aainstitutetext: Departamento de Física, Universidade Federal de Pernambuco,
50670-901, Recife, Pernambuco, Brazilbbinstitutetext: Van Swinderen Institute
for Particle Physics and Gravity, University of Groningen, 9747 Groningen,
Netherlandsccinstitutetext: NIKHEF, Science Park 105, 1098 XG Amsterdam,
Netherlands

# Remarks on holographic models of the Kerr-AdS5 geometry

Julián Barragán Amado<EMAIL_ADDRESS>a Bruno Carneiro da Cunha
<EMAIL_ADDRESS>b,c and Elisabetta Pallante<EMAIL_ADDRESS>
###### Abstract

We study the low-temperature limit of scalar perturbations of the Kerr-AdS5
black-hole for generic rotational parameters. We motivate the study by
considering real-time holography of small black hole backgrounds. Using the
isomonodromic technique, we show that corrections to the extremal limit can be
encoded in the monodromy parameters of the Painlevé V transcendent, whose
expansion is given in terms of irregular chiral conformal blocks. After
discussing the contribution of the intermediate states to the quasi-normal
modes, we perform a numerical analysis of the low-lying frequencies. We find
that the fundamental mode is perturbatively stable at low temperatures for
small black holes and that excited perturbations are superradiant, as expected
from thermodynamical considerations. We close by considering the holographic
interpretation of the unstable modes and the decaying process.

###### Keywords:

Black Hole Scattering, Gauge/Gravity Correspondence, Holographic Models.

##  1 Introduction

Holography has been considered a tool to study strongly-coupled phenomena in
high energy physics for some time now. In particular, the description of
strongly-coupled plasma put forward by Policastro:2001yc ; Baier:2007ix  and
Bhattacharyya:2008jc ; Bhattacharyya:2008ji ; Bhattacharyya:2008mz  laid
ground to a very precise procedure to study systems such as finite temperature
deformations of conformal field theories based on ${\cal N}=4$ SYM from
gravitational physics. It is particularly successful in dealing with the
hydrodynamical limit of such systems, and quantities like the entropy to
viscosity ratio Policastro:2001yc  and energy loss calculations Gubser:2006bz
; Liu:2006ug .

More recently, it has been stressed the necessity of modeling rotation and
vorticity into the holographic picture McInnes:2018hid ; McInnes:2020mos ,
which depends on the holographic consideration of the generic rotating, vacuum
black hole in anti-de Sitter (AdS) backgrounds – Kerr-AdS5 black hole for
short Hawking:1998kw ; Hawking:1999dp . Usually, the study relies on
thermodynamical arguments about the behavior of fluctuations in the black hole
background. For instance, in Arefeva:2020jvo  the authors analyzed the
propagation of a string in this background to estimate energy losses in a
rotating fluid. Hydrodynamical features and the effect of rotation in large
five-dimensional Anti-de Sitter black holes were also studied in
Garbiso:2020puw .

For a given field theory, the holographic description is based on the Ward
identities associated to the conformal group currents deHaro:2000vlm . The
departure from a given ultraviolet fixed point in the renormalization group
flow is implemented by the (inwards) radial evolution on the currents using
Einstein equations from the asymptotic boundary AdS structure. In this paper,
we study the validity of the description for generic rotation parameters at
low temperatures. The motivation for this is threefold.

First and foremost, the usefulness of holography to study the infrared (IR)
behavior of the associated field theories depends on the gravitational
perturbations at low temperatures. As we will revisit below, large black holes
have a lower bound in the temperature, so, this study will focus on small
black holes in the low temperature regime. At high temperatures, these black
holes were studied by using the hydrodynamical approach in
Bhattacharyya:2008ji , which, among other things, recovers (flat space)
Navier-Stokes equations by assuming that the mean free path of the components
of the fluid is small when compared to the AdS scale. This hints at the fact
that the dual field theory is in the strong coupling regime. This construction
is at odds with holography, which starts from the assumption that the field
theory has small corrections to the ultraviolet (UV) free or weakly
interacting conformal point. More specifically, the available examples of
holography usually come from theories that are weakly coupled at the UV, and
become strongly coupled at the IR. We will make an unwarranted – although not
unprecedented – assumption that the description based on conformal currents is
still valid in the latter regime. We also remark that it is an open question
whether this generic view of holography can be applied to the particular case
of QCD-like asymptotically free theories with a mass gap. In the best case
scenario, the holographic interpretation in the rest of this paper will refer
to generic asymptotically conformal systems, not necessarily subject to the
analysis in Bochicchio:2017sgq .

Secondly, a more comprehensive study of real-time holography Skenderis:2008dg
for the Kerr-AdS5 black hole at low temperatures is desirable. Rotation mixes
time and angular coordinates, and induces a “non-flat” asymptotic structure of
the black hole background. The first fact poses problems for the analytic
continuation of quantities computed in Euclidean signature and the second
gives rise to trace anomalies which are heavily dependent on the particular
field theory dual considered. We note however, that in a “classical limit”
where the AdS radius is much larger than the Planck length, the quantities
computed at different asymptotic structures, in our case essentially the one
arising from global AdS $\mathbb{R}\times S^{3}$ and the one arising from the
Poincaré patch $\mathbb{R}^{1,3}$ are relatable by a conformal transformation.
Also, the repercussions of the AdS instabilities found in Green:2015kur , of
which the scalar field linear unstable modes found by the authors in Barragan-
Amado:2018pxh  are a particular example, are expected to be of fundamental
relevance to the understanding of holography, and the study presented here
seemed like a natural step further.

The third reason is somewhat technical. As argued in Hijano:2015zsa , black
hole perturbations should be described by a particular limit of the four-
dimensional conformal block Dolan:2003hv . As argued in Amado:2017kao  (see
also daCunha:2016crm ), the pertinent perturbation can be computed using two-
dimensional chiral conformal blocks. Semi-classical aspects of the latter
which are relevant to the discussion were anticipated in Litvinov:2013sxa ,
and a full fledged solution was presented in the seminal work of
Gamayun:2013auu , also uncovering a relation between semiclassical and $c=1$
conformal blocks, both related to the Painlevé VI tau function.

The low temperature black hole considered here necessitates a careful
consideration of the confluence limit, which is given in terms of the Painlevé
V tau function Lisovyy:2018mnj . Albeit in this paper we are focused on the
study of the quasi-normal modes (QNMs) for small black holes $r_{+}\ll 1$, the
resulting relation holds for generic five-dimensional Kerr black holes in the
near-extremal limit. On the one hand, this is to our knowledge a new
application of the Painlevé V transcendent – see also daCunha:2015ana ;
CarneirodaCunha:2019tia 111See Novaes:2018fry  for the parallel story in four-
dimensional Kerr-de Sitter black holes, using the Painlevé VI transcendent..
On the other hand, the appearance of the two-dimensional conformal blocks may
be seen from one point of view as a calculational tool, but, as we will argue,
the connection between these and four-dimensional blocks merits further
investigation.

While the writing of this work was in course, Aminov:2020yma  came out with a
similar study of (four-dimensional) quasi-normal modes of black holes using
the properties of conformal blocks. The authors of Aminov:2020yma  employ the
usual relation between Fuchsian equations and semiclassical conformal blocks
as the starting point, and their relation to Seiberg-Witten theory
Alday:2009aq  has been once more studied in Nekrasov:2020qcq . We note,
however, that there are advantages and drawbacks to each approach. The
expansion in terms of semiclassical conformal blocks is more direct and leads
directly to an asymptotic expansion of the accessory parameter (called $K_{0}$
below) of the associated differential equation (44). The $c=1$ expansion used
here implements an explicit solution to the Riemann-Hilbert problem of finding
both accessory parameters $t_{0}$ and $K_{0}$ in terms of monodromy data. As
we saw in an earlier paper Barragan-Amado:2018pxh , an asymptotic expansion
for the quasi-normal modes in the five-dimensional Kerr-AdS black hole is more
effectively obtained using $c=1$ conformal blocks.

With all this in mind, we have structured the paper as follows. In Sec. 2 we
review the parameters of the Kerr-AdS5 black hole, as well as its asymptotic
structure and conserved charges – mass and angular momenta. We spend some time
analyzing the holographic interpretation of the asymptotic structure, in
particular the interplay between the $\mathbb{R}\times S^{3}$ (global
structure) and $\mathbb{R}^{1,3}$ (Poincaré patch), in a “near the
ultraviolet” description which allows us to identify the state in the putative
field theory corresponding to the black hole and to digress about the fate of
perturbations within the scope of linear analysis.

Sec. 3 comprises the bulk of the results. We consider scalar perturbations of
the Kerr-AdS5 black hole and show that the low temperature limit of the QNMs
is given by zeros of the Painlevé V tau function. Using asymptotic expansions
of the Painlevé V transcendent for small parameter, we compute the QNMs
frequencies for small black holes. We then turn to discuss the CFT description
of the perturbations, and find that the contributions to the fundamental QNM
are indeed given by the vacuum conformal block, at least for small black
holes. The higher modes are correspondingly given by special blocks with
internal dimensions given by degenerate “heavy” operators, in a chiral version
of the analysis in Fitzpatrick:2015foa . We use the expansions of the Painlevé
tau function and the accessory parameter to derive the expansion of the
fundamental QNM frequency for small black holes, including the first
correction for non-zero temperature and we compare the asymptotic expression
with the numerical result. Higher modes are studied numerically in the same
regime, and we find indeed the unstable modes of Hawking:2000vt , provided the
superradiance condition is met.

In Sec. 4 we discuss the holographic consequences of the analysis, focussing
on the unstable modes. We see that the naive extrapolation of the near-
ultraviolet result, pointing to the irrelevance of the perturbation, is
misguided, and that the instabilities arising from the same perturbations as
treated in Sec. 3 may change considerably the fate of the state as one
approaches the infrared. We also find that the spatial dependence of the
decaying modes may be confined to small angles as the mass of the state
increases. We close in Sec. 5 with some remarks on the results and their
interpretation.

There are two Appendices. Appendix A gives a self-contained derivation of the
conserved charges for pure gravity in asymptotically AdS spaces and emphasizes
the role of the conformal structure, as a special subcase of Hollands:2005wt .
In Appendix B we review the Fredholm determinant formulation of the Painlevé
VI transcendent derived from the generic isomonodromic tau function proposed
in Gavrylenko:2016zlf , and its dependence on the relevant monodromy
parameters is made explicit.

##  2 The Kerr-AdS5 black hole

The Kerr-${\rm AdS_{5}}$ black hole is represented by the metric
Hawking:1998kw

$\displaystyle ds^{2}=$
$\displaystyle-\dfrac{\Delta_{r}}{\rho^{2}}\left(dt-\dfrac{a_{1}\sin^{2}\theta}{1-a_{1}^{2}}d\phi_{1}-\dfrac{a_{2}\cos^{2}\theta}{1-a_{2}^{2}}d\phi_{2}\right)^{2}+\dfrac{\Delta_{\theta}\sin^{2}\theta}{\rho^{2}}\left(a_{1}\,dt-\dfrac{(r^{2}+a_{1}^{2})}{1-a_{1}^{2}}d\phi_{1}\right)^{2}$
$\displaystyle+\dfrac{1+r^{2}}{r^{2}\rho^{2}}\left(a_{1}a_{2}\,dt-\dfrac{a_{2}(r^{2}+a_{1}^{2})\sin^{2}\theta}{1-a_{1}^{2}}d\phi_{1}-\dfrac{a_{1}(r^{2}+a_{2}^{2})\cos^{2}\theta}{1-a_{2}^{2}}d\phi_{2}\right)^{2}$
$\displaystyle+\dfrac{\Delta_{\theta}\cos^{2}\theta}{\rho^{2}}\left(a_{2}\,dt-\dfrac{(r^{2}+a_{2}^{2})}{1-a_{2}^{2}}d\phi_{2}\right)^{2}+\dfrac{\rho^{2}}{\Delta_{r}}dr^{2}+\dfrac{\rho^{2}}{\Delta_{\theta}}d\theta^{2},$
(1)

where we set the AdS radius to one and

$\displaystyle\Delta_{r}=\dfrac{1}{r^{2}}(r^{2}+a_{1}^{2})(r^{2}+a_{2}^{2})(1+r^{2})-2M,$
$\displaystyle\Delta_{\theta}=1-a_{1}^{2}\cos^{2}\theta-
a_{2}^{2}\sin^{2}\theta,\quad\quad\
\rho^{2}=r^{2}+a_{1}^{2}\cos^{2}\theta+a_{2}^{2}\sin^{2}\theta,$ (2)

and $a_{1}$ and $a_{2}$ are two independent rotation parameters. The mass
relative to the ${\rm AdS}_{5}$ vacuum solution and the angular momenta were
the subject of some discussion Awad:2000aj ; Gibbons:2004ai ; Hollands:2005wt
; Olea:2006vd . To add to the confusion, we include our own derivation in
Appendix A. The values obtained using our procedure match those of
Gibbons:2004ai

$\displaystyle{\cal M}=\frac{\pi
M(2(1-a_{1}^{2})+2(1-a_{2}^{2})-(1-a_{1}^{2})(1-a_{2}^{2}))}{4(1-a_{1}^{2})^{2}(1-a_{2}^{2})^{2}},$
(3) $\displaystyle{\cal J}_{1}=\frac{\pi
M}{2(1-a_{2}^{2})}\frac{a_{1}}{(1-a_{1}^{2})^{2}},\quad\quad{\cal
J}_{2}=\frac{\pi M}{2(1-a_{1}^{2})}\frac{a_{2}}{(1-a_{2}^{2})^{2}}.$ (4)

In the following it will be useful to redefine the black hole parameters in
terms of the roots of $\Delta_{r}$

$\Delta_{r}=\frac{1}{r^{2}}(r^{2}-r_{0}^{2})(r^{2}-r_{-}^{2})(r^{2}-r_{+}^{2})$
(5)

which, for a regular, finite charged, dressed horizon black hole satisfy
$r_{0}^{2}<-1<0<r_{-}^{2}<r_{+}^{2}$. Each non-zero root $r_{i}$ can be
associated to a Killing horizon, although only those of $r_{-}$ and $r_{+}$
are physical. To each horizon we can associate the temperature $T_{k}$ and
angular velocities $\Omega_{1,k}$ and $\Omega_{2,k}$, $k=0,+,-$, given by

$\displaystyle\Omega_{1,k}=\dfrac{a_{1}(1-a_{1}^{2})}{r^{2}_{k}+a_{1}^{2}}\qquad\Omega_{2,k}=\dfrac{a_{2}(1-a_{2}^{2})}{r^{2}_{k}+a_{2}^{2}}$
$\displaystyle
T_{k}=\dfrac{r^{2}_{k}\Delta_{r}^{\prime}(r_{k})}{4\pi(r^{2}_{k}+a_{1}^{2})(r^{2}_{k}+a_{2}^{2})}=\frac{r_{k}}{2\pi}\frac{(r_{k}^{2}-r_{i}^{2})(r_{k}^{2}-r_{j}^{2})}{(r_{k}^{2}+a_{1}^{2})(r_{k}^{2}+a_{2}^{2})},$
(6)

where the prime stands for the derivative with respect to $r$. We note than
$T_{+}>0$, $T_{-}<0$ and $T_{0}$ is purely imaginary.

###  2.1 Asymptotic structure

The asymptotic structure of the metric as $r\rightarrow\infty$ is involved for
non-zero rotation parameters $a_{1}$ and $a_{2}$. By the suitable change of
coordinates

$\displaystyle(1-a_{1}^{2})\bar{r}^{2}\sin^{2}\bar{\theta}=(r^{2}+a_{1}^{2})\sin^{2}\theta,$
(7)
$\displaystyle(1-a_{2}^{2})\bar{r}^{2}\cos^{2}\bar{\theta}=(r^{2}+a_{2}^{2})\cos^{2}\theta,$
(8)
$\displaystyle\bar{t}=t,\quad\quad\bar{\phi}_{1}=\phi_{1}+a_{1}t,\quad\quad\bar{\phi}_{2}=\phi_{2}+a_{2}t,$
(9)

we arrive, after some manipulation, at the useful formulas

$\displaystyle 1+\bar{r}^{2}=\frac{1-a_{1}^{2}\cos^{2}\theta-
a_{2}^{2}\sin^{2}\theta}{(1-a_{1}^{2})(1-a_{2}^{2})}(1+r^{2}),$ (10)
$\displaystyle\bar{r}^{2}(1-a_{1}^{2}\sin^{2}\bar{\theta}-a_{2}^{2}\cos^{2}\bar{\theta})=r^{2}+a_{1}^{2}\sin^{2}\theta+a_{2}^{2}\cos^{2}\theta,$
(11)
$\displaystyle\frac{d\bar{r}^{2}}{1+\bar{r}^{2}}+\bar{r}^{2}d\bar{\theta}^{2}=\rho^{2}\left(\frac{dr^{2}}{\Delta_{r}}+\frac{d\theta^{2}}{\Delta_{\theta}}\right),$
(12)

so that, for large $\bar{r}$, the metric becomes

$ds^{2}=-(1+\bar{r}^{2})d\bar{t}^{2}+\frac{d\bar{r}^{2}}{1+\bar{r}^{2}}+\bar{r}^{2}(d\bar{\theta}^{2}+\sin^{2}\bar{\theta}\,d\bar{\phi}_{1}^{2}+\cos^{2}\bar{\theta}\,d\bar{\phi}_{2}^{2})\\\
+\frac{2M}{\bar{\Delta}_{\theta}^{2}\bar{r}^{2}}\left(\frac{d\bar{r}^{2}}{\bar{r}^{4}}+\frac{1}{\bar{\Delta}_{\theta}}(d\bar{t}-a_{1}\sin^{2}\bar{\theta}\,d\bar{\phi}_{1}-a_{2}\cos^{2}\bar{\theta}\,d\bar{\phi}_{2})^{2}\right)+\ldots$
(13)

where the ellipsis denotes subleading terms in $\bar{r}$ – see the Appendix in
Arefeva:2020jvo , and

$\bar{\Delta}_{\theta}=1-a_{1}^{2}\sin^{2}\bar{\theta}-a_{2}^{2}\cos^{2}\bar{\theta}.$
(14)

In the first line of (13) we recognize the metric of vacuum global AdS.

