0}$ $\displaystyle=\int
dr\,\left(Q_{0}\phi_{0}^{*}\mathcal{H}_{0}\phi_{0}+\phi_{0}^{*}\mathcal{H}_{0}Q_{0}\phi_{0}\right)$
(H.9) $\displaystyle=\int
dr\,\left(\phi_{0}^{*}\left(\widehat{Q}_{0}\mathcal{H}_{0}+\mathcal{H}_{0}Q_{0}\right)\phi_{0}\right).$
(H.10)

Thus we see that a sufficient condition for $\delta S_{0}$ to vanish is

$\widehat{Q}_{0}\mathcal{H}_{0}=-\mathcal{H}_{0}Q_{0}.$ (H.11)

Since $Q_{0}$ is a first order operator, it can be written in the form

$Q_{0}=a(r)\partial_{r}b(r)$ (H.12)

for some functions $a(r)$ and $b(r)$. It follows that $\widehat{Q}_{0}$ can be
written

$\widehat{Q}_{0}=-b(r)\partial_{r}a(r)=-\frac{1}{g(r)}Q_{0}g(r),$ (H.13)

where $g(r)=a(r)/b(r)$. If

$\left[Q_{0},g(r)\mathcal{H}_{0}\right]=0,$ (H.14)

we can write

$\widehat{Q}_{0}\mathcal{H}_{0}=-\frac{1}{g(r)}Q_{0}g(r)\mathcal{H}_{0}=-\mathcal{H}_{0}Q_{0},$
(H.15)

so (H.11) is met. Thus if we can find some operator $Q_{0}$ that commutes with
$g(r)\mathcal{H}_{0}$ for the same $g(r)$ that appears in (H.13), then
$\delta\phi_{0}=Q_{0}\phi_{0}$ is a symmetry of the action $S_{0}$.

It is worth pausing to note that at this stage, the ladder structure has not
been used at all, and as noted previously, $n$ is set to zero simply for
concreteness. If we somehow encounter any operator $Q_{n}$ obeying (H.13) and
(H.14), then we have an associated symmetry of the action $S_{n}$. Having a
ladder structure may make it easier to find such a $Q_{n}$ but is in no way
necessary.

If we have such an operator $Q_{0}$, we can then define

$Q_{n}=\mathcal{D}^{+}_{n-k}Q_{n-k}\mathcal{D}^{-}_{n},$ (H.16)

assuming $n$ and $k$ are such that this recursion terminates at $Q_{0}$. We
also assume $\left(\mathcal{D}^{\pm}_{n}\right)^{*}=\mathcal{D}^{\pm}_{n}$.
Under what circumstances is $\delta\phi_{n}=Q_{n}\phi_{n}$ a symmetry of the
action $S_{n}$? The same procedure as in the $n=0$ case leads us to the
sufficient condition

$\widehat{Q}_{n}\mathcal{H}_{n}=-\mathcal{H}_{n}Q_{n}.$ (H.17)

Of course, if

$\widehat{Q}_{n}=-\frac{1}{g(r)}Q_{n}g(r)$ (H.18)

and

$\left[Q_{n},g(r)\mathcal{H}_{n}\right]=0,$ (H.19)

then (H.17) is satisfied, but since $Q_{n}$ is a higher order operator, we
have

$\widehat{Q}_{n}=\widehat{\mathcal{D}}^{-}_{n}\widehat{Q}_{n-k}\widehat{\mathcal{D}}^{+}_{n-k},$
(H.20)

and there is no guarantee that

$\widehat{Q}_{n}=-\frac{1}{g(r)}Q_{n}g(r)$ (H.21)

for any $g(r)$. However, we only require the condition

$\widehat{\mathcal{D}}^{\pm}_{n}=-\frac{1}{f(r)}\mathcal{D}^{\mp}_{n\pm
k}f(r),$ (H.22)

where $f(r)$ is the function from (H.5). Since the ladder operators are first
order, an expression of this form will always hold, but the requirement on
$f(r)$ is key, as it implies

