Tidal resonances for fuzzballs

We study the gravitational tidal response of D1D5, Top Star and (1,0,n) strata horizonless geometries. We find that the tidal interactions in fuzzball geometries, unlike in the case of black holes, exhibits a sequence of resonant peaks associated to the existence of metastable bound states. The spectrum of resonant frequencies is computed by semi-analytical and numerical methods.


Introduction
Black holes (BH) are objects which have always been of extreme interests for physicists since their "discovery" as solutions of Einstein's equations at the beginning of the past century.For many years they have been at the center of an intense theoretical research which could have been considered a mathematical amusement until their indirect detection [1,2].Nowadays they have become real physical objects [3] and there are reasonable hopes that their properties could be measured experimentally via the detection of gravitational waves [4][5][6][7][8].
The main feature of BH's is specified in their name: they are black.They absorb the impinging radiation.Nothing can escape the horizon which cloakes a classical space-time singularity.This is essentially the content of the weak cosmic censorship conjecture [9][10][11][12].But even if the geometry outside the horizon shows no pathologies the same cannot be said for the part inside the horizon.The solution to the latter problems might require quantum mechanics which, in turn, raises more problems like the information paradox see for example [13] and references therein.These problems lead to a more radical viewpoint questioning the existence of an horizon.Or, in other words, is it possible to distinguish BH's from exotic objects more massive than neutron stars but without an horizon?These objects have been collectively dubbed Exotic Compact Objects (ECO).This category comprises gravastars, wormholes, firewalls and fuzzballs just to name a few, see [14][15][16] for recent reviews.ECO's can be distinguished from BH's by their photon sphere shapes [17][18][19], multipolar structure [20][21][22][23][24][25], quasi normal modes (QNM) and their characteristic echo based gravitational waves (GW) emission [26].See [27] for a review.
The internal structure of a celestial object (a BH or an ECO) can be obtained studying their reaction under a gravitational perturbation.The mathematical tool to do so is the tidal Love/dissipation number (TLN) which were introduced in [28], see also [29][30][31].They were shown to vanish for Schwarzschild and Kerr BH's [30][31][32] but they are non trivial in higher dimensions, non asymptotically flat spaces, in alternative theories of gravity [33][34][35][36][37][38] and for BH-like compact objects [39,40].The TLN response of a gravity object codified the mixing between the solutions of the wave equation for a choice of boundary conditions, and it is imprinted in the Post Newtonian (PN) expansion of the gravitational wave signal radiated by a lighter mass inspiraling around it [41].It shows up at order 5PN beyond the quadrupole approximation, making hard its experimental measure.Aim of this work is to show that, in the case of fuzzballs, the TLN response exhibit resonant peaks in the line of frequencies amplifying the signal in a significant way.The study of TLN resonances will be the central focus of this paper.To carry on this task we will exploit the correspondence recently developed in [42][43][44][45][46][47] that relates gravity to N " 2 supersymmetric SU(2) gauge theories and/or two-dimensional conformal field theories (CFT), see also [41,[48][49][50][51][52][53].This correspondence exploits the results of Seiberg-Witten (SW) [54], localization [55][56][57] and AGT duality [58], to provide a combinatorial description of the wave functions describing the gravity systems at linear order.In this framework, the Post-Newtonian expansion of the gravitational wave emitted by a particle orbiting around a BH is described by a an instanton sum in a quiver gauge theory, or alternatively as a linear combination of conformal blocks of a two-dimensional CFT [41].The tidal response is given by a ratio of Gamma functions depending on a single holomorphic function: the quantum SW period apu, qq with Coulomb branch parameter u and coupling q parametrizing the orbital number and the wave frequency in gravity [47].
In this paper, we apply these ideas to the study of the tidal response for D1D5 [59], Topological Star [60,61] and D1D5p superstrata geometries of type (1,0,s) [62,63].Perturbations around these geometries are described by a differential equation of the Heun type and can be mapped to a SU p2q gauge theory with matter transforming in the fundamental representation [44,45,50,52].The crucial difference with respect to the case of BH's is the presence of poles in the connection matrix associated to the real frequencies where the argument of a Gamma function, in the tidal function, becomes a negative integer.We compare the spectrum of resonant frequencies with that of QNM's, i.e. solutions of the wave equation satisfying regular boundary conditions at the origin and outgoing ones at infinity.QNM's exist for discrete of choices of frequencies the complex plane.In the case of BH's, where the effective potential has only a maximum (at the photonsphere), they have typically a significantly large imaginary part leading to fast damping modes.Fuzzballs instead allow also for QNM frequencies with small imaginary part describing metastable bound states leaving near the minimum of the effective potential.We find that tidal interactions always blow up at these slowly damping QNM frequencies, providing a gravitational wave picture of the fuzzball interior.This is the organization of the paper: in Section 2 we review the gauge-gravity correspondence and illustrate the ideas in a toy model of the wave dynamics.In Section 3, 4 and 5 we discuss the cases of: D1D5 fuzzball, the (1,0,s) Strata and Top Star geometries respectively.In each case we study the spectrum of tidal resonances and compare them against that of QNM's.

