††institutetext: Dipartimento di Fisica, Università di Roma “Tor Vergata" &
Sezione INFN Roma2, Via della Ricerca Scientifica 1, 00133, Roma, Italy

# Charge (in)stability and superradiance of Topological Stars

Andrea Cipriani  Carlo Di Benedetto  Giorgio Di Russo111Address after February
9th: Institut de Physique Théorique, Université Paris-Saclay, CEA, Orme des
Merisiers, Gif-sur-Yvette, 91191 CEDEX, France Alfredo Grillo  Giuseppe Sudano

###### Abstract

We study linear massive scalar charged perturbations of Topological Stars in
the fuzzball and in the black hole (Black String) regimes. The objects that
naturally couple to the electric 3-form field strength of these solutions are
charged strings, wound around the compact direction. We explore the
possibility of instabilities of these solutions, in analogy with the charge
instability already highlighted for other non-BPS geometries like JMaRT. This
issue is addressed by calculating quasi-normal mode frequencies with a variety
of techniques: WKB approximation, direct integration, Leaver method and by
exploiting the recently discovered correspondence between black hole/fuzzball
perturbation theory and quantum Seiberg-Witten curves. All mode frequencies we
find have negative imaginary parts, implying an exponential decay in time.
This suggests a linear stability of Topological Stars also in this new
scenario. In addition, we study the charge superradiance for the Black String.
We compute the amplification factor with the numerical integration method and
a quantum Seiberg-Witten motivated definition including instantonic
corrections.

##  1 Introduction

Black holes (BH) are nowadays regarded as the ideal playground to focus on
when attempting to solve the lasting clash between Quantum Mechanics and
General Relativity. As a matter of fact, a complete Quantum Gravity theory
should be able to puzzle out some inconsistencies regarding BH thermodynamical
properties, e. g. the microscopic origin of their entropy and the information
paradox Mathur:2009hf ; Bena:2022rna . One of the major successes of String
Theory has been the counting of microstates for a supersymmetric BH, in the
limit of vanishing gravitational coupling Sen:1995in ; Strominger:1996sh . On
the other hand, the resolution of the information paradox also requires to
know the structure of these microstates at finite coupling. In this respect,
one suggestion is represented by the ‘fuzzball proposal’ Lunin:2001fv ;
Mathur:2005zp : singular and horizon-full BHs should be thought as the
superposition of quantum states, some of which admit a classical description
as smooth and horizonless gravity solutions with the same conserved quantities
(mass, electric charge and angular momentum) of the BH. Remarkably, these
geometries emit radiation preserving quantum information.

The identification of some (a)typical microstate geometries has been rather
successful for supersymmetric BHs Giusto:2004id ; Giusto:2004ip ;
Giusto:2011fy ; Turton:2012ny ; Bena:2015bea ; Bena:2016agb ; Bena:2016ypk ;
Bena:2017xbt ; Bena:2017upb ; Bianchi:2016bgx ; Bianchi:2017bxl . This task
has been more challenging for non-BPS or even non-extremal BHs (which is the
ultimate goal if one aims to find microstate geometries for astrophysical
BHs). For instance, JMaRT solitons Jejjala:2005yu , the first known non-BPS
smooth horizonless geometries, are not viable microstate geometries: a BH with
the same charges would be over-rotating, thus its horizon would collapse
leading to a naked singularity. Moreover, due to the presence of an ergo-
region without horizon, JMaRT solitons are both affected by an ergo-region
instability Cardoso:2005gj  and a charge instability Bianchi:2023rlt . Very
recent developments in the fuzzball programme for non-BPS geometries also
include microstrata Ganchev:2021pgs ; Ganchev:2023sth  and Schwarzschild-like
topological solitons Bah:2022yji .

In this paper, we focus on a recently discovered smooth horizonless non-BPS
geometry called Topological Star (TS) Bah:2020ogh ; Bah:2020pdz ;
Heidmann:2021cms ; Bah:2021owp ; Heidmann:2022zyd . This is a static solution
of Einstein-Maxwell theory in 5 dimensions, whose 4-dimensional reduction is a
BH solution of Einstein-Maxwell-Dilaton theory. A TS is characterized by a
non-trivial topology and its Gregory-Laflamme instability Gregory:1993vy  is
avoided by turning on electromagnetic fluxes. Specifically, the magnetic field
of a TS is sourced by a magnetic monopole, whereas the electric field is
sourced by a string wound around the compact direction. A double Wick rotation
or an analytical continuation of the parameters Bah:2021irr  maps the TS to a
black object with the same mass and charges. This is called Black String (BS)
and is a two-charge non-rotating instance of a BH in the Cvetic-Youm class
Cvetic:1996xz ; Cvetic:1997uw .

The study of BHs is not only appealing for theoretical reasons: since a few
years ago, new observational techniques have been yielding outstanding
results. LIGO and Virgo interferometers LIGOScientific:2016aoc ;
LIGOScientific:2016lio  have been able to detect a signal of gravitational
waves coming from the late phase (ring-down) of the merger of two BHs.
Moreover, the Event Horizon Telescope EventHorizonTelescope:2019dse  has
provided the first direct images of the light ring and the accretion disk of a
BH. On grounds of these outcomes, one might hope to directly test novel ideas
in BH physics or even Quantum Gravity, through direct observations. In
particular, these may allow to discriminate between BHs and fuzzball
geometries due to their different dynamical properties. In addition, the
fuzzball conjecture could be tested as opposed to alternative models
Cardoso:2016rao ; Cardoso:2017cqb ; Cardoso:2019rvt , collectively called
Exotic Compact Objects (ECO). A non-exhaustive list of the latter also
includes boson stars Schunck:2003kk , gravastars Mazur:2001fv ;
Danielsson:2017riq , firewalls Almheiri:2012rt  and wormholes
Solodukhin:2005qy  (also see Balasubramanian:2022gmo ; Balasubramanian:2022lnw
; Climent:2024trz  for recent progress).

The dynamical aspects of gravitational objects can be examined by probing them
with particles, waves or strings Bianchi:2017sds . BHs and fuzzballs can be
discriminated by identifying observables with different behaviours for the two
types of geometries: the shape of the light rings Bianchi:2018kzy ;
Bianchi:2020des ; Bianchi:2020yzr , the spectrum of Quasi-Normal Modes (QNMs)
Chowdhury:2007jx ; Chakrabarty:2019ujg ; Bena:2020yii ; Ikeda:2021uvc , the
Tidal Love Number (TLN) Consoli:2022eey ; Bianchi:2023sfs ; DiRusso:2024hmd
and the multi-polar structure Bena:2020see ; Bianchi:2020bxa ; Bena:2020uup ;
Bianchi:2020miz ; Mayerson:2020tpn ; Bah:2021jno . Superradiance Brito:2015oca
is another significant phenomenon in this context. It consists in the
amplification of wave perturbations scattering off BHs, which thereby lose
energy, charge and angular momentum. Superradiance has been introduced and
mostly studied Press:1972zz ; Teukolsky:1974yv  for rotating BHs, where it can
be considered as the wave counterpart of the Penrose process; nonetheless it
has been also highlighted to occur for static charged BHs DiMenza:2014vpa ;
Benone:2015bst ; Baake:2016oku . Remarkably, superradiance can only take place
for black objects and not for fuzzball geometries which, instead, preserve
information.

The purpose of the present investigation is to explore the dynamical behaviour
of TS and BS under charged linear scalar perturbations. A single TS is not a
good candidate as a microstate geometry for astrophysical BHs 222Indeed, TSs
usually constitute the building blocks of more complicated multi-bubble
solutions Bah:2021owp . (e.g., it is magnetically charged and it has been
proven to be a ‘spherical spacetime mirror’ for photons Heidmann:2022ehn ).
Nevertheless, because of the large symmetry of the TS metric, the dynamics of
particles or waves in their background is separable. Relevant information
about their dynamical features can be easily extracted: TSs, then, represent
very useful tools to learn or to test the typical behaviour expected for more
general smooth horizonless geometries.

The response of TS under linear scalar massive or massless perturbations has
been already studied in Heidmann:2022ehn ; Heidmann:2023ojf ; Bianchi:2023sfs
. In particular, photon geodesics, QNMs and TLNs in the TS background have
been analyzed, allowing to argue for the linear stability of TSs. Moreover,
being their dynamical TLN real, TSs have been proved to undergo deformations
only and not dissipation, consistently with the absence of a horizon.

In the aforementioned works, probes have been chosen to be chargeless or at
most to have Kaluza-Klein momentum along the compact direction. In the present
paper, instead, we investigate the possibility of instabilities when the
probes are charged under the electric field of a TS, in analogy with the
charge instability pointed out for JMaRT geometries Bianchi:2023rlt . The
electric gauge field of a TS is a 2-form, sourced by an electrically charged
string, wound around the compact direction. Differently from the previous
treatments, probes are not point-like particles, but closed strings wound
around the compact direction. These could be identified as some of the winding
modes of TS’s electric string, after they slide off the bolt during the
discharge mechanism.

We will make use of the correspondence between BH/fuzzball perturbation theory
and quantum Seiberg - Witten (qSW) curves of $\mathcal{N}=2$ SYM theory with
gauge group SU(2) Aminov:2020yma . Besides providing a further tool to compute
QNMs Bianchi:2021xpr ; Bianchi:2021mft ; Bianchi:2022qph ; Bianchi:2023rlt ;
Bianchi:2023sfs  this correspondence has allowed to devise a definition for
the dynamical TLN Consoli:2022eey ; Bianchi:2023sfs ; DiRusso:2024hmd , by
taking into account instantonic corrections. In all these developments, a
prominent role has been played by the Heun connection formulae Bonelli:2021uvf
; Bonelli:2022ten , recently found exploiting the Alday-Gaiotto-Tachikawa
(AGT) duality Alday:2009aq . In particular, the radial equation for waves in
TS/BS background is a Confluent Heun Equation (CHE), which can be mapped to
the CHE arising from a qSW curve with three massive hypermultiplets.
Alternative semi-analytical procedures for the computation of QNMs are based
on integrability techniques Fioravanti:2021dce ; Fioravanti:2020udo ;
Rossi:2022mwx .

From the QNM spectrum, we infer that TSs are stable also when probed with
scalar charged perturbations. Indeed, only modes with negative imaginary part
have been found. We cannot rule out instabilities under vector or tensor
perturbations as well as non-linear instabilities Eperon:2016cdd  also
discovered for BPS geometries Giusto:2004id ; Giusto:2004ip . Some further
insights will be reported in a future work DiRusso:2024xy .

Being static geometries, TSs and BSs cannot give rise to rotation
superradiance. We investigate, instead, the possibility of charge
superradiance. No amplitude increases have been discovered for TSs at any
frequencies. This is consistent with the fact that information is preserved by
fuzzball geometries. By contrast, BSs can be well-suited to examine charge
superradiance. We show the presence of an enhancement of wave amplitudes in a
certain range of frequencies for BSs, through the analysis of the
amplification factor. On grounds of the BH/fuzzball perturbation theory - qSW
curves correspondence, we have added the instantonic corrections to the
definition of this observable. The results have been compared with a
corresponding numerical integration. The absence of mirror mechanisms, though,
excludes the possibility of superradiant instabilities Press:1972zz .

