††institutetext: Dipartimento di Fisica, Università di Roma “Tor Vergata” &
Sezione INFN Roma2, Via della ricerca scientifica 1, 00133, Roma, Italy

# 2-charge circular fuzz-balls and their perturbations

Massimo Bianchi  Giorgio Di Russo

###### Abstract

We perform a detailed study of perturbations around 2-charge circular fuzz-
balls and compare the results with the ones obtained in the case of ‘small’
BHs. In addition to the photon-sphere modes that govern the prompt ring-down,
we find a new branch of long-lived QNMs localised inside the photon-sphere at
the (meta)stable minimum of the radial effective potential. The latter are
expected to dominate late time signals in the form of ‘echoes’. Moreover,
contrary to ‘small’ BHs, some ‘static’ tidal Love numbers are non-zero and
independent of the mass, charges and angular momentum of the fuzz-ball. We
rely on the recently established connection between BH or fuzz-ball
perturbation theory and quantum Seiberg-Witten curves for ${\cal N}=2$ SYM
theories, which in turn are related to Liouville CFT via the AGT
correspondence. We test our results against numerical results obtained with
Leaver’s method of continuous fractions or Breit-Wigner resonance method for
direct integration and with the WKB approximation based on geodesic motion. We
also exclude rotational super-radiance, due to the absence of an ergo-region,
and absorption, due to the absence of a horizon.

##  1 Introduction

Discriminating fuzz-balls from BHs is a challenging endeavour Guo:2017jmi ;
Bena:2020see ; Bianchi:2020bxa ; Bena:2020uup ; Bianchi:2020miz ;
Mayerson:2020tpn ; Ikeda:2021uvc ; Bah:2021jno ; Fransen:2022jtw ;
Cano:2022wwo . Recently significant progress has been made in the analysis of
BH and fuzz-ball perturbation theory relying on the remarkable connection with
quantum Seiberg-Witten (SW) curves for ${\cal N}=2$ SYM theories
Aminov:2020yma ; Bianchi:2021xpr ; Bianchi:2021mft ; Bonelli:2021uvf ;
Bonelli:2022ten ; Consoli:2022eey . Unfortunately this approach requires
separability and integrability of the relevant dynamics. Since most micro-
state geometries only admit time translation isometry, little progress can be
made in this direction unless one restricts one’s attention onto very special
classes of micro-states. Circular profiles in the two-charge system stand out
as a very convenient playground to test ideas and address long-standing issues
Lunin:2001jy ; Mathur:2009hf . Indeed both geodesic motion and wave equation
are separable in this background and the study of Quasi Normal Modes (QNMs),
absorption and tidal deformability can be tackled with renewed strength thanks
to the availability of new powerful techniques borrowed from SYM.

Moreover varying the radius $a_{f}$ of the circular profile one can
interpolate between a ‘small’ BH111The BH is ‘small’ in that it has zero
horizon area in the supergravity approximation. ($a_{f}=0$) and flat space-
time ($a_{f}\rightarrow\infty$). One can envisage an average of the circular
fuzz-ball observables and a comparison with the small BH ones. This would be a
first step towards a much more laborious analysis for generic micro-states of
‘large’ (BPS) BHs with at least three charges in $D=5$ or four charges in
$D=4$.

Our toy model exposes new features that in principle should allow to
discriminate circular fuzz-balls from (small) BHs. As we will see the absence
of a horizon, replaced by a smooth cap in the geometry, produces a new set of
long-lived QNMs that correspond to meta-stable states trapped in the
‘potential well’ inside the photon-sphere that slowly leak out to infinity by
‘quantum’ tunnelling through the barrier222We put ‘quantum’ in quotes since we
actually deal with ’classical’ waves!. In binary mergers these long-lived QNMs
should appear in the echoes Abedi:2016hgu ; Cardoso:2017cqb ; Oshita:2018fqu ;
Wang:2018gin ; Barack:2018yly ; Wang:2019rcf ; Cardoso:2019rvt ;
Barausse:2020rsu  that dominate the late stage of the GW signal and should be
‘easily’ recognisable from the photon-sphere QNMs that dominate the prompt
ring-down phase.

Morever, despite the presence of angular momentum in the two independent
spatial hyperplanes, the solution has no ergo-region ($g_{tt}<0$ always), so
no (near) super-radiant modes are present in the spectrum. The absorption
cross section is zero, while it is non-vanishing for the small BH. We also
point out differences in the tidal deformability both in the static
($\omega=0$) and non-static ($\omega\neq 0$) case. Such a comparison between
‘small’ BHs and ‘stringy’ micro-states is very intriguing and, combined with
analysis of chaotic behaviour in scattering amplitudes involving highly
excited string states Gross:2021gsj ; Rosenhaus:2021xhm ; Firrotta:2022cku ;
Bianchi:2022mhs  or in BH dynamics Shenker:2013pqa ; Balasubramanian:2016ids ;
Saraswat:2021ong  should pave the way to further our understanding of the
string-BH correspondence Horowitz:1996nw ; Damour:1999aw .

The plan of the paper is as follows.

We will start by reanalysing massive probes in the 2-charge small BH
background. After discussing geodesic motion we pass to consider scalar wave
equations that we initially solve in the WKB approximation. We then establish
the dictionary with the quantum SW curve for ${\cal N}=2$ SYM theory with
gauge group $SU(2)$ and no hypers. In turn this is equivalent to a (modified)
Mathieu equation (a Doubly Reduced Doubly Confluent Heun Equation or DRDCHE
for short) whose solution is analogous to the one for (massive) probes in the
D3-brane background. The angular part is elementary in that it is simply given
in terms of spherical harmonics. Finally we compare the results with the ones
obtained using numerical methods based on Leaver’s continuous fractions.

We then address the same issues in the D1/D5 $\approx$ D3/D3’ circular fuzz-
ball of radius $a_{f}$. After recalling the form of the metric, we study both
geodesic motion and scalar wave equation and show separability. The new
feature is the dependence of the separation constant $K^{2}$ on the
combination $a_{f}\omega$ very much as in rotating Kerr BHs. Both for the
radial and the angular dynamics the relevant ODE is a Reduced Confluent Heun
Equation or RCHE for short that can be identified as the quantum SW curve for
${\cal N}=2$ SYM theory with gauge group $SU(2)$ and $N_{f}=(2,0)$ hypers. We
study perturbations and find a new class of QNMs that are localised at the
(meta)stable minimum of the radial effective potential that differ from the
‘standard’ ones, known as the light-ring or photon-sphere modes, localised at
the unstable maximum of the effective potential. In order to illustrate the
analysis we consider two particular cases $\ell\sim m_{\psi}$ (with
$m_{\phi}=0$) and $\ell\sim m_{\phi}+..$ (with $m_{\psi}=0$) corresponding to
‘motion’ in two special hyperplanes, $\theta=0$ and $\theta=\pi/2$
respectively. We check the results against direct integration and Breit-Wigner
resonances, especially tailored for the long-lived modes.

We also study tidal deformability and show that some static Tidal Love Numbers
(TLNs) are non-vanishing and independent of the fuzzball parameters in a
certain normalisation. This is a striking result that may be taken as another
smoking gun for the existence of some micro-structure at the would-be horizon
scale.

Finally we argue for the absence of instabilities or super-radiance due to the
absence of an ergo-region, since $g_{tt}<0$ everywhere, and the lack of
absorption due to the absence of a horizon where to dump the incoming waves.

We conclude with a summary of our results and an outlook.

Two appendices contain a brief introduction to the AGT corrspondence and a
collection of tables of QNMs for fuzz-balls.

##  2 2-charge systems

The 2-charge (BPS) system is a very useful play-ground to test ideas and
conjectures concerning the fate of (charged) BHs in string theory and arguably
in any consistent theory of quantum gravity.

There is a variety of ways of realising this system: from momentum-winding
states to bound states of D1- and D5-branes (or equivalently D3-D3’). While
the string theory description is always reliable / consistent, in fact
equivalent in different frames, the validity of the low-energy supergravity
approximation lead one to focus on the D-brane picture.

The degeneracy of (micro)states with fixed charges $P$ and $W$ grows as

$d(P,W)=\exp{4\pi\sqrt{PW}}$ (1)

even though in the supergravity description the solution has no event horizon,
one can argue for the presence of a ‘stretched horizon’ Sen:2004dp  and for
this reason the solution it is often dubbed a ’small BH’ Cano:2018hut . Adding
one more charge $K$ allows to reproduce the asymptotic behaviour of ‘large’
(BPS) BHs in $D=5$ with

$d(P,W,K)=\exp{4\pi\sqrt{KPW}}=e^{S_{BH}}$ (2)

and explain the microscopic origin of the BH entropy Strominger:1996sh .
Unfortunately, despite significant progress Bena:2016ypk , only a small
fraction of the 3-charge micro-states are explicitly known Cvetic:2022fnv  and
even for them probing the geometry with massless/ massive particles or waves
turns out to be rather laborious due to the lack of separability/integrability
with flat space-time asymptotics333With AdS boundary conditions, the $(1,0,n)$
system looks separable. Bianchi:2018kzy ; Bena:2019azk .

For this reason, we focus on 2-charge states in $D=5$ in the D-brane
formulation. We choose the ‘string’ profile to be circular Lunin:2001jy ;
Bianchi:2017sds  to achieve integrability while keeping one free parameter
(the radius $a_{f}$) that should allow to test averaging and other effects in
the relevant (reduced) ensemble of micro-states. The most general profile can
oscillate in 8 transverse directions. The relevant CFT is an ${\cal N}=(4,4)$
$\sigma$-model with target space $(T^{4})^{N}/S_{N}$ with $N=PW$ that admits a
marginal deformations interpolating from the ‘free’ orbifold point (at weak
coupling), whereby the theory is integrable, to the ‘near’-horizon limit (at
strong coupling), whereby holography connects the CFT to Type IIB strings on
$AdS_{3}\times S^{3}\times T^{4}$. One can replace $T^{4}$ with $K3$ slightly
changing the description but with similar properties.

One can also interpolate between the non-singular horizonless geometry and the
singular small BHs varying the parameters of the profile. In particular, for
the circular profile we focus on, this corresponds to taking $a_{f}\rightarrow
0$. This will be crucial for our investigation and for testing (our)
conjectures on how the ‘small’ BH behaviour is reproduced by the micro-states.
Moreover finding ‘classical’ observables that discriminate (small) BHs from
2-charge (circular) fuzz-balls may suggest that a similar situation should
hold true for far more interesting large (BPS) BHs in $D=5$ (3-charge) and
$D=4$ (4-charge). Hopefully one may expect a similar analysis could carry on
to neutral non-BPS BHs and their micro-states Bah:2022yji . Once again one
should keep in mind that the explicitly known solutions are not particularly
amenable to the study of perturbations. Very much as for super-strata (and
even more so for multi-center solutions) they only admit reduced isometry,
typically time-translation and maybe one rotation. Another interesting
possibility is the breaking of equatorial symmetry Fransen:2022jtw  that is
not incompatible with integrability.

##  3 BH and fuzz-ball PT: a lightning review

BH perturbation theory dates back to the early ’60’s when Regge, Wheeler
Regge:1957td  and then Zerilli Zerilli  studied linearised wave equations
around Schwarzschild geometry444According to K. S. Thorne KSThorne ,
‘singularity’ at that time meant the ‘horizon’, the original emphasys was on
’wormhole’, a term coined by J. A. Wheeler … as ‘black-hole’, or ‘hair, or
‘S-matrix’.

Later on Teukolsky Teuko , studied the same problem in Kerr geometry. While in
the spherically symmetric case, the angular part completely decouples from the
radial part, in the rotating case, the dynamics quite surprisingly separates
but the separation constant, a.k.a. Carter constant ChandraBH , depends on the
frequency of the perturbation and is not exactly known i. e.
$K^{2}=\ell(\ell+1)+{\cal O}(a_{J}\omega)$ where555We denote the ratio $J/M$
by $a_{J}$, since we will use $a$ and $a_{D}$ to denote the fundamental
periods of the SW curve and $a_{f}$ to denote the radius of the circular
profile later on. $a_{J}=J/M$.

###  3.1 Massive scalar perturbations

For (massive) scalar perturbations of a BH or a fuzzball, one has

${\Box}\Phi=\mu_{0}^{2}\Phi$

where
${\Box}=\sqrt{||g||}^{-1}\partial_{\mu}(\sqrt{||g||}g^{\mu\nu}\partial_{\mu}\Phi)$
is the scalar wave operator and $\mu_{0}=2\pi/\lambda$ is its inverse Compton
wave-length666One may object that the notion of Compton wave-length
$\lambda=\hbar/mc$ involves Planck constant which looks odd for ‘classical’
waves. Yet for KK excitations $\lambda\approx 2\pi R/n$ and $\mu\approx n/R$,
since $\hbar$ cancels.. Setting

$\Phi=e^{-i\omega t}e^{im_{\phi}\phi}R(r)S(\theta)$

allows in all the cases of interest here to separate the dynamics.

In particular for Schwarzschild BHs one finds ChandraBH

$R^{\prime\prime}(r)+{2(r-M)\over
r(r-2M)}R^{\prime}(r)+\Bigg{[}{r^{2}\omega^{2}\over(r-2M)^{2}}-{\ell(\ell+1)\over
r(r-2M)}\Bigg{]}R(r)=0$

and

${1\over\sin\theta}\partial_{\theta}\Big{[}\sin\theta\partial_{\theta}S(\theta)\Big{]}-{m_{\phi}^{2}\over\sin^{2}\theta}S(\theta)=-K^{2}S(\theta)$

where $K^{2}=K_{0}^{2}=\ell(\ell+1)$ and
$S_{\ell,m_{\phi}}(\theta)e^{im_{\phi}\phi}=Y_{\ell,m_{\phi}}(\theta,\phi)$
are the familiar spherical harmonics.

Both radial and angular equations can be put in canonical Schrödinger-like
form

$\Phi^{\prime\prime}(y)+Q_{BH}(y)\Phi(y)=0$ (3)

Notice that for the radial equation one has

$Q_{r,w}(r)=f(r)[\omega^{2}-V_{BH}(r)]$

where $f(r)=\left(1+{1\over{r\over 2M}-1}\right)^{2}$ plays the role of an
effective/running mass and $V_{BH}(r)$ depends on $\ell$ but neither on
$\omega$ nor on $m_{\phi}$ due to spherical symmetry.

For rotating BHs and fuzz-balls $V_{eff}$ may depend on $\omega$ both
explicitly and implicitly through the dependence of the ‘conserved’ quantities
(like $K^{2}$) on $\omega$.

In order to reduce the mass to a constant (one), a new radial variable
$\tilde{r}$, known as ‘tortoise’ coordinate, can be introduced that satisfies

$d\tilde{r}=\sqrt{f(r)}dr$

in this way the ‘horizon’, if present, is mapped to $\tilde{r}=-\infty$ while
at spatial infinity $\tilde{r}\approx r$. For Schwarzschild BHs, the
‘tortoise’ coordinate coincides with the time for radial fall, viz.

$\tilde{r}=r+2M\log\left({r\over 2M}-1\right)=t_{rad-fall}$

Bringing the resulting equation into canonical form introduces further terms
in the effective potential

$\tilde{V}_{BH}(\tilde{r})=V_{BH}(r(\tilde{r}))+\Delta V_{BH}(\tilde{r})$

For Schwarzschild BHs one has the Regge-Wheeler (Zerilli) potential ChandraBH

$\tilde{V}_{RWZ}(\tilde{r})={(r-2M)\Big{[}\ell(\ell+1)r+2M\Big{]}\over r^{4}}$

For rotating (Kerr-Newman) BHs the separation constant
$K^{2}=\ell(\ell+1)+{\cal O}(a_{J}^{2}\omega^{2}$) depends on $\omega$ and
$a_{J}=J/M$ and the definition of the tortoise coordinate is ambiguous.

For non-extremal geometries both angular and radial equations boil down to
Confluent Heun Equations (CHE) with two regular singular points (at the
horizons) and an irregular singular point (at infinity).

Until recently, neither solutions nor connection formulae for Heun Equation
and its cousins were known in closed form. As a consequence the most efficient
approach to the solution with prescribed boundary conditions (needed e.g. for
QNMs) was based on numerical methods developed by Leaver Leaver1 , relying on
continuous fractions that allow to solve (numerically) 3-term recursion
relations. When these involve more than 3 terms, Gaussian elimination may be
employed to subsequently reduce the number of terms Leaver2 . A relatively
good estimate is provided by the WKB approximation that connects with geodesic
motion in the eikonal limit (for large $\ell$) that can be used as a ‘seed’
for numerics.

In particular, QNMs Iyer:1986vv  can be related to a special class of
geodesics representing closed (un)stable (‘circular’) orbits around the
compact object (BH / fuzz-ball). They are referred to as ‘critical’ in that
they are determined by the simultaneous vanishing of the radial momentum
$P_{r}^{2}=Q(r)$ and its derivative (‘zero force’):

$Q(r)=Q^{\prime}(r)=0$ (4)

Note that the zero-force condition is compatible with both a maximum
(classically unstable) and a minimum (classically stable). When replacing
‘particles’ with waves, the minimum may become meta-stable in that the wave
may tunnel the potential barrier represented by the photon-sphere and leak out
to infinity.

This means that one should expect (at least) two sets of QNMs for Exotic
compact objects (ECOs) or fuzz-balls. For massive probes there might be
(innermost) stable circular orbits (ISCO) where light particles can be trapped
and emit electro-magnetic radiation that ’pollutes’ (in that it represent an
environmental effect) the photon-sphere and other features of the BH in vacuo
EventHorizonTelescope:2019dse ; Johnson:2019ljv ; Kapec:2022dvc ;
Bianchi:2020yzr .

The photon-sphere modes, that dominate the prompt ring-down of the remnant of
a binary merger, correspond to unstable modes localised around the tip of the
potential. These are the ‘standard’ ones that are present also in the small BH
limit at $a_{f}=0$ as we will see. The two conditions allow to determine the
critical radius $r_{c}$ and the critical impact parameter $b_{c}=(K/E)_{c}$ in
terms of the other parameters.

In the ‘classically-allowed’ regions where $Q_{r,w}$ is positive and
large777We denote by $Q_{r,w}$ the square radial momentum in the wave equation
to distinguish it from $Q_{r,geo}$., the wave equation can be solved in a
semiclassical approximation:

$\psi(r)={1\over\sqrt[4]{Q_{r,w}(r)}}e^{\pm
i\int^{r}\sqrt{Q_{r,w}(r^{\prime})}dr^{\prime}}$ (5)

This approximation fails near the zeros of $Q_{r,w}$ called $r_{\pm}$ where
the (radial) momentum vanishes and should be amended in the ‘classically-
forbidden’ regions where $Q_{r,w}$ is negative. The matching of the solution
at two consecutive inversion points implies the Bohr-Sommerfeld (BS)
quantization condition:

$\int_{r_{-}}^{r_{+}}\sqrt{Q_{r,w}(r)}dr=\pi\left(n+{1\over 2}\right)$ (6)

with $n$ a non negative integer a.k.a. overtone number. This leads to standard
energy quantisation or, in our case where everything is classical, to the
complex frequency of the (meta)stable modes localised at a local minimum of
the potential, if present in the form of an outer or internal stable light-
ring Bianchi:2020yzr .