TLN vs QNMs
Black holes and ECO's are often described by integrable geometries where the angular and radial motions of the gravitational perturbations can be separated and described in terms of a ordinary differential equations of Schrödinger type with z a radial or angular variable.For example, asymptotically AdS 4 Kerr-Newman BH's are described by Heun equations of type (2.1) with four regular singularities located at the BH horizons.Asymptotically flat BH's in four and five dimensions and a large family of D-brane bound states and fuzzballs are instead described by confluent Heun equations obtained by colliding two or more singularities at a point (typically zero or infinity).
The general solution can be always written as with Ψ α pzq given in terms of Heun functions and c α determined by the boundary conditions at the origin (the horizon for BH's).We will always set the boundary at z " 1, the radial infinity at z " 0, and choose the solutions Ψ α pzq such that with D the dimension and ℓ the orbital quantum number of the wave.The limit ω{z, z Ñ 0 describes the region (near zone) where distances are large with respect to the size of the object but still much smaller than the gravitational wavelength.In this limit, centrifugal forces dominate over the rest of the interactions.The growing and decreasing components, Ψ ´and Ψ `respectively, can be viewed accordingly as the "source " and "response" terms of the perturbation and the tidal response function Lpωq can be defined as The real and imaginary parts of Lpωq compute the dynamical Love and dissipation numbers of the geometry.The coefficients c ˘will be computed using the Heun connection formulae derived in [46,47], linking the behaviour of the Heun functions near the origin z " 1 to the radial infinity z " 0. QNM's on the other hand, are defined as outgoing modes in the opposite asymptotic region where z !ω.They are in general complex, and exist for discrete choices of the frequencies.

Tidal response resonances vs QNMs
We will show that for compact objects Lpωq has an infinite number of poles where the tidal response of the geometry blows up.This sequence of resonances will be related to the existence of metastable states (QNM's) confined in the interior of the photon sphere.They are characterised by QNM frequencies with small imaginary parts describing bound states living near the minimum of the effective potential.QNM frequencies will be computed by direct integration of the differential equation.They correspond to zeros of the Wronskian, Σ QNM pωq, computed between the solution Ψ in pzq satisfying regular boundary conditions at the origin and Ψ out pzq satisfying outgoing boundary conditions at infinity Σ QNM pωq " Ψ in pz ˚qΨ 1  out pz ˚q ´Ψ1 in pz ˚qΨ out pz ˚q " 0 (2.5) with z ˚an arbitrary point.We remind that the Wronskian is constant, so the result does not depend on the choice of z ˚.Since both the origin and infinity are always singular points, boundary conditions have to be imposed slightly off these points.The two solutions are approximated around these points by a series expansion that are used to fix the boundary conditions and are then extrapolated numerically to the interior point z ˚using Mathematica.
Alternatively, QNM frequencies can be estimated relying on a WKB approximation of the solution.In this framework the frequencies ω n are determined by the quantization condition with r 1,2 two zeros of Qpr, ω n q around its maximum.We notice that, in general, the zeros r 1,2 depend on ω n , so equation (2.6) is highly non-trivial, but it can still be solved numerically.

