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

# More on the SW-QNM correspondence

Massimo Bianchi a Dario Consoli a Alfredo Grillo a Josè Francisco Morales
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We exploit the recently proposed correspondence between gravitational
perturbations and quantum Seiberg-Witten curves to compute the spectrum of
quasi-normal modes of asymptotically flat Kerr Newman black holes and
establish detailed gauge/gravity dictionaries for a large class of black
holes, D-branes and fuzzballs in diverse dimensions. QNM frequencies obtained
from the quantum periods of $SU(2)$ ${\cal N}=2$ SYM with $N_{f}=3$ flavours
are compared against numerical results, WKB (eikonal) approximation and
geodetic motion showing remarkable agreement. Starting from the master example
relating quasi-normal modes of Kerr-Newman black holes in AdS4 to $SU(2)$
gauge theory with $N_{f}=4$, we illustrate the procedure for some simple toy-
models that allow analytic solutions. We also argue that the AGT version of
the gauge/gravity correspondence may give precious hints as to the
physical/geometric origin of the quasi-normal modes/Seiberg-Witten connection
and further elucidate interesting properties (such as tidal Love numbers and
grey-body factors) that can help discriminating black holes from fuzzballs.

††preprint: PREPRINT

##  1 Introduction

Compact gravitating objects, such as black holes (BHs), D-branes and micro-
state geometries (‘fuzz-balls’) are often characterised by a set of Quasi-
Normal Modes (QNMs) Bianchi:2021xpr  that govern the linear response to
external perturbations. In the eikonal (WKB) approximation, the complex QNM
frequencies can be written as

$\omega_{QNM}\approx\omega_{c}(\ell)-i(2n+1)\lambda$ (1)

with $\omega_{c}(\ell)$ the frequencies of the (unstable) ‘circular’ orbits
forming the so-called photon-sphere, $\lambda$ the Lyapunov exponent, encoding
the damping time of the wave and quantifying the chaotic behaviour of
geodesics near the photon-sphere, and $n$ the so-called ‘overtone’ number
Cardoso:2008bp ; Mashhoon:1985cya ; Schutz:1985km ; Iyer:1986np ; Yang:2012he
; Bianchi:2020des ; Bianchi:2020yzr ; ToVSapQNM .

The interest in accurate values of $\omega_{QNM}$ is two-fold. On the one hand
they dominate the Gravitational-Wave ring-down signal in binary mergers and
may help discriminating BHs from fuzzballs or other Exotic Compact Objects
Bianchi:2020bxa ; Bianchi:2020miz ; Bena:2020see ; Bena:2020uup ; Bah:2021jno
. On the other hand, due to the choice of boundary conditions, QNMs solve non
self-adjoint spectral problems, such as Regge-Wheeler-Zerilli or Teukolsky
equations RegWheel ; Zerilli:1970se ; Teukolsky:1972my , and form an over-
complete set that play a crucial role in the study of BH perturbations. Alas
even for the simplest (spherically symmetric) case (e.g. Schwarzschild BHs)
accurate values of $\omega_{QNM}$ can only be computed via numerical methods
Leaver:1985ax ; Leaver:1990zz .

Quite recently, attempting exact WKB quantization techniques Mironov:2009uv ;
Zenkevich:2011zx ; Bourgine:2017aqk ; Fioravanti:2019vxi ; Grassi:2018bci ;
Grassi:2019coc , a new astonishing gauge-gravity connection between the QNM
spectral problem and quantum Seiberg-Witten (SW) curves Seiberg:1994rs ;
Seiberg:1994aj ; Nekrasov:2009rc  for $\mathcal{N}=2$ SYM theories was
suggested and tested in the case of Kerr BHs in 4-d Grassi . It is worth
mentioning that similar problems have been studied using the monodromy
properties of differential equations yielding similar results Novaes:2014lha ;
CarneirodaCunha:2015hzd ; CarneirodaCunha:2015qln ; Amado:2017kao ;
Lencses:2017dgf ; BarraganAmado:2018zpa ; Novaes:2018fry ;
CarneirodaCunha:2019tia ; Amado:2020zsr ; Bershtein:2021uts ;
BarraganAmado:2021uyw ; Cavalcante:2021scq ; daCunha:2021jkm ; Amado:2021erf .

The QNM-SW correspondence was extended in Bianchi:2021xpr  to several gravity
systems including BHs in higher dimensions, D-branes, their bound-states and
fuzzballs (smooth horizonless micro-state geometries). Moreover, exploiting
the AGT correspondence Alday:2009aq , wave functions for Kerr BHs were related
to correlators in two-dimensional CFTs involving degenerate fields, thus
providing a new tool to study other interesting observables of the gravity
solution such as Love numbers, absorption coefficients and grey body factors
BonTanzetc .

Aim of this paper is to apply these ideas to a large class of gravity
backgrounds and to develop some numerical, WKB, and geodetic motion methods
that allow to test the QNM results. QNMs are obtained as solutions of the wave
equation with outgoing boundary conditions outside the photon-sphere and
ingoing in the interior. For a geometry with enough isometries, the equation
can be separated into ordinary Schrödinger like differential equations
describing the wave propagation in the radial and angular directions. The
equations can be put in the canonical form viz.

${d^{2}\Psi\over dz^{2}}+Q(z)\Psi=0$ (2)

with $Q(z)$ a rational function. We find that the $Q$-functions characterising
many BH and brane solutions in various dimensions can assume the form

$Q(z)={P_{2n+2}(z)\over\Delta_{n+3}(z)^{2}}$ (3)

with $P_{2n+2}(z)$ and $\Delta_{n+3}(z)$ polynomials of order $(2n+2)$ and
$(n+3)$ respectively. The same differential equation describes the dynamics of
an $SU(2)^{n}$ linear $\mathcal{N}=2$ quiver theory in the Nekrasov-
Shatashvilli (NS) $\Omega$-background given by setting $\epsilon_{1}=\hbar$,
$\epsilon_{2}=0$. More precisely, the Q-function defines the quadratic
differential $\phi_{2}(z)=Q(z)dz^{2}$ of the $\Omega$-deformed version of the
SW Gaiotto:2009we . Zeroes of $P_{2n+2}(z)$ specify the positions of the
branch points of the associated SW curve, while those of $\Delta_{n+3}(z)$
encode the gauge couplings. Identifying the two $Q$’s one can establish a
dictionary between the parameters describing the gravity solution
(radial/angular variable $z$, mass $\mathcal{M}$, charge $\mathcal{Q}$,
angular momentum $\mathcal{J}$, frequency $\omega$ and conserved ‘quantum’
numbers $\ell$, $m$’s) and the gauge theory parameters ($z_{SW}$, the RG scale
$\Lambda$, hypermultiplet masses $m_{f}$ and Coulomb branch moduli $u_{a}$).

Quite remarkably, a large class of BHs and brane systems can be described in
terms of SW geometries associated to $\mathcal{N}=2$ SYM with a single $SU(2)$
gauge group, i.e. $n=1$, and $N_{f}$ hypers in the fundamental (doublet)
representation. For instance, the spectral problems of AdS Kerr-Newman (KN)
BHs in four dimensions lead to Heun equations with four regular singularities
that can be mapped to $SU(2)$ gauge theory with four fundamentals. The
asymptotically flat KN BH non-extremal and extremal solutions arise from the
general case after confluence of one or two pairs of singularities and lead to
$SU(2)$ gauge theories with $N_{f}=3$ and $N_{f}=2$ fundamentals,
respectively.

The common feature of all the solutions is the presence of a photon-sphere (or
a photon-halo in the rotating case), associated to degenerate choices of the
frequencies where two zeroes of $Q(z)$ coincide. In the gauge theory picture
this corresponds to points in the moduli space where two branch points collide
and the elliptic geometry degenerates in the absence of a NS
$\Omega$-background. The singularity is smoothed out by quantum corrections
once $\epsilon_{1}=\hbar$ is turned on. QNMs are associated to solutions of
the exact WKB quantization condition

$a_{\gamma}=\oint_{\gamma}\lambda=(n_{\gamma}+\nu)\hbar$ (4)

with $\gamma$ the degenerating cycle in the classical limit and $n_{\gamma}$
an integer and $\nu=0,{\textstyle\frac{1}{2}}$ depending on $\gamma$. For
example, radial and angular equations will be associated to degenerations of
$a_{D}$ and $a$-cycles respectively, with $-n_{r}$ parametrizing the overtone
and $n_{\theta}=\ell$ the orbital number. We find that $\nu=0$ and $\nu=1/2$
for quantization of the $a_{D}$ and $a$ cycles respectively. The period
$a_{\gamma}$ admits an ‘instanton’ expansion (in powers of $\Lambda$) up to
one-loop terms ($\log\Lambda$) as well as a ’semi-classical’ expansion (in
powers of ${\hbar}$).

Aim of the present paper is to exploit the new gauge-gravity connection to
compute the QNMs of asymptotically flat KN BHs in 4-d. In order to test our
results we compare them with the numerical results obtained via continuous
fractions à la Leaver Leaver:1985ax ; Leaver:1990zz , geodetic motion and WKB
approximation, finding reassuring agreement. We will also establish detailed
gauge/gravity dictionaries for several gravity solutions including AdS KN BHs
in 4-d, D3-branes and their bound-states Cvetic:1995uj , CCLP solutions of
Einstein-Maxwell gravity in 5-d CCLP1 ; CCLP2 , circular D1-D5 fuzz-balls
Lunin:2001fv  and regular JMaRT solutions in 6-d JMaRT ; GMS1 ; GMS2 .

We will mostly focus on massless minimally-coupled scalar
perturbations111Generalization to vector and tensor modes is straightforward,
though tedious and does not add much to the general features of the QNMs..
Extending our analysis to generic micro-state geometries, in the spirit of the
fuzzball proposal, seems hard due to the lack of isometries that prevents from
writing down (de-)coupled ODE’s for the QNMs. The use of the AGT
correspondence, on the other hand, may provide additional information in
discriminating BHs from fuzzballs. We intend to explore these observables for
D-branes and fuzzballs in the near future.

The plan of the paper is as follows. In Section 2 we study QNM solutions of
AdS KN scalar wave equations using semi-classical methods based on WKB and
geodetic motion. In Section 3, we review the quantum SW geometry and establish
the QNM-SW correspondence for the master AdS KN example. We illustrate the
algorithm in a handful of toy models and examples where analytic solutions for
QNMs can be found and related to quantum periods of free gauge theories. The
AGT version of the QNM-SW correspondence introduced in BonTanzetc  is briefly
reviewed and extended to the case of $SU(2)$ gauge theory with $N_{f}=4$
flavours. In Section 4 we introduce a numerical method based on Leaver’s
continuous fraction approximation that accounts also for extremal cases. In
Section 5 we compare the results for QNM frequencies of KN BHs obtained via
WKB, SW and numerical methods. Section 6 contains the detailed QNM-SW
dictionary for various BHs and brane systems, including the prototypical
D3-brane discussed in Bianchi:2021xpr . In Section 7 we draw some conclusions.
We relegate some technical details and some tables and plots of results to
Appendices A, B and C.

##  2 The AdS Kerr-Newman solution: wave equation vs geodetic motion

We consider gravity solutions surrounded by photon-spheres. QNMs in these
geometries can be defined as solutions of the wave equation with outgoing
boundary conditions at infinity and ingoing boundary conditions at the horizon
or, for smooth horizonless geometries, regularity in the interior of the
photon-sphere. For concreteness we focus on massless scalar perturbations but
higher spin (viz. vector and tensor) perturbations of the geometry can be
studied with similar techniques.

In this section we consider the case of KN (charged rotating) BH solution in
AdS4 and derive semi-classical formulae for the QNMs using WKB methods and
geodetic motion (see Suzuki:1998vy ; Zhidenko:2003wq ; Giammatteo:2005vu  for
previous studies of QNMs in Kerr-AdS spacetimes).

###  2.1 The wave equation

The KN-AdS BH solution is characterised by the mass $\mathcal{M}$, the angular
momentum parameter $a_{{}_{\mathcal{J}}}=\mathcal{J}/\mathcal{M}$, the
electrical charge ${\cal Q}$ and the AdS size $L$. The line element in Boyer-
Lindquist coordinates reads Caldarelli:1999xj

$ds^{2}=-\frac{\Delta_{r}[dt-
a_{{}_{\mathcal{J}}}\,d\phi(1-\chi^{2})]^{2}}{\alpha_{L}^{2}\rho^{2}}+\frac{\Delta_{\chi}[a_{{}_{\mathcal{J}}}\,dt-d\phi(a_{{}_{\mathcal{J}}}^{2}+r^{2})]^{2}}{\alpha_{L}^{2}\rho^{2}}+\rho^{2}\left(\frac{dr^{2}}{\Delta_{r}}+\frac{d\chi^{2}}{\Delta_{\chi}}\right)$
(5)

where $\chi=\cos\theta$ and

$\displaystyle\Delta_{r}$
$\displaystyle=(r^{2}+a_{{}_{\mathcal{J}}}^{2})\left(1+\frac{r^{2}}{L^{2}}\right)-2\mathcal{M}r+\mathcal{Q}^{2}\,,\quad\Delta_{\chi}=(1-\chi^{2})\left(1-\frac{a_{{}_{\mathcal{J}}}^{2}\chi^{2}}{L^{2}}\right)\,,$
(6) $\displaystyle\rho^{2}$
$\displaystyle=r^{2}+a_{{}_{\mathcal{J}}}^{2}\chi^{2}\,,\qquad\alpha_{L}=1-\frac{a_{{}_{\mathcal{J}}}^{2}}{L^{2}}$

We are interested on QNMs arising from scalar perturbations of the metric. On
AdS they are described by the wave equation

$(\square-
M_{\Phi}^{2})\Phi=\left[{1\over\sqrt{g}}\partial_{M}\left(\sqrt{g}\,g^{MN}\partial_{N}\right)-M_{\Phi}^{2}\right]\Phi=0$
(7)

with222According to holography, this massive scalar field is dual to scalar
operators of conformal dimension $\Delta=1$ or $\Delta=2$, as
$M_{\Phi}^{2}L^{2}=\Delta(\Delta-3)$ in $AdS_{4}/CFT_{3}$.

$M_{\Phi}^{2}=-2/L^{2}$ (8)

QNMs on asymptotically flat spaces can be obtained by sending $L\to\infty$ and
are described by massless scalar waves.

For the KN-AdS BH metric (5), the wave equation (7) can be separated into
radial and angular equations. Denoting by $z$ the radial or angular variable,
the individual equation for the radial or angular function $\phi(z)$ takes the
generic form

$\phi^{\prime\prime}(z)+q_{1}(z)\,\phi^{\prime}(z)+q_{0}(z)\phi(z)=0$ (9)

By writing

$\phi(z)=e^{-{1\over 2}\int^{z}q_{1}(z^{\prime})dz^{\prime}}\Psi(z)$ (10)

one can bring (9) to the canonical form (2) with

$Q(z)=q_{0}(z)-{q_{1}(z)^{2}\over 4}-{q_{1}^{\prime}(z)\over 2}$ (11)

Explicitly, taking

$\Phi(t,r,\chi,\phi)=e^{{-\rm i}(\omega
t-m_{\phi}\phi)}\frac{R(r)S(\chi)}{\sqrt{\Delta_{r}\Delta_{\chi}}}$ (12)

the wave equation separates into two equations of type (2) with

$\displaystyle Q_{r}$
$\displaystyle=\frac{1}{\Delta_{r}^{2}}\left[\alpha_{L}^{2}\left(\omega(a_{{}_{\mathcal{J}}}^{2}{+}r^{2}){-}a_{{}_{\mathcal{J}}}m_{\phi}\right)^{2}-\Delta_{r}(K^{2}{+}r^{2}M_{\Phi}^{2}){-}{\textstyle\frac{1}{2}}\Delta_{r}\Delta_{r}^{\prime\prime}{+}{\textstyle\frac{1}{4}}\Delta_{r}^{\prime}{}^{2}\right]$
(13) $\displaystyle Q_{\chi}$
$\displaystyle=\frac{1}{\Delta_{\chi}^{2}}\left[{-}\alpha_{L}^{2}\left(a_{{}_{\mathcal{J}}}\omega(1{-}\chi^{2}){-}m_{\phi}\right)^{2}{+}\Delta_{\chi}(K^{2}{-}a_{{}_{\mathcal{J}}}^{2}\chi^{2}M_{\Phi}^{2}){-}{\textstyle\frac{1}{2}}\Delta_{\chi}\Delta_{\chi}^{\prime\prime}{+}{\textstyle\frac{1}{4}}\Delta_{\chi}^{\prime}{}^{2}\right]$

and $K^{2}$ a separation constant. It is easy to check that for $M_{\Phi}$
given by (8) the $r^{6}$ terms in the numerator of (13) exactly cancel, so
that the numerator of both the radial $Q_{r}(r)$ and angular $Q_{\chi}(\chi)$
are given by polynomials of order four. The resulting $Q$-characteristic
functions will be put in correspondence with that describing the dynamics of
$SU(2)$ gauge theory with $N_{f}=(2,2)$ fundamentals.

Finally, the radial wave function $R(r)$ should be supplemented with the
boundary conditions

$R(r)\underset{r\to\infty}{\sim}e^{{\rm i}\,\omega\,r}\,,\quad
R(r)\underset{r\to r_{H}}{\sim}e^{-{\rm i}\,\omega\,(r-r_{H})}$ (14)

with $\omega=-P_{t}$ the frequency of the wave and $r_{H}$ the horizon. On the
other hand, the angular wave-function $S(\chi)$ satisfies boundary conditions
arising from periodicity and regularity at $\chi=\pm 1$.

###  2.2 WKB approximation

With the AdS-KN case in mind, let us consider the individual wave equations
for radial and angular dynamics

$\Psi^{\prime\prime}(z)+Q(z)\,\Psi(z)=0$ (15)

that can be both viewed as the Schrödinger equation for a particle subject to
a potential $V$ with $Q=E-V$ and $E$ the energy. In the limit of large
frequencies the equation can be solved in a semiclassical approximation by
writing

$\Psi(z)={1\over\sqrt{\varphi^{\prime}(z)}}\left(C_{1}e^{{\rm
i}\varphi(z)}+C_{2}\,e^{-{\rm i}\varphi(z)}\right)$ (16)

with

$\varphi(z)=\int^{z}\sqrt{Q(z^{\prime})}\leavevmode\nobreak\ dz^{\prime}$ (17)

The approximation breaks down near the zeroes $z_{\pm}$ (inversion points) of
$Q(z)$ where $\varphi^{\prime}(z)$ vanishes. The matching between the
solutions on the two sides of the inversion points requires that the frequency
$\omega$ satisfy the Bohr-Sommerfeld quantization condition

$\int_{z_{-}}^{z_{+}}\sqrt{Q(z)}\,dz={\rm}\pi\left(n+{\textstyle\frac{1}{2}}\right)\quad.$
(18)

with $n$ a non-negative integer. In the semiclassical limit where inversion
points collide the integral can be approximated as

$\int_{z_{-}}^{z_{+}}\sqrt{Q(z)}dz\approx\int_{z_{-}}^{z_{+}}\sqrt{Q(z_{c})+{Q^{\prime\prime}(z_{c})\over
2}(z-z_{c})^{2}}\,dz\approx{{\rm i}\pi
Q(z_{c})\over\sqrt{2Q^{\prime\prime}(z_{c})}}$ (19)

where $z_{c}\in\left[z_{-},z_{+}\right]$ is the extremum inside the
integration contour, i.e. $Q^{\prime}(z_{c})=0$. Applying (19) to the radial
and angular equations, one finds

$\displaystyle Q_{s}^{\prime}(z_{c}^{s})$ $\displaystyle=$ $\displaystyle 0$
$\displaystyle{Q_{s}(z_{c}^{s})\over\sqrt{2Q_{s}^{\prime\prime}(z_{c}^{s})}}$
$\displaystyle=$ $\displaystyle-{\rm
i}\left(n_{s}+{\textstyle\frac{1}{2}}\right)$ (20)

with $s=r,\theta$. To make easier the comparison against standard results in
the QNM literature we use $\theta$ (rather than $\chi$) variable in the WKB
analysis and introduce the separation constant $A$ instead of $K^{2}$

$K^{2}=A-m_{\phi}^{2}+\alpha_{L}^{2}(a_{{}_{\mathcal{J}}}\omega-m_{\phi})^{2}$
(21)

The equation $Q_{\theta}^{\prime}(\theta_{c})=0$ can be solved by taking
$\theta_{c}=\pi/2$. The remaining equations can be solved for the critical
radius $r_{c}$, the frequency $\omega$ and the separation constant $A$ by
giving to $\omega$ and $A$ small imaginary parts. More precisely, we write

$\omega=\omega_{c}+{\rm i}\omega_{I}\qquad,\qquad A=A_{c}+{\rm i}A_{I}$ (22)

with $|\omega_{I}|\ll|\omega_{c}|$ and $|A_{I}|\ll|A_{c}|$ and solve equations
(20) order by order in $\omega_{I}$ and $A_{I}$. To leading order, using that
$Q_{r}^{\prime\prime}(r_{c})>0$ and $Q_{\theta}^{\prime\prime}(\theta_{c})<0$,
one finds

$\displaystyle\partial_{r}Q_{r}(r_{c},\omega_{c},A_{c})=Q_{r}(r_{c},\omega_{c},A_{c})=0$
$\displaystyle
B_{\theta}(\omega_{c},A_{c})={Q_{\theta}(\omega_{c},A_{c})\over\sqrt{-2Q_{\theta}^{\prime\prime}(\omega_{c},A_{c})}}=n_{\theta}+{\textstyle\frac{1}{2}}$
(23)

with all functions evaluated at $\theta_{c}=\pi/2$ and

$n_{\theta}=\ell-|m_{\phi}|$ (24)

Equations (23) can be solved for $r_{c}$, $\omega_{c}$ and $A_{c}$. The
imaginary parts follows from the expansion of (20) to linear order in
$\omega_{I}$, $A_{I}$. One finds

$\displaystyle\omega_{I}=-{\left(n_{r}+{\textstyle\frac{1}{2}}\right)\sqrt{2Q_{r}^{\prime\prime}}\over\partial_{\omega}Q_{r}-\partial_{A}Q_{r}{\partial_{\omega}B_{\theta}\over\partial_{A}B_{\theta}}}\qquad,\qquad
A_{I}=-\omega_{I}{\partial_{\omega}B_{\theta}\over\partial_{A}B_{\theta}}$
(25)

with all functions evaluated at $\omega_{c}$, $A_{c}$, $r_{c}$ and
$\theta_{c}$.