This paper is organised as follows. In Section 2 we introduce the TS solution
and discuss the coupling of its electric field with electrically charged
strings. We write the wave equation for charged linear scalar perturbations
and separate the radial and the angular dynamics. In Section 3 we review the
theory of quantum Seiberg-Witten curves for $\mathcal{N}=2$ super Yang-Mills
with gauge group $SU(2)$; we perform the matching with the perturbation theory
in TS background and define the quantization condition for the computation of
QNMs. Section 4 illustrates Leaver method to compute QNMs for the TS. The
frequencies of the charged QNMs of a TS calculated with the latter methods are
contained in Section 5 and compared with those arising from a numerical
integration. Section 6 investigates charge superradiance for the BS. We
compute the near-extremal near-superradiant QNMs and the prompt ring-down
modes. The employed techniques are: WKB approximation, numerical integration,
Leaver method and the correspondence with qSW curves. Finally, the
amplification factor is defined and corrected with the instantonic
contributions. The outcome is compared with a numerical prediction. In Section
7 we draw our conclusions and suggest possible outlooks. In Appendices we
report further details. Appendix A is focused on the electric field generated
by an electrically charged wire in a spacetime of generic dimension. In
Appendix B we discuss the measure units and the dimensions of the observables
we study. Appendix C describes the numerical method used to compute the
amplification factor.

##  2 Charged scalar perturbations in a Topological Star background

In this Section we present the solution of a TS. We probe this geometry with
charged strings with respect to its electric field. Accordingly, we devise a
shift of variables in the scalar wave equation, that takes into account the
stringy nature and the charge of the probes.

###  2.1 The background

TSs are smooth horizonless static solutions of Einstein-Maxwell theory in 5
dimensions, described by the action Bah:2020ogh ; Bah:2020pdz

$S_{5}=\int\mathrm{d}^{5}x\,\sqrt{-\det{g}}\left({\frac{R}{2\kappa_{5}^{2}}}-{\frac{1}{2}}\bigl{|}F^{(m)}\bigr{|}^{2}-{\frac{1}{2}}\bigl{|}F^{(e)}\bigr{|}^{2}\right)$
(1)

The corresponding equations of motion are333In this paper,
$\varepsilon_{01234}=+1$.

$\mathrm{d}\star F^{(m)}=\mathrm{d}\star F^{(e)}=0\quad,\quad
R_{\mu\nu}=\kappa_{5}^{2}\left(T_{\mu\nu}-{\frac{1}{3}}g_{\mu\nu}{T_{\alpha}}^{\alpha}\right)$
(2)

where $T_{\mu\nu}$ is the stress energy tensor

$T_{\mu\nu}={F^{(m)}}_{\mu\alpha}{{F^{(m)}}_{\nu}}^{\alpha}-{\frac{1}{4}}g_{\mu\nu}{F^{(m)}}_{\alpha\beta}{F^{(m)}}^{\alpha\beta}+{\frac{1}{2}}\Bigg{[}{F^{(e)}}_{\mu\alpha\beta}{{F^{(e)}}_{\nu}}^{\alpha\beta}-{\frac{1}{6}}g_{\mu\nu}{F^{(e)}}_{\alpha\beta\gamma}{F^{(e)}}^{\alpha\beta\gamma}\Bigg{]}$
(3)

The metric of a TS is given by

$\mathrm{d}s^{2}=-f_{s}(r)\,\mathrm{d}t^{2}+{\frac{\mathrm{d}r^{2}}{f_{b}(r)f_{s}(r)}}+r^{2}\,(\mathrm{d}\theta^{2}+\sin^{2}\theta\,\,\mathrm{d}\phi^{2})+f_{b}(r)\,\mathrm{d}y^{2}$
(4)

where

$f_{s}(r)=1-\frac{r_{s}}{r},\quad f_{b}(r)=1-\frac{r_{b}}{r}$ (5)

In (4), $t$ and $(r,\theta,\phi)$ are the time and the spherical coordinates
of the 4-dimensional spacetime, while $y$ is a further compact coordinate such
that $y\sim y+2\pi R_{y}$. The metric of a TS is then static, asymptotically
$\mathbb{R}^{1,3}\times S^{1}_{y}$ and spherically symmetric in 4
dimensions444This gravity solution can be seen as the harmonic superposition
of a Schwarzschild String (4-dimensional Schwarzschild BH $\times S^{1}_{y}$)
and a static Bubble of Nothing (an Euclidean Schwarzschild solution with a
time direction). Introduced to describe an instability of the Kaluza-Klein
vacuum Witten:1981gj , the Bubble of Nothing has been recently studied to
understand the stability of non-SUSY vacua Ooguri:2017njy ;
GarciaEtxebarria:2020xsr ; Dibitetto:2020csn ; Bomans:2021ara  and in the
framework of the Cobordism Conjecture Delgado:2023uqk ..

Three singularities can be identified for (4):

  * •

$r=0$ is a full-fledged curvature singularity;

  * •

$r=r_{s}$ is a horizon, being such that the component $g_{tt}$ of the metric
changes sign;

  * •

$r=r_{b}$ is the bolt singularity, where the compact $y$-circle degenerates to
a point.

In correspondence of the latter singularity, the spacetime smoothly terminates
at a cap. According to the location of the singularities we can distinguish
two kinds of solutions:

  * •

TS for $r_{b}>r_{s}>0$: the horizon and the singularity are pushed out of the
geometry, which is then smooth and horizonless;

  * •

BS for $r_{s}\geq r_{b}>0$: the spacetime is horizon-full.

The two different regimes are linked by the double analytical continuation or
exchange of parameters

$(t,y)\leftrightarrow(it,iy)\quad\mathrm{or}\quad r_{s}\leftrightarrow r_{b}$

The presence of electromagnetic fluxes prevents the gravitational collapse of
the solution. The electric and magnetic field strengths are respectively given
by

$F^{(e)}=\frac{Q}{r^{2}}\,\mathrm{d}r\wedge\mathrm{d}t\wedge\mathrm{d}y\quad,\quad
F^{(m)}=P\sin\theta\,\mathrm{d}\theta\wedge\mathrm{d}\phi$ (6)

The equations of motion (2) are solved if:

$P^{2}+Q^{2}=\frac{3r_{b}r_{s}}{2\kappa_{5}^{2}}$ (7)

The source for the field strength $F^{(e)}$ is a charged line wound along the
compact direction $y$. Indeed this is the object that naturally couples to the
2-form gauge potential:

$B^{(e)}=-\frac{Q}{r}\,\mathrm{d}t\wedge\mathrm{d}y$ (8)

In order to understand the nature of the parameter $Q$ in (6) and (8), in
Appendix A a reference to what happens to the case of an infinitely extended
and uniformly charged wire is discussed. The final result is that $Q$ can be
interpreted as the linear charge density of this wire and this implies, in
particular units discussed in the Appendix B, that $[Q]=[L]$, consistently
with (7).

The absence of Gregory-Laflamme Gregory:1993vy  and thermodynamical
instabilities for the TS further requires that Bah:2021irr  (the analogous
condition for the BS arises after $r_{b}\leftrightarrow r_{s}$)

$r_{s}<r_{b}<2r_{s}$ (9)

Finally, (4) admits a 4-dimensional reduction to the Einstein-Maxwell-Dilaton
theory. This allows to deduce the corresponding ADM mass

$\mathcal{M}=\frac{2\pi}{\kappa_{4}^{2}}(2r_{s}+r_{b})$ (10)

This expression holds both for the TS and for the BS. Remarkably, TSs are non-
BPS fuzzball geometries, since extremality is obtained for $r_{b}=r_{s}$
(hence just for one particular instance of BS).

###  2.2 Stringy probes in the Topological Star background

We want to investigate the response of TS background to charged probes. Both
the source and the probes of the gauge potential (8) are strings wound along
the compact direction555Stringy probes have been also helpful to explore the
phenomenon of tidal scrambling in fuzzball geometries Martinec:2020cml ;
Ceplak:2021kgl .. We expect the probe strings to originate from the discharge
mechanism of the background. In particular, TS geometry smoothly caps off at
$r=r_{b}$ hence the winding modes of the source can slip off the bolt
singularity with no obstacles. However, we work in the probe limit.

Let us focus on the instance of rigid probes, which namely propagate without
oscillations. Under this assumption, the embedding coordinates of their
worldsheet in the TS geometry are

$X^{M}(\tau,\sigma)=(X^{\mu}(\tau),\,Y=y^{0}+n\sigma
R_{y})\quad,\quad\sigma\in[0,2\pi]$ (11)

The spacetime indices are split as $M=(\mu,y)$, with $\mu=(t,r,\theta,\phi)$.
($\tau$, $\sigma$) are the (dimensionless) worldsheet coordinates of the
string, $n$ is an integer and corresponds to the winding number.

The dynamics of the string in TS background is governed by the non-linear
sigma model Callan:1985ia

$S=\frac{1}{4\pi\alpha^{\prime}}\int_{\Sigma}\mathrm{d}\sigma\mathrm{d}\tau\sqrt{-h}\Big{[}G_{MN}(X)\partial_{a}X^{M}\partial_{b}X^{N}h^{ab}+B_{MN}(X)\partial_{a}X^{M}\partial_{b}X^{N}\varepsilon^{ab}\Big{]}$
(12)

In this action, $\Sigma$ represents the worldsheet of the string,
$h^{ab}=\mathrm{diag}(-+)$ is the worldsheet metric, $\varepsilon_{ab}$ is the
2-dimensional antisymmetric tensor (such that $\varepsilon_{\tau\sigma}=1$).
Moreover $G_{MN}$ and $B_{MN}$ are the metric (4) and the gauge potential (8)
of the background (the dilaton is missing in the TS solution).