Gauge gravity correspondence
The Heun equation arises also in the study of gauge theories on curved spacetimes.For example, the dynamics of the N " 2 supersymmetric SU p2q gauge theory with four hypermultiplets living in a Nekrasov-Shatashvili [64] Ω-background with parameters ϵ 1 " 1, ϵ 2 " 0 is codified in the quantum differential equation with q the gauge coupling and P L pxq " px ´m1 qpx ´m2 q , P R pxq " px ´m3 qpx ´m4 q Here u parametrises the Coulomb branch and m i the masses.This gauge theory can be described by pairs of D4 branes stretched between two NS5 branes or between a NS5 and infinity on the left and on the right.The zeros of P L,R pxq, P pxq give the positions of the D4-branes in the three regions and parametrise the four masses and the Coulomb branch.We will label the hypermultiplet content by N f " p2, 2q and its decoupling limits (where some masses are sent to infinity) by N f " pN L , N R q. Equation (2.7) can be always brought to the form (2.1) by taking

Ψpzq
(2.9) leading to ´`m 1 `m2 The gauge theory can be alternatively described by a mirrored version of (2.7) given by after the identifications The use of this second picture will become clear later.Using any of the two equivalent descriptions, the dynamics of the gauge theory is codified into a single holomorphic function apu, qq, known as the Seiberg Witten quantum period.To compute it, it is convenient to Fourier transform (2.7), or (2.12), and bring it into the form of a difference equation [65,66].
The integrability condition of this difference equation can be written in the infinite fraction form [67] Σ TLN paq " qM pa `1q P pa `1q ´qMpa`2q (2.14) can be easily solved for apu, qq order by order in q.The starting point of the recursion is the free theory q " 0, where a « qÑ0 ?u.In this limit, the wave equations reduce to where c ˘, c3,4 are some constants.They are determined by requiring regularity at the origin z " 1.
Expanding the hypergeometric functions around z " 1, assuming that the masses are real1 and m 1 `m2 ą 0, m 3 `m4 ą 0, one finds that regularity at the origin requires (2.17) This fixes the ratios between the two asymptotic coefficients around z " 0 to be Turning on q, the differential equation becomes a Heun equation and the c's coefficients become the components of the connection matrix relating two basis of Heun functions.The crucial observation in [47] is that the ratio between the c's coefficients in the Heun case is given again by (2.18) with a « qÑ0 ?u replaced by the quantum SW period apu, qq.The whole non-triviality of the Heun connection matrix is codified into a single holomorphic function apu, qq.
The confluence limits of the Heun equation can be studied similarly by decoupling some masses in the gauge theory description.Notice that this operation breaks the Left-Right symmetry, so which description you use makes a difference.We find it convenient to use Lpωq or r Lpωq to describe the tidal response for theories with N L ą N R or N L ă N R .For example, the N f " p2, 1q is obtained in the limit q Ñ 0, m 4 Ñ 8, keeping finite their product qm 4 Ñ ´q.In this limit (2.7) keeps its form with the replacement P L pxq Ñ px ´m3 q.The differential equation becomes a confluent Heun equation.A further decoupling of the m 3 mass produces a N f " p2, 0q theory, described by a reduced confluent Heun equation.Similarly one can decouple the m 1,2 masses, using the formulae with the tilde leading to N f " p1, 2q and N f " p0, 2q gauge theories.
We will be mainly interested in the poles of the tidal functions.They appear for frequencies such that the argument of some of the gamma functions in (2.18) becomes a negative integer.In the case of BH's this never happens, since the masses are always purely imaginary and a is real.We find instead that the tidal response of fuzzball geometries exhibits typically an infinite sequence of resonances corresponding to solutions of We find that typically only one of the two equations admits real solutions.To find the frequencies, we will plug a obtained from (2.19) into (2.14) and look numerically for the zeros of Σ TLN pωq along the ω-line.

Toy wave dynamics
In this Section we illustrate the ideas of tidal resonances and their connection with QNM's in a toy model of the wave dynamics introduced in [52].We consider the quantum mechanics and ω 2 the particle energy.The general solution of (2.1) can be written as We consider the two extreme cases where the boundary of the space at x " 0 represents either a BH (ingoing wave only) or a fuzzball (perfectly reflecting mirror).They correspond to the choice of boundary conditions We can view the B (decreasing) and C (growing) components in the intermediate region in (2.21) as describing the amplitude of the "response" of the geometry to a "source" perturbation.The TLN is defined as the ratio of the response and source coefficients Lpωq " Bpωq Cpωq (2.24) Matching functions and derivatives at x " L one finds The important difference we observe already at this level, is that the BH TLN has no poles since β " ik 1 has no real solutions; while L Fuzz pωq exhibits an infinite number of poles located at the solutions of the trascendental equation: QNM's instead are defined as solutions involving only outgoing waves at infinity, i.e.D " 0. Matching functions and derivatives at x " L, L `∆ one finds that QNM frequencies are determined by the eigenvalue equations Fuzz : We notice that the equations defining the poles of L Fuzz pωq in (2.25) are very different in general from those in (2.27) defining QNM's.Still in the limit β∆ Ñ 8, the two conditions coincide.Indeed in this limit the right hand side of the first equation in (2.27) reduces to ´1, leading again to (2.26).The resonances of the tidal function are therefore associated to the QNM's describing the metastable states confined inside the cavity.