###  2.3 Geodetic motion

QNM frequencies in the semi-classical approximation can be alternatively
derived from the geodetic motion of massless particles near the photon-sphere
of the gravitating object. In the Hamiltonian formalism, geodetic motion is
described by the Hamilton-Jacobi equations

$\displaystyle\dot{x}^{M}={\partial{\cal H}\over\partial
P_{M}}\qquad,\qquad\dot{P}_{M}=-{\partial{\cal H}\over\partial x^{M}}$ (26)

with

$\mathcal{H}={\textstyle\frac{1}{2}}g^{MN}P_{M}P_{N}=0$ (27)

For the AdS KN metric, the Hamiltonian can be written in the separable form

$2{\cal H}=\Delta_{r}(P_{r}^{2}-Q_{\rm
r,geo})+\Delta_{\theta}(P_{\theta}^{2}-Q_{\theta,{\rm geo}})$ (28)

with

$\displaystyle Q_{r,{\rm geo}}(r)$
$\displaystyle=\frac{\alpha_{L}^{2}[\omega(r^{2}+a_{{}_{\cal
J}}^{2})-a_{{}_{\cal
J}}m_{\phi}]^{2}-\Delta_{r}[A{\,-\,}m_{\phi}^{2}{\,+\,}\alpha_{L}^{2}(a_{{}_{\mathcal{J}}}\omega{\,-\,}m_{\phi})^{2}]}{\Delta_{r}^{2}}$
(29) $\displaystyle Q_{\theta,\rm geo}(\theta)$
$\displaystyle=\frac{\Delta_{\theta}\,\sin^{2}\theta\,[A{\,-\,}m_{\phi}^{2}{\,+\,}\alpha_{L}^{2}(a_{{}_{\mathcal{J}}}\omega{\,-\,}m_{\phi})^{2}]-\alpha_{L}^{2}(m_{\phi}{\,-\,}a_{{}_{\cal
J}}\omega\sin^{2}\theta)^{2}}{\Delta_{\theta}^{2}}$

The null equation ${\cal H}=0$ reduces to the one-dimensional conditions

$P_{r}(r)^{2}-Q_{\rm
r,geo}(r)=P_{\theta}(\theta)^{2}-Q_{\rm\theta,geo}(\theta)=0$ (30)

The characteristic $Q$-functions match those in the wave equations in the
eikonal limit where the last two terms in (13) can be discarded and
$Q_{\theta}\approx\sin^{2}\theta\,Q_{\chi}(\cos\theta)$. Similarly the null
condition (30) match radial and angular wave equations after quantization of
momenta, i.e. $P_{s}\approx-{\rm i}\partial_{s}\ln\Psi_{s}({x}^{s})$.

Zeroes of the $Q$-functions are then associated to inversion points of the
geodetic motion. A double zero of $Q_{r}$ signals the existence of a photon
sphere, i.e. null circular orbits. These geodesics are known to exist for a
given range of parameters (depending on the angular momenta $K$ and
$P_{\phi}$, the radius $r_{c}$ and of the frequency $\omega_{c}$), such that

$\displaystyle Q_{\rm r,geo}(r_{c},\omega_{c},A_{c})=Q_{\rm
r,geo}^{\prime}(r_{c},\omega_{c},A_{c})=0$ (31)

These are precisely the equations defining the extremum and the real part of
the QNM frequency in the WKB approximation. On the other hand, the imaginary
part of the frequency can be related to the radial velocity of a freely
falling geodesics at the photon sphere333Here we used $\partial_{A}{\cal H}=0$
to rewrite ${\Delta_{\theta}\over\Delta_{r}}=-{\partial_{A}Q_{r,\rm
geo}\over\partial_{A}Q_{\theta,\rm geo}}$.

${dr\over dt}={{\partial{\cal H}\over\partial P_{r}}\over{\partial{\cal
H}\over\partial P_{t}}}={2P_{r}\over\partial_{\omega}Q_{r,\rm
geo}{+}{\Delta_{\theta}\over\Delta_{r}}\partial_{\omega}Q_{\theta,\rm
geo}}={2\sqrt{Q_{r,\rm geo}}\over\partial_{\omega}Q_{r,\rm
geo}-{\partial_{A}Q_{r,\rm geo}\over\partial_{A}Q_{\theta,\rm
geo}}\partial_{\omega}Q_{\theta,\rm
geo}}\approx{\,-\,}2\lambda(r{\,-\,}r_{c})$ (32)

with

$\lambda={\sqrt{Q_{r,\rm geo}^{\prime\prime}\over
2}\over\partial_{\omega}Q_{r,\rm geo}-{\partial_{A}Q_{r,\rm
geo}\over\partial_{A}Q_{\theta,\rm geo}}\partial_{\omega}Q_{\theta,\rm geo}}$
(33)

$\lambda$ is known as the Lyapunov exponent and quantifies the chaotic
behavior of nearly critical geodesics around the photon-sphere Cardoso:2008bp
; Yang:2012he ; Bianchi:2020des ; Bianchi:2020yzr ; ToVSapQNM . We write

$\omega_{\rm geo}=\omega_{c}-{\rm i}\lambda(2n_{\rm geo}+1)$ (34)

that agrees with (25) in the limit of large charges where

$\partial_{A}Q_{r}{\partial_{\omega}B_{\theta}\over\partial_{A}B_{\theta}}\approx\partial_{A}Q_{r,\rm
geo}{\partial_{\omega}Q_{\theta,\rm geo}\over\partial_{A}Q_{\theta,\rm geo}}$
(35)

where all functions are understood evaluated at
$\omega_{,}A_{c},r_{c},\theta_{c}$.

###  2.4 Example: Kerr-Newman BH

Let us illustrate the WKB formulas for the case of asymptotically flat Kerr-
Newman and Schwarzschild BHs. The $Q$-characteristic functions are obtained
from (29) by sending the radius of AdS to infinity, i.e. $\alpha_{L}\to 1$,
$M_{\Phi}\to 0$, leading to

$\displaystyle Q_{r,{\rm geo}}(r)$
$\displaystyle=\frac{[\omega(r^{2}+a_{{}_{\cal J}}^{2}){-}a_{{}_{\cal
J}}m_{\phi}]^{2}}{\Delta_{r}^{2}}-\frac{A{-}2m_{\phi}a_{{}_{\cal
J}}\omega{+}a_{{}_{\cal J}}^{2}\omega^{2}}{\Delta_{r}}$ (36) $\displaystyle
Q_{\theta,\rm geo}(\theta)$ $\displaystyle=A+a_{{}_{\cal
J}}^{2}\omega^{2}\cos^{2}\theta-{m_{\phi}^{2}\over\sin^{2}\theta}$

#### Schwarzschild:

Setting $a_{{}_{\cal J}}=0$ and $\mathcal{Q}=0$ one finds the formulas for the
Schwarzschild BH

$\displaystyle Q_{r,{\rm geo}}={\omega^{2}r^{3}-A(r-2\mathcal{M})\over
r(r-2\mathcal{M})^{2}}\quad,\quad Q_{\theta,\rm
geo}(\theta)=A-{m_{\phi}^{2}\over\sin^{2}\theta}\quad,\quad B_{\theta}$
$\displaystyle={A-m_{\phi}^{2}\over 2m_{\phi}}$ (37)

Plugging this into (23) and (25) and solving for $\omega$, $A$, $r_{c}$ one
finds

$r_{c}=3\mathcal{M}\quad,\quad
A=|m_{\phi}|(|m_{\phi}|{+}1{+}2n_{\theta})\quad,\quad\mathcal{M}\,\omega_{\rm
Sch}={\sqrt{A_{c}}\over 3\sqrt{3}}-{\rm i}{2n_{r}+1\over 6\sqrt{3}}$ (38)

##  3 Quantum Seiberg-Witten curves for ${\cal N}=2$ SYM with flavours

In this section we first extend the QNM-SW dictionary (and its AGT version) to
the general case of $SU(2)$ ${\cal N}=2$ SYM theories with $N_{f}=4$
hypermultiplets that will be later on associated to the wave equation of
scalar metric perturbations of KN-AdS BHs. We then move on to theories with
less than four hypermultiplets, which are relevant for other geometries under
consideration, and show how SW quantization works in some toy examples that
allows for exact solutions.

###  3.1 The classical Seiberg-Witten curve

Figure 1: Brane configuration for ${\cal N}=2$ SYM with $SU(2)$ gauge group:
the QFT lives on a stack of two D4-branes (horizontal lines) suspended between
two non dynamical NS5-branes (vertical lines). The four external D4 flavour-
branes extending to infinity provide the matter hypermultiplets of the theory.

To get some intuition on the gauge/gravity dictionary, let us start by
considering the classical SW curve for an $SU(2)$ gauge theory with $N_{f}=4$
hypermultiplets with masses $m_{i}$ in flat space

$q\,y^{2}P_{L}(x)+yP_{0}(x)+P_{R}(x)=0$ (39)

with

$P_{0}(x)=x^{2}{-}u{+}q\,p_{0}(x)\,,\quad
P_{R}(x)=(x{-}m_{1})(x{-}m_{2})\,,\quad P_{L}(x)=(x{-}m_{3})(x{-}m_{4})\,\,$
(40)

where $q=e^{2\pi{\rm i}\tau}$ is the gauge coupling,
$u={\textstyle\frac{1}{2}}\langle{\rm tr}\varphi^{2}\rangle$ the Coulomb
branch modulus and $p_{0}(x)$ a quadratic polynomial in $x$ determined below
in (52).

The SW curve (39) can be derived from the brane configuration shown in Fig 1
Witten:1997sc . The zeroes of $P_{0}(x)$ are associated to the positions of
the colour D4-branes, while those of $P_{L/R}(x)$ to the positions of the
flavour D4-branes. The degrees of $P_{0}(x)$ and $P_{L/R}(x)$ are given by the
rank of the colour and flavour group respectively. Finally the distance
between the NS5-branes is proportional to the square inverse gauge coupling.
Solving for $y$ one finds

$y_{\pm}={1\over 2qP_{L}}\left(-P_{0}\pm\sqrt{P_{0}^{2}-4qP_{L}P_{R}}\right)$
(41)

The elliptic curve can be viewed as a double cover of the complex plane with
branch points $e_{i}$ defined by

$P_{0}^{2}-4qP_{L}P_{R}=\prod_{i=1}^{4}(x-e_{i})$ (42)

The periods of the elliptic curve are defined as

$a=\oint_{\alpha}\lambda_{0}\qquad,\qquad a_{D}=\oint_{\beta}\lambda_{0}$ (43)

with $\alpha$ and $\beta$ the two fundamental cycles and

$\lambda_{0}={\textstyle\frac{1}{2}}(\lambda_{+}-\lambda_{-})\qquad{\rm
with}\qquad\lambda_{\pm}={1\over 2\pi{\rm i}}x\,\partial_{x}\ln
y_{\pm}(x)\,dx$ (44)

the SW differentials. The dynamics is coded in the analytic pre-potential
${\cal F}(a;q,m_{i})$ in that the ${\cal N}=2$ vector multiplet Lagrangian
${\cal L}=\int d^{4}\theta{\cal F}(\Phi;q,m_{i})$ and

$2\pi{\rm i}\,a_{D}=-{\partial{\cal F}\over\partial a}(a;q,m_{i})$ (45)

At $q=0$ the four branch points collide in pairs at $\pm\sqrt{u}$, the
$\alpha$-cycle shrinks to zero and the $a$-period becomes

$a\approx{1\over 2\pi{\rm i}}\oint_{\alpha}{xP_{0}^{\prime}(x)\over
P_{0}(x)}\,dx=\sqrt{u}+\ldots$ (46)

Alternatively, collecting powers of $x$, the curve (39) can be written as

$A(y)\,x^{2}+B(y)\,x+C(y)=0$ (47)

and the periods as

$a=2\int_{\tilde{e}_{1}}^{\tilde{e}_{2}}\lambda_{0}\qquad,\qquad
a_{D}=2\int_{\tilde{e}_{2}}^{\tilde{e}_{3}}\lambda_{0}$ (48)

with $\tilde{e}_{i}$ the zeroes of $B^{2}{\,-\,}4AC$ and

$\lambda_{0}={x_{+}-x_{-}\over 4\pi{\rm i}}\,{dy\over
y}={\sqrt{B^{2}-4AC}\over 4\pi{\rm i}Ay}\,dy$ (49)

where $x_{\pm}$ are the solutions of (47).

###  3.2 The quantum curve

In the presence of a non-trivial $\Omega$-background, $\epsilon_{1}=\hbar$,
$\epsilon_{2}=0$, the dynamics of the gauge theory is described by a quantum
curve obtained from the classical one after replacing $x$, $y$ by operators
satisfying the commutation relation

$[\hat{x},\ln\hat{y}]=\hbar$ (50)

The quantum curve follows from (39) by distributing the powers of $y$
symmetrically Zenkevich:2011zx

$\left[q\,\hat{y}^{1\over 2}\,P_{L}(\hat{x})\,\hat{y}^{1\over
2}+P_{0}(\hat{x})+\,\hat{y}^{-{1\over 2}}\,P_{R}(\hat{x})\,\hat{y}^{-{1\over
2}}\right]U=0$ (51)

with $P_{0}$, $P_{L}$, $P_{R}$ given in (40) and

$p_{0}(x)=x^{2}{-}(x{+}\tfrac{\hbar}{2})\sum_{i}m_{i}+u+\sum_{i<j}m_{i}m_{j}+\tfrac{\hbar^{2}}{2}$
(52)

This function is determined by requiring that the quantum SW differential,
defined later in (71), behaves at large $x$ as Poghossian:2010pn ;
Fucito:2011pn

$\lambda_{+}=\sum_{n=0}^{\infty}{\langle{\rm tr}\varphi^{n}\rangle\over
x^{n}}=2+\frac{2u}{x^{2}}+\ldots$ (53)

Using $\hat{x}\hat{y}=\hat{y}(\hat{x}+\hbar)$ to bring all the dependence on
$\hat{y}$ to the left, and setting $\hat{x}=\hbar\,y\,\partial_{y}$, one can
view (51) as an ordinary differential equation in the $y$-variable

$\left[qy^{2}P_{L}(\hat{x}{+}{\textstyle\frac{\hbar}{2}})+yP_{0}(\hat{x})+P_{R}(\hat{x}{-}{\textstyle\frac{\hbar}{2}})\right]U(y)=\left[A(y)\hat{x}^{2}+B(y)\,\hat{x}+C(y)\right]U(y)=0$
(54)

with

$\displaystyle A$ $\displaystyle=$ $\displaystyle(1{+}y)(1{+}qy)\qquad,\qquad
B={-}m_{1}{-}m_{2}-\hbar+qy\left[y(\hbar{-}m_{3}{-}m_{4}){-}\sum_{i}m_{i}\right]$
(55) $\displaystyle C$ $\displaystyle=$
$\displaystyle(m_{1}{+}{\textstyle\frac{\hbar}{2}})(m_{2}{+}{\textstyle\frac{\hbar}{2}}){-}uy{+}qy\Bigg{[}u+\sum_{i<j}m_{i}m_{j}{-}{\textstyle\frac{\hbar}{2}}\sum_{i}m_{i}{+}{\textstyle\frac{\hbar^{2}}{2}}{+}y\left(m_{3}{-}{\textstyle\frac{\hbar}{2}}\right)\left(m_{4}{-}{\textstyle\frac{\hbar}{2}}\right)\Bigg{]}$

The differential equation (54) can be cast in canonical from by taking

$U(y)={1\over\sqrt{y}}\,e^{-\frac{1}{2\hbar}\,\int^{y}\frac{B(y^{\prime})}{y^{\prime}A(y^{\prime})}\,dy^{\prime}}\Psi(y)$
(56)

with

$Q_{\rm
SW}(y)={4\,C\,A{-}B^{2}{+}2\,\hbar\,y(B\,A^{\prime}{-}A\,B^{\prime}){+}\hbar^{2}\,A^{2}\over
4\,\hbar^{2}\,y^{2}\,A^{2}}$ (57)

###  3.3 The gauge/gravity dictionary

In this section we establish the gauge/gravity dictionary using as a working
example the AdS KN metric. To this aim, we first write $Q_{\rm SW}(y)$ in the
form444At infinity $Q\simeq\delta_{4}/y^{2}$.

$Q_{\rm
SW}(y)=\sum_{i=1}^{3}\frac{\delta_{i}}{(y-y_{i})^{2}}+{\nu_{1}+qy(\delta_{4}{-}\delta_{1}{-}\delta_{2}{-}\delta_{3})\over
y(1+y)(1+yq)}$ (58)

where $y_{i}=\\{0,-1,{\,-\,}1/q\\}$,

$\displaystyle{{\delta}}_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{4}-\frac{(m_{1}-m_{2})^{2}}{4\hbar^{2}}\quad,\quad{{\delta}}_{2}=\frac{1}{4}-\frac{(m_{1}+m_{2})^{2}}{4\hbar^{2}}$
$\displaystyle{{\delta}}_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{4}-\frac{(m_{3}+m_{4})^{2}}{4\hbar^{2}}\quad,\quad{{\delta}}_{4}=\frac{1}{4}-\frac{(m_{3}-m_{4})^{2}}{4\hbar^{2}}$
(59)

and

$4\hbar^{2}\nu_{1}=(q{-}1)(\hbar^{2}{+}4u){+}2(m_{1}^{2}{+}m_{2}^{2}){+}2q\left[2m_{3}m_{4}{+}(m_{1}{+}m_{2})(m_{3}{+}m_{4}){-}\hbar\sum_{i}m_{i}\right]$
(60)

To compare with gravity, one can consider an arbitrary change of variables
$y\to y(z)$. The differential equation in the $z$-variable can be brought
again to canonical form with the new characteristic function given by

$Q(z)=Q_{\rm SW}(y)\,y^{\prime}(z)^{2}+{y^{\prime\prime\prime}(z)\over
2y^{\prime}(z)}-\frac{3}{4}\left[{y^{\prime\prime}(z)\over
y^{\prime}(z)}\right]^{2}$ (61)

For example, performing an $SL(2,\mathbb{C})$ transformation

$y={z_{24}\over z_{12}}{z-z_{1}\over z-z_{4}}\qquad,\qquad
q={z_{12}z_{34}\over z_{24}z_{13}}$ (62)

that maps the points $(0,-1,-1/q,\infty)$ to arbitrary points $z_{i}$, one
finds

$Q(z)={P_{4}(z)\over\Delta_{4}(z)^{2}}$ (63)

that corresponds to a Heun equation with four regular singularities. This
matches precisely the form of the radial and angular wave equations (13) for
AdS KN BH provided

$q={z_{12}z_{34}\over
z_{24}z_{13}}\qquad,\qquad\delta_{i}={P_{4}(z_{i})\over\Delta_{4}^{\prime}(z_{i})^{2}}\qquad,\qquad{{\nu}}_{1}=\frac{z_{12}z_{14}}{z_{24}}\underset{z=z_{1}}{\rm
Res}\,Q(z)$ (64)

where left and right hand sides of the equations are given in terms of gauge
and gravity variables respectively. In particular the $\delta_{i}$ determine
the masses $m_{i}$, while $\nu_{1}$ expresses $u$ in terms of gravity
variables.