The conjugate momenta to the spacetime coordinates in (12) read

$P_{M}=\frac{\partial L}{\partial\dot{X}^{M}}$ (13)

where $\dot{X}^{M}=\partial_{\tau}X^{M}$ and the Lagrangian
$L=L\bigl{[}X^{M},\partial_{a}X^{M}\bigr{]}$ is such that

$S=\int\mathrm{d}\tau\,L$

The effect of the gauge field on the expression of the conjugate momenta can
be studied by splitting the Lagrangian as

$L=L_{G}+L_{B}$ (14)

with

$L_{G}=\frac{1}{4\pi\alpha^{\prime}}\int_{0}^{2\pi}\mathrm{d}\sigma\sqrt{-h}\,G_{MN}(X)\partial_{a}X^{M}\partial_{b}X^{N}h^{ab}$
$L_{B}=\frac{1}{4\pi\alpha^{\prime}}\int_{0}^{2\pi}\mathrm{d}\sigma\sqrt{-h}\,B_{MN}(X)\partial_{a}X^{M}\partial_{b}X^{N}\varepsilon^{ab}$

We expect the latter term to shift $P_{M}=\frac{\partial
L_{G}}{\partial\dot{X}^{M}}$, i.e. the momentum in absence of the coupling
with the electric field. In particular, the gauge potential (8) and the
embedding coordinates (11) imply that

$L_{B}=\frac{nR_{y}}{\alpha^{\prime}}\frac{Q}{r}\dot{X}^{t}$ (15)

where the integral in the coordinate $\sigma$ has been easily performed since
no dependence on $\sigma$ is left in the integrand. Being (15) only dependent
on $\dot{X}^{t}$ (and not on the other $\dot{X}$s) all conjugate momenta are
left invariant by the coupling with the gauge field, except

$P_{t}\rightarrow P_{t}+\frac{qQ}{r}$ (16)

or equivalently in terms of the energy $E$ ($P_{t}=-E$)

$E\rightarrow E-\frac{qQ}{r}$ (17)

In these expressions we have implicitly used the definition

$q=\frac{nR_{y}}{\alpha^{\prime}}$ (18)

The presence of a compact direction also leads to a shift in the mass of the
probe (compared to point-like particles in 4 dimensions):

$\mu^{2}\rightarrow\mu^{2}+f_{b}(r)q^{2}+\frac{p^{2}}{f_{b}(r)}$ (19)

The warping factor (5) $f_{b}=1-\frac{r_{b}}{r}$ arises from the metric (4):
indeed, the mass of the string depends on the actual length of the
compactification radius which is modulated by the $g_{yy}$ component of the
metric. In (19), besides (18), we have made use of the definition

$p=\frac{\hbar m}{R_{y}}$ (20)

where $m$ is the integer number representing Kaluza-Klein momentum units.

###  2.3 Charged scalars in the Topological Star background

At the linear level, scalar perturbations decouple from vector and tensor
modes. Therefore, the wave equation for a massive scalar in the background of
a TS is simply the Klein-Gordon equation666Keep in mind that, according to the
measure units of Appendix B, the mass parameter in this equation is actually
$\tilde{\mu}^{2}=\frac{\mu^{2}}{\hbar^{2}}$. This also applies to the momentum
$p$ appearing in (22): it should be $\tilde{p}=p/\hbar$. In the following we
will omit the tilde in both instances.

$\Box\Phi=\mu^{2}\Phi$ (21)

The radial and angular dynamics can be separated by imposing the ansatz

$\Phi=e^{i(-\omega t+py+m_{\phi}\phi)}R(r)S(\theta)$ (22)

In this expression, the exponential contains the conserved conjugate momenta:
for instance $\omega$ is linked to the energy introduced in (17), through the
definition $E=\hbar\omega$.
The solutions $S(\theta)$ of the angular equation are the spherical harmonics
in $S^{2}$. The radial equation, instead, reads

$(r-r_{b})(r-r_{s})R^{\prime\prime}(r)+(2r-r_{b}-r_{s})R^{\prime}(r)+\Bigg{[}r^{2}\left(\frac{r\omega^{2}}{r-r_{s}}-\frac{rp^{2}}{r-r_{b}}-\mu^{2}\right)-\ell(\ell+1)\Bigg{]}R(r)=0$
(23)

The charged scalar modes analyzed in this paper can be seen as scalar
excitations of the stringy probes of Subsection 2.2. The coupling to the
electric field and the string nature of the probe can be taken into account by
replacing the frequency and the mass according to (17) and (19). Thus, in the
radial equation we perform the following shifts

$\omega\rightarrow\omega-\frac{qQ}{r}\quad,\quad\mu^{2}\rightarrow\mu^{2}+q^{2}\left(1-\frac{r_{b}}{r}\right)+\frac{p^{2}}{(1-\frac{r_{b}}{r})}$
(24)

After these substitutions, the radial equation (23) can be recast in the
canonical (Schrödinger) form

$\psi^{\prime\prime}(r)+Q_{W}(r)\psi(r)=0$ (25)

by introducing the function $\psi(r)$, related to the original one by

$R(r)=\frac{\psi(r)}{\sqrt{(r-r_{s})(r-r_{b})}}$ (26)

In our instance, the characteristic $Q-$function $Q_{W}(r)$ appearing in (25)
is

$\displaystyle Q_{W}(r)=$
$\displaystyle\frac{A_{4}r^{4}+A_{3}r^{3}+A_{2}r^{2}+A_{1}r+A_{0}}{4(r-r_{b})^{2}(r-r_{s})^{2}}$
(27) $\displaystyle A_{4}=$ $\displaystyle-4(\omega^{2}-\mu^{2}-p^{2}-q^{2})$
$\displaystyle A_{3}=$ $\displaystyle
4(p^{2}r_{s}+q^{2}(2r_{b}+r_{s})+(r_{b}+r_{s})\mu^{2}-2qQ\omega-
r_{b}\omega^{2})$ $\displaystyle A_{2}=$ $\displaystyle
4(\ell(\ell+1)+q^{2}(-Q^{2}+r_{b}^{2}+2r_{b}r_{s})+r_{b}r_{s}\mu^{2}-2qQr_{b}\omega)$
$\displaystyle A_{1}=$ $\displaystyle
4(\ell(\ell+1)(r_{b}+r_{s})+q^{2}r_{b}(r_{b}r_{s}-Q^{2}))$ $\displaystyle
A_{0}=$ $\displaystyle r_{b}^{2}-2(1+2\ell(\ell+1))r_{b}r_{s}+r_{s}^{2}$

From the expression (27) of the $Q$-function, we infer that the radial
dynamics of charged scalar perturbations in TS background is described by a
Confluent Heun Equation (CHE); this is a Fuchsian differential equation with
two regular singularities at $r=r_{b}$ and $r=r_{s}$ and an irregular one at
$r=\infty$.

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

In this section, following the notation of Bianchi:2021mft , we provide a
short overview of how to construct quantum Seiberg-Witten (qSW) curves for
$\mathcal{N}=2$ theories with an $SU(2)$ gauge group. Subsequently, we discuss
how to compute the quantum periods $(a,a_{D})$, which are instrumental in
deriving exact quantization conditions for QNM frequencies and characterize
the response of the geometry to tidal deformations.

###  3.1 Quantum Seiberg-Witten curves

The classical SW curves for $SU(2)$ gauge theory with $N_{f}=3$ fundamental
hypermultiplets with masses $m_{i}$ in flat space are:

$\Lambda^{2}y^{2}P_{L}(x)+yP_{0}(x)+P_{R}(x)=0$ (28)

with

$P_{L}(x){=}x{-}m_{3},\qquad
P_{0}(x){=}x^{2}{-}u{+}\Lambda\left(x{+}\frac{1}{2}{-}m_{1}{-}m_{2}{-}m_{3}\right),\qquad
P_{R}(x){=}(x{-}m_{1})(x{-}m_{2})$ (29)

In these definitions $\Lambda=e^{2\pi i\tau}$, $\tau$ is the gauge coupling
and $u$ is the Coulomb branch (CB) modulus. In presence of a non trivial
$\Omega-$background, $\epsilon_{1}=1$, $\epsilon_{2}=0$, the dynamics of the
gauge theory is described by the quantum curve which can be obtained by the
classical one by promoting the variables $x$ and $y$ to operators satisfying
the commutation relation:

$[\hat{x},\ln{\hat{y}}]=1$ (30)

The qSW curve follows from the classical one (29)

$\left[\Lambda\hat{y}^{1/2}P_{L}(\hat{x})\hat{y}^{1/2}+P_{0}(\hat{x})+\hat{y}^{-1/2}P_{R}(\hat{x})\hat{y}^{-1/2}\right]f(z)=0$
(31)

This gauge theory can be understood in terms of D4-branes forming pairs
stretched between two NS5-branes or between an NS5 and infinity, positioned
both to the left and right. The locations of the zeros of $P_{L,R}(x)$,
$P_{0}(x)$ determine the positions of the D4-branes in these three regions,
serving as parameters for the four masses and defining the CB.

Equation (31) can be put in normal form by redefining the wave function as

$f(z)=e^{-\frac{qz}{2}}z^{\frac{m_{1}+m_{2}}{2}}(1+z)^{-\frac{1+m_{1}+m_{2}}{2}}\Psi(z)$
(32)

in such a way that (31) becomes

$\Psi^{\prime\prime}(z)+Q_{12}(z)\Psi(z)=0$ (33)

with

$Q_{12}(z){=}{-}\frac{\Lambda^{2}}{4}{+}\frac{1{-}(m_{1}{-}m_{2})^{2}}{4z^{2}}{+}\frac{1{-}(m_{1}{+}m_{2})^{2}}{4(1{+}z)^{2}}{-}\frac{m_{3}\Lambda}{z}{-}\frac{1{-}2m_{1}^{2}{-}2m_{2}^{2}{+}4u{+}2\Lambda(m_{1}{+}m_{2}{-}1)}{4z(1{+}z)}$
(34)

###  3.2 Gauge/gravity dictionaries

The identification between $Q_{W}$ (27) and $Q_{12}$ (34) allows us to
construct the dictionary between the parameters of the gravity solution and of
the gauge theory side. For the TS we choose:

$z=\frac{r-r_{b}}{r_{b}-r_{s}}$ (35)

in such a way that $r=\infty\leftrightarrow z=\infty$, $r=r_{b}\leftrightarrow
z=0$ and $r=r_{s}\leftrightarrow z=-1$. This is suitable for the present case
because the values $z<0$ are not physical, exactly like for the case with
$r<r_{b}$. By doing this change of variables, and setting $\mu=p=0$ for
simplicity, we get the following dictionary

$\displaystyle\hskip 42.67912ptm_{1}=m_{2}=-\frac{(qQ-
r_{s}\omega)}{\sqrt{\frac{r_{b}}{r_{s}}-1}},\quad
m_{3}=\frac{-q^{2}r_{s}-2qQ\omega+(r_{b}+2r_{s})\omega^{2}}{2i\sqrt{\omega^{2}-q^{2}}}$
(36) $\displaystyle
u=\left(\ell{+}\frac{1}{2}\right)^{2}{-}q^{2}[Q^{2}{+}(r_{b}{-}r_{s})r_{s}]{+}4qQr_{s}\omega{-}3r_{s}^{2}\omega^{2}{-}i\sqrt{\omega^{2}{-}q^{2}}(r_{b}{-}r_{s})\left(1{+}2\frac{qQ{-}r_{s}\omega}{\sqrt{\frac{r_{b}}{r_{s}}{-}1}}\right)$
$\displaystyle\hskip
142.26378pt\Lambda=-2\text{i}(r_{b}-r_{s})\sqrt{\omega^{2}-q^{2}}$

We choose to impose $p=\mu=0$ also for the BS. In this instance,

$z=\frac{r-r_{s}}{r_{s}-r_{b}}$ (37)

so that the dictionary is the following:

$m_{1}=-m_{2}=\frac{i(r_{s}\omega-qQ)}{\sqrt{1-\frac{r_{b}}{r_{s}}}},\quad
m_{3}=\frac{-q^{2}r_{s}-2qQ\omega+(r_{b}+2r_{s})\omega^{2}}{2i\sqrt{\omega^{2}-q^{2}}}$
(38)
$u=\left(\ell+\frac{1}{2}\right)^{2}-q^{2}Q^{2}+2qQ(r_{b}+r_{s})\omega-\omega^{2}(r_{b}^{2}+r_{b}r_{s}+r_{s}^{2})+i(r_{b}-r_{s})\sqrt{\omega^{2}-q^{2}}$
$\Lambda=-2i(r_{s}-r_{b})\sqrt{\omega^{2}-q^{2}}$