D1-D5 circular fuzzball
The D1D5 circular fuzzball is a smooth solution of minimal six-dimensional supergravity specified by a circular profile of radius a.The six-dimensional metric is given by with We have introduced the auxiliary parameter ϵ " 0, 1 that interpolates between asymptotically AdS 3 ˆS3 and flat geometries respectively.We consider a massive scalar perturbation satisfying The scalar wave equation is separable using the ansatz Φ " RprqSpχqe ip´ωt`m ψ ψ`m ϕ ϕq (3.4)The angular and radial equations read `ϵω 2 p2L 2 `a2 ϵq´Aq ‰ " 0 with χ " cos θ.The two equations can be written in the normal form (2.1) with `pL 2 ω ´am ϕ q 2 ´a2 pρ 2 `a2 q 2 In figure 2 we display the effective potential V pρq governing the radial motion, for some values of the parameters.We restrict ourselves to the case m ϕ " 0, where Q ρ pρq " ω 2 ´V pρq.We observe that for a small enough the potential for, both the asymptotically flat and the near horizon geometry, has always a minimum so it allows for the existence of metastable bound states.This is not the case for asymptotically flat D1D5 fuzzball with a large.To build the gauge gravity dictionary, it is convenient to move the singularities to the points 0, 1 and 8 and write Q in the form For the angular equation one finds the gauge gravity dictionary z χ " 1 χ 2 , Spχq " Ψpzq a p1 ´zq (3.9) while radial variables are given by z " a 2 ρ 2 `a2 , Rprq " Ψpzq a p1 ´zq , u " 1 4 r1`A`L 2 pµ 2 ´2ϵω 2 q`a 2 ϵpµ 2 ´ϵω 2 qs (3.10) The characteristic polynomials are P L pxq " px ´m1 qpx ´m2 q , P R pxq " 1 P pxq " x 2 ´u `q , M pxq " px ´m1 ´1 2 qpx ´m2 ´1 2 q (3.11) In the following we consider in turn the cases of asymptotically AdS 3 ˆS3 ϵ " 0 and asymptotically flat ϵ " 1 geometries.The crucial difference between the two cases is that the wave equation for ϵ " 0 is of hypergeometric type while that for ϵ " 1 is given by a reduced confluent Heun equation with an irregular singularity at z " 0.