$\widehat{\mathcal{D}}^{\pm}_{n}\mathcal{H}_{n}=-\frac{1}{f(r)}\mathcal{D}^{\mp}_{n\pm
k}f(r)\mathcal{H}_{n}=-\mathcal{H}_{n\mp k}\mathcal{D}^{\mp}_{n\pm k}.$ (H.23)

Thus (H.22) allows us to use (H.4) to push $\mathcal{H}_{n}$ to the left past
the series of $\widehat{\mathcal{D}}^{+}$ operators down to $\widehat{Q}_{0}$,
where it will have become $\mathcal{H}_{0}$, and we can use (H.14) to move it
past $\widehat{Q}_{0}$. Then we can use (H.22) again to keep moving
$\mathcal{H}_{0}$ to the right past the $\widehat{\mathcal{D}}^{-}$ operators,
until it becomes $\mathcal{H}_{n}$ again, having picked up an overall minus
sigh from the step where it was pushed past $\widehat{Q}_{0}$.

To see a specific example, we take $n=1$ and $k=1$ and compute

$\displaystyle\widehat{Q}_{1}\mathcal{H}_{1}$
$\displaystyle=\widehat{\mathcal{D}}^{-}_{1}\widehat{Q}_{0}\widehat{\mathcal{D}}^{+}_{0}\mathcal{H}_{1}$
$\displaystyle=-\widehat{\mathcal{D}}^{-}_{1}\widehat{Q}_{0}\mathcal{H}_{0}\mathcal{D}^{-}_{1}$
$\displaystyle=\widehat{\mathcal{D}}^{-}_{1}\mathcal{H}_{0}Q_{0}\mathcal{D}^{-}_{1}$
$\displaystyle=-\mathcal{H}_{1}\mathcal{D}^{+}_{0}Q_{0}\mathcal{D}^{-}_{1}$
$\displaystyle=-\mathcal{H}_{1}Q_{1}.$ (H.24)

Let us now consider the two cases we addressed in this work. For the
Reissner–Nordström black hole, we replace $n$ with $\ell$. The specific
dependence of $V_{\ell}(r)$ on $\ell$ in the form $\ell(\ell+1)$ made the
lowering operator for $\ell=0$ commute with $H_{0}$, so, using (2.37) and
(2.31), we have

$Q_{0}=D_{0}^{-}=\Delta\partial_{r},\quad g(r)=f(r)=\Delta.$ (H.25)

Since $f(r)=g(r)$, in this case we do actually satisfy (H.18) and (H.19). This
can also be seen directly from the expression

$Q_{\ell}=(-1)^{\ell}\Delta^{-(\ell+1)/2}\left(\Delta^{3/2}\partial_{r}\right)^{2\ell+1}\Delta^{-\ell/2},$
(H.26)

where all derivatives act on everything to the right.

For the conformally coupled scalar in de Sitter, $n$ is replaced by $\gamma$
and $\ell$, though as we did in the main text, we shall suppress the $\gamma$
index. As with the black hole, the dependence of $V_{\ell}$ on $\ell$ in the
form $\ell(\ell+1)$ gave us a $Q_{0}$ from the bottom of the ladder. Using
(G.12) and (G.9), we have

$Q_{0}=\widetilde{D}_{0}^{-}=-\frac{\Delta}{r^{3}}\partial_{r}r,\quad
g(r)=f(r)=\frac{\Delta}{r^{4}}.$ (H.27)

Once again, the equality of $f(r)$ and $g(r)$ makes this case more tractable,
and we have a compact expression for $Q_{\ell}$:

$Q_{\ell}=(-1)^{\ell+1}\left(\frac{\Delta}{r^{2}}\right)^{-(\ell+1)/2}r^{\ell}\left(\frac{\Delta^{3/2}}{r^{4}}\partial_{r}\right)^{2\ell+1}\left(\frac{\Delta}{r^{2}}\right)^{-\ell/2}r^{\ell+1}.$
(H.28)