In order to apply the procedure in Appendix A to compute the mass and angular
momenta we take $\tilde{\Omega}=\Omega=1/\sqrt{1+\bar{r}^{2}}$, and write (13)
as follows

$ds^{2}=\frac{1}{\Omega^{2}}\left(-d\bar{t}^{2}+\frac{d\Omega^{2}}{1-\Omega^{2}}+(1-\Omega^{2})d\Omega_{3}\right)+\\\
2M\frac{\Omega^{2}}{\bar{\Delta}_{\theta}^{2}}\left(d\Omega^{2}+\frac{1}{\bar{\Delta}_{\theta}}(d\bar{t}-a_{1}\sin^{2}\bar{\theta}\,d\bar{\phi}_{1}-a_{2}\cos^{2}\bar{\theta}\,d\bar{\phi}_{2})^{2}\right)+\ldots,$
(15)

where
$d\Omega_{3}=d\bar{\theta}^{2}+\sin^{2}\bar{\theta}\,d\bar{\phi}_{1}^{2}+\cos^{2}\bar{\theta}\,d\bar{\phi}_{2}^{2}$
is the usual metric of the 3-sphere. Again, the ellipsis at the end of (15)
refers to subleading terms, now of order ${\cal O}(\Omega^{3})$. Like in
Fefferman-Graham development Fefferman:1985aa , the structure of the expansion
can be explained by the equations of motion. The induced metric $\bar{g}_{ab}$
at the conformal infinity $\Omega=0$ is given by the first line of (15),
without the $\Omega^{-2}$ factor. The $\Omega$ dependence of this term fixes,
up to order $\Omega^{4}$, the conformal structure at infinity – see (136). The
term in the second line of (15) defines the asymptotic charges and thus
$h_{ab}=\Omega^{2}\gamma_{ab}$. Its form is explained in terms of
$n^{a}=\bar{g}^{ab}\nabla_{b}\Omega$, the unit normal to the boundary, and the
vector field

$\chi^{a}=\frac{\partial}{\partial\bar{t}}+a_{1}\frac{\partial}{\partial\bar{\phi}_{1}}+a_{2}\frac{\partial}{\partial\bar{\phi}_{2}}=\frac{\partial}{\partial
t}.$ (16)

This vector field generates an isometry of $\bar{g}_{ab}$ and of the AdS
metric. It is actually an isometry of the Kerr-AdS metric, and the Killing
vector becomes null at the horizons $r=r_{0},r_{-},r_{+}$. In terms of the
induced metric $\bar{g}_{ab}$, one has

$\bar{g}_{ab}\chi^{a}\chi^{b}=-1+(1-\Omega^{2})(a_{1}^{2}\sin^{2}\bar{\theta}+a_{2}^{2}\cos^{2}\bar{\theta})=-\bar{\Delta}_{\theta}+{\cal
O}(\Omega^{2}).$ (17)

We can thus write

$h_{ab}=2M\frac{\Omega^{2}}{\bar{\Delta}_{\theta}^{2}}(n_{a}n_{b}+\frac{1}{\bar{\Delta}_{\theta}}\bar{\chi}_{a}\bar{\chi}_{b})+{\cal
O}(\Omega^{3})$ (18)

with $\bar{\chi}_{a}=\bar{g}_{ab}\chi^{b}$. The leading contribution to the
metric correction $h_{ab}$ is actually invariant under the conformal structure
chosen. Indeed, if we take $\Omega^{\prime}=\upomega\Omega$, with
$\upomega\neq 0$, then the quantities above transform as

$n^{\prime}_{a}=\upomega
n_{a}+\Omega\nabla_{a}\omega,\quad\quad\bar{g}^{\prime}_{ab}=\upomega^{2}\bar{g}_{ab},\quad\quad\bar{\chi}^{\prime}_{a}=\upomega^{2}\bar{\chi}_{a},\quad\quad\bar{\Delta}^{\prime}=\upomega^{2}\bar{\Delta}_{\theta},$
(19)

and it can be checked that $h^{\prime}_{ab}=h_{ab}$ in (18), up to terms of
order $\Omega^{3}$. In the following, we will refer to the leading part in
(18) as just $h_{ab}$. It is traceless with respect to the pure AdS metric,
and transverse

$\nabla_{a}h^{ab}=0,$ (20)

where the covariant derivative is the one compatible with the pure AdS metric.

The true Fefferman-Graham parameter $\varrho$, defined in Fefferman:1985aa ,
can be computed as an expansion in $\Omega$:

$\varrho=\Omega+\frac{1}{4}\Omega^{3}+\frac{1}{4}\left(\frac{1}{2}+\frac{M}{\bar{\Delta}_{\theta}^{2}}\right)\Omega^{5}+{\cal
O}(\Omega^{7}),$ (21)

not to be confused with $\rho$ in (2). We have the following structure for the
metric in terms of $\varrho$, following the procedure in deHaro:2000vlm ,

$ds^{2}=\frac{d\varrho^{2}}{\varrho^{2}}+\frac{1}{\varrho^{2}}\left(-d\bar{t}^{2}+d\Omega_{3}\right)+\frac{1}{2}\left(-d\bar{t}^{2}-d\Omega_{3}\right)\\\
+\varrho^{2}\left(\left(\frac{1}{16}+\frac{M}{2\bar{\Delta}_{\theta}^{2}}\right)(-d\bar{t}^{2}+d\Omega_{3})+\frac{2M}{\bar{\Delta}_{\theta}^{3}}(d\bar{t}-a_{1}^{2}\sin^{2}\bar{\theta}\,d\bar{\phi}_{1}-a_{2}^{2}\cos^{2}\bar{\theta}\,d\bar{\phi}_{2})^{2}\right)+\ldots,$
(22)

again up to ${\cal O}(\varrho^{3})$ terms. We recognize the induced metric
$\bar{g}_{ab}$, corresponding to the induced line element
$-d\bar{t}^{2}+d\Omega_{3}$. The particular conformal structure of
$\mathbb{R}\times S^{3}$ is not invariant under the transformation
$\Omega\rightarrow\upomega\Omega$. By a suitable choice of $\upomega$ (see
(30)), one could change the induced metric to the flat Minkowski one
$\mathbb{R}^{3,1}$. In the following, we will refer to the $\mathbb{R}\times
S^{3}$ induced metric as $\bar{g}_{ab}$, or global AdS structure, and to the
flat $\mathbb{R}^{3,1}$ as $\hat{g}_{ab}$, or flat AdS structure.

###  2.2 Holography and hydrodynamical issues

In global AdS, the holographic stress-energy tensor can be read directly from
(22) above and has the expression Arefeva:2020jvo

$T^{h}_{ab}=\frac{1}{4\pi
G_{N,5}}\left(\frac{1}{16}\bar{g}_{ab}+\frac{2M}{\bar{\Delta}_{\theta}^{2}}\left(u_{(a}u_{b)}+\frac{1}{4}\bar{g}_{ab}\right)\right)$
(23)

whose $M$-dependent term satisfies the conformally invariant, perfect fluid
relation with energy density $\bar{\rho}$ and pressure $\bar{P}$

$4\pi G_{N,5}\bar{\rho}=12\pi G_{N,5}\bar{P}=3M/2\bar{\Delta}_{\theta}^{2},$
(24)

hence $\bar{P}=\bar{\rho}/3$, and the holographic stress-energy tensor is
traceless. The normalized fluid velocity is
$u^{a}=\chi^{a}/\sqrt{\bar{\Delta}_{\theta}}$ and it is divergence-free
$\bar{\nabla}_{a}u^{a}=0$, where $\bar{\nabla}_{a}$ is the covariant
derivative associated to the metric $\bar{g}_{ab}$. Despite depicting a
classical energy profile, and being dependent on the choice of conformal
structure, (23) is by construction a valid approximation to the stress-energy
tensor of the state in the field theory in the UV limit, which holographically
corresponds to $\bar{r}\rightarrow\infty$.

The acceleration of $u^{a}$ is

$u^{a}\bar{\nabla}_{a}u^{b}=-\frac{a_{1}^{2}-a_{2}^{2}}{\bar{\Delta}_{\theta}}\sin\bar{\theta}\cos\bar{\theta}\,\frac{\partial}{\partial\bar{\theta}}=-\frac{1}{4\rho}\bar{\nabla}^{b}\rho=\frac{1}{2\bar{\Delta}_{\theta}}\bar{\nabla}^{b}\bar{\Delta}_{\theta},$
(25)

and it follows that $T^{h}_{ab}$ in (23) is conserved with respect to the
metric $\bar{g}_{ab}$, i.e. $\bar{\nabla}^{a}T^{h}_{ab}=0$. The term
independent of $M$ in (23) corresponds to the four-dimensional gravitational
conformal anomaly deHaro:2000vlm , and it can be disregarded for large enough
$M$. The total energy associated to the fluid is

$4\pi G_{N,5}E=4\pi
G_{N,5}\int_{\mathrm{S}^{3}}d\Omega_{3}\,\bar{\rho}=\frac{3\pi^{2}M}{(1-a_{1}^{2})(1-a_{2}^{2})},$
(26)

which can be compared with the total ADM mass ${\cal M}$ in (4).

The perfect fluid form of the holographic stress-energy tensor for the Kerr-
AdS black hole in (23) mirrors the result for four-dimensional backgrounds
from Cardoso:2013pza , and is in fact expected on general grounds. The
conformally invariant nature of (23) hints at an underlying conformally
invariant field theory dual. Following the holographic renormalization group
flow towards the IR, the corrections to the metric will in general break this
conformal invariance, which is interpreted as the departure from a conformal
fixed point in the field theory. If we think of the field theory as free or
weakly coupled at the UV, the flow towards the IR may bring the theory to a
strongly coupled regime, resulting in, among other things, corrections to the
perfect fluid form of its stress-energy tensor (23). In the spirit of
holographic renormalization group flow, we will assume that the radial
evolution given by Einstein equations will give the proper picture for the
dual field theory as the energy scale is brought down.

For high-temperatures, one can interpret the results of Bhattacharyya:2007vs
and Bhattacharyya:2008ji  as the equivalence between the general relativity
equations and the Navier-Stokes equations for the fluid mechanics. In our
study, we want to consider low temperatures, outside the scope of their
analysis. This necessarily leads us to consider small black holes, because
large black holes in AdS are hot. To see this latter point, let us recall
that, when written in terms of $r_{+}$, $a_{1}$ and $a_{2}$, the temperature
of the black hole outer horizon (6) is

$2\pi
T_{+}=r_{+}\left(\frac{r_{+}^{2}+1}{r_{+}^{2}+a_{1}^{2}}+\frac{r_{+}^{2}+1}{r_{+}^{2}+a_{2}^{2}}-\frac{1}{r_{+}^{2}}\right),$
(27)

which, for constant $r_{+}$, is a monotonically decreasing function of $a_{1}$
and $a_{2}$. However, for $a_{1},a_{2}<1$, $T_{+}$ is always larger than
$T_{\mathrm{min}}=(2r_{+}^{2}-1)/(2\pi r_{+})$. We can therefore conclude that
there cannot be a zero-temperature black hole for $r_{+}^{2}>1/2$.

There are basically two ways of dealing with this issue. One is to forgo our
misgivings and simply take the calculations we will be performing as
definitions of the field theory system in the infrared regime for small
$T_{+}$. The other is to view the correspondence between the gravitational and
the field theoretical system as dynamical, and give up on the equilibrium view
of the fluid, from which the equivalence between mean free path and $r_{+}$
was deduced. Either choice will have interesting consequences.

Dwelling in the second choice for a moment, if the fluid is no longer in
equilibrium, there is no reason to restrict the analysis to time-independent
configurations. The conformal transformation to conformally flat boundary
coordinates mixes time and space and induces time-dependence for the dynamics
of the state. Using the (classical) conformal symmetry of the fluid, let us
then perform the transformation of $\bar{g}_{ab}$ to conformally flat
coordinates. The transformation

$\hat{t}=\frac{1}{2}\tan\frac{\bar{t}+\bar{\chi}}{2}+\frac{1}{2}\tan\frac{\bar{t}-\bar{\chi}}{2},\qquad\hat{r}=\frac{1}{2}\tan\frac{\bar{t}+\bar{\chi}}{2}-\frac{1}{2}\tan\frac{\bar{t}-\bar{\chi}}{2},$
(28)

where

$\cos\bar{\chi}=\sin\bar{\theta}\cos\bar{\phi}_{1},\qquad\tan\hat{\theta}=\frac{\cos\bar{\theta}}{\sin\bar{\theta}\sin\bar{\phi}_{1}}$
(29)

and $\hat{\phi}=\bar{\phi}_{2}$, turns the metric into

$d\hat{s}^{2}=\frac{-d\hat{t}^{2}+d\hat{r}^{2}+\hat{r}^{2}(d\hat{\theta}^{2}+\sin^{2}\hat{\theta}\,d\hat{\phi}^{2})}{(1+\frac{1}{4}(\hat{t}+\hat{r})^{2})(1+\frac{1}{4}(\hat{t}-\hat{r})^{2})}.$
(30)

We recall that we use bars to refer to coordinates in the $\mathbb{R}\times
S^{3}$ manifold ($\\{\bar{t},\bar{\theta},\bar{\phi}_{1},\bar{\phi}_{2}\\}$)
and hatted coordinates for Minkowski space $\mathbb{R}^{1,3}$
($\\{\hat{t},\hat{r},\hat{\theta},\hat{\phi}\\}$). The flat metric will be
henceforth referred to as $\hat{g}_{ab}$.

We can use the conformal symmetry of the leading order perturbation $h_{ab}$
in (19) to derive the Poincaré coordinates, flat AdS version of the
holographic stress-energy tensor, in the same approximation as (23). The
perturbation is explicitly given by the same formula as (18)

$h_{ab}=2M\frac{z^{2}}{\hat{\Delta}^{2}}(n_{a}n_{b}+\frac{1}{\hat{\Delta}}\hat{\chi}_{a}\hat{\chi}_{b})$
(31)

where $n_{a}=(dz)_{a}$, $\hat{\chi}_{a}=\hat{g}_{ab}\chi^{b}$, and

$\hat{\Delta}=-\hat{g}_{ab}\chi^{a}\chi^{b}\\\
=(1+\tfrac{1}{4}(\hat{t}+\hat{r})^{2})(1+\tfrac{1}{4}(\hat{t}-\hat{r})^{2})-(1+\tfrac{1}{4}(\hat{t}^{2}-\hat{r}^{2}))^{2}a_{1}^{2}-\hat{r}^{2}(a_{1}^{2}\cos^{2}\hat{\theta}+a_{2}^{2}\sin^{2}\hat{\theta}).$
(32)

The flat-space version of the stress-energy tensor
$\hat{T}^{h}_{ab}=\upomega^{-2}T^{h}_{ab}$, with

$\upomega^{2}=(1+\tfrac{1}{4}(\hat{t}+\hat{r})^{2})(1+\tfrac{1}{4}(\hat{t}-\hat{r})^{2})$
(33)

is conserved with respect to the flat metric $\hat{g}_{ab}$ (see Appendix D in
Wald:1984 ), and it reads

$\hat{T}^{h}_{ab}=\hat{\rho}\hat{u}_{a}\hat{u}_{b}+\hat{P}(\hat{g}_{ab}+\hat{u}_{a}\hat{u}_{b}),\qquad\hat{\rho}=3\hat{P}=\frac{1}{4\pi
G_{N,5}}\frac{3M}{2\hat{\Delta}^{2}},$ (34)

and $\hat{u}^{a}=\chi^{a}/\sqrt{\hat{\Delta}}$ is the normalized velocity of
the fluid with respect to $\hat{g}_{ab}$.

Figure 1: Three snapshots of the energy density profile (34), as sagittal
projection. The free propagation shows two “blobs” of radiation traveling
along the $z$ axis and meeting close to the origin as $t\rightarrow 0$. The
two parameters $a_{1}$ and $a_{2}$ control the spread of each blob away from
the symmetry axis and the astigmatism as the two blobs meet.

In order to study the near ultraviolet behavior in the presence of the
perturbation, we turn to the effect of the asymptotic form of the correction
$h_{ab}$. The pure AdS metric encodes the conformal structure – at least of
its conformal part. One can then interpret the effect of $h_{ab}$ as encoding
the departure from the ultraviolet fixed point as one turns on the
renormalization group flow towards the infrared.

