# A new method for exact results on Quasinormal Modes of Black Holes

Davide Fioravanti  fioravanti at bo.infn.it Sezione INFN di Bologna and
Dipartimento di Fisica e Astronomia, Università di Bologna
Via Irnerio 46, 40126 Bologna, Italy  Daniele Gregori  daniele.gregori6 at
unibo.it also visiting NORDITA.  Sezione INFN di Bologna and Dipartimento di
Fisica e Astronomia, Università di Bologna
Via Irnerio 46, 40126 Bologna, Italy

###### Abstract

We develop a new method for writing simple exact equations characterizing
gravity solutions among which black holes and in particular the quasinormal
modes. More precisely, we derive the full system of functional and
Thermodynamic Bethe Ansatz non linear integral equations of quantum
integrability. In particular, we prove that the Quasinormal Modes verify
different equivalent exact quantization conditions and identify them with
Bethe roots. We numerically solve the integral equation and compare the
results with other methods. Eventually, we can definitely certify its
simplicity, accuracy and effectiveness. Furthermore, this method connects
different unexpected fields and paves the way for innovative ways of
investigations in gravity and gauge theories.

##  I I. Introduction, plan and goals

Thanks to the extraordinary discovery of gravitational waves in 2015
LIGOScientific:2016 , a new window for our understanding of the Universe has
opened. In fact, it has become possible to make progress in also fundamental
physics, testing General Relativity (GR) in extreme regimes and in particular
to discriminate between GR Black Holes (BHs) and Exotic Compact Objects (ECOs)
or Fuzzballs appearing in Modified Theories of Gravity or String Theory. This
is possible importantly by analysing the Quasinormal Modes (QNMs) that
characterize the linear perturbations of the metric and the fields, for
instance in the ringdown phase of BHs merging Cardoso:2017cqb ;
BianchiConsoliGrilloMorales:2021 ; Mayerson:2020 . Computing QNMs numerically
has been until now typically quite laborious and this has been also due to the
difficulties in developing exact analytic characterizations of QNMs. In this
direction, a significant improvement has been realized very recently as QNMs
have been identified by exact quantization conditions on dual periods of some
$\mathcal{N}=2$ supersymmetric gauge, i.e. deformed Seiberg-Witten (SW),
theories AminovGrassiHatsuda:2020 ; SeibergWitten:1994b ;
NekrasovShatashvili:2009 . On the latter we have in fact some exact control
and then this surprising SW-QNM correspondence – so dubbed in
BianchiConsoliGrilloMorales:2021b  – has already allowed to find many new
theoretical and computational results for BHs and other spacetime geometries
(cf. for instance CasalsCosta:2021 ; Hatsuda:2020Teukolsky ;
BonelliIossaLichtigTanzini:2021 ). An explanation of this correspondence has
been constructed in a rather general case BonelliIossaLichtigTanzini:2021  by
exploiting another correspondence between $\mathcal{N}=2$ gauge theory and
Conformal Field Theory AldayGaiottoTachikawa:2010 . However, we are going to
show that it is possible to explain the SW-QNM correspondence by analysing
closely the Ordinary Differential Equations (ODEs) describing the
perturbations in gravitational physics. We are able to do this on the basis of
our previous works FioravantiGregori:2019 ; FioravantiPoghossian:2019 , where
we have connected the $\mathcal{N}=2$ gauge theories to quantum integrable
theories, in particular the gauge periods to the Baxter’s $Q$ and $T$
functions. To this aim we have started from the ODEs characterizing the
periods and developed further the elegant ODE/IM correspondence between ODEs
and Integrable Models (IM) DoreyTateo1998 ; BazhanovLukyanovZamolodchikov2001
; DoreyTateo1999 ; FioravantiRossi:2021 .

Here we prove that QNMs are nothing but the zeros of the $Q$ function (Bethe
roots) and then find an entirely new set of functional and integral (non-
linear) equations for them (in particular quantization conditions). In
specific, we derive a new quantization condition involving the solution of a
celebrated non-linear integral equation, the Thermodynamic Bethe Ansatz (TBA)
ones. As a result, this way turns out to be a very simple and powerful way to
compute QNMs. In the light of these findings, we understand and develop here
the lines of generality in our construction FioravantiGregori:2019 ;
FioravantiPoghossian:2019  to put it into correspondence with gravitational
physics.