##  Appendix I Identities for General First Order Differential Operators

Consider a linear differential operator given by

$\delta_{i}=A(t,r)x^{i}+B(t,r)\partial_{x^{i}}+C(t,r)x^{i}\partial_{t}+D(t,r)x^{i}\partial_{r}.$
(I.1)

If $c$ is an $n$-index traceless symmetric tensor, then

$c_{i_{1}\cdots i_{n}}\delta_{i_{1}}\cdots\delta_{i_{n}}f(t,r)=c_{i_{1}\cdots
i_{n}}x^{i_{1}}\cdots
x^{i_{n}}\mathcal{D}_{n-1}\mathcal{D}_{n-2}\cdots\mathcal{D}_{1}\mathcal{D}_{0}f(t,r),$
(I.2)

where

$\mathcal{D}_{j}=\mathcal{E}+\frac{j}{r}D(t,r),$ (I.3)

with

$\mathcal{E}=A(t,r)+\left(B(t,r)+\frac{1}{r}D(t,r)\right)\partial_{r}+C(t,r)\partial_{t}.$
(I.4)

In addition, we find, suppressing the $t$ and $r$ arguments to save space,

$\begin{split}&\delta^{ij}\delta_{i}\delta_{j}\left(c_{i_{1}\cdots
i_{n}}x^{i_{1}}\cdots x^{i_{n}}f\right)=c_{i_{1}\cdots i_{n}}x^{i_{1}}\cdots
x^{i_{n}}\bigg{[}-\frac{n(n+1)}{r^{2}}B^{2}f\\\
&\quad+r^{2}\left(\mathcal{E}+\frac{n+3}{r^{2}}B+\frac{n+1}{r}D\right)\circ\left(\mathcal{E}+\frac{n}{r^{2}}B+\frac{n}{r}D\right)f\bigg{]}.\end{split}$
(I.5)

In the black hole case, with $\delta_{i}=\delta_{K_{i}}$, we have

$A(t,r)=\frac{\Delta^{\prime}}{2r},\quad
B(t,r)=-\frac{r\Delta^{\prime}}{2},\quad C(t,r)=0,\quad
D(t,r)=\frac{1}{2r^{2}}\partial_{r}\left(r^{2}\Delta\right),$ (I.6)

so we find

$\mathcal{E}=\frac{\sqrt{\Delta}}{r}\partial_{r}\sqrt{\Delta}.$ (I.7)

We can express (I.2) as

$c_{i_{1}\cdots i_{n}}\delta_{i_{1}}\cdots\delta_{i_{n}}f(t,r)=c_{i_{1}\cdots
i_{n}}x^{i_{1}}\cdots
x^{i_{n}}\frac{1}{r^{n}\Delta^{(n+1)/2}}\left(\Delta^{3/2}\partial_{r}\right)^{n}\sqrt{\Delta}f(t,r),$
(I.8)

and (LABEL:eq:delta_id2) as

$\begin{split}\delta^{ij}\delta_{i}\delta_{j}\left(c_{i_{1}\cdots
i_{n}}x^{i_{1}}\cdots x^{i_{n}}f\right)&=c_{i_{1}\cdots i_{n}}x^{i_{1}}\cdots
x^{i_{n}}\left[\Delta^{3/2}\partial_{r}^{2}\left(\sqrt{\Delta}f\right)+2n\Delta^{3/2}\partial_{r}\left(\frac{\sqrt{\Delta}}{r}f\right)\right.\\\
&\quad\left.-\frac{n(n+1)}{4}\left(\left(\Delta^{\prime}\right)^{2}-\frac{4\Delta^{2}}{r^{2}}\right)f\right].\end{split}$
(I.9)

In the de Sitter case with generic coupling, for $\delta_{i}=\delta_{K_{i}}$
or $\delta_{P_{i}}$, we have