One should note at this point that there is an ambiguity in choosing
$\hat{\phi}=\bar{\phi}_{2}$ in passing from the $\mathbb{R}\times S^{3}$
conformal structure to Minkowski space $\mathbb{R}^{3,1}$. The Kerr-AdS black
hole background can be interpreted as the resulting CFT (thermal) state
obtained by turning expectation values for the operators comprising the
generators of the Cartan subalgebra of the conformal group $\mathrm{SO}(4,2)$

$\frac{\partial}{\partial\bar{t}}\sim
M_{01},\qquad\frac{\partial}{\partial\bar{\phi}_{1}}\sim
M_{23},\qquad\frac{\partial}{\partial\bar{\phi}_{2}}\sim M_{45}$ (35)

where the right-hand side makes use of an explicit representation of the
conformal group as six-dimensional “Lorentz” generators of $\mathbb{R}^{2,4}$.
In choosing $\hat{\phi}=\bar{\phi}_{2}$, we explicitly chose $M_{45}$ as the
angular momentum generator in the four-dimensional picture. One could as well
choose $M_{23}$, which would amount to setting $\hat{\phi}=\bar{\phi}_{1}$.
The effect of this choice would be simply to interchange $a_{1}$ with $a_{2}$
in quantities such as (32) and (34). We will come back to this point in Sec.
4.

A comparatively simple and preliminary calculation is to study the effect on
the conformal dimension $\Delta$, as read from the two-point function of two
scalar primary operators as we depart from the conformal point in the
ultraviolet. In general, one expects a number of these primary operators to
share the same conformal dimension, and the presence of the background does
not lift this degeneracy. As it is well-known, the primary operators satisfy a
conformal Casimir Ward identity, which can be seen as the scalar wave equation
in AdS. To give the calculation a purpose independent of the background
conformal structure chosen at infinity, we will introduce the
$\mathrm{SO}(4,2)$ invariant quantity $\sigma$:

$\sigma=\frac{z_{i}^{2}+z_{j}^{2}+|{\bf x}_{i}-{\bf
x}_{j}|^{2}}{2z_{i}z_{j}},$ (36)

which is related to the invariant $\mathrm{AdS}$ geodesic distance $\ell$ by
$\sigma=\cosh\ell$. As it is well-known, the pure AdS scalar propagator

$G(\sigma)=\langle\Phi(z_{i},{\bf x}_{i})\Phi(z_{j},{\bf
x}_{j})\rangle=2^{\Delta}K\sigma^{-\Delta}{{}_{2}F_{1}}(\tfrac{1}{2}\Delta,\tfrac{1}{2}\Delta+\tfrac{1}{2};\Delta-1;\sigma^{-2}),$
(37)

reproduces the conformal two-point function for two local primary operators
with conformal dimension $\Delta$ sitting at the same value of the cut-off
parameter $z=z_{i}=z_{j}$. As one considers large values of $z$, however, the
simple CFT expression $|{\bf x}_{i}-{\bf x}_{j}|^{-2\Delta}$ no longer holds,
since keeping the coordinate $z$ fixed (and large) is akin to keeping fixed
the length of the flux tube between the insertions Polyakov:2000jg , which is
not usually achieved in QFT calculations. At any rate, as seen in
Ferrara:1973vz  – see also Amado:2017kao , this dependence can be trivialized
for pure AdS by a suitable choice of the $z$ coordinate.

The correction $h_{ab}$ to the pure AdS metric $g_{ab}$ at spatial infinity
modifies this dependence (indices are raised with the vacuum metric):

$\bar{\nabla}^{2}G=\nabla^{2}G-{g}^{ab}\,{g}^{cd}({\nabla}_{a}h_{db}-{\nabla}_{d}h_{ab}){\nabla}_{c}G-h^{ab}\,{\nabla}_{a}\,{\nabla}_{b}G=\Delta(\Delta-4)G,$
(38)

where ${g}_{ab}$ and ${\nabla}_{a}$ are the vacuum AdS metric and covariant
derivative, and we are assuming non-coincident insertion points. The last term
$\Delta(\Delta-4)$ is equal to the AdS mass squared, which is equal to the
$\mathrm{SO}(4,2)$ quadratic Casimir. Continuing with the calculation of (38),
we have, due to the tracelessness of $h_{ab}$,

$g^{ab}g^{cd}(\nabla_{a}h_{db}-\nabla_{d}h_{ab})=\nabla_{a}h^{ac},$ (39)

where indices are raised with the pure $\mathrm{AdS}_{5}$ metric $g^{ab}$. By
means of the formulas (123) and (125), with the choice of $\tilde{\Omega}=z$
so that $\tilde{g}_{ab}=\hat{g}_{ab}$ in Appendix A, we relate this divergence
to

$\nabla_{a}h^{ac}=z^{4}\left(\hat{\nabla}_{a}\hat{h}^{ac}-\frac{4Mz}{\hat{\Delta}^{2}}\hat{n}^{c}\right)=0,$
(40)

which vanishes owing to the tracelessness of $h_{ab}$ and the relation between
the acceleration of $u^{a}$ and the gradient of $\hat{\Delta}$. The term
$h^{ab}\nabla_{a}\nabla_{b}G$ in (38) gives less trouble, and the resulting
equation is

$\left(\sigma^{2}-1-z^{4}\hat{h}^{ab}\nabla_{a}\sigma\nabla_{b}\sigma\right)\frac{d^{2}G}{d\sigma^{2}}+5\sigma\frac{dG}{d\sigma}=\Delta(\Delta-4)G,$
(41)

where $\hat{h}^{ab}=\hat{g}^{ac}h_{cd}\hat{g}^{db}$ has its indices raised by
the flat metric $\hat{g}_{ab}$. Using the formula for $\sigma$ given in terms
of the flat coordinates and $z$,

$z^{4}\hat{h}^{ab}\nabla_{a}\sigma\nabla_{b}\sigma=\frac{2M}{\hat{\Delta}^{2}(\mathbf{x}_{i})}|\mathbf{x}_{ij}|^{4}+{\cal
O}(z^{2},|\mathbf{x}_{ij}|^{6}),$ (42)

the simple Ansatz $G\propto\sigma^{-\Delta^{\prime}}$ then captures the
asymptotic behavior of the solution, with

$\displaystyle\Delta^{\prime}$
$\displaystyle=\Delta+\frac{\Delta(\Delta+1)}{2(\Delta-2)}\left(\frac{z^{4}}{|\mathbf{x}_{ij}|^{4}}+\frac{2Mz^{4}}{\hat{\Delta}^{2}(\mathbf{x}_{i})}\right)+\ldots$
(43) $\displaystyle=\Delta_{\mathrm{AdS}}\left(1+\frac{1}{4\pi
G_{N,5}}\frac{2(\Delta+1)}{3(\Delta-2)}\hat{\rho}z^{4}\right)+\ldots$

where we can read the anomalous dimension as the correction to the pure AdS
value $\Delta_{\mathrm{AdS}}$, defined by the expansion of the vacuum
propagator (37), that is, in the absence of the black hole.

Note that the effect of the background, at least perturbatively, is to
increase the conformal dimension of the perturbation for $\Delta>2$, which
applies in particular to the marginal case $\Delta=4$. For $z$ infinitesimally
close to zero and $\Delta$ sufficiently close to $4$, (43) tells us that, as
the correction to the AdS value $\Delta_{\mathrm{AdS}}$ increases,
$\Delta^{\prime}$ crosses the marginal value of $4$ and the corresponding
perturbation becomes irrelevant, and as such should not alter substantially
the fate of the flow as one approaches the infrared limit. On the other hand,
the fact that corrections to the bare AdS value do become large at $z^{4}\sim
M^{-1}$ can be seen as hint that the theory is becoming increasingly strongly
coupled towards the IR. As we will see in the following, the remark about the
fluctuations not substantially altering the fate of the state as it flows will
have to be revised.

##  3 Scalar perturbations

A complete holographic interpretation of a scalar perturbation requires the
analysis of the field in the exact Kerr-AdS5 background, which we will
consider in the low temperature regime. Let us start with generic
considerations on the scalar field. The Klein-Gordon equation in Kerr-AdS5 is
separable in terms of two (Fuchsian) ordinary differential equations, which
can be cast in the standard form:

$\begin{gathered}\frac{d^{2}y}{dz^{2}}+p(z)\frac{dy}{dz}+q(z)y(z)=0,\\\
p(z)=\frac{1-\theta_{0}}{z}+\frac{1-\theta_{t}}{z-t_{0}}+\frac{1-\theta_{1}}{z-1},\\\
q(z)=\frac{(\theta_{0}+\theta_{t}+\theta_{1}+\theta_{\infty}-2)(\theta_{0}+\theta_{t}+\theta_{1}-\theta_{\infty}-2)}{4z(z-1)}-\frac{t_{0}(t_{0}-1)K_{0}}{z(z-1)(z-t_{0})},\end{gathered}$
(44)

where the set $\\{\theta_{0},\theta_{t},\theta_{1},\theta_{\infty}\\}$ are
called the single monodromy parameters and $\\{t_{0},K_{0}\\}$ are called the
accessory parameters. For the Klein-Gordon field mode
$\Phi_{n,\ell,m_{1},m_{2}}$, with

$\Phi_{n,\ell,m_{1},m_{2}}(t,r,\theta,\phi_{1},\phi_{2})=e^{-i\omega
t+im_{1}\phi_{1}+im_{2}\phi_{2}}R_{n,\ell,m_{1},m_{2}}(r)S_{\ell,m_{1},m_{2}}(\theta),$
(45)

the radial system’s single monodromy parameters are

$\theta_{\mathrm{Rad},0}=\theta_{-},\qquad\theta_{\mathrm{Rad},t}=\theta_{+},\qquad\theta_{\mathrm{Rad},1}=2-\Delta,\qquad\theta_{\mathrm{Rad},\infty}=\theta_{0}$

where

$\theta_{k}=\dfrac{i}{2\pi}\left(\dfrac{\omega-
m_{1}\Omega_{k,1}-m_{2}\Omega_{k,2}}{T_{k}}\right),\quad
k=\pm,0,\qquad\theta_{\infty}=2-\Delta,$ (46)

and the respective accessory parameters are

$t_{\mathrm{Rad},0}=z_{0}=\frac{r_{+}^{2}-r_{-}^{2}}{r_{+}^{2}-r_{0}^{2}},$
(47)
$4z_{0}(z_{0}-1)K_{\mathrm{Rad},0}=-\frac{C_{\ell}+r_{-}^{2}\Delta(\Delta-4)-\omega^{2}}{r_{+}^{2}-r_{0}^{2}}-(z_{0}-1)[(\theta_{-}+\theta_{+}-1)^{2}-\theta_{0}^{2}-1]\\\
-z_{0}\left[2(\theta_{+}-1)(1-\Delta)+(2-\Delta)^{2}-2\right].$ (48)

The angular system’s single monodromy parameters are given by

$\displaystyle\theta_{\mathrm{Ang},0}=m_{1},\qquad\theta_{\mathrm{Ang},t}=m_{2},\qquad\theta_{\mathrm{Ang},1}=2-\Delta,\qquad\theta_{\mathrm{Ang},\infty}\equiv\varsigma=\omega+a_{1}m_{1}+a_{2}m_{2},$
(49)

and the accessory parameters are

$t_{\mathrm{Ang},0}=u_{0}=\frac{a_{2}^{2}-a_{1}^{2}}{a_{2}^{2}-1},$ (50)
$4u_{0}(u_{0}-1)K_{\mathrm{Ang},0}=-\frac{C_{\ell}-\omega^{2}-a_{1}^{2}\Delta(\Delta-4)}{1-a_{2}^{2}}-u_{0}\left[(m_{2}+\Delta-1)^{2}-m_{2}^{2}-1\right]\\\
-(u_{0}-1)\left[(m_{1}+m_{2}+1)^{2}-\varsigma^{2}-1\right].$ (51)

In previous work Amado:2017kao ; Barragan-Amado:2018pxh , the authors stressed
how the eigenvalue problem for the angular equation and the QNM problem can be
solved in terms of the composite monodromy parameters associated with the
differential equation (44). In particular, the transformation between the
accessory parameters $\\{t_{0},K_{0}\\}$ and the composite monodromy
parameters $\\{\sigma_{0t},\sigma_{t1}\\}$ is defined in terms of the Painlevé
VI transcendent tau function:

$\displaystyle\tau(\theta_{0},\theta_{t},\theta_{1},\theta_{\infty};\sigma_{0t},\sigma_{t1};t_{0})=0,$
(52) $\displaystyle\frac{\partial}{\partial
t}\log\tau(\theta_{0},\theta_{t}-1,\theta_{1},\theta_{\infty}+1;\sigma_{0t}-1,\sigma_{t1}-1;t_{0})-\frac{(\theta_{t}-1)\theta_{1}}{2(t_{0}-1)}-\frac{(\theta_{t}-1)\theta_{0}}{2t_{0}}=K_{0}.$
(53)

The expansion of the tau function for small $t_{0}$ is given in Appendix B,
and the reader is referred to Gavrylenko:2016zlf  for further details.

As derived in Amado:2017kao , the quantization condition for the angular
eigenfunctions can be cast in terms of the monodromy parameters:

$\sigma_{\mathrm{Ang},0t}=-\ell,\qquad\ell\in\mathbb{N},$ (54)

from which we can compute the small $u_{0}$ expansion of the separation
constant $C_{\ell}$:

$C_{\ell}=\omega^{2}+\ell(\ell+2)-\varsigma^{2}-\frac{a_{1}^{2}+a_{2}^{2}}{2}\left(\ell(\ell+2)-\varsigma^{2}-\Delta(\Delta-4)\right)\\\
-\frac{\left(a_{1}^{2}-a_{2}^{2}\right)\left(m_{1}^{2}-m_{2}^{2}\right)}{2\ell(\ell+2)}\left(\ell(\ell+2)-\varsigma^{2}+(\Delta-2)^{2}\right)\\\
-\frac{(a_{1}^{2}-a_{2}^{2})^{2}}{1-a^{2}_{2}}\bigg{[}\frac{\left(\ell(\ell+2)+m_{2}^{2}-m_{1}^{2}\right)\left(\ell(\ell+2)+(\Delta-2)^{2}-\varsigma^{2}\right)}{2\ell(\ell+2)}\\\
-\frac{13}{32}\ell(\ell+2)+\frac{1}{32}\left(5+14(m^{2}_{1}+\varsigma^{2})-18(m^{2}_{2}+(\Delta-2)^{2})\right)\\\
-\frac{\left((m_{1}+1)^{2}-m_{2}^{2}\right)\left((1-m_{1})^{2}-m^{2}_{2}\right)\left((\Delta-1)^{2}-\varsigma^{2}\right)\left((\Delta-3)^{2}-\varsigma^{2}\right)}{32(\ell-1)(\ell+3)}\\\
+\frac{\left((m^{2}_{1}-m^{2}_{2})((\Delta-2)^{2}-\varsigma^{2})+8\right)^{2}-64-2(m^{2}_{1}+m^{2}_{2})((\Delta-2)^{2}-\varsigma^{2})^{2}}{32\ell(\ell+2)}\\\
-\frac{2(m^{2}_{1}-m^{2}_{2})^{2}((\Delta-2)^{2}+\varsigma^{2})}{32\ell(\ell+2)}-\frac{\left(m^{2}_{1}-m^{2}_{2}\right)^{2}\left((\Delta-2)^{2}-\varsigma^{2}\right)^{2}}{64}\left(\frac{1}{(\ell+2)^{3}}-\frac{1}{\ell^{3}}\right)\bigg{]}\\\
+\mathcal{O}\left(\left(\frac{a_{1}^{2}-a_{2}^{2}}{1-a_{2}^{2}}\right)^{3}\right).$
(55)

The radial QNMs can also be determined with similar techniques. As seen in
Barragan-Amado:2018pxh , the purely ingoing boundary condition at infinity
requires that the composite monodromy parameter satisfies:

$R_{n,\ell,m_{1},m_{2}}(r)\sim\begin{cases}(r-r_{+})^{\frac{1}{2}\theta_{+}},\quad
r\rightarrow r_{+};\\\ r^{2-\Delta},\quad
r\rightarrow\infty,\end{cases}\quad\Longrightarrow\quad\cos\pi\sigma_{1t}=\cos\pi(\theta_{t}+\theta_{1}).$
(56)

This condition can be conveniently written in terms of the expansion parameter
$s$ that enters the tau function expansion (144). From the explicit
representation for the monodromy matrices (see also Its:2016jkt ), we have

$\sin^{2}\pi\sigma_{0t}\cos\pi\sigma_{1t}=\cos\pi\theta_{0}\cos\pi\theta_{\infty}+\cos\pi\theta_{t}\cos\pi\theta_{1}\\\
-\cos\pi\sigma_{0t}(\cos\pi\theta_{0}\cos\pi\theta_{1}+\cos\pi\theta_{t}\cos\pi\theta_{\infty})\\\
-\frac{1}{2}(\cos\pi\theta_{\infty}-\cos\pi(\theta_{1}-\sigma_{0t}))(\cos\pi\theta_{0}-\cos\pi(\theta_{t}-\sigma_{0t}))s\\\
-\frac{1}{2}(\cos\pi\theta_{\infty}-\cos\pi(\theta_{1}+\sigma_{0t}))(\cos\pi\theta_{0}-\cos\pi(\theta_{t}+\sigma_{0t}))s^{-1}.$
(57)

Note that this equation is of the form $A-Bs-Cs^{-1}=0$, and by direct
calculation it can be checked that

$A=B+C+\sin^{2}\pi\sigma_{0t}(\cos\pi(\theta_{t}+\theta_{1})-\cos\pi\sigma_{1t}).$

Then, due to the quantization condition (56), the roots are readily seen to be
$s=1$ and $s=C/B$, with the last one compatible with the parametrization of
the monodromy matrices. The parameter $s$ can be written explicitly in terms
of the monodromy parameters as

$s=\frac{\sin\frac{\pi}{2}(\theta_{t}+\theta_{0}+\sigma)\sin\frac{\pi}{2}(\theta_{t}-\theta_{0}+\sigma)}{\sin\frac{\pi}{2}(\theta_{t}+\theta_{0}-\sigma)\sin\frac{\pi}{2}(\theta_{t}-\theta_{0}-\sigma)}\frac{\sin\frac{\pi}{2}(\theta_{1}+\theta_{\infty}+\sigma)\sin\frac{\pi}{2}(\theta_{1}-\theta_{\infty}+\sigma)}{\sin\frac{\pi}{2}(\theta_{1}+\theta_{\infty}-\sigma)\sin\frac{\pi}{2}(\theta_{1}-\theta_{\infty}-\sigma)}.$
(58)

This expression for $s$, when substituted in the tau function with the
$\theta_{k}$ parameters of the radial equation, will allow us through (53) to
constrain the frequencies $\omega$, and thus determine the QNMs of the scalar
perturbation.