When the two inversion points are the ends of a classically-forbidden region
and (almost) coincide (for critical geodesics), one should ‘analytically’
continue BS formula and the integral can be approximated as follows:

$\int_{r_{-}}^{r_{+}}\sqrt{Q_{r,w}(r)}dr\sim\int_{r_{-}}^{r_{+}}\sqrt{Q_{r,w}(r_{c})+{Q^{\prime\prime}_{r,w}(r_{c})\over
2}(r-r_{c})^{2}}dr\sim{i\pi
Q_{r,w}(r_{c})\over\sqrt{2Q^{\prime\prime}_{r,w}(r_{c})}}$ (7)

since $Q^{\prime}_{r,w}(r_{c})=0$. The resulting frequencies acquire an
imaginary part, i.e. $\omega=\omega_{R}+i\omega_{I}$ and characterise what are
called quasi-normal modes (QNMs).

In the WKB approximation one finds

$\omega_{QNM}^{WKB}=\omega_{c}(\ell,...)-i(2n+1)\lambda$ (8)

where $\omega_{c}$ is the angular velocity of the un-stable circular orbits
and $\lambda$ is the Lyapunov exponent governing the chaotic behaviour of
near-critical geodesics around the photon-sphere. It can be determined in a
variety of ways, the simplest being probably using ‘radial’ geodesic deviation
around $r=r_{c}$

${dr\over dt}\approx\lambda(r-r_{c})$ (9)

Notice that the bound on chaos $\lambda\leq 2\pi K_{B}T_{BH}/\hbar$ set by
Maldacena:2015waa  can be violated near extremality and must be replaced with
a more general bound Bianchi:2020des ; Bianchi:2020yzr .

In addition to the unstable orbits, some BHs admit stable circular orbits for
massive probes outside the photon-sphere that are known as ISCOs (Innermost
Stable Circular Orbits) ChandraBH . The image produced by the ETH represents a
combination of photon-sphere or photon-halo, in the case of rotating objects
since $r_{c}$ depends on the value the angular momentum $J$ and $J_{z}$ of the
probe, and emission from plasma populating the ISCOs, that depend on the
angular momentum $J$ and $J_{z}$ of the probe, too
EventHorizonTelescope:2019dse ; Johnson:2019ljv ; Kapec:2022dvc ;
Bianchi:2020yzr .

Moreover, non-singular ECOs such as fuzz-balls or generic micro-states may
also admit (meta)stable orbits inside the photon-sphere. A set of long-lived
metastable modes localised around the relative minimum may arise.

Assuming for simplicity that the radial equation can be put in the form

$\psi^{\prime\prime}+f(\omega^{2}-V_{BH})\psi=0$ (10)

with $V_{BH}$ the radial effective potential. Neglecting the (mild) dependence
of $V_{BH}$ on $\omega$888Otherwise one has to define $\Omega_{\pm}$:
$\omega^{2}-V_{BH}(...,\omega)=(\omega-\Omega_{+})(\omega-\Omega_{-})$. and
treating $f$ as an ‘effective’ mass, rather than introducing a tortoise radial
coordinate $d\xi=\sqrt{f(r)}dr$, in the WKB approximation one finds

$\omega_{QNM}^{meta}=\omega^{meta}_{R}-i\omega^{meta}_{I}$ (11)

with

$(\omega^{meta}_{R})^{2}=V_{BH}(r_{min})+\left(n+{1\over
2}\right)\sqrt{2V_{BH}^{\prime\prime}(r_{min})\over f(r_{min})}$ (12)

and

$\omega^{meta}_{I}={\pi\over{\cal T}}|T|^{2}$ (13)

where

$|T|^{2}=\exp\Bigg{[}-2\int_{r_{-}}^{r_{+}}dr\sqrt{f(r)(V_{BH}(r)-\omega^{2})}\Bigg{]}$

is the tiny probability of tunnelling ‘under’ the photon-sphere and

${\cal T}={2\pi\over\omega_{R}}={d\over v}$

gives an estimate of the time in between two bounces on the ‘wall’ of width
$d\approx{r_{+}}-{r_{-}}$. We will show that this indeed happens for circular
fuzz-balls when $a_{f}\underset{\sim}{<}L$ is large enough. For small
$a_{f}<<L$, the effective potential is too steep to accommodate any level at
all.

###  3.2 Other observables

Other observables, such as the absorption cross-section, super-radiance
amplification factor, echoes and tidal deformability require detailed
knowledge of the connection formulae that can be derived using the remarkable
correspondence between BH/fuzz-ball PT and quantum SW curves for ${\cal N}=2$
theories with $SU(2)$ gauge group and $N_{f}\leq 4$ fundamental flavours
Aminov:2020yma ; Bianchi:2021xpr ; Bianchi:2021mft ; Bonelli:2021uvf ;
Bonelli:2022ten ; Consoli:2022eey . In fact the AGT correspondence (after
Alday, Gaiotto, Tachikawa Alday:2009aq ) allows to gain further insights
Bianchi:2021xpr ; Bianchi:2021mft ; Bonelli:2021uvf ; Bonelli:2022ten ;
Consoli:2022eey  as we will see later on and in appendix A.

In canonical form and for real $\omega$ one has $Q=Q^{*}$ so that the system
admits a conserved ‘current’ 999Caveat in Liouville CFT ${k}$’s, ${p}$’s are
usually purely imaginary! Bonelli:2021uvf ; Bonelli:2022ten ; Consoli:2022eey
.

${\cal J}={\rm
Im}[\Psi^{*}\partial_{z}\Psi]=k_{H}(|C^{(H)}_{+}|^{2}-|C^{(H)}_{-}|^{2})=k_{\infty}(|C^{(\infty)}_{+}|^{2}-|C^{(2)}_{-}|^{2})$
${\cal J}_{abs}=-k_{H}|C^{(H)}_{-}|^{2}\quad,\quad{\cal
J}_{ref}=+k_{H}|C^{(H)}_{+}|^{2}$ ${\cal
J}_{in}=-k_{\infty}|C^{(\infty)}_{-}|^{2}\quad,\quad{\cal
J}_{out}=+k_{\infty}|C^{(\infty)}_{+}|^{2}$

The absorption probability is defined by

${\cal P}_{abs}(\omega)={|{\cal J}_{abs}(\omega)|\over|{\cal
J}_{in}(\omega)|}={k_{H}|C^{(H)}_{-}|^{2}\over
k_{\infty}|C^{(\infty)}_{-}|^{2}}={|C^{(\infty)}_{-}|^{2}-|C^{(\infty)}_{+}|^{2}\over|C^{(\infty)}_{-}|^{2}}=1-{|C^{(\infty)}_{+}|^{2}\over|C^{(\infty)}_{-}|^{2}}$

where the latter form relies on current conservation.

Echoes are present when the reflectivity ${\cal R}$ at the would-be horizon
(‘cap’) is nonzero ${\cal R}\neq 0$.

On the other hand, in the presence of an ergo-region, one can compute the
super-radiance amplification factor

$Z(\omega)={|{\cal J}_{out}(\omega)|\over|{\cal
J}_{in}(\omega)|}-1={k_{H}|C^{(H)}_{-}|^{2}\over
k_{\infty}|C^{(\infty)}_{-}|^{2}}$

that is positive below the super-radiant threshold

$\omega_{SR}\leq m_{\phi}\Omega_{H}$

where $\Omega_{H}$ is the angular velocity at the horizon and $m_{\phi}$ is
the azimuthal number. Super-radiance amplifies the intensity of a wave and
allows to extract energy and angular momentum from a rotating or charged BH
Brito:2015oca  very much as Penrose process Penrose:1971uk  uses particles to
perform the same task, even in smooth horizonless geometries with an ergo-
region Jejjala:2005yu ; Cardoso:2005gj ; Bianchi:2019lmi ; Chakrabarty:2019ujg
. The end-point of this ergo-region instability is expected to be a smooth
horizonless BPS geometries Giusto:2004ip .

The small imaginary parts of the super-radiant modes of near extremal rotating
BHs can be computed by means of the QNM-qSW correspondence Bianchi:2022wku .
Quite remarkably at extremality, one can also take advantage of generalised
symmetry under Couch-Torrence conformal inversions CouchTorr ; Cvetic:2020kwf
; Cvetic:2021lss ; Bianchi:2021yqs ; Bianchi:2022wku  that exchanges horizon
with infinity and keeps the photon-sphere/halo fixed! Bianchi:2021yqs ;
Bianchi:2022wku

###  3.3 BH/fuzz-ball perturbations and quantum Seiberg-Witten curves

We will now briefly describe how BH/fuzz-ball perturbations can be studied by
means of quantum SW curves of ${\cal N}=2$ SYM SW9407087 ; SW9408099 . We will
rely on the Hanany-Witten Hanany:1996ie  setup in which (linear quiver)
theories are realised on D4-branes suspended in between NS5-branes. For our
purposes it is sufficient to consider $N_{c}=2$ i.e. $SU(2)$ gauge group,
after $U(1)$ decoupling, and $N_{f}\leq 4$, in fact $N_{L},N_{R}\leq 2$. The
latter correspond to ‘flavour’ D4-brane infinitely extended to the Left or to
the Right of the two NS5-branes Hanany:1996ie .

In particular the CHE that encompasses all BH perturbations in $D=4$ can be
shown to correspond to $N_{f}=3=(2,1)$ or $N_{f}=2=(1,1)$ for the radial
equation in the extremal case (a.k.a. DCHE i.e. Doubly CHE) Aminov:2020yma ;
Bianchi:2021xpr ; Bianchi:2021mft ; Bonelli:2021uvf ; Bonelli:2022ten ;
Consoli:2022eey ; BACCP1702.01016 ; BACCP1812.08921 . For circular fuzz-balls
both radial and angular equations, coupled to one another by Carter constant,
boil down to RCHE that can be shown to correspond to $N_{f}=(2,0)$, reducing
to $N_{f}=(0,0)$ DRDCHE for the radial equation in the small BH limit whereby
$a_{f}\rightarrow 0$.

The ‘classical’ SW curve SW9407087 ; SW9408099 , whose (fundamental) periods
$a$ and $a_{D}$ encode the gauge-invariant Coulomb branch parameter

$u=\langle{\rm Tr}\phi^{2}\rangle=a^{2}+...$

can be written as

$qy^{2}\prod_{f=1}^{N_{L}}(x-m_{f})+y\prod_{u=1}^{2}(x-e_{r})+\prod_{f=1}^{N_{R}}(x-m_{f})=0$
(14)

where $N_{c}=2$ is the number of colour branes, $N_{L}+N_{R}=N_{f}$ is the
number of fundamental hypers (‘flavour’ branes), $q=\Lambda^{\beta}$ with
$\beta=4-N_{f}$ the one-loop $\beta$-function coefficient and $m_{f}$ are the
masses of the hypers.

The quantum version obtains after turning on an $\Omega$ background à la
Nekrasov-Shatashvili (NS) $(\varepsilon_{1}=\hbar,\varepsilon_{2}=0)$
Marshakov:2009gn ; Nekrasov:2009rc ; Mironov:2009uv ; Z1103.4843 ;
BF1711.07935 ; GM1806.01407 ; GGM1908.07065 . As a result

$y\rightarrow\hat{y}=y\quad,\quad x\rightarrow\hat{x}=\hbar y{d\over dy}$ (15)

In this way one obtains a second order differential equation

$\left[A(y)\hat{x}^{2}+B(y)\hat{x}+C(y)\right]U(y)=0$

with (ir)regular singularities determined by the choice of $q$, $u$ and
$m_{f}$ that can be put in canonical form

$\Psi^{\prime\prime}(y)+Q_{SW}(y)\Psi(y)=0$ (16)

or in other standard forms, depending on the number and nature of the
singularities.

After ‘quantum’ deformation, the SW differential reads

$\lambda_{+}(x)=-{x\over 2\pi i}{1\over W(x)}{dW(x)\over dx}$ (17)

where $W(x)$ can obtained recursively, using the difference equation thet
follows from $\hat{y}=e^{-\hbar\partial_{x}}$

$qP_{L}(x-{\hbar\over 2})P_{R}(x+{\hbar\over
2})W(x)W(x-\hbar)+P_{0}(x)W(x)+1=0$ (18)

with

$P_{L}(x)=\prod_{f=1}^{N_{L}}(x-m_{f})\quad,\quad
P_{R}(x)=\prod_{f=1}^{N_{R}}(x-m_{f})\quad,\quad
P_{0}(x)=\prod_{u=1}^{2}(x-e_{r})$ (19)

For $SU(2)$ (rather than $U(2)$) one sets $e_{1}+e_{2}=0$ and $e_{1}e_{2}=-u$
(the gauge invariant Coulomb branch parameter). The quantum deformed basic
period is given by

$a(u,q,m_{f};\hbar)=\oint\lambda_{+}(x)dx$ (20)

Quite remarkably for $N_{f}=4$ (i.e. $N_{L}=N_{R}=2$) one gets an equation
with 4 regular singular points that can be written in the form of Heun
Equation (HE) and governs scalar perturbations101010Dual to operators of
dimension $\Delta=1,2$ on the boundary. with ‘mass’ ${\mu}^{2}L^{2}=-2$ around
Kerr-Newman BHs in AdS4. In order to study BH/fuzz-ball PT with flat
asymptotics, one can decouple flavours one after the other by performing the
double scaling limit

$q_{N_{f}}\rightarrow 0\quad,\quad m_{M_{f}}\rightarrow\infty\quad{\rm
with}\quad q_{N_{f}}m_{N_{f}}=q_{N_{f}{-}1}\quad{\rm fixed}$ (21)

In the spherically symmetric case the angular equation decouples completely
from the radial equations and the ‘separation’ constant is simply
$K^{2}=K_{0}^{2}=\ell(\ell+{D-3})$.

When the BH/ECO/fuzz-ball carries spin (e.g. $J=Ma_{J}$ for Kerr) the
separation constant depends on the combination $a_{J}\omega$ that intertwines
angular and radial equations. For the angular dynamics, in the qSW approach,
the relevant quantisation condition involves the $a$-cycle viz.

$a^{\theta}={\ell\over{D-3}}+{1\over 2}$ (22)

in units of $\hbar=\varepsilon_{1}(=1)$, while $\varepsilon_{2}=0$. Since the
dictionary yields

$u^{\theta}={1\over 4}(1+K^{2})$ (23)

knowledge of $u(a)=a^{2}+...$ from the quantum Matone relation Matone:1995rx ;
Flume:2004rp

$u=-q{\partial{\cal F}\over\partial q},$ (24)

where ${\cal F}(a;m_{f};q;\hbar)$ is the quantum-deformed analytic pre-
potential Nekrasov:2009rc ; MM0910.5670 ; Z1103.4843 , is tantamount to
knowledge of $K^{2}$ as a function of the principal quantum number $\ell$, of
the ‘azimuthal’ quantum number(s), related to the masses $m^{\theta}_{f}$, as
well as of $q_{\theta}\approx a_{J}\omega$. In this sense, using qSW curves à
la NS solves completely the problem of finding the so-called spheroidal
harmonics and their eigen-values $K^{2}$BCC0511111 !

The analysis of the radial equation is significantly more laborious since the
relevant quantisation condition involves the $a_{D}$-cycle viz.

$2\pi ia_{D}=-{\partial{\cal F}\over\partial a}=-2a\log q+...=2\pi
i(n+\delta)$ (25)

where ${\cal F}(a,m_{f},q,\hbar)$ is the NS prepotential, while $n$ is known
as the overtone number and $\delta$ is a (mass-dependent) shift. This follows
from the identification of the ‘vanishing’ cycle in the classical limit. For
the prompt ring-down QNMs the ‘vanishing’ cycle corresponds to the photon-
sphere. For the long-lived QNMs the ‘vanishing’ cycle corresponds to the
‘internal’ classically-stable circular orbit.

In practice one has to find the dictionary

$u=U(\ell,m_{f},...Q,M,a_{J},\omega)$ (26)

and invert the relevant quantum Matone relation (24) in order to express $a$
as a function of $u$ and find ${\cal F}(a;m_{f};q;\hbar)$ (up to the one-loop
term, that can be computed by other means Nekrasov:2009rc ; MM0910.5670 ;
Z1103.4843 ). Finally one has to impose the quantisation on $a_{D}$ as an
‘eigenvalue’ equation for $\omega$, whose solutions are functions of $n$,
$\ell$, $m$’s, $M$, $Q$, $J$’s. In Bianchi:2021xpr ; Bianchi:2021mft ;
Bianchi:2022wku  we checked agreement among qSW, Leaver and WKB for a wide
variety of BHs and (bound-states of) D-branes.

Momentarily we will extend the analysis to the simplest (BPS) fuzz-ball, the
one with circular profile in $D=5$. Although the BH limit has vanishing
horizon area the degeneracy of the micro-states is a signal of a non-trivial
structure at the would-be horizon that we would like to study in detail
looking for new signatures and for guidance on how to perform ensemble
averages.

As in Bianchi:2021xpr ; Bianchi:2021mft ; Bianchi:2021yqs ; Bianchi:2022wku ,
we focus on scalar perturbations, extension to spin $s=1,2$ is straightforward
but more technical and tedious111111Since the backgrounds we consider are BPS,
in principle one can relate linearised scalar fluctuations to vector and
tensor ones by means of the unbroken supersymmetry..

##  4 Probing and re-probing small BHs with massive particles and waves

In order to set the stage for our investigation and for later comparison, we
will start by discussing (massive) neutral scalar perturbations of the ‘small’
BH solution of the D1/D5 branes system. Sometimes we switch to the T-dual
D3/D3’ description that makes more transparent the decoupling of the $T^{4}$
part of the geometry for neutral probes (i.e. those carrying only momentum
along $S^{1}$ and no KK momentum on $T^{4}$).

The metric (in the ‘democratic’ D3/D3’ frame) is Lunin:2001jy ; Mathur:2009hf
:

$ds^{2}{=}{-}H^{-1}(dt^{2}{-}dz^{2}){+}H\Big{[}dr^{2}{+}r^{2}(d\theta^{2}{+}\cos^{2}\theta
d\psi^{2}{+}\sin^{2}\theta d\phi^{2})\Big{]}{+}\sqrt{H_{1}\over
H_{5}}dz^{2}_{1,2}{+}\sqrt{H_{5}\over H_{1}}dz^{2}_{3,4}$ (27)

where

$H(r)=\sqrt{H_{1}(r)H_{5}(r)}\quad,\quad H_{i}(r)=1+{L_{i}^{2}\over r^{2}}$
(28)

where $L_{1}^{2}=32\pi^{2}g_{s}N_{1}(\alpha^{\prime})^{3}$ and
$L_{5}^{2}=g_{s}N\alpha^{\prime}$. Other Type IIB supergravity fields acquire
non-trivial profiles, but they will play no role in our present analysis.