In this letter, after introducing QNMs in section II, in section III we
explain the Integrability/Gauge/Gravity correspondence in the simplest case of
the D3 brane spacetime, corresponding to our previous treatment of $SU(2)$
$N_{f}=0$ gauge theory via the self-dual Liouville theory
FioravantiGregori:2019 . To show the generality of the Integrability/Gravity
correspondence, in section IV we develop the same theory for the spacetime
given by the intersection of D3 branes, a generalization of extremal Reissner-
Nordström (RN) BHs, corresponding to $SU(2)$ $N_{f}=2$ gauge theory Ikeda:2021
; BianchiConsoliGrilloMorales:2021b  and a new IM (which reproduces known IMs
in some limits, cf. below). In section V we highlight the reasons for
applicability of our method and conclude that different features of many
Integrability/Gauge/Gravity cases can be treated in future perspective.

##  II II. Brief overview on QNMs

We recall the definition of quasinormal modes following Nollert:1999 . A
linear perturbation of a BH is a solution $\Phi(t,x)$ of some linear PDE
derived from the equations for the fields and metric. Its Laplace transform
$\hat{\Psi}(s,x)$, with $\Re s>0$, satisfies some ODE with non-homogeneous
term $\mathcal{I}(s,x)$, combination of $\Phi$ and its time derivative at
initial time. The corresponding homogeneous equation is exactly the ODE we are
going to study in the next sections, which we write here for general potential
$U(x)$

$\left\\{-\frac{\partial^{2}}{\partial
x^{2}}+U(x)+s^{2}\right\\}\Psi(s,x)=0\,.$ (1)

As usual, $x$ is the so called tortoise coordinate, such that the BH horizon
is put at $x\to-\infty$ and spacetime infinity at $x\to+\infty$. For decaying
potentials, the bounded solutions behave at $x\to\pm\infty$ respectively as

$\displaystyle\Psi_{\pm}(s,x)$ $\displaystyle\sim e^{\mp sx}\,\,.$ (2)

Via those known as Green functions, we can write the solution of the non-
homogenous equation $\hat{\Psi}(s,x)$ and then take its inverse Laplace
transform to obtain the original perturbation as

$\displaystyle\Phi(t,x)=$ (3)
$\displaystyle\sum_{n}e^{s_{n}t}\text{Res}\left(\frac{1}{W(s)}\right)\Biggr{|}_{s_{n}}\int_{-\infty}^{\infty}\Psi_{-}(s_{n},x_{<})\Psi_{+}(s_{n},x_{>})\mathcal{I}(s_{n},x^{\prime})\,dx^{\prime}$

where $x_{<}=\text{min}(x^{\prime},x)\,,x_{<}=\text{max}(x^{\prime},x)\,$. The
crucial point for us is that the perturbation is a sum over the residues of
the inverse wronskian of the regular solutions (2):

$W(s_{n})=W[\Psi_{+},\Psi_{-}]=0\,.$ (4)

Besides, condition (4) means that at these special points the two solutions
(2) (in general independent) become linearly dependent. By setting $s=i\omega$
we recover the usual intuitive definition of QNMs as the frequencies of plane
wave solutions both incoming at the horizon and outgoing at infinity. However,
as well explained in Nollert:1999 , this last definition is not mathematically
rigorous, since it would lead to diverging boundary conditions. Instead, the
QNMs $\omega_{n}$ have an imaginary part $\Im\omega_{n}<0$, so that the
perturbation they describe is damped to zero as $t\to\infty$.

##  III III. An example: the D3 brane

The D3 brane is described by the line element

$ds^{2}=H(r)^{-\frac{1}{2}}(-dt^{2}+d\mathbf{x}^{2})+H(r)^{\frac{1}{2}}(dr^{2}+r^{2}d\Omega_{5}^{2})\,,$
(5)

where $\mathbf{x}$ are the longitudinal coordinates, $H(r)=1+L^{4}/r^{4}$and
$d\Omega_{5}^{2}$ denotes the metric of the transverse round $S^{5}$-sphere
BianchiConsoliGrilloMorales:2021 . The ODE which describes the scalar field
perturbation of the D3 brane is BianchiConsoliGrilloMorales:2021 ;
GubserHashimoto:1998

$\frac{d^{2}\phi}{dr^{2}}+\left[\omega^{2}\left(1+\frac{L^{4}}{r^{4}}\right)-\frac{(l+2)^{2}-\frac{1}{4}}{r^{2}}\right]\phi=0\,.$
(6)

Upon the change of variables

$r=Le^{\frac{y}{2}}\,\quad\omega L=-2ie^{\theta}\,\quad P=\frac{1}{2}(l+2)\,,$
(7)

the equation reduces to the generalized Mathieu equation

$-\frac{d^{2}}{dy^{2}}\psi+\left[e^{2\theta}(e^{y}+e^{-y})+P^{2}\right]\psi=0\,.$
(8)