###  3.1 The low-temperature limit

At small $r_{+}$, one can check from (27) that the requirement of positive
temperature limits the range of available $a_{1}$ and $a_{2}$, which will be
taken to be of order $r_{+}$. Bearing this in mind, we define the parameters

$\begin{gathered}\epsilon=\frac{r_{+}^{2}-r_{-}^{2}}{2r_{+}^{2}},\qquad\alpha_{1}=\frac{a_{1}}{r_{+}},\qquad\alpha_{2}=\frac{a_{2}}{r_{+}},\qquad\alpha^{2}_{+}=\frac{a_{1}^{2}+a_{2}^{2}}{2r_{+}^{2}},\quad\quad\alpha^{2}_{-}=\frac{a_{1}^{2}-a_{2}^{2}}{2r_{+}^{2}},\end{gathered}$
(59)

with the understanding that $\alpha_{+}$ will be of order $1$. We will see
below that the value of $\epsilon$ will be constrained by temperature
requirements, whereas $\alpha_{-}$ is completely determined from $\epsilon$,
$\alpha_{+}$ and $r_{+}$.

The radii and mass parameters are in turn given in terms of $\epsilon$, and
$\alpha_{\pm}$ by

$\displaystyle r_{-}^{2}=(1-2\epsilon)r_{+}^{2},\quad\quad-
r_{0}^{2}=1+2(1+\alpha^{2}_{+}-\epsilon)r_{+}^{2},$ (60) $\displaystyle
2M=-(1+r_{0}^{2})(1+r_{+}^{2})(1+r_{-}^{2})=2r_{+}^{2}(1+r_{+}^{2})(1+\alpha^{2}_{+}-\epsilon)(1+(1-2\epsilon)r_{+}^{2}).$
(61)

Given that the black hole charges are determined by three parameters, the
following relation between $r_{+}$, $\epsilon$, $\alpha_{+}$ and $\alpha_{-}$
holds

$(\alpha^{2}_{+}-(1-2\epsilon)r_{+}^{2})^{2}-\alpha_{-}^{4}=(1-2\epsilon)(1+r_{+}^{2})(1+(1-2\epsilon)r_{+}^{2}),$
(62)

so that, for fixed $r_{+}$ and $\epsilon$, the set of available
$\alpha_{+}^{2}$ and $\alpha_{-}^{2}$ spans a hyperbola. The requisite of
positive temperature will truncate the hyperbola well before
$\alpha_{+}^{2}<1/r_{+}^{2}$, which corresponds to $a_{1},a_{2}<1$. We will
assume that $a_{1}>a_{2}$, i.e. $\alpha_{-}^{2}$ positive.

Let us now use $z_{0}=(r_{+}^{2}-r_{-}^{2})/(r_{+}^{2}-r_{0}^{2})$ as
extremality parameter. We can see that it is proportional to the temperature
by employing (6)

$2\pi
T_{+}=\frac{r_{+}(r_{+}^{2}-r_{-}^{2})(r_{+}^{2}-r_{0}^{2})}{(r_{+}^{2}+a_{1}^{2})(r_{+}^{2}+a_{2}^{2})}=z_{0}\frac{(1+(3+2\alpha^{2}_{+}-2\epsilon)r_{+}^{2})^{2}}{2r_{+}^{3}(1+\alpha_{+}^{2}-\epsilon)(1+(1-2\epsilon)r_{+}^{2})},$
(63)

and it vanishes with $\epsilon$:

$z_{0}=\frac{r_{+}^{2}-r_{-}^{2}}{r_{+}^{2}-r_{0}^{2}}=\frac{2\epsilon
r_{+}^{2}}{1+(3+2\alpha_{+}^{2}-2\epsilon)r_{+}^{2}},\qquad\epsilon=z_{0}\frac{1+(3+2\alpha_{+}^{2})r_{+}^{2}}{2(1+z_{0})r_{+}^{2}}.$
(64)

The parametrization of $T_{+}$ in terms of $z_{0}$ is relevant for the near-
extremal limit of the monodromy parameters (46), to be discussed below.
Dwelling on this a little further, consider the parametrization of $T_{+}$ in
terms of $\epsilon$ from (63)

$2\pi
T_{+}=\frac{\epsilon}{r_{+}}\frac{1+(3+2\alpha^{2}_{+}-2\epsilon)r_{+}^{2}}{(1+\alpha_{+}^{2}-\epsilon)(1+(1-2\epsilon)r_{+}^{2})}.$
(65)

We find that the temperature is generically proportional to $\epsilon$, but
small temperatures are defined relative to $r_{+}$. This means that, for small
black holes, it is not sufficient to require small $\epsilon$, but also that
$\epsilon\ll r_{+}$. In addition, we will take $\alpha_{\pm}$ to be of order
one, following the argument at the beginning of the Section.

With the parametrization above, the $\theta_{\pm}$ parameters defined by (46)
have the asymptotic form:

$\theta_{\pm}=\pm\frac{i\Lambda}{z_{0}}+\frac{\theta_{\star}}{2},$ (66)

where $\Lambda$ and $\theta_{\star}$ will have their expansions in $r_{+}$
given below. Low temperature ($\epsilon\ll 1$), small $r_{+}$ expansions of
the relevant quantities are

$\theta_{0}=\omega+(m_{1}\alpha_{1}+m_{2}\alpha_{2})r_{+}-3(1+\alpha_{+}^{2}-\epsilon)\omega
r_{+}^{2}+\ldots$ (67)
$\Lambda=-(2(1+\alpha_{2}^{2})-\epsilon(1-\alpha_{2}^{2}))\frac{m_{1}\alpha_{1}}{2}r_{+}^{2}-(2(1+\alpha_{1}^{2})-\epsilon(1-\alpha_{1}^{2}))\frac{m_{2}\alpha_{2}}{2}r_{+}^{2}\\\
+(2(1+\alpha_{+}^{2})-(3+\alpha_{+}^{2})\epsilon)\omega r_{+}^{3}+\ldots$ (68)
$\theta_{\star}=-i(2(1-\alpha_{2}^{2})-\epsilon(1-3\alpha_{2}^{2}))\frac{m_{1}\alpha_{1}}{4}-i(2(1-\alpha_{1}^{2})-\epsilon(1-3\alpha_{1}^{2}))\frac{m_{2}\alpha_{2}}{4}\\\
+i(2+\epsilon)(1+\alpha_{+}^{2})\frac{\omega r_{+}}{2}+\ldots$ (69)

where $\alpha_{i}=a_{i}/r_{+}$, and the ellipsis stands for terms of higher
order in $r_{+}$ and $\epsilon$, which will not be relevant for the following
analysis. Note that the leading behavior of $\Lambda$ and $\theta_{\star}$
with $r_{+}$ depends on whether $m_{1}$ and $m_{2}$ vanish or not. In
particular, if $m_{1}=m_{2}=0$, then $\Lambda$ vanishes as $r_{+}^{3}$ and
$\theta_{\star}$ vanishes as $r_{+}$. Otherwise, they vanish as $r_{+}^{2}$
and have finite limits, respectively. This behavior does not depend on whether
$\epsilon$, which is proportional to the temperature, is zero or not.

The quantization conditions (53) can be applied, along with (58) to find the
QNMs’ frequencies. For generic temperatures, the study was performed in
Barragan-Amado:2018pxh . The low-temperature limit is a confluence limit of
the Heun equation, given that both $\theta_{+}$ and $\theta_{-}$ diverge as
$z_{0}\rightarrow 0$. The tau function defined in (139) will have to be
studied in this limit.

####  3.1.1 The confluent limit of the tau function

The Plemelj operator $\mathsf{D}$ which composes the Fredholm determinant
(139) has a fairly straightforward confluence limit. We will assume that the
$\sigma$ and $\kappa$ parameters, defined in (144), have finite limits as
$z_{0}\rightarrow 0$, which will be verified a posteriori. In this case, the
hypergeometric functions entering the parametrices $\mathsf{D}$ can be
expanded in an asymptotic series for small $z_{0}$ yielding confluent
hypergeometric functions

$\Psi(-\sigma,\theta_{+},\theta_{-};z_{0}/z)=\Psi^{(0)}_{c}(-\sigma,\theta_{\star};i\Lambda/z)+z_{0}\Psi^{(1)}_{c}(-\sigma,\psi_{+},\psi_{-};i\Lambda/z)+\ldots$
(70)

where the confluent parametrix is, defining
$\theta_{\star}=\psi_{+}+\psi_{-}$:

$\displaystyle\Psi^{(0)}_{c}(-\sigma,\theta_{\star};i\Lambda/z)=\begin{pmatrix}\phi_{c}(-\sigma,\theta_{\star};i\Lambda/z)&\chi_{c}(-\sigma,\theta_{\star};i\Lambda/z)\\\
\chi_{c}(\sigma,\theta_{\star};i\Lambda/z)&\phi_{c}(\sigma,\theta_{\star};i\Lambda/z)\end{pmatrix},$
$\displaystyle\phi_{c}(\pm\sigma,\theta_{\star};i\Lambda/z)={{}_{1}F_{1}}(\tfrac{-\theta_{\star}\pm\sigma}{2};\pm\sigma;-i\Lambda/z),$
(71)
$\displaystyle\chi_{c}(\theta_{\star},\pm\sigma;-i\Lambda/z)=\pm\frac{-\theta_{\star}\pm\sigma}{2\sigma(1\pm\sigma)}\,\frac{i\Lambda}{z}\,{{}_{1}F_{1}}(1+\tfrac{-\theta_{\star}\pm\sigma}{2},2\pm\sigma;-i\Lambda/z).$
(72)

It follows that the operator $\mathsf{D}$ is analytic in $z_{0}$ and $\Lambda$

$\mathsf{D}(z_{0},i\Lambda)=\mathsf{D}_{0}(i\Lambda)+z_{0}\mathsf{D}_{1}(i\Lambda)+z_{0}^{2}\mathsf{D}_{2}(i\Lambda)+\ldots,$
(73)

with each term computed recursively in terms of confluent hypergeometric
functions. The first term $\mathsf{D}_{0}(i\Lambda)$ defines the Fredholm
determinant expression for the Painlevé V transcendent defined in
Lisovyy:2018mnj . As we will see, we will not need the corrections
$\mathsf{D}_{1},\mathsf{D}_{2},\ldots$ for our analysis.

Let us now study the behavior of the $s$ parameter appearing in (139) as
$z_{0}\rightarrow 0$. The expansion of the Painlevé VI tau function has the
schematic structure (147)

$\tau=z_{0}^{A_{0}}(1-z_{0})^{A_{1}}(1+A_{1,-1}Y^{-1}z_{0}+A_{1,0}z_{0}+A_{1,1}Yz_{0}+{\cal
O}(z_{0}^{2})),$ (74)

where $A_{0},A_{1},A_{1,0},A_{1,\pm 1}$ are monodromy-dependent coefficients
and $Y=\kappa z_{0}^{\sigma}$, with $\kappa$ defined in (144). The series,
save from the $z_{0}^{A_{0}}$ term, is analytic in $z_{0}$ and meromorphic in
$Y$. Finding the non-trivial zero for $Y$ – the first condition in (53) – can
then be achieved perturbatively (assuming $\Re\sigma>0$)

$Y=\kappa z_{0}^{\sigma}=-A_{1,-1}z_{0}\chi_{VI}(z_{0}),$ (75)

where $\chi_{VI}(z_{0})$ is given as a series in $z_{0}$. The left hand side
can be written as

$\kappa z_{0}^{\sigma}=s\Pi z_{0}^{\sigma}\frac{\Gamma(1+\tfrac{i}{2}\Lambda
z_{0}^{-1}+\frac{\sigma}{2})}{\Gamma(1+\tfrac{i}{2}\Lambda
z_{0}^{-1}-\frac{\sigma}{2})}$ (76)

with $\Pi$ defined as the confluent limit of the product of Gamma functions in
(144)

$\Pi=\frac{\Gamma^{2}(1-\sigma)}{\Gamma^{2}(1+\sigma)}\frac{\Gamma(1+\tfrac{1}{2}(\theta_{\star}+\sigma))}{\Gamma(1+\tfrac{1}{2}(\theta_{\star}-\sigma))}\frac{\Gamma(1+\tfrac{1}{2}(\theta_{1}-\theta_{\infty}+\sigma))\Gamma(1+\tfrac{1}{2}(\theta_{1}+\theta_{\infty}+\sigma))}{\Gamma(1+\tfrac{1}{2}(\theta_{1}-\theta_{\infty}-\sigma))\Gamma(1+\tfrac{1}{2}(\theta_{1}+\theta_{\infty}-\sigma))}.$
(77)

The right hand-side of (75) can now be expanded

$A_{1,-1}z_{0}\chi_{VI}(z_{0})=A^{V}_{1,-1}z_{0}\left(\frac{\sigma}{2}+\frac{i\Lambda}{z_{0}}\right)\left(\chi_{V}(i\Lambda)+{\cal
O}(\Lambda^{n}z^{m}_{0})\right)$ (78)

where $\chi_{V}(i\Lambda)$ is the series analogous to (75) that we find from
the Painlevé V tau function. The correction terms to the series are, as argued
above, of the form $\Lambda^{n}z_{0}^{m}$ and hence subleading. Using the
property of the gamma function in the left hand side to cancel the
$\sigma/2+i\Lambda/z_{0}$ term, we have the expansion

$\frac{\Gamma(\tfrac{i}{2}\Lambda
z_{0}^{-1}+\frac{\sigma}{2})}{\Gamma(1+\tfrac{i}{2}\Lambda
z_{0}^{-1}-\frac{\sigma}{2})}=\left(\frac{i\Lambda}{z_{0}}\right)^{\sigma-1}\left(1+{\cal
O}\left(\frac{z_{0}^{2}}{\Lambda^{2}}\right)\right),$ (79)

for the argument of the Gamma function not close to the negative real axis –
which should be expected for real positive frequencies. Combining the
expansions, we find that in the confluent limit, (75) is replaced by

$\kappa z^{\sigma}=s\Pi(i\Lambda)^{\sigma}\left(1+{\cal
O}(z_{0}^{2}\Lambda^{-2})\right).$ (80)

In conclusion, we have that in order to compute the first non-trivial $z_{0}$
correction, it suffices to consider the Painlevé V expansion. Corrections to
the monodromy parameter are of order $z^{2}_{0}$, and hence less relevant for
the confluent limit.

Let us now review the type of correction we have for the tau function away
from the confluence point, where it coincides with the Painlevé V
transcendent. The relevant expansions in $\Lambda$ and $z_{0}$ are:

  1. 1.

Expansion of the $s$ parameter, which is given in terms of powers of
$-iz_{0}/\Lambda$.

  2. 2.

Expansion of the parametrices $\Psi_{c}^{(i)}$ (70), which are given in terms
of hypergeometric functions with monomials in $z_{0}^{n}(i\Lambda)^{m}$.

It is not difficult to see that, in the small $\Lambda$ limit, the first type
of expansion will dominate over the second. Disregarding the contributions of
the second type, we have exactly the same type of expansion as the Painlevé V.
This means that the first non-trivial correction to the accessory parameter
problem of the confluent Heun equation comes in the monodromy parameter, not
in the tau function expansion. In short, the first non-trivial correction is
obtained by expanding the $\kappa$ parameter.