###  4.1  Massive Geodesics

The mass shell condition for massive neutral particles in Hamiltonian
formalism reads Chervonyi:2013eja ; Bianchi:2017sds :

$\mathcal{H}={1\over 2}g^{\mu\nu}P_{\mu}P_{\nu}=-{1\over 2}\mu_{0}^{2}$ (29)

with

$2\mathcal{H}=-H\mathcal{E}^{2}+H^{-1}\Big{[}P_{r}^{2}+{P_{\theta}^{2}\over
r^{2}}+{J_{\psi}^{2}\over r^{2}\cos^{2}\theta}+{J_{\phi}^{2}\over
r^{2}\sin^{2}\theta}\Big{]}+\sqrt{H_{5}\over
H_{1}}\left(P_{1}^{2}{+}P_{2}^{2}\right)+\sqrt{H_{1}\over
H_{5}}\left(P_{3}^{2}{+}P_{4}^{2}\right)$ (30)

where $\mathcal{E}^{2}=E^{2}-P_{z}^{2}$ and the conserved momenta are:

$P_{t}={\partial\mathcal{L}\over\partial\dot{t}}=-E\quad,\quad
P_{z}={\partial\mathcal{L}\over\partial\dot{z}}\quad,\quad
P_{\psi}={\partial\mathcal{L}\over\partial\dot{\psi}}=J_{\psi}\quad,\quad
P_{\phi}={\partial\mathcal{L}\over\partial\dot{\phi}}=J_{\psi}\quad,\quad
P_{I}={\partial\mathcal{L}\over\partial\dot{Z^{I}}}$ (31)

The problem is separable and the remaining dynamical momenta are

$\displaystyle P_{r}^{2}=Q_{r,geo}(r)=H_{1}H_{5}\mathcal{E}^{2}-{K^{2}\over
r^{2}}-H_{5}\left(P_{1}^{2}+P_{2}^{2}\right)-H_{1}\left(P_{3}^{2}+P_{4}^{2}\right)-H\mu_{0}^{2}$
(32) $\displaystyle
P_{\theta}^{2}=Q_{\theta,geo}(\theta)=K^{2}-{J_{\psi}^{2}\over\cos\theta^{2}}-{J_{\phi}^{2}\over\sin^{2}\theta}$

where $K^{2}$ is the separation constant, associated to the total angular
momentum. For later use one can define an impact parameter $b=K/\mathcal{E}$.

Due to spherical symmetry, geodesics are planar. For large $b$ the geodesics
are open and correspond to small deflections angles. For small $b$ the
geodesics end on the horizon $r=0$. For a critical value $b=b_{c}$ the
geodesics wind around the small BH and end on a limiting cycle $r=r_{c}$, that
represents a critical unstable orbit. In order to determine $(r_{c},b_{c})$ or
equivalently $(r_{c},K_{c})$ one has to solve the critical conditions
$Q_{r,geo}(r_{c},K_{c})=Q^{\prime}_{r,geo}(r_{c},K_{c})=0$.

For simplicity, we will take $L_{1}=L_{5}=L$, that allow to separate the
dynamics even for massive probes. Solving the criticality conditions, one
finds

$\displaystyle
r^{2}_{c}={L^{2}{\mu}^{2}+K_{c}^{2}-2L^{2}\mathcal{E}^{2}\pm\sqrt{(2L^{2}\mathcal{E}^{2}-L^{2}{\mu}^{2}-K_{c}^{2})^{2}-4L^{4}\mathcal{E}^{2}(\mathcal{E}^{2}-{\mu}^{2})}\over
2(\mathcal{E}^{2}-{\mu}^{2})}$ (33)
$\displaystyle(2L^{2}\mathcal{E}^{2}-L^{2}{\mu}^{2}-K_{c}^{2})^{2}=4L^{4}\mathcal{E}^{2}(\mathcal{E}^{2}-{\mu}^{2})$

where $\mu^{2}=\mu_{0}^{2}+\sum_{i}P_{i}^{2}$. The only acceptable solution of
the previous system is:

$K_{c}=L\left(\mathcal{E}+\sqrt{\mathcal{E}^{2}-{\mu}^{2}}\right)\quad,\quad
r_{c}=L\sqrt[4]{{\mathcal{E}^{2}\over\mathcal{E}^{2}-{\mu}^{2}}}$ (34)

that corresponds to the un-stable orbits forming the photon-sphere (or light-
ring) when $\mu=0$. Notice that there are neither stable circular orbits
(ISCOs) outside nor inside the photon-sphere, contrary to the 4-d case. While
no large radius ISCOs are expected since for large $r$ the potential behaves
as $1/r^{2}$, that does not admit such stable orbits, there are none even at
‘small’ $r$ where relativistic corrections become important. As a result we do
not expect any additional (long-lived / meta-stable) QMNs beyond the
‘standard’ prompt ring-down modes.

In order to address this and related issues we will now consider wave
propagation and scattering in the background of a small BH.

###  4.2 Wave equation

The wave equation for massive scalar probe in the D1-D5 background (27) is of
the form:

$(\Box-\mu_{0}^{2})\Phi=0$ (35)

The ansatz

$\Phi=\exp\left(-i\omega
t+iP_{z}z+im_{\phi}\phi+im_{\psi}\psi+i\sum_{I=1}^{4}P_{I}Z^{I}\right)R(r)S(\theta).$
(36)

allows to separate the dynamics. The (polar) angular equation

$\Bigg{[}{1\over\sin 2\theta}\partial_{\theta}(\sin
2\theta\partial_{\theta})-{m_{\psi}^{2}\over\cos^{2}\theta}-{m_{\phi}^{2}\over\sin^{2}\theta}\Bigg{]}S(\theta)=-\ell(\ell+2)S(\theta)$
(37)

is solved by the spherical harmonics on $S^{3}$ which are related to the
characters of $SU(2)$ Bianchi:2017sds . The radial equation becomes:

$R^{\prime\prime}(r){+}{3\over
r}R^{\prime}(r){+}\Bigg{[}H^{2}\tilde{\omega}^{2}{-}{\ell(\ell+2)\over
r^{2}}{-}H_{5}(P_{1}^{2}{+}P_{2}^{2}){-}H_{1}(P_{3}^{2}{+}P_{4}^{2}){-}\mu_{0}^{2}H\Bigg{]}R(r){=}0$
(38)

where $\tilde{\omega}^{2}=\omega^{2}-P_{z}^{2}$. As for the geodesics, we will
set $L_{1}=L_{5}=L$ from now on, without much loss of generality. The
resulting radial equation can be put in canonical form setting
$R(r)=r^{-3/2}\Psi(r)$:

$\displaystyle\Psi^{\prime\prime}(r)+Q_{r,w}(r)\Psi(r)=0$ (39) $\displaystyle
Q_{r,w}(r)={(\tilde{\omega}^{2}-{\mu}^{2})r^{4}+\Big{[}2L^{2}\tilde{\omega}^{2}-L^{2}{\mu}^{2}+{1\over
4}-(\ell+1)^{2}\Big{]}r^{2}+L^{4}\tilde{\omega}^{2}\over r^{4}}$

###  4.3 WKB approximation

Eq (39) can be viewed as the Schrödinger equation for a non-relativistic
particle with an ‘effective mass’ $m_{eff}=f/2$ and ‘energy’ ${\cal
E}=\omega^{2}$ in a potential i.e.

$Q_{r,w}=f(\omega^{2}-U)=2m_{eff}({\cal E}-V)$ (40)

The ‘effective mass’ can be set to unity by the introduction of a ‘tortoise’
coordinate $\xi$ such that $f=2m_{eff}=(d\xi/dr)^{2}$ and a further
redefinition of $\Psi$ necessary to bring the equation back to canonical form.

As already mentioned, there are several observables that can be extracted from
solving the wave equation. In particular the QNMs correspond to outgoing waves
at infinity that satisfying ingoing boundary conditions at the (zero-area)
horizon $r=0$.

In the WKB approximation, it is reasonable to assume that QNMs satisfy
$|\omega_{I}|\ll|\omega_{R}|$ so that (6) can be solved perturbatively in
$\omega_{I}$. At zeroth order, we compute
$\omega_{R}\approx\omega_{c}(\ell,...)$ and plugging the expression of
$\omega_{R}$ in (7) we find:

$\omega^{QNM}_{WKB}={(\ell+1)^{2}-{1\over 4}+L^{2}{\mu}^{2}\over
2L\sqrt{(\ell+1)^{2}-{1\over 4}}}-i(2n+1){\Big{[}(\ell+1)^{2}-{1\over
4}-L^{2}{\mu}^{2}\Big{]}^{3\over 2}\Big{[}(\ell+1)^{2}-{1\over
4}+L^{2}{\mu}^{2}\Big{]}^{1\over 2}\over 4L\Big{[}(\ell+1)^{2}-{1\over
4}\Big{]}^{2}}$ (41)

As already observed, the imaginary part is related to the Lyapunov exponent
governing the chaotic behaviour of nearly critical geodesics around the light-
ring (photon-sphere). The numerical values of QNM in WKB approximation for the
lowest overtone ($n=0$) and for various $\ell$ are displayed in the first
column of Table 1.

###  4.4 Massive D1/D5-SW-Mathieu dictionary

The radial wave equation (39) for the massive scalar field on a D1/D5
background is mutatis mutandis the same as on a D3 background Gubser:1998iu ,
that can be mapped to the quantum SW curve of pure SU(2) gauge theory, or
equivalently to the (modified) Mathieu equation, as shown in Bianchi:2021xpr ;
Bianchi:2021mft . Setting

$r={\hbar\tilde{\omega}L^{2}\over
2}\sqrt{y}=L\sqrt[4]{{\tilde{\omega}^{2}\over\tilde{\omega}^{2}-{\mu}^{2}}}e^{iz}$

the relevant potentials are:

$\displaystyle Q_{SW}(y)={4qy^{2}+(\hbar^{2}-4u)y+4\over 4\hbar^{2}y^{3}}$
(42) $\displaystyle Q_{M}(z)=\alpha-2\beta\cos(2z)$

The gauge/gravity/Mathieu dictionary reads:

$\alpha={4u\over\hbar^{2}}=(\ell+1)^{2}+L^{2}{\mu}^{2}-2L^{2}\tilde{\omega}^{2}\quad,\quad\beta={4\sqrt{q}\over\hbar^{2}}=\tilde{\omega}L^{2}\sqrt{\tilde{\omega}^{2}-{\mu}^{2}}$
(43)

In the Mathieu ‘version’ $\alpha$ plays the role of energy eigen-value and the
symmetry $z\leftrightarrow-z$ is nothing but the generalised Couch-Torrence
symmetry $r\leftrightarrow r^{2}_{c}/r$ CouchTorr  that exchanges infinity and
horizon keeping the photon sphere fixed. As a result Mathieu equation admits
fundamental even and odd solutions.

Moreover the potential is periodic under $z\rightarrow z+\pi$, so that a non-
trivial solution of Mathieu equation $w(z)$ is quasi-periodic:

$w(z+\pi)=e^{\pi i\nu}w(z)$ (44)

with $\nu(\alpha,\beta)$ known as Floquet exponent, that in turn can be
related to the quantum $a$-period at weak coupling and to the quantum
$a_{D}$-period at strong coupling. At weak coupling, one finds:

$a(u,q)={\hbar\over 2}\nu(\alpha,\beta)$ (45)

In the weak coupling limit ($\beta\ll\alpha$) the energy eigenvalue is:

$\displaystyle\alpha=\nu^{2}+{\beta^{2}\over
2(\nu^{2}-1)}+{(5\nu^{2}+7)\beta^{4}\over 32(\nu^{2}-1)^{3}(\nu^{2}-4)}+...$
(46) $\displaystyle u=a^{2}+{2\over(4a^{2}-1)}q+{7+20a^{2}\over
2(a^{2}-1)(4a^{2}-1)^{3}}q^{2}+...$

Inverting this relation i.e. finding $\nu=2a$ in terms of $u$ is subtle. One
has to pay particular attention when – as in the case at hand –
$\alpha=4u\approx n^{2}=\nu^{2}+...$ is close to a square integer, so that
$\nu=n+\delta n(q)$. The standard formula does not work since it is derived
with the tacit assumption that the poles of the quantum SW differential
$\lambda_{+}(x)$ be different from one another. When two poles coalesce, the
residues must be computed with extra care and the expression of $a(u)$
acquires a different form.

The photon-sphere can be more conveniently described in the opposite limit
$\alpha\sim\pm 2\beta$ ($u\sim\pm 2\sqrt{q}$) that corresponds to the strong
coupling regime of the gauge theory where the $a_{D}$-cycle degenerates. In
this limit the energy eigenvalue appears:

$\displaystyle\alpha=2\beta+2s\sqrt{-\beta}-\frac{s^{2}+1}{8}-\frac{s^{3}+3s}{128\sqrt{-\beta}}+\frac{5s^{4}+34s^{2}+9}{4096\beta}-\frac{33s^{5}+410s^{3}+405s}{131072(-\beta)^{3/2}}+$
(47)
$\displaystyle-\frac{63s^{6}+1260s^{4}+2943s^{2}+486}{1048576\beta^{2}}-\frac{527s^{7}+15617s^{5}+69001s^{3}+41607s}{33554432(-\beta)^{5/2}}+...$

with $s=2\nu$ and the Floquet exponent $\nu$ is now linked with the
$a_{D}$-cycle. The WKB quantization condition translates into $s=2n+1$ where
$n$ is the overtone. If we plug the dictionary in the previous expression, the
latter becomes an equation for $\omega$ which can be numerically solved in
order to find QNM frequencies. See second column in Table 1.

###  4.5 Leaver’s numerical method

In order to confirm the results obtained with WKB approximation and with qSW,
one can solve the differential equation exploiting the method of continuous
fraction formulated by Leaver Leaver1 ; Leaver2 .

The radial equation (39) can be transformed in such a way as to show two
regular and one irregular singular point by setting121212The two choices of
sign cover two different allowed ranges in $r$.:

$r=L\sqrt[4]{{\tilde{\omega}^{2}\over\tilde{\omega}^{2}-{\mu}^{2}}}\left(1+2y\pm
2\sqrt{y(1+y)}\right)$ (48)

leading to

$y(1+y)\Psi^{\prime\prime}(y)+{1+2y\over
2}\Psi^{\prime}(y)+\Big{[}2\beta-\alpha+16\beta y(1+y)\Big{]}\Psi(y)=0$ (49)

that can be put in the standard form of a CHE for $W(z)=e^{\pm
4i\sqrt{\beta}z}\Psi(z)$ with $z=-y$. The ansatz for the solution is:

$\Psi(y)=e^{i\omega
y}y^{\sigma_{+}}(1+y)^{\sigma_{-}}\sum_{n=0}^{\infty}c_{n}\left({y\over
1+y}\right)^{n}$ (50)

The constants $\sigma_{+}$ and $\omega$ are determined by the behaviours near
the horizon and at infinity respectively, while $\sigma_{-}$ is fixed by
requiring that the recursion relation involves only three terms. We find:

$\omega^{2}=16\beta\quad,\quad\sigma_{+}=0,{1\over
2}\quad,\quad\sigma_{-}=-\sigma_{+}-{1\over 2}$ (51)

The two choices for $\sigma_{+}$ generate two different branches for QNMs,
indeed we will see that $\sigma_{+}=0$ is associated to even overtones while
$\sigma_{+}=1/2$ with odd ones. The recurrence relation is:

$\displaystyle\alpha_{n}c_{n+1}+\beta_{n}c_{n}+\gamma_{n}c_{n-1}=0$ (52)
$\displaystyle\alpha_{n}=(n+\sigma_{+}+1)(n+\sigma_{+}+1/2)$
$\displaystyle\beta_{n}=-{1\over
4}-2n^{2}-\alpha+2\beta+n(-1+8i\sqrt{\beta}-4\sigma_{+})-\sigma_{+}(1+2\sigma_{+})+2i\sqrt{\beta}(1+4\sigma_{+})$
$\displaystyle\gamma_{n}=(n+\sigma_{+})(n+\sigma_{+}-1/2)$

where we considered $c_{-1}=0$. The QNM frequencies associated to the overtone
$n$ can be obtained by solving numerically the equation:

$\beta_{n}={\alpha_{n-1}\gamma_{n}\over\beta_{n-1}-{\alpha_{n-2}\gamma_{n-1}\over\beta_{n-2}-...}}+{\alpha_{n}\gamma_{n+1}\over\beta_{n+1}-{\alpha_{n+1}\gamma_{n+2}\over\beta_{n+2}-...}}$
(53)

Results coming from all the techniques that we exposed are collected in Table
1.

| WKB | Seiberg-Witten | Leaver | Mathieu
---|---|---|---|---
$\ell=0$ | 0.433013-0.25 I | 0.498509-0.278493 I | 0.498509-0.263183 I | 0.501666-0.245563 I
$\ell=1$ | 0.968246-0.25 I | 0.999627-0.255004 I | 0.999643-0.253669 I | 0.999883-0.253497 I
$\ell=2$ | 1.47902-0.25 I | 1.50114-0.250172 I | 1.49988-0.251682 I | 1.4999-0.251675 I
$\ell=3$ | 1.98431-0.25 I | 2.00112-0.249572 I | 1.99994-0.250958 I | 1.99995-0.250958 I
$\ell=4$ | 2.48747-0.25 I | 2.50072-0.249845 I | 2.49997-0.250617 I | 2.49997-0.250617 I
$\ell=5$ | 2.98957-0.25 I | 2.98907-0.249864 I | 2.99998-0.25043 I | 2.99998-0.25043 I
$\ell=6$ | 3.49106-0.25 I | 3.50865-0.2499 I | 3.49999-0.250317 I | 3.49999-0.250317 I
$\ell=7$ | 3.99218-0.25 I | 4.00383-0.249924 I | 3.99999-0.250243 I | 3.99999-0.250243 I
$\ell=8$ | 4.49305-0.25 I | 4.50955-0.24994 I | 4.49999-0.250192 I | 4.49999-0.250192 I
$\ell=9$ | 4.99375-0.25 I | 5.00698-0.249951 I | 5.-0.250156 I | 5.-0.250156 I
$\ell=10$ | 5.49432-0.25 I | 5.49877-0.24996 I | 5.5-0.250129 I | 5.5-0.250129 I
Table 1: Prompt Ringdown modes of $D1-D5$ small BH for $n=0$, $M=0$ and $L=1$

###  4.6 Absorption probability and cross section

Using the connection formulae for Mathieu equation or equivalently for the
DRDCHE one can determine the grey-body factor of small BHs as a function of
$\omega$ and $\ell$. The analysis is very similar to the one for D3-branes
Gubser:1998iu .

Setting131313The new $z$ is different from the one in Mathieu equation.

$r={2\sqrt{z}\over\sqrt{\tilde{\omega}^{2}-{\mu}^{2}}}$

the wave equation can be written in the form of a DRDCHE

$W^{\prime\prime}(z)+\left({1\over z}-{p\over z^{2}}+{\varepsilon\over
z^{3}}\right)W(z)=0$ (54)

with $p=u-{1\over 4}$ and $\varepsilon=q={\tilde{\omega}^{4}L^{4}\over
16}\left(1-{\mu^{2}\over\tilde{\omega}^{2}}\right)$

The DRDCHE has two irregular singularities: one at $z=0$, corresponding to the
‘horizon’ at $r=0$, and the other one at $z\rightarrow\infty$, corresponding
to asymptotically flat space-time at $r=\infty$.