$A(t,r)=0,\quad B(t,r)=Le^{\pm t/L}\frac{\sqrt{\Delta}}{r^{2}},\quad
C(t,r)=\pm e^{-t/L}\frac{r}{\sqrt{\Delta}},\quad D(t,r)=0,$ (I.10)

with the plus signs for the $\delta_{K_{i}}$ and the minus signs for the
$\delta_{P_{i}}$. Then (I.2) and (LABEL:eq:delta_id2) greatly simplify:

$\displaystyle c_{i_{1}\cdots i_{n}}\delta_{i_{1}}\cdots\delta_{i_{n}}f(t,r)$
$\displaystyle=c_{i_{1}\cdots i_{n}}x^{i_{1}}\cdots x^{i_{n}}\left(Le^{\pm
t/L}\frac{\sqrt{\Delta}}{r^{2}}\partial_{r}\pm
e^{-t/L}\frac{r}{\sqrt{\Delta}}\partial_{t}\right)^{n}f(t,r)$ (I.11)
$\displaystyle\begin{split}\delta^{ij}\delta_{i}\delta_{j}\left(c_{i_{1}\cdots
i_{n}}x^{i_{1}}\cdots x^{i_{n}}f\right)&=c_{i_{1}\cdots i_{n}}x^{i_{1}}\cdots
x^{i_{n}}\left[r^{2}\left(\pm e^{\pm
t/L}\frac{r}{\sqrt{\Delta}}\partial_{t}+Le^{\pm
t/L}\frac{\sqrt{\Delta}}{r^{2}}\partial_{r}\right)^{2}f\right.\\\
&\quad\left.+\left(2n+3\right)Le^{\pm t/L}\frac{\sqrt{\Delta}}{r}\left(\pm
e^{-t/L}\frac{r}{\sqrt{\Delta}}\partial_{t}+Le^{\pm
t/L}\frac{\sqrt{\Delta}}{r^{2}}\partial_{r}\right)f\right].\end{split}$ (I.12)

Lastly, in the de Sitter case with conformal coupling, for
$\delta_{i}=\delta_{\widetilde{K}_{i}}$, we have

$A(t,r)=-\frac{1}{L},\quad B(t,r)=L,\quad C(t,r)=0,\quad D(t,r)=-\frac{r}{L},$
(I.13)

and we find

$\displaystyle c_{i_{1}\cdots i_{n}}\delta_{i_{1}}\cdots\delta_{i_{n}}f(t,r)$
$\displaystyle=c_{i_{1}\cdots i_{n}}x^{i_{1}}\cdots
x^{i_{n}}\frac{L^{n}r^{n+1}}{\Delta^{(n+1)/2}}\left(\frac{\Delta^{3/2}}{r^{4}}\partial_{r}\right)^{n}\frac{\sqrt{\Delta}}{r}f(t,r).$
(I.14)
$\displaystyle\begin{split}\delta^{ij}\delta_{i}\delta_{j}\left(c_{i_{1}\cdots
i_{n}}x^{i_{1}}\cdots x^{i_{n}}f\right)&=c_{i_{1}\cdots i_{n}}x^{i_{1}}\cdots
x^{i_{n}}\left[\frac{L^{2}}{r^{4}}\Delta^{3/2}\bigg{(}\partial_{r}^{2}\left(\sqrt{\Delta}f\right)\right.\\\
&\quad\left.+\frac{2n}{r^{(n+1)/2}}\partial_{r}\left(r^{(n-1)/2}\sqrt{\Delta}f\right)\bigg{)}-n(n+1)\frac{L^{2}}{r^{2}}f\right].\end{split}$
(I.15)

##  Appendix J More Ways to Climb Ladders

Though the focus of this work has been on operators that act on the radial
$\phi(r)$ solutions, we have seen that we can act with various (C)KVs to move
around the space of the full $\Phi(t,r,\theta,\varphi)$ solutions. In this
appendix, we shall explore this further.