The final step needed before the calculation can be performed is solving for
the quantization condition for the radial equation, expressed in terms of the
monodromy parameters in (58). It approaches, as $z_{0}\rightarrow 0$,

$s=e^{\mp
i\pi\sigma}\frac{\sin\frac{\pi}{2}(\theta_{\star}+\sigma)}{\sin\frac{\pi}{2}(\theta_{\star}-\sigma)}\frac{\sin\frac{\pi}{2}(\theta_{1}+\theta_{\infty}+\sigma)\sin\frac{\pi}{2}(\theta_{1}-\theta_{\infty}+\sigma)}{\sin\frac{\pi}{2}(\theta_{1}+\theta_{\infty}-\sigma)\sin\frac{\pi}{2}(\theta_{1}-\theta_{\infty}-\sigma)}+\ldots,$
(81)

for $\Re\Lambda>0$ or $\Re\Lambda<0$, respectively. In this case the
convergence is exponential, up to terms of order $e^{-2\pi|\Lambda|/z_{0}}$.
Again, corrections are (exponentially) suppressed in terms of $|\Lambda|$.

####  3.1.2 Quasi-normal modes

With these premises, the solution for the QNMs follows from the radial
quantization condition, expressed in terms of $s$ in (81), and the analysis is
actually closely related to that of Barragan-Amado:2018pxh . In the
$z_{0}\rightarrow 0$ limit, the conditions to solve become

$\displaystyle\tau_{V}(\theta_{\star},\theta_{1},\theta_{\infty};\sigma,s;i\Lambda)=0,$
(82) $\displaystyle
c_{0}=i\Lambda\frac{d}{du}\log\hat{\tau}(\theta_{\star}-1,\theta_{1},\theta_{\infty}+1;\sigma-1,s;i\Lambda)+\frac{1}{4}((\sigma-1)^{2}-(\theta_{\star}-1)^{2})$
(83)

with $\hat{\tau}=\det(\mathbbold{1}-\mathsf{A}\Phi\mathsf{D}_{0}\Phi^{-1})$
and
$\tau_{V}(i\Lambda)=\tilde{C}(i\Lambda)^{-\frac{1}{4}(\sigma^{2}-\frac{1}{2}\theta_{\star}^{2})}e^{-\frac{i}{2}\Lambda\theta_{1}}\hat{\tau}$
is the Painlevé V tau function. The Plemelj operator $\mathsf{A}$ is defined
in Appendix B and $\mathsf{D}_{0}$ is in (73). The accessory parameter $c_{0}$
is the limit as $z_{0}\rightarrow 0$ of the radial accessory parameter
$z_{0}K_{0}$, given by (48) in our application.

The procedure is now to tackle the condition in (82), which can be inverted
for the zero $\tau_{V}=0$ in order to solve for $\kappa z_{0}^{\sigma}$ as a
series in $\Lambda$. The result is

$s\Pi(i\Lambda)^{\sigma}\left(1+{\cal
O}(z_{0}^{2})+\ldots\right)=i\frac{(\sigma+\theta_{\star})((\sigma+\theta_{1})^{2}-\theta_{\infty}^{2})}{8\sigma^{2}(\sigma-1)^{2}}\Lambda\,\chi(i\Lambda)$
(84a) where $\chi(i\Lambda)$ is given as a series in $\Lambda$
$\chi(i\Lambda)=1+\chi_{1}(i\Lambda)+\chi_{2}(i\Lambda)^{2}+...,$ (84b) with
$\chi_{1}=(\sigma-1)\frac{\theta_{\star}(\theta_{1}^{2}-\theta_{\infty}^{2})}{\sigma^{2}(\sigma-2)^{2}},$
(84c) and
$\chi_{2}=\frac{\theta_{\star}^{2}(\theta_{1}^{2}-\theta_{\infty}^{2})^{2}}{64}\left(\frac{5}{\sigma^{4}}-\frac{1}{(\sigma-2)^{4}}-\frac{2}{(\sigma-2)^{2}}+\frac{2}{\sigma(\sigma-2)}\right)\\\
-\frac{(\theta_{1}^{2}-\theta_{\infty}^{2})^{2}+2\theta_{\star}^{2}(\theta_{1}^{2}+\theta_{\infty}^{2})}{64}\left(\frac{1}{\sigma^{2}}-\frac{1}{(\sigma-2)^{2}}\right)\\\
+\frac{(1-\theta_{\star}^{2})(\theta_{1}^{2}-(\theta_{\infty}-1)^{2})(\theta_{1}^{2}-(\theta_{\infty}+1)^{2})}{128}\left(\frac{1}{(\sigma+1)^{2}}-\frac{1}{(\sigma-3)^{2}}\right).$
(84d)

The higher order terms can be computed recursively. For our application, we
use the quantization condition (81) and, after using $\sin(\pi
z)\Gamma(z)\Gamma(1-z)=\pi$ a number of times, (84a) now reads

$\chi(i\Lambda)=e^{-\frac{i}{2}\pi\sigma}\Theta\Lambda^{\sigma-1}\left(1+{\cal
O}(z_{0}^{2})+\ldots\right)$ (85)

where

$\Theta=i\frac{\Gamma^{2}(2-\sigma)\Gamma(\tfrac{1}{2}(\sigma-\theta_{\star}))\Gamma(\tfrac{1}{2}(\sigma-\theta_{1}+\theta_{\infty}))\Gamma(\tfrac{1}{2}(\sigma-\theta_{1}-\theta_{\infty}))}{\Gamma^{2}(\sigma)\Gamma(\tfrac{1}{2}(2-\sigma-\theta_{\star}))\Gamma(\tfrac{1}{2}(2-\sigma-\theta_{1}+\theta_{\infty}))\Gamma(\tfrac{1}{2}(2-\sigma-\theta_{1}-\theta_{\infty}))}.$
(86)

The treatment of the second condition (83) is analogous, substituting the
value of $\chi(i\Lambda)$ into the logarithmic derivative and solving for
$c_{0}$ as a function of $\Lambda$. The result, assuming $\Re\sigma>0$, is

$c_{0}=k_{0}+k_{1}(i\Lambda)+k_{2}(i\Lambda)^{2}+\ldots+k_{n}(i\Lambda)^{n}+\ldots,$
(87a) with the first three terms in the expansion given by
$k_{0}=\frac{(\sigma-1)^{2}-(\theta_{\star}-1)^{2}}{4},\quad\quad
k_{1}=\frac{\theta_{1}-1}{2}+\frac{\theta_{\star}}{2}+\frac{\theta_{\star}(\theta_{1}^{2}-\theta_{\infty}^{2})}{4\sigma(\sigma-2)},$
(87b)
$k_{2}=\frac{1}{32}+\frac{\theta_{\star}^{2}(\theta_{1}^{2}-\theta_{\infty}^{2})^{2}}{64}\left(\frac{1}{\sigma^{3}}-\frac{1}{(\sigma-2)^{3}}\right)+\frac{(1-\theta_{\star}^{2})(\theta_{\infty}^{2}-\theta_{1}^{2})^{2}+2\theta_{\star}^{2}(\theta_{\infty}^{2}+\theta_{1}^{2})}{32\sigma(\sigma-2)}\\\
-\frac{(1-\theta_{\star}^{2})((\theta_{\infty}-1)^{2}-\theta_{1}^{2})((\theta_{\infty}+1)^{2}-\theta_{1}^{2})}{32(\sigma+1)(\sigma-3)}.$
(87c)

Each term in the expansion of $\chi$ and $c_{0}$ is a meromorphic function of
$\sigma$, symmetric under $\sigma\rightarrow 2-\sigma$. There are single poles
at $\sigma=3,4,5,\ldots$, with the pole of order $\sigma=n$ only seen at
$k_{n-1}$ term, and a pole of higher order near $\sigma=2$, which is present
at all orders save for $k_{0}$. The asymptotics of the $\chi_{n}$ (84a) and
$k_{n}$ (87) terms as $\sigma\simeq 2$, reads:

$(\sigma-2)\chi_{n}\simeq
k_{n}=-\mathbf{C}_{n-1}\frac{\theta_{\star}^{n}(\theta_{1}^{2}-\theta_{\infty}^{2})^{n}}{8^{n}(\sigma-2)^{2n-1}}+\ldots,\quad\quad
n\geq 2,$ (88)

where the terms left out are of lower order in $\sigma-2$. The relation
$k_{n}=(\sigma-2)\chi_{n}$ no longer holds as one considers less divergent
terms $(\sigma-2)^{-m}$, $m<2n-1$. Finally, $\mathbf{C}_{n}$ is the $n$-th
Catalan number:

$\mathbf{C}_{n}=\frac{1}{n+1}\begin{pmatrix}2n\\\
n\end{pmatrix}=1,1,2,5,14,\ldots$ (89)

This remark will be useful when we discuss the CFT description and the
fundamental QNM in the following.

###  3.2 QNMs and CFT description of perturbations

Before delving into the QNMs, let us discuss the conformal aspects of the
perturbations, both in the four-dimensional picture, and the “two-dimensional”
point of view encoded in the preceding analysis. If we take the massless
scalar field (at $\Delta=4$) to represent the scalar sector of gravitational
perturbations in AdS, then the gravitational QNMs can be understood as
resonances created by perturbations of the CFT state represented by the
stress-energy tensor (23). The holographic view of the scattering process has
been considered in a great number of papers over the years, with the ideas
listed here discussed in Hijano:2015zsa  and daCunha:2016crm . The starting
point is that scattering amplitudes are encoded in the 4-point function

$\langle{\cal V}_{\mathrm{Heavy}}(\infty){\cal V}_{\mathrm{Light}}({\bf
x}_{1}){\cal V}_{\mathrm{Light}}({\bf x}_{2}){\cal
V}_{\mathrm{Heavy}}(0)\rangle$ (90)

with ${\cal V}_{\mathrm{Heavy}}$ vertex operators associated with the
background charges – the mass and angular momenta of the black hole in our
case – and ${\cal V}_{\mathrm{Light}}$ associated with the local insertion of
the scalar perturbations. In a Euclidean setting the positions of the heavy
operators are unambiguous, but in our Lorentz setting we will take $0$ and
$\infty$ to denote past and future infinity respectively. The correlation
functions such as in (90) have a rich history in the early development of
CFTs, notably the seminal work of Ferrara:1971vh ; Ferrara:1972xq ;
Ferrara:1973vz ; Polyakov:1974gs . The “partial wave” decomposition studied
there can be thought of as a higher dimensional conformal block, in which the
quantum numbers of the intermediate states mirror the decomposition of the
(scalar) wave equation in terms of $\mathrm{SO}(4,2)$ quantum numbers
associated to the generators (35).

For pure AdS, it has been known for some time that these higher dimensional
conformal blocks can be factorized in terms of “two-dimensional” classical
conformal blocks (hypergeometric functions) Dolan:2000ut ; Dolan:2003hv ;
Dolan:2011dv . Now, we want to view the conditions (53) relating the
parameters of the differential equation to monodromy data as a deformed
version of this factorization, valid for the case with non-zero background
charges (35). Given that the tau function expansion is given in terms of
conformal blocks Gamayun:2012ma , there are indeed some analogies, although in
this case the two-dimensional CFT is chiral and has unit central charge $c=1$.
Furthermore, $z_{0}$ and $u_{0}$ are independent, and in principle complex,
variables.

The latter point may be a consequence of Liouville exponentiation
Iorgov:2014vla , and indeed there is a semiclassical view also relating the
accessory parameter of the Heun equation to the Painlevé transcendent
Litvinov:2013sxa . In the latter view, the accessory parameter also appears as
the logarithmic derivative of the two-dimensional conformal block:

$\langle
V_{\Delta_{1}}(\infty)V_{\Delta_{2}}(1)\Pi_{\sigma}V_{\Delta_{3}}(t)V_{\Delta_{4}}(0)\rangle$
(91)

where $V_{\Delta_{i}}(z_{i})$ denotes each vertex operator, and $\Pi_{\sigma}$
the projection onto the Verma module constructed on the primary state, with
Liouville momentum parametrized by $\sigma$. The conformal dimension
$\Delta_{i}$ of each vertex operator $V_{\Delta_{i}}(z)$ is given in terms of
the Liouville momentum $P_{i}$ and the Liouville coupling constant $b$ as

$\Delta_{i}=\frac{c-1}{24}+P_{i}^{2},\qquad c=1+6Q^{2}=1+6(b+b^{-1})^{2}.$
(92)

In this picture, all operators appearing in (53) are “heavy” in the sense that
they have large Liouville momenta:

$P_{0}=\frac{\theta_{-}}{b},\qquad P_{t}=\frac{\theta_{+}}{b},\qquad
P_{1}=\frac{2-\Delta}{b},\qquad P_{\infty}=\frac{\theta_{0}}{b},$ (93)

which not only provide the Liouville interpretation for the single monodromy
parameters $\theta_{i}$, but also hint at a deeper connection. If the black
hole absorbs a quantum of energy and angular momenta given by $\omega$,
$m_{1}$ and $m_{2}$, respectively, then the increase in its entropy is given
by

$\delta S=\frac{\omega-
m_{1}\Omega_{1,+}-m_{2}\Omega_{2,+}}{b^{2}T_{+}}=2\pi\frac{\theta_{+}}{b^{2}},$
(94)

where we recovered Liouville theory’s $\hbar=b^{2}$. Thus, the entropy gained
in the process is, up to the Liouville coupling, given by the Liouville
momentum. This gives support to the interpretation of $V_{\Delta_{i}}$, at
least for the inner and outer horizon, as associated to a thermal state in the
underlying dual theory. By assuming the latter to be unitary and modular
invariant, we can use Cardy’s formula Cardy:1986ie

$\frac{S}{2\pi}\simeq\sqrt{\frac{c}{6}\left(L_{0}-\frac{c}{24}\right)}$ (95)

for the entropy of such CFT at conformal dimension $L_{0}=\Delta_{i}$.
Assuming the classical limit $b\rightarrow 0$, and solving for
$L_{0}=\Delta_{i}$, we have again the interpretation of $\delta S_{i}/2\pi b$
as the Liouville momentum.

The construction outlined above is suitable for the generic case considered in
Barragan-Amado:2018pxh , but it works in a completely analogous way for the
confluence limit we considered in Section 3.1.1. In the latter case, the two
primary vertex operators associated to the inner and outer horizons merge into
a Whittaker operator Gaiotto:2009ma . The associated conformal block is then
an irregular one Nagoya:2015cja , instead of (91). The argument presented here
about the nature of the intermediate states does have a direct parallel,
though, and for further details we refer to the discussion in
CarneirodaCunha:2019tia .

###  3.3 Fundamental QNM

The calculation of the fundamental mode is technically very similar to the
corresponding one in Barragan-Amado:2018pxh , where one solves for the first
eigenfrequency $\omega_{1,0,0,0}$ by applying the conditions (53) given the
quantization of the angular and radial monodromy parameters (54) and (56).
From the discussion above, the corresponding task at low temperature involves
solving the equations (85) and (87).

As suggested by Fig. 2, for generic values of $M$, $a_{1}$ and $a_{2}$ a
smooth zero-temperature limit for the fundamental QNM frequency exists. At
small $r_{+}$, we can work out asymptotic formulas, with

$i\Lambda=i\lambda
r_{+}^{3},\qquad\theta_{\star}=i\phi_{\star}r_{+},\qquad\sigma=2-\nu
r_{+}^{2},$ (96)

where the values of $\lambda$ and $\phi_{\star}$ can be read from (68) and
(69), by setting $m_{1}=m_{2}=0$. We find

$\phi_{\star}=\frac{1}{2}(2+\epsilon)(1+\alpha_{+}^{2})\Delta+\ldots,\qquad\lambda=(2(1+\alpha_{+}^{2})-(3+\alpha_{+}^{2})\epsilon)\Delta+\ldots,$
(97)

where we used the fact that $\omega$ is approximately $\Delta$ in the
$r_{+}\rightarrow 0$ limit. As a matter of fact, we will assume that

$\theta_{0}=\omega-3(1+\alpha_{+}^{2}-\epsilon)\omega
r_{+}^{2}+\ldots=\Delta-\beta r_{+}^{2},$ (98)

where we introduced the $\beta$ parameter to measure the difference between
$\Delta$ and the monodromy parameter $\theta_{0}$. Note also that the
expansion of $\theta_{0}$ implies that $\beta$ is related to the correction to
the QNM frequency at small $r_{+}$

$\omega=\Delta-(\beta-3(1+\alpha_{+}^{2}-\epsilon)\Delta)r_{+}^{2}+{\cal
O}(\epsilon^{2},r_{+}^{4}).$ (99)

Finally, we remark that, at low temperature we have $\epsilon\ll r_{+}\ll 1$,
and (62) implies that $\alpha_{1}$ and $\alpha_{2}$ are related to $r_{+}$,

$\alpha_{1}^{2}\alpha_{2}^{2}=(1-2\epsilon)((1+r_{+}^{2})^{2}-2r_{+}^{2}\alpha_{+}^{2})+{\cal
O}(\epsilon^{2}).$ (100)

We will now consider the conditions (82) and (83), encoded in the equations
(85) and (87), as equations determining $\beta$ and $\nu$ for small $r_{+}$.
First, the $\Theta$ parameter in (85) has the small $r_{+}$ expansion for
$\ell=0$:

$\Theta=\frac{\phi_{\star}}{2r_{+}^{3}\nu^{2}}\frac{(\nu+\beta)}{(\nu-\beta)}(\Delta-1)\left(1+i\frac{\nu}{\phi_{*}}r_{+}+{\cal
O}(\epsilon r_{+}^{2},r_{+}^{2})\right).$ (101)

As anticipated by the argument in the preceding sections, the $\epsilon$
corrections are accompanied by a factor of $r_{+}^{2}$ and are always
subdominant in the small $r_{+}$ limit.