###  3.4 Quantum periods and exact quantization

The quantum periods $(a,a_{D})$ can be computed exactly in $\hbar$ and
perturbatively in $q$ using localization Nekrasov:2002qd . Alternatively they
can be derived by solving the difference equation following from the quantum
SW (51) curve in the $x$ rather than in the $y$-plane, i.e. setting
$\hat{y}=e^{-\hbar\partial_{x}}$ Poghossian:2010pn ; Fucito:2011pn

$\left[q\,P_{L}(x-\tfrac{\hbar}{2})\,\hat{y}+P_{0}(x)+\,P_{R}(x+\tfrac{\hbar}{2})\,\hat{y}^{-1}\right]\widetilde{U}(x)=0$
(65)

Introducing the functions

$W(x)={1\over
P_{R}(x+{\textstyle\frac{\hbar}{2}})}{\widetilde{U}(x)\over\widetilde{U}(x+\hbar)}\quad,\quad
M(x)=P_{L}(x-{\textstyle\frac{\hbar}{2}})P_{R}(x-{\textstyle\frac{\hbar}{2}})$
(66)

equation (65) can be cast in the form

$q\,M(x)\,W(x)W(x-\hbar)+P_{0}(x)W(x)+1=0$ (67)

That can be recursively solved order by order in $q$. In the small $q$ limit
one can write $W(x)$ as a continuous fraction

$W(x)=-{1\over P_{0}(x)+q\,M(x)W(x-\hbar)}=-{1\over P_{0}(x)-{q\,M(x)\over
P_{0}(x-\hbar)-{q\,M(x-\hbar)\over P_{0}(x-2\hbar)-\ldots}}}$ (68)

so that

$W(x)=-{1\over P_{0}(x)}\left(1+{q\,M(x)\over
P_{0}(x)P_{0}(x-\hbar)}+O(q^{2})\right)$ (69)

It is easy to see that to order $q^{k}$, the function $W(x)$ has poles at
points $x_{n}^{+}=\sqrt{u}+n\hbar$ or $x_{n}^{-}=-\sqrt{u}-n\hbar$, with
$0\leq n\leq k$. The quantum period $a(u)$ can therefore be written as a sum
over residues

$a(u)=\oint_{\alpha}\lambda_{+}=2\pi{\rm i}\sum_{n=0}^{\infty}{\rm
Res}_{\sqrt{u}+n\hbar}\lambda_{+x}(x)$ (70)

of the $\hbar$-deformed SW differential

$\lambda_{+}(x)=-{x\over 2\pi{\rm i}}d\ln W(x)$ (71)

that coincides with (44) at $\hbar{\,=\,}0$. Inverting (70) one finds $u(a)$
order by order in $q$. The Nekrasov-Shatashvili (NS) prepotential

${\cal
F}_{NS}(a,q,m_{i},\hbar)=\lim_{(\epsilon_{1}\epsilon_{2})\to(\hbar,0)}\epsilon_{1}\epsilon_{2}\log
Z_{\rm Nekrasov}(a,\epsilon_{1},\epsilon_{2},q,m_{i})$ (72)

is then obtained from the quantum version of the Matone relation Matone:1995rx
; Flume:2004rp

$u=-q\,{\partial{\cal F}_{NS}(a,\hbar,q)\over\partial q}$ (73)

after integrating upon $q$. The integration $q$-independent constant is
obtained from the one-loop prepotential, see appendix A for details. Dropping
the dependence on $m_{i}$ and $\hbar$ one writes

${\cal F}_{NS}(a,q)={\cal F}_{\rm tree}(a,q)+{\cal F}_{\rm 1{-}loop}(a)+{\cal
F}_{\rm inst}(a,q)$ (74)

The tree level and instanton prepotentials are obtained after integration over
$q$ of $u(a)$. One finds

$\displaystyle{\cal F}_{\rm tree}$ $\displaystyle=$ $\displaystyle-a^{2}\log
q$ $\displaystyle{\cal F}_{\rm inst}$ $\displaystyle=$ $\displaystyle
q\Bigg{[}-{4a^{2}{\,+\,}3\hbar^{2}\over 8}{-}{2m_{1}m_{2}m_{3}m_{4}\over
4a^{2}{+}\hbar^{2}}{+}{\textstyle\frac{\hbar}{2}}\sum_{i}m_{i}{-}{\textstyle\frac{1}{2}}\sum_{i<j}m_{i}m_{j}\Bigg{]}+O(q^{2})$
$\displaystyle{\partial{\cal F}_{\rm 1{-}loop}\over\partial a}$
$\displaystyle=$
$\displaystyle\hbar\log\left[\frac{\Gamma^{2}(1+\tfrac{2a}{\hbar})}{\Gamma^{2}(1-\tfrac{2a}{\hbar})}\prod_{i=1}^{4}{\Gamma\left({1\over
2}{+}{m_{i}-a\over\hbar}\right)\over\Gamma\left({1\over
2}{+}{m_{i}+a\over\hbar}\right)}\right]$ (75)

Finally the $a_{D}$-period is given by

$a_{D}(u)=-{1\over 2\pi{\rm i}}{\partial{\cal F}_{NS}\over\partial a}$ (76)

Higher order terms in the instanton expansion in powers of $q$ can be obtained
systematically.

QNM frequencies are obtained by imposing WKB exact quantization conditions on
a specific SW quantum period

$a_{\gamma}(u)=\oint_{\gamma}\lambda=\hbar\left(n_{\gamma}+\nu\right)$ (77)

defined such that $\gamma$ shrinks to zero size at the classical level and
$\nu=0,{\textstyle\frac{1}{2}}$ depending on $\gamma$. We find that $\nu=0$
and $\nu={\textstyle\frac{1}{2}}$ for quantization of the $a_{D}$ and $a$
cycles respectively. The cycle $\gamma$ is determined by using the QNM/SW
dictionary to map the colliding zeroes of the characteristic function $Q_{\rm
geo}(z)$ governing the geodetic motion to colliding branch points in the SW
gauge theory picture.

###  3.5 Theories with $N_{f}<4$

Theories with less fundamentals can be found by decoupling some of the hypers
by sending their mass $m\to\infty$ and $q\to 0$, keeping finite the product
$\tilde{q}=-mq$ parametrizing the gauge coupling of the new theory that will
be renamed $q$ for simplicity555Every time a flavour in $P_{R}$ decouples, the
$y$ variable must be rescaled $\tilde{y}=-y/m$ in order to keep the SW curve
as in eq (54).. The general case will be labelled by
$N_{f}=({N}_{L},{N}_{R})$666In all cases we consider $N_{L},N_{R}\leq 2$ which
leads to second order differential equations. and $N_{f}={N}_{L}+{N}_{R}\leq
3$. The curve is given as in (54) with

$P_{L}(x)=\prod_{i=3}^{2+{N}_{L}}(x-m_{i})\,,\qquad
P_{R}(x)=\prod_{i=1}^{{N}_{R}}(x-m_{i})$ (78)

and

$P_{0}(x)=x^{2}-u+q\,\delta_{N_{f},\,3}\left(x-\sum_{i}m_{i}+\frac{\hbar}{2}\right)+q\,\delta_{N_{f},\,2}$
(79)

The correct expression for the NS prepotential can be obtained from (3.4)
performing the limit $q\to 0$, $m\to\infty$. For example for $N_{f}=(1,2)$ one
finds

$\displaystyle{\cal F}_{\rm tree}$
$\displaystyle=-a^{2}\log\left(-\frac{q}{\hbar}\right)\,,\qquad{\cal F}_{\rm
inst}=q\Bigg{[}\frac{1}{2}\sum_{i=1}^{3}m_{i}+\frac{2m_{1}m_{2}m_{3}}{4a^{2}{-}\hbar^{2}}-\frac{\hbar}{2}\Bigg{]}+O(q^{2})$
(80) $\displaystyle{\partial{\cal F}_{\rm 1{-}loop}\over\partial a}$
$\displaystyle=\hbar\log\left[\frac{\Gamma^{2}(1+\frac{2a}{\hbar})}{\Gamma^{2}(1-\frac{2a}{\hbar})}\prod_{i=1}^{3}{\Gamma\left({1\over
2}{+}{m_{i}-a\over\hbar}\right)\over\Gamma\left({1\over
2}{+}{m_{i}+a\over\hbar}\right)}\right]$

where the extra contributions in the tree-level component of the prepotential
is produced by the one-loop term in the decoupling limit. Similarly the
characteristic $Q$-function is given by (58) with

$\displaystyle A$ $\displaystyle=1{+}y\,,\qquad B=qy^{2}+qy-m_{1}-m_{2}-\hbar$
(81) $\displaystyle C$
$\displaystyle=-qy^{2}(m_{3}-{\textstyle\frac{\hbar}{2}})-uy+qy\Bigg{[}{\textstyle\frac{\hbar}{2}}-\sum_{i=1}^{3}m_{i}\Bigg{]}+(m_{1}+{\textstyle\frac{\hbar}{2}})(m_{2}+{\textstyle\frac{\hbar}{2}})$

###  3.6 Examples at $q=0$

It is instructive to illustrate the various ingredients of the gauge/gravity
dictionary in the simple case of gravity backgrounds related to free gauge
theories ($q=0$). Setting $q=0$ in the $N_{f}=(2,2)$ curve (39) one finds

$\left[y\,P_{0}(\hat{x})+P_{R}(\hat{x}-{\textstyle\frac{\hbar}{2}})\right]U(y)=0$
(82)

with $\hat{x}=\hbar\,y\,\partial_{y}$ and

$P_{0}(x)=x^{2}-u\qquad,\qquad P_{R}(x)=(x-m_{1})(x-m_{2})$ (83)

Writing

$\Psi(y)=y^{-\frac{m_{1}+m_{2}}{2\hbar}}(1+y)^{\frac{\hbar+m_{1}+m_{2}}{2\hbar}}\leavevmode\nobreak\
U(y)$ (84)

one can bring the differential equation (82) to canonical form with

$Q_{\textup{free}}(y)=\frac{\hbar^{2}-(m_{1}-m_{2})^{2}}{4\hbar^{2}y^{2}}+\frac{\hbar^{2}-(m_{1}+m_{2})^{2}}{4\hbar^{2}(y+1)^{2}}+\frac{2(m_{1}^{2}+m_{2}^{2})-4u-\hbar^{2}}{4\hbar^{2}y(y+1)}$
(85)

and

$\nu_{1}=\frac{m_{1}^{2}+m_{2}^{2}}{2\hbar^{2}}-\frac{u}{\hbar^{2}}-\frac{1}{4}$
(86)

The solutions to the canonical equation are hypergeometric functions

$\displaystyle\Psi(y)=$
$\displaystyle\,d_{1}\,y^{\frac{\hbar+m_{1}-m_{2}}{2\hbar}}(1{\,+\,}y)^{\frac{\hbar+m_{1}+m_{2}}{2\hbar}}\,{}_{2}F_{1}(\tfrac{1}{2}+\tfrac{m_{1}-\sqrt{u}}{\hbar},\tfrac{1}{2}+\tfrac{m_{1}+\sqrt{u}}{\hbar},1+\tfrac{m_{1}-m_{2}}{\hbar}|-y)$
(87) $\displaystyle+$ $\displaystyle\,d_{2}\,y^{\frac{\hbar-
m_{1}+m_{2}}{2\hbar}}(1{\,+\,}y)^{\frac{\hbar+m_{1}+m_{2}}{2\hbar}}\,{}_{2}F_{1}(\tfrac{1}{2}+\tfrac{m_{2}-\sqrt{u}}{\hbar},\tfrac{1}{2}+\tfrac{m_{2}+\sqrt{u}}{\hbar},1+\tfrac{m_{2}-m_{1}}{\hbar}|-y)$

In appendix B we show how one can recover these solutions from the difference
equation (65).

Two special cases of analytic solutions: spherical harmonics and inverted
hydrogen atom will be discussed in the following, while static and super-
radiant modes of KN BHs will be discussed later on.

####  3.6.1 Spherical harmonics

Spherical harmonics $Y_{\ell m}(\theta,\varphi)$ are defined as eigenfunctions
of the Laplacian on the 2-sphere

$\nabla^{2}_{S^{2}}Y_{\ell m}(\theta,\varphi)=-\ell(\ell+1)Y_{\ell
m}(\theta,\varphi)$ (88)

Writing $Y_{\ell m}(\theta,\varphi)=e^{im\varphi}U_{\ell m}(\chi)$, the
equation can be written as

$\left[\frac{\partial}{\partial\chi}\left((1-\chi^{2})\frac{\partial}{\partial\chi}\right)+\left(\ell(\ell+1)-\frac{m^{2}}{1-\chi^{2}}\right)\right]{U}_{\ell
m}(\chi)=0$ (89)

where $\chi=\cos\theta$. With generic boundary conditions this equation admits
two solutions

$U_{\ell m}(\chi)=c_{1}\,P_{\ell\,m}(\chi)+c_{2}\,Q_{\ell\,m}(\chi)$ (90)

where

$\displaystyle P_{\ell m}(\chi)$
$\displaystyle=\frac{1}{\Gamma(1-m)}\left(\frac{1+\chi}{1-\chi}\right)^{\frac{m}{2}}{}_{2}F_{1}\left(-\ell,\ell+1;1-m;\frac{1-\chi}{2}\right)$
(91) $\displaystyle Q_{\ell m}(\chi)$ $\displaystyle=\frac{\pi}{2\sin\pi
m}\left[\cos\pi m\,P_{\ell
m}(\chi)-\frac{\Gamma(\ell+m+1)}{\Gamma(\ell-m+1)}\,P_{\ell,\,-m}(\chi)\right]$

are Legendre associated functions of the first and second kind respectively.
Requiring regularity at $\chi=1$ and $\chi=-1$, one finds that $c_{2}=0$ and
the solution reduces to the Legendre associated polynomials $P_{\ell m}(\chi)$
with $\ell$, $m$ integers and $\ell\geq|m|$.

Now let us see how this result is recovered from the WKB exact quantization of
the SW period. Writing

${U}_{\ell m}(\chi)={1\over\sqrt{1-\chi^{2}}}\Psi(\chi)$ (92)

the equation (89) can be brought to the canonical form with

$Q_{\chi}(\chi)=\frac{(1-m^{2})(1+\chi^{2})}{2\left(1{\,-\,}\chi^{2}\right)^{2}}+\frac{2\ell(\ell{\,+\,}1){\,+\,}1{\,-\,}m^{2}}{2(1-\chi^{2})}$
(93)

The gauge gravity dictionary reads777Here and below we always fix the sign
ambiguities in the dictionary as will.

$q=0\quad,\quad\frac{u}{\hbar^{2}}=(\ell+\tfrac{1}{2})^{2}\quad,\quad
m_{1}=0\quad,\quad\frac{m_{2}}{\hbar}=|m|\quad;\quad y=-\frac{1}{2}(1-\chi)$
(94)

At large $\ell,m$, the conditions
$Q_{\chi}(\chi_{c})=Q_{\chi}^{\prime}(\chi_{c})=0$ are solved by $\chi_{c}=0$
and $\ell=|m|$. Translating back into the gauge variables one finds that
$a_{\gamma}=\sqrt{u}-m_{2}=a-m_{2}\approx 0$ in the classical limit. The cycle
$\gamma$ contains then the $a$-cycle and the pole of $\lambda$ at $x=m_{2}$.
Turning on $\hbar$ one finds

$a_{\gamma}=\sqrt{u}-m_{2}=(\ell-|m|+{\textstyle\frac{1}{2}})\hbar$ (95)

in agreement with (77) for $n_{\gamma}=\ell-|m|$, or equivalently

$a=\hbar\left(\ell+\frac{1}{2}\right)$ (96)

####  3.6.2 ‘Inverted’ hydrogen atom

Another simple toy-model admitting QNMs is the “inverted hydrogen atom”,
obtained by flipping the sign of the hydrogen atom effective potential, i.e.
the hydrogen-like potential with repulsive charges and an imaginary angular
momentum

$V(r)=\frac{\mu}{r}-\frac{\lambda^{2}+\tfrac{1}{4}}{r^{2}}$ (97)

where $\mu$ and ${\lambda}^{2}$ are taken positive. The general solution to
the differential equation

$-\frac{1}{r^{2}}\frac{\partial^{2}}{\partial
r^{2}}\left[r^{2}\psi(r)\right]+\left(\frac{\mu}{r}-\omega^{2}-\frac{{\lambda}^{2}+\tfrac{1}{4}}{r^{2}}\right)\psi(r)=0$
(98)

is given by

$\displaystyle\psi(r)=$ $\displaystyle c_{1}e^{-i\omega
r}(r\omega)^{-\frac{3}{2}-i{\lambda}}\,_{1}F_{1}(\tfrac{1}{2}-i{\lambda}-\frac{i\mu}{2\omega},1-2i{\lambda}|2i\omega
r)+$ (99) $\displaystyle+c_{2}e^{-i\omega
r}(r\omega)^{-\frac{3}{2}+i{\lambda}}\,_{1}F_{1}(\tfrac{1}{2}+i{\lambda}-\frac{i\mu}{2\omega},1+2i{\lambda}|2i\omega
r)$

We look for solutions with in-going boundary conditions at $r=0$888In this
toy-model $r=0$ plays the role of the ‘horizon’. and outgoing at $r=\infty$.
The in-going wave requirement at $r=0$ leads to $c_{2}=0$ while the outgoing
behaviour at $\infty$ boils down to the quantization condition

$\frac{1}{2}-i{\lambda}+\frac{i\mu}{2\omega}=-n$ (100)

or equivalently to

$\omega=\frac{\mu}{2{\lambda}+2i(n+\tfrac{1}{2})}$ (101)

Now let us see how to re-derive this result from the exact SW quantization.
First, writing $\psi=r^{-2}\Psi_{\rm grav}(r)$, one can bring (98) to
canonical form with

$Q(r)=\omega^{2}-\frac{\mu}{r}+\frac{{\lambda}^{2}+\tfrac{1}{4}}{r^{2}}$ (102)

This function can be mapped to the $Q$-function of $N_{f}=(1,1)$ free theory
$q=0$

$Q_{\textup{SW}}(y)=-\frac{m_{1}}{y^{3}\hbar^{2}}+\frac{\hbar^{2}-4u}{4y^{2}\hbar^{2}}-\frac{1}{4y^{4}\hbar^{2}}$
(103)

Comparing the two $Q$-functions, one finds the gauge/gravity dictionary

$q=0\quad,\quad\frac{u}{\hbar^{2}}=-{\lambda}^{2}\quad,\quad\frac{m_{1}}{\hbar}=-\frac{i\mu}{2\omega}\quad,\quad
r=-\frac{i}{2\hbar\omega y}$ (104)

To understand which cycle shrinks at the classical level, we consider the
geodetic motion near the “photon-sphere” defined by the critical conditions

$Q_{\rm geo}(r_{c},\omega_{c})=Q_{\rm geo}^{\prime}(r_{c},\omega_{c})=0$ (105)

with

$Q_{\rm geo}(r)=\omega^{2}-\frac{\mu}{r}+\frac{{\lambda}^{2}}{r^{2}}$ (106)

The solution reads

$r_{c}\approx{2\lambda^{2}\over\mu}\,,\qquad\omega_{c}\approx{\mu\over
2\lambda}$ (107)

or in the gauge theory variables $m_{1}\approx-\sqrt{u}$. We conclude that the
vanishing period is $a_{\gamma}=\sqrt{u}+m_{1}=a+m_{1}$, i.e. the cycle
including the $a$-cut and the poles at $m_{1}$. Turning on $\hbar$ one finds
the exact WKB quantization condition

$a_{\gamma}=\oint_{\gamma}\lambda_{SW}=\sqrt{u}+m_{1}=\hbar(n+\tfrac{1}{2})$
(108)

in agreement with (101).