Near horizon geometry
We start by considering the simple case ϵ " 0. For this choice, the gauge theory representing the fuzzball perturbations is free q " 0 and both angular and radial equations take a hypergeometric form with regular singularities at 0, 1, 8. Their solutions can therefore be written in terms of hypergeometric functions.In particular, the angular function Spχq can be written in terms of 5d spherical harmonics with separation constant The tidal response function Lpωq is given by (2.18) with resonant frequencies located at the solutions of (2.19) .Using the gauge gravity dictionary (3.10), one finds that the solutions with ω real exist only for those special values of m 1 leading to ω poles n " a L 2 p2n `2a `1 `mψ `mϕ q , n " 0, 1, 2, . . .
Finally we observe that in the massless limit µ Ñ 0, the sequences of zeros and poles accidentally fall on top of each other leading to a finite result for L. Indeed setting µ " 0 from the very beginning and expanding for small z one finds the logarithmic divergent tidal response Rpzq " Lpaq " ´śℓ a polynomial function in ω with no resonances.This can be observed in the left plot of figure 3 where no resonances show up for ϵ " 0.  2: TLN resonance vs QNM frequencies for D1D5 fuzzballs with L " ϵ " 1, ℓ " m ψ " 10, m ϕ " µ " 0, a " 0.1.
For ϵ " 1 the singularity at z " 0 is irregular and the solutions Ψ α are given in terms of the reduced confluent Heun functions rather than hypergeometric ones.The tidal response is described by Lpωq in (2.18), with apqq determined by the continuous fraction equation (2.14).We refer the reader to [50] for recent investigation of the Love number of circular D1D5 fuzzballs.To determine the resonant frequencies we first plug a " ´n ´1 2 ´m1 , n " 1, 2, . . .(3.18) into (2.14)truncated to a given instanton number q k .We also set a χ " 1 `ℓ 2 in the continuous fraction equation associated to the angular equation.We then use the dictionaries (3.9) and (3.10) to express the angular and radial equations as a system of two algebraic equations and solve it numerically for A and ω.For example, truncating to zero instanton order, one finds a « ?u `. . .that after using the gauge gravity dictionary (3.10) leads to A « ℓpℓ `2q `. . .and ω n « a L2 p2n `ℓ `mψ `mϕ `2q `. . .(3.19)We notice that at this order the result coincides with that of the near horizon geometry given by (3.14).This is not surprising, since for a small the minimum of the effective potential is located deep inside the near horizon geometry which provides a good description of the physics of metastable states.Higher instanton corrections can be incorporated by keeping more and more terms in the continuous fraction equations.We find two solutions for the m 1 -dependent pole equation for each n and no one for the equation involving m 2 .
The results converge very fast, reaching four-five digits accuracy already at three or four instanton order for a " 0.1.The results for the frequencies ω for some specific choices of parameters are displayed in table 1-2.The results are compared against those based on the WKB method and direct numerical integration of the differential equation 2 .The match between QNM frequencies and poles of the tidal response is impressive.The agreement can be also appreciated in the right plot of figure 3 where we display the characteristic functions |Σ QNM | 2 and |Σ TLN | 2 defined by (2.5) and (2.14) respectively.QNM frequencies and resonant peaks of the tidal response correspond to the zeros of these two functions and they perfectly match.
In the opposite limit, let us say a " 1 (in units of L), the effective potential has no minima, so metastable states are not present.For such choices, the resonances of the tidal interactions still exist but they are no longer related to QNM's.It would be interesting to understand what kind of information about the internal structure of the fuzzball can be extracted from the spectrum of resonances in this case.
4 Superstrata (1,0,s) Superstrata [62,63] generalize the D1D5 circular geometries by introducing a third charge and allowing a more general string profile.The class we consider here, labelled by (1,0,s), is characterized by two real positive numbers a, b and an integer s.We focus on the near horizon region, where the geometry is integrable [68], and therefore radial and angular motion can be separated.The geometry is asymptoticaly AdS 3 ˆS3 and it is described by the metric ds The solution is specified by three independent parameters chosen between R y , L, a and b satisfying the relation (4.2).Starting from the ansatz Φ " RprqSpχqe ip´ωt`mφφ`m ψ ψq ( the massive wave equation separates into [68] B The angular equation is solved by spherical harmonics in five dimensions with The radial equation leads to a Fuchsian differential equation with two regular singularities and one irregular one of degree 1 `rs{2s.We consider here the cases of s=0,1,2 leading to hypergeometric or Heun equations.The tidal response of superstrata geometries are holographically related to two-point functions in the dual two-dimensional CFT living on the boundary of AdS 3 at the intersection of D1D5 branes.Resonances of the tidal response therefore correspond to poles of the two point function and have been recently studied in [69].

p1, 0, 0q stratum
The p1, 0, 0q geometry describes a D1D5 two charge state with a string profile partially wrapping an internal circle.Its size and orientation is specified by the parameters a and b.The case b " 0 corresponds to the D1D5 circular profile studied in the last section.The important difference with respect to the circular case is that for b ‰ 0, the wave dynamics is separable only for ϵ " 0, so we restrict ourselves to this choice.The (1,0,0) near horizon geometry is described again by a free N f " p2, 0q gauge theory The analysis of the tidal response follows mutatis mutandis that of the circular D1D5 case given in Subsection 3.1 and all formulae in that section still hold but now gauge and gravity variables are related to each other by the dictionary z " a 2 ρ 2 `a2 , Rpzq " Resonances are found by solving in ω, leading to Like in the near horizon D1D5 case, no resonances show up when µ " 0 due to an accidental cancellation between the numerator and denominator of the response function.