The connection formula for the wave ingoing into the horizon at $z=0$ to the
waves at $z=\infty$ reads Bonelli:2022ten

$z^{3\over 4}e^{2i\sqrt{\varepsilon\over
z}}W^{DRDC}_{0}(p,\varepsilon;\sqrt{z})={z^{1\over 4}\over 2\pi
i}\sum_{\sigma=\pm}{\Gamma(1-2\sigma a(q))^{2}\over-2\sigma
a(q)}\varepsilon^{{1\over 4}+\sigma a(q)}e^{-{\sigma\over 2}\partial_{a}{\cal
F}}\times$
$\left[e^{2i\sqrt{z}}W^{DRDC}_{0}\left(p,\varepsilon;\sqrt{\varepsilon\over
z}\right)+ie^{-2\pi i\sigma a(q)}e^{-2i\sqrt{z}}W^{DRDC}_{0}\left(p,e^{2\pi
i}\varepsilon;-\sqrt{\varepsilon\over z}\right)\right]$

where

$W^{DRDC}_{0}(p,\varepsilon;x)=1+i{3+16p\over
16\sqrt{\varepsilon}}\sqrt{z}+{\cal O}(z)$ (55)

is the DRDC Heun function, ${\cal F}={\cal F}(a,q,m_{f};\hbar)$ is the NS pre-
potential and $a(q)$ the fundamental (quantum-deformed) period. The wave at
infinity can be written as

$\Psi_{\infty}\approx
B(\omega)e^{-i\sqrt{\tilde{\omega}^{2}-\mu^{2}}r}+C(\omega)e^{+i\sqrt{\tilde{\omega}^{2}-\mu^{2}}r}$
(56)

with (using the connection formulae)

$B(\omega)=A\sin(\pi a_{D}-2\pi a)\quad,\quad C(\omega)=A\sin(\pi a_{D})$ (57)

so that the absorption probability is given by

$P(\omega)=1-{|\sin(\pi a_{D})|^{2}\over|\sin(\pi a_{D}-2\pi a)|^{2}}$ (58)

Since for $\omega$ real $a$ (half of the Floquet exponent $\nu$) is real and
$a_{D}=i\tau$ is purely imaginary, the expression simplifies to

$P_{abs}(\omega)=1-{\sinh^{2}(\pi\tau)\over\sinh^{2}(\pi\tau)+\sin^{2}(2\pi
a)}={\sin^{2}(2\pi a)\over\sinh^{2}(\pi\tau)+\sin^{2}(2\pi a)}$ (59)

that clearly satisfies $0\leq P_{abs}(\omega)\leq 1$. The computation of $a$
or directly $\eta=\exp{2\pi ia}$, as in Gubser:1998iu  is (almost)
straightforward in terms of residues of the quantum SW differential or using
cum grano salis the continuous fractions formula for the ‘energy’ eigenvalue
of Mathieu equation, summarised in (46).

The computation of $a_{D}$ or $\chi=\exp{\pi ia_{D}}$, as in Gubser:1998iu ,
is more laborious. Following Gubser:1998iu  one can write

$\chi=\exp{\pi
ia_{D}}=q^{-a}{\Gamma(a+\sqrt{u}+1)\Gamma(a-\sqrt{u}+1)\over\Gamma(-a+\sqrt{u}+1)\Gamma(-a-\sqrt{u}+1)}{v(-a)\over
v(+a)}$ (60)

where

$v(+a)=\sum_{n=0}^{\infty}A_{n}(a)q^{n}$ (61)

In turn, the coefficients are given by nested sums 141414To avoid confusion
with other $a$’s, we call $B_{p}$ what was called $a_{p}$ in Gubser:1998iu
and denote their $r$ by $\sqrt{u}$.

$A_{n}(a)=\sum_{p_{1}=0}^{\infty}\sum_{p_{2}=2}^{\infty}...\sum_{p_{n}=2}^{\infty}B_{p_{1}+a}B_{p_{1}+p_{2}+a}...B_{\Sigma_{i}p_{i}+a}$
(62)

where

$B_{a}={1\over[(a+1)^{2}-u][(a+2)^{2}-u]}$ (63)

that can be computed recursively. Starting from $A_{0}=1$ one finds

$A_{1}(a)=S[1](a)\quad,\quad A_{2}(a)={1\over 2}S^{2}[1](a)-{1\over
2}S[2](a)-S[1,1](a)...$ (64)

where the ‘loop variables’ are defined as

$S[k_{0},k_{1},...k_{n}](a)=\sum_{p=-\infty}^{+\infty}\prod_{j=0}^{n}B^{k_{j}}_{p+j}(a)$
(65)

with the understanding that $B_{p}(a)=0$ for $p<0$, so that

$\displaystyle S[1]$
$\displaystyle={2z+3\over(4{u}-1)[(1+z)^{2}-{u}]}+{\psi(z-\sqrt{u}+1)-\psi(z+\sqrt{u}+1)\over\sqrt{u}(4{u}-1)}$
(66) $\displaystyle S[2]$
$\displaystyle={35-70{u}+8{u}^{2}+84z-100{u}z+16{u}^{2}z+70z^{2}-56{u}z^{2}+20z^{3}-16{u}z^{3}\over(4{u}-1)^{3}[{u}-(z+1)^{2}]^{2}}+$
$\displaystyle+{(20{u}{-}1)(\psi(1{-}\sqrt{u}{+}z)-\psi(1{+}\sqrt{u}{+}z))\over
2\sqrt{u}^{3}(4{u}{-}1)^{3}}+{(4{u}{+}1)(\psi^{(1)}(1{-}\sqrt{u}{+}z)+\psi^{(1)}(1{+}\sqrt{u}{+}z))\over
2{u}(4{u}{-}1)^{2}}$ $\displaystyle S[1,1]$
$\displaystyle={(10{u}-1)(\psi(1-\sqrt{u}+z)-\psi(1+\sqrt{u}+z))\over
4\sqrt{u}^{3}({u}-1)(4{u}-1)^{2}}{-}{\psi^{(1)}(2-\sqrt{u}+z)+\psi^{(1)}(2+\sqrt{u}+z)\over
4{u}(4{u}-1)}+$
$\displaystyle-{4u^{3}(3+2z)+u^{2}(35-26z-36z^{2}-8z^{3})-u(109+143z+65z^{2}+10z^{3})+2(2+z)^{2}\over
4({u}-1)(4{u}-1)^{2}[{u}-(z+2)^{2}][{u}-(z+1)^{2}]^{2}}$

Setting $\mu=0$ for simplicity and plugging in the expressions of $a$ and
$a_{D}$ in terms of $q={\omega^{4}L^{4}\over 16}$ and $u={1\over
4}(\ell+1)^{2}-{1\over 2}\omega^{2}L^{2}$ one finds an expansion of the form

$P_{\ell}(\omega)={4\pi^{2}\delta_{\ell}q^{\ell+1}\over(\ell!)^{4}(\ell+1)^{2}}\sum_{n=0}^{\infty}\sum_{k=0}^{n}P^{(\ell)}_{n,k}q^{n/2}\log^{k}(\hat{q})$
(67)

with $\hat{q}=qe^{4\gamma_{E}}$ with $\gamma_{E}\approx 0.577216=-\psi(1)$ the
Euler-Mascheroni constant and $\delta_{\ell}=1$ for $\ell\neq 0$, while
$\delta_{0}=256/225$ due to a subtle cancellation between powers of $\sqrt{q}$
in $a_{D}$ or rather in $2i\sin\pi a_{D}=\chi-\chi^{-1}$.

With this proviso, the coefficient $P_{0,0}^{(\ell)}=1$ for any $\ell$ the
first few other coefficients are:

$\displaystyle P^{(0)}_{1,0}={499\over 60}\quad,\quad P^{(0)}_{1,1}=-{68\over
15}$ (68) $\displaystyle P^{(0)}_{2,0}={115477\over 1600}-{2224\pi^{2}\over
225}\quad,\quad P^{(0)}_{2,1}=-{3604\over 75}\quad,\quad
P^{(0)}_{2,2}={856\over 75}$ $\displaystyle
P^{(0)}_{3,0}={230199619-59928064\pi^{2}-33945600\zeta(3)\over
432000}\quad,\quad P^{(0)}_{3,1}=-{7(3595071-504832\pi^{2})\over 54000}$
$\displaystyle P^{(0)}_{3,2}={170222\over 1125}\quad,\quad
P^{(0)}_{3,3}=-{75616\over 3375}$ $\displaystyle P^{(1)}_{1,0}={31\over
3}\quad,\quad P^{(1)}_{1,1}=-2$ (69) $\displaystyle P^{(1)}_{2,0}={4763\over
72}-{4\pi^{2}\over 3}\quad,\quad P^{(1)}_{2,1}=-{67\over 3}\quad,\quad
P^{(1)}_{2,2}=2$ $\displaystyle P^{(1)}_{3,0}={17399\over
48}-16\pi^{2}-{32\zeta(3)\over 3}\quad,\quad P^{(1)}_{3,1}=-{1379\over
9}+{8\pi^{2}\over 3}\quad,\quad P^{(1)}_{3,2}=24\quad,\quad
P^{(1)}_{3,3}=-{4\over 3}$ $\displaystyle P^{(2)}_{1,0}={649\over
72}\quad,\quad P^{(2)}_{1,1}=-{4\over 3}$ (70) $\displaystyle
P^{(2)}_{2,0}={985567\over 20736}-{16\pi^{2}\over 27}\quad,\quad
P^{(2)}_{2,1}=-{337\over 27}\quad,\quad P^{(2)}_{2,2}={8\over 9}$
$\displaystyle P^{(2)}_{3,0}={22584863\over 116640}-{466\pi^{2}\over
81}-{256\zeta(3)\over 81}\quad,\quad
P^{(2)}_{3,1}=-{15843885-184329\pi^{2}\over 233280}\quad,\quad
P^{(2)}_{3,2}={233\over 27}$ $\displaystyle P^{(2)}_{3,3}=-{32\over 81}$

Note that the coefficients do not enjoy uniform transcendentally, as for
D3-branes Gubser:1998iu . Note also that the leading dependence on $\omega$
comes from $q^{\ell+1}\approx\omega^{4\ell+4}$.

Using a generalisation of optical theorem to $d=D-1$ non-compact transverse
spatial directions, see Gubser:1998iu ; Gubser:1997qr , that relies on partial
wave expansion in terms of the relevant spherical harmonics (Gegenbauer
polynomials, related to characters of $SU(2)$ Bianchi:2017sds ) and Bessel
functions one has (for massless probes $k=\omega$)

$\sigma_{\ell}(\omega)={2^{d-2}\pi^{{d\over 2}-1}\over
3\omega^{5}}(\ell+{d\over 2}-1)\Gamma({d\over
2}-1)\left({}^{\ell+d-3}_{\quad\ell\quad}\right)(1-|S_{\ell}|^{2})$ (71)

Replacing $1-|S_{\ell}|^{2}=P_{\ell}$ and putting $d=4$ for for D3/D3’ or
D1/D5 $d=4$ one finally finds

$\sigma_{\ell}(\omega)={4\pi\over\omega^{3}}(\ell+1)^{2}P_{\ell}(\omega)$ (72)

with the absorption probably computed above. It is worth recalling how crucial
this kind of analysis was for substantiating the AdS/CFT correspondence in its
early days. Yet the role of the Couch-Torrence symmetry CouchTorr
$z\leftrightarrow-z$ or $r\leftrightarrow L^{2}/r$ that exchanges the horizon
with flat infinity, keeping the photon-sphere fixed, has not been fully
exploited in this context Bianchi:2021yqs ; Bianchi:2022wku .

###  4.7 Tidal Love numbers

Another interesting observable is tidal deformability. For static waves
$\omega=0$ and in a spherically symmetric case, such as a small BH, the tidal
Love number Love1909.0008 ; Bonelli:2021uvf ; Cvetic:2021vxa ;
Pereniguez:2021xcj ; Consoli:2022eey  is determined by the ratio of the term
decaying at infinity (‘response’) wrt the one growing at infinity (‘source’).

For $\omega=0$ and $\mu=0$ the two behaviours are

$\Psi(r)\approx r^{\ell+{3\over 2}}\quad{\rm and}\quad\Psi\approx
r^{-\ell-{1\over 2}}$ (73)

In fact the exact solution is

$\Psi(r)=A_{\ell}r^{\ell+{3\over 2}}+B_{\ell}r^{-\ell-{1\over 2}}$ (74)

where $A_{\ell}$ plays the role of the ‘source’ and $B_{\ell}$ of the
‘response’. The static Love number is ${\cal L}_{\ell}=B_{\ell}/A_{\ell}$.
Imposing regularity at the horizon ($r=0$) implies $B_{\ell}=0$ and as a
consequence

${\cal L}^{smallBH}_{\ell,stat}=0$ (75)

Similarly to Kerr-Newman BHs in 4-d and extremal BHs in 5-d Pereniguez:2021xcj
.

In the dynamical case $\omega\neq 0$ the definition of tidal Love numbers is
somewhat ambiguous, since the wave behaves as $e^{\pm i\omega r}$ at infinity.
In the (quasi)static limit $\omega r<<1$ and $M/r<<1$151515$M$ represents the
mass/length scale of the compact object. ($M<<r<<1/\omega$) the wave behaves
according to

$\Psi(r)\approx r^{\lambda_{-}}{\cal R}(\omega r)+r^{\lambda_{+}}{\cal
S}(\omega r)$

where ${\lambda_{\pm}}$ depend on $\omega$ and $\ell$. One can generalise the
notion of Love number and define a Love ‘function’ Consoli:2022eey

${\cal L}(\omega r)={{\cal R}(\omega r)\over{\cal S}(\omega r)}$

that can be derived using the by-now available connection formulae. In the
case of $N_{f}=(1,2)$ q-SW curves, setting $\omega=0$ one finds
Consoli:2022eey

${\cal L}(0)={\cal
L}_{0}=L+i\Lambda={\Gamma(-2a)\Gamma(1+m_{1}+a)\Gamma(1+m_{2}+a)\over\Gamma(2a)\Gamma(1+m_{1}-a)\Gamma(1+m_{2}-a)}$
(76)

where the imaginary part $\Lambda$ account for dissipative effects.

In order to put (49) in canonical $N_{f}=(1,2)$ q-SW form, we have to perform
several steps.

First we have to redefine $\Psi(y)\rightarrow y^{-1/4}(1+y)^{-1/4}\Psi(y)$, so
that

$\Psi^{\prime\prime}(y)+{256\beta y^{4}+512\beta
y^{3}+4(1-4\alpha+72\beta)y^{2}+4(1-4\alpha+8\beta)y+3\over
16y^{2}(1+y)^{2}}\Psi(y)=0$ (77)

Furthermore if we map $y=-1/z$ and expand in partial fractions we obtain:

$\displaystyle\Psi^{\prime\prime}(z)$ $\displaystyle+\left({16\beta\over
z^{4}}+{1-4\alpha+8\beta\over 4z^{2}}+{3\over
16(z-1)^{2}}+{1-4\alpha+8\beta\over 4z}+{-1+4\alpha-8\beta\over
4(z-1)}\right)\Psi(z)=0$ (78)

which corresponds to $Q_{2,1}(z)$ in Consoli:2022eey  that can be written in
terms of the CFT data (see Appendix A) as

$\displaystyle\Psi^{\prime\prime}(z)$ $\displaystyle+\left({-}{q^{2}\over
4z^{4}}{+}{qc\over z^{3}}{+}{1-4u-2q+4k_{0}q\over
4z^{2}}{+}{\delta_{k_{0}}\over(z-1)^{2}}+\right.$ (79)
$\displaystyle\left.+{1-4u-2q+4k_{0}q+4\delta_{k_{0}}-4\delta_{p_{0}}\over
4z}+{-1+4u+2q-4k_{0}q-4\delta_{k_{0}}+4\delta_{p_{0}}\over
4(z-1)}\right)\Psi(z)=0$

For D1-D5 small BH the dictionary is thus:

$q{=}{-}8i\sqrt{\beta}\quad,\quad c=0\quad,\quad
u=\alpha+2i\sqrt{\beta}-2\beta\quad,\quad\delta_{k_{0}}{=}\delta_{p_{0}}{=}{3\over
16}$ (80)

which implies $m_{1}=-1/2$, $m_{2}=0$ and $m_{4}=1/2$, while $m_{3}$ is
undetermined and can be decoupled (taken to infinity). Mapping to
$N_{f}=(1,2)$ q-SW, the $a$-cycle can be written as:

$a=\sqrt{u}-{1\over 8\sqrt{u}}q-{1+2u\over 128u^{3/2}}q^{2}-{1+2u\over
1024u^{5/2}}q^{3}+{5+7u-8u^{2}-6u^{3}\over 32768(u-1)u^{7/2}}q^{4}+...$ (81)

In order to obtain the Love function, we have to plug in (76) the dictionary
(146), (80) and (81) together with (43).

In the static limit, using the duplication formula for $\Gamma$ functions, we
obtain:

$\mathcal{L}_{0}={\Gamma(-2a)\Gamma({1\over
2}+a)\Gamma(1+a)\over\Gamma(2a)\Gamma({1\over
2}-a)\Gamma(1-a)}=-2^{-4a}=-2^{-4\sqrt{(\ell+1)^{2}+L^{2}\mu^{2}}}$ (82)

Notice that a non-zero result is in disagreement with the ‘rigorously’ static
case. As observed in Consoli:2022eey  the two limits do not commute. Computing
the ‘dynamical’ Love number with $\omega\neq 0$ via the ‘Love function’
Consoli:2022eey  (defined for $L<<r<<1/\omega$) by imposing regularity at the
horizon and then taking the ratio of the coefficient of the term decaying at
infinity (as $r^{-\ell-{1\over 2}-\Delta\ell(\omega)}$) and of the term
growing at infinity (as $r^{+\ell+{3\over 2}+\Delta\ell(\omega)}$) yields the
above non-zero result. The mathematical reason is that the singularity
structure of the equation changes though in the ‘intermediate’ region
$L<<r<<1/\omega$ one can still keep only the $1/r^{2}$ term in the potential.
Physics suggests that the relevant Love number be the dynamical one since a
‘static’ perturbation with $\omega=0$ exactly is hard to conceive in an
astrophysical setting.

Since the small BH has no ergo-region we do not expect super-radiance for
neutral probes. Yet one can study charge super-radiance Brito:2015oca ;
Aalsma:2022knj . This is beyond the scope of the present investigation.

##  5 Probing and re-probing circular fuzz-ball with particles and waves

We are ready to discuss the main topic of the present investigation: scalar
perturbations of a fuzz-ball in $D=5$ with circular profile of radius $a_{f}$.

The metric of the D1/D5 circular fuzz-ball (in the democratic D3/D3’ frame)
reads Lunin:2001jy :

$\displaystyle ds^{2}=$ $\displaystyle
H^{-1}\Big{[}-(dt+\omega_{\phi}d\phi)^{2}+(dz+\omega_{\psi}d\psi)^{2}\Big{]}+H\Big{[}(\rho^{2}+a_{f}^{2}\cos^{2}\theta)\left({d\rho^{2}\over\rho^{2}+a_{f}^{2}}+d\theta^{2}\right)+$
(83) $\displaystyle+\rho^{2}\cos^{2}\theta
d\psi^{2}+(\rho^{2}+a_{f}^{2})\sin^{2}\theta
d\phi^{2}\Big{]}+\left({H_{1}\over H_{5}}\right)^{1\over
2}dz^{2}_{1,2}+\left({H_{5}\over H_{1}}\right)^{1\over 2}dz^{2}_{3,4}$

where

$\displaystyle H=\sqrt{H_{1}H_{5}}\quad,\quad
H_{i}=1+{L_{i}^{2}\over\rho^{2}+a_{f}^{2}\cos^{2}\theta}$ (84)
$\displaystyle\omega_{\phi}={a_{f}L_{1}L_{5}\sin^{2}\theta\over\rho^{2}+a_{f}^{2}\cos^{2}\theta}\quad,\quad\omega_{\psi}={a_{f}L_{1}L_{5}\cos^{2}\theta\over\rho^{2}+a_{f}^{2}\cos^{2}\theta}$

and the spheroidal coordinates are related to the standard cartesian
coordinates by

$x_{1}+ix_{2}=\rho\cos\theta e^{i\psi}\quad,\quad
x_{3}+ix_{4}=\sqrt{\rho^{2}+a_{f}^{2}}\sin\theta e^{i\phi}$ (85)

so that

$\rho\cos\theta=r\cos\tilde{\theta}\quad,\quad\sqrt{\rho^{2}+a_{f}^{2}}\sin\theta=r\sin\tilde{\theta}$
(86)

with $r$ and $\tilde{\theta}$ the usual radial and polar angular coordinates.