###  3.7 The AGT picture

We would like to conclude this section with another precious tool available in
the study of $\mathcal{N}=2$ SYM: the AGT correspondence Alday:2009aq . Thanks
to this correspondence, one can relate the characteristic function
$Q_{SW}(y)$, the wave function $\Psi(y)$ and the gauge partition function $Z$
of $\mathcal{N}=2$ SYM to correlators of two-dimensional Conformal Field
Theories (CFTs) BonTanzetc . In this section, we summarize the main
ingredients of the dictionary in view of its application to the study of QNMs
and other observables of BHs, D-branes and fuzz-balls. For $SU(2)$ SYM with
$N_{f}=4$, which underlies all the examples analysed here, one can consider a
Liouville theory with background charge $Q$ and central charge $c$ given by

$c=1+6\,Q^{2}\quad,\quad Q=b+{1\over b}\quad,\quad
b=\sqrt{\epsilon_{1}\over\epsilon_{2}}$ (109)

and denote by $V_{\alpha_{i}}=e^{2\alpha_{i}\phi}$ the chiral operators of
dimensions

$h_{i}=\alpha_{i}(Q-\alpha_{i})$ (110)

where $i=1,\dots,4$, corresponding to the four flavours. According to AGT, the
gauge partition function is related to the four-point function

$Z=e^{{\cal F}\over\epsilon_{1}\epsilon_{2}}=\left\langle
V_{\alpha_{1}}(y_{1})V_{\alpha_{2}}(y_{2})V_{\alpha_{3}}(y_{3})V_{\alpha_{4}}(y_{4})\right\rangle$
(111)

with

$q={y_{12}y_{34}\over y_{24}y_{13}}$ (112)

We consider the NS limit, $\epsilon_{1}=\hbar$, $\epsilon_{2}\to 0$, leading
to

$b\to 0\quad,\quad\alpha_{i}\to\infty\quad,\quad b\,\alpha_{i}={\rm finite}$
(113)

The characteristic function $Q_{\rm SW}(y)$ is identified with the ratio999The
coefficients $c_{2,3}$ are determined by matching the asymptotic $Q_{\rm
SW}(y)\approx h_{4}/y^{2}$ at infinity, that boils down to the conditions
$\displaystyle\sum_{i=1}^{3}c_{i}=0\qquad,\qquad\sum_{i=1}^{3}(h_{i}+c_{i}y_{i})=h_{4}$
(114)

$\displaystyle Q_{\rm SW}(y)$ $\displaystyle=$ $\displaystyle
b^{2}{\left\langle
T(y)\prod_{i=1}^{4}V_{\alpha_{i}}(y_{i})\right\rangle\over\left\langle\prod_{i=1}^{4}V_{\alpha_{i}}(y_{i})\right\rangle}=b^{2}\sum_{i=1}^{3}\left({h_{i}\over(y-y_{i})^{2}}+{c_{i}\over
y-y_{i}}\right)$ (115)

with $y_{i}=\\{0,-1,-1/q,\infty\\}$, $h_{i}$ the dimensions and

$b\,\alpha_{1,2}={\textstyle\frac{1}{2}}+{m_{1}\mp m_{2}\over
2\hbar}\qquad,\qquad b\,\alpha_{3,4}={\textstyle\frac{1}{2}}+{m_{3}\pm
m_{4}\over 2\hbar}$ (116)

We notice that in the double-scaling limit (113) the combinations
$\delta_{i}=b^{2}h_{i}$ and $\nu_{i}=b^{2}c_{i}$ are kept finite. Finally the
wave function is associated to the five-point correlation function BonTanzetc

$\Psi(y)=\left\langle V_{\alpha_{12}}(y)V_{\alpha_{1}}(y_{1})\ldots
V_{\alpha_{4}}(y_{4})\right\rangle$ (117)

involving the insertion of a degenerate field with
$\alpha_{12}=-{\textstyle\frac{b}{2}}$. The function $\Psi(y)$ satisfies the
BPZ equation Belavin:1984vu

$\displaystyle\Psi^{\prime\prime}(y)+b^{2}\sum_{i=1}^{4}\left[{h_{i}\over(y-y_{i})^{2}}+{\partial_{y_{i}}\over(y-y_{i})}\right]\Psi(y)=0$
(118)

that follows from the fact that ${\cal
O}_{12}=\left(L_{-1}^{2}+b^{2}L_{-2}\right)V_{\alpha_{12}}$ is a null state,
and its insertion inside any correlator leads to a vanishing result. In the
limit $\hbar\to 0$, $\Delta_{12}\ll\Delta_{i}$, so the insertion of the
degenerate field modifies the correlator only slightly leading to
$\partial_{y_{i}}\Psi(y)\approx c_{i}\Psi(y)$ with $c_{i}$ some constants. One
finds then again a differential equation in the canonical form that can be
mapped to the equation for the QNMs in gravity after proper identification of
the parameters.

##  4 Numerical analysis

In section 5 we will compute the spectrum of QNMs for Kerr Newman black holes
using geodetic motion and SW techniques. In order to test those results, in
this section we apply (and extend) the method of continuous fractions
introduced by Leaver in Leaver:1985ax ; Leaver:1990zz  to find numerical
estimates of the frequencies for differential equations of the form (15) with

$Q(z)={P_{4}(z)\over\Delta_{2}(z)^{2}}$ (119)

This will be the relevant case for the study of QNMs of KN solutions in the
section 5. We will also show that the same equation describes the extremal
case after a proper variable redefinition.

###  4.1 Radial equation

The radial wave equation of the KN solution shows two regular
singularities101010For notational simplicity we assume $z_{\pm}$ to be real in
general. Extension to complex $z_{\pm}$ is straightforward. at $z=z_{\pm}$ and
an irregular singularity at $z=\infty$. We look for a solution away from the
singularities, i.e. for $z>z_{+}>z_{-}$. We start from the ansatz

$\Phi(z)=e^{\nu
z}(z{-}z_{-})^{\sigma_{-}}(z{-}z_{+})^{\sigma_{+}}\sum_{n=0}^{\infty}{c}_{n}\left(\frac{z{-}z_{+}}{z{-}z_{-}}\right)^{n}$
(120)

The constants $\sigma_{+}$, $\nu$ are determined by requiring that the ansatz
solves the differential equation near $z_{+}$ and infinity viz.

$\nu^{2}=-\,{P^{(4)}_{4}(z_{+})\over
4!}\quad,\quad\sigma_{+}(\sigma_{+}{-}1){+}\frac{P_{4}(z_{+})}{\delta}=0$
(121)

with $\delta=z_{+}-z_{-}$. The boundary conditions on $z_{+}$ and infinity
select which solution of the above conditions should be used, e.g. if $z_{+}$
is an horizon to get the QNMs frequencies we have to choose $\sigma_{+}$ such
that the wave is incoming, i.e. $\text{Im }\nu>0$. On the other hand
$\sigma_{-}$ is conveniently fixed by requiring that the recursion involves
only three terms. One finds

$\sigma_{-}=-\sigma_{+}-\nu\,\delta-{P^{{}^{\prime\prime\prime}}_{4}(z_{+})\over
12\,\nu}$ (122)

Plugging the ansatz into the wave equation one finds the recursive relation

$\displaystyle\alpha_{n}\,{c}_{n+1}+\beta_{n}\,{c}_{n}+\gamma_{n}\,{c}_{n-1}=0$
(123)

with $c_{-1}=0$ and

$\displaystyle\alpha_{n}$ $\displaystyle=-\delta(1+n)(n+2\sigma_{+})$ (124)
$\displaystyle\beta_{n}$
$\displaystyle=2\delta\left[(n+\sigma_{+})(n-\delta\nu-\sigma_{-})+\left(1-\sigma_{+}\right)\sigma_{+}\right]-P_{4}^{\prime}(x_{+})$
$\displaystyle\gamma_{n}$
$\displaystyle=P_{4}^{\prime}(x_{+}){-}\frac{1}{2}\delta
P_{4}^{\prime\prime}(x_{+}){-}\delta\left[\left(\delta\nu{+}2(\sigma_{-}{+}\sigma_{+})\right)\nu\delta{+}(n{-}\sigma_{-}{-}\sigma_{+})(n{-}\sigma_{-}{+}\sigma_{+}{-}1)\right]$

Finally QNM frequencies $\omega_{n}$ associated to the overtone $n$ can be
obtained by truncating the recursion to a chosen level (taken to be large) and
solving numerically the equation

$\beta_{n}=\frac{\alpha_{n-1}\,\gamma_{n}}{\beta_{n-1}-\frac{\alpha_{n-2}\,\gamma_{n-1}}{\beta_{n-2}-\ldots}}+\frac{\alpha_{n}\gamma_{n+1}}{\beta_{n+1}+\frac{\alpha_{n+1}\gamma_{n+2}}{\beta_{n+2}\ldots}}$
(125)

viewed as an equation for $\omega_{n}$.

###  4.2 Angular equation

The angular differential equation for KN BHs has the same singularity
structure as the radial one, but now we look for a regular solution in the
interval $z_{-}\leq z\leq z_{+}$. To this aim it is convenient to change the
ansatz (120) with

$\Phi(z)=e^{\nu(z-z_{-})}(z{-}z_{-})^{\sigma_{-}}(z_{+}{-}z)^{\sigma_{+}}\sum_{n=0}^{\infty}c_{n}^{\chi}(z{-}z_{-})^{n}$
(126)

The exponents $\sigma_{\pm}$ are determined by requiring that the ansatz
solves the differential equation and is regular in $z_{\pm}$, while $\nu$ is
chosen such that the recursion involves only three terms

$\sigma_{\pm}=\frac{1}{2}+\frac{1}{2}\sqrt{1-\frac{4P_{4}(z_{\pm})}{\delta^{2}}}\quad,\quad\nu^{2}=-\,{P^{(4)}_{4}(z_{-})\over
4!}$ (127)

with coefficients

$\displaystyle\alpha_{n}^{\chi}$
$\displaystyle=\delta^{2}(1+n)(n+2\sigma_{-})$ (128)
$\displaystyle\beta_{n}^{\chi}$
$\displaystyle=\delta\left[2\left(n+\sigma_{-}\right)\left(1+\delta\nu-\sigma_{+}-\sigma_{-}\right)-n(n+1)\right]+P_{4}^{\prime}\left(z_{-}\right)$
$\displaystyle\gamma_{n}^{\chi}$
$\displaystyle=2\delta\nu\left(1+\delta\nu-\sigma_{+}-\sigma_{-}-n\right)-\frac{\delta}{6}P_{4}^{(3)}\left(z_{-}\right)$

The recurrence relation can be solved for the separation constant. The angular
overtone number $n$ turns out to be identified with the difference $\ell-|m|$
as in (95).

###  4.3 Extremal geometries

The methods in the previous sections do not apply straightforwardly to
extremal geometries in $D=4$ or higher dimensions. In the extremal case, the
wave equation has two irregular singular points (infinity and the horizon),
and therefore the above ansatz should be modified. One can expand the solution
around a regular point Onozawa:1995vu , however this typically yields a
recurrence relation with four or five terms.

Here we circumvent the problem by mapping the $N_{f}=(1,1)$ gauge theory to
the $N_{f}=(1,2)$ theory studied in the previous subsections. This can be done
whenever the masses of all hypers coincide.

The gauge theory variable $Y$ in the $N_{f}=(1,2)$ theory is related to $y$ in
the $N_{f}=(1,1)$ theory with $m_{3}=m_{1}$ via

$y=-q^{-{\textstyle\frac{1}{2}}}\left(\sqrt{Y}\pm\sqrt{1+Y}\right)^{2}$ (129)

with lowercase and capital parameters identified as

$Q=-4\sqrt{q}\quad,\quad
U=u+2\left(m_{1}-\frac{\hbar}{2}\right)\sqrt{q}\quad,\quad M_{1}=0\quad,\quad
M_{2}=\frac{\hbar}{2}\quad,\quad M_{3}=m_{1}$ (130)

##  5 QNMs of Kerr-Newman BH

In this section we compute the spectrum of QNM frequencies for Kerr-Newman BHs
using the three methods: geodetic motion, SW and continuous fractions
(numerical). We also discuss two special cases of analytic solutions: static
and super-radiant modes related to free gauge theories. The metric and
gauge/gravity dictionary for the Kerr-Newman BH solution can be obtained from
the previous AdS case by sending $L\to\infty$.

###  5.1 Gauge/gravity dictionary

The line element of the KN metric in Boyer-Lindquist coordinates reads
Caldarelli:1999xj

$ds^{2}=-\frac{{\Delta_{r}}}{\rho^{2}}(dt-
a_{{}_{\mathcal{J}}}\sin^{2}\theta\,d\phi)^{2}+\frac{\sin^{2}\theta}{\rho^{2}}\left[a_{{}_{\mathcal{J}}}\,dt-(r^{2}{+}a_{{}_{\mathcal{J}}}^{2})\,d\phi\right]^{2}+\frac{\rho^{2}dr^{2}}{{\Delta_{r}}}+\rho^{2}d\theta^{2}$
(131)

where

$\displaystyle\Delta_{r}$ $\displaystyle=r^{2}-2{\cal M}r+{\cal
Q}^{2}+a_{{}_{\mathcal{J}}}^{2}\,,\qquad\rho^{2}=r^{2}+a_{{}_{\mathcal{J}}}^{2}\cos^{2}\theta\,,$
(132)

These BHs posses two generally distinct horizons located at the zeroes of
$\Delta_{r}$

$r_{\pm}={\cal M}{\pm}\sqrt{{\cal M}^{2}{-}a_{{}_{\mathcal{J}}}^{2}{-}{\cal
Q}^{2}}$ (133)

The wave equation can be separated using the ansatz

$\Phi(t,r,\theta,\phi)=e^{{-\rm i}(\omega
t-m_{\phi}\phi)}\frac{R(r)S(\chi)}{\sqrt{(1-\chi^{2})\Delta_{r}}}$ (134)

with $\chi=\cos\theta$. The radial and angular $Q$-functions read

$\displaystyle Q_{r}$
$\displaystyle=\frac{\left(\omega(r^{2}+a_{{}_{\mathcal{J}}}^{2})-a_{{}_{\mathcal{J}}}m_{\phi}\right)^{2}-\Delta_{r}\left(a_{{}_{\mathcal{J}}}^{2}\omega^{2}-2a_{{}_{\mathcal{J}}}\omega
m_{\phi}+A\right)+\frac{1}{4}\left(\Delta_{r}^{{}^{\prime}}{}^{2}-2\Delta_{r}^{{}^{\prime\prime}}\Delta_{r}\right)}{\Delta_{r}^{2}}$
(135) $\displaystyle Q_{\chi}$
$\displaystyle=\frac{(1-\chi^{2})(a_{{}_{\mathcal{J}}}^{2}\omega^{2}\chi^{2}+A)-m_{\phi}^{2}+1}{(1-\chi^{2})^{2}}$

where $A$ is the separation constant and $m_{\phi}$ the azimuthal angular
momentum of the incoming particle. Both differential equations on the gravity
side can be matched with that of $SU(2)$ gauge theory with $N_{f}=(1,2)$
fundamentals. The dictionary for the radial equation reads

$\displaystyle{q\over\hbar}$ $\displaystyle=2\,{\rm
i}\,\omega\,(r_{+}{-}r_{-})\,,$ (136) $\displaystyle{u\over\hbar^{2}}$
$\displaystyle=A+2m_{\phi}a_{{}_{\mathcal{J}}}\omega-\omega^{2}\left(3a_{{}_{\mathcal{J}}}^{2}+4r_{+}\left(r_{-}+r_{+}\right)\right)+\left(\frac{1}{2}+i\left(r_{+}-r_{-}\right)\omega\right)^{2}$
$\displaystyle{m_{1}\over\hbar}$ $\displaystyle={m_{3}\over\hbar}=-{\rm
i}(r_{+}+r_{-})\omega\quad,\quad{m_{2}\over\hbar}=-\frac{{\rm
i}\left[\left(r_{-}^{2}+r_{+}^{2}+2a_{{}_{\mathcal{J}}}^{2}\right)\omega-2a_{{}_{\mathcal{J}}}m_{\phi}\right]}{r_{+}-r_{-}}\,;\quad$
$\displaystyle y$ $\displaystyle=-{r{-}r_{-}\over r_{+}{-}r_{-}}$

whereas for the angular equation one has

$\displaystyle{q^{\chi}\over\hbar}$
$\displaystyle=4a_{{}_{\mathcal{J}}}\omega\quad,\quad{u^{\chi}\over\hbar^{2}}=a_{{}_{\mathcal{J}}}\omega\left(a_{{}_{\mathcal{J}}}\omega-2m_{\phi}+2\right)+A+\frac{1}{4}$
(137) $\displaystyle{m_{2}^{\chi}\over\hbar}$
$\displaystyle=m_{\phi}\quad,\quad m^{\chi}_{1}=m^{\chi}_{3}=0\quad,\quad
y^{\chi}=-\frac{1-\chi}{2}$

The extremal limit is obtained by sending $r_{-}\to r_{+}$. This corresponds
to taking $q\to 0$ and $m_{2}\to\infty$ while their product
$q_{\text{ext}}=-q\,m_{2}$ remains finite. The resulting theory has
$N_{f}=(1,1)$ fundamentals with

$\displaystyle{q_{\text{ext}}\over\hbar^{2}}$
$\displaystyle=4\,\omega\left[{a_{{}_{\cal
J}}}\,m_{\phi}-\omega\,(r_{+}^{2}+a_{{}_{\cal
J}}^{2})\right]\quad,\quad{u\over\hbar^{2}}=A+\frac{1}{4}+2m_{\phi}a_{{}_{\mathcal{J}}}\omega-(3a_{{}_{\cal
J}}^{2}+8r_{+}^{2})\omega^{2}\,,$ (138) $\displaystyle{m_{1}\over\hbar}$
$\displaystyle={m_{3}\over\hbar}=-2{\rm i}r_{+}\omega\,\quad;\quad
y_{\text{ext}}=-\frac{y}{m_{2}}=\frac{{\rm
i}}{2\hbar}\frac{r-r_{+}}{(r_{+}^{2}+a_{{}_{\cal J}}^{2})\omega-a_{{}_{\cal
J}}m_{\phi}}$

###  5.2 SW vs Numerical vs WKB

To showcase the computation of QNMs using the three methods, let us consider
an explicit KN solution with

${\cal M}=1\quad,\quad{\cal Q}=0.5\quad,\quad a_{{}_{\cal
J}}=0.3\quad,\quad\ell=m_{\phi}=2$ (139)

The simplest computation is given by the eikonal WKB or geodetic motion. Real
parts of the QNM frequency and separation constants are given by solving (23)
with $Q_{\rm geo}$ given by (36). Imaginary parts follow from (25).