We first need one more tool. For both the black hole and de Sitter space, we
have a triplet of KVs $J_{i}$ corresponding to spatial rotations. In the usual
way, we can construct operators that raise and lower $m$. We take

$J_{\pm}^{\mu}\partial_{\mu}=\left(J_{1}^{\mu}\pm
iJ_{2}^{\mu}\right)\partial_{\mu}=\pm ie^{\pm i\phi}\partial_{\theta}-e^{\pm
i\phi}\cot\theta\,\partial_{\varphi}.$ (J.1)

Then in the de Sitter case, we find

$\delta_{J_{\pm}}\Phi_{\gamma\ell
m}(t,r,\theta,\varphi)\propto\Phi_{\gamma\ell,m\pm 1}(t,r,\theta,\varphi).$
(J.2)

(For the black hole case, simply drop the $\gamma$ index and the $t$
argument.)

The power of these new operators comes from the fact that $\Phi_{\gamma\ell
m}=0$ for $\ell<m$. Focusing on the black hole for now, we see that (2.16)
shows that for $m=\ell$, the effect of $\delta_{K_{3}}$ on $\Phi_{\ell\ell}$
is solely to raise $\ell$, as the second term vanishes. We can thus map any
$\Phi_{\ell m}$ to $\Phi_{\ell+1,m}$, up to an overall constant, with the
following sequence of operators

$\left(\delta_{J_{-}}\right)^{\ell-m}\delta_{K_{3}}\left(\delta_{J_{+}}\right)^{\ell-m}\Phi_{\ell
m}\propto\Phi_{\ell+1,m},$ (J.3)

or perhaps more clearly,

$(\ell,m)\xrightarrow{\left(\delta_{J_{+}}\right)^{\ell-m}}(\ell+1,\ell)\xrightarrow{\delta_{K_{3}}}(\ell+1,\ell)\xrightarrow{\left(\delta_{J_{-}}\right)^{\ell-m}}(\ell+1,m).$
(J.4)

Unfortunately, in the black hole case there is no way to map $\Phi_{\ell m}$
to $\Phi_{\ell-1,m}$ using any combination of (C)KVs.

Moving to de Sitter space, we find from (3.9) that setting
$\gamma=\gamma_{\pm}$ makes the $F^{+}_{\gamma\ell}$ term vanish, so the
effect of $\delta_{K_{3}}$ is to lower $\ell$ alone. This gives us a way to
map any quasinormal mode $\Phi_{\gamma\ell m}$ to $\Phi_{\gamma,\ell-1,m}$. We
lower $\gamma=\gamma_{\pm}+2n$ to $\gamma_{\pm}$ with $n$ applications of
$\delta^{ij}\delta_{K_{i}}\delta_{K_{j}}$, lower $\ell$ with $\delta_{K_{3}}$,
and then raise $\gamma$ back up with $n$ applications of
$\delta^{ij}\delta_{P_{i}}\delta_{P_{j}}$:

$\left(\delta^{ij}\delta_{P_{i}}\delta_{P_{j}}\right)^{n}\delta_{K_{3}}\left(\delta^{kl}\delta_{K_{k}}\delta_{K_{l}}\right)^{n}\Phi_{\gamma_{\pm}+2n,\ell
m}\propto\Phi_{\gamma_{\pm}+2n,\ell-1,m},$ (J.5)

or,

$(\gamma_{\pm}+2n,\ell,m)\xrightarrow{\left(\delta^{kl}\delta_{K_{k}}\delta_{K_{l}}\right)^{n}}(\gamma_{\pm},\ell,m)\xrightarrow{\delta_{K_{3}}}(\gamma_{\pm},\ell-1,m)\xrightarrow{\left(\delta^{ij}\delta_{P_{i}}\delta_{P_{j}}\right)^{n}}(\gamma_{\pm}+2n,\ell-1,m).$
(J.6)