Figure 2: The fundamental QNM frequency $\omega_{1,0,0,0}$ as a function of
the outer horizon temperature $T_{+}$ for a Kerr-AdS5 black hole with fixed
$r_{+}=0.03$ and $\alpha_{+}^{2}=\sqrt{2}$. We note that the temperature has
the effect of increasing both the real and imaginary parts of the frequency,
up to the maximal value $2\pi(T_{+})_{\mathrm{max}}\simeq 8.7$, corresponding
to $r_{-}=0$, or $\epsilon=1/2$ in (65).

The accessory parameter $c_{0}$ as in (87) and the $\chi$ function defined in
(85) both receive contributions from all orders in $z_{0}$ in the small
$r_{+}$ limit. This is due to the proximity of $\sigma$ to the critical value
$2$, as further clarified in the next subsection. Contributions to $c_{0}$ and
$\chi$ can be resummed using the generating formula for the Catalan numbers

$1+x+2x^{2}+5x^{3}+14x^{4}+\ldots=\sum_{n=0}^{\infty}\mathbf{C}_{n}x^{n}=\frac{1-\sqrt{1-4x}}{2x}.$
(102)

The result for $\chi$ can then be written as

$\chi=\frac{1}{2}+\frac{\phi_{\star}(\theta_{0}^{2}-\theta_{1}^{2})}{8\nu^{2}}\lambda+\frac{1}{2}\sqrt{1+\frac{\phi_{\star}(\theta_{0}^{2}-\theta_{1}^{2})}{2\nu^{2}}\lambda}-\frac{(\theta_{0}^{2}-\theta_{1}^{2})^{2}\lambda^{2}}{64\nu^{2}\sqrt{1+\frac{\phi_{\star}(\theta_{0}^{2}-\theta_{1}^{2})}{2\nu^{2}}\lambda}}r_{+}^{2}+{\cal
O}(r_{+}^{4}),$ (103)

where $\theta_{1}=2-\Delta$. At this order in $r_{+}$, we can substitute
$\theta_{0}^{2}-\theta_{1}^{2}=4(\Delta-1)+{\cal O}(r_{+}^{2})$. The first
non-trivial correction in $\epsilon$ can be obtained by just considering the
first correction in the parameters $\lambda$ and $\phi_{\star}$. The expansion
for the accessory parameter $c_{0}$ in the right-hand side of (87a) can also
be resummed in the same way:

$c_{0}=\frac{i}{2}\phi_{\star}r_{+}+\frac{1}{4}\phi_{\star}^{2}r_{+}^{2}-\frac{\nu
r_{+}^{2}}{2}\sqrt{1+\frac{\phi_{\star}(\theta_{0}^{2}-\theta_{1}^{2})}{2\nu^{2}}\lambda}+\frac{i}{2}(\theta_{1}-1)\lambda
r_{+}^{3}+{\cal O}(r_{+}^{4}).$ (104)

The same parameter $c_{0}$ is defined as $\lim_{z_{0}\rightarrow 0}z_{0}K_{0}$
and can be computed from (48) and (55) by taking the appropriate
$r_{-}\rightarrow r_{+}$ limit. In the $\epsilon\ll r_{+}\ll 1$ limit and for
$\ell=0$, this provides for the left-hand side of equation (87a),

$c_{0}=\frac{i}{2}\phi_{\star}r_{+}+\frac{1}{4}\phi_{\star}^{2}r_{+}^{2}-\frac{1+\alpha_{+}^{2}}{2}\Delta(\Delta+2)r_{+}^{2}\\\
+\frac{1}{2}\Delta(\Delta+2)\epsilon
r_{+}^{2}+\frac{i}{2}(\theta_{1}-1)\lambda r_{+}^{3}+{\cal
O}(\epsilon^{2},\epsilon r_{+}^{4},r_{+}^{4}),$ (105)

which, equated to the right-hand side of (104), leads to an equation for $\nu$
as in (96)

$\sqrt{\nu^{2}+2\lambda\phi_{\star}(\Delta-1)}=(1+\alpha_{+}^{2})\Delta(\Delta+2)-\Delta(\Delta+2)\epsilon+\ldots,$
(106)

where the terms left out scale with $\epsilon r_{+}^{2}$ and $\epsilon^{2}$.
Taking into account (100), this equation can be solved to yield, at first non-
trivial order in $\epsilon$ and $r_{+}$,

$\nu=(1+\alpha_{+}^{2})\Delta\sqrt{\Delta^{2}+8}\left(1-\frac{\Delta^{2}+2\Delta+6}{(1+\alpha_{+}^{2})(\Delta^{2}+8)}\epsilon\right)+{\cal
O}(\epsilon^{2},r_{+}^{2},\epsilon r_{+}^{2})$ (107)

and likewise for $\chi$

$\chi=\frac{(\Delta+2+\sqrt{\Delta^{2}+8})^{2}}{4(\Delta^{2}+8)}\left(1-\frac{(\Delta+2)(\sqrt{\Delta^{2}+8}-\Delta-2)}{(1+\alpha_{+}^{2})(\Delta^{2}+8)}\,\epsilon\right)+{\cal
O}(\epsilon^{2},r_{+}^{2},\epsilon r_{+}^{2}).$ (108)

Finally, by substituting (107) and (108) into (85), we can compute $\beta$

$\beta=(1+\alpha_{+}^{2})(\Delta(\Delta+2)+2i\Delta(\Delta-1)r_{+})-\Delta(\Delta+2)\epsilon\\\
-i(3+\alpha_{+}^{2})\Delta(\Delta-1)\epsilon r_{+}+{\cal
O}(r_{+}^{2},r_{+}^{2}\log r_{+},\epsilon r_{+}^{2},\epsilon^{2}).$ (109)

Figure 3: The expression for the fundamental mode $\omega_{1,0,0,0}$ given by
(110) compared with numerical results for $\epsilon=10^{-6}$ and
$\alpha_{+}^{2}=\sqrt{2}$.

We remark that now, due to the presence of the $\Lambda^{\sigma-1}$ term in
(85), there will be non-analytical corrections of the type $r_{+}^{2}\log
r_{+}$ which, however, are subdominant for small $r_{+}$. Finally, using the
relation between $\beta$ and $\omega$ given by (99), we find the first
corrections to the fundamental QNM frequency as a function of $r_{+}$ and
$\epsilon$,

$\omega_{1,0,0,0}=\Delta-(1+\alpha_{+}^{2})\Delta(\Delta-1)r_{+}^{2}-2i(1+\alpha_{+}^{2})\Delta(\Delta-1)r_{+}^{3}\\\
+\Delta(\Delta-1)\epsilon
r_{+}^{2}+i(3+\alpha_{+}^{2})\Delta(\Delta-1)\epsilon r_{+}^{3}\\\ +{\cal
O}(r_{+}^{4},r_{+}^{4}\log r_{+}^{2},\epsilon
r_{+}^{4},\epsilon^{2}r_{+}^{2})$ (110)

where the subscript stands for $n,\ell,m_{1}$ and $m_{2}$ quantum numbers.

The imaginary part of the frequency controls the relaxation times of the
perturbation in the dual CFT. Equation (110) is valid for $\epsilon\ll
r_{+}\ll 1$, with $\alpha_{+}$ constant and of order ${\cal O}(1)$. In this
limit, the effect of non-zero temperature is accounted for by the
$\epsilon$-dependent terms, which, in agreement with Fig. 2, induce an
increase of the real and imaginary parts of the frequency, as $\epsilon$
increases. Despite having a “reinforcing” effect, making perturbations longer-
lived, the temperature does not have a strong enough influence to induce
instabilities. For perturbations above the unitarity bound $\Delta>1$, the
fundamental mode is stable at zero or small enough temperature. These results
complement the finite-temperature result of Barragan-Amado:2018pxh .

####  3.3.1 Catalan numbers and intermediate CFT levels

The two-dimensional version of (90), depicting the perturbation of the black
hole background, was studied in Fitzpatrick:2015foa , where it was shown that
the vacuum block, where the internal dimension is zero, was given by the
generating function for the Catalan numbers in the heavy-light semiclassical
limit, where two of the insertions have high conformal dimensions
$\Delta_{i}\propto c\rightarrow\infty$, with $c$ being the central charge.

There is indeed a clear parallel to what is being counted in the fundamental
mode calculations in the last subsection. The finite-temperature version of
the calculation outlined in Barragan-Amado:2018pxh  does have this hierarchy
between the vertex operators, since the ones associated with the inner and
outer horizons do become “light” in the small black hole regime, and the
residue at $\sigma=2$ does indeed correspond to the identity operator as
intermediate state in (90).

Yet, there are subtleties. We have already remarked that (91) is a chiral
conformal block. A difference from Fitzpatrick:2015foa  is that in principle
all operators in (91) are “heavy”, in the sense that their conformal
dimensions scaling with $c$. The parametric hierarchy between operators arises
from the parameter $r_{+}$, which can be made small. It is a combined effect
of the position of the insertion and the scaling of the dimensions of the
“light” operators. The corresponding calculation using the Virasoro algebra
seems straightforward, but lies outside the scope of this work.

Another difference comes from the confluence limit taken as the temperature
goes to zero. In this case the limiting form of the Liouville momenta for the
inner and outer horizon is given by (66), and the conformal dimensions of the
operators associated to $\theta_{\pm}$ are in fact much larger than $c$. The
“light” operator is the resulting operator coming from the leading term of the
OPE for the operators related to the inner and outer horizons:

$V_{P_{3}}(t)V_{P_{4}}(0)\sim V_{P}(0)+\ldots$ (111)

where we label the operators by the Liouville momentum, and $P=P_{3}+P_{4}$.
As the first descendant of the resulting module $[V_{P}]$ one finds the
Whittaker operator Gaiotto:2009ma ; Nagoya:2015cja

$V_{\theta_{\star}/b,i\Lambda}(0)\sim:e^{2\theta_{\star}/b\phi_{L}+i\Lambda\partial\phi_{L}}:$
(112)

which is labelled by the confluence parameters. We used the Feigin-Fuchs
parametrization of the Liouville field $\phi_{L}$ (see Nagoya:2015cja  or
CarneirodaCunha:2019tia ) in the right-hand side of (112) to compare our
notation with the literature. In this case, the parameters in the regime
investigated here do warrant the “light” requisite of Fitzpatrick:2015foa
since they are proportional to $r_{+}$ and $r_{+}^{3}$, respectively, as can
be checked in (68) and (69). In light of this, the results obtained in Sec.
3.3 point to an analogous simplification of the conformal block in terms of
the Catalan number generating function when the two light operators in the
analysis of Fitzpatrick:2015foa  are substituted by an irregular light
operator. For more on the relation between irregular conformal blocks and the
Painlevé V tau function we recommend Lisovyy:2018mnj .

###  3.4 Higher quasi-normal modes

Higher QNMs, with $\ell\geq 1$, allow for non-zero values of $m_{1}$ and
$m_{2}$. In turn, these multiply the angular velocities $\Omega_{i,+}$, which
are given in our parametrization by

$\Omega_{i,+}=\frac{a_{i}(1-a_{i}^{2})}{r_{+}^{2}+a_{i}^{2}}=\frac{\alpha_{i}(1-\alpha_{i}^{2}r_{+}^{2})}{r_{+}(1+\alpha_{i}^{2})},$
(113)

with $\alpha_{i}=a_{i}/r_{+}$, $i=1,2$. As explained before, at small $r_{+}$
the range of allowed $a_{i}$ is capped by $r_{+}$, so it makes sense to
consider fixed values of $\alpha_{i}\lesssim 1$. Then in the small $r_{+}$
regime, the velocities $\Omega_{i,\pm}\sim 1/r_{+}$ become very large and the
monodromy parameter (46)

$\theta_{+}=\frac{i}{2\pi}\left(\frac{\omega-
m_{1}\Omega_{1,+}-m_{2}\Omega_{2,+}}{T_{+}}\right)$ (114)

varies so that its imaginary part can become negative. Note that $r_{+}$ can
alternatively be thought of as the ratio between the outer horizon and the AdS
radius.

As a matter of fact, it has been long established from heuristic arguments
that combinations of $\omega,m_{1}$ and $m_{2}$ such that $\Im\theta_{+}<0$
display stimulated emission – the Zeldovich-Starobinski effect – see
Brito:2015oca  for a detailed account. When $\Im\theta_{+}$ is negative, we
have the superradiance window and unstable modes are bound to appear. The
instability was anticipated by the thermodynamical analysis Hawking:1999dp
and first observed numerically for four-dimensional black holes in
Cardoso:2004hs . The effect is, however, tiny and hard to study using
conventional numerical methods, even at finite temperature Cardoso:2006wa .
Our results in Barragan-Amado:2018pxh  provided supporting evidence for the
positivity of the imaginary part of the $\ell=1,m_{1}=1,m_{2}=0$ QNM frequency
in the small $r_{+}$ limit, based on the quantization conditions (54) and
(56).

Figure 4: The $\ell=1$ modes for $a_{12}=a_{1}^{2}-a_{2}^{2}=0.001$ as a
function of $r_{+}$ and fixed, very small temperature $T_{+}\sim 10^{-8}$. The
modes $m_{1}=1,m_{2}=0$ (dark red) and $m_{1}=0,m_{2}=1$ (light red) display a
small positive imaginary part for values where $\Im\theta_{+}<0$ (inset).

The low temperature version of the calculation for QNMs in Barragan-
Amado:2018pxh  follows an analogous strategy, based on (85) and (87). Since
the final analytical expression for the imaginary part of the $\ell\geq 1$ QNM
frequencies is not particularly illuminating, we will limit ourselves to
present some numerical results and discuss the qualitative features of the
analyzed modes. The numerical analysis was conducted using an arbitrary-
precision implementation of the Painlevé VI and V tau functions with a Fourier
truncation of the operators $\mathsf{A}$, $\mathsf{D}$ and $\mathsf{D}_{0}$ at
$N=32$ levels, using the Arb library at $96$-digit precision222The
implementation of the Painlevé VI and V tau functions in Julia programming
language can be obtained in  https://github.com/strings-ufpe/painleve. The
authors thank O. Lisovyy for clarification on the details of the truncation in
a private communication.. The library is designed to control numerical
uncertainties by performing calculations with intervals of complex numbers. In
all of the analysis the intervals were too small to be of significance, of
order $~{}10^{-12}$, and they will be omitted from the analysis.

In Fig. 4 we display a typical plot of QNMs frequencies as a function of
$r_{+}$ for $\ell=1$. One sees that modes with either $m_{1}$ or $m_{2}$
positive are unstable for small values of $r_{+}$ and moderate values of
$a_{1}$ and $a_{2}$, of which the example shown is characteristic. From the
superradiant condition $\Im\theta_{+}<0$, one sees that there are two
competing factors determining whether the mode will be unstable or not: we
have the positive contribution from the real part of the frequency and the
negative contribution from the angular velocity. Now, as one turns on the
rotation parameters $a_{1}$ and $a_{2}$, the real part of the frequency gets
negative corrections that overwhelm the growth in angular velocity,
effectively reducing the superradiant window. The “kink” in the zoomed in
inset as $\Im\theta_{+}$ crosses to positive values comes from the change of
sign of $\Lambda$ in (66). In the zero temperature limit, this change of sign
will induce a discontinuity in the $s$ value obtained from the quantization
condition (81).

Figure 5: Stable (top, blue) and unstable (bottom, red) modes for $\ell=2$ and
$a_{1}^{2}-a_{2}^{2}=0.001$ as functions of $r_{+}$. Like in the $\ell=1$
case, the positive imaginary part is very small and restricted to the
condition $\Im\theta_{+}<0$.

From the conformal block point of view, the calculation of the QNMs is truly
perturbative in the CFT levels, with the internal momentum centered at
$\sigma=2+\ell-\nu_{\ell}r_{+}^{2}$. It is interesting to note that these
should correspond in the $r_{+}\rightarrow 0$ limit to degenerate “heavy”
Liouville operators at integer $\sigma\simeq 2+\ell$. As a matter of fact, the
latter values for $\sigma$ do indeed correspond to poles in the conformal
block expansion at level $t_{0}^{\ell+1}$. It would be interesting to
investigate whether this alternative approach could lead to a simplified
analytic study of these modes.

Figure 6: The frequencies of the unstable modes for $\ell=1$ as a function of
$r_{+}$ for various values of $a_{12}=a_{1}^{2}-a_{2}^{2}$. Note that the
superradiant window induced by the condition $\Im\theta_{+}<0$ becomes smaller
with increasing $a_{12}$ due to the larger reduction of the real part of the
eigenfrequency.