It corresponds to the conformal self-dual Liouville theory with momentum $P$
and rapidity $\theta$ FioravantiGregori:2019 ; ZamolodchikovMemorial .
Instead, to obtain the $\mathcal{N}=2$ $SU(2)$ pure ($N_{f}=0$) gauge theory
with Omega background in the Nekrasov Shatashvili limit $\epsilon_{2}\to 0$,
$\epsilon_{1}=\hslash\neq 0$, we need to use the change of parameters $\omega
L=-2i\frac{\Lambda_{0}}{\hslash}\,,\quad\frac{1}{8}(l+2)^{2}=\frac{u}{\hslash^{2}}$,
where $u$ is the Coulomb branch modulus and $\Lambda_{0}$ the instanton
coupling FioravantiGregori:2019 . Here we will just state the necessary
results of the ODE/IM correspondence procedure for the Liouville Field Theory
and refer to FioravantiGregori:2019  for details and the proper 2D statistical
interpetation. In fact, we want to explain ODE/IM correspondence for a more
general case in section III. In general, the starting point of the ODE/IM
method is the definition of a $Q$ function as the wronskian of two regular
solutions $\psi_{+,0},\psi_{-,0}$ at $y\to\pm\infty$, respectively:

$Q(\theta,P)=W[\psi_{+,0},\psi_{-,0}](\theta,P)\,.$ (9)

Of course, $\psi_{+,0}(y)=\psi_{-,0}(-y)$ and cf. next section for physical
interpretation. Then we derived the fundamental $QQ$ functional relation

$Q(\theta+i\pi/2)Q(\theta-i\pi/2)=1+Q(\theta)^{2}\,,$ (10)

from which we derived all the theory. Here and in the following we can omit
the dependence on $P$ as it stays fixed. Crucially, the QNMs condition (4)
translates into

$Q(\theta_{n})=0\,,$ (11)

namely the zeros of the Baxter’s $Q$ function which are the Bethe roots
Bazhanov:1996dr . Now, we prove that condition (11) is equivalent to the
quantization condition of the dual gauge period

$A_{D}(a,\Lambda_{0,n},\hslash)=i\pi n\,,\qquad n\in\mathbb{Z}\,.$ (12)

as conjectured by AminovGrassiHatsuda:2020 . For it was already proposed in
GrassiGuMarino  on a numerical basis this relevant relation

$Q(a,\Lambda_{0},\hslash)=i\frac{\sinh
A_{D}(a,\Lambda_{0},\hslash)}{\sinh\frac{2\pi i}{\hslash}a}\,,$ (13)

involving the period $a$ and the dual one
$A_{D}=\partial{\mathcal{F}}/{\partial a}$. Actually, it can be proven within
ODE/IM or checked numerically by easily computing the l.h.s. with TBA below
(15). Instead, the computation of $A_{D}$ in (12,13) relies on the expansion
of the prepotential $\mathcal{F}$ in powers (number of instantons) of
$\Lambda_{0}^{4}$ Nekrasov:2002 ; NekrasovOkounkov:2003 : the period $a$ is
related to the moduli parameter $u$ (or $P$) through the Matone’s relation
Matone:1995 ; FlumeFucitoMoralesPoghossian:2004 . In this respect, only the
first instanton contributions are easily accessible and summing them up
(naively) is accurate as long as $|\Lambda_{0}|/\hslash\ll 1$. Thus, this make
hard to access QNMs values $|\Lambda_{0,n}|/\hslash\gg 1$.

On the contrary, we found in general this procedure to an exact result, which
resums all instantons. Let us define the $Y(\theta,P)=Q^{2}(\theta,P)$
function and derive from (10) the $Y$-system

$Y(\theta+i\pi/2)Y(\theta-i\pi/2)=\Bigl{(}1+Y(\theta)\Bigr{)}^{2}\,.$ (14)

Eventually, we solve it explicitly (up to quadratures) via a Thermodynamic
Bethe Ansatz (TBA) integral equation for the pseudoenergy
$\varepsilon(\theta)=-\ln Y(\theta)$:

$\displaystyle\varepsilon(\theta)$
$\displaystyle=\frac{16\sqrt{\pi^{3}}}{\Gamma(\frac{1}{4})^{2}}e^{\theta}-2\int_{-\infty}^{\infty}\frac{\ln\left[1+\exp\\{-\varepsilon(\theta^{\prime})\\}\right]}{\cosh(\theta-\theta^{\prime})}\frac{d\theta^{\prime}}{2\pi}\,.$
(15)

In this $P$ does not appear explicitly, but fixes the solution by its
asymptotic linear behaviour $\varepsilon(\theta_{0},P)\simeq+8P\theta$, $P>0$,
at $\theta\to-\infty$. Eventually, the $QQ$ system (10) characterizes the QNMs
as $Y(\theta_{n}-i\pi/2)=-1$, i.e. the TBA quantization condition

$\varepsilon(\theta_{n^{\prime}}-i\pi/2)=-i\pi(2n^{\prime}+1)\,,\qquad
n^{\prime}\in\mathbb{Z}\,$ (16)

which can be easily implemented by using the TBA (15) as table I shows. These
values match very well with those obtained by the standard method of continued
fractions by Leaver and is consistent with the ($l\to\infty$) WKB
approximation (geodetic method) Leaver:1985 ; BianchiConsoliGrilloMorales:2021
.