###  3.3 Quantum Seiberg-Witten periods

The dynamics of the gauge theory is codified in terms of a single holomorphic
function which is the qSW period $a$. In order to compute it, we have to
Fourier transform (31) bringing it in the form of a difference equation
Poghossian:2010pn ; Fucito:2011pn , whose integrability condition can be
written as a continued fraction:

$P_{0}(a)=\frac{M(a+1)}{P_{0}(a+1)-\frac{M(a+2)}{P_{0}(a+2)-...}}+\frac{M(a)}{P_{0}(a-1)-\frac{M(a-1)}{P_{0}(a-1)-...}}$
(39)

with

$M(x)=\Lambda P_{L}\left(x-\frac{1}{2}\right)P_{R}\left(x-\frac{1}{2}\right)$
(40)

Equation (39) with (29) can be solved for $u(a,\Lambda)$ order by order in
$\Lambda$, getting

$\displaystyle u(a,\Lambda)=$ $\displaystyle
a^{2}+\Lambda\left[\frac{1}{2}(1-m_{1}-m_{2}-m_{3})-\frac{2m_{1}m_{2}m_{3}}{4a^{2}-1}\right]+\frac{\Lambda^{2}}{128(a^{2}-1)}\left[4a^{2}-5+\right.$
$\displaystyle\left.+4(m^{2}_{1}+m^{2}_{2}+m^{2}_{3})-\frac{48(m_{1}^{2}m_{2}^{2}+m_{2}^{2}m_{3}^{2}+m_{1}^{2}m_{3}^{2})}{4a^{2}-1}+\frac{64m^{2}_{1}m^{2}_{2}m^{2}_{3}(20a^{2}+7)}{(4a^{2}-1)^{3}}\right]+$
$\displaystyle+\dots$ (41)

We can invert this series in order to obtain

$\displaystyle a(u,\Lambda)=$
$\displaystyle\sqrt{u}+\frac{\Lambda}{4\sqrt{u}}\left(\frac{4m_{1}m_{2}m_{3}}{4u-1}+m_{1}+m_{2}+m_{3}-1\right)+$
(42)
$\displaystyle-\frac{\Lambda^{2}}{256\sqrt{u}}\left(\frac{1024m_{1}^{2}m_{2}^{2}m_{3}^{2}}{(4u-1)^{3}}-\frac{256m_{1}m_{2}\left(m_{1}\left(m_{2}m_{3}-2\right)-2\left(m_{2}+m_{3}-1\right)\right)m_{3}}{(1-4u)^{2}}+\right.$
$\displaystyle\left.+\frac{8\left(m_{2}+m_{3}+m_{1}\left(1-4m_{2}m_{3}\right)-1\right){}^{2}}{u}+\frac{\left(4m_{1}^{2}-1\right)\left(4m_{2}^{2}-1\right)\left(4m_{3}^{2}-1\right)}{u-1}+\right.$
$\displaystyle\left.+\frac{64\left(\left(\left(1{-}12m_{3}^{2}\right)m_{2}^{2}{+}4m_{3}m_{2}{+}m_{3}^{2}\right)m_{1}^{2}{+}4m_{2}m_{3}\left(m_{2}{+}m_{3}{-}1\right)m_{1}{+}m_{2}^{2}m_{3}^{2}\right)}{4u{-}1}{+}4\right)+$
$\displaystyle{+}\dots$ (43)

The Nekrasov-Shatashvili (NS) prepotential $\mathcal{F}_{NS}(a,\Lambda)$ is
obtained by integrating the quantum version of the Matone relation
Matone:1995rx ; Flume:2004rp

$u=-\Lambda\frac{\partial\mathcal{F}_{NS}(a,\Lambda)}{\partial\Lambda}$ (44)

Furthermore, we have to add a $\Lambda$-indipendent constant obtained from the
one-loop prepotential, so that

$\mathcal{F}_{NS}(a,\Lambda)=\mathcal{F}_{tree}(a,\Lambda)+\mathcal{F}_{1-loop}(a)+\mathcal{F}_{inst}(a,\Lambda)$
(45)

where

$\displaystyle\mathcal{F}_{\text{tree}}(a,\Lambda)=-a^{2}\log\Lambda$
$\displaystyle\mathcal{F}_{\text{inst}}(a,\Lambda)=\Lambda\frac{1-m_{2}-m_{3}+4a^{2}(m_{1}+m_{2}+m_{3}-1)+m_{1}(4m_{2}m_{3}-1)}{2(4a^{2}-1)}+\dots$
(46) $\displaystyle\frac{\partial\mathcal{F}_{\text{one-loop}}(a)}{\partial
a}=\log\left[\frac{\Gamma^{2}(1+2a)}{\Gamma^{2}(1-2a)}\prod_{i=1}^{3}\frac{\Gamma\left(\frac{1}{2}+m_{i}-a\right)}{\Gamma\left(\frac{1}{2}+m_{i}+a\right)}\right]$

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

$a_{D}=-\frac{1}{2\pi i}\frac{\partial\mathcal{F}_{NS}}{\partial a}$ (47)

QNM frequencies are obtained by imposing the exact quantization condition
Bonelli:2022ten ; Consoli:2022eey :

$a_{D}=n\quad,\quad n=0,1,2,...$ (48)

once expressed in terms of the gravity variables via the dictionary. The
latter is solved in $\omega$ in the complex plane.

##  4 Leaver method

In this section, we describe a semi-analytical method used to solve the radial
differential equation involving a continued fraction, introduced by Leaver
Leaver:1985ax ; Leaver:1990zz . This method has been applied in
Bianchi:2023sfs ; Heidmann:2023ojf  to the study of QNMs for neutral scalar
waves in TS geometry.

For generality, let us implement this technique on the (1,2)-SW curve
(33)-(34), once we redefine:

$\Psi(z)=\sqrt{z(1+z)}R(z)$ (49)

so that

$z(1+z)R^{\prime\prime}(z)+(1+2z)R^{\prime}(z)+\Big{[}-\frac{\Lambda^{2}z^{2}}{4}+\frac{\Lambda(\Lambda+4m_{3})z}{4}-\frac{(m_{1}-m_{2})^{2}}{4z}+\frac{(m_{1}+m_{2})^{2}}{4(1+z)}+$
(50) $+\frac{1}{4}(1-4u+2\Lambda(1-m_{1}-m_{2}-2m_{3}))\Big{]}R(z)=0$

The coordinate $z$ is the coordinate that originates at the cap in the case
where the geometry is smooth $(r_{b}>r_{s})$, while it originates at the
horizon in the case of a BS:

$z=\begin{cases}\frac{r-r_{b}}{r_{b}-r_{s}}&\quad\mathrm{if}\ r_{b}>r_{s}\\\
\frac{r-r_{s}}{r_{s}-r_{b}}&\quad\mathrm{if}\ r_{s}>r_{b}\end{cases}$ (51)

Once the appropriate gauge/gravity dictionary is chosen, the Frobenius
exponents for the regular singular points are chosen in such a way that at
$z=0$ they encode the information that the solution is regular in the case of
smooth geometry or that the wave is ingoing in the case of a BS, while at
infinity, where the singularity is irregular, the wave is outgoing in both
cases. The other Frobenius exponent at $z=-1$ is chosen such that the
recurrence relation is three-termed. Therefore, the correct ansatz is:

$R_{L}(z)=e^{-\frac{\Lambda
z}{2}}z^{-\frac{m_{1}-m_{2}}{2}}(1+z)^{\frac{m_{1}-m_{2}-2(1+m_{3})}{2}}\sum_{n=0}^{\infty}c_{n}\left(\frac{z}{1+z}\right)^{n}$
(52)

If we plug (52) in (50), we obtain a three terms recursion relation of the
form:

$\alpha_{n}c_{n+1}+\beta_{n}c_{n}+\gamma_{n}c_{n-1}=0\quad,\quad n\geq 0$ (53)

where the coefficients are

$\displaystyle\alpha_{n}=$ $\displaystyle(1+n)(1+n-m_{1}+m_{2})$ (54)
$\displaystyle\beta_{n}=$
$\displaystyle{-}\left(m_{3}{+}n\right)(\Lambda{+}2n){-}m_{2}\left(\Lambda{+}m_{3}{+}2n{+}1\right){+}m_{1}\left(m_{2}{+}m_{3}{+}2n{+}1\right){-}m_{3}{-}2n{-}u{-}\frac{3}{4}$
$\displaystyle\gamma_{n}=$ $\displaystyle(n+m_{2}+m_{3})(n-m_{1}+m_{3})$

and where we chose $c_{-1}=0$. The QNMs associated to the overtone $n$ are
obtained by truncating the recursion and solving numerically the continuous
fraction:

$\beta_{n}=\frac{\alpha_{n}\gamma_{n}}{\beta_{n-1}-\frac{\alpha_{n-2}\gamma_{n-1}}{\beta_{n-2}-\dots}}+\frac{\alpha_{n}\gamma_{n+1}}{\beta_{n+1}-\frac{\alpha_{n+1}\gamma_{n+2}}{\beta_{n+2}-\dots}}$
(55)

viewed as an equation for the QNMs $\omega$.

The two methods exposed in Section 3 and in the current one can be applied in
both cases of TSs ($r_{b}>r_{s}$) and BSs ($r_{s}>r_{b}$).

##  5 Charged QNMs of Topological Star

In this Section we analyze the case of TSs. We compute the QNMs with qSW
technique and Leaver method; furthermore, we obtain the same quantities also
with a direct integration and then we compare the results obtained with these
different methods.

###  5.1 Computation of QNMs

The dictionary (36) is the fundamental ingredient required for implementing
the Leaver method (54), (55) and the SW quantization condition (3.3), (47),
(48). As demonstrated in Figure 1 and Tables 1, 2 and 3, only stable modes are
present. In the last column, by $SW_{s}$ we mean that the number of
instantonic corrections used in the SW procedure is equal to $s$.

The resulting QNM frequencies have also been validated against those obtained
using a direct integration method, which involves matching numerical solutions
in different integration regions. Starting from the cap, in accordance with
the solution $(\ref{RCAPts})$, we impose a regular boundary condition and
integrate the equation up to a specified extraction point. From this point,
following the solution $(\ref{RINF})$, we impose outgoing waves and continue
integration until reaching infinity. The two solutions can be matched by
requiring the vanishing of the Wronskian, which then becomes an eigenvalue
equation for QNM frequencies. In Table 2, the column associated to the
numerical integration is empty because this algorithm fails to be predictive
when the QNMs are highly damped, that is when their imaginary parts are close
to 1. Indeed in this scenario we need a huge number of subleading corrections
at the cap and at infinity to estimate QNMs with high imaginary parts.

As it can be seen, the results of the QNMs corresponding to the different
methods are in agreement. We expect that a larger number of instantonic
corrections in the qSW method will produce estimates closer to the ones
provided by numerical integration and Leaver method.