Let us consider now the exact SW quantization. The radial and angular
equations can be mapped to a $SU(2)$ theory with $N_{f}=(1,2)$ flavours. We
start from the expression (80) for the NS prepotential up to four-instantons
and invert the Matone relation (73) to compute $a$ as a function of $u$. The
vanishing cycles for the radial and angular equations being $a_{D}$ and
$a^{\chi}$, respectively, we impose the quantization conditions

$a_{D}=-\frac{1}{2\pi\rm{i}}\frac{\partial{\cal F}}{\partial a}=\hbar
n_{r}\quad,\quad a^{\chi}-m_{2}^{\chi}=\hbar\left(n_{\chi}+\frac{1}{2}\right)$
(140)

The two equations can be solved numerically for $\omega$ and $A$. We use
Mathematica software, and to find the roots we look for solutions around
$\omega_{\rm geo}$ and $A_{\rm geo}$. Comparing against the WKB/geodetic
results we find

$n_{r}=-n_{\rm Num}=-(n_{r,geo}+1)\quad,\quad n_{\chi}=\ell-|m|$ (141)

where $n_{r,geo}$ starts from zero. Finally we compare the results with those
provided by a continuous fraction numerical method à la Leaver. The results of
the three methods, along the relative error w.r.t. the numerical results for
the Real and Imaginary part of the frequencies, are shown in the following
tables

$\begin{array}[]{||c||c|c||c|c||}\hline\cr
n_{\text{Num}}=1&\omega&A_{\text{lm}}&\text{Error}_{\%}(\omega_{\text{Re}})&E_{\%}(\omega_{\text{Im}})\\\
\hline\cr\text{Geo}&0.555-0.0957i&5.99+0.0024i&2.42&0.524\\\
\hline\cr\text{SW}_{2}&0.589-0.105i&5.99.+0.000651i&3.54&8.8\\\
\hline\cr\text{SW}_{4}&0.557-0.0927i&5.99+0.00135i&2.07&3.68\\\
\hline\cr\text{Num}&0.569-0.0963i&5.99+0.00141i&\text{}&\text{}\\\
\hline\cr\end{array}$ (142)

and then for $n_{r}=2$

$\begin{array}[]{||c||c|c||c|c||}\hline\cr
n_{\text{Num}}=2&\omega&A_{\text{lm}}&\text{Error}_{\%}(\omega_{\text{Re}})&E_{\%}(\omega_{\text{Im}})\\\
\hline\cr\text{Geo}&0.555-0.287i&5.99+0.0072i&0.0917&1.7\\\
\hline\cr\text{SW}_{2}&0.545-0.3i&5.99+0.00212i&1.66&2.55\\\
\hline\cr\text{SW}_{4}&0.558-0.294i&5.99+0.00408i&0.634&0.495\\\
\hline\cr\text{Num}&0.554-0.292i&5.99+0.00417i&\text{}&\text{}\\\
\hline\cr\end{array}$ (143)

In figures 2, 3 and 4 we display the results for the QNMs for various choices
of the charge and angular momentum obtained using each of the three methods
described in this paper. We set $\ell=m_{\phi}=2$, the mass $\cal M$ to one
and find an amazing agreement between the three methods, even for low values
of the energies and angular momenta, wherein the geodetic motion approximation
is not expected to work. Tables collecting the data in the plots are presented
in the appendix for the convenience of the reader.

Figure 2: QNMs of a RN BH for $n_{\text{Num}}=1$ (left) and $n_{\text{Num}}=2$
(right), $\ell=m_{\phi}=2$, with $a=0$, $\mathcal{M}=1$. $\mathcal{Q}$ varies
between $0$ and $0.9$.

Figure 3: QNMs of a KN BH for $n_{\text{Num}}=1$ (left) and $n_{\text{Num}}=2$
(right), $\ell=m_{\phi}=2$, with $\mathcal{Q}=0.5$, $\mathcal{M}=1$,
$a_{{}_{\mathcal{J}}}$ varies between $0$ and $0.4$.

###  5.3 Analytic solutions

In this subsection we show two special cases of KN solutions related to
perturbative gauge theories

  * •

static: $\omega=0$

  * •

super-radiant: $\omega=m_{\phi}\Omega_{\phi}$ with
$\Omega_{\phi}={a_{{}_{\mathcal{J}}}\over\mathcal{M}^{2}+a_{{}_{\mathcal{J}}}^{2}}$
the angular velocity at the horizon

where analytic solutions can be found.

####  5.3.1 Static Kerr-Newman wave

Following the recent analysis for Kerr BH BonTanzetc , here we consider a
static wave ($\omega=0$) in the KN BH that, according to the gauge/gravity
dictionary (136), is mapped to a gauge theory with $q=0$. It is easy to see
that mixing terms depend on $a_{{}_{\mathcal{J}}}$ only through the
combination $a_{{}_{\mathcal{J}}}\omega$, thus for $\omega=0$ angular and
radial equations completely decouple and the separation constant simply
becomes $A=\ell(\ell+1)$. The radial equation reduces to

$R^{{}^{\prime\prime}}(r)+\frac{a_{{}_{\mathcal{J}}}^{2}m_{\phi}^{2}-A\,(r-r_{+})(r-r_{-})+\tfrac{1}{4}(r_{+}-r_{-})^{2}}{(r-r_{+})^{2}(r-r_{-})^{2}}\,R(r)=0$
(144)

where $\Delta_{r}=(r-r_{+})(r-r_{-})$. The gauge/gravity dictionary reads (see
136)

$q=0\quad,\quad\frac{u}{\hbar^{2}}=(\ell+\tfrac{1}{2})^{2}\quad,\quad
m_{1}=m_{3}=0\quad,\quad\frac{m_{2}}{\hbar}=\frac{2im_{\phi}\,a_{{}_{\mathcal{J}}}}{r_{+}-r_{-}}$
(145)

The general solution can be written as

$\frac{R(r)}{\sqrt{\Delta_{r}}}=c_{1}\,P_{\ell,\mu}\left(1-\frac{2\left(r-r_{-}\right)}{r_{+}-r_{-}}\right)+c_{2}\,Q_{-\ell-1,\mu}\left(1-\frac{2\left(r-r_{-}\right)}{r_{+}-r_{-}}\right)$
(146)

with $P_{\ell,\mu}$ and $Q_{\ell,\mu}$ associated Legendre functions defined
in (91) and

$\mu=-\frac{m_{2}}{\hbar}=-\frac{2{\rm i}m_{\phi}a_{{}_{\cal
J}}}{r_{+}-r_{-}}$ (147)

For large $r$ the two solutions above behave like $r^{\ell}$ and $r^{-\ell-1}$
respectively. The static Love number is defined as the ratio of the
coefficients of these terms Binnington:2009bb . For a BH, one has to impose
in-going boundary conditions at $r=r_{+}$. In this limit the contribution of
$Q_{-1-\ell,\mu}$ diverges so regularity requires $c_{2}=0$, leading to a zero
static Love number as expected Binnington:2009bb ; Damour:2009vw ;
LeTiec:2020bos .

####  5.3.2 Near super-radiant modes

Figure 4: QNMs of a KN BH for $n_{\text{Num}}=1$ (left) and $n_{\text{Num}}=2$
(right), $\ell=m_{\phi}=2$, with $a_{{}_{\mathcal{J}}}=0.5$, $\mathcal{M}=1$,
$\mathcal{Q}$ varies between $0$ and $0.4$.

In this subsection we consider near super-radiant modes, also known as zero-
damping modes (ZDMs), which are close to the super-radiant threshold frequency
$\omega_{{}_{SR}}$ Brito:2015oca  and the imaginary part of the frequency is
almost vanishing. These modes are produced by near extremal BH mergers
Starobinskil:1974nkd ; Teukolsky:1974yv ; Yang:2013uba .

As a concrete example let’s consider again the KN BH, whose QNM-SW dictionary
is given in (136). Near super-radiant modes are defined by taking

$\omega=\omega_{{}_{SR}}+\nu{\delta}$ (148)

with $\delta=r_{+}-r_{-}$ the distance between the inner and outer horizon
taken to be very small and

$\omega_{{}_{SR}}=\frac{m_{\phi}\,a_{{}_{\mathcal{J}}}}{r_{+}^{2}+a_{{}_{\mathcal{J}}}^{2}}=m_{\phi}\Omega_{\phi}$
(149)

the super-radiant frequency.

In order to compute $\nu$ one has to construct the QNM solutions in the near
extremal limit. Far from the horizon $r\gg r_{+}\gg\delta$ the radial equation
reduces to

$R{{}^{\prime\prime}}(r)+\left[\omega_{{}_{SR}}^{2}+\frac{4\omega_{{}_{SR}}^{2}r_{+}}{r-r_{+}}+\frac{(6r_{+}^{2}+a_{{}_{\mathcal{J}}}^{2})\omega_{{}_{SR}}^{2}-A}{(r-r_{+})^{2}}\right]R(r)=0\,.$
(150)

The solutions to this equation are confluent hypergeometric functions.
Requiring the absence of incoming waves one finds

$R(r)=c_{\infty}e^{i\omega_{{}_{SR}}(r-r_{+})}(r-r_{+})^{\frac{1}{2}+\alpha}U\left(\tilde{A};\tilde{B};z_{r}\right)$
(151)

where $c_{\infty}$ is a constant and $U$ is the Tricomi confluent
hypergeometric function111111Related to the Kummer confluent hypergeometric
functions by
$U(\tilde{A};\tilde{B};z)=\frac{\Gamma(1-\tilde{B})}{\Gamma(\tilde{A}-\tilde{B}+1)}{}_{1}F_{1}(\tilde{A};\tilde{B};z)+\frac{\Gamma(\tilde{B}-1)z^{1-\tilde{B}}}{\Gamma(\tilde{A})}{}_{1}F_{1}(\tilde{A}-\tilde{B}+1;2-\tilde{B};z)$
(152) , while

$\tilde{A}=\frac{1}{2}+\alpha-2i\omega_{{}_{SR}}r_{+}\,,\qquad\tilde{B}=1+2\alpha\,,\qquad
z_{r}=-2i\omega_{{}_{SR}}(r-r_{+})$ (153)

with

$\alpha=\sqrt{A+\frac{1}{4}-(a_{{}_{\mathcal{J}}}^{2}+6r_{+}^{2})\,\omega_{{}_{SR}}^{2}}$
(154)

On the other hand, the radial equation in the near horizon limit can be
approximated by taking $r=r_{+}+\tau\delta$ and sending $\delta\to 0$. In the
variable $\tau$ one finds

$\displaystyle R{{}^{\prime\prime}}(\tau)+Q(\tau)R(\tau)=0\,,$ (155)

with

$Q(\tau)=\frac{\left(\nu(r_{+}^{2}{+}a_{{}_{\mathcal{J}}}^{2})+2r_{+}\omega_{{}_{SR}}\tau\right)^{2}+\tfrac{1}{4}-\tau(\tau+1)(\alpha^{2}+4r_{+}^{2}\omega_{{}_{SR}}^{2}-\tfrac{1}{4})}{\tau^{2}(\tau+1)^{2}}$
(156)

The solution can be written in terms of ordinary hypergeometric functions.
Imposing in-going boundary conditions at the horizon one finds

$R(\tau)=c_{H}\tau^{\frac{1}{2}-\frac{ia_{{}_{\mathcal{J}}}\nu}{\Omega_{\phi}}}(1+\tau)^{\frac{1}{2}-\frac{ia_{{}_{\mathcal{J}}}\nu}{\Omega_{\phi}}+2ir_{+}\omega_{{}_{SR}}}{}_{2}F_{1}\left(\bar{A},\bar{B};\bar{C};-\tau\right)$
(157)

where $c_{H}$ is a constant and

$\bar{A}=\frac{1}{2}-\alpha-\frac{2{\rm
i}a_{{}_{\mathcal{J}}}\nu}{\Omega_{\phi}}+2{\rm
i}r_{+}\omega_{{}_{SR}}\,,\qquad\bar{B}=\frac{1}{2}+\alpha-\frac{2{\rm
i}a_{{}_{\mathcal{J}}}\nu}{\Omega_{\phi}}+2{\rm
i}r_{+}\omega_{{}_{SR}}\,,\qquad\bar{C}=1-\frac{2{\rm
i}a_{{}_{\mathcal{J}}}\nu}{\Omega_{\phi}}$ (158)

By expanding (151) near the horizon $r\approx r_{+}$ one finds

$R(r)\sim(r-r_{+})^{\frac{1}{2}-\alpha}\left[1+(-2{\rm
i}\omega_{{}_{SR}})^{-2\alpha}(r-r_{+})^{2\alpha}\frac{\Gamma(2\alpha)\Gamma(\bar{C}-\bar{B})}{\Gamma(-2\alpha)\Gamma(\bar{C}-\bar{A})}+\ldots\right]$
(159)

while far away from the horizon $r\gg r_{+}$, (157) reduces to

$R(r)\sim(r-r_{+})^{\frac{1}{2}-\alpha}\left[1+\delta^{2\alpha}(r-r_{+})^{2\alpha}\frac{\Gamma(-2\alpha)\Gamma(\bar{B})\Gamma(\bar{C}-\bar{A})}{\Gamma(2\alpha)\Gamma(\bar{A})\Gamma(\bar{C}-\bar{B})}+\ldots\right]$
(160)

by matching (159) and (160) one finds

$\left(-2{\rm
i}\omega_{{}_{SR}}\,\delta\right)^{-2\alpha}\frac{\Gamma(2\alpha)^{2}\Gamma(\bar{A})\Gamma(\bar{C}-\bar{B})^{2}}{\Gamma(-2\alpha)^{2}\Gamma(\bar{B})\Gamma(\bar{C}-\bar{A})^{2}}=1$
(161)

Since $\delta\sim 0$ and ${\rm Re}\,\alpha>0$, the factor $\delta^{-\alpha}$
in the left hand side diverges, so it has to be compensated by a pole of
$\Gamma(\bar{B})$ in the denominator i.e.

$\bar{B}=-n+(-2{\rm i}\,\omega_{{}_{SR}}\delta)^{2\alpha}\eta$ (162)

with

$\eta=\frac{(-1)^{n}}{n!}\frac{\Gamma(-2\alpha)^{2}\Gamma\left(\bar{C}-\bar{A}\right)^{2}}{\Gamma(2\alpha)^{2}\Gamma\left(\bar{A}\right)\Gamma\left(\bar{C}-\bar{B}\right)^{2}}$
(163)

leading to Hod:2008se ; Hod:2012bw

$\displaystyle\omega$ $\displaystyle=$
$\displaystyle\omega_{{}_{SR}}+{\delta}\left[{\Omega_{\phi}\,r_{+}\,\omega_{{}_{SR}}\over
a_{{}_{\mathcal{J}}}}-{\rm
i}(n+{\textstyle\frac{1}{2}}+\alpha){\Omega_{\phi}\over
2a_{{}_{\mathcal{J}}}}\right]+\ldots$ (164) $\displaystyle=$
$\displaystyle\Omega_{\phi}m_{\phi}(1+4\pi\,T_{\text{BH}}\,r_{+})-2\pi{\rm
i}\,T_{\text{BH}}\,(\alpha+n+\tfrac{1}{2})+O\left(T_{BH}^{1+2\alpha}\right)$

This is consistent with the WKB analysis which states that the imaginary part
of the quasi-normal frequencies is given by the Lyapunov exponent (1) that for
near extremal rotating BHs the photon-sphere corresponding to co-rotating
impinging photons coalesces with the horizon and $\lambda$ is proportional to
the BH temperature Bianchi:2020des ; Maldacena:2015waa .

The resulting near super-radiant frequencies can be easily re-derived from
exact SW quantization. In the limit $\delta\to 0$, the gauge coupling vanishes
and the mass $m_{2}$ diverges unless we choose $\omega$ as in (148). In this
particular limit the radial dictionary (136) reduces to

$\displaystyle\frac{q}{\hbar}=2i\omega_{{}_{SR}}\delta\,,\qquad\frac{u}{\hbar^{2}}=\alpha^{2}\,,\qquad\frac{m_{1}}{\hbar}=\frac{m_{3}}{\hbar}=-2i\omega_{{}_{SR}}r_{+}\,,\qquad\frac{m_{2}}{\hbar}=2i[r_{+}\omega_{{}_{SR}}{-}\nu(a_{{}_{\mathcal{J}}}^{2}{+}r_{+}^{2})]$
(165)

where we kept the leading terms in $\delta$. For $q$ small, instanton
contributions can be discarded but the 1-loop term must be kept and the
quantization of $a_{D}$ reduces to

$\exp\left(-\frac{2\pi{\rm
i}\,a_{D}}{\hbar}\right)=\left(-{q\over\hbar}\right)^{\tfrac{2\sqrt{u}}{\hbar}}\frac{\Gamma^{2}(1+\tfrac{2\sqrt{u}}{\hbar})}{\Gamma^{2}(1-\tfrac{2\sqrt{u}}{\hbar})}\prod_{i=1}^{3}{\Gamma\left({1\over
2}{+}{m_{i}-\sqrt{u}\over\hbar}\right)\over\Gamma\left({1\over
2}{+}{m_{i}+\sqrt{u}\over\hbar}\right)}=1$ (166)

which is precisely equation (161) written in terms of the gauge variables.

##  6 Other examples

Gauge/ gravity dictionaries similar to the one we have found for KN (AdS) can
be found for other asymptotically flat or AdS gravity solution that admit a
photon-sphere. We will consider in turn: D3-branes, BPS BHs from four
intersecting stacks of D3-branes Cvetic:1995uj , five-dimensional
(asymptotically flat) CCLP BHs CCLP1 ; CCLP2 , including their extremal limits
BMPV , D1D5 circular fuzzballs Lunin:2001fv , JMaRT smooth horizonless
solutions JMaRT , including their BPS limit, known as GMS solutions GMS1 ;
GMS2 . The derivation of their QNMs following the steps described for KN (AdS)
is straightforward but laborious and is beyond the scope of the present
investigation.

###  6.1 D3-branes

D3-branes arise as solutions in type IIB supergravity and are a very
interesting example since they are dual to pure $SU(2)$ SYM with
$N_{f}=(0,0)$. The dilaton-axion field of type IIB is decoupled from the
metric and its equation of motion corresponds to the differential equation for
a scalar perturbation on the D3 background, it can be mapped to the Mathieu
equation by performing the change of variables $r=L\,e^{{\rm i}z}$
Cvetic:2000xz ; Gubser:1998iu . The spectrum of QNMs for this system has been
studied first in Kurita:2002he  for small $\omega$ and then in general using
SW curves in Bianchi:2021xpr . The metric reads

$ds^{2}=H(r)^{-{1\over 2}}(-dt^{2}+d{\bf x}^{2})+H(r)^{{1\over
2}}(dr^{2}+r^{2}d\Omega_{5}^{2})$ (167)

the radial equation can be put in canonical form with

$Q(r)={4\omega^{2}(r^{4}+L^{4})-r^{2}(4\ell(\ell+4)+15)\over 4r^{4}}\ $ (168)

and the gauge/gravity dictionary reads

$\frac{q}{\hbar^{4}}={\omega^{4}\,L^{4}\over
16}\,,\qquad{u\over\hbar^{2}}=\left(\frac{\ell}{2}+1\right)^{2}\,;\qquad\hbar^{2}y={4r^{2}\over\omega^{2}L^{4}}$
(169)

We refer the reader to Bianchi:2021xpr  for the detailed study of QNMs of
D3-branes, including a successful comparison of the three methods
(WKB/geodesics, quantum SW curve, Leaver’s continuous fraction) to compute
them.

###  6.2 Intersecting D3 branes BHs in four dimensions

Other examples of 4-dimensional (BPS) BHs can be found in type IIB
supergravity considering the intersection of four stacks of D3-branes. These
geometries possess four different charges $\mathcal{Q}_{i}$ which, if taken
equal, lead to an extremal Reissner-Nordström BH. The line element reads
Cvetic:1995uj

$ds^{2}=-f(r)\,dt^{2}+f(r)^{-1}\left[dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right)\right]$ (170)

where

$f(r)=\prod_{i=1}^{4}\left(1+\frac{\mathcal{Q}_{i}}{r}\right)^{-\frac{1}{2}}$
(171)

Thanks to spherically symmetry the angular wave-equation can be solved in
terms of spherical harmonics, while the radial equation in canonical form is
defined by

$Q_{r}=\frac{\omega^{2}\prod_{i=1}^{4}\left(\mathcal{Q}_{i}+r\right)-\ell(\ell+1)r^{2}}{r^{4}}$
(172)

The wave equation can be mapped to the quantum SW curve for $SU(2)$ with
$N_{f}=(1,1)$ and

$\displaystyle{q\over\hbar^{2}}$ $\displaystyle\text{ =
}-4\omega^{2}\sqrt{\Sigma_{4}}\,,\quad{u\over\hbar^{2}}\text{ =
}\left(\ell+\frac{1}{2}\right)^{2}-\omega^{2}\left(\Sigma_{2}+2\sqrt{\Sigma_{4}}\right)$
(173) $\displaystyle{m_{1}\over\hbar}$ $\displaystyle\text{ =
}\frac{i\omega\Sigma_{3}}{2\sqrt{\Sigma_{4}}}\,,\quad{m_{3}\over\hbar}\text{ =
}\frac{i\omega\Sigma_{1}}{2}\,;\quad
y\hbar=-\frac{ir}{2\omega\sqrt{\Sigma_{4}}}$

where

$\Sigma_{n}=\sum_{i_{1}<\cdots<i_{n}}^{4}\mathcal{Q}_{i_{1}}\cdots\mathcal{Q}_{i_{n}}$
(174)

The entropy of the system is proportional to $\sqrt{\Sigma_{4}}$. In the limit
where $\Sigma_{4}\to 0$ the gauge coupling $q$ vanishes and $m_{1}$ diverges,
so that one fundamental decouples and the resulting theory is $N_{f}=(1,0)$.