$n$ | $l$ | TBA | Leaver | WKB
---|---|---|---|---
$0$ | $0$ | $\underline{1.369}12\,-\underline{0.504}048i$ | $\underline{1.369}72-\underline{0.504}311i$ | $1.41421\,-0.5i$
$0$ | $1$ | $\underline{2.091}18\,-\underline{0.501}788i$ | $\underline{2.091}76-\underline{0.501}811i$ | $2.12132\,-0.5i$
$0$ | $2$ | $\underline{2.80}57\,-\underline{0.50100}9i$ | $\underline{2.80}629-\underline{0.50100}0i$ | $2.82843\,-0.5i$
$0$ | $3$ | $\underline{3.517}23\,-\underline{0.5006}49i$ | $\underline{3.517}83-\underline{0.5006}34i$ | $3.53553\,-0.5i$
$0$ | $4$ | $\underline{4.227}28\,-\underline{0.5004}53i$ | $\underline{4.227}90-\underline{0.5004}38i$ | $4.24264\,-0.5i$
Table 1: Comparison of QNMs of the D3 brane from TBA (15) (through (16) with
$n^{\prime}=0$), Leaver (continued fractions) method and WKB (geodetic)
approximation ($L=1$).

We note that the physical condition $\Im\omega<0$ becomes by (7) $-\pi/2+2\pi
n<\Im\theta<\pi/2+2\pi n$, for $n\in\mathbb{Z}$. However, the TBA (15) is
valid only for the fundamental strip $|\Im\theta|<\pi/2$. In fact, in this
region we find directly the QNMs for overtone number $n=0=n^{\prime}$. We
expect that analytically continuing the TBA by using the $Y$-system (14) in
the other strips $|\Im(\theta-2\pi in)|<\pi/2$, we would obtain the other
overtone numbers. We leave more details on this for future work.

Within our set-up of functional and integral equations for entire functions in
$\theta$ (integrability), we can find other quantization conditions on the
roots $\theta_{n}$ (QNMs). For instance, the $TQ$ relation
FioravantiGregori:2019

$T(\theta)Q(\theta)=Q(\theta-i\pi/2)+Q(\theta+i\pi/2)\,$ (17)

means $Q(\theta_{n}-i\pi/2)+Q(\theta_{n}+i\pi/2)=0$. This and the $QQ$
relation (10) actually fixes $Q(\theta_{n}+i\pi/2)Q(\theta_{n}-i\pi/2)=1$ and
then

$Q(\theta_{n}\pm i\pi/2)=\pm i$ (18)

are fixed, too. Again (10) around $\theta_{n}$ forces $Q(\theta+i\pi/2)=i\pm
Q(\theta)+\dots$ and $Q(\theta-i\pi/2)=-i\pm Q(\theta)+\dots$ up to smaller
corrections (dots). Therefore, (17) imposes

$T(\theta_{n})=\pm 2\,.$ (19)

On equating the invariant trace of the monodromy matrix in the Floquet and
ODE/IM bases (the coefficients in the latter involve the $T$ function), we
obtain a nice equality involving the period $a$ ($=\nu$, Floquet index)
FioravantiGregori:2019

$T(\theta)=2\cos 2\pi a\,.$ (20)

In conclusion, condition (11) means that also the period $a$ is quantised

$a(\theta_{n})=\frac{n}{2}\,,\qquad n\in\mathbb{Z}\,\,.$ (21)

This is exactly the condition used by BianchiConsoliGrilloMorales:2021 . Yet,
here we have fixed the general limits of its validity as relying on specific
forms of the $TQ$ and $QQ$ systems (17) and (10) respectively: it does not
work in general, but we will see in the next section the specific conditions
for its validity.