In Fig. 5 we display the $\ell=2$ case. Again, the unstable modes correspond
to positive $m_{1}$ and/or $m_{2}$, and the region of instability is
determined by the superradiant window $\Im\theta_{+}<0$. The comparison to
$\ell=1$ – see also Fig. 6 – shows that the decay rate of the most unstable
mode Hawking:1999dp , i.e., the one with largest
$m_{1}\Omega_{1,+}+m_{2}\Omega_{2,+}$ decreases with $\ell$. Indeed, Fig. 7
shows that the maximum of the decay rate as a function of $r_{+}$ does
decrease from $\ell=1$ to $\ell=4$, making those higher $\ell$ states longer-
lived. If this trend continues for larger $\ell$, it would mean that the
decaying profile of the scalar emissions from the black hole is a combined
effect of the angular dependence of the coupling between the black hole and
the scalar perturbations on the one hand and the half-life of the perturbation
as a function of $\ell$, and the radial eigenvalue $n$ on the other.

Figure 7: The most unstable mode $m_{1}=\ell$ as a function of $r_{+}$ for
$\ell=1,2,3,4$. Note that the smaller decay rate for larger values of $\ell$
is accompanied by a larger width of the superradiant window. The black hole is
expected to decay through different values of $\ell$, $m_{1}$ and $m_{2}$
depending on the values of its charges.

One should also note from Fig. 6 that the superradiant window appears to
decrease as $a_{12}=a_{1}^{2}-a_{2}^{2}$ increases, indicating that, as far as
the contributions to $\theta_{+}$ are concerned, the negative correction to
the real part of the eigenfrequency in (114) dominates over the effect of the
increased angular rotation. So, as $a_{12}$ increases, the real part of
$\Im\theta_{+}$ turns positive. In fact, around the value $a_{12}\simeq 0.045$
there is no superradiance for $\ell=1$, as can be inferred from Fig. 6. This
is in contrast with the fragmentation phenomenon, which is expected at
sufficiently distorted – i.e, highly rotating – five-dimensional black holes
as argued in Emparan:2003sy .

##  4 Holographic decay

Given the results of the preceeding sections for the QNMs and their
instability, we may reflect on the fate of the corresponding state in the
putative dual CFT. The instabilities signal that the state associated to the
black hole will decay. The particular features of the decay, such as its rate
and final products, will of course depend on the coupling between the black
hole and the perturbation fields, which in holography can be read from the
stress-energy tensor. For the scalar type of perturbations considered here we
can deduce an interaction Hamiltonian of the sort ${\cal
H}_{\mathrm{int}}=\lambda h^{ab}T_{ab}$. As we learned from the discussion
above, the imaginary part of the QNMs frequencies remains fairly constant with
$\ell>0$, with the most unstable mode at given $\ell$ being the one with the
largest $m_{1}\Omega_{1,+}+m_{2}\Omega_{2,+}$.

Let us first consider the case where the field theory lives in the global
boundary $\mathbb{R}\times S^{3}$. The energy density profile can be read from
(23)

$4\pi
G_{N,5}\bar{\rho}=\frac{3M}{2\bar{\Delta}_{\theta}^{2}}=\frac{3M}{2}\left(1+2(a_{1}^{2}\sin^{2}\bar{\theta}+a_{2}^{2}\cos^{2}\bar{\theta})+\ldots\right).$
(115)

We interpret the expansion in the right-hand side to be the contribution of
the higher $S^{3}$ spherical harmonics. From (115) we conclude that the
generation of the $\ell$-th mode will be dampened by a factor of
$a_{i}^{\ell}$. Since $a_{1}$ and $a_{2}$ are parametrically small, of order
$r_{+}$, the corresponding five-dimensional spheroidal harmonics can be
approximated by their zero rotation counterparts, i.e, the three-dimensional
spherical harmonics Lindblom:2017maa

$Y_{\ell}^{m_{1},m_{2}}(\bar{\theta},\bar{\phi}_{1},\bar{\phi}_{2})=\sqrt{\frac{\ell+1}{2\pi^{2}}}\sqrt{\frac{((\ell+m_{1}+m_{2})/2)!((\ell-
m_{1}-m_{2})/2)!}{((\ell+m_{1}-m_{2})/2)!((\ell-m_{1}+m_{2})/2)!}}\times\\\
(\sin\bar{\theta})^{m_{1}}(\cos\bar{\theta})^{m_{2}}P_{\frac{1}{2}(\ell-
m_{1}-m_{2})}^{(m_{1},m_{2})}(\cos
2\bar{\theta})e^{im_{1}\bar{\phi}_{1}+im_{2}\bar{\phi}_{2}},$ (116)

where $P_{n}^{(a,b)}(z)$ are the Jacobi polynomials. At given $\ell$, the most
unstable mode has the energy dependence proportional to the $tt$-component of
the scalar stress-energy tensor, in turn roughly proportional to the absolute
value squared of the field itself

$\bar{\rho}_{\mathrm{fluc.}}\propto\left|\Phi_{\ell,m_{1}=\ell,m_{2}=0}\right|^{2}=\left|e^{-i\omega(\ell)\bar{t}}Y_{\ell}^{\ell,0}\right|^{2}\propto
e^{2\Im\omega(\ell)\bar{t}}\sin^{2\ell}\bar{\theta},$ (117)

which favors localization at the $\bar{\theta}\simeq\pi/2$ plane. In this
expression $\Im\omega(\ell)$ corresponds to the imaginary part of the
eigenfrequency $\omega_{1,\ell,\ell,0}$, which is assumed positive for the
unstable mode.

In the case where the dual theory lives in flat coordinates, we can in
principle just apply the transformation (29) to translate from the global
results above. However, our analysis in Sec. 3 was made by assuming
$a_{1}>a_{2}$, which in global coordinates is not a restriction because one
can interchange the azimuthal variables $\bar{\phi}_{1}$ and $\bar{\phi}_{2}$.
When we transform to flat coordinates, we choose one of them to map to the
coordinate $\hat{\phi}$, and we have to consider the two choices separately.

For the case $a_{1}>a_{2}$, we have $\hat{\phi}=\bar{\phi}_{2}$ as in Sec. 3,
and the energy profile for the fluctuation reads

$\bar{\rho}_{\mathrm{fluc.}}\propto
e^{2\Im\omega(\ell)\bar{t}}\left(\frac{(1+\tfrac{1}{4}(\hat{t}^{2}-\hat{r}^{2}))^{2}+\hat{r}^{2}\cos^{2}\hat{\theta}}{(1+\tfrac{1}{4}(\hat{t}+\hat{r})^{2})(1+\tfrac{1}{4}(\hat{t}-\hat{r})^{2})}\right)^{\ell/2}$
(118)

where

$\bar{t}=\arctan\frac{\hat{t}+\hat{r}}{2}+\arctan\frac{\hat{t}-\hat{r}}{2}.$
(119)

The case $a_{1}<a_{2}$, can be realized by setting
$\hat{\phi}=\bar{\phi}_{1}$, and the coordinate transformation changes from
(29) to

$\sin\bar{\theta}=\sin\hat{\chi}\sin\hat{\theta},$ (120)

which corresponds to interchanging $a_{1}\leftrightarrow a_{2}$ in the profile
(32). The energy dissipated in the perturbation mode is now

$\bar{\rho}_{\mathrm{fluc.}}\propto
e^{2\Im\omega(\ell)\bar{t}}\left(\frac{\hat{r}^{2}\sin^{2}\hat{\theta}}{(1+\tfrac{1}{4}(\hat{t}+\hat{r})^{2})(1+\tfrac{1}{4}(\hat{t}-\hat{r})^{2})}\right)^{\ell/2},$
(121)

with $\bar{t}$ given by (119).

We note that, with the second choice, there is a tendency for the decay to
occur in the $\hat{x}-\hat{y}$ plane ($\sin^{2}\hat{\theta}\simeq 1$),
following the maximum of $\bar{\rho}_{\mathrm{fluc.}}$. As we discussed above,
the specific value of $\ell$ which will be preferred by the decay will depend
on the details of the state and the coupling. In Fig. 7 we see that, even
though the maximum of the imaginary part of the eigenfrequency decreases with
$\ell$, larger values of $r_{+}$ tend to favor larger values of $\ell$ for the
decay. This leads us to infer that, at low temperatures, the localization
effect of the decay products resulting from (121) increases with the mass
${\cal M}$. We also point out the difference between the time dependence of
the decay modes depending whether we consider global coordinates
$\mathbb{R}\times S^{3}$ (bar coordinates) or conformally flat coordinates
$\mathbb{R}^{3,1}$ (hatted coordinates). Whereas in global coordinates the
fluctuations grow exponentially without bound, at least in the linear analysis
we do here, the growth in conformally flat coordinates is capped as
$\hat{t}\rightarrow\infty$ due to (119).

##  5 Discussion

In this paper we analyzed the holographic aspects of the generically rotating
Kerr-AdS5 black hole in Lorentzian signature, focussing on the low-temperature
limit $T_{+}\simeq 0$ and small black holes $r_{+}\ll 1$. We reviewed the
definition of asymptotic charges, discussed the geometrical meaning of the
regularization involved in the definition of mass, and determined the
holographic stress-energy tensor associated to the black hole in the dual
field theory – whose thermal state is characterized by non-zero vacuum
expectation values for all generators of the Weyl subgroup of the conformal
group $\mathrm{SO}(4,2)$. Finally, we computed the first correction to the
boundary-to-boundary scalar propagator due to the presence of the black hole
and found that it makes scalar perturbations with dimension close to the
marginal value ($\Delta=4$) irrelevant, thus suggesting that such
perturbations should not substantially change the fate of the background as
the theory flows to the IR.

In order to further analyze this question, we set out to treat scalar
perturbations by means of the exact solution constructed in Barragan-
Amado:2018pxh , adapted to the low-temperature limit. We found instabilities
in the QNMs for all values of the angular eigenvalue $\ell>0$, as anticipated
by Hawking:2000vt , using thermodynamical arguments. For $\ell=0$, the first
order correction in the temperature has an enhancement effect which increases
the decay time of the perturbation, but it is not strong enough to induce
instabilities. We have also studied numerically the dependence of the $\ell>0$
instabilities on the black hole parameters as we vary $\ell$. These
instabilities of AdS space are expected from general arguments Green:2015kur ,
and we attempted a holographic interpretation by studying the instabilities of
the thermal field theory state corresponding to the black hole and the spatial
dependence of its decay products. We found qualitative hints that the ejecta
tend to collimate at the axis of highest rotation as the black hole mass
increases.

We have also shown that the general procedure of Barragan-Amado:2018pxh ,
which solved for the non-local boundary conditions related to QNMs in terms of
monodromy data of the Heun differential equations involved, can be used
effectively to find QNMs in the zero temperature limit. This limit led us to
consider the monodromy parameters to be determined by the Painlevé V
transcendent, obtained from the confluence limit of the Painlevé VI. The
importance of the expansion of the Painlevé transcendents in terms of $c=1$
Virasoro conformal blocks, as studied in Lisovyy:2018mnj , provides us with
the same interesting factorization of the four-dimensional conformal blocks in
terms of (semi-classical) two-dimensional ones, as first anticipated by
Dolan:2003hv . The new ingredient here is that the two-dimensional conformal
blocks are of the irregular type, as also appeared in the black hole
perturbation context in CarneirodaCunha:2019tia . The semiclassical conformal
blocks associated to the radial perturbations seem to arise from a unitary
theory, and the Vertex operators associated to the inner and outer horizons
can be interpreted as thermal states, with their degeneracy equating the
entropy of the scalar perturbation with quantum numbers given by
$\omega,m_{1}$ and $m_{2}$ as it is absorbed by the black hole. Finally, the
study of QNMs at small values of $r_{+}$ showed the relevance of short
Virasoro representations for the intermediate states of the conformal blocks,
again anticipated in a different context by Fitzpatrick:2015foa .

The motivation for considering low-temperature black holes stemmed from the
necessity for pushing the hydrodynamical analogy of Bhattacharyya:2007vs , if
we want to understand the holographic interpretation of instabilities in AdS
space. We have seen that the phase diagram of these states in the dual theory
has a richer set of associated phenomena than what is expected from the
“naive” near-UV considerations, with instabilities in the higher modes and
collimation of decay products. Tools usually used in holographic
renormalization-group flow are not really suitable to determine the fate of
the state as one flows to the infrared. We can use the same methods proposed
here to study what happens at larger values of $r_{+}$, but the appearance of
instabilities as well as fragmentation issues Emparan:2003sy  – the latter
absent here – can further contribute to complicate the issue of determining
the fate of the decay process.

Finally, we stress that the isomonodromic method used here for the numerical
analysis overcomes a major hurdle arising in usual Frobenius matching. At low
temperatures, the latter method suffers from large spatial oscillations of the
modes near the outer horizon $r_{+}$. The isomonodromic method has no such
hindrance. Moreover, our investigation also shows that the mode expansion used
in Frobenius matching is not suitable to study the fundamental mode. Indeed,
in the finite temperature case, we have shown that the monodromy parameters,
and thus the actual value of the QNM frequency, receive contributions from all
levels in the Frobenius expansion. These contributions can be interpreted, in
the two-dimensional conformal block attribution, as coming from all
descendants of the CFT primaries associated to the intermediate states. We
verified that these contributions can be resummed in terms of the generating
function of the Catalan numbers. Also, short representations of the CFT
Virasoro algebra are relevant for the calculation of QNMs at higher levels. We
defer the study of other interesting black hole limits, as well as higher spin
perturbations Barragan-Amado:2020pad  to future work.

## Acknowledgements

We apologize for any omission in this important and long standing field of
research. The authors would like to thank Dmitry Melnikov, Monica Guica, Oleg
Lisovyy for comments and suggestions along the way. Finally, we thank the
referee for pointing out the necessity of further comparison between (110) and
our numerical results, motivating a revision of the fundamental mode frequency
in a previous version of this paper.

##  Appendix A Asymptotic charges of AdS spaces

Let the asymptotic metric be given by

$g_{ab}=\frac{1}{\tilde{\Omega}^{2}}\tilde{g}_{ab}+h_{ab}$ (122)

where $h_{ab}$ is understood to be small with respect to the first term in the
$\tilde{\Omega}\rightarrow 0$ limit, in a manner that will become precise at
the end of the Appendix. As we will see, for asymptotically AdS spaces,
$\tilde{\Omega}$ has spacelike gradient and vanishes at the conformal
boundary. The boundary structure is dependent on the particular choice of
$\tilde{\Omega}$, and the two choices considered in this paper are
$\tilde{g}_{ab}$ is either the flat metric (Poincaré patch), called
$\hat{g}_{ab}$ in the main text, or the $\mathbb{R}\times S^{3}$ metric
(global coordinates), referred to by $\bar{g}_{ab}$ in the main text. Let us
introduce the covariant derivates $\nabla_{a}$, associated with $g_{ab}$ and
$\tilde{\nabla}_{a}$, associated with $\tilde{g}_{ab}$. If $\chi^{a}$ is a
vector field, we can use Leibniz rule to show that the difference
$\nabla_{a}\chi^{b}-\tilde{\nabla}_{a}\chi^{b}$ is a linear operator on
$\chi^{a}$, therefore, for any two covariant derivatives, we have

$\nabla_{a}\chi^{b}=\tilde{\nabla}_{a}\chi^{b}+{C^{b}}_{ac}\chi^{c},$ (123)

with the difference between the connections given by the conditions that
$\nabla_{a}$ is compatible with $g_{ab}$ and $\tilde{\nabla}_{a}$ with
$\tilde{g}_{ab}$. To first order in $h_{ab}$:

$\displaystyle{C^{b}}_{ac}$
$\displaystyle=\frac{1}{2}g^{bd}(\tilde{\nabla}_{a}g_{dc}+\tilde{\nabla}_{c}g_{ad}-\tilde{\nabla}_{d}g_{ac})$
(124)
$\displaystyle=\frac{1}{2}\tilde{\Omega}^{2}(\tilde{g}^{bd}-\tilde{\Omega}^{2}h^{bd})(\tilde{\nabla}_{a}(\frac{1}{\tilde{\Omega}^{2}}\tilde{g}_{dc}+h_{dc})+\tilde{\nabla}_{c}(\frac{1}{\tilde{\Omega}^{2}}\tilde{g}_{ad}+h_{ad})-\tilde{\nabla}_{d}(\frac{1}{\tilde{\Omega}^{2}}\tilde{g}_{ac}+h_{ac})),$
(125)

with indices in the right hand side now and hereafter raised with
$\tilde{g}^{ab}$, so that $h^{ab}\equiv\tilde{g}^{ac}h_{cd}\tilde{g}^{db}$.
Expanding up to quadratic terms in $h_{ab}$,

${C^{b}}_{ac}=-\frac{1}{\tilde{\Omega}}(\delta_{c}^{b}\tilde{\nabla}_{a}\tilde{\Omega}+\delta_{a}^{b}\tilde{\nabla}_{c}\tilde{\Omega}-\tilde{g}_{ac}\tilde{\nabla}^{b}\tilde{\Omega})+\tilde{\Omega}(h_{c}^{b}\tilde{\nabla}_{a}\tilde{\Omega}+h_{a}^{b}\tilde{\nabla}_{c}\tilde{\Omega}-\tilde{g}_{ac}h^{bd}\tilde{\nabla}_{d}\tilde{\Omega})\\\
+\tfrac{1}{2}\tilde{\Omega}^{2}(\tilde{\nabla}_{a}h_{c}^{b}+\tilde{\nabla}_{c}h_{a}^{b}-\tilde{\nabla}^{b}h_{ac}).$
(126)