###  6.3 CCLP five-dimensional BHs

CCLP metrics describe rotating solutions of Einstein-Maxwell theory in $d=5$,
with mass ${\cal M}$, charge ${\cal Q}$, and angular momentum parameters
$\ell_{1}$, $\ell_{2}$. The line element reads CCLP1 ; CCLP2

$\displaystyle ds^{2}$
$\displaystyle=-dt^{2}-\frac{2\mathcal{Q}\,\omega_{2}}{\Sigma}(dt-\omega_{1})+\Delta_{t}(dt-\omega_{1})^{2}+\Sigma\left(d\theta^{2}+\frac{r^{2}dr^{2}}{\Delta_{r}}\right)+$
(175)
$\displaystyle+d\psi^{2}\cos^{2}\theta\,(r^{2}+\ell_{2}^{2})+d\phi^{2}\sin^{2}\theta\,(r^{2}+\ell_{1}^{2})$

with the one-forms $\omega_{1,2}$ given by

$\displaystyle\omega_{1}$
$\displaystyle=\ell_{2}\,\cos^{2}\theta\,d\psi+\ell_{1}\,\sin^{2}\theta\,d\phi\quad,\quad\omega_{2}=\ell_{1}\,\cos^{2}\theta\,d\psi+\ell_{2}\,\sin^{2}\theta\,d\phi$
(176)

and the functions $\Delta_{r}$, $\Delta_{t}$ and $\Sigma$ given by

$\displaystyle\Delta_{r}$
$\displaystyle=(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2})\,,\quad\Delta_{t}=\frac{2\mathcal{M}\Sigma-\mathcal{Q}^{2}}{\Sigma^{2}}\,,\quad\Sigma=r^{2}+\ell_{1}^{2}\cos^{2}\theta+\ell_{2}^{2}\sin^{2}\theta$
(177)

These geometries, much like KN BH’s, possess two horizons located at

$r_{\pm}^{2}=\widehat{{\cal M}}\pm\sqrt{\widehat{{\cal M}}^{2}-\widehat{{\cal
Q}}^{2}}$ (178)

with

$\widehat{{\cal M}}={\cal
M}-\frac{\ell_{1}^{2}+\ell_{2}^{2}}{2}\quad,\quad\widehat{{\cal Q}}={\cal
Q}+\ell_{1}\ell_{2}$ (179)

The wave-equation for a scalar perturbation in the CCLP background can be
separated into radial and angular equations. Introducing the variables
$z=r^{2}$ and $\xi=\cos^{2}\theta$, these can be brought into canonical form
with

$\displaystyle Q_{z}$
$\displaystyle=\frac{z\left(\mathcal{L}_{\mathcal{M}}^{2}-\mathcal{L}_{\mathcal{Q}}^{2}\right)+2\mathcal{L}_{\mathcal{Q}}\left(\widehat{\mathcal{Q}}\,\mathcal{L}_{\mathcal{M}}+\widehat{\mathcal{M}}\,\mathcal{L}_{\mathcal{Q}}\right)-\Delta_{z}\left(K^{2}-\omega^{2}(z+2\mathcal{M})+4\right)+\Delta_{z}^{\prime}{}^{2}}{4\Delta_{z}^{2}}$
(180) $\displaystyle Q_{\xi}$
$\displaystyle=\frac{(1-\xi)\xi\left(K^{2}+\omega^{2}\xi\left(\ell_{1}^{2}-\ell_{2}^{2}\right)\right)+1-(1-\xi)m_{\psi}^{2}-\xi
m_{\phi}^{2}}{4(1-\xi)^{2}\xi^{2}}$

where $\Delta_{z}=(z-z_{+})(z-z_{-})$, $K^{2}$ is the separation constant,
$m_{\phi}$, $m_{\psi}$ are the projections of the total angular momentum along
two orthogonal 2-planes, and

$\mathcal{L}_{\mathcal{M}}=\ell_{1}m_{\phi}+\ell_{2}m_{\psi}-2\mathcal{M}\,\omega\,,\quad\mathcal{L}_{\mathcal{Q}}=\ell_{1}m_{\psi}+\ell_{2}m_{\phi}+\mathcal{Q}\,\omega$
(181)

Both equations can be mapped to $SU(2)$ gauge theory with $N_{f}=(0,2)$
flavours. The gauge/gravity dictionary for the radial wave equation reads

${q\over\hbar^{2}}=-\frac{\omega^{2}}{4}({z}_{+}{-}{z}_{-})\,,\;\;{u\over\hbar^{2}}=\frac{1{+}K^{2}{-}\omega^{2}(z_{+}{+}2\mathcal{M})}{4}\,,\;\;{m_{1,2}\over\hbar}={-}\frac{i}{2}\frac{\mathcal{L}_{\mathcal{M}}{\mp}\mathcal{L}_{\mathcal{Q}}}{\sqrt{z_{+}}{\pm}\sqrt{z_{-}}}\,;\;\;y={z{-}z_{-}\over
z_{-}{-}z_{+}}$ (182)

The dictionary for the angular part reads

${q^{\xi}\over\hbar^{2}}=\frac{\omega^{2}}{4}\left(\ell_{1}^{2}-\ell_{2}^{2}\right)\,,\quad{u^{\xi}\over\hbar^{2}}=\frac{1}{4}\left(1+K^{2}+\omega^{2}\ell_{1}^{2}\right)\,,\quad{m^{\xi}_{1,2}\over\hbar}=\frac{m_{\phi}\pm
m_{\psi}}{2}\,;\quad y^{\xi}=-\xi$ (183)

Similarly to KN BH’s, imposing the extremality condition $z_{-}=z_{+}$ (or
$\widehat{\cal M}^{2}=\widehat{\cal Q}^{2}$) leads to the decoupling of the
flavour associated to $m_{2}$. As $m_{2}\to\infty$ we keep
$q_{\text{ext}}=-q\,m_{2}$ finite, after rescaling $y_{\text{ext}}=-y/m_{2}$
one is left with an $N_{f}=(0,1)$ theory with

$\displaystyle\frac{q_{\text{ext}}}{\hbar^{3}}$ $\displaystyle=-\frac{{\rm
i}\,\omega^{2}\sqrt{z_{+}}\left[(\ell_{1}+\ell_{2})(m_{\phi}+m_{\psi})-(2\mathcal{M}-\mathcal{Q})\,\omega\right]}{4}\,,\qquad{u\over\hbar^{2}}=\frac{1{+}K^{2}{-}\omega^{2}(\widehat{{\cal
M}}{+}2\mathcal{M})}{4}$ (184) $\displaystyle{m_{1}\over\hbar}$
$\displaystyle=\frac{(\ell_{1}-\ell_{2})(m_{\phi}-m_{\psi})-(2\mathcal{M}+\mathcal{Q})\omega}{4{\rm
i}\sqrt{z_{+}}}\,;\qquad\frac{y_{\text{ext}}}{\hbar^{2}}={z-z_{+}\over{\rm
i}\sqrt{z_{+}}\left(\mathcal{L}_{\mathcal{M}}+\mathcal{L}_{\mathcal{Q}}\right)}$

A particular case is the BMPV BH which is obtained by imposing also the BPS
condition ${\cal M}={\cal Q}$ (which implies $\ell_{2}=-\ell_{1}$) BMPV . The
radial dictionary (184) reduces to

$\displaystyle{q_{\text{ext}}\over\hbar^{3}}=\frac{i{\cal
M}\sqrt{z_{+}}\omega^{3}}{4}\,,\quad{u\over\hbar^{2}}$
$\displaystyle=\frac{1{+}K^{2}{-}\omega^{2}(3\mathcal{M}-\ell_{1}^{2})}{4}\,,\quad{m_{1}\over\hbar}=\frac{i\left[\ell_{1}(m_{\psi}-m_{\phi})+\tfrac{3\omega\mathcal{M}}{2}\right]}{2\sqrt{z_{+}}}$
(185)

The dual gauge theory of the angular equation of a BMPV BH is still
$N_{f}=(0,2)$, the dictionary follows from (183) and reads

${q^{\xi}\over\hbar^{2}}=0\,,\quad{u^{\xi}\over\hbar^{2}}=\frac{1}{4}\left(1+K^{2}+\omega^{2}\ell_{1}^{2}\right)\,,\quad{m^{\xi}_{1,2}\over\hbar}=\frac{m_{\phi}\pm
m_{\psi}}{2}\,;\quad y^{\xi}=-\xi$ (186)

Notice that the angular equation relates to free $SU(2)$ SYM, therefore the
exact solutions are known and $K^{2}=\ell(\ell+2)$.

At variant with KN BH’s, there is another extremal limit that leads to pure
$SU(2)$ SYM with no flavour obtained by imposing the vanishing of $z_{+}$, i.
e. $2{\cal M}{\,=\,}\ell_{1}^{2}+\ell_{2}^{2}$ and ${\cal
Q}{\,=\,}-\ell_{1}\ell_{2}$. In this limit the mass $m_{1}$ diverges and while
$\hat{q}_{\text{ext}}=-q_{\text{ext}}\,m_{1}$ is finite. The resulting
geometry is associated to a gauge theory with $N_{f}=(0,0)$ and

$\frac{\hat{q}_{\text{ext}}}{\hbar^{4}}=\frac{\omega^{2}}{16}(\mathcal{L}_{\mathcal{M}}^{2}-\mathcal{L}_{\mathcal{Q}}^{2})\,,\quad{u\over\hbar^{2}}=\frac{1{+}K^{2}{-}(\ell_{1}^{2}+\ell_{2}^{2})\omega^{2}}{4}\,;\quad\hat{y}_{\text{ext}}\,\hbar^{2}=\frac{4z}{\mathcal{L}_{\mathcal{M}}^{2}-\mathcal{L}_{\mathcal{Q}}^{2}}$
(187)

As for the D3D3D3D3 BHs vanishing entropy corresponds to a theory with fewer
flavours.

###  6.4 D1D5 fuzzball

Next we consider a D1D5 circular fuzzball with radius $a_{f}$ and equal
charges ${\cal Q}_{1}={\cal Q}_{5}=L^{2}$. The smooth horizonless metric is
given by Lunin:2001fv

$\displaystyle ds^{2}$
$\displaystyle=H_{f}^{-1}\left[(d{v}+\omega_{\psi}\,d\psi)^{2}-(dt+\omega_{\phi}\,d\phi)^{2}\right]+$
(188)
$\displaystyle+H_{f}\left[d\phi^{2}\sin^{2}\theta(\rho^{2}+a_{f}^{2})+\frac{\Sigma_{f}}{\rho^{2}+a_{f}^{2}}\left[d\rho^{2}+(\rho^{2}+a_{f}^{2})d\theta^{2}\right]+\rho^{2}d\psi^{2}\cos^{2}\theta\right]$

with

$\omega_{\phi}=\frac{L^{2}a_{f}\sin^{2}\theta}{\Sigma_{f}}\,,\quad\omega_{\psi}=\frac{L^{2}a_{f}\cos^{2}\theta}{\Sigma_{f}}\,,\quad
H_{f}=1+\frac{L^{2}}{\Sigma_{f}}\,,\quad\Sigma_{f}=\rho^{2}+a_{f}^{2}\cos^{2}\theta$
(189)

Setting

$\Phi=e^{-i\omega t+iP_{v}v+im_{\phi}\phi+im_{\psi}\psi}R(\rho)S(\chi)$ (190)

the wave equation can be separated, brought into canonical form (15) and
matched to that of $SU(2)$ gauge theory with $N_{f}=(0,2)$ fundamentals. The
$Q$-functions read

$\displaystyle Q_{\text{D1D5},\rho}$
$\displaystyle=\frac{\left(a_{f}^{2}-\rho^{2}\right)^{2}+4\left[\rho^{2}\mathcal{L}_{\phi}^{2}-\left(a_{f}^{2}+\rho^{2}\right)\left(\mathcal{L}_{\psi}^{2}+\rho^{2}\left(1+K^{2}-\left(2L^{2}+\rho^{2}\right)\tilde{\omega}^{2}\right)\right)\right]}{4\rho^{2}\left(a_{f}^{2}+\rho^{2}\right)^{2}}$
(191) $\displaystyle Q_{\text{D1D5},\chi}$
$\displaystyle=\frac{\left(\chi^{2}+1\right)^{2}-4\left[\chi^{2}m_{\phi}^{2}+\left(1-\chi^{2}\right)\left(m_{\psi}^{2}-\chi^{2}\left(1+K^{2}+\tilde{\omega}^{2}a_{f}^{2}\chi^{2}\right)\right)\right]}{4\chi^{2}\left(1-\chi^{2}\right)^{2}}$

where $\chi=\cos\theta$ and we defined

$\mathcal{L}_{\phi}=a_{f}\,m_{\phi}-L^{2}\omega\quad,\quad\mathcal{L}_{\psi}=a_{f}\,m_{\psi}-L^{2}P_{v}\quad,\quad\tilde{\omega}^{2}=\omega^{2}-P_{v}^{2}$
(192)

The gauge/gravity dictionary for the radial equation reads

$\displaystyle{q\over\hbar^{2}}=\frac{a_{f}^{2}\tilde{\omega}^{2}}{4}\,,\quad{u\over\hbar^{2}}=\frac{1+K^{2}+\tilde{\omega}^{2}\left(a_{f}^{2}-2L^{2}\right)}{4}\,,\quad{m_{1,2}\over\hbar}=\frac{{\cal
L}_{\phi}\mp{\cal L}_{\psi}}{2a_{f}}\,;\quad y=\frac{\rho^{2}}{a_{f}^{2}}$
(193)

while for the angular equation one finds

$\displaystyle{q^{\chi}\over\hbar^{2}}=\frac{a_{f}^{2}\tilde{\omega}^{2}}{4}\quad,\quad{u^{\chi}\over\hbar^{2}}=\frac{1+K^{2}+\tilde{\omega}^{2}a_{f}^{2}}{4}\quad,\quad{m_{1,2}^{\chi}\over\hbar}=\frac{m_{\phi}\pm
m_{\psi}}{2}\quad;\quad y^{\chi}=-\chi^{2}$ (194)

In the BH limit $a_{f}=0$ the gauge coupling goes to zero while both masses
diverge ($q_{\text{BH}}=m_{1}m_{2}\,q$ is finite), the resulting theory is
$N_{f}=(0,0)$ with radial dictionary

$\displaystyle\frac{q_{\text{BH}}}{\hbar^{4}}=\left(\frac{L\tilde{\omega}}{2}\right)^{4}\quad,\quad{u\over\hbar^{2}}=\frac{1+K^{2}-2\tilde{\omega}^{2}L^{2}}{4}\quad;\quad\hbar^{2}\,y_{\text{BH}}=\frac{4\rho^{2}}{\tilde{\omega}^{2}L^{4}}$
(195)

As for D3-branes the wave equation in this case can be solved exactly.

###  6.5 JMaRT and GMS geometries

JMaRT solutions are (non-)BPS smooth horizonless geometries with three charges
${\cal Q}_{1}$, ${\cal Q}_{5}$ and ${\cal Q}_{P}$ and two angular momenta
${\cal J}_{\phi}$ and ${\cal J}_{\psi}$. The explicit form of metric and the
other field profiles can be found in the original paper JMaRT . The charges
and the angular momenta are given by

$\mathcal{M}=\sum_{i}\frac{c_{i}^{2}+s_{i}^{2}}{2}\,M\quad,\quad\mathcal{Q}_{i}=Mc_{i}s_{i}\quad,\quad\mathcal{J}_{\phi}=-m\frac{\mathcal{Q}_{1}\mathcal{Q}_{5}}{R_{v}}\quad,\quad\mathcal{J}_{\psi}=n\frac{\mathcal{Q}_{1}\mathcal{Q}_{5}}{R_{v}}$
(196)

with $i=1,5,P$ and $c_{i}=\cosh\delta_{i}$ ,$s_{i}=\sinh\delta_{i}$ ‘boost’
parameters satisfying $c_{i}^{2}-s_{i}^{2}=1$. The charges can be parametrized
as

$m-n={j+j^{-1}\over s+s^{-1}}\quad,\quad m+n={j-j^{-1}\over s-s^{-1}}$ (197)

with

$j=\sqrt{a_{2}\over a_{1}}\leq 1\quad,\quad s=\sqrt{s_{1}s_{5}s_{P}\over
c_{1}c_{5}c_{P}}$ (198)

and

$\mathcal{Q}_{1}={g_{s}\alpha^{\prime 3}\over
V_{4}}n_{1}\quad,\quad\mathcal{Q}_{5}=g_{s}\alpha^{\prime}n_{5}\quad,\quad\mathcal{Q}_{P}=\frac{g_{s}^{2}\alpha^{\prime
4}n_{P}}{R_{v}^{2}V_{4}}$ (199)

Note that the volume of the internal 4-torus is ${\cal
V}(T^{4})=(2\pi)^{4}V_{4}$ while regularity requires
$n_{P}={n\,m\,n_{1}\,n_{5}}$121212For simplicity we consider the non-orbifold
case..

The scalar wave equation was separated in JMaRT  we will mostly follow their
notations. As we will show both radial and angular equations can be mapped to
the differential equation associated to an $N_{f}=(0,2)$ theory.

Setting

$\Phi(t,y,r,\theta,\psi,\phi)=e^{-i\omega
t-iP_{v}{v}+im_{\psi}\psi+im_{\phi}\phi}H(r)\Theta(\theta)$ (200)

the radial and angular equations for $H(r)$ and $\Theta(\theta)$ separate.
Using

$z=\frac{r^{2}-r_{+}^{2}}{r_{+}^{2}-r_{-}^{2}}\quad,\quad$ (201)

where

$r_{\pm}^{2}=\frac{M-a_{1}^{2}-a_{2}^{2}}{2}\pm\frac{1}{2}\sqrt{(M-a_{1}^{2}-a_{2}^{2})^{2}-4a_{1}^{2}a_{2}^{2}}$
(202)

The radial equation reads

$4\,\frac{d}{dz}\left[z(z+1)\frac{dH}{dz}\right]+\left[1-\nu^{2}+\kappa^{2}z+\frac{\alpha^{2}}{z+1}-\frac{\beta^{2}}{z}\right]H(z)=0$
(203)

Performing the redefinition

$H(z)={F(z)\over\sqrt{z(z+1)}}$ (204)

we get the radial $Q$-function in canonical form

$Q_{\text{JMaRT}}(z)=\frac{\kappa^{2}z^{3}+z^{2}(\kappa^{2}-\nu^{2}+1)+z(\alpha^{2}+\beta^{2}-\nu^{2}+1)+1+\beta^{2}}{4z^{2}(z+1)^{2}}$
(205)

where

$\displaystyle\kappa^{2}=\tilde{\omega}^{2}(r_{+}^{2}-r_{-}^{2})$ (206)
$\displaystyle\nu^{2}=1+{{K^{2}}}+\tilde{\omega}^{2}(r_{+}^{2}+Ms_{1}^{2}+Ms_{5}^{2})+M(\omega
c_{P}+P_{v}s_{P})$ $\displaystyle\alpha=\omega R_{v}\sigma+P_{v}R_{v}\tau-
m_{\phi}n-m_{\psi}m$ $\displaystyle\beta=P_{v}R_{v}\tau+m_{\phi}m-m_{\psi}n$

with $\tilde{\omega}^{2}=\omega^{2}-P_{v}^{2}$ and

$\sigma={c_{1}^{2}c_{5}^{2}c_{P}^{2}-s_{1}^{2}s_{5}^{2}s_{P}^{2}\over
c_{1}s_{1}c_{5}s_{5}}>0\quad,\quad\tau={(c_{1}^{2}c_{5}^{2}-s_{1}^{2}s_{5}^{2})c_{P}s_{P}\over
c_{1}s_{1}c_{5}s_{5}}>0$ (207)

From eq. (205) we can find the dictionary with an $N_{f}=(0,2)$ theory viz.