##  IV IV. General theory: intersecting D3 branes and beyond

To illustrate how our method develops in general, we consider the spacetime
given by the intersection of four stacks of D3-branes in type IIB
supergravity. These geometries are characterised by four different charges
$\mathcal{Q}_{i}$ which, if all equal, lead to an extremal RN BH. In isotropic
coordinates the line element writes

$ds^{2}=-f(r)dt^{2}+f(r)^{-1}[dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})]\,,$ (22)

with $f(r)=\prod_{i=1}^{4}\left(1+\mathcal{Q}_{i}/r\right)^{-\frac{1}{2}}\,.$
The ODE describing the scalar perturbation is, with
$\Sigma_{k}=\sum_{i_{1}<...<i_{k}}^{4}\mathcal{Q}_{i_{1}}\cdots\mathcal{Q}_{i_{k}}$
Ikeda:2021 ; BianchiConsoliGrilloMorales:2021b

$\frac{d^{2}\phi}{dr^{2}}+\left[-\frac{(l+\frac{1}{2})^{2}-\frac{1}{4}}{r^{2}}+\omega^{2}\sum_{k=0}^{4}\frac{\Sigma_{k}}{r^{k}}\right]\phi=0\,.$
(23)

Changing variables as $r=\sqrt[4]{\Sigma_{4}}e^{y}$ and

$\displaystyle\omega\sqrt[4]{\Sigma_{4}}$
$\displaystyle=-ie^{\theta}\,\,\,M_{j}=\frac{1}{2}\frac{\Sigma_{2j-1}}{\sqrt[4]{\Sigma_{4}}^{2j-1}}e^{\theta}\,\,\,P^{2}=(l+\frac{1}{2})^{2}-\omega^{2}\Sigma_{2}\,,$
(24)

($j=1,2$) the ODE takes the form

$-\frac{d^{2}}{dy^{2}}\psi+\left[e^{2\theta}(e^{2y}+e^{-2y})+2e^{\theta}(M_{1}e^{y}+M_{2}e^{-y})+P^{2}\right]\psi=0\,.$
(25)

It generalizes that for the Perturbed Hairpin IM FateevLukyanov:2005  and may
have its own 2D statistical field theory interpetation. Connection with
$SU(2)$ $N_{f}=2=(1,1)$ gauge theory is realized by flavour masses
$m_{j}=\hslash M_{j}$, modulus $u=P^{2}/\hslash^{2}$ and instanton coupling
$\Lambda_{2}=4\hslash e^{\theta}$ ItoKannoOkubo:2017 ;
BianchiConsoliGrilloMorales:2021b .

Now, we develop the ODE/IM procedure for this general model by extending the
porcedure of FioravantiGregori:2019 . The regular solutions at $y\to\pm\infty$
are determined by

$\displaystyle\psi_{\pm,0}(y)$ $\displaystyle\simeq
2^{-\frac{1}{2}-M_{\pm}}e^{-(\frac{1}{2}+M_{\pm})\theta\mp(\frac{1}{2}+M_{\pm})y}e^{-e^{\theta\pm
y}}\,,\quad y\to\pm\infty\,,$ (26)

with $M_{+}=M_{1}$ and $M_{-}=M_{2}$. Equation (25) enjoys the discrete
symmetries

$\displaystyle\Omega_{\pm}$
$\displaystyle:\,\,\theta\to\theta+i\pi/2\,,\,\,y\to y\pm
i\pi/2\,,\,\,M_{1}\to\mp M_{1}\,,\,\,M_{2}\to\pm M_{2}\,,$ (27)

which are consistent with the brane dictionary (24). (The brane parameters
vary as $\Sigma_{1}\to\pm
i\Sigma_{1}\,,\Sigma_{2}\to-\Sigma_{2}\,,\Sigma_{3}\to\mp
i\Sigma_{3}\,,\Sigma_{4}\to\Sigma_{4}$.) Thanks to these symmetries , one can
define other independent solutions $\psi_{-,k}=\Omega_{-}^{k}\psi_{-,0}\,,$
$\psi_{+,k}=\Omega_{+}^{k}\psi_{+,0}$. We also have the invariance properties
$\Omega_{+}^{k}\psi_{-,0}=\psi_{-,0}\,,$
$\Omega_{+}^{k}\psi_{+,0}=\psi_{+,0}$. These solutions are normalized such
that their wronskians are $W[\psi_{-,k+1},\psi_{-,k}]=-i\exp\\{(-1)^{k}i\pi
M_{2}\\}$, $W[\psi_{+,k+1},\psi_{+,k}]=i\exp\\{(-1)^{k}i\pi M_{1}\\}$. One can
now define a Baxter’s $Q$ function as the wronskian

$Q_{+,+}(\theta,P)=W[\psi_{+,0},\psi_{-,0}](\theta,P,M_{1},M_{2})\,.$ (28)

(We will use the notation $Q_{\pm,\pm}=Q(\theta,\pm M_{1},\pm M_{2})$,
$Q_{\pm,\mp}=Q(\theta,\pm M_{1},\mp M_{2})$.) By the properties of wronskians,
we can write the linear connection relations as