Raising $\ell$ is more involved but can be done, unlike in the
Reissner–Nordström case. We must first lower $\gamma$ to $\gamma_{\pm}$ and
$m$ to zero with $\delta^{ij}\delta_{K_{i}}\delta_{K_{j}}$ and
$\delta_{J_{-}}$ (in either order). We can then lower $\ell$ to zero with
$\delta_{K_{3}}$. Then we use

$c^{\ell+1,m}_{i_{1}\cdots
i_{\ell+1}}\delta_{P_{i_{1}}}\cdots\delta_{P_{i_{\ell+1}}}$ (J.7)

to jump from $(\gamma_{\pm},0,0)$ all the way to $(\gamma_{\pm},\ell,m)$.
Lastly, we use $\delta^{ij}\delta_{P_{i}}\delta_{P_{j}}$ to raise $\gamma$
back to its original value. We can represent this as

$\left(\delta^{ij}\delta_{P_{i}}\delta_{P_{j}}\right)^{n}\left(c^{\ell+1,m}_{i_{1}\cdots
i_{\ell+1}}\delta_{P_{i_{1}}}\cdots\delta_{P_{i_{\ell+1}}}\right)\left(\delta_{K_{3}}\right)^{\ell}\left(\delta^{ij}\delta_{K_{i}}\delta_{K_{j}}\right)^{n}\left(\delta_{J_{-}}\right)^{m}\Phi_{\gamma_{\pm}+2n,\ell
m}\propto\Phi_{\gamma_{\pm}+2n,\ell+1,m},$ (J.8)

or

$\begin{split}&(\gamma_{\pm}+2n,\ell,m)\xrightarrow{\left(\delta_{J_{-}}\right)^{m}}(\gamma_{\pm}+2n,\ell,0)\xrightarrow{\left(\delta^{ij}\delta_{K_{i}}\delta_{K_{j}}\right)^{n}}(\gamma_{\pm},\ell,0)\xrightarrow{\left(\delta_{K_{3}}\right)^{\ell}}\\\
&(\gamma_{\pm},0,0)\xrightarrow{c^{\ell+1,m}_{i_{1}\cdots
i_{\ell+1}}\delta_{P_{i_{1}}}\cdots\delta_{P_{i_{\ell+1}}}}(\gamma_{\pm},\ell+1,m)\xrightarrow{\left(\delta^{ij}\delta_{P_{i}}\delta_{P_{j}}\right)^{n}}(\gamma_{\pm}+2n,\ell+1,m).\end{split}$
(J.9)

Thus in the de Sitter case, we can move from any quasinormal mode to any other
via a series of operations, all built on KVs.