Conserved quantities associated with vector fields $\xi^{a}$ are defined as
the integral of the $(d-2)$-form

$Q[\xi]_{a_{1}\ldots a_{d-2}}=-\frac{1}{2\kappa}\epsilon[g]_{aba_{1}\ldots
a_{d-2}}g^{ac}\nabla_{c}\xi^{b},$ (127)

with $\epsilon[g]$ the volume form associated with $g_{ab}$333For convenience,
a term equal to the volume of the $d-2$ sphere dividing $Q[\xi]$ is omitted..
The contravariant derivative can also be computed to first order in $h_{ab}$:

$g^{ac}\nabla_{c}\xi^{b}=\tilde{\Omega}^{2}(\tilde{g}^{ac}-\tilde{\Omega}^{2}h^{ac})(\tilde{\nabla}_{c}\xi^{b}+{C^{b}}_{cd}\xi^{d})$
(128)

or

$g^{ac}{C^{b}}_{cd}=-\tilde{\Omega}(\delta_{d}^{b}\tilde{\nabla}^{a}\tilde{\Omega}+\tilde{g}^{ab}\tilde{\nabla}_{d}\tilde{\Omega}-\delta^{a}_{d}\tilde{\nabla}^{b}\tilde{\Omega})+\tilde{\Omega}^{3}(h^{b}_{d}\tilde{\nabla}^{a}\tilde{\Omega}+h^{ab}\tilde{\nabla}_{d}\tilde{\Omega}-h^{a}_{d}\tilde{\nabla}^{b}\tilde{\Omega})\\\
+\tilde{\Omega}^{3}(\delta_{d}^{b}h^{ac}\tilde{\nabla}_{c}\tilde{\Omega}+h^{ab}\tilde{\nabla}_{d}\tilde{\Omega}-\delta_{d}^{a}h^{bc}\tilde{\nabla}_{c}\tilde{\Omega})+\tfrac{1}{2}\tilde{\Omega}^{4}(\tilde{\nabla}^{a}h_{d}^{b}+\tilde{\nabla}_{d}h^{ab}-\tilde{\nabla}^{b}h_{d}^{a}),$
(129)

Now, the vector fields we are interested in are Killing vector fields of the
induced metric at the conformal boundary $\tilde{\Omega}=0$. These will
satisfy:

$\xi^{d}\tilde{\nabla}_{d}\tilde{\Omega}=0,\quad\quad\tilde{\nabla}^{a}\xi^{b}+\tilde{\nabla}^{b}\xi^{a}=0,$
(130)

denoting the vanishing Lie derivatives with respect to $\xi^{a}$ of
$\tilde{\Omega}$ and $\tilde{g}^{ab}$, respectively. Note that we are assuming
that $\xi^{a}$ and $\tilde{\nabla}^{a}\tilde{\Omega}$ are orthogonal, which is
suitable for the asymptotically AdS case.

Wrapping it all up,

$g^{ac}\nabla_{c}\xi^{b}=2\tilde{\Omega}\xi^{[a}\tilde{\nabla}^{b]}\tilde{\Omega}+\tilde{\Omega}^{2}\tilde{\nabla}^{[a}\xi^{b]}-2\tilde{\Omega}^{3}(\xi^{d}h_{d}^{[a}\tilde{\nabla}^{b]}\tilde{\Omega}+\xi^{[a}h^{b]c}\tilde{\nabla}_{c}\tilde{\Omega})\\\
+\tilde{\Omega}^{4}(\xi^{d}\tilde{\nabla}^{[a}h_{d}^{b]}-h^{d[a}\tilde{\nabla}_{d}\xi^{b]}+\tfrac{1}{2}\mathsterling_{\xi}h^{ab})+{\cal
O}(h^{2})$ (131)

where $\mathsterling_{\xi}h^{ab}$ is the Lie derivative of $h^{ab}$ with
respect to $\xi^{a}$. The volume form is readily computed:

${\bm{\epsilon}}[g]=\frac{1}{\tilde{\Omega}^{d}}{\bm{\epsilon}}[\tilde{g}](1+\tfrac{1}{2}\tilde{\Omega}^{2}h_{a}^{a}+{\cal
O}(h^{2}))$ (132)

and so the charge density can be expanded

$Q[\xi]_{a_{1}\ldots
a_{d-2}}=-\frac{1}{2\kappa}\epsilon[\tilde{g}]_{aba_{1}\ldots
a_{d-2}}\frac{1}{\tilde{\Omega}^{d-1}}(2\xi^{[a}\tilde{\nabla}^{b]}\tilde{\Omega}+\tilde{\Omega}\tilde{\nabla}^{[a}\xi^{b]}\\\
-2\tilde{\Omega}^{2}(\xi^{d}h_{d}^{[a}\tilde{\nabla}^{b]}\tilde{\Omega}+\xi^{[a}h^{b]c}\tilde{\nabla}_{c}\tilde{\Omega}-\tfrac{1}{2}h_{c}^{c}\xi^{[a}\tilde{\nabla}^{b]}\tilde{\Omega})\\\
+\tilde{\Omega}^{3}(\xi^{d}\tilde{\nabla}^{[a}h_{d}^{b]}-h^{d[a}\tilde{\nabla}_{d}\xi^{b]}+\tfrac{1}{2}h_{c}^{c}\tilde{\nabla}^{[a}\xi^{b]})+{\cal
O}(h^{2}))$ (133)

By integrating ${\mathbf{Q}}[\xi]$ in the conformal boundary
$\tilde{\Omega}\rightarrow 0$ limit we can find the conserved quantity.
Supposing a falloff behavior of the stress-energy tensor, the Einstein
equation for negative cosmological constant (and $\ell_{AdS}=1$) near infinity
is written as:

$R_{ac}=-(d-1){g_{ac}},$ (134)

which, when written in terms of $\tilde{g}_{ab}$, yields, up to terms of order
$h_{ab}$:

$\tilde{R}_{ac}+(d-2)\frac{\tilde{\nabla}_{a}\tilde{\nabla}_{c}\tilde{\Omega}}{\tilde{\Omega}}-(d-1)\tilde{g}_{ac}\tilde{g}^{bd}\frac{\tilde{\nabla}_{b}\tilde{\Omega}\tilde{\nabla}_{d}\tilde{\Omega}}{\tilde{\Omega}^{2}}+\tilde{g}_{ac}\tilde{g}^{bd}\frac{\tilde{\nabla}_{b}\tilde{\nabla}_{d}\tilde{\Omega}}{\tilde{\Omega}}=-(d-1)\frac{\tilde{g}_{ac}}{\tilde{\Omega}^{2}}.$
(135)

Requiring that the term proportional to $\tilde{\Omega}^{-2}$ vanishes at the
boundary implies that $n^{a}=\tilde{g}^{ab}\tilde{\nabla}_{b}\tilde{\Omega}$
is normalized space-like according to $\tilde{g}_{ab}$ in the
$\tilde{\Omega}\rightarrow 0$ limit. We note, however, that the order ${\cal
O}(\tilde{\Omega}^{-1})$ term will set the (conformal, Ricci) geometry at
infinity through

$\tilde{R}_{ac}-\frac{1}{2(d-1)}\tilde{g}_{ac}\tilde{R}=-(d-2)\frac{1}{\tilde{\Omega}}\tilde{\nabla}_{a}\tilde{\nabla}_{c}\tilde{\Omega}.$
(136)

Therefore the conformal geometry chosen for the four-dimensional manifold at
infinity will influence the asymptotics of $\tilde{\Omega}$, and thus the
actual value of the asymptotic charges.

We may consider now two cases for $\mathbf{Q}[\xi]$, depending on $\xi^{a}$:

  1. 1.

$\xi^{a}$ is the time translation operator, and therefore orthogonal to the
surface of integration used to define the conserved quantity;

  2. 2.

$\xi^{a}$ generates rotations, so it is tangent to the surface of integration;

In the first case, the term $\xi^{[a}\tilde{\nabla}^{b]}\tilde{\Omega}$ is a
bi-vector normal to the surface of integration and $\mathbf{Q}[\xi]$ diverges
in the $\tilde{\Omega}\rightarrow 0$ limit. One usually deals with this
divergence by adding a boundary (Gibbons-Hawking-York) term to the action to
regularize it, although the procedure of fixing the conformal structure, much
in the spirit of what has been done in this calculation, was defended in
Witten:2018lgb . For the perturbed conformal metric, the vector
$\tilde{\nabla}^{a}\tilde{\Omega}$ approaches the unit vector at spatial
infinity $n^{a}$ by:

$n^{a}=(1+N)^{-1/2}\tilde{\nabla}^{a}\tilde{\Omega},\quad\quad
N=\tilde{\Omega}^{2}h_{ab}\tilde{\nabla}^{a}\tilde{\Omega}\tilde{\nabla}^{b}\tilde{\Omega}.$
(137)

The second term in brackets in the first line of (133) is
$\tilde{\Omega}\tilde{\nabla}^{[a}\xi^{b]}$ which we will also take to vanish,
either because $\xi^{a}$ is covariantly constant with respect to
$\tilde{g}_{ab}$, as in the case of time translation, or because the
derivative is tangent to the surface of integration, as in the case of
rotations. The next terms will converge in the $\tilde{\Omega}\rightarrow 0$
limit if $h_{ab}=\tilde{\Omega}^{d-3}\gamma_{ab}$. Substituting in (133) and
taking the limit we have:

$Q[\xi]_{a_{1}\ldots
a_{d-2}}=\frac{1}{\kappa}\epsilon[\tilde{g}]_{aba_{1}\ldots
a_{d-2}}((d-1)n^{[a}\gamma^{b]}_{c}\xi^{c}-2\xi^{[a}\gamma^{b]}_{c}n^{c}+(\gamma^{c}_{c}+N)\xi^{[a}n^{b]})+{\cal
O}(\tilde{\Omega}).$ (138)

One can then conclude that the value for generic charges in asymptotically AdS
spaces stems not only from the conformal structure encoded in $\tilde{g}$, but
also from the notion of “normalized direction in RG flow”, encoded by the
normalization of $n^{a}$.

##  Appendix B The Fredholm determinant formulation of the Painlevé VI
transcendent

This is a review of the Fredholm determinant formulation of the Painlevé VI,
borrowing heavily from Gavrylenko:2016zlf

$\tau(t)=\mathrm{const}\cdot
t^{\frac{1}{4}(\sigma^{2}-\theta_{0}^{2}-\theta_{t}^{2})}(1-t)^{-\frac{1}{2}\theta_{t}\theta_{1}}\det(\mathbbold{1}-\mathsf{A}\Phi(t)\mathsf{D}\Phi(t)^{-1}),$
(139)

where the Plemelj operators $\mathsf{A},\mathsf{D}$ act on the space of pairs
of square-integrable functions defined on ${\cal C}$, a circle on the complex
plane with radius $R<1$:

$(\mathsf{A}g)(z)=\oint_{\cal C}\frac{dz^{\prime}}{2\pi
i}A(z,z^{\prime})g(z^{\prime}),\quad(\mathsf{D}g)(z)=\oint_{\cal
C}\frac{dz^{\prime}}{2\pi i}D(z,z^{\prime})g(z^{\prime}),\quad
g(z)=\begin{pmatrix}f_{+}(z)\\\ f_{-}(z)\end{pmatrix},$ (140)

with kernels given, for $|t|<R$, explicitly by

$\begin{gathered}A(z,z^{\prime})=\frac{\Psi(\sigma,\theta_{1},\theta_{\infty};z)\Psi^{-1}(\sigma,\theta_{1},\theta_{\infty};z^{\prime})-\mathbbold{1}}{z-z^{\prime}},\\\
D(z,z^{\prime})=\frac{\mathbbold{1}-\Psi(-\sigma,\theta_{t},\theta_{0};t/z)\Psi^{-1}(-\sigma,\theta_{t},\theta_{0};t/z^{\prime})}{z-z^{\prime}}.\end{gathered}$
(141)

The operators $\mathsf{A}$ and $\mathsf{D}$ can be thought of as projecting
the analytic and principal parts of a generic function defined on the circle
${\cal C}$ into the space generated by functions with definite monodromy. The
parametrix $\Psi$ and the “gluing” matrix $\Phi$ are

$\leavevmode\resizebox{422.77661pt}{}{$\displaystyle\Psi(\alpha_{1},\alpha_{2},\alpha_{3};z)=\begin{pmatrix}\phi(\alpha_{1},\alpha_{2},\alpha_{3};z)&\chi(\alpha_{1},\alpha_{2},\alpha_{3};z)\\\
\chi(-\alpha_{1},\alpha_{2},\alpha_{3};z)&\phi(-\alpha_{1},\alpha_{2},\alpha_{3};z)\end{pmatrix},\quad\Phi(\kappa,\sigma;t)=\begin{pmatrix}t^{-\sigma/2}\kappa^{-1/2}&0\\\
0&t^{\sigma/2}\kappa^{1/2}\end{pmatrix}$},$ (142)

with $\phi$ and $\chi$ given in terms of Gauss’ hypergeometric function – note
the overall minus sign with respect to the conventions in Gavrylenko:2016zlf :

$\displaystyle\begin{gathered}\phi(\alpha_{1},\alpha_{2},\alpha_{3};z)={{}_{2}F_{1}}(\tfrac{1}{2}(\alpha_{1}-\alpha_{2}+\alpha_{3}),\tfrac{1}{2}(\alpha_{1}-\alpha_{2}-\alpha_{3});\alpha_{1};z)\\\
\chi(\alpha_{1},\alpha_{2},\alpha_{3};z)=\frac{\alpha_{3}^{2}-(\alpha_{1}-\alpha_{2})^{2}}{4\alpha_{1}(1+\alpha_{1})}z\,{{}_{2}F_{1}}(1+\tfrac{1}{2}(\alpha_{1}-\alpha_{2}+\alpha_{3}),1+\tfrac{1}{2}(\alpha_{1}-\alpha_{2}-\alpha_{3});2+\alpha_{1};z).\end{gathered}$

(143)

Finally, $\kappa$ is a known function of the monodromy parameters:

$\kappa=s\frac{\Gamma^{2}(1-\sigma)}{\Gamma^{2}(1+\sigma)}\frac{\Gamma(1+\tfrac{1}{2}(\theta_{t}+\theta_{0}+\sigma))\Gamma(1+\tfrac{1}{2}(\theta_{t}-\theta_{0}+\sigma))}{\Gamma(1+\tfrac{1}{2}(\theta_{t}+\theta_{0}-\sigma))\Gamma(1+\tfrac{1}{2}(\theta_{t}-\theta_{0}-\sigma))}\times\\\
\frac{\Gamma(1+\tfrac{1}{2}(\theta_{1}+\theta_{\infty}+\sigma))\Gamma(1+\tfrac{1}{2}(\theta_{1}-\theta_{\infty}+\sigma))}{\Gamma(1+\tfrac{1}{2}(\theta_{1}+\theta_{\infty}-\sigma))\Gamma(1+\tfrac{1}{2}(\theta_{1}-\theta_{\infty}-\sigma))},$
(144)

and, finally, the parameter $s$ is given in terms of the monodromy parameters
$\\{\sigma_{0t},\sigma_{t1}\\}$:

$s=\frac{(w_{1t}-2p_{1t}-p_{0t}p_{01})-(w_{01}-2p_{01}-p_{0t}p_{1t})\exp({\pi
i\sigma_{0t}})}{(2\cos\pi(\theta_{t}-\sigma_{0t})-p_{0})(2\cos\pi(\theta_{1}-\sigma_{0t})-p_{\infty})},$
(145)

where

$\begin{gathered}p_{i}=2\cos\pi\theta_{i},\quad p_{ij}=2\cos\pi\sigma_{ij},\\\
w_{0t}=p_{0}p_{t}+p_{1}p_{\infty},\quad
w_{1t}=p_{1}p_{t}+p_{0}p_{\infty},\quad
w_{01}=p_{0}p_{1}+p_{t}p_{\infty}.\end{gathered}$ (146)

The expansion for the tau function at small $t$ can be computed directly from
its definition:

$\tau(t)=Ct^{\frac{1}{4}(\sigma^{2}-\theta_{0}^{2}-\theta_{t}^{2})}(1-t)^{\frac{1}{2}\theta_{1}\theta_{t}}\left(1+\left(\frac{\theta_{1}\theta_{t}}{2}+\frac{(\theta_{0}^{2}-\theta_{t}^{2}-\sigma^{2})(\theta_{\infty}^{2}-\theta_{1}^{2}-\sigma^{2})}{8\sigma^{2}}\right)t\right.\\\
-\frac{(\theta_{0}^{2}-(\theta_{t}-\sigma)^{2})(\theta_{\infty}^{2}-(\theta_{1}-\sigma)^{2})}{16\sigma^{2}(1+\sigma)^{2}}\kappa
t^{1+\sigma}\\\
\left.-\frac{(\theta_{0}^{2}-(\theta_{t}+\sigma)^{2})(\theta_{\infty}^{2}-(\theta_{1}+\sigma)^{2})}{16\sigma^{2}(1-\sigma)^{2}}\kappa^{-1}t^{1-\sigma}+\ldots\right).$
(147)

The accessory parameter $K_{0}$, defined by (53) as essentially the logarithm
derivative of $\tau$, has an analogous expansion for small $t$ and can be seen
in Barragan-Amado:2018pxh .

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