$\displaystyle ie^{i\pi M_{1}}\psi_{-,0}$
$\displaystyle=Q_{-,+}(\theta+i\pi/2)\psi_{+,0}-Q_{+,+}(\theta)\psi_{+,1}$
(29) $\displaystyle ie^{i\pi M_{1}}\psi_{-,1}$
$\displaystyle=Q_{-,-}(\theta+i\pi)\psi_{+,0}-Q_{+,-}(\theta+i\pi/2)\psi_{+,1}$

and taking their wronskian we obtain the $QQ$ system

$\displaystyle
Q_{+,-}(\theta+\frac{i\pi}{2})Q_{-,+}(\theta-\frac{i\pi}{2})=e^{-i\pi(M_{1}-M_{2})}+Q_{-,-}(\theta)Q_{+,+}(\theta)\,.$
(30)

One can define a $Y$ function as
$Y_{2,+,+}(\theta)=\exp\\{i\pi(M_{1}-M_{2})\\}Q_{+,+}(\theta)Q_{-,-}(\theta)\,$and
the $Y$ system is then

$\displaystyle
Y_{+,-}(\theta+\frac{i\pi}{2})Y_{-,+}(\theta-\frac{i\pi}{2})=[1+Y_{+,+}(\theta)][1+Y_{-,-}(\theta)]\,.$
(31)

In gravity variables it reads

$\displaystyle
Y(\theta+\frac{i\pi}{2},-i\Sigma_{1},-\Sigma_{2},i\Sigma_{3})Y(\theta-\frac{i\pi}{2},-i\Sigma_{1},-\Sigma_{2},i\Sigma_{3})$
(32)
$\displaystyle=[1+Y(\theta,\Sigma_{1},\Sigma_{2},\Sigma_{3})][1+Y(\theta,-\Sigma_{1},\Sigma_{2},-\Sigma_{3})]$

and can be inverted into the TBA

$\displaystyle\varepsilon_{\pm,\pm}(\theta)$
$\displaystyle=[f_{0,+}\mp\frac{i\pi}{2}(\frac{\Sigma_{1}}{\Sigma_{4}^{1/4}}-\frac{\Sigma_{3}}{\Sigma_{4}^{3/4}})]e^{\theta}-\varphi\ast(\bar{L}_{\pm\pm}+\bar{L}_{\mp\mp})(\theta)$
(33) $\displaystyle\bar{\varepsilon}_{\pm,\pm}(\theta)$
$\displaystyle=[\bar{f}_{0,+}\pm\frac{\pi}{2}(\frac{\Sigma_{1}}{\Sigma_{4}^{1/4}}+\frac{\Sigma_{3}}{\Sigma_{4}^{3/4}})]e^{\theta}-\varphi\ast(L_{\pm\pm}+L_{\mp\mp})(\theta)$

where we defined $\varepsilon(\theta)=-\ln
Y(\theta,\Sigma_{1},\Sigma_{2},\Sigma_{3},\Sigma_{4})$,
$\bar{\varepsilon}(\theta)=\varepsilon(\theta,i\Sigma_{1},-\Sigma_{2},-i\Sigma_{3},\Sigma_{4})$,
$L=\ln[1+\exp\\{-\varepsilon\\}]$, $\varphi(\theta)=(2\pi\cosh(\theta))^{-1}$.
The forcing term is obtained from the relation $Q_{+,+}(\theta)=-ie^{i\pi
M_{1}}\lim_{y\to+\infty}\psi_{-,0}(y)/\psi_{+,1}(y)$ (following from (29)) and
we define $f_{0,\pm}=c_{0,+,\pm}+c_{0,-,\mp}$ and
$c_{0,\pm,\pm}=\int_{-\infty}^{\infty}\Bigl{[}\sqrt{2\cosh(2y)\pm\frac{\Sigma_{1}}{\sqrt[4]{\Sigma_{4}}}e^{y}\pm\frac{\Sigma_{3}}{\sqrt[4]{\Sigma_{4}^{3}}}e^{-y}+\frac{\Sigma_{2}}{\sqrt{\Sigma_{4}}}}-\text{reg.}\Bigr{]}\,dy\,\,$
, which in turn can be expressed either through a triple power series for
small parameters or as an elliptic integral. We have to numerically input $l$
in the TBA with the boundary condition at $\theta\to-\infty$:
$\varepsilon_{\pm,\pm}(\theta)\simeq 4P\theta\simeq 4(l+1/2)\theta$, also
following from the asymptotic of the ODE (25) (the precision improves by
adding also the constant at the subleading order). This TBA is a
generalization of those found in FateevLukyanov:2005 ; Imaizumi:2021  for the
Perturbed Hairpin IM and the $N_{f}=2$ gauge theory with equal masses
$m_{1}=m_{2}$ respectively, although we shall pay particular attention to the
change of variables from gravity or gauge to integrability: this results in
different TBA equations as first noted in FioravantiGregori:2019 . Through
this TBA we find again the QNMs to be given by the Bethe roots condition