Figure 1: Plots of the modulus of the Leaver difference equation (55) in the complex $\omega$-plane for $r_{b}=1$, $r_{s}=0.8$, $\ell=2$, $Q=2$, $q=0.5$, $p=0$, $\mu=0$. The three plots at the bottom are zoomed-in snapshots of the regions around the first three zeros of the Leaver function. $\ell$ | Numerical integration | Leaver | $SW_{4}$
---|---|---|---
$0$ | $1.34819-0.218003\,i$ | $1.34819-0.218003\,i$ | $1.33277-0.261032\,i$
$1$ | $1.73838-0.126107\,i$ | $1.73838-0.126107\,i$ | $1.72053-0.117156\,i$
$2$ | $2.18231-0.0797013\,i$ | $2.18231-0.0797013\,i$ | $2.17005-0.0762405\,i$
$3$ | $2.63417-0.0517219\,i$ | $2.63417-0.0517219\,i$ | $2.6258-0.060272\,i$
Table 1: TS QNMs for different values of $\ell$ and fixed overtone $n=0$ and $r_{b}=1$, $r_{s}=0.8$, $Q=2$, $q=0.5$, $p=\mu=0$. $n$ | Numerical integration | Leaver | $SW_{2}$
---|---|---|---
$1$ | $-$ | $2.36322-0.330499\,i$ | $2.24082-0.293755\,i$
$2$ | $-$ | $2.60406-0.633878\,i$ | $2.83614-0.700827\,i$
$3$ | $-$ | $2.88498-0.955453\,i$ | $2.89907-0.926271\,i$
Table 2: TS QNMs for fixed $\ell=2$, $r_{b}=1$, $r_{s}=0.8$, $Q=2$, $q=0.5$, $p=\mu=0$ and for various overtone numbers. $n$ | Numerical integration | Leaver | $SW_{4}$
---|---|---|---
$0$ | $5.81013-0.000491187\,i$ | $5.81013-0.000491187\,i$ | $5.81895-0.0139528\,i$
$1$ | $5.97759-0.0411436\,i$ | $5.97759-0.0411436\,i$ | $5.96178-0.0271379\,i$
$2$ | $6.13468-0.17184\,i$ | $6.13468-0.17184\,i$ | $6.12144-0.14779\,i$
$3$ | $-$ | $6.31023-0.343684\,i$ | $6.30019-0.385102\,i$
Table 3: TS QNMs for fixed $\ell=10$, $r_{b}=1$, $r_{s}=0.8$, $Q=2$, $q=0.5$,
$p=\mu=0$ and for various overtone numbers.

##  6 Charged QNMs and Superradiance of Black String

In this Section we pass to examine the BSs. We explain why BSs can produce
charge superradiance and we also show that in TSs case, instead, this
phenomenon cannot take place. At this point, we can introduce the near
extremal, near superradiant limit together with the probe limit, arriving at
computing QNMs with the usual two methods of qSW and Leaver and now also with
the matching asymptotic expansion. Prompt Ring Down modes are expected as well
and indeed they are computed, in addition to the two persistent methods, with
a WKB approximation. At the end of this Section, the amplification factor of
superradiance for BSs is computed, keeping in count also the first instantonic
corrections.

###  6.1 Charge superradiance

TS and BS are static and charged geometries and hence they can only give rise
to charge superradiance. Focusing on BS, we have to determine first of all the
behaviour of the radial wave function $\Psi(z)$ at the boundaries of the
spacetime under consideration, that is $z=0$ and $z=\infty$. By solving (33)
with $Q_{12}$ given by (34) adapted at the two boundaries in exam, we get

$\Psi(z)\simeq\begin{cases}\mathcal{O}z^{\frac{1}{2}-i\,\,\frac{qQ-\omega
r_{s}}{\sqrt{1-\frac{r_{b}}{r_{s}}}}}+\mathcal{T}z^{\frac{1}{2}+i\,\,\frac{qQ-\omega
r_{s}}{\sqrt{1-\frac{r_{b}}{r_{s}}}}},\qquad\qquad\qquad z=0\\\
\mathcal{R}e^{i(r_{s}-r_{b})\,z\,\,\sqrt{\omega^{2}-q^{2}}}+\mathcal{I}e^{-i(r_{s}-r_{b})\,z\,\,\sqrt{\omega^{2}-q^{2}}},\quad
z\to+\infty\\\ \end{cases}$ (56)

where, following the convention of Brito:2015oca , $\mathcal{I}$ and
$\mathcal{R}$ are, respectively, the amplitudes of incident and reflected
waves (defined at spatial infinity $z=\infty$), while $\mathcal{T}$ and
$\mathcal{O}$ are the amplitudes of transmitted and outgoing waves across the
boundary $z=0$ (the horizon, in this geometry).
The probability currents associated to the two asymptotic solutions are:

$\displaystyle
J_{0}=\frac{2i(-qQ+r_{s}\omega)}{\sqrt{1-\frac{r_{b}}{r_{s}}}}(|\mathcal{O}|^{2}-|\mathcal{T}|^{2})$
(57) $\displaystyle
J_{\infty}=2\,i\,(r_{b}-r_{s})\sqrt{\omega^{2}-q^{2}}(|\mathcal{I}|^{2}-|\mathcal{R}|^{2})$
(58)

and they must be equal due to the conservation and this implies:

$|\mathcal{I}|^{2}=|\mathcal{R}|^{2}+\frac{(qQ-
r_{s}\omega)\sqrt{r_{s}}}{(r_{s}-r_{b})^{\frac{3}{2}}\sqrt{\omega^{2}-q^{2}}}(|\mathcal{O}|^{2}-|\mathcal{T}|^{2})$
(59)

Notice that if $\omega<\omega_{SR}=\frac{qQ}{r_{s}}$ then the wave is
superradiantly amplified ($|\mathcal{I}|^{2}<|\mathcal{R}|^{2}$) 777For a BH
the boundary conditions at the horizon involve only ingoing waves, so that
$\mathcal{O}=0$.. For this reason $\omega_{SR}$ will be denoted as
superradiance threshold.
This result is in completely agreement with the one we could obtain by using
the classical laws of BH thermodynamics, as firstly understood by Bekenstein
PhysRevD.7.949 . Indeed, the reduction from 5 to 4 dimensions for BS gives
rise to a RN BH with radius of the horizon equal to $r_{s}$. The first law of
thermodynamics for this kind of BH reads

$\delta M=\frac{T_{H}}{4}\delta A_{H}+\Phi_{H}\delta\mathcal{Q}$ (60)

where $T_{H}$, $A_{H}$, $\Phi_{H}$ are the temperature, the area and the
electric potential at the horizon, hence $\Phi_{H}=\mathcal{Q}/r_{s}$. If we
send a monochromatic wave with frequency $\omega$ and charge $q$, the
variation of the mass of the background in terms of its electric charge is

$\delta M=\delta\mathcal{Q}\frac{\omega}{q}$ (61)

and so, plugging in (60), we get

$\delta\mathcal{Q}=\frac{T_{H}q}{4}\frac{\delta A_{H}}{\omega-q\Phi_{H}}$ (62)

In order to have charge superradiance ($\delta\mathcal{Q}<0$), since $\delta
A_{H}\geq 0$ for the second law of BH thermodynamics, we must have
$\omega-q\Phi_{H}<0$. This is exactly what we found out in (59).

It is worth noticing that in the case of TS $(r_{b}>r_{s})$, it is not
possible to have superradiant amplification. This circumstance arises from the
fact that smooth geometries resolve the information paradox and therefore the
amplitude of the incident wave on our compact object will be equal to the
reflected one. Once the regularity is imposed at the cap of the geometry
$r=r_{b}$, the amplification curve obtained by numerically integrating the
radial wave equation is zero. This will be shown in Figure 2 together with the
amplification factor of BS.

###  6.2 Near extremal and near superradiant modes

In this subsection we will analytically point out some peculiar modes,
referred as near extremal, near superradiant modes, that will be computed by
exploiting some matching asymptotic expansion technique.
Firstly notice that in the near extremal limit ($r_{s}\to r_{b}$) the theory
decouples ($\Lambda\to 0$) and, in order to avoid divergences arising in the
dictionary (38), we must fix $\omega=\omega_{SR}$. In this way the dictionary
becomes:

$m_{1}=m_{2}=\Lambda=0,\quad u=\left(\ell+\frac{1}{2}\right)^{2},\quad
m_{3}=-\frac{iq}{2}\sqrt{Q^{2}-r_{s}^{2}}$ (63)

In order to reproduce results known in literature Consoli:2022eey ;
Starobinskil:1974nkd , we consider the approximation $m_{3}\simeq 0$, which is
consistent with $q\simeq 0$ 888This parameter can be small since it is
proportional to $R_{y}$., called probe limit.

Let us now apply the technique previously cited. The differential equation
(33) with $Q_{12}$ in (34) in the decoupling limit $\Lambda=0$ can be solved
in terms of hypergeometric functions. The solution that guarantees ingoing
waves at the horizon $z=0$ is

$\Psi(z)=z^{\frac{1}{2}(1-m_{1}+m_{2})}(1+z)^{\frac{1}{2}(1+m_{1}+m_{2})}{}_{2}F_{1}\Big{[}\frac{1}{2}+m_{2}-\sqrt{u},\frac{1}{2}+m_{2}+\sqrt{u},1-m_{1}+m_{2},-z\Big{]}$
(64)

The behaviour of the previous solution at infinity is

$\Psi(z)\sim
z^{\frac{1}{2}-\sqrt{u}}\left(1+{z^{2\sqrt{u}}\Gamma(\frac{1}{2}-m_{1}-\sqrt{u})\Gamma(\frac{1}{2}+m_{2}-\sqrt{u})\Gamma(2\sqrt{u})\over\Gamma(\frac{1}{2}-m_{1}+\sqrt{u})\Gamma(\frac{1}{2}+m_{2}+\sqrt{u})\Gamma(-2\sqrt{u})}\right)$
(65)

At $z=\infty$ the differential equation becomes

$\Psi^{\prime\prime}(z)+\left(-{m_{3}\Lambda\over z}-{\Lambda^{2}\over
4}+{1-4u\over 4z^{2}}\right)\Psi(z)=0$ (66)

whose general solution is

$\Psi(z)=\sum_{\alpha=\pm}A_{\alpha}e^{{\alpha\Lambda z\over
2}}(z\Lambda)^{{1\over 2}{+}\sqrt{u}}U\Big{[}{1\over 2}{-}\alpha
m_{3}{+}\sqrt{u},1{+}2\sqrt{u},{-}\alpha\Lambda z\Big{]}$ (67)

In order to have only outgoing waves at infinity, we are forced to set
$A_{+}=0$ (and $A_{-}=1$ without loss of generality). After that we can expand
the previous (67) solution at the horizon

$\Psi(z)\sim z^{{1\over
2}-\sqrt{u}}\left(1+{z^{2\sqrt{u}}\Lambda^{2\sqrt{u}}\Gamma({1\over
2}+m_{3}+\sqrt{u})\Gamma(-2\sqrt{u})\over\Gamma\left(\frac{1}{2}+m_{3}-\sqrt{u}\right)\Gamma(2\sqrt{u})}\right)$
(68)