## References

  * [1] L. Hui, A. Joyce, R. Penco, L. Santoni, and A. R. Solomon, “Ladder symmetries of black holes. Implications for love numbers and no-hair theorems,” JCAP 01 no. 01, (2022) 032, arXiv:2105.01069 [hep-th].
  * [2] J. Ben Achour, E. R. Livine, S. Mukohyama, and J.-P. Uzan, “Hidden symmetry of the static response of black holes: applications to Love numbers,” JHEP 07 (2022) 112, arXiv:2202.12828 [gr-qc].
  * [3] T. Katagiri, M. Kimura, H. Nakano, and K. Omukai, “Vanishing Love of Black Holes in General Relativity: From Spacetime Conformal Symmetry of a Two-dimensional Reduced Geometry,” arXiv:2209.10469 [gr-qc].
  * [4] W. H. Press, “Time Evolution of a Rotating Black Hole Immersed in a Static Scalar Field,” APJ 175 (July, 1972) 243.
  * [5] K. Martel and E. Poisson, “Gravitational perturbations of the Schwarzschild spacetime: A Practical covariant and gauge-invariant formalism,” Phys. Rev. D 71 (2005) 104003, arXiv:gr-qc/0502028.
  * [6] H. Fang and G. Lovelace, “Tidal coupling of a Schwarzschild black hole and circularly orbiting moon,” Phys. Rev. D 72 (2005) 124016, arXiv:gr-qc/0505156.
  * [7] T. Damour and O. M. Lecian, “On the gravitational polarizability of black holes,” Phys. Rev. D80 (2009) 044017, arXiv:0906.3003 [gr-qc].
  * [8] T. Damour and A. Nagar, “Relativistic tidal properties of neutron stars,” Phys. Rev. D 80 (2009) 084035, arXiv:0906.0096 [gr-qc].
  * [9] T. Binnington and E. Poisson, “Relativistic theory of tidal Love numbers,” Phys. Rev. D80 (2009) 084018, arXiv:0906.1366 [gr-qc].
  * [10] B. Kol and M. Smolkin, “Black hole stereotyping: Induced gravito-static polarization,” JHEP 02 (2012) 010, arXiv:1110.3764 [hep-th].
  * [11] P. Landry and E. Poisson, “Gravitomagnetic response of an irrotational body to an applied tidal field,” Phys. Rev. D 91 no. 10, (2015) 104026, arXiv:1504.06606 [gr-qc].
  * [12] P. Landry and E. Poisson, “Tidal deformation of a slowly rotating material body. External metric,” Phys. Rev. D 91 (2015) 104018, arXiv:1503.07366 [gr-qc].
  * [13] N. Gürlebeck, “No-hair theorem for Black Holes in Astrophysical Environments,” Phys. Rev. Lett. 114 no. 15, (2015) 151102, arXiv:1503.03240 [gr-qc].
  * [14] R. A. Porto, “The effective field theorist’s approach to gravitational dynamics,” Phys. Rept. 633 (2016) 1–104, arXiv:1601.04914 [hep-th].
  * [15] E. Poisson, “Gravitomagnetic Love tensor of a slowly rotating body: post-Newtonian theory,” arXiv:2007.01678 [gr-qc].
  * [16] A. Le Tiec and M. Casals, “Spinning Black Holes Fall in Love,” arXiv:2007.00214 [gr-qc].
  * [17] A. Le Tiec, M. Casals, and E. Franzin, “Tidal Love Numbers of Kerr Black Holes,” Phys. Rev. D 103 (2021) 084021, arXiv:2010.15795 [gr-qc].
  * [18] H. S. Chia, “Tidal Deformation and Dissipation of Rotating Black Holes,” arXiv:2010.07300 [gr-qc].
  * [19] W. D. Goldberger, J. Li, and I. Z. Rothstein, “Non-conservative effects on Spinning Black Holes from World-Line Effective Field Theory,” arXiv:2012.14869 [hep-th].
  * [20] L. Hui, A. Joyce, R. Penco, L. Santoni, and A. R. Solomon, “Static response and Love numbers of Schwarzschild black holes,” JCAP 04 (2021) 052, arXiv:2010.00593 [hep-th].
  * [21] P. Charalambous, S. Dubovsky, and M. M. Ivanov, “On the Vanishing of Love Numbers for Kerr Black Holes,” arXiv:2102.08917 [hep-th].
  * [22] P. Charalambous, S. Dubovsky, and M. M. Ivanov, “Hidden Symmetry of Vanishing Love Numbers,” Phys. Rev. Lett. 127 no. 10, (2021) 101101, arXiv:2103.01234 [hep-th].