${q\over\hbar^{2}}=\frac{\kappa^{2}}{4}\quad,\quad{u\over\hbar^{2}}=\frac{\nu^{2}-\kappa^{2}}{4}\quad,\quad{m_{1}\over\hbar}=\frac{\alpha+i\beta}{2}\quad,\quad{m_{2}\over\hbar}=\frac{\alpha-i\beta}{2}\quad;\quad
y_{SW}=z$ (208)

The angular equation reads

$\frac{1}{\sin 2\theta}\frac{d}{d\theta}\left(\sin
2\theta\frac{d\Theta}{d\theta}\right)+\left[K^{2}+\tilde{\omega}^{2}(a_{2}^{2}\cos^{2}\theta+a_{1}^{2}\sin^{2}\theta)-{m_{\psi}^{2}\over\cos^{2}\theta}-{m_{\phi}^{2}\over\cos^{2}\theta}\right]\Theta=0$
(209)

that can be put in canonical form by setting $\xi=\cos^{2}\theta$ and

$\Theta=\frac{S(\xi)}{\sqrt{\xi(1-\xi)}}$ (210)

the $Q$-function reads

$Q_{\text{JMaRT}}^{\theta}={1+K^{2}\xi(1-\xi)-m^{2}_{\phi}\xi-m^{2}_{\psi}(1-\xi)+\tilde{\omega}^{2}\xi(1-\xi)[a_{2}^{2}\xi+a_{1}^{2}(1-\xi)]\over
4\xi^{2}(1-\xi)^{2}}$ (211)

Again, the above potential matches the one of the $N_{f}=(0,2)$ theory with
the following dictionary

${q^{\xi}\over\hbar^{2}}=-\frac{(a_{1}^{2}{-}a_{2}^{2})\tilde{\omega}^{2}}{4}\quad,\quad{u^{\xi}\over\hbar^{2}}=\frac{1{+}K^{2}}{4}+\frac{a_{2}^{2}\tilde{\omega}^{2}}{4}\quad,\quad{m_{1,2}^{\xi}\over\hbar}=\frac{m_{\phi}{\pm}m_{\psi}}{2}\quad;\quad
y_{SW}^{\xi}=-\xi$ (212)

The GMS geometry GMS1 ; GMS2  can be obtained from the JMaRT geometry setting
$m=n+1$.

It is worth mentioning that the analysis performed in Cardoso:2005gj ;
Chakrabarty:2015foa ; Eperon:2016cdd ; Chakrabarty:2019ujg  on the spectrum of
QNM in either the eikonal limit or when $\tilde{\omega}=0$ corresponds, in the
gauge theory, to $q$ going to infinity or zero, respectively. As for the
extremal KN case discussed in section 5.3.2 an analytic derivation of the QNMs
can be provided, which completely agrees with the quantization of the $a_{D}$
cycle on the gauge theory side, by solving the wave equation in different
regions and imposing matching conditions and the correct boundary conditions.

##  7 Conclusions and outlook

Let us conclude by summarising the results reached by the present
investigation and speculating about the origin of the QNM-SW gauge/gravity
correspondence.

After reviewing the three available approaches i.e. WKB/geodesics, numerical
methods à la Leaver and quantum SW curves, we have exploited them to compute
the QNMs of massless scalar perturbations of KN BHs with arbitrary charge
$\mathcal{Q}$ and angular momentum
$\mathcal{J}=a_{{}_{\mathcal{J}}}\mathcal{M}$. The values are displayed in the
plots in figures 2, 3, 4 and in the tables in Appendix C. The agreement with
the numerical results is remarkable. Even more remarkable is the agreement
with the geodesics/WKB approach for low values of the charges where the semi-
classical approximation is not fully justified.

Moreover we have illustrated the procedure for various toy models: the
’inverted hydrogen atom’, ’spherical harmonics’ , static KN waves, and near
super-radiant modes of quasi-extremal KN BHs. These cases are related to free
gauge theories and admit an analytic solution allowing us to streamline the
prescription to identify the cycle $\gamma$ whose period has to be quantized
with the one shrinking to zero size in the classical limit $\hbar\rightarrow
0$.

We have also shown how different gauge theories, e.g. with the same $SU(2)$
group but different number of flavours, may capture the same QNMs. We
exploited this to rewrite differential equations with two irregular singular
points (aka doubly-confluent Heun equation DCHE) as differential equations
with two regular singular points and one irregular singular point (aka
confluent Heun equation CHE). The prototypical case with $N_{f}=4$ that
capture the QNMs of KN-AdS BH is governed by the ’standard’ Heun equation with
four regular singular points.

Finally we have established the detailed gauge / gravity dictionaries for
various classes of BHs, branes and fuzz balls in diverse dimensions. Quite
surprisingly the elliptic geometry associated to an $SU(2)$ gauge group
combined with various choices of flavours is sufficient to encompass systems
ranging from (intersecting) D3-branes and their BPS bound-states, CCLP BHs in
$D=5$ and their extremal limits, D1-D5 circular fuzz balls, JMaRT smooth
horizonless geometries and their BPS limit aka GMS. The computation of their
QNMs following the steps described earlier on looks feasible though somewhat
laborious. Each case deserves a dedicated analysis.

In this work we exploit the mathematical equivalence among a priori completely
unrelated systems, a four-dimensional gauge theory and a BH (or a brane)
solution. M-theory may provide a hint on the physical and geometrical origin
of this correspondence. Gauge theories with $\mathcal{N}=2$ supersymmetry on a
NS background can be realized by wrapping M5-branes on Riemann surfaces that
coincide with the SW curves themselves embedded on a non-commutative space
Fucito:2011pn . It is tempting to speculate that the very same Riemann surface
representing a homological 2-cycle in a local CY geometry, wrapped by a stack
of M2-branes may provide the M-theory description of the corresponding BHs.
The two systems are related by electromagnetic duality in eleven dimensions.
The near photon-sphere dynamics would be captured by some CFT that might well
be the AGT dual of the $\mathcal{N}=2$ SYM theory, i.e. the reduction of the
M5 brane theory along the four-dimensional gauge theory spacetime effectively
compactified by the presence of the $\Omega$-background. For extremal Kerr and
KN BHs a form of the holographic correspondence between near-horizon extremal
Kerr (NHEK) and 2-d chiral CFT Guica:2008mu ; Compere:2012jk  has already been
exploited in the study of (metric) perturbations in Kerr-AdS and its near-
horizon geometry Dias:2012pp . Further investigation is required to push these
ideas onto firmer grounds.

We remark that the methods of the present investigation allow to compute not
only the spectrum of QNMs but also other physical properties of these
geometries, that can help to discriminate between BHs and smooth horizonless
compact objects.

## Acknowledgments

We would like to thank A. Aldi, C. Argento, G. Bonelli, V. Cardoso, G. Di
Russo, D. Fioravanti, M. Firrotta, F. Fucito, A. Grassi, T. Ikeda, C. Iossa,
M. Mariño, D. Panea Lichtig, P. Pani, G. Raposo, R. Savelli, A. Tanzini and Y.
Zenkevich for interesting discussions and valuable suggestions.

##  Appendix A The one-loop prepotential

The one-loop prepotential for $SU(2)$ gauge theory with $N_{f}=4$ fundamentals
is given by

$\displaystyle{\cal F}_{\rm 1{-}loop}(a)$ $\displaystyle=$
$\displaystyle\epsilon_{1}\,\epsilon_{2}\log{\prod_{i=1}^{4}\Gamma_{2}\left(m_{i}-a+{\epsilon\over
2}\right)\Gamma_{2}\left(m_{i}+a+{\epsilon\over
2}\right)\over\Gamma_{2}(2a+\epsilon)\Gamma_{2}(-2a+\epsilon)}$ (213)

with $\Gamma_{2}$ the Barnes double gamma function defined by the
integral131313In the following and in the main text we used the shorthand
notation $\Gamma_{2}(x)$ when it is not necessary to specify its dependence on
the $\epsilon_{i}$. The related function
$\gamma_{\epsilon_{1},\epsilon_{2}}(x)=\Gamma_{2}(x+\epsilon)$ is often used
in the literature.

$\log\Gamma_{2}(x|\epsilon_{1},\epsilon_{2})={d\over
ds}\left({\Lambda^{s}\over\Gamma(s)}\int_{0}^{\infty}{dt\over
t}{t^{s}\,e^{-xt}\over(1-e^{-\epsilon_{1}t})(1-e^{-\epsilon_{2}t})}\right)\Big{|}_{s=0}$
(214)

and with $\epsilon=\epsilon_{1}+\epsilon_{2}$. In the limit $\epsilon_{2}\to
0$ one finds

$\displaystyle\lim_{\epsilon_{2}\to
0}\epsilon_{2}\partial_{x}\log\Gamma_{2}(x|\epsilon_{1},\epsilon_{2})$
$\displaystyle=$ $\displaystyle-{d\over
ds}\left({\Lambda^{s}\over\Gamma(s)}\int_{0}^{\infty}{dt\over
t}{t^{s}\,e^{-xt}\over 1-e^{-\epsilon_{1}t}}\right)\Big{|}_{s=0}=-{d\over
ds}\left[\epsilon_{1}^{1-s}\Lambda^{s}\zeta_{s}\left({x\over\epsilon_{1}}\right)\right]_{s=0}$
(215) $\displaystyle=$
$\displaystyle-\epsilon_{1}\log\Gamma\left({x\over\epsilon_{1}}\right)-x\log\left({\epsilon_{1}\over\Lambda}\right)+{\epsilon_{1}\over
2}\log\left({2\pi\epsilon_{1}\over\Lambda}\right)$

with $\zeta_{s}$ the Hurtwitz zeta function. Plugging this into (213) and
setting $\epsilon_{1}=\hbar$ one finds the NS one-loop prepotential

$\displaystyle{\partial{\cal F}_{\rm 1{-}loop}(a)\over\partial a}$
$\displaystyle=$ $\displaystyle
2\hbar\log{\Gamma\left(1+{2a\over\hbar}\right)\over\Gamma\left(1-{2a\over\hbar}\right)}+\hbar\sum_{i=1}^{4}\log{\Gamma\left({1\over
2}+{m_{i}-a\over\hbar}\right)\over\Gamma\left({1\over
2}+{m_{i}+a\over\hbar}\right)}$ (216)

##  Appendix B $U(y)$ vs ${\widetilde{U}}(x)$ at $q=0$

In this appendix we show how the wave functions $U(y)$ and
${\widetilde{U}}(x)$ are related in the free gauge theory case where all the
calculations can be performed analytically.

We first solve (82) at $q=0$

$U^{\prime\prime}(y)+\frac{y\hbar-
m_{1}-m_{2}}{y(y+1)\hbar}\,U^{\prime}(y)+\frac{(m_{1}+\tfrac{\hbar}{2})(m_{2}+\tfrac{\hbar}{2})-uy}{y^{2}(y+1)\hbar^{2}}\,U(y)=0$
(217)

The general solutions are hypergeometric functions

$\displaystyle U(y)=$
$\displaystyle\,d_{1}\,y^{\frac{1}{2}+\frac{m_{1}}{\hbar}}\,{}_{2}F_{1}(\tfrac{1}{2}+\tfrac{m_{1}-\sqrt{u}}{\hbar},\tfrac{1}{2}+\tfrac{m_{1}+\sqrt{u}}{\hbar},1+\tfrac{m_{1}-m_{2}}{\hbar}|-y)$
(218) $\displaystyle+$
$\displaystyle\,d_{2}\,y^{\frac{1}{2}+\frac{m_{2}}{\hbar}}\,{}_{2}F_{1}(\tfrac{1}{2}+\tfrac{m_{2}-\sqrt{u}}{\hbar},\tfrac{1}{2}+\tfrac{m_{2}+\sqrt{u}}{\hbar},1+\tfrac{m_{2}-m_{1}}{\hbar}|-y)$

where $d_{1,2}$ are integration constants. Let us see consider now the
difference equation (68). For $q=0$, the difference equation reduces to

$W(x)=-\frac{1}{P_{0}(x)}$ (219)

leading to

$\widetilde{U}(x+\hbar)=-{P_{0}(x)\over
P_{R}(x+{\textstyle\frac{\hbar}{2}})}\widetilde{U}(x)$ (220)

The solution can be written in terms of Gamma functions as

$\widetilde{U}(x)=\frac{(-1)^{\frac{x}{\hbar}}\Gamma(\tfrac{x-\sqrt{u}}{\hbar})\Gamma(\tfrac{x+\sqrt{u}}{\hbar})}{\Gamma(\tfrac{1}{2}+\tfrac{x-m_{1}}{\hbar})\Gamma(\frac{1}{2}+\frac{x-m_{2}}{\hbar})}$
(221)

up to an overall constant. The connection with the wave function $U(y)$ is
given by the Laplace transform of $\widetilde{U}(x)$, defined as

$U(y)=\frac{1}{2\pi i}\int_{\gamma}y^{\frac{x}{\hbar}}\,\widetilde{U}(x)\,dx$
(222)

with $\gamma$ a contour enclosing the two series of poles: $x=\sqrt{u}-\hbar
n$ and $x=-\sqrt{u}-\hbar n$. Computing the residues and summing over $n$ one
finds

$\frac{(-y)^{\frac{\sqrt{u}}{\hbar}}\Gamma(\frac{2\sqrt{u}}{\hbar}){}_{2}F_{1}\left(\tfrac{1}{2}{+}\tfrac{m_{1}{-}\sqrt{u}}{\hbar},\tfrac{1}{2}{+}\tfrac{m_{2}{-}\sqrt{u}}{\hbar},1{-}\tfrac{2\sqrt{u}}{\hbar}|{-}y^{-1}\right)}{\Gamma(\frac{1}{2}{+}\frac{\sqrt{u}-m_{1}}{\hbar})\Gamma(\frac{1}{2}{+}\frac{\sqrt{u}-m_{2}}{\hbar})}+(\sqrt{u}\to{-}\sqrt{u})$
(223)

that after using hypergeometric identities matches precisely (218) for a
constant $d_{1}=d_{2}$.

##  Appendix C Tables

In this appendix we show the QNMs of scalar perturbations of the metric with
$\ell=m_{\phi}=2$ and $n=0,1$. We start with the Reissner-Nordström BH, then
move to the generic Kerr-Newman BH with ${\cal Q}=0.5\,{\cal M}$ and
$a_{{}_{\mathcal{J}}}=0.5\,{\cal M}$. The BH mass ${\cal M}$ is always set to
one.

In the tables we show the results of the computation performed using three
methods: the geodesic approximation (described in section 2.3), then the
quantization of the WKB cycle (using the SW prepotential with fourth
instantons contributions, as shown in section 3.5), and finally Leaver’s
numerical method (illustrated in section 4).

###  C.1 Reissner-Nordström BH

$n_{\text{geo}}=0$

$\begin{array}[]{||c||c|c|c||}\hline\cr\mathcal{Q}&\omega_{\text{geo}}&\omega_{\text{SW},4}&\omega_{\text{Num}}\\\
\hline\cr 0&0.471-0.0962i&0.476-0.0958i&0.484-0.0968i\\\ \hline\cr
0.1&0.472-0.0963i&0.477-0.0959i&0.484-0.0968i\\\ \hline\cr
0.3&0.479-0.0967i&0.484-0.0966i&0.491-0.0972i\\\ \hline\cr
0.6&0.504-0.0978i&0.511-0.0987i&0.517-0.0983i\\\ \hline\cr
0.9&0.567-0.0963i&0.581-0.106i&0.582-0.0966i\\\ \hline\cr\end{array}$ (224)

$n_{\text{geo}}=1$

$\begin{array}[]{||c||c|c|c||}\hline\cr\mathcal{Q}&\omega_{\text{geo}}&\omega_{\text{SW},4}&\omega_{\text{Num}}\\\
\hline\cr 0&0.471-0.289i&0.467-0.297i&0.464-0.296i\\\ \hline\cr
0.1&0.472-0.289i&0.467-0.297i&0.465-0.296i\\\ \hline\cr
0.3&0.479-0.29i&0.475-0.298i&0.472-0.297i\\\ \hline\cr
0.6&0.504-0.294i&0.502-0.299i&0.499-0.299i\\\ \hline\cr
0.9&0.567-0.289i&0.567-0.292i&0.567-0.293i\\\ \hline\cr\end{array}$ (225)

###  C.2 Kerr-Newman BH: ${\cal Q}=0.5\,{\cal M}$

$n_{\text{geo}}=0$

$\begin{gathered}\begin{array}[]{||c||c|c|c||}\hline\cr
a_{{}_{\mathcal{J}}}&\omega_{\text{Geo}}&\omega_{\text{SW},4}&\omega_{\text{Num}}\\\
\hline\cr 0&0.493-0.0974i&0.497-0.105i&0.506-0.0979i\\\ \hline\cr
0.1&0.511-0.0972i&0.514-0.103i&0.524-0.0977i\\\ \hline\cr
0.2&0.531-0.0967i&0.533-0.0997i&0.545-0.0972i\\\ \hline\cr
0.3&0.555-0.0957i&0.553-0.0952i&0.569-0.0963i\\\ \hline\cr
0.4&0.582-0.0942i&0.577-0.089i&0.596-0.0947i\\\ \hline\cr\end{array}\\\
\begin{array}[]{||c||c|c|c||}\hline\cr
a_{{}_{\mathcal{J}}}&A_{\text{geo}}&A_{\text{SW},4}&A_{\text{Num}}\\\
\hline\cr 0&6.&6.&6.\\\ \hline\cr
0.1&6.+0.000248i&6.+0.000142i&6.+0.000146i\\\ \hline\cr
0.2&6.+0.00103i&6.+0.000586i&6.+0.000605i\\\ \hline\cr
0.3&5.99+0.0024i&6.+0.00135i&6.+0.00141i\\\ \hline\cr
0.4&5.99+0.00442i&5.99+0.00243i&5.99+0.00259i\\\
\hline\cr\end{array}\end{gathered}$ (226)

$n_{\text{geo}}=1$

$\begin{gathered}\begin{array}[]{||c||c|c|c||}\hline\cr
a_{{}_{\mathcal{J}}}&\omega_{\text{Geo}}&\omega_{\text{SW},4}&\omega_{\text{Num}}\\\
\hline\cr 0&0.493-0.292i&0.49-0.299i&0.487-0.299i\\\ \hline\cr
0.1&0.511-0.292i&0.51-0.298i&0.507-0.298i\\\ \hline\cr
0.2&0.531-0.29i&0.532-0.297i&0.529-0.296i\\\ \hline\cr
0.3&0.555-0.287i&0.558-0.294i&0.554-0.292i\\\ \hline\cr
0.4&0.582-0.283i&0.588-0.289i&0.584-0.287i\\\ \hline\cr\end{array}\\\
\begin{array}[]{||c||c|c|c||}\hline\cr
a_{{}_{\mathcal{J}}}&A_{\text{geo}}&A_{\text{SW},4}&A_{\text{Num}}\\\
\hline\cr 0&6.&6.&6.\\\ \hline\cr
0.1&6.+0.000248i&6.+0.000142i&6.+0.000146i\\\ \hline\cr
0.2&6.+0.00103i&6.+0.000586i&6.+0.000605i\\\ \hline\cr
0.3&5.99+0.0024i&6.+0.00135i&6.+0.00141i\\\ \hline\cr
0.4&5.99+0.00442i&5.99+0.00243i&5.99+0.00259i\\\
\hline\cr\end{array}\end{gathered}$ (227)