The two solutions (65) and (68) match if:

$\Lambda^{2\sqrt{u}}-{\Gamma(1+2\sqrt{u})^{2}\Gamma({1\over
2}-m_{1}-\sqrt{u})\Gamma\left(\frac{1}{2}+m_{2}-\sqrt{u}\right)\Gamma\left(\frac{1}{2}+m_{3}-\sqrt{u}\right)\over\Gamma(1-2\sqrt{u})^{2}\Gamma\left(\frac{1}{2}-m_{1}+\sqrt{u}\right)\Gamma\left({1\over
2}+m_{2}+\sqrt{u}\right)\Gamma\left(\frac{1}{2}+m_{3}+\sqrt{u}\right)}=0$ (69)

Since the first term in (69) is going to zero, in order to satisfy the
equation we require that one of the $\Gamma-$functions in the denominator of
the second term must diverge. The correct choice is

${1\over 2}+m_{2}+\sqrt{u}=-n\quad,\quad n=0,1,2,...$ (70)

Using the dictionary (38) and replacing
$\omega=\omega_{SR}-\nu\sqrt{r_{s}-r_{b}}$, we can find the correction to the
superradiant frequency by expanding for small $r_{s}-r_{b}$:

$\omega_{match}={qQ\over r_{s}}-i{\ell+1+n\over
r_{s}^{3/2}}\sqrt{r_{s}-r_{b}}$ (71)

The observant reader will have noticed that the matching condition (69)
coincides with the quantization of the cycle $a_{D}$ as written in (48), where
only the tree-level and the 1-loop contributions are present. So, starting
from (69), in order to consider the istantonic corrections, let us replace
$\sqrt{u}=a$ with the $a$-cycle defined as (42) and we also add the
instantonic contribution of the prepotential (3.3), getting at the end

$\Lambda^{2a}-{\Gamma(1+2a)^{2}\Gamma({1\over
2}-m_{1}-a)\Gamma(\frac{1}{2}+m_{2}-a)\Gamma(\frac{1}{2}+m_{3}-a)\over\Gamma(1-2a)^{2}\Gamma(\frac{1}{2}-m_{1}+a)\Gamma({1\over
2}+m_{2}+a)\Gamma(\frac{1}{2}+m_{3}+a)}e^{\partial_{a}F_{inst}(\Lambda)}=0$
(72)

The results obtained with the matching asymptotic expansion (71), with SW
quantization (72) and with Leaver continued fraction (54),(55), using the
dictionary (38), are displayed in Tables 4 and 5.

$r_{b}$ | $\omega_{match}$ | Leaver | $SW_{4}$
---|---|---|---
$0.9$ | $1-0.948683\,i$ | $1.51407-1.10179\,i$ | $1.51404-1.10182\,i$
$0.95$ | $1-0.67082\,i$ | $1.06253-0.7356\,i$ | $1.06253-0.7356\,i$
$0.99$ | $1-0.3\,i$ | $1.01209-0.305512\,i$ | $1.01209-0.305512\,i$
$0.999$ | $1-0.0948683\,i$ | $1.0012-0.0950405\,i$ | $1.0012-0.0950407\,i$
Table 4: Near extremal near superradiant modes corresponding to the first overtone number $n=0$ for $r_{s}=1$, $\ell=2$, $q=0.5$, $Q=2$, $\mu=p=0$. $r_{b}$ | $\omega_{match}$ | Leaver | $SW_{4}$
---|---|---|---
$0.9$ | $1-1.26491\,i$ | $1.4123-1.39\,i$ | $1.41232-1.38995\,i$
$0.95$ | $1-0.894427\,i$ | $1.07965-1.00857\,i$ | $1.07965-1.00857\,i$
$0.99$ | $1-0.4\,i$ | $1.01595-0.409877\,i$ | $1.01596-0.409876\,i$
$0.999$ | $1-0.126491\,i$ | $1.0016-0.126801\,i$ | $1.0016-0.126801\,i$
Table 5: Near extremal near superradiant modes corresponding to the second
overtone number $n=1$ for $r_{s}=1$, $\ell=2$, $q=0.5$, $Q=2$, $\mu=p=0$.

###  6.3 Prompt Ring Down modes

As extensively shown in Bianchi:2022qph ; Bianchi:2023sfs ; DiRusso:2024hmd ,
in general, within smooth and horizonless solutions, one can distinguish
between unstable and metastable modes, corresponding to frequencies near the
maximum or minimum of the effective potential, respectively. Unstable
frequencies are also referred to as prompt ring down modes, bearing in mind
the analogy with BH mergers. Given the shape of the effective potential of the
BS, which exhibits only one unstable light ring, we expect such modes to be
present in the spectrum of QNMs and we aim to provide a WKB estimate of their
frequency values. Exploiting the results shown in Bianchi:2021mft , prompt
ring down frequencies must satisfy the Bohr-Sommerfeld quantization condition

$\int_{z_{-}}^{z_{+}}\sqrt{Q(z)}dz=\pi\left(n+\frac{1}{2}\right)$ (73)

with $n$ the overtone number and $z_{\pm}$ the two turning points. In the
semiclassical limit the turning points collide and the integral can be
approximated as

$\int_{z_{-}}^{z_{+}}\sqrt{Q(z)}dz\simeq\int_{z_{-}}^{z_{+}}\sqrt{Q(z_{c})+\frac{Q^{\prime\prime}(z_{c})}{2}(z-z_{c})^{2}}dz\simeq\frac{i\pi
Q(z_{c})}{\sqrt{2Q^{\prime\prime}(z_{c})}}$ (74)

where $z_{c}\in[z_{-},z_{+}]$ the critical point $Q^{\prime}(z_{c})=0$. So we
are left with

$\frac{Q(z_{c})}{\sqrt{2Q^{\prime\prime}(z_{c})}}=-i\left(n+\frac{1}{2}\right)$
(75)

The equation can be solved perturbatively by substituting the information that
$\omega=\omega_{\text{R}}+i\omega_{\text{I}}$ and expanding for small
$\omega_{\text{I}}$. It is easy to see that $\omega_{\text{R}}$ is simply the
critical frequency at the unstable photon sphere. The results are shown in
Tables 6 and 7 and are compared with the results provided by the Leaver method
and SW quantization.

$\ell$ | WKB | Leaver | $SW_{4}$
---|---|---|---
$0$ | $0.884096-0.402014\,i$ | $1.04019-0.582305\,i$ | $1.0416-0.571598\,i$
$1$ | $1.31439-0.144228\,i$ | $1.36621-0.13088\,i$ | $1.37756-0.138392\,i$
$2$ | $1.68218-0.13711\,i$ | $1.71123-0.133916\,i$ | $1.71388-0.108494\,i$
$3$ | $2.05574-0.135935\,i$ | $2.07588-0.134619\,i$ | $2.06643-0.135559\,i$
$4$ | $2.43315-0.135379\,i$ | $2.44854-0.134684\,i$ | $2.44221-0.138805\,i$
$5$ | $2.81279-0.134979\,i$ | $2.82523-0.134556\,i$ | $2.82053-0.143357\,i$
$6$ | $3.19382-0.134657\,i$ | $3.20425-0.134375\,i$ | $3.20389-0.143435\,i$
$7$ | $3.57575-0.134387\,i$ | $3.58472-0.134186\,i$ | $3.58429-0.144436\,i$
$8$ | $3.9583-0.134157\,i$ | $3.96617-0.134008\,i$ | $3.9698-0.145597\,i$
$9$ | $4.34129-0.13396\,i$ | $4.34831-0.133845\,i$ | $4.35221-0.144408\,i$
$10$ | $4.72462-0.133788\,i$ | $4.73094-0.133697\,i$ | $4.73747-0.146576\,i$
Table 6: Prompt ring down modes for the first overtone number $n=0$ and $r_{s}=1$, $r_{b}=0.8$, $q=0.5$, $Q=2$, $p=\mu=0$ for various $\ell$. $r_{b}$ | $WKB$ | Leaver | $SW_{4}$
---|---|---|---
$0.5$ | $1.67435-0.180019\,i$ | $1.72041-0.170672\,i$ | $1.70821-0.156853\,i$
$0.6$ | $1.67723-0.166767\,i$ | $1.71766-0.159428\,i$ | $1.7079-0.144709\,i$
$0.7$ | $1.67991-0.152547\,i$ | $1.71465-0.147257\,i$ | $1.70806-0.130406\,i$
$0.8$ | $1.68218-0.13711\,i$ | $1.71123-0.133916\,i$ | $1.71388-0.108494\,i$
$0.9$ | $1.6836-0.120151\,i$ | $1.70704-0.119066\,i$ | $1.71739-0.121633\,i$
Table 7: Prompt ring down modes for the first overtone number $n=0$ and
$r_{s}=1$, $\ell=2$, $q=0.5$, $Q=2$, $p=\mu=0$ for various $r_{b}$.

###  6.4 Amplification factor

The general solution at infinity (67) has to be matched with (65), encoding
ingoing boundary conditions at the horizon. This happens if

$\displaystyle A_{+}{=}$
$\displaystyle\frac{\Gamma\left(\frac{1}{2}{-}m_{3}{-}\sqrt{u}\right)\Gamma\left(\frac{1}{2}{-}m_{3}{+}\sqrt{u}\right)}{\Gamma\left(\frac{1}{2}{-}m_{3}{+}\sqrt{u}\right)\Gamma\left(\frac{1}{2}+m_{3}{-}\sqrt{u}\right){-}({-}1)^{2\sqrt{u}}\Gamma\left(\frac{1}{2}{-}m_{3}{-}\sqrt{u}\right)\Gamma\left(\frac{1}{2}{+}m_{3}{+}\sqrt{u}\right)}\times$
(76)
$\displaystyle\Bigg{[}\frac{\Gamma\left(2\sqrt{u}\right)\Gamma\left(\frac{1}{2}{-}m_{1}{-}\sqrt{u}\right)\Gamma\left(\frac{1}{2}{+}m_{2}{-}\sqrt{u}\right)\Gamma\left(\frac{1}{2}{+}m_{3}{-}\sqrt{u}\right)}{\Gamma\left({-}2\sqrt{u}\right)}+$
$\displaystyle{-}\frac{\Lambda^{2\sqrt{u}}\Gamma\left({-}2\sqrt{u}\right)\Gamma\left(\frac{1}{2}{-}m_{1}{+}\sqrt{u}\right)\Gamma\left(\frac{1}{2}{+}m_{2}{+}\sqrt{u}\right)\Gamma\left(\frac{1}{2}{+}m_{3}{+}\sqrt{u}\right)}{\Gamma\left(2\sqrt{u}\right)}\Bigg{]}$
$\displaystyle A_{-}{=}$
$\displaystyle\frac{\Gamma\left(\frac{1}{2}{+}m_{3}{-}\sqrt{u}\right)\Gamma\left(\frac{1}{2}{+}m_{3}{+}\sqrt{u}\right)}{\Gamma\left(\frac{1}{2}{-}m_{3}{+}\sqrt{u}\right)\Gamma\left(\frac{1}{2}{+}m_{3}{-}\sqrt{u}\right){-}({-}1)^{2\sqrt{u}}\Gamma\left(\frac{1}{2}{-}m_{3}{-}\sqrt{u}\right)\Gamma\left(\frac{1}{2}{+}m_{3}{+}\sqrt{u}\right)}\times$
(77)
$\displaystyle\Bigg{[}\frac{\Gamma\left(-2\sqrt{u}\right)\Lambda^{2\sqrt{u}}\Gamma\left(\frac{1}{2}{-}m_{1}+\sqrt{u}\right)\Gamma\left(\frac{1}{2}{+}m_{2}+\sqrt{u}\right)\Gamma\left(\frac{1}{2}{-}m_{3}{+}\sqrt{u}\right)}{\Gamma\left(2\sqrt{u}\right)}+$
$\displaystyle-\frac{\Gamma\left(2\sqrt{u}\right)\Gamma\left(\frac{1}{2}-m_{1}-\sqrt{u}\right)\Gamma\left(\frac{1}{2}+m_{2}-\sqrt{u}\right)\Gamma\left(\frac{1}{2}-m_{3}-\sqrt{u}\right)}{\Gamma\left(-2\sqrt{u}\right)}\Bigg{]}$