  * [23] A. A. Starobinskiǐ, “Amplification of waves during reflection from a rotating “black hole”,” Soviet Journal of Experimental and Theoretical Physics 37 (July, 1973) 28.
  * [24] L. Hui, A. Joyce, R. Penco, L. Santoni, and A. R. Solomon, “Near-zone symmetries of Kerr black holes,” JHEP 09 (2022) 049, arXiv:2203.08832 [hep-th].
  * [25] D. Anninos, S. A. Hartnoll, and D. M. Hofman, “Static Patch Solipsism: Conformal Symmetry of the de Sitter Worldline,” Class. Quant. Grav. 29 (2012) 075002, arXiv:1109.4942 [hep-th].
  * [26] A. Lopez-Ortega, “Quasinormal modes of D-dimensional de Sitter spacetime,” Gen. Rel. Grav. 38 (2006) 1565–1591, arXiv:gr-qc/0605027.
  * [27] G. S. Ng and A. Strominger, “State/Operator Correspondence in Higher-Spin dS/CFT,” Class. Quant. Grav. 30 (2013) 104002, arXiv:1204.1057 [hep-th].
  * [28] D. L. Jafferis, A. Lupsasca, V. Lysov, G. S. Ng, and A. Strominger, “Quasinormal quantization in de Sitter spacetime,” JHEP 01 (2015) 004, arXiv:1305.5523 [hep-th].
  * [29] Z. Sun, “Higher spin de Sitter quasinormal modes,” JHEP 11 (2021) 025, arXiv:2010.09684 [hep-th].
  * [30] S. Chandrasekhar and S. L. Detweiler, “The quasi-normal modes of the Schwarzschild black hole,” Proc. Roy. Soc. Lond. A 344 (1975) 441–452.
  * [31] K. D. Kokkotas and B. G. Schmidt, “Quasinormal modes of stars and black holes,” Living Rev. Rel. 2 (1999) 2, arXiv:gr-qc/9909058.
  * [32] E. Berti, “Black hole quasinormal modes: Hints of quantum gravity?,” Conf. Proc. C 0405132 (2004) 145–186, arXiv:gr-qc/0411025.
  * [33] E. Berti, V. Cardoso, and A. O. Starinets, “Quasinormal modes of black holes and black branes,” Class. Quant. Grav. 26 (2009) 163001, arXiv:0905.2975 [gr-qc].
  * [34] B. Raffaelli, “Hidden conformal symmetry on the black hole photon sphere,” JHEP 03 (2022) 125, arXiv:2112.12543 [gr-qc].
  * [35] S. Hadar, D. Kapec, A. Lupsasca, and A. Strominger, “Holography of the photon ring,” Class. Quant. Grav. 39 no. 21, (2022) 215001, arXiv:2205.05064 [gr-qc].
  * [36] D. Kapec, A. Lupsasca, and A. Strominger, “Photon Rings Around Warped Black Holes,” arXiv:2211.01674 [gr-qc].
  * [37] A. Kehagias, D. Perrone, and A. Riotto, “Quasinormal Modes and Love Numbers of Kerr Black Holes from AdS2 Black Holes,” arXiv:2211.02384 [hep-th].
  * [38] M. G. Eastwood, “Higher symmetries of the Laplacian,” Annals Math. 161 (2005) 1645–1665, arXiv:hep-th/0206233.
  * [39] Z. Sun, “A note on the representations of $\text{SO}(1,d+1)$,” arXiv:2111.04591 [hep-th].
  * [40] G. Compton and I. A. Morrison, “Hidden symmetries for transparent de Sitter space,” Class. Quant. Grav. 37 no. 12, (2020) 125001, arXiv:2003.08023 [gr-qc].
  * [41] J. D. Bekenstein, “Novel “no-scalar-hair” theorem for black holes,” Phys. Rev. D 51 no. 12, (1995) R6608.
  * [42] W. Magnus, F. Oberhettinger, and R. P. Soni, “Formulas and Theorems for the Special Functions of Mathematical Physics,” Formulas and Theorems for the Special Functions of Mathematical Physics, Springer-Verlag (1966) .
  * [43] M. A. Rakha, A. K. Rathie, and P. Chopra, “On some new contiguous relations for the Gauss hypergeometric function with applications,” Computers and Mathematics with Applications 61 (2011) 620–629.
  * [44] J. Stalker, “Complex Analysis - Fundamentals of the Classical Theory of Functions,” Complex Analysis - Fundamentals of the Classical Theory of Functions, Birkhauser (1998) .