It is easy to check that in the limit $a_{f}\rightarrow 0$ one gets back the
small BH metric, while for $a_{f}\neq 0$ the solution is non-singular and
horizon-less Lunin:2001jy . Notice however that, contrary to the small BH, the
circular fuzz-ball carries angular momentum. This will play a role in the
following.

The generalized momenta are:

$P_{t}=-{\dot{t}+\omega_{\phi}\dot{\phi}\over H}=-E\quad,\quad
P_{z}={\dot{z}+\omega_{\psi}\dot{\psi}\over H}\quad,\quad
P_{\rho}={H(\rho^{2}+a_{f}^{2}\cos^{2}\theta)\over\rho^{2}+a_{f}^{2}}\dot{\rho}$
(87) $P_{\theta}=H(\rho^{2}+a_{f}^{2}\cos^{2}\theta)\dot{\theta}\quad,\quad
P_{\phi}=-{\omega_{\phi}\over
H}\dot{t}+\Bigg{[}H(\rho^{2}+a_{f}^{2})\sin^{2}\theta-{\omega_{\phi}^{2}\over
H}\Bigg{]}\dot{\phi}=J_{\phi}$ $P_{\psi}={\omega_{\psi}\over
H}\dot{z}+\left(H\rho^{2}\cos^{2}\theta+{\omega_{\psi}^{2}\over
H}\right)\dot{\psi}=J_{\psi}\quad,\quad P_{1,2}=\left({H_{1}\over
H_{5}}\right)^{1\over 2}\dot{z}_{1,2}\quad,\quad P_{3,4}=\left({H_{5}\over
H_{1}}\right)^{1\over 2}\dot{z}_{3,4}$

where the only non conserved ‘dynamical’ momenta are $P_{\rho}$ and
$P_{\theta}$. If we invert the previous relations, we obtain:

$\dot{t}=HE-{\omega_{\phi}(J_{\phi}+E\omega_{\phi})\over
H(\rho^{2}+a_{f}^{2})\sin^{2}\theta}\quad,\quad\dot{\phi}={J_{\phi}+E\omega_{\phi}\over
H(\rho^{2}+a_{f}^{2})\sin^{2}\theta}$ (88)
$\dot{z}=HP_{z}-{\omega_{\psi}(J_{\psi}-P_{z}\omega_{\psi})\over
H\rho^{2}\cos^{2}\theta}\quad,\quad\dot{\psi}={J_{\psi}-P_{z}\omega_{\psi}\over
H\rho^{2}\cos^{2}\theta}$ $\dot{\rho}={\rho^{2}+a_{f}^{2}\over
H(\rho^{2}+a_{f}^{2}\cos^{2}\theta)}P_{\rho}\quad,\quad\dot{\theta}={1\over
H(\rho^{2}+a_{f}^{2}\cos^{2}\theta)}P_{\theta}$
$\dot{z}_{1,2}=\left({H_{5}\over H_{1}}\right)^{1\over
2}P_{1,2}\quad,\quad\dot{z}_{3,4}=\left({H_{1}\over H_{5}}\right)^{1\over
2}P_{3,4}$

###  5.1 Massless Geodesics

For zero KK momenta along the internal torus $P_{I}=0$, massless geodesics in
Hamiltonian form are determined by

$P_{\rho}^{2}(\rho^{2}+a_{f}^{2})+{\mathcal{L}_{\psi}^{2}\over\rho^{2}}-{\mathcal{L}_{\phi}^{2}\over\rho^{2}+a_{f}^{2}}-\mathcal{E}^{2}(\rho^{2}+a_{f}^{2}+L_{1}^{2}+L_{5}^{2})+P_{\theta}^{2}+{J_{\psi}^{2}\over\cos^{2}\theta}+{J_{\phi}^{2}\over\sin^{2}\theta}+\mathcal{E}^{2}a_{f}^{2}\sin^{2}\theta=0$
(89)

where

$\mathcal{E}^{2}=E^{2}-P_{z}^{2}\quad,\quad\mathcal{L}_{\phi}^{2}=(J_{\phi}a_{f}-EL_{1}L_{5})^{2}\quad,\quad\mathcal{L}_{\psi}^{2}=(J_{\psi}a_{f}-P_{z}L_{1}L_{5})^{2}$
(90)

The conjugate momenta $E=-P_{t}$, $P_{z}$, $J_{\phi}=P_{\phi}$, and
$J_{\psi}=P_{\psi}$ are conserved but the spherical symmetry $SO(3)\times
SO(3)$ of the small BH is broken to $SO(2)_{\phi}\times SO(2)_{\psi}$ by the
very presence of the circular profile. Despite the reduced isometry, very much
as for Kerr BHs, separation is possible à la Carter Chervonyi:2013eja ;
Bianchi:2017sds :

$\displaystyle
P_{\rho}^{2}=Q_{R}(\rho)=\mathcal{E}^{2}\left(1+{L_{1}^{2}+L_{5}^{2}\over\rho^{2}+a_{f}^{2}}\right)+{\mathcal{L}_{\phi}^{2}\over(\rho^{2}+a_{f}^{2})^{2}}-{\mathcal{L}_{\psi}^{2}\over\rho^{2}(\rho^{2}+a_{f}^{2})}-{K^{2}\over\rho^{2}+a_{f}^{2}}$
(91) $\displaystyle
P_{\theta}^{2}=Q_{A}(\theta)=K^{2}-{J_{\psi}^{2}\over\cos^{2}\theta}-{J_{\phi}^{2}\over\sin^{2}\theta}-\mathcal{E}^{2}a_{f}^{2}\sin^{2}\theta$

Notice that with this choice, the separation constant $K^{2}$ is
positive171717This is not so for other ‘standard’ choices e.g.
$A=K^{2}-a_{f}^{2}{\cal E}^{2}$. for real $\mathcal{E}$ and plays the role of
the total angular momentum including frame-dragging effects. Notwithstanding
the separation of the radial dynamic from the polar angular one, geodesics are
non-planar in general. In fact they typically give rise to quite non-trivial
motion in any direction. Some simplification arises when the motion takes
place in a given ‘hyperplane’ with $\theta=\theta_{0}$. We will call these
geodesics ‘shear-free’, borrowing the terminology from the familiar case of
Kerr BHs ChandraBH . We will start with the two simplest cases: $\theta=0$ and
$\theta=\pi/2$.

####  5.1.1 $\theta=0$ and $J_{\phi}=0$

For simplicity but without significant loss of generality, we set $P_{z}=0$
and $L_{1}=L_{5}=L$. In the hyper-plane $\theta=0$, orthogonal to the plane of
the circle, the angular equation forces one to take $J_{\phi}=0$, so that
$K^{2}=J_{\psi}^{2}$. The potential reads:

$Q_{R}(y)=E^{2}\Bigg{[}1-\left({\beta^{2}\over y}-{2\lambda^{2}\over
1+y}-{\lambda^{4}\over(1+y)^{2}}\right)\Bigg{]}=E^{2}\Big{[}1-\mathcal{V}_{eff}(y)\Big{]}$
(92)

where we defined

$y={\rho^{2}\over a_{f}^{2}}\quad,\quad\beta={K\over a_{f}E}={b\over
a_{f}}={|J_{\psi}|\over a_{f}E}\quad,\quad\lambda={L\over a_{f}}$ (93)

Since

$\mathcal{V}_{eff}(y)\sim{\beta^{2}-2\lambda^{2}\over
y}+\mathcal{O}\left(y^{-2}\right)\quad\text{as}\quad y\rightarrow\infty$ (94)

in order to have a photon-sphere one must consider $\beta^{2}>2\lambda^{2}$.
Critical geodesics are related to double roots of $Q_{R}$

$\mathcal{V}_{eff}(y)=1\quad,\quad\mathcal{V}^{\prime}_{eff}(y)=0$ (95)

considered as a system in $y_{c}$ and $\beta_{c}$. The solutions are:

$y_{c,+}={1\over
2}(\lambda^{2}-2+\lambda\sqrt{\lambda^{2}-8})\quad,\quad\beta_{c,+}^{2}={8(11\lambda-14\lambda^{3}+2\lambda^{5}+\sqrt{\lambda^{2}-8}(1-6\lambda^{2}+2\lambda^{4})\over(\lambda+\sqrt{\lambda^{2}-8})^{3}}$
(96)

and

$y_{c,-}={1\over
2}(\lambda^{2}-2-\lambda\sqrt{\lambda^{2}-8})\quad,\quad\beta_{c,-}^{2}={8(11\lambda-14\lambda^{3}+2\lambda^{5}-\sqrt{\lambda^{2}-8}(1-6\lambda^{2}+2\lambda^{4})\over(\lambda-\sqrt{\lambda^{2}-8})^{3}}$
(97)

for $\lambda>2\sqrt{2}$ we have in general $y_{c,+}>y_{c,-}$ which are
respectively a maximum and a minumum (see Fig. 1).

Figure 1: Left pannel: plot of the effective potential for $\lambda=3$ which
implies $y_{c,+}=5$ and $\beta_{c,+}=5\sqrt{5}/2$. Yellow line correspond to
$\beta=\beta_{c,+}$, green line to $\beta=5.7>\beta_{c,+}$ and blue line to
$\beta=5.5<\beta_{c,+}$. Right pannel: for $\lambda=3$ the minimum is located
at $y_{c,-}=2$ and $\beta_{c,-}=4\sqrt{2}$. Yellow line correspond to
$\beta=\beta_{c,-}$, green line to $\beta=6>\beta_{c,-}$ and blue line to
$\beta=5.5<\beta_{c,-}$

Notice that, when both are present, the maximum at $y=y_{c,+}$ corresponds to
the photon-sphere (light-ring) comprising unstable circular orbits, while the
minimum at $y=y_{c,-}$ to a internal stable circular orbit.

####  5.1.2 $\theta=\pi/2$ and $J_{\psi}=0$

If we consider geodesics in the hyper-plane $\theta=\pi/2$, where the circular
profile is located, we are forced to set $J_{\psi}=0$, so the angular equation
reduces to:

$\beta^{2}=\beta_{\phi}^{2}+1\quad,\quad\beta_{\psi}={J_{\psi}\over a_{f}E}=0$
(98)

Using the notation introduced in (93), the radial function reads

$Q_{R}=E^{2}\Bigg{[}1-\left({\beta_{\phi}^{2}+1-2\lambda^{2}\over(1+y)}-{(\lambda^{2}-\beta_{\phi})^{2}\over(1+y)^{2}}\right)\Bigg{]}=E^{2}\Big{[}1-\mathcal{V}_{eff}(y)\Big{]}$
(99)

The presence of the photon-sphere requires $\beta_{\phi}^{2}>2\lambda^{2}-1$
i.e. $\beta_{\phi}>+\sqrt{2\lambda^{2}-1}$ (co-rotating) or
$\beta_{\phi}<-\sqrt{2\lambda^{2}-1}$ (counter-rotating). The critical regime
is encoded in the system $Q_{R}=Q^{\prime}_{R}=0$, whose solutions are:

$\beta^{corot}_{\phi,c}=2\lambda-1\quad,\quad
y^{corot}_{c}=\lambda(\lambda-2)$ (100)
$\beta^{counter}_{\phi,c}=-2\lambda-1\quad,\quad
y^{counter}_{c}=\lambda(\lambda+2)$ $\beta_{\phi,c}=1\quad y_{c}=-\lambda^{2}$

In addition we must require $y_{c}>0$ that excludes the last case and imposes
$\lambda\geq 2$ for the co-rotating solution. See Fig. 2.

Figure 2: Left pannel: plot of the effective potential for $\lambda=3$ which
implies $y_{c}^{corot}=3$ and $\beta_{c}^{corot}=5$. Yellow line corresponds
to $\beta=\beta_{c}^{corot}$, green line to $\beta=5.3>\beta_{c}^{corot}$ and
blue line to $\beta=4.7<\beta_{c}^{corot}$. Right pannel: for $\lambda=3$ we
have $y_{c}^{counter}=15$ and $\beta_{c}^{counter}=-7$. Yellow line
corresponds to $\beta_{c}^{counter}$, green line to
$\beta=-7.3<\beta_{c}^{counter}$ and blue line to
$\beta=-6.7<\beta_{c}^{counter}$

The peculiar nature of the geometry in the $\theta=\pi/2$ plane, where space
‘ends’ with a cap at $\rho=0$, representing a circle of radius $a_{f}$ in that
plane, allows for ‘critical’ geodesics at $y=0$. Indeed it is easy to check
that setting $\rho=0$ and $P_{\rho}=0$ one finds a solution for

$E={2a_{f}J_{\phi}\over L^{2}+a_{f}^{2}}=a_{f}\dot{\phi}$ (101)

Such kind of geodesics correspond to meta-stable modes as shown by the Fig. 3
for different values of $\lambda$.

Figure 3: Plot of the effective potential allowing the special geodesics at
$\rho=0$ with $E={2a_{f}J_{\phi}\over L^{2}+a_{f}^{2}}$ for different values
of $\lambda$

.

####  5.1.3 Radial fall and general geodesics

Due to its angular momenta, the circular fuzz-ball drags particles even if
they have $J_{\phi}=J_{\psi}=0$, since in this case
$K^{2}=P_{\theta}^{2}+a_{f}^{2}\omega^{2}\sin^{2}\theta\geq 0$ (strictly if
$\theta\neq 0$).

For $\theta=0$ (constant) so that $P_{\theta}=0$ and $J_{\phi}=J_{\psi}=0$,
$K^{2}=0$ one has ‘polar radial fall’. It is easy to check that in this case
$\dot{\rho}=E$ ‘free fall’ (in ‘proper’ time). Since $\dot{t}=HE$ in this
case, and for $L_{1}=L_{5}=L$ one finds

$t-t_{0}=\rho+{L^{2}\over a}\arctan{\rho\over a}$ (102)

In general (for $L_{1}=L_{5}=L$)

$P_{\rho}=\dot{\rho}{\rho^{2}+a_{f}^{2}\cos^{2}\theta+L^{2}\over\rho^{2}+a_{f}^{2}}$
(103)

and

$P_{\theta}=\dot{\theta}(\rho^{2}+a_{f}^{2}\cos^{2}\theta+L^{2})$ (104)

so much so that

${d\rho\over(\rho^{2}+a_{f}^{2})P_{\rho}}=\pm{d\theta\over P_{\theta}}$ (105)

For $J_{\phi}=J_{\psi}=0$,
$P_{\theta}=\sqrt{K^{2}-a_{f}^{2}E^{2}\sin^{2}\theta}$, while

$P^{2}_{\rho}=E^{2}\left(1+{L^{2}\over\rho^{2}+a_{f}^{2}}\right)^{2}-{K^{2}\over\rho^{2}+a_{f}^{2}}$

Combining the two, one finds

${d\rho\over\sqrt{E^{2}(\rho^{2}+a_{f}^{2}+L^{2})^{2}-K^{2}(\rho^{2}+a_{f}^{2})}}=\pm{d\theta\over\sqrt{K^{2}-a_{f}^{2}E^{2}\sin^{2}\theta}}$
(106)

both can be reduced to incomplete elliptic integrals of the first kind. For
the polar angle one finds

$\int^{\theta}_{\theta_{0}}{d\theta^{\prime}\over
K\sqrt{1-{a_{f}^{2}E^{2}\over K^{2}}\sin^{2}\theta^{\prime}}}={1\over
K}\Big{[}F\left(\theta;{a_{f}E\over K}\right)-F\left(\theta_{0};{a_{f}E\over
K}\right)\Big{]}$ (107)

while for the radial dynamics one has

$\int^{\rho}_{\rho_{0}}{d\rho^{\prime}\over\sqrt{E^{2}\rho^{4}{+}[2E^{2}(a_{f}^{2}{+}L^{2}){-}K^{2}]\rho^{2}{+}E^{2}(a_{f}^{2}{+}L^{2})^{2}{-}K^{2}a_{f}^{2}}}=$
(108) $={1\over
E\rho_{+}}\int_{\rho_{0}\over\rho_{-}}^{\rho\over\rho_{-}}{dx\over\sqrt{(1-x^{2})\Big{[}1{-}\left({\rho_{-}\over\rho_{+}}\right)^{2}x^{2}\Big{]}}}={1\over
E\rho_{+}}\Big{[}F\left({\rho\over\rho_{-}};{\rho_{-}\over\rho_{+}}\right)-F\left({\rho_{0}\over\rho_{-}};{\rho_{-}\over\rho_{+}}\right)\Big{]}$

with

$\rho^{2}_{\pm}=-a_{f}^{2}-L^{2}+{K^{2}\over
2E^{2}}\left(1\pm\sqrt{1-4{E^{2}L^{2}\over K^{2}}}\right)$

####  5.1.4  Shear-free geodesics

Shear-free geodesics ChandraBH  with $\theta=\theta_{0}$, are possible when
$P_{\theta}=0$ i.e.

$K^{2}_{s-free}={J_{\phi}^{2}\over\sin^{2}\theta_{0}}+{J_{\psi}^{2}\over\cos^{2}\theta_{0}}+a_{f}^{2}E^{2}\sin^{2}\theta_{0}$
(109)

and $P_{\theta}^{\prime}=0$ i.e.

$0=-{J_{\phi}^{2}\over\sin^{3}\theta_{0}}\cos\theta_{0}+{J_{\psi}^{2}\over\cos^{3}\theta_{0}}\sin\theta_{0}+a_{f}^{2}E^{2}\sin\theta_{0}\cos\theta_{0}$
(110)

The radial momentum does not simplify much and leads to incomplete elliptic
integrals of the first kind, as above.

Barring the special cases $\theta_{0}=0$ or $\pi/2$, that we have already
analysed in detail, and setting $\xi=\cos^{2}\theta_{0}$,
$\beta^{2}=K^{2}/a_{f}^{2}{\cal E}^{2}$,
$\beta_{\phi}^{2}=J_{\phi}^{2}/a_{f}^{2}{\cal E}^{2}$ and
$\beta_{\psi}^{2}=J_{\psi}^{2}/a_{f}^{2}{\cal E}^{2}$ one finds

${\beta_{\phi}^{2}\over(1-\xi)^{2}}-{\beta_{\psi}^{2}\over\xi^{2}}=1$ (111)

and

${\beta_{\phi}^{2}\over 1-\xi}+{\beta_{\psi}^{2}\over\xi}+1-\xi=\beta^{2}$
(112)

Solving the first for $\xi=f(\beta_{\phi},\beta_{\psi})\in(0,1)$181818This
condition imposes restrictions on the allowed values of
$\beta_{\phi},\beta_{\psi}$. and plugging into the second yields the special
values of $\beta_{s{-}f}$ in terms of $\beta_{\phi},\beta_{\psi}$ i.e. of
$K^{2}_{s{-}f}$ in terms of $J^{2}_{\phi}$, $J^{2}_{\psi}$ and $a_{f}^{2}{\cal
E}^{2}$.