At infinity, the behaviour of (67) is

$\Psi(z)\sim Y_{+}e^{\frac{\Lambda z}{2}}z^{m_{3}}{+}Y_{-}e^{-\frac{\Lambda
z}{2}}z^{-m_{3}}$ (78)

$Y_{+}=-(-1)^{\frac{1}{2}+m_{3}-\sqrt{u}}\Lambda^{m_{3}}A_{+}\quad,\quad
Y_{-}=\Lambda^{-m_{3}}A_{-}$ (79)

The amplification factor is defined as Brito:2015oca ; Consoli:2022eey

$Z=\Bigl{|}\frac{Y_{+}}{Y_{-}}\Bigr{|}^{2}-1$ (80)

Introducing the quantity

$Q_{\rm{grav}}=\frac{qQ-\omega r_{s}}{\sqrt{1-\frac{r_{b}}{r_{s}}}}$ (81)

in the near-decoupling limit ($\Lambda\ll 1$), the dictionary (38) becomes

$m_{1}=-m_{2}=-\text{i}Q_{\text{grav}},\qquad
m_{3}=\frac{q}{2}\sqrt{r_{s}^{2}-Q^{2}},\qquad
u=\left(\ell+\frac{1}{2}\right)^{2}$ (82)

Recalling we also work in the probe limit $q\simeq 0$, we impose
$m_{3}\rightarrow 0$.

In these approximations, the amplification factor can be written as:

$Z\sim\left|\frac{1-\Sigma}{1+\Sigma}\right|^{2}-1\sim-4\text{Re}\Sigma$ (83)

where

$\Sigma=\frac{\Lambda^{2\sqrt{u}}\Gamma\left({1\over
2}+\sqrt{u}\right)\Gamma\left({1\over
2}-m_{1}+\sqrt{u}\right)\Gamma\left({1\over
2}+m_{2}+\sqrt{u}\right)\Gamma\left(-2\sqrt{u}\right)^{2}}{\Gamma\left({1\over
2}-\sqrt{u}\right)\Gamma\left({1\over
2}-m_{1}-\sqrt{u}\right)\Gamma\left({1\over
2}+m_{2}-\sqrt{u}\right)\Gamma\left(2\sqrt{u}\right)^{2}}$ (84)

Plugging in the previous equation (84) the approximated dictionary (82) and
expanding for small $Q_{grav}$ we obtain:

$\Sigma=\frac{i(-1)^{\ell}2^{-1-4\ell}Q_{grav}\Lambda^{2\ell+1}\Gamma\left(1+\ell\right)^{4}}{\Gamma(2+2\ell)^{2}\Bigl{[}\left({\frac{1}{2}}\right)_{\ell}\Bigr{]}^{2}}$
(85)

Figure 2: Comparison among different BS amplification factors: the red line
represent the curve obtained by numerical integration while blue, yellow and
green curves represent (83) with (86) and with 0, 1 and 2 instantonic
corrections respectively. The chosen parameters are $r_{s}=1$, $r_{b}=0.8$,
$\ell=2$, $q=0.5$, $Q=2$, $p=\mu=0$. The orange line refers to the
amplification factor of TS with the same parameters as BS with $r_{b}$ and
$r_{s}$ exchanged.

In order to take the instantonic corrections into account we have to perform
the replacement $\ell\rightarrow a-\frac{1}{2}$:

$\Sigma=\frac{\pi^{2}(-1)^{a}2^{1-8a}Q_{grav}\Lambda^{2a}\Gamma\left(a+\frac{1}{2}\right)^{2}}{\Gamma(a)^{2}\Gamma(a+1)^{2}}$
(86)

Unfortunately, as it is shown in Figure 2, the first instantonic corrections
do not get closer to the numerical curve (see Appendix C), stating that the
convergence would require a larger number of instantonic corrections. In the
same Figure, we display the amplifaction factor for TS and it is clear that it
is zero within the numerical precision of the software employed in the
calculation (Wolfram Mathematica).

The superradiance here observed for the charged non-rotating BS is not linked
to the superradiant instability highlighted in Press:1972zz ; Cardoso:2005gj ;
Cardoso:2005vk : indeed, the BS lacks the reflection mechanism needed to trap
and enhance the outgoing perturbation.

##  7 Conclusions and Outlook

Starting from the suggestion of Bianchi:2023sfs , we have performed a detailed
analysis of linear massive scalar charged perturbations both for Topological
Stars (TSs) and for Black Strings (BSs). The probe used in order to accomplish
this goal is a stringy object winding around the compact dimension $y$.

The wave equation describing the dynamics of this problem is a CHE with two
regular singularities (at the cap $r=r_{b}$ and at the horizon $r=r_{s}$) and
one irregular singularity at infinity. Several methods have been implemented
with the aim of computing QNMs. One of them consists in exploiting the
correspondence with qSW curves for $\mathcal{N}=2$ SYM with $SU(2)$ gauge
group and $N_{f}=(1,2)$ fundamental hypermultiplets and another one is the
Leaver method. These two have been used both for TSs and BSs, while other
techniques have been peculiarly employed for the two types of solutions.

For TSs, a direct numerical integration is used, as well. The QNMs determined
with the three approaches are reported in Tables 1, 2 and 3. Concerning BSs,
the matching asymptotic expansion for near extremal and near superradiant
modes is used, while for prompt ring down modes a WKB approximation is
exploited. The near extremal and near superradiant QNMs computed with the
previous three approaches are reported in Tables 4 and 5, while the prompt
ring down modes are presented in Tables 6 and 7. We find good agreement among
the applied methods.

As it can be seen, for all the cases $\text{Im}\omega<0$, confirming the
stability with respect to linearized charged and scalar perturbations. Despite
TS and JMaRT soliton are both non-BPS smooth horizonless geometries, the
former seems not to be affected by charged linear instability. Then, as argued
in Bianchi:2023rlt , the culprit of JMaRT ergoregion and charge instability
should not to be identified with non-extremality, but with the over-rotation.

In addition, we studied the charge superradiance of TSs and BSs. As exposed in
Brito:2015oca , rotational superradiance occurs for geometries with a horizon
inside an ergoregion. Our focus, instead, has been on static backgrounds which
then can only give rise to a charge superradiance - very much as Reissner -
Nordström Brito:2015oca ; DiMenza:2014vpa ; Benone:2015bst ; Baake:2016oku  \-
due to the presence of a horizon and a charged probe with respect to the
electromagnetic fluxes of the solutions999We are aware of the fact that
superradiance can also take place for horizonless geometries Brito:2015oca ,
leading to superradiant instabilities. Nonetheless, as highlighted in this
paper, this does not happen for TSs.. Our results confirm charge superradiance
for BSs; TSs, instead, do not superradiate, as expected for fuzzball
geometries which conserve information. This is reported in Figure 2. The first
instantonic corrections are taken into consideration for the computation of
the amplification factor of BS. As it can be argued from the Figure 2, we
expect that the convergence is reached by increasing the number of instantonic
corrections. This agrees with results shown in DiRusso:2024hmd .

We have also had the opportunity to study the Tidal Love Numbers of TSs and
BSs in the case of charged probes. Focusing on the former, we find that these
numbers exhibit the same peculiarities of the ones for the case of neutral
probes as analysed in DiRusso:2024hmd , that is there are poles in
correspondence of metastable QNMs.

We have established the stability of these geometries only for linear scalar
perturbations. As a future development, we aim to investigate the stability of
modes of higher spins (e.g. vector bosons and gravitons) or in the non-linear
case.

## Acknowledgments

We are grateful to M. Bianchi and J. F. Morales for their invaluable support
through helpful discussions and insightful comments on the manuscript. We also
thank A. Argenzio, Y. F. Bautista, I. Bena, V. Bevilacqua, D. Bini, G.
Bonelli, V. Collazuol, V. Cuozzo, S. De Angelis, A. De Santis, G. Dibitetto,
G. Di Ubaldo, R. Dulac, A. Evangelista, D. Fioravanti, G. Frittoli, F. Fucito,
C. Gambino, C. Gasbarra, A. Grassi, L. Grimaldi, C. Iossa, F. Margari, A.
Ruiperez, L. Tabarroni, A. Tanzini, A. Tokareva for fruitful scientific
exchanges. G.D.R. thanks IPhT Paris-Saclay for the kind hospitality during
completion of this work.