###  C.3 Kerr-Newman BH: $a_{{}_{\cal J}}=0.5\,{\cal M}$

$n_{\text{geo}}=0$

$\begin{gathered}\begin{array}[]{||c||c|c|c||}\hline\cr\mathcal{Q}&\omega_{\text{geo}}&\omega_{\text{SW},4}&\omega_{\text{Num}}\\\
\hline\cr 0&0.572-0.0929i&0.561-0.0807i&0.586-0.0935i\\\ \hline\cr
0.1&0.574-0.0929i&0.563-0.0807i&0.587-0.0935i\\\ \hline\cr
0.2&0.578-0.0929i&0.567-0.0808i&0.592-0.0934i\\\ \hline\cr
0.3&0.586-0.0927i&0.574-0.0809i&0.6-0.0933i\\\ \hline\cr
0.4&0.598-0.0924i&0.585-0.0811i&0.612-0.093i\\\ \hline\cr\end{array}\\\
\begin{array}[]{||c||c|c|c||}\hline\cr\mathcal{Q}&A_{\text{geo}}&A_{\text{SW},4}&A_{\text{Num}}\\\
\hline\cr 0&5.98+0.00672i&5.99+0.00357i&5.99+0.00392i\\\ \hline\cr
0.1&5.98+0.00674i&5.99+0.00357i&5.99+0.00393i\\\ \hline\cr
0.2&5.98+0.00679i&5.99+0.00359i&5.99+0.00396i\\\ \hline\cr
0.3&5.98+0.00687i&5.99+0.00363i&5.99+0.00401i\\\ \hline\cr
0.4&5.98+0.00699i&5.99+0.00367i&5.99+0.00408i\\\
\hline\cr\end{array}\end{gathered}$ (228)

$n_{\text{geo}}=1$

$\begin{gathered}\begin{array}[]{||c||c|c|c||}\hline\cr\mathcal{Q}&\omega_{\text{geo}}&\omega_{\text{SW},4}&\omega_{\text{Num}}\\\
\hline\cr 0&0.572-0.279i&0.578-0.287i&0.573-0.283i\\\ \hline\cr
0.1&0.574-0.279i&0.579-0.287i&0.575-0.283i\\\ \hline\cr
0.2&0.578-0.279i&0.584-0.287i&0.58-0.283i\\\ \hline\cr
0.3&0.586-0.278i&0.593-0.286i&0.588-0.283i\\\ \hline\cr
0.4&0.598-0.277i&0.606-0.285i&0.601-0.281i\\\ \hline\cr\end{array}\\\
\begin{array}[]{||c||c|c|c||}\hline\cr\mathcal{Q}&A_{\text{geo}}&A_{\text{SW},4}&A_{\text{Num}}\\\
\hline\cr 0&5.98+0.0202i&5.99+0.011i&5.99+0.0116i\\\ \hline\cr
0.1&5.98+0.0202i&5.99+0.0111i&5.99+0.0117i\\\ \hline\cr
0.2&5.98+0.0204i&5.99+0.0111i&5.99+0.0117i\\\ \hline\cr
0.3&5.98+0.0206i&5.99+0.0112i&5.99+0.0119i\\\ \hline\cr
0.4&5.98+0.021i&5.99+0.0114i&5.99+0.0121i\\\
\hline\cr\end{array}\end{gathered}$ (229)

## References

  * (1) M. Bianchi, D. Consoli, A. Grillo and J. F. Morales, _QNMs of branes, BHs and fuzzballs from Quantum SW geometries_ , 2105.04245.
  * (2) V. Cardoso, A. S. Miranda, E. Berti, H. Witek and V. T. Zanchin, _Geodesic stability, Lyapunov exponents and quasinormal modes_ , _Phys. Rev. D_ 79 (2009) 064016, [0812.1806].
  * (3) B. Mashhoon, _Stability of charged rotating black holes in the eikonal approximation_ , _Phys. Rev. D_ 31 (1985) 290–293.
  * (4) B. F. Schutz and C. M. Will, _BLACK HOLE NORMAL MODES: A SEMIANALYTIC APPROACH_ , _Astrophys. J. Lett._ 291 (1985) L33–L36.
  * (5) S. Iyer and C. M. Will, _Black Hole Normal Modes: A WKB Approach. 1. Foundations and Application of a Higher Order WKB Analysis of Potential Barrier Scattering_ , _Phys. Rev. D_ 35 (1987) 3621.
  * (6) H. Yang, D. A. Nichols, F. Zhang, A. Zimmerman, Z. Zhang and Y. Chen, _Quasinormal-mode spectrum of Kerr black holes and its geometric interpretation_ , _Phys. Rev. D_ 86 (2012) 104006, [1207.4253].
  * (7) M. Bianchi, A. Grillo and J. F. Morales, _Chaos at the rim of black hole and fuzzball shadows_ , _JHEP_ 05 (2020) 078, [2002.05574].
  * (8) M. Bianchi, D. Consoli, A. Grillo and J. F. Morales, _Light rings of five-dimensional geometries_ , _JHEP_ 03 (2021) 210, [2011.04344].
  * (9) T. Ikeda, M. Bianchi, D. Consoli, A. Grillo, J. F. Morales, P. Pani et al., _Black-hole microstate spectroscopy: Ringdown, quasinormal modes, and echoes_ , _Phys. Rev. D_ 104 (2021) 066021, [2103.10960].
  * (10) M. Bianchi, D. Consoli, A. Grillo, J. F. Morales, P. Pani and G. Raposo, _Distinguishing fuzzballs from black holes through their multipolar structure_ , _Phys. Rev. Lett._ 125 (2020) 221601, [2007.01743].
  * (11) M. Bianchi, D. Consoli, A. Grillo, J. F. Morales, P. Pani and G. Raposo, _The multipolar structure of fuzzballs_ , _JHEP_ 01 (2021) 003, [2008.01445].
  * (12) I. Bena and D. R. Mayerson, _Multipole Ratios: A New Window into Black Holes_ , _Phys. Rev. Lett._ 125 (2020) 22, [2006.10750].
  * (13) I. Bena and D. R. Mayerson, _Black Holes Lessons from Multipole Ratios_ , _JHEP_ 03 (2021) 114, [2007.09152].
  * (14) I. Bah, I. Bena, P. Heidmann, Y. Li and D. R. Mayerson, _Gravitational Footprints of Black Holes and Their Microstate Geometries_ , 2104.10686.
  * (15) T. Regge and J. A. Wheeler, _Stability of a Schwarzschild singularity_ , _Phys. Rev._ 108 (1957) 1063–1069.
  * (16) F. J. Zerilli, _Effective potential for even parity Regge-Wheeler gravitational perturbation equations_ , _Phys. Rev. Lett._ 24 (1970) 737–738.
  * (17) S. A. Teukolsky, _Rotating black holes - separable wave equations for gravitational and electromagnetic perturbations_ , _Phys. Rev. Lett._ 29 (1972) 1114–1118.
  * (18) E. W. Leaver, _An Analytic representation for the quasi normal modes of Kerr black holes_ , _Proc. Roy. Soc. Lond. A_ 402 (1985) 285–298.
  * (19) E. W. Leaver, _Quasinormal modes of Reissner-Nordstrom black holes_ , _Phys. Rev. D_ 41 (1990) 2986–2997.
  * (20) A. Mironov and A. Morozov, _Nekrasov Functions and Exact Bohr-Zommerfeld Integrals_ , _JHEP_ 04 (2010) 040, [0910.5670].
  * (21) Y. Zenkevich, _Nekrasov prepotential with fundamental matter from the quantum spin chain_ , _Phys. Lett. B_ 701 (2011) 630–639, [1103.4843].
  * (22) J.-E. Bourgine and D. Fioravanti, _Quantum integrability of $\mathcal{N}=2$ 4d gauge theories_, _JHEP_ 08 (2018) 125, [1711.07935].
  * (23) D. Fioravanti and D. Gregori, _Integrability and cycles of deformed ${\cal N}=2$ gauge theory_, _Phys. Lett. B_ 804 (2020) 135376, [1908.08030].
  * (24) A. Grassi and M. Mariño, _A Solvable Deformation of Quantum Mechanics_ , _SIGMA_ 15 (2019) 025, [1806.01407].
  * (25) A. Grassi, J. Gu and M. Mariño, _Non-perturbative approaches to the quantum Seiberg-Witten curve_ , _JHEP_ 07 (2020) 106, [1908.07065].
  * (26) N. Seiberg and E. Witten, _Electric - magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory_ , _Nucl. Phys. B_ 426 (1994) 19–52, [hep-th/9407087].
  * (27) N. Seiberg and E. Witten, _Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD_ , _Nucl. Phys. B_ 431 (1994) 484–550, [hep-th/9408099].
  * (28) N. A. Nekrasov and S. L. Shatashvili, _Quantization of Integrable Systems and Four Dimensional Gauge Theories_ , in _16th International Congress on Mathematical Physics_ , pp. 265–289, 8, 2009.  0908.4052.  DOI.
  * (29) G. Aminov, A. Grassi and Y. Hatsuda, _Black Hole Quasinormal Modes and Seiberg-Witten Theory_ , 2006.06111.
  * (30) F. Novaes and B. Carneiro da Cunha, _Isomonodromy, Painlevé transcendents and scattering off of black holes_ , _JHEP_ 07 (2014) 132, [1404.5188].
  * (31) B. Carneiro da Cunha and F. Novaes, _Kerr Scattering Coefficients via Isomonodromy_ , _JHEP_ 11 (2015) 144, [1506.06588].
  * (32) B. Carneiro da Cunha and F. Novaes, _Kerr–de Sitter greybody factors via isomonodromy_ , _Phys. Rev. D_ 93 (2016) 024045, [1508.04046].
  * (33) J. B. Amado, B. Carneiro da Cunha and E. Pallante, _On the Kerr-AdS/CFT correspondence_ , _JHEP_ 08 (2017) 094, [1702.01016].
  * (34) M. Lencsés and F. Novaes, _Classical Conformal Blocks and Accessory Parameters from Isomonodromic Deformations_ , _JHEP_ 04 (2018) 096, [1709.03476].
  * (35) J. Barragán Amado, B. Carneiro Da Cunha and E. Pallante, _Scalar quasinormal modes of Kerr-AdS 5_, _Phys. Rev. D_ 99 (2019) 105006, [1812.08921].
  * (36) F. Novaes, C. Marinho, M. Lencsés and M. Casals, _Kerr-de Sitter Quasinormal Modes via Accessory Parameter Expansion_ , _JHEP_ 05 (2019) 033, [1811.11912].
  * (37) B. Carneiro da Cunha and J. a. P. Cavalcante, _Confluent conformal blocks and the Teukolsky master equation_ , _Phys. Rev. D_ 102 (2020) 105013, [1906.10638].
  * (38) J. B. Amado, B. Carneiro da Cunha and E. Pallante, _Vector perturbations of Kerr-AdS 5 and the Painlevé VI transcendent_, _JHEP_ 04 (2020) 155, [2002.06108].
  * (39) M. Bershtein, P. Gavrylenko and A. Grassi, _Quantum spectral problems and isomonodromic deformations_ , 2105.00985.
  * (40) J. Barragán Amado, B. Carneiro da Cunha and E. Pallante, _Remarks on holographic models of the Kerr-AdS 5 geometry_, _JHEP_ 05 (2021) 251, [2102.02657].
  * (41) J. a. P. Cavalcante and B. C. da Cunha, _Scalar/Dirac perturbations of the Reissner-Nordström black hole and Painlevé transcendents_ , 2109.06929.
  * (42) B. C. da Cunha and J. a. P. Cavalcante, _Teukolsky master equation and Painlevé transcendents: Numerics and extremal limit_ , _Phys. Rev. D_ 104 (2021) 084051, [2105.08790].
  * (43) J. B. Amado, B. C. da Cunha and E. Pallante, _QNMs of scalar fields on small Reissner-Nordström-AdS 5 black holes_, 2110.08349.
  * (44) L. F. Alday, D. Gaiotto and Y. Tachikawa, _Liouville Correlation Functions from Four-dimensional Gauge Theories_ , _Lett. Math. Phys._ 91 (2010) 167–197, [0906.3219].
  * (45) G. Bonelli, C. Iossa, D. P. Lichtig and A. Tanzini, _Exact solution of Kerr black hole perturbations via CFT 2 and instanton counting_, 2105.04483.
  * (46) D. Gaiotto, _N=2 dualities_ , _JHEP_ 08 (2012) 034, [0904.2715].
  * (47) M. Cvetic and D. Youm, _Dyonic BPS saturated black holes of heterotic string on a six torus_ , _Phys. Rev. D_ 53 (1996) 584–588, [hep-th/9507090].
  * (48) Z. W. Chong, M. Cvetic, H. Lu and C. N. Pope, _General non-extremal rotating black holes in minimal five-dimensional gauged supergravity_ , _Phys. Rev. Lett._ 95 (2005) 161301, [hep-th/0506029].
  * (49) Z. W. Chong, M. Cvetic, H. Lu and C. N. Pope, _Five-dimensional gauged supergravity black holes with independent rotation parameters_ , _Phys. Rev. D_ 72 (2005) 041901, [hep-th/0505112].
  * (50) O. Lunin and S. D. Mathur, _Metric of the multiply wound rotating string_ , _Nucl. Phys. B_ 610 (2001) 49–76, [hep-th/0105136].
  * (51) V. Jejjala, O. Madden, S. F. Ross and G. Titchener, _Non-supersymmetric smooth geometries and D1-D5-P bound states_ , _Phys. Rev. D_ 71 (2005) 124030, [hep-th/0504181].
  * (52) S. Giusto, S. D. Mathur and A. Saxena, _Dual geometries for a set of 3-charge microstates_ , _Nucl. Phys. B_ 701 (2004) 357–379, [hep-th/0405017].
  * (53) S. Giusto, S. D. Mathur and A. Saxena, _3-charge geometries and their CFT duals_ , _Nucl. Phys. B_ 710 (2005) 425–463, [hep-th/0406103].
  * (54) H. Suzuki, E. Takasugi and H. Umetsu, _Perturbations of Kerr-de Sitter black hole and Heun’s equations_ , _Prog. Theor. Phys._ 100 (1998) 491–505, [gr-qc/9805064].
  * (55) A. Zhidenko, _Quasinormal modes of Schwarzschild de Sitter black holes_ , _Class. Quant. Grav._ 21 (2004) 273–280, [gr-qc/0307012].
  * (56) M. Giammatteo and I. G. Moss, _Gravitational quasinormal modes for Kerr anti-de Sitter black holes_ , _Class. Quant. Grav._ 22 (2005) 1803–1824, [gr-qc/0502046].
  * (57) M. M. Caldarelli, G. Cognola and D. Klemm, _Thermodynamics of Kerr-Newman-AdS black holes and conformal field theories_ , _Class. Quant. Grav._ 17 (2000) 399–420, [hep-th/9908022].
  * (58) E. Witten, _Solutions of four-dimensional field theories via M theory_ , _Nucl. Phys. B_ 500 (1997) 3–42, [hep-th/9703166].
  * (59) R. Poghossian, _Deforming SW curve_ , _JHEP_ 04 (2011) 033, [1006.4822].
  * (60) F. Fucito, J. F. Morales, D. R. Pacifici and R. Poghossian, _Gauge theories on $\Omega$-backgrounds from non commutative Seiberg-Witten curves_, _JHEP_ 05 (2011) 098, [1103.4495].
  * (61) N. A. Nekrasov, _Seiberg-Witten prepotential from instanton counting_ , _Adv. Theor. Math. Phys._ 7 (2003) 831–864, [hep-th/0206161].
  * (62) M. Matone, _Instantons and recursion relations in N=2 SUSY gauge theory_ , _Phys. Lett. B_ 357 (1995) 342–348, [hep-th/9506102].
  * (63) R. Flume, F. Fucito, J. F. Morales and R. Poghossian, _Matone’s relation in the presence of gravitational couplings_ , _JHEP_ 04 (2004) 008, [hep-th/0403057].
  * (64) A. A. Belavin, A. M. Polyakov and A. B. Zamolodchikov, _Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory_ , _Nucl. Phys. B_ 241 (1984) 333–380.
  * (65) H. Onozawa, T. Mishima, T. Okamura and H. Ishihara, _Quasinormal modes of maximally charged black holes_ , _Phys. Rev. D_ 53 (1996) 7033–7040, [gr-qc/9603021].
  * (66) T. Binnington and E. Poisson, _Relativistic theory of tidal Love numbers_ , _Phys. Rev. D_ 80 (2009) 084018, [0906.1366].
  * (67) T. Damour and A. Nagar, _Relativistic tidal properties of neutron stars_ , _Phys. Rev. D_ 80 (2009) 084035, [0906.0096].
  * (68) A. Le Tiec, M. Casals and E. Franzin, _Tidal Love Numbers of Kerr Black Holes_ , _Phys. Rev. D_ 103 (2021) 084021, [2010.15795].
  * (69) R. Brito, V. Cardoso and P. Pani, _Superradiance: New Frontiers in Black Hole Physics_ , _Lect. Notes Phys._ 906 (2015) pp.1–237, [1501.06570].
  * (70) A. A. Starobinskil and S. M. Churilov, _Amplification of electromagnetic and gravitational waves scattered by a rotating ”black hole”_ , _Sov. Phys. JETP_ 65 (1974) 1–5.
  * (71) S. A. Teukolsky and W. H. Press, _Perturbations of a rotating black hole. III - Interaction of the hole with gravitational and electromagnet ic radiation_ , _Astrophys. J._ 193 (1974) 443–461.
  * (72) H. Yang, A. Zimmerman, A. Zenginoğlu, F. Zhang, E. Berti and Y. Chen, _Quasinormal modes of nearly extremal Kerr spacetimes: spectrum bifurcation and power-law ringdown_ , _Phys. Rev. D_ 88 (2013) 044047, [1307.8086].
  * (73) S. Hod, _Quasinormal resonances of near-extremal Kerr-Newman black holes_ , _Phys. Lett. B_ 666 (2008) 483–485, [0810.5419].
  * (74) S. Hod, _Resonance spectrum of near-extremal Kerr black holes in the eikonal limit_ , _Phys. Lett. B_ 715 (2012) 348–351, [1207.5282].
  * (75) J. Maldacena, S. H. Shenker and D. Stanford, _A bound on chaos_ , _JHEP_ 08 (2016) 106, [1503.01409].
  * (76) J. C. Breckenridge, R. C. Myers, A. W. Peet and C. Vafa, _D-branes and spinning black holes_ , _Phys. Lett. B_ 391 (1997) 93–98, [hep-th/9602065].
  * (77) M. Cvetic, H. Lu and J. F. Vazquez-Poritz, _Absorption by extremal D3-branes_ , _JHEP_ 02 (2001) 012, [hep-th/0002128].
  * (78) S. S. Gubser and A. Hashimoto, _Exact absorption probabilities for the D3-brane_ , _Commun. Math. Phys._ 203 (1999) 325–340, [hep-th/9805140].
  * (79) Y. Kurita and M.-a. Sakagami, _Quasinormal modes of D3-brane black holes_ , _Phys. Rev. D_ 67 (2003) 024003, [hep-th/0208063].
  * (80) V. Cardoso, O. J. C. Dias, J. L. Hovdebo and R. C. Myers, _Instability of non-supersymmetric smooth geometries_ , _Phys. Rev. D_ 73 (2006) 064031, [hep-th/0512277].
  * (81) B. Chakrabarty, D. Turton and A. Virmani, _Holographic description of non-supersymmetric orbifolded D1-D5-P solutions_ , _JHEP_ 11 (2015) 063, [1508.01231].
  * (82) F. C. Eperon, H. S. Reall and J. E. Santos, _Instability of supersymmetric microstate geometries_ , _JHEP_ 10 (2016) 031, [1607.06828].
  * (83) B. Chakrabarty, D. Ghosh and A. Virmani, _Quasinormal modes of supersymmetric microstate geometries from the D1-D5 CFT_ , _JHEP_ 10 (2019) 072, [1908.01461].
  * (84) M. Guica, T. Hartman, W. Song and A. Strominger, _The Kerr/CFT Correspondence_ , _Phys. Rev. D_ 80 (2009) 124008, [0809.4266].
  * (85) G. Compère, _The Kerr/CFT correspondence and its extensions_ , _Living Rev. Rel._ 15 (2012) 11, [1203.3561].
  * (86) O. J. C. Dias, J. E. Santos and M. Stein, _Kerr-AdS and its Near-horizon Geometry: Perturbations and the Kerr/CFT Correspondence_ , _JHEP_ 10 (2012) 182, [1208.3322].