Otherwise, one can solve for $\beta_{\phi}$ and $\beta_{\psi}$ in terms of
$\xi$ and $\beta$ and find

$\displaystyle\beta_{\phi}^{2}=(1-\xi)^{2}(\beta^{2}+2\xi-1)\quad$
$\displaystyle\sim\quad J_{\phi}^{2}=\sin^{4}\theta_{0}[K^{2}+a_{f}^{2}{\cal
E}^{2}(2\cos^{2}\theta_{0}-1)]$ (113)
$\displaystyle\beta_{\psi}^{2}=\xi^{2}(\beta^{2}+2\xi-2)\quad$
$\displaystyle\sim\quad J_{\psi}^{2}=\cos^{4}\theta_{0}[K^{2}-2a_{f}^{2}{\cal
E}^{2}\sin^{2}\theta_{0}]$

For $K^{2}>2a_{f}^{2}{\cal E}^{2}$ ($\beta>\sqrt{2}$) $\theta_{0}$ can take
any value. For $K^{2}<2a_{f}^{2}{\cal E}^{2}$ ($\beta<\sqrt{2}$) $\theta_{0}$
is constrained by $\theta_{0}<\arccos\left(1-{K^{2}\over 2a_{f}^{2}{\cal
E}^{2}}\right)$.

For small $a_{f}$, $|\beta_{\phi}|,|\beta_{\psi}|,\beta>>1$ and one finds

$(1-\xi)|\beta_{\psi}|\approx\pm\xi|\beta_{\phi}|\quad\sim\quad\xi\approx{|\beta_{\psi}|\over|\beta_{\psi}|\pm|\beta_{\phi}|}$
(114)

(only the + sign is acceptable) and then

$|\beta_{\phi}||\beta_{\psi}|+\beta_{\psi}^{2}-\beta^{2}\xi\approx
0\quad\sim\quad\beta^{2}=\left[|\beta_{\phi}||\beta_{\psi}|+\beta_{\psi}^{2}\right]{|\beta_{\psi}|+|\beta_{\phi}|\over|\beta_{\psi}|}=[|\beta_{\psi}|+|\beta_{\phi}|]^{2}$
(115)

or

$\beta=|\beta_{\phi}|+|\beta_{\psi}|\quad\sim\quad K=|J_{\phi}|+|J_{\psi}|$
(116)

Clearly as soon as $a_{f}\neq 0$ the condition gets modified.

Shear-free geodesics will play a role in the study of wave perturbations with
$\ell=|m_{\phi}|+|m_{\psi}|$ which is the minimal value of $\ell$ for given
$m_{\phi}$ and $m_{\psi}$, since in general $\ell=|m_{\phi}|+|m_{\psi}|+2r$
with $r=0,1,...$.

###  5.2 Wave equation

Let us now pass and consider the Klein-Gordon equation for massless scalar
waves in the D1-D5 fuzz-ball background. The wave equation can be separated as
follows:

$\displaystyle\Bigg{\\{}{1\over\rho}\partial_{\rho}\Big{[}\rho(\rho^{2}+a_{f}^{2})\partial_{\rho}\Big{]}+\tilde{\omega}^{2}(\rho^{2}+a_{f}^{2})\left(1+{L_{1}^{2}+L_{5}^{2}\over\rho^{2}+a_{f}^{2}}\right)+{\mathcal{L}_{\phi}^{2}\over\rho^{2}+a_{f}^{2}}-{\mathcal{L}_{\psi}^{2}\over\rho^{2}}-K^{2}\Bigg{\\}}R(\rho)=0$
(117)
$\displaystyle\Big{[}\nabla_{S_{3}}^{2}-\tilde{\omega}^{2}a_{f}^{2}\sin^{2}\theta+K^{2}\Big{]}S(\theta)=0$

where

$\nabla_{S_{3}}^{2}={1\over\sin(2\theta)}\partial_{\theta}\Big{[}\sin(2\theta)\partial_{\theta}\Big{]}-{m_{\phi}^{2}\over\sin^{2}\theta}-{m_{\psi}^{2}\over\cos^{2}\theta}$
(118)

Let’s transform the radial equation after introducing:

${\rho^{2}\over a_{f}^{2}}=y\quad,\quad{L_{1}^{2}+L_{5}^{2}\over
a_{f}^{2}}=\lambda\quad,\quad{{\lambda}^{2}_{\phi,\psi}}={\mathcal{L}_{\phi,\psi}^{2}\over
a_{f}^{2}}$ (119)

so that the differential equation in canonical form becomes:

$\psi^{\prime\prime}(y)+\Bigg{[}{1{-}{{\lambda}^{2}_{\psi}}\over
4y^{2}}+{1{-}{{\lambda}^{2}_{\phi}}\over
4(1+y)^{2}}+{2{+}K^{2}{-}{{\lambda}^{2}_{\phi}}{-}{{\lambda}^{2}_{\psi}}{-}a_{f}^{2}\tilde{\omega}^{2}\lambda\over
4(1+y)}+{{-}2{-}K^{2}{+}{{\lambda}^{2}_{\phi}}{+}{{\lambda}^{2}_{\psi}}{+}a_{f}^{2}\tilde{\omega}^{2}(1{+}\lambda)\over
4y}\Bigg{]}\psi(y)=0$ (120)

If we compare with the (2,0) quantum SW curve Bianchi:2021xpr ;
Bianchi:2021mft ; Bonelli:2022ten ; Consoli:2022eey :

$\psi^{\prime\prime}(y)+\Bigg{[}{1-\left({m_{1}-m_{2}\over\hbar}\right)^{2}\over
4y^{2}}+{1-\left({m_{1}+m_{2}\over\hbar}\right)^{2}\over
4(1+y)^{2}}+{1-2\left({m_{1}^{2}+m_{2}^{2}\over\hbar^{2}}\right)+{4u\over\hbar^{2}}\over
4(1+y)}+{-1+2\left(m_{1}^{2}+m_{2}^{2}\over\hbar^{2}\right)+{4(q-u)\over\hbar^{2}}\over
4y}\Bigg{]}\psi(y)=0$ (121)

The dictionary between radial equation and $(2,0)$ q-SW curve reads:

${m_{1,2}\over\hbar}={{\lambda}_{\phi}\mp\lambda_{\psi}\over
2}={\mathcal{L}_{\phi}\mp\mathcal{L}_{\psi}\over
2a_{f}}\quad,\quad{u\over\hbar^{2}}={1-\tilde{\omega}^{2}(L_{1}^{2}+L_{5}^{2})+K^{2}\over
4}\quad,\quad q={\hbar^{2}\tilde{\omega}^{2}a_{f}^{2}\over 4}$ (122)

The angular equation in the coordinate $y=-\cos^{2}\theta=-\xi$ and in
canonical form becomes

$\psi^{\prime\prime}(y)+\Bigg{[}{2+K^{2}-m_{\phi}^{2}-m_{\psi}^{2}\over
4(1+y)}+{1-m_{\psi}^{2}\over 4y^{2}}+{1-m_{\phi}^{2}\over
4(1+y)^{2}}+{-2-K^{2}+a_{f}^{2}\tilde{\omega}^{2}+m_{\phi}^{2}+m_{\psi}^{2}\over
4y}\Bigg{]}\psi(y)=0$ (123)

The dictionary between angular equation and $(2,0)$ q-SW reads

${m_{1,2}^{\theta}\over\hbar}={m_{\phi}\pm m_{\psi}\over
2}\quad,\quad{u_{\theta}\over\hbar^{2}}={1+K^{2}\over 4}\quad,\quad
q_{\theta}={\hbar^{2}\tilde{\omega}^{2}a_{f}^{2}\over 4}$ (124)

Quite remarkably $q_{\theta}=q_{R}$.

Let us focus on the latter first.

####  5.2.1 Angular equation

In the (2,0) flavour case the SW curve is given by

$P_{L}(x)=(x-m_{1,A})(x-m_{2,A})\quad,\quad P_{R}(x)=1\quad,\quad
P_{0}(x)=x^{2}-u_{A}+q$ (125)

The quantum period $a_{A}(u_{A})$ can then be written as:

$\displaystyle a_{A}(u_{A})$ $\displaystyle=\oint_{\alpha}\lambda_{+}=2\pi
i\sum_{n=0}^{\infty}Res_{\sqrt{u_{A}}+n}\lambda_{+}(x)$ (126) $\displaystyle
a_{A}(u_{A})$ $\displaystyle=\sqrt{u_{A}}+{1-4m_{1,A}m_{2,A}-4u_{A}\over
4\sqrt{u_{A}}(4u_{A}-1)}q+\Bigg{\\{}{2-3u_{A}\over
64(u_{A}-1)u_{A}^{3/2}}+{3(m_{1,A}^{2}+m_{2,A}^{2})\over
16\sqrt{u_{A}}(1-5u_{A}+4u_{A}^{2})}+$
$\displaystyle+{m_{1,A}m_{2,A}(1-12u_{A})\over
4(1-4u_{A})^{2}u_{A}^{3/2}}-{m_{1,A}^{2}m_{2,A}^{2}\Big{[}2+5u_{A}(12u_{A}-7)\Big{]}\over
4(u_{A}-1)u^{3/2}(4u_{A}-1)^{3}}\Bigg{\\}}q^{2}+$
$\displaystyle+\Bigg{\\{}{5u_{A}-2-5u_{A}^{2}\over
256(u_{A}-1)^{2}u_{A}^{5/2}}+{3(m_{1,A}^{2}+m_{2,A}^{2})(1-15u_{A}+20u_{A}^{2})\over
64u_{A}^{3/2}(1-5u_{A}+4u_{A}^{2})^{2}}+$
$\displaystyle+{m_{1,A}m_{2,A}(m_{1,A}^{2}+m_{2,A}^{2})(560u_{A}^{3}-1120u_{A}^{2}+497u_{A}-27)\over
16(u_{A}-1)^{2}u_{A}^{3/2}(4u_{A}-9)(4u_{A}-1)^{3}}+$
$\displaystyle+{m_{1,A}m_{2,A}(54-861u_{A}+6397u_{A}^{2}-13100u_{A}^{3}+10480u_{A}^{4}-2880u_{A}^{5})\over
64(u_{A}-1)^{2}u_{A}^{5/2}(4u_{A}-9)(4u_{A}-1)^{3}}+$
$\displaystyle-{3m_{1,A}^{2}m_{2,A}^{2}(2-39u_{A}+371u_{A}^{2}-840u_{A}^{3}+560u_{A}^{4})\over
16(4u_{A}-1)^{4}(u_{A}-1)^{2}u_{A}^{5/2}}+$
$\displaystyle+{m_{1,A}^{3}m_{2,A}^{3}(18-413u_{A}+4705u_{A}^{2}-15260u_{A}^{3}+18480u_{A}^{4}-6720u_{A}^{5})\over
4(u_{A}-1)^{2}u_{A}^{5/2}(4u_{A}-9)(4u_{A}-1)^{5}}\Bigg{\\}}q^{3}+{\cal
O}(q^{4})$

Inverting this relation, one finds:

$\displaystyle u_{A}=a_{A}^{2}+{4a_{A}^{2}+4m_{1,A}m_{2,A}-1\over
2(4a_{A}^{2}-1)}q+\Bigg{[}{1\over
32(a_{A}^{2}-1)}-{3(m_{1,A}^{2}+m_{2,A}^{2})\over
8-40a_{A}^{2}+32a_{A}^{4}}+{(7+20a_{A}^{2})m_{1,A}^{2}m_{2,A}^{2}\over
2(a_{A}^{2}-1)(4a_{A}^{2}-1)^{3}}\Bigg{]}q^{2}+$ (127)
$\displaystyle+\Bigg{[}{5m_{1,A}m_{2,A}\over
4(16a_{A}^{6}{-}56a_{A}^{4}{+}49a_{A}^{2}{-}9)}{-}{m_{1,A}m_{2,A}(m_{1,A}^{2}{+}m_{2,A}^{2})(17{+}28a_{A}^{2})\over(4a_{A}^{2}{-}1)^{3}(4a_{A}^{4}{-}13a_{A}^{2}{+}9)}{+}{4m_{1,A}^{3}m_{2,A}^{3}(29{+}232a_{A}^{2}{+}144a_{A}^{4})\over(4a_{A}^{2}{-}1)^{5}(4a_{A}^{4}{-}13a_{A}^{2}{+}9)}\Bigg{]}q^{3}$
$\displaystyle\quad+{\cal O}(q^{4})$ (128)

The quantization condition for $a_{A}$ reads $a_{A}=n_{A}+{1\over 2}$. Using
the dictionary with the angular equation, we find:

$K^{2}=4u_{A}-1=\ell(\ell+2)+4\sum_{k=1}^{\infty}u_{k}q^{k}$ (129)

where we identified $n_{A}=\ell/2$ and $u_{k}$ are the coefficients of the
instanton expansion for $u_{A}$ (127).

####  5.2.2 Radial equation

The radial equation can be put in the standard form of a RCHE Bonelli:2022ten

${d^{2}W\over dz^{2}}+\left({\gamma\over z}+{\delta\over z-1}\right){dW\over
dz}+{\beta z-\zeta\over z(z-1)}W=0$

with gauge / Heun dictionary

$u=\zeta-\beta+{1\over 4}(\gamma+\delta-1)\>,\quad
q=\beta\>,\quad{\lambda}_{\psi}=\gamma\>,\quad{\lambda}_{\phi}=\delta$

Recall that $q_{R}=q_{A}={1\over 4}\tilde{\omega}^{2}a_{f}^{2}$ while
$u_{R}=u_{A}-{1\over 2}\tilde{\omega}^{2}L^{2}$ and
${\lambda}_{\psi}=m_{\psi}-P_{z}{L^{2}\over a_{f}}$,
${\lambda}_{\phi}=m_{\phi}-\omega{L^{2}\over a_{f}}$.

The connection from $z=0$ (regular) to $z=\infty$ (irregular) (e.g.
$z=-\rho^{2}/a_{f}^{2}$) reads Bonelli:2022ten

$\sqrt{\pi}z^{+{1\over 4}+{\gamma+\delta\over 2}}e^{\pm i\delta{\pi\over
2}}W^{RC}_{0}(\zeta,\beta,\gamma,\delta;z)=$
$\sum_{\sigma=\pm}Ae^{i\pi\sigma(a-a_{D})}e^{2i\sqrt{\beta
z}}W^{RC}_{\infty}(\zeta,\beta,\gamma,\delta;{1\over\sqrt{z}})+i\sum_{\sigma=\pm}Ae^{-i\pi\sigma(a+a_{D})}e^{-2i\sqrt{\beta
z}}W^{RC}_{\infty}(\zeta,e^{2\pi i}\beta,\gamma,\delta;{1\over\sqrt{z}})$

where we used

${a(\zeta)\Gamma(2\sigma a(\zeta))^{2}\Gamma(\gamma)(e^{i\pi}\beta)^{-{1\over
4}-\sigma a(\zeta)}e^{-{1\over 2}\partial_{a_{0}}{\cal F}+{\sigma\over
2}\partial_{a}{\cal F}}\over\Gamma\left({\gamma-\delta+1\over 2}+\sigma
a(\zeta)\right)\Gamma\left({\gamma+\delta-1\over 2}+\sigma
a(\zeta)\right)}=Ae^{i\pi\sigma(a-a_{D})}$

with $A$ independent of $\sigma$, $2\pi ia_{D}=-\partial_{a}{\cal F}$ with
${\cal F}(a,q,m_{f};\hbar)$ NS pre-potential and

$W^{RC}_{0}(\zeta,\beta,\gamma,\delta;z)=1-{\zeta\over\gamma}z+{\gamma(\beta-\zeta)+\zeta(\zeta-\delta)\over
2\gamma(\gamma+1)}z^{2}+{\cal O}(z^{3})$

is the Reduced Confluent Heun function.

All the relevant observables can be extracted from this formula. Prompt-ring
down modes correspond to absence of the (ingoing) term $e^{-2i\sqrt{\beta z}}$
which is equivalent to the quantisation of the cycle $a_{D}-a$. In practice
one has to compute the quantum period $a(u)$, invert it and use quantum Matone
relation Matone:1995rx ; Flume:2004rp  to find ${\cal F}(a,q,m_{f};\hbar)$ and
then $a_{D}={-1\over 2\pi i}\partial{\cal F}/\partial a$. Expressing $a_{D}$
and $a$ in terms of $u$, that in turn is given in terms of $u_{A}$ and
$\tilde{\omega}L$, one finds an eigenvalue equation for $\omega_{QNM}$. The
numerical results are collected in appendix B.

Due the absence of a horizon the absorption probability is zero. This is
tantamount to saying that for real $\omega$ the reflected and incident waves
from infinity have the same amplitude. Indeed

$\displaystyle A_{I}=iA\sum_{\sigma=\pm}\sigma
e^{-i\pi\sigma(a+a_{D})}=2A\sin\pi(a+a_{D})$ (130) $\displaystyle
A_{R}=A\sum_{\sigma=\pm}\sigma e^{i\pi\sigma(a-a_{D})}=-2iA\sin\pi(a-a_{D})$

and for real $\omega$, $q=\omega^{2}a_{f}^{2}/4$ is real and so is $a$, while
$a_{D}=i\tau$ is purely imaginary, thus

$\displaystyle{|A_{I}|^{2}\over 4|A|^{2}}=$
$\displaystyle|i\sinh\pi\tau\cos\pi a+\cosh\pi\tau\sin\pi a|^{2}=$ (131)
$\displaystyle=|-i\sinh\pi\tau\cos\pi a+\cosh\pi\tau\sin\pi
a|^{2}={|A_{R}|^{2}\over 4|A|^{2}}$

####  5.2.3 Breit-Wigner resonance and direct integration Methods

The existence of a metastable minimum inside the photon-sphere implies the
presence of a new class of quasi-bound states whose imaginary part, which is
related to the tunnelling probability through the barrier, is in general very
small. Since the imaginary part of the QNMs is related to the inverse of the
life-time, we expect that this family of modes will dominate the late time of
the ring-down phase (long-lived modes). As pioneered by Chandrasekhar and
Ferrari ChanFerra  in the study of gravitational wave scattering on ultra-
compact stars, quasi bound states will present as Breit-Wigner resonances in
the scattering amplitude. More recently the resonance method has been used to
compute long lived modes of AdS BHs Berti:2009wx  and of ultra-compact
configurations like constant density stars and gravastars Cardoso:2014sna
whose effective potentials share astonishing analogies with the one for
circular fuzz-ball, including the absence of a horizon. To implement these
technique, our starting point is the choice of physically motivated boundary
conditions. The general asymptotic behaviour at infinity reads:

$R(\rho)\sim
B(\omega)e^{-i\omega\rho}+C(\omega)e^{i\omega\rho}\qquad\qquad\rho\rightarrow\infty$
(132)

The boundary condition $B=0$ describes purely outgoing wave at infinity i.e.
prompt ring down modes, while the condition $C=0$ is associated to states
localized in proximity of the minimum of the effective potential and
exponentially decaying at spatial infinity, i.e. quasi-bound or long-lived
modes. Close to the ‘cap’ at $\rho=0$, we impose regularity of the radial wave
function:

$R(\rho)\sim\rho^{|m_{\psi}|}\qquad\qquad\rho\rightarrow 0$ (133)

Starting from $\rho=0$, we numerically integrated until “infinity”
$\rho=\Lambda>>L,a_{f}$ searching for exponentially decaying solutions
($C=0$). Then we proceeded by studying the minima of the square modulus of the
numerically integrated radial function which at this level ($\rho=\Lambda$) is
a function $S(\omega)$ only of the frequency $\omega$, that may be considered
(almost) real, since we expect $\omega_{R}>>|\omega_{I}|$. The minima of
$|S(\omega)|^{2}$ will represent the real parts of long-live modes. Now, since
the characteristic frequencies $\omega_{0}=\omega_{R}+i\omega_{I}$ are
obtained by imposing $S(\omega)=0$ and since $\omega_{R}>>|\omega_{I}|$, we
get

$S(\omega_{R}+i\omega_{I})\simeq S(\omega_{R})+i\omega_{I}{dS(\omega)\over
d\omega}(\omega_{R})=0$ (134)

Expanding around $\omega_{R}$, restricting the argument of $S(\omega)$ to be
real and using the previous expansion, we obtain:

$S(\omega)\simeq S(\omega_{R})\left[1-{\omega-\omega_{R}\over
i\omega_{I}}\sim\omega-\omega_{R}-i\omega_{I}\right]$ (135)

So near the real $\omega$-axis:

$|S(\omega)|^{2}\simeq{\cal K}[(\omega-\omega_{R})^{2}+\omega_{I}^{2}]$ (136)

Using a parabolic fit near the minimum of $|S(\omega)|^{2}$ we extracted the
value of the concavity ${\cal K}$, so we obtained a reliable approximation of
$\omega_{I}$ by simply inverting (136). The whole procedure can be
satisfactorily described as in Fig. 4, while the numerical values of long-
lived modes are collected in appendix B. Direct integration method consists in
numerically finding the complex roots of the function $S(\omega_{0})=0$.
Usually direct integration approach is very efficient and reliable in
computing quasi-bound states since the boundary condition $C=0$ can be imposed
on the leading behaviour of the radial function at infinity. Conversely prompt
ring down modes, that would require $B=0$ as boundary condition, require to
determine the subleading exponentially suppressed behaviour of radial function
at infinity which could be contaminated by numerical error Pani:2013pma .