$\bar{\varepsilon}_{+,+}(\theta_{n^{\prime}}-i\pi/2)=-i\pi(2n^{\prime}+1)\,,\quad
Q_{+,+}(\theta_{n})=0$ (34)

and we show in tables II and III their agreement with continued fraction
(Leaver) method and WKB approximation ($l\to\infty$) Leaver:1985 ; Ikeda:2021
; BianchiRusso:2021 . For $\Sigma_{1}\neq\Sigma_{3}$ and $\Sigma_{4}\neq 1$
the Leaver method is not applicable, at least in its original version, as the
recursion produced by the ODE involves more than $3$ terms (cf.
BianchiConsoliGrilloMorales:2021 ; Leaver:1985 ) and thus also for this reason
the TBA method may be regarded as convenient. However, there is a recent
development, the so-called matrix Leaver method, which is still applicable
Leaver:1990 ; KumarBannonPribytokRodgers:2020 .

$n$ | $l$ | TBA | Leaver | WKB
---|---|---|---|---
$0$ | 1 | $\underline{0.86}9623\,-\underline{0.372}022i$ | $\underline{0.86}8932-\underline{0.372}859i$ | $0.89642-0.36596i$
$0$ | 2 | $\underline{1.477}990\,-\underline{0.368}144i$ | $\underline{1.477}888-\underline{0.368}240i$ | $1.4940-0.36596i$
$0$ | 3 | $\underline{2.080}200\,-\underline{0.3670}76i$ | $\underline{2.080}168-\underline{0.3670}97i$ | $2.0916-0.36596i$
$0$ | 4 | $\underline{2.6803}63\,-\underline{0.3666}37i$ | $\underline{2.6803}50-\underline{0.3666}42i$ | $2.6893-0.36596i$

Table 2: Comparison of QNMs obtained from TBA (33), Leaver method (through
(34) with $n^{\prime}=0$) and WKB approximation ($\Sigma_{1}=\Sigma_{3}=0.2$,
$\Sigma_{2}=0.4$, $\Sigma_{4}=1$.)

$n$ | $l$ | TBA | Leaver | WKB
---|---|---|---|---
$0$ | 1 | $0.896681\,-0.40069i$ | N.A. | $0.93069-0.39458i$
$0$ | 2 | $1.5308\,-0.39676i$ | N.A. | $1.5511-0.39458i$
$0$ | 3 | $2.15708\,-0.395689i$ | N.A. | $2.1716-0.39458i$
$0$ | 4 | $2.78077\,-0.39525i$ | N.A. | $2.7921-0.39458i$

Table 3: Comparison of QNMs obtained from TBA (33), (through (34) with
$n^{\prime}=0$) and WKB approximation ($\Sigma_{1}=0.1$, $\Sigma_{2}=0.2$,
$\Sigma_{3}=0.3$, $\Sigma_{4}=1$). Since $\Sigma_{1}\neq\Sigma_{3}$ the Leaver
method is not applicable (N.A.) in its original version.

From ODE/IM we have evidence that generalizations of formula (13) can be
derived (some are already available for the limit $N_{f}=1$
GrassiHaoNeitzke:2021 ). In this way we expect that in general the
integrability Bethe roots condition (34), which we have shown to follow
straightforwardly from BHs physics, corresponds in gauge theory to the
quantization of the $A_{D}$ period conjectured in AminovGrassiHatsuda:2020 .

The ODE/IM construction goes further because the presence of the irregular
singularity of (25) at $y\to\pm\infty$ (Stokes phenomenon) allow us to define
$T$ functions

$T_{+,+}(\theta)=-iW[\psi_{-,-1},\psi_{-,1}]\,,\,\,\tilde{T}_{+,+}(\theta)=iW[\psi_{+,-1},\psi_{+,1}]\,.$
(35)

Then, on expanding $\psi_{\pm,1}$ in terms of $\psi_{\pm,0}$, $\psi_{\pm,-1}$,
similarly to (29), the $TQ$ relations follow