## Appendix

##  Appendix A Electric field of a charged wire

Consider an infinitely extended and uniformly charged wire along the $y$
direction in $\mathbb{R}^{1,D-1}$. The metric of this space is

$\mathrm{d}s^{2}=-\mathrm{d}t^{2}+\mathrm{d}y^{2}+\mathrm{d}r^{2}+r^{2}\mathrm{d}\Omega_{D-3}^{2}$
(87)

where $\mathrm{d}\Omega_{D-3}^{2}$ is the metric of a $(D-3)$ \- sphere of
unitary radius. The transverse space $\mathcal{M}^{D-2}$ has the topology of
$\mathbb{R}^{+}\times S^{D-3}$. For a $(D-3)$-sphere embedded in a
$(D-2)$-dimensional (flat) space, we introduce $D-3$ angles
$\theta_{1},\dots,\theta_{D-3}$ and only one of them is azimuthal
($\theta_{D-3}\in[0,2\pi)$), while all the other ones are polar
($\theta_{1},\dots,\theta_{D-4}\in[0,\pi)$). The induced metric is

$\mathrm{d}\Omega^{2}_{D-3}=\sum_{j=1}^{D-3}\left(\prod_{i=1}^{j-1}\sin^{2}{\theta_{i}}\right)\mathrm{d}\theta_{j}^{2}$
(88)

In the transverse space the problem presents a radial symmetry, so the
electric gauge potential sourced by the wire is

$B^{(e)}=-\Phi(r)\,\mathrm{d}t\wedge\mathrm{d}y$ (89)

Hence, the electric field strength is

$F^{(e)}=\mathrm{d}B^{(e)}=E(r)\,\mathrm{d}r\wedge\mathrm{d}t\wedge\mathrm{d}y$
(90)

The Hodge dual is

$\star
F^{(e)}=-E(r)\,r^{D-3}\sqrt{|g_{S^{D-3}}|}\,\mathrm{d}\theta_{1}\wedge\dots\wedge\mathrm{d}\theta_{D-3}$
(91)

where $g_{S^{D-3}}$ is the determinant of (88). By Maxwell’s equations101010In
this expression we have used that, for a generic $k$-form $A_{k}$ in a
D-dimensional spacetime,  $\star\star A_{k}=(-)^{s}(-)^{k(D-k)}A_{k}$ (92) For
(87), $s=1$ and $\mathrm{d}\star F^{(e)}$ is a $(D-2)$-form. ,

$\mathrm{d}\star F^{(e)}=-\star j^{(e)}$ (93)

and the 2-form electric current is
$j^{(e)}=-\frac{\rho}{\epsilon_{0}}\mathrm{d}t\wedge\mathrm{d}y$, with $\rho$
the electric volume density, so

$\star
j^{(e)}=\frac{\rho}{\epsilon_{0}}\,r^{D-3}\,\sqrt{|g_{S^{D-3}}|}\,\mathrm{d}r\wedge\mathrm{d}\theta_{1}\wedge\dots\wedge\mathrm{d}\theta_{D-3}$
(94)

Due to the fact that $\rho$ is the electric volume density, then
$\rho=\mathcal{Q}/(L_{y}V^{D-2})$, where $\mathcal{Q}$ is the electric charge
along the length $L_{y}$ of the wire and $V^{D-2}$ is the volume of the
transverse space. So $\rho=\lambda/V^{D-2}$, with $\lambda$ linear electric
density, and consequently

$\int_{\mathcal{M}^{D-2}}\mathrm{d}\star
F^{(e)}=-\int_{\mathcal{M}^{D-2}}\star
j^{(e)}=-\frac{\lambda}{\epsilon_{0}V^{D-2}}V^{D-2}=-\frac{\lambda}{\epsilon_{0}}$
(95)

On the other side, thanks to Stokes’ theorem, we can write

$\int_{\mathcal{M}^{D-2}}\mathrm{d}\star F^{(e)}=\int_{S^{D-3}}\star F^{(e)}$
(96)

where we are assuming that the transverse space $\mathcal{M}^{D-2}$ has as
boundary the ($D-3$)-sphere at fixed $r$ embedded in it. So

$\int_{S^{D-3}}\star
F^{(e)}=-E(r)r^{D-3}\int_{S^{D-3}}\sqrt{|g_{S^{D-3}}|}\,\mathrm{d}\theta_{1}\wedge\dots\wedge\mathrm{d}\theta_{D-3}=-V(S^{D-3})E(r)r^{D-3}$
(97)

Equalling (95) with (97), we get

$E(r)=\frac{\lambda}{V(S^{D-3})\epsilon_{0}r^{D-3}}$ (98)

It is known that

$V(S^{D-3})=\frac{2\pi^{\frac{D-2}{2}}}{\Gamma\left(\frac{D-2}{2}\right)}$
(99)

and so

$E(r)=\frac{\lambda\Gamma\left(\frac{D-2}{2}\right)}{2\epsilon_{0}\pi^{\frac{D-2}{2}}r^{D-3}}$
(100)

Notice that for $D=4$ we get the well known result of standard
Electromagnetism in $1+3$ dimensions.
Comparing (100) with $Q/r^{2}$ of (4) 111111We stress that, even if the probe
has finite length while now we have considered an infinite length of the wire,
the quantity $F^{(e)}$ is dimensionally the same., we realize that the
parameter $Q$ has the dimensions of a linear charge density and in measure
units such that

$\frac{1}{4\pi\epsilon_{0}}=1$ (101)

they are the same.

##  Appendix B Measure units

We choose to work in measure units such that

$c=G_{5}=\frac{1}{4\pi\varepsilon_{0}}=1$ (102)

In this framework, from Schwarzschild-Tangherlini metric Tangherlini:1963bw ,
we gain

$[M]=[L]^{D-3}=[L]^{2}$

Consequently, the force has dimension $[F]=[M]\,[L]^{-1}=[L]$ and, from (100),
$[F]=[q][E]=[q]^{2}[L]^{-3}$, we get that the dimension of the electric charge
is

$[q]=[L]^{2}$

Finally, the dimension of the linear charge density is

$[Q]=[q][L]^{-1}=[L]$

The fact that $[M]=[L]^{2}$ and $[Q]=[L]$ is consistent with the conditions
(7) and (10). Notice that in these measure units it is inconsistent to set
$\hbar=1$ and indeed, due to the fact that $[E]=[M]=[L]^{2}$, then

$[\hbar]=[E][L]=[L]^{3}$

Let us notice that according to these measure units, the tension of a string
has dimension

$[T]=[M][L]^{-1}=[L]$ (103)

hence the constant $\alpha^{\prime}$ is such that

$[\alpha^{\prime}]=[T]^{-1}=[L]^{-1}$ (104)

In this way, the action (12) has the correct dimension $[S]=[\hbar]=[L]^{3}$.
Moreover the parameter $q$ defined in (18) has dimension

$[q]=[L]^{2}$ (105)

as expected for a charge and consistently with (19).

##  Appendix C Numerical amplification factor

In this Appendix, we briefly describe the numerical procedure implemented in
Mathematica used to plot the amplification curve. The starting point consists
in finding the leading and a sufficient number of subleading behaviours at
infinity and at the cap. At infinity, the radial wave function behaves as:

$R_{\infty}(r)\simeq
e^{i\epsilon\,r\sqrt{\omega^{2}-q^{2}}}r^{-1-\frac{i\epsilon(q^{2}r_{s}+2qQ\omega-(r_{b}+2r_{s})\omega^{2})}{2\sqrt{\omega^{2}-q^{2}}}}\sum_{n=0}^{N_{\infty}}d_{n}r^{-n}$
(106)

where $\epsilon=\pm 1$ encodes both the behaviour at infinity and where we set
$\mu=p=0$ for simplicity. The first coefficient is:

$\displaystyle\frac{d_{1}}{d_{0}}=$
$\displaystyle\frac{q^{2}\left(2r_{b}+r_{s}\right)-\omega^{2}r_{b}-2qQ\omega}{4\left(q^{2}-\omega^{2}\right)}-\frac{1}{8\epsilon\left(\omega^{2}-q^{2}\right)^{3/2}}\Big{[}i(4q^{3}Q\omega\left(2r_{b}+3r_{s}\right)+$
(107)
$\displaystyle-2\omega^{2}\left(q^{2}\left(3r_{b}r_{s}+2r_{b}^{2}+6r_{s}^{2}\right)+2\ell(\ell+1)\right)-4qQ\omega^{3}\left(r_{b}+2r_{s}\right)+\omega^{4}\left(4r_{b}r_{s}+3r_{b}^{2}+8r_{s}^{2}\right)+$
$\displaystyle+q^{4}\left(3r_{s}^{2}-4Q^{2}\right)+4q^{2}\ell(\ell+1))\Big{]}$
$\displaystyle\dots$

For a TS $(r_{b}>r_{s})$ we have to impose regularity at the cap, so the
correct ansatz is:

$R_{c}(r)\simeq\sum_{n=0}^{N_{c}}c_{n}(r-r_{b})^{n}$ (108)

where

$\displaystyle\frac{c_{1}}{c_{0}}=$ $\displaystyle-\frac{r_{b}(qQ-
r_{b}\omega)^{2}-(r_{b}-r_{s})\ell(\ell+1)}{(r_{b}-r_{s})^{2}}$ (109)
$\displaystyle\frac{c_{2}}{c_{0}}=$
$\displaystyle\frac{1}{4(r_{b}-r_{s})^{4}}\Big{[}3\omega^{2}r_{b}^{2}\left(2q^{2}Q^{2}r_{b}^{2}+r_{s}\left(r_{b}-r_{s}\right)\right)-2qQ\omega
r_{b}\left(r_{b}^{2}\left(2q^{2}Q^{2}+1\right)+r_{b}r_{s}-2r_{s}^{2}\right)+$
$\displaystyle{+}q^{2}\left(q^{2}Q^{4}r_{b}^{2}{+}Q^{2}\left(r_{b}{-}r_{s}\right)\left(2r_{b}{+}r_{s}\right){+}r_{b}\left(r_{b}{-}r_{s}\right)^{3}\right){-}\ell^{2}\left(r_{b}{-}r_{s}\right)\left(r_{b}\left(2\left(qQ{-}\omega
r_{b}\right)^{2}{+}1\right){-}r_{s}\right){+}$
$\displaystyle{-}2\ell\left(r_{b}{-}r_{s}\right)\left(r_{b}\left(qQ{-}\omega
r_{b}\right)^{2}{+}r_{b}{-}r_{s}\right){-}4qQ\omega^{3}r_{b}^{5}{+}\ell^{4}\left(r_{b}{-}r_{s}\right)^{2}{+}2\ell^{3}\left(r_{b}{-}r_{s}\right)^{2}{+}\omega^{4}r_{b}^{6}\Big{]}$
$\displaystyle\dots$

Otherwise for a BS the boundary conditions at the horizon require an ingoing
wave:

$R_{H}(r)=(r-r_{s})^{\frac{i(qQ-
r_{s}\omega)}{\sqrt{1-\frac{r_{b}}{r_{s}}}}}\sum_{n=0}^{N_{H}}\tilde{c}_{n}(r-r_{s})^{n}$
(110)

where

$\displaystyle\frac{\tilde{c}_{1}}{\tilde{c}_{0}}=$
$\displaystyle\frac{1}{\left(r_{b}{-}r_{s}\right)^{2}\left(i\sqrt{1{-}\frac{r_{b}}{r_{s}}}{-}2qQ{+}2\omega
r_{s}\right)}\Big{[}\left(r_{b}{-}r_{s}\right)\left(\omega
r_{s}{-}qQ\right){+}i\sqrt{1{-}\frac{r_{b}}{r_{s}}}(q^{2}\left(Q^{2}r_{b}{+}r_{s}\left(r_{b}{-}r_{s}\right)^{2}\right){+}$
(111) $\displaystyle{+}2qQ\omega
r_{s}\left(r_{s}-2r_{b}\right)+\omega^{2}r_{s}^{2}\left(3r_{b}-2r_{s}\right)+\ell(\ell+1)\left(r_{s}-r_{b}\right))\Big{]}$
$\displaystyle\dots$

The numerical integration is performed starting from the horizon (or the cap)
and proceeding up to infinity. If we denote $A_{\text{in}}$ and
$A_{\text{out}}$ as the zero-order coefficients $d_{0}$ of the asymptotically
expanded wave at infinity (106), respectively corresponding to $\epsilon=-1$
and $\epsilon=1$, and if we compare this function with its numerical
counterpart along with their derivatives, we obtain expressions for
$A_{\text{in}}$ and $A_{\text{out}}$. The final step consists in plotting the
quantity $Z=\left|\frac{A_{\text{out}}}{A_{\text{in}}}\right|^{2}-1$ as a
function of $\omega$.

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