Figure 4: Plot of the function $|S(\omega)|^{2}$ considered as a real function
of $\omega$ for $a_{f}=0.01$ and angular quantum numbers $\ell=1$,
$m_{\phi}=0$, $m_{\psi}=\pm 1$. Resonances are visible when $\omega_{R}\simeq
0.04+n*0.02$. In the inset we plot the behaviour near the first minimum which
shows a parabolic-like shape.

###  5.3 Tidal Love numbers

As for (small) BHs we can study the tidal Love number of circular fuzz-balls.

For static waves $\omega=0$ the problem becomes spherically symmetric, since
the angular momentum of the fuzz-ball $a_{f}$ appears only in combination with
$\omega$. As usual the tidal Love number is determined by the ratio of the
coefficient of term decaying at infinity (‘response’) wrt the one growing at
infinity (‘source’).

For $\omega=0$ and $\mu=0$ the radial equation in the variable
$y=\rho^{2}/a_{f}^{2}$ is formally identical to the angular equation in the
variable $z=\cos^{2}\theta$ up to a sign change $y=-z$

$\Psi^{\prime\prime}(z)+\left[{1-m_{\phi}^{2}\over
4(1-z)^{2}}+{1-m_{\psi}^{2}\over
4z^{2}}-{K^{2}+2-m_{\phi}^{2}-m_{\psi}^{2}\over 4z(z-1)}\right]\Psi(z)=0$
(137)

Setting $\Psi(z)=z^{1+|m_{\psi}|\over 2}(1-z)^{1+|m_{\phi}|\over 2}F(z)$ leads
to a hypergeomtric equation with

$c=1+|m_{\psi}|\quad,\quad a={|m_{\psi}|+|m_{\phi}|-\ell\over 2}\quad
b=1+{|m_{\psi}|+|m_{\phi}|+\ell\over 2}=a+\ell+1$ (138)

Notice that

$\ell=|m_{\psi}|+|m_{\phi}|+2r$ (139)

with $r=0,1,2,....$ ($r=0$ is the wave counterpart of the shear-free
geodesics) so that $a=-r$ and $b=\ell-r+1\geq r+1$ and $c$ is an integer.

The ‘regular’ solution (at $z=0$ and $z=1$) of the angular equation are the
spherical harmonics on $S^{3}$

$S_{\ell}(\theta)e^{im_{\phi}\phi}e^{im_{\psi}\psi}=D^{(\ell/2)}_{m_{1},m_{2}}(2\theta,\phi,\psi)=d^{(\ell/2)}_{m_{1},m_{2}}(2\theta)e^{i(m_{1}+m_{2})\phi}e^{i(m_{1}-m_{2})\psi}$
(140)

which in turn are related to the characters of $SU(2)$ that are polynomial in
$z$.

Using the relation with the hypergeometric equation, the ‘regular’ solution
reads

$\Psi_{reg}(z)=z^{1+|m_{\psi}|\over 2}(1-z)^{1+|m_{\phi}|\over
2}F(-r,\ell-r+1;1+|m_{\psi}|,z)$ (141)

with $F_{r}(z)=F(-r,\ell-r+1;1+|m_{\psi}|,z)$ a degree $r$ (Jacobi) polynomial
in $z=-\rho^{2}/a_{f}^{2}$. The other independent solution is irregular at
both $z=0$ (‘cap’) and $z=1$ (unphysical) since it develops a logarithmic
behaviour191919Indeed the second solution is of the form
$G(z)=\log(z)F_{r}(z)+\sum_{k=-m_{\psi}}^{\infty}c_{k}z^{k}$. due to the
integer difference between the Frobenius exponents:
$\Delta\alpha_{0}=|m_{\psi}|$ and $\Delta\alpha_{1}=|m_{\phi}|$.

Going back to the ‘physical’ perturbation described by
$R(\rho)={\Psi\over\sqrt{z(1-z)}}$ and imposing regularity at $z=0$, i.e.
choosing $R_{reg}(z)={\Psi_{reg}(z)\over\sqrt{z(1-z)}}$ fixes completely the
behaviour at $\infty$

$R_{reg}(z)\sim z^{r+{|m_{\phi}|+|m_{\phi}|\over 2}}\left(1-{1\over
z}\right)^{|m_{\phi}|\over 2}{F(-r,\ell-r+1;1+|m_{\psi}|,z)\over
z^{r}}=z^{{\ell\over 2}}+...+\kappa^{\ell}_{m_{\phi},m_{\psi}}z^{-{\ell\over
2}-1}$ (142)

where $\kappa^{\ell}_{m_{\phi},m_{\psi}}$ is the coefficient of the term of
order $z^{-\ell-1}$ in the expansion of the factor

$f(z)=(1-{1\over z})^{|m_{\phi}|\over 2}{F(-r,\ell-r+1;1+|m_{\psi}|,z)\over
z^{r}}$ (143)

One easily finds

$\kappa^{\ell}_{m_{\phi},m_{\psi}}={(-1)^{m_{\psi}+m_{\phi}+r+1}\Gamma\left(1+\frac{|m_{\phi}|}{2}\right)\over\Gamma\left(-r-|m_{\psi}|-\frac{|m_{\phi}|}{2}\right)\Gamma(|m_{\phi}|+|m_{\psi}|+r+2}\times$
(144)
${}_{3}F_{2}\left(-r,\frac{|m_{\phi}|}{2}+|m_{\psi}|+r+1,|m_{\phi}|+|m_{\psi}|+r+1;|m_{\psi}|+1,|m_{\phi}|+|m_{\psi}|+r+2;1\right)$

that looks finite and non-zero at least for $m_{\phi}$ odd, quite
independently of the value of $m_{\psi}$, while it vanishes for $m_{\phi}$
even.

More remarkably, $\kappa^{\ell}_{m_{\phi},m_{\psi}}$ is totally independent of
$a_{f}$ and $L$ so even for ‘typical’ fuzz-balls with $a_{f}<<L$ one would
have a non-zero result (for $m_{\phi}$ odd)202020We thank Daniel Mayerson for
enlightening discussion on this point..

It is often customary to define TLN from an asymptotic expansion in terms of
the rescaled radial variables $\hat{r}=r/(r_{+}{-}r_{-})$ or $\tilde{r}=r/2M$
in $D=4$ or $\hat{r}^{2}=r^{2}/(r_{+}^{2}{-}r_{-}^{2})$ or
$\tilde{r}^{2}=r^{2}/2M$ in $D=5$. In our case this would suggest introducing
$\tilde{\rho}=\rho L/a$. As a result the TLN would rescale as
$\tilde{\kappa}^{\ell}_{m_{\phi},m_{\psi}}=\kappa^{\ell}_{m_{\phi},m_{\psi}}a_{f}^{2\ell+2}/L^{2\ell+2}$
and would smoothly go to zero as $a_{f}\rightarrow 0$.

Anyway it seems that contrary to small BHs

${\cal L}^{fuzz}_{stat}\neq 0$ (145)

Also notice that contrary to ‘standard’ $D=5$ BHs Pereniguez:2021xcj
212121Similar results have been obtained by Daniel Mayerson for BMPV BHs. We
thank him for sharing his insights. no log terms are possible since they would
be incompatible with regularity at $z=0$ and they cannot be generated by
analytic continuation of $R_{reg}(z)=z^{|m_{\psi}|\over
2}(1-z)^{|m_{\phi}|\over 2}F_{r}$ to infinity since $F_{r}$ is a polynomial.
This is another feature that distinguishes (circular) fuzz-balls from (small)
BHs even in the rigorously static limit.

| Angular quantum numbers | Static Love Number
---|---|---
$\ell=0$ |  $m_{\phi}=0$, $m_{\psi}=0$, $r=0$ | 0
$\ell=1$ |  $m_{\phi}=1$, $m_{\psi}=0$, $r=0$ | $-{1\over 8}$
|  $m_{\phi}=1$, $m_{\psi}=1$, $r=0$ | 0
$\ell=2$ |  $m_{\phi}=2$, $m_{\psi}=0$, $r=0$ | 0
|  $m_{\phi}=1$, $m_{\psi}=1$, $r=0$ | $-{1\over 16}$
|  $m_{\phi}=0$, $m_{\psi}=2$, $r=0$ | 0
|  $m_{\phi}=0$, $m_{\psi}=0$, $r=1$ | 0
$l=3$ |  $m_{\phi}=3$, $m_{\psi}=0$, $r=0$ | ${3\over 128}$
|  $m_{\phi}=2$, $m_{\psi}=1$, $r=0$ | 0
|  $m_{\phi}=1$, $m_{\psi}=2$, $r=0$ | $-{5\over 128}$
|  $m_{\phi}=0$, $m_{\psi}=3$, $r=0$ | 0
|  $m_{\phi}=1$, $m_{\psi}=0$, $r=1$ | ${7\over 128}$
|  $m_{\phi}=0$, $m_{\psi}=1$, $r=1$ | 0
Table 2: Some TLNs for different choices of angular quantum numbers. Note that
TLNs are non-zero for $m_{\phi}$ odd.

This non-zero result in the ‘rigorously’ static case should be compared with
the $\omega\rightarrow 0$ limit of the ‘dynamical’ Love number with
$\omega\neq 0$ via the ‘Love function’ (defined for $L<<r<<1/\omega$) by
imposing regularity at the horizon (or at the ‘cap’) and then taking the ratio
of the coefficient of the ‘response’ term decaying at infinity (as
$\rho^{-\ell-2-\Delta\ell(\omega)}$) and of the ‘source’ term growing at
infinity (as $\rho^{+\ell+\Delta\ell(\omega)}$). Once again and for the same
reasons as before, the two limits in general do not commute Consoli:2022eey .
We collect some values of the TLNs for different choises of angular quantum
numbers in Table 2

##  6 Conclusions and outlook

We have studied scalar perturbations of circular fuzz-balls and compared the
results for physical observable such as QNMs, TLNs and grey-body factor with
the ones for a small BH obtained in the limit $a_{f}\rightarrow 0$.

Let’s summarise our results:

  * •

The circular fuzz-ball presents two branches of QNMs: the prompt-ring down,
a.k.a. photon-sphere, modes localised at the maximum of the radial effective
potential and long-lived modes, a.k.a. quasi-bound modes, localised at a local
(metastable) minimum of the radial effective potential; small BH only admit
prompt-ring down modes that match with those of the circular fuzz-ball when
$a_{f}<<L$.

  * •

The circular fuzz-ball has zero absorption cross section, due to the absence
of a horizon. On the contrary, small BHs, very much as D3-branes Gubser:1998iu
have a non-zero $\sigma_{abs}(\omega)$ notwithstanding the vanishing area of
the horizon. We computed $P_{abs}(\omega)$ and then $\sigma_{abs}(\omega)$
relying on a combination of properties of (modified) Mathieu equation, that is
the qSW curve for pure ${\cal N}=2$ SYM theory, and instanton techniques à la
Nekrasov-Shatashvili.

  * •

Even in the rigorously static case $\omega=0$ the circular fuzz-ball expose
non-zero TLNs when one of the two azimuthal ‘quantum’ numbers ($m_{\phi}$) is
odd. The (adimensional) results are independent of $a_{f}$ and $L$. A
rescaling by $(a_{f}/L)^{2\ell+2}$ is suggested by similar analyses for BHs.
On the contrary small BHs have zero TLNs in the rigorously static case, while
the static limit $\omega\rightarrow 0$ of the Tidal Love function, introduced
in Consoli:2022eey  produces a non-zero results.

  * •

In both cases absence of an ergo-region (since $g_{tt}>0$ always and
everywhere) prevents super-radiance to take place. Yet charge super-radiance
is expected to take place at the small BH ‘horizon’ for probes with charge $q$
when $\omega<q\Phi_{H}$, with $\Phi_{H}$ the ‘electric’ potential at the
horizon222222We plan to study this and similar effects in the near future
MBCdBGdR .

As expected and in some sense required for a correct description of the small
BH ‘micro states’, the differences tend to disappear for small $a_{f}$, except
for the adimensional TLNs. Yet the absence of any absorption that is replaced
by the presence of echoes, associated to long-lived quasi-bound modes,
produces observable features that should allow to ‘classically’ discriminate
fuzz balls from BHs. In particular ‘ensemble average’ may lead to measurable
deviations in the GW signal of fuzz-ball mergers as compared with BH mergers.
These can be hinted at by studying the evolution in the time domain of a
gaussian shell along the lines of Ikeda:2021uvc  in the single micro-state or
‘averaged’ fuzz-ball.

For simplicity we worked with (massive) scalar perturbations, while the
ultimate goal would be to study GW or EM waves. However in the BPS context we
performed our analysis, residual supersymmetry transformation should allow to
map spin 0 to spin 1 and possibly spin 2 and reduce the problem to the correct
identification of the Killing spinors. This applies to more realistic micro-
state geometries based on multi-center ansatze that admit only reduced
isometry, whereby vector or tensor perturbations would really be hard to
analyse.

Our approach is largely based on integrability so it is hard to apply to such
micro states. One aspect that could still be addressed is the breaking of
equatorial symmetry emphasised in Fransen:2022jtw  that is present in a class
of BHs, known as Rasheed-Larsen BHs Larsen:1999pp ; Rasheed:1995zv , that lead
to separable dynamics. It would be interesting to apply our techniques in this
or similar cases.

Integrability is known to take place also in some 3-charge superstrata
Bena:2016ypk  with AdS asymptotics, corresponding to a decoupling limit that
only focus on the ‘cap’ region. A preliminary analysis shows that this
description could only capture the long-lived / quasi-bound modes, that are in
fact exactly stable in this approximation due to the infinite wall at the
boundary of AdS, but not the prompt ring-down modes. Establishing a connection
between TLNs as response to a source and response functions in the AdS context
Bena:2019azk  could elucidate the meaning of the non-vanishing adimensional
TLNs that we found here. Last but not least one would like to explore similar
properties of neutral micro-states (possibly with ‘large’ dipole moments)
Bah:2022yji ; Mayerson:2022ekj ; Ganchev:2022vrv .

## Acknowledgements

We would like to thank P. Arnaudo, G. Bonelli, G. Bossard, D. Capocci, D.
Consoli, M. Firrotta, F. Fucito, A. Grassi, A. Grillo, A. Hashimoto, C. Iossa,
D. Mayerson, J. F. Morales, N. Nekrasov, D. Panea-Lichtig, P. Pani, R.
Poghosyan, F. Riccioni, R. Savelli, A. Tanzini, for interesting discussions
and clarifications. M. B. would like to thank Ecole Polytechnique for the kind
hospitality while this work was being completed. We thank the MIUR PRIN
contract $2020KR4KN2$ String Theory as a bridge between Gauge Theories and
Quantum Gravity and the INFN project $ST\&FI$ String Theory and Fundamental
Interactions for partial support.

##  Appendix A The AGT picture

AGT duality Alday:2009aq  relates 4-d $\mathcal{N}=2$ quiver theories in a
non-commutative $\Omega$-background à la Nekrasov-Shatasvili Nekrasov:2009rc ;
MM0910.5670 ; Z1103.4843  to 2-d Liouville CFT. Denoting by
${\epsilon_{1}},{\epsilon_{2}}$ the quantum deformation parameters and by
$b=\sqrt{\frac{\epsilon_{1}}{\epsilon_{2}}}$, the central charge of the CFT is
given by

$c=1+6Q^{2}\quad{\rm with}\quad Q=b+\frac{1}{b}$

AGT relates conformal blocks of the 2-d Liouville CFT to (ratioes of) quiver
partition functions

$\mathcal{C}_{p_{0}\cdots
p_{n+1}}^{\alpha_{1}\cdots\alpha_{n+1}}\left(\\{z_{i}\\}\right)\prod_{j=1}^{n}z_{j}^{-\Delta_{p_{j}}+\Delta_{j}+\Delta_{p_{j}+1}}=\frac{Z_{\text{inst}}\left(\\{\vec{a}_{i}\\},\\{q_{i}\\}\right)}{Z_{U(1)}(\\{q_{i}\\})}$

the dictionary relates insertion points $z_{i}$ to (ratios of) gauge couplings
$q_{i}$, the dimension $\Delta_{i}$ of the chiral vertex operators to masses
$m_{f}$ or scalar vev’s $a_{r}$. Finally the BPZ (Belavin-Polyakov-
Zamolodchikov) equation satisfied by the conformal blocks is related to the
quantum SW curve, which in turn plays the role of wave equation in a BH or
fuzz-ball background, the central theme of our analysis.