(1,0,1) microstates
Microstate geometries of type (1,0,1) are described by an interacting SU p2q gauge theory with N f " p0, 2q fundamentals.The wave equation can be put into the normal form (2.1) with ´`m 3 `m4  with l given by (4.9).The characteristic polynomials are P L pxq " 1 , P R pxq " px ´m3 qpx ´m4 q P pxq " x 2 ´u `q , M pxq " px ´m3 ´1 2 qpx ´m4 ´1 2 q (4.12) The tidal response is described by r Lpqq given in (2.18).As in the case of the near horizon D1D5 circular geometry a small mass is introduced in order to split the sequences of poles and zeros of r L. We use µ as a regulator, assuming it is always small but not zero.The wave dynamics is described again by a reduced confluent Heun equation, but now the irregular singularity is located at z " 8. On the other hand the point z " 0 describing the space-time infinity, corresponds now to a regular singularity.
Assuming m ψ " m 3 `m4 ą 0 and noticing that m 3 ă m 4 , one finds the resonances at the frequencies satisfying the quantization conditions At zero instanton this leads to Higher instanton contributions can be incorporated as well.The convergence rate of the instanton series is shown in figure 5.A 0.1% of accuracy is reached already with four instantons for a " 0.1 or six for a " 1.In tables 4 we display the results for the resonant frequencies and their associate QNM's for some choices of the gravity parameters.Again a perfect agreement is found between resonances of the tidal response and QNM frequencies.3: (1,0,1) TLN resonances for ℓ " m ψ " 2, m ϕ " µ " 0, R y " b " 1: Left) a " 0.1 and Right) a " 1 ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲

(1,0,2) microstates
Perturbations of the (1,0,2) geometry are described by a confluent Heun equation related to a N f " p1, 2q gauge theory.The wave equation can be written in the form (2.1) with Q12 " m 2 q z `1 4 ´1 4 pm 3 ´m4 q 2 z 2 `1 4 ´1 4 pm 3 `m4 q 2 pz ´1q 2 `1 2 `m3 q `m4 q ´m2 The characteristic polynomials are P L pxq " px ´m2 q , P R pxq " px ´m3 qpx ´m4 q P pxq " x 2 ´u `q `x `1 2 ´m2 ´m3 ´m4 ˘, M " px ´mi ´1 2 q (4.18) The effective potentials for various choices of a is displayed in figure 6.Real resonances in the tidal response are obtained at the zeros of the equation in the second line of (2. 19) with the choice of a minus sign.At zero instanton, one finds ω n " 2a 2 R y p2a 2 `b2 q « m ϕ `cpℓ `mψ `2n `2q Including higher instanton contributions one finds the results listed in table 4. Again a perfect agreement is observed between tidal response resonances and QNM's.The match is also illustrated in figure 7.

Topological stars
Topological star (TS) are horizonless and asymptotically flat solutions of the Einstein-Maxwell theory in 5 dimensions.The solution is spherically symmetric so the angular with p the internal momentum and ∆prq " pr ´rs qpr ´rb q , r s ă r b ă 2r s (5. 2 The geometry is smooth and ends on r b .The massless scalar wave equation on this geometry can be put into the form (2.1) with ´m3 q z `1´pm 1 ´m2 q 2 4z 2 p1 `zq `1´pm 1 `m2 q 2 4zp1 `zq 2 `1´4u`2qp1´m 1 ´m2 q 4zp1 `zq The resulting equation is of confluent like type and describes the dynamics of a supersymmetric SU(2) gauge theory with N f " p2, 1q fundamentals and characteristic polynomials P L pxq " px ´m1 qpx ´m2 q , P R pxq " px ´m3 q P pxq " x 2 ´u `q ˜x `1 2 ´3 ÿ i"1 m i ¸, M pxq " The tidal response is given by L in (2.18) and the poles are located at frequencies such that The results for some choices of the parameters are shown in table 5 and figure 8.We observe again that the QNM's with a small imaginary part always correspond to resonances in the tidal response of the gravity object.The opposite is not true, since some of the resonances of the tidal response have no counterpart in the spectrum of QNM's.It would be interesting to understand the physical origin of these resonances.

Figure 1 :
Figure 1: Effective potential V pxq for the BH (left) and fuzzball (right) cases