$\displaystyle T_{+,+}(\theta)Q_{+,+}(\theta)$ $\displaystyle=e^{i\pi
M_{2}}Q_{+,-}(\theta-\frac{i\pi}{2})+e^{-i\pi
M_{2}}Q_{+,-}(\theta+\frac{i\pi}{2})$ (36)
$\displaystyle\tilde{T}_{+,+}(\theta)Q_{+,+}(\theta)$ $\displaystyle=e^{i\pi
M_{1}}Q_{-,+}(\theta-\frac{i\pi}{2})+e^{-i\pi
M_{1}}Q_{-,+}(\theta+\frac{i\pi}{2})\,.$

As above (20), we equate the invariant trace of the monodromy matrix in the
Floquet and ODE/IM bases (for the coefficients in the latter the $T$s are
needed) and obtain equalities involving now also the masses $M_{1}$ and
$M_{2}$

$\displaystyle 2\cos 2\pi\nu+2\cos 2\pi M_{1}$
$\displaystyle=\tilde{T}_{+,+}(\theta)\tilde{T}_{-,+}(\theta+i\frac{\pi}{2})$
(37) $\displaystyle 2\cos 2\pi\nu+2\cos 2\pi M_{2}$
$\displaystyle=T_{+,+}(\theta)T_{+,-}(\theta+i\frac{\pi}{2})\,.$

This generalizes a simpler formula conjectured in FateevLukyanov:2005  and
should give a proof method for the one guessed in FioravantiPoghossian:2019 .
Besides, also in this case the period $a=\nu$, the Floquet index.

Only for equal masses $M_{1}=M_{2}\equiv M$ we can make considerations on $TQ$
and $QQ$ systems (36,30) like in section III and conclude that

$T_{+,+}(\theta_{n})T_{-,-}(\theta_{n})=4\,,$ (38)

which generalizes (19). Then, we use
$T_{+,-}(\theta+i\frac{\pi}{2})=T_{+,+}(\theta)$,
$\tilde{T}_{-,+}(\theta+i\frac{\pi}{2})=\tilde{T}_{+,+}(\theta)$ and (38) in
(37) (invariant upon changing sign of $M$) and conclude

$[\cos 2\pi\nu+\cos 2\pi M]_{\theta=\theta_{n}}=\pm 2\,.$ (39)

Still for equal masses, there is the extra symmetry
$\psi_{+,0}(y)=\psi_{-,0}(-y)$ as in the previous $N_{f}=0$ case. It exchanges
infinity ($y=\infty$) and the (analogue) horizon ($y=-\infty$), leaving the
photon-sphere ($y=0$) fixed as in BianchiRusso:2021  and imposes the identity
$\tilde{T}_{+,+}(\theta)=T_{+,+}(\theta)$.

##  V V. Perspectives

From the generality of the case of previous section it is manifest that our
method applies to many other theories both to the angular and the radial
problems BianchiConsoliGrilloMorales:2021b . We can list, for instance, the
$SU(2)$ $N_{f}=(2,0)$ gauge theory and the associated gravity counterparts,
like D1D5 fuzzballs and CCLP 5D BHs, as well as to $SU(2)$ $N_{f}=3$ gauge
theory and the general asymptotically flat Kerr-Newman BH
BianchiConsoliGrilloMorales:2021b . Instead of both irregular singularities as
above, in these cases the ODE shows a regular and an irregular point, as in
the original correspondence for conformal minimal models DoreyTateo1998 : the
associated integrability structure may share some similarity with the latter
case. Another simple but tricky case happens when all the singularities are
regular, e.g. $SU(2)$ $N_{f}=4$ or asymptotically AdS BHs, since then the ODE
has only regular singularities and no Stokes sectors. Actually, our procedure
should still apply though trivialize as in DTS  for the supersymmetric XXZ
spin chain. Yet, more recently these regular equations were discovered to
appear (as they should) as ODEs in the spectral parameter, bi-spectral to
another irregular ODE FioravantiRossi:2021 : this bi-spectrality map is a
route to be followed.

In any case, the TBA method seems in many cases convenient with respect to
other methods, such as the original Leaver or SW methods, which cannot always
be easily applied. The latter for instance needs a clever numerical re-
summation procedure AminovGrassiHatsuda:2020 .

Eventually, we note that much of the BH theory seems to go in parallel to the
ODE/IM correspondence construction and its 2D statistical field theory
interpretation, beyond the determination of QNMs: as an example, the
absorption coefficient seems a ration of $Q$s. To this aim it is relevant to
understand the rôle of excited states of ODE/IM BLZexc ; Fexc  which preserve
the form of the functional equations, but change the integral ones.

###### Acknowledgements.

Acknowledgements We thank M. Bianchi, D. Consoli, A. Grillo, F. Morales for
suggestions, and H. Shu, K. Zarembo for discussions. DG thanks NORDITA for
warm hospitality.

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