In order to briefly illustrate the procedure232323See Consoli:2022eey  for a
more detailed description in similar contexts., consider the $n+3$-point
function of chiral vertex operators $V_{\alpha_{i}}=e^{2\alpha_{i}\phi}$ with
$\Delta_{i}=\alpha_{i}\left(Q-\alpha_{i}\right)$

${\cal
G}(z_{i},\bar{z}_{i})=\langle\prod_{i=0}^{n+2}V_{\alpha_{i}}(z_{i})\rangle$

and expand it in terms of conformal blocks

${\cal G}(z_{i},\bar{z}_{i})=\sum_{p_{1}\cdots p_{n}}\langle
p_{0}|V_{\alpha_{0}}(z_{0})|p_{1}\rangle\cdots\langle
p_{n}|V_{\alpha_{n+2}}(z_{n+2})|p_{n+1}\rangle=\sum_{p_{1}\cdots
p_{n}}\left|\mathcal{C}_{p_{0}\cdots
p_{n+1}}^{\alpha_{1}\cdots\alpha_{n+1}}\right|^{2}$

For definiteness choose $n=2$, related to an $SU(2)\times SU(2)$ quiver. The
NS partition function, to be identified with the wave-function, is associated
to conformal block involving level 2 degenerate field with
$\alpha_{3}=-{b}/{2}$

$\Psi(\\{z_{i}\\})=\mathcal{C}_{p_{0}\cdots
p_{3}}^{\alpha_{1}\cdots\alpha_{3}}\left(\\{z_{i}\\}\right)$

Since $\left(L_{-1}^{2}+b^{2}L_{2}\right)V_{\alpha_{3}}\sim 0$ is null,
$\Psi(\\{z_{i}\\})$ satisfies the BPZ equation

$\Psi^{\prime\prime}(\\{z_{i}\\})+b^{2}\sum_{i\neq
1}^{4}\left[\frac{\Delta_{i}}{(z-z_{i})^{2}}+\frac{1}{z-z_{i}}\partial_{z_{i}}\right]\Psi(\\{z_{i}\\})=0$

Setting $z_{0}=\infty$, $z_{1}=1$, $z_{2}=q$, $z_{3}=z$, $z_{4}=0$ and
performing a semi-classical double scaling limit $b\rightarrow 0$, i.e.
$\epsilon_{1}=0$ in the NS $\Omega$-background, with fixed

$b^{2}\Delta_{i}=\delta_{i}\,,\quad
b^{2}c_{i}=\nu_{i}\,,\quad\partial_{z_{i}}\Psi(y;\\{z_{i}\\})=c_{i}\Psi(y;\\{z_{i}\\})$

one gets the quantum SW curve / differential equation for $\Psi(z)$
with242424There are many possible choices of signs in the following
expressions.

$\displaystyle\delta_{p_{0}}={1\over 4}-p_{0}^{2}\quad,$ $\displaystyle\quad
p_{0}={m_{1}-m_{2}\over 2}$ (146) $\displaystyle\delta_{k_{0}}={1\over
4}-k_{0}^{2}\quad,$ $\displaystyle\quad k_{0}={1+m_{1}+m_{2}\over 2}$
$\displaystyle\delta_{p_{3}}={1\over 4}-p_{3}^{2}\quad,$ $\displaystyle\quad
p_{3}={m_{3}-m_{4}\over 2}$ $\displaystyle\delta_{k}={1\over 4}-k^{2}\quad,$
$\displaystyle\quad k={1-m_{3}-m_{4}\over 2}$ $\displaystyle c=k+p_{3}\quad,$
$\displaystyle\quad c^{\prime}=p_{0}-k_{0}$

##  Appendix B Tables

$n=0$ | WKB | Seiberg-Witten
---|---|---
$a_{f}=0$ (BH) | 0.433013-0.25 I | 0.498509 - 0.278493 I
$a_{f}=0.01$ | 0.433037 - 0.249966 I | 0.498493 - 0.263181 I
$a_{f}=0.02$ | 0.43311 - 0.249862 I | 0.498476 - 0.263177 I
$a_{f}=0.05$ | 0.433622 - 0.249139 I | 0.498352 - 0.263152 I
$a_{f}=0.1$ | 0.435459 - 0.246543 I | 0.497917 - 0.263058 I
$a_{f}=0.15$ | 0.438544 - 0.242183 I | 0.497212 - 0.262893 I
$a_{f}=0.2$ | 0.442909 - 0.236058 I | 0.496261 - 0.262649 I
$a_{f}=0.25$ | 0.448583 - 0.228314 I | 0.495096 - 0.262321 I
$a_{f}=0.3$ | 0.455564 - 0.219488 I | 0.493746 - 0.261908 I
Table 3: Prompt ringdown modes : $l=0\Rightarrow m_{\phi}=m_{\psi}=0$ $n=0$ | WKB | Seiberg-Witten
---|---|---
$a_{f}=0$ (BH) | 0.968246-0.25 I | 0.999627 - 0.255004 I
$a_{f}=0.01$ | 0.968272 - 0.249978 I | 0.999742 - 0.253996 I
$a_{f}=0.02$ | 0.968352 - 0.249912 I | 0.999769 - 0.253954 I
$a_{f}=0.05$ | 0.968913 - 0.249446 I | 0.999962 - 0.253661 I
$a_{f}=0.1$ | 0.970928 - 0.247753 I | 1.00064 - 0.252607 I
$a_{f}=0.15$ | 0.974335 - 0.244826 I | 1.00176 - 0.250821 I
$a_{f}=0.2$ | 0.979216 - 0.240485 I | 1.00326 - 0.248265 I
$a_{f}=0.25$ | 0.9857 - 0.234407 I | 1.0051 - 0.244889 I
$a_{f}=0.3$ | 0.993986 - 0.226022 I | 1.00719 - 0.240641 I
Table 4: Prompt ringdown modes: $l=1\Rightarrow m_{\phi}=0,m_{\psi}=\pm 1$ $n=0$ | WKB | Direct Integration | Breit-Wigner
---|---|---|---
$a_{f}=0.01$ | $0.0380185{-}7.57895E{-}15I$ | $0.039992{-}(****)E{-}14I$ | $0.039992{-}(****)I$
$a_{f}=0.02$ | $0.0759967{-}3.34991E{-}12I$ | $0.079936{-}2.06087E{-}12I$ | $0.079936{-}3.31768E{-}12I$
$a_{f}=0.05$ | $0.18929{-}1.17547E{-}8I$ | $0.199001{-}8.11612E{-}10I$ | $0.199001{-}8.39096E{-}10I$
$a_{f}=0.1$ | $0.373622{-}7.24191E{-}6I$ | $0.392034{-}5.38521E{-}7I$ | $0.392034{-}5.86285E{-}7I$
$a_{f}=0.15$ | $0.548311{-}0.000376738I$ | $0.573191{-}0.0000267682I$ | $0.573191{-}0.0000288871I$
$a_{f}=0.2$ | $0.709157{-}0.00680183I$ | $0.73646{-}0.000489566I$ | $0.73646{-}0.000490324I$
$a_{f}=0.25$ | $0.852547{-}0.0643601I$ | $0.875631{-}0.00358531I$ | $0.876019{-}0.00362734I$
Table 5: Long-lived modes modes: $l=1\Rightarrow m_{\phi}=0,m_{\psi}=\pm 1$ $n=0$ | WKB | Seiberg-Witten
---|---|---
$a_{f}=0$ (BH) | 0.968246-0.25 I | 0.999627 - 0.255004 I
$a_{f}=0.01$ | 0.970767 - 0.249338 I | 1.00222 - 0.253381 I
$a_{f}=0.02$ | 0.973331 - 0.248641 I | 1.00471 - 0.252735 I
$a_{f}=0.05$ | 0.981289 - 0.246332 I | 1.01225 - 0.250681 I
$a_{f}=0.1$ | 0.995494 - 0.241681 I | 1.02498 - 0.246864 I
$a_{f}=0.15$ | 1.01099 - 0.235877 I | 1.03789 - 0.242534 I
$a_{f}=0.2$ | 1.0279 - 0.22874 I | 1.05093 - 0.237671 I
$a_{f}=0.25$ | 1.04637 - 0.220116 I | 1.06405 - 0.23227 I
$a_{f}=0.3$ | 1.06652 - 0.209993 I | 1.0772 - 0.226339 I
Table 6: Prompt ringdown modes: $l=1\Rightarrow m_{\phi}=1,m_{\psi}=0$ $n=0$ | WKB | Seiberg-Witten
---|---|---
$a_{f}=0$ (BH) | 0.968246-0.25 I | 0.999627 - 0.255004 I
$a_{f}=0.01$ | 0.965767 - 0.250629 I |  $0.997258-0.25462$ I
$a_{f}=0.02$ | 0.963329 - 0.251226 I |  $0.994793-0.255212$ I I
$a_{f}=0.05$ | 0.956253 - 0.252833 I |  $0.98746-0.256884$ I
$a_{f}=0.1$ | 0.94521 - 0.254956 I |  $0.97545-0.259339$ I
$a_{f}=0.15$ | 0.935027 - 0.256475 I |  $0.963716-0.261413$ I
$a_{f}=0.2$ | 0.925625 - 0.257482 I |  $0.952263-0.263141$ I
$a_{f}=0.25$ | 0.916931 - 0.258053 I |  $0.941093-0.264559$ I
$a_{f}=0.3$ | 0.908883 - 0.258258 I |  $0.930206-0.265698$ I
Table 7: Prompt ringdown modes: $l=1\Rightarrow m_{\phi}=-1,m_{\psi}=0$ $n=0$ | WKB | Seiberg-Witten
---|---|---
$a_{f}=0$ (BH) | 1.47902-0.25 I | 1.50114 - 0.250172 I
$a_{f}=0.01$ | 1.47904 - 0.249988 I | 1.49856 - 0.24844 I
$a_{f}=0.02$ | 1.47908 - 0.249954 I | 1.49857 - 0.248423 I
$a_{f}=0.05$ | 1.47941 - 0.249712 I | 1.49862 - 0.248304 I
$a_{f}=0.1$ | 1.48058 - 0.248846 I | 1.49881 - 0.247884 I
$a_{f}=0.15$ | 1.48252 - 0.247396 I | 1.49913 - 0.247201 I
$a_{f}=0.2$ | 1.48525 - 0.245354 I | 1.4996 - 0.24628 I
$a_{f}=0.25$ | 1.48874 - 0.242712 I | 1.50026 - 0.245149 I
$a_{f}=0.3$ | 1.49301 - 0.239465 I | 1.50114 - 0.243834 I
Table 8: Prompt ringdown modes: $l=2\Rightarrow m_{\phi}=m_{\psi}=0$: the first non-shear free $n=0$ | WKB | Seiberg-Witten
---|---|---
$a_{f}=0$ (BH) | 1.47902-0.25 I | 1.50114 - 0.250172 I
$a_{f}=0.01$ | 1.47906 - 0.249975 I |  $1.49859-0.248427$ I
$a_{f}=0.02$ | 1.4792 - 0.249902 I |  $1.49868-0.248369$ I
$a_{f}=0.05$ | 1.48014 - 0.249383 I |  $1.49933-0.247964$ I
$a_{f}=0.1$ | 1.48355 - 0.247482 I |  $1.50165-0.246489$ I
$a_{f}=0.15$ | 1.48935 - 0.244131 I |  $1.50552-0.243924$ I I
$a_{f}=0.2$ | 1.49779 - 0.238996 I |  $1.51094-0.240099$ I
$a_{f}=0.25$ | 1.50922 - 0.231446 I |  $1.5179-0.234761$ I
$a_{f}=0.3$ | 1.52428 - 0.22021 I |  $1.52629-0.227565$ I I
Table 9: Prompt ringdown modes: $l=2\Rightarrow m_{\phi}=0,m_{\psi}=\pm 2$ $n=0$ | WKB | Direct Integration | Breit-Wigner
---|---|---|---
$a_{f}=0.01$ | $0.0589341{-}3.59432E{-}21I$ | $0.0599879{-}(****)I$ | $0.0599879-(****)I$
$a_{f}=0.02$ | $0.117801{-}2.74315E{-}17I$ | $0.119904{-}(****)I$ | $0.119904-(****)I$
$a_{f}=0.05$ | $0.293332{-}4.42517E{-}12I$ | $0.298503{-}(****)E{-}9I$ | $0.298503-1.80738E{-}9I$
$a_{f}=0.1$ | $0.578388{-}5.65627E{-}8I$ | $0.588085{-}4.99815E{-}8I$ | $0.588085-5.23269E{-}8I$
$a_{f}=0.15$ | 0.847314-0.0000191774 I | $0.8601{-}8.91*10^{-7}I$ | $0.8601-9.37417E{-}7I$
$a_{f}=0.2$ | 1.093-0.00135584 I | $1.10614{-}0.0000603124I$ | $1.10614-0.0000698527I$
$a_{f}=0.25$ | 1.30917-0.0361528 I | $1.31713{-}0.0013159I$ | $1.31716-00135213I$
Table 10: Long-lived modes: $l=2\Rightarrow m_{\phi}=0,m_{\psi}=\pm 2$ $n=0$ | WKB | Seiberg-Witten
---|---|---
$a_{f}=0$ (BH) | 1.47902-0.25 I | 1.50114 - 0.250172 I
$a_{f}=0.01$ | 1.48155 - 0.249561 I |  $1.50118-0.247959$ I
$a_{f}=0.02$ | 1.48413 - 0.249091 I |  $1.50382-0.247455$ I
$a_{f}=0.05$ | 1.49219 - 0.247476 I |  $1.51193-0.245834$ I
$a_{f}=0.1$ | 1.50678 - 0.244051 I |  $1.52602-0.242725$ I
$a_{f}=0.15$ | 1.52294 - 0.239567 I |  $1.54077-0.239004$ I
$a_{f}=0.2$ | 1.54084 - 0.233792 I |  $1.55616-0.234501$ I
$a_{f}=0.25$ | 1.56074 - 0.226362 I |  $1.57214-0.228992$ I
$a_{f}=0.3$ | 1.58298 - 0.216656 I |  $1.58864-0.222206$ I
Table 11: Prompt ringdown modes: $l=2\Rightarrow m_{\phi}=1,m_{\psi}=\pm 1$ $n=0$ | WKB | Direct Integration | Breit-Wigner
---|---|---|---
$a_{f}=0.01$ |  $0.05823{-}2.5985E{-}23$ I | $0.059988{-}(****)I$ | $0.059988{-}(****)I$
$a_{f}=0.02$ |  $0.116394{-}1.97546E{-}19$ I | $0.119904{-}(****)I$ | $0.119904{-}(****)I$
$a_{f}=0.05$ |  $0.289845{-}3.12586E{-}14$ I | $0.298504{-}(****)I$ | $0.298504{-}(****)I$
$a_{f}=0.1$ |  $0.571628{-}3.81488E{-}10$ I | $0.588116{-}1.101E{-}9I$ | $0.588116{-}1.59509E{-}9I$
$a_{f}=0.15$ |  $0.837718{-}1.22639E{-7}$ I | $0.860343{-}3.91098E{-}7I$ | $0.860343{-}4.06279E{-}7I$
$a_{f}=0.2$ |  $1.081246{-}8.20825E{-}6$I | $1.10727{-}0.0000262723I$ | $1.10727{-}0.0000278005I$
$a_{f}=0.25$ |  $1.296233{-}0.000206853$ I | $1.32108{-}0.000633244I$ | $1.32108{-}0.000633721I$
Table 12: Long-lived modes: $l=2\Rightarrow m_{\phi}=1,m_{\psi}=\pm 1$ $n=0$ | WKB | Seiberg-Witten
---|---|---
$a_{f}=0$ (BH) | 1.47902-0.25 I | 1.50114 - 0.250172 I
$a_{f}=0.01$ | 1.47655 - 0.250407 I | 1.49598 - 0.248916 I
$a_{f}=0.02$ | 1.47412 - 0.250783 I | 1.49342 - 0.24937 I
$a_{f}=0.05$ | 1.46716 - 0.251731 I | 1.48594 - 0.250635 I
$a_{f}=0.1$ | 1.45652 - 0.252742 I | 1.47411 - 0.252442 I
$a_{f}=0.15$ | 1.44703 - 0.25308 I | 1.46308 - 0.253881 I
$a_{f}=0.2$ | 1.43864 - 0.252765 I | 1.45287 - 0.254947 I
$a_{f}=0.25$ | 1.43132 - 0.251796 I | 1.4435 - 0.255624 I
$a_{f}=0.3$ | 1.42503 - 0.250146 I | 1.43497 - 0.255878 I
Table 13: Prompt ringdown modes: $l=2\Rightarrow m_{\phi}=-1,m_{\psi}=\pm 1$

$n=0$ | WKB | Direct integration | Breit-Wigner
---|---|---|---
$a_{f}=0.01$ |  $0.0382363{-}3.25139E{-}24$ I |  $0.0399947{-}(****)$ I | $0.0399947{-}(****)I$
$a_{f}=0.02$ |  $0.0764463{-}2.41336E{-}20$ I |  $0.0799573{-}(****)$ I | $0.0799573{-}(****)I$
$a_{f}=0.05$ |  $0.190657{-}3.37968E{-}15$ I |  $0.199335{-}(****)$ I | $0.199335{-}(****)I$
$a_{f}=0.1$ |  $0.378054{-}3.08729E{-}11$ I | $0.394708{-}4.92541E{-}11I$ | $0.394708{-}5.52827E{-}11I$
$a_{f}=0.15$ |  $0.559017{-}7.26757E{-9}$ I | $0.582304{-}1.60485E{-}8I$ | $0.582303{-}2.46023E{-}8I$
$a_{f}=0.2$ |  $0.730548{-}3.7315E{-}7$ I |  $0.758577{-}8.83296E{-}7$ I | $0.758577{-}9.40976E{-}7I$
$a_{f}=0.25$ |  $0.88991{-}7.99721E{-}6$ I |  $0.920289{-}0.0000176402$ I | $0.920289{-}0.0000181988I$
Table 14: Long-lived modes: $l=2\Rightarrow m_{\phi}=-1,m_{\psi}=\pm 1$ $n=0$ | WKB | Seiberg-Witten
---|---|---
$a_{f}=0$ (BH) | 1.47902-0.25 I | 1.50114 - 0.250172 I
$a_{f}=0.01$ | 1.48405 - 0.249139 I | 1.50378 - 0.247486 I
$a_{f}=0.02$ | 1.48914 - 0.248246 I | 1.50906 - 0.246522 I
$a_{f}=0.05$ | 1.50476 - 0.245359 I | 1.52514 - 0.243616 I
$a_{f}=0.1$ | 1.53208 - 0.239781 I | 1.55263 - 0.238765 I
$a_{f}=0.15$ | 1.56113 - 0.233085 I | 1.58057 - 0.23374 I
$a_{f}=0.2$ | 1.59211 - 0.225054 I | 1.60861 - 0.227994 I
$a_{f}=0.25$ | 1.62519 - 0.215449 I | 1.63688 - 0.220948 I
$a_{f}=0.3$ | 1.66056 - 0.20407 I | 1.66572 - 0.212378 I
Table 15: Prompt Ringdown modes: $l=2\Rightarrow m_{\phi}=2,m_{\psi}=0$ $n=0$ | WKB | Seiberg-Witten
---|---|---
$a_{f}=0$ (BH) | 1.47902-0.25 I | 1.50114 - 0.250172 I
$a_{f}=0.01$ | 1.47405 - 0.25083 I | 1.49339 - 0.249399 I
$a_{f}=0.02$ | 1.46913 - 0.251629 I | 1.48826 - 0.250345 I
$a_{f}=0.05$ | 1.45471 - 0.253855 I | 1.4732 - 0.253122 I
$a_{f}=0.1$ | 1.43172 - 0.257048 I | 1.44913 - 0.257473 I
$a_{f}=0.15$ | 1.40991 - 0.259679 I | 1.42631 - 0.261374 I
$a_{f}=0.2$ | 1.38921 - 0.26183 I | 1.40466 - 0.26475 I
$a_{f}=0.25$ | 1.36951 - 0.263571 I | 1.3841 - 0.267553 I
$a_{f}=0.3$ | 1.35075 - 0.264959 I | 1.3645 - 0.269752 I
Table 16: Prompt Ringdown modes: $l=2\Rightarrow m_{\phi}=-2,m_{\psi}=0$

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