Scaling similarities and quasinormal modes of D0 black hole solutions

We study the gravity solution dual to the D0 brane quantum mechanics, or BFSS matrix model, in the ’t Hooft limit. The classical physics described by this gravity solution is invariant under a scaling transformation, which changes the action with a specific critical exponent, sometimes called the hyperscaling violating exponent. We present an argument for this critical exponent from the matrix model side, which leads to an explanation for the peculiar temperature dependence of the entropy in this theory, S ∝ T9/5. We also present a similar argument for all other Dp-brane geometries. We then compute the black hole quasinormal modes. This involves perturbing the finite temperature geometry. These perturbations can be easily obtained by a mathematical trick where we view the solution as the dimensional reduction of an AdS2+9/5× S8 geometry.


Introduction and Motivation
The duality between the BFSS [1] matrix model and the D0 brane near horizon geometry is one of the simplest examples of holographic dualities.This is a rich system with different energy regimes [2].In this paper, we will concentrate on the first interesting regime as we go down in energies.Namely, we consider the large N theory in the range of temperatures where g 2 is the overall coupling playing the role of ℏ in the Matrix quantum mechanics and has dimensions of (energy) 3 .Here λ 1/3 is an energy scale that sets the effective coupling of the matrix model (also called the 't Hooft coupling).In this regime, the matrix model is strongly coupled and the system is described by the near horizon geometry of a ten dimensional charged black hole, whose precise geometry we will discuss later.This is a simple example of a quantum mechanics system (as opposed to a quantum field theory) which has a holographic dual described by Einstein gravity (as opposed to a more exotic gravity theory).In the temperature range (1), this quantum many-body system is displaying a quantum critical behavior which we will call a "scaling similarity" [3].The geometry is such that under a time rescaling t → γt and a suitable rescaling of the radial dimension, the action changes by S → γ − θS, with θ a certain scaling exponent.Since the action changes, this is not a symmetry of the quantum theory.However, at large N we can just do classical physics in the bulk, and this transformation becomes an actual symmetry of the equations of motion.We describe this symmetry in detail and compute the scaling exponent θ from the matrix model side, see also [4,5,6].What we call a "scaling similarity" has also been called hyperscaling violation [7,8], and the exponent θ = −θ, where θ is the hyperscaling violating exponent defined in [7,8] 1 .Other Dp brane metrics display a similar critical behavior [3,9].We also explain how this scaling similarity can be used to classify the perturbations propagating on the geometry, which are dual to operators in the quantum theory.
An interesting property of this strongly coupled many-body system is its response to perturbations.In the bulk, these are described by black hole excitations called quasinormal modes [10], see [11] for a review.Each quasinormal mode decays with a particular complex frequency proportional to the black hole temperature.Their imaginary part arises because the perturbations fall into the black hole.This is viewed as thermalization in the quantum system.Notably, even though the system has N2 degrees of freedom, the quasinormal mode frequencies are independent of N .In addition, the number of modes below any given frequency is also independent of N .This is an important feature of any quantum system that has a black hole description within Einstein gravity.
The quasinormal modes are the fingerprint of the black hole.Their frequencies are fixed by the geometry around the black hole and are independent of the initial perturbation.So they are among the definite predictions that the duality makes.Namely, a simulation of this model in either a classical or quantum computer should reproduce these modes.Finding them would constitute evidence that the quantum system gives rise to black holes.In fact, we could say it is evidence similar in spirit to the evidence we have from LIGO observations [12]-we are observing some vibrations whose waveforms (or frequencies) are computed by solving Einstein's equations.
The computation of quasinormal modes involves expanding all fields to quadratic order around the background 2 .This seems a somewhat daunting task.However, we are helped by a few tricks.First we note that the solution itself can be formally viewed as the dimensional reduction from an AdS 2+ θ × S 8 solution in 10 + θ dimensions where θ is a fractional number equal to the action scaling exponent mentioned above [14].Then, we can use the spectrum of dimensions of operators derived in [15] to determine the masses for the fields.In fact, we will also mention a quick trick to get the spectrum by going to eleven dimensions.In addition, the finite temperature solution is the usual AdS 2+ θ black brane.Then we can simply write the wave equation for a massive scalar field on this background.Solving this equation numerically we find the quasinormal modes.
As a side remark, in appendix B, we provide a quick rederivation of the one loop eight derivative correction to the thermodynamics of the black hole which was originally obtained in [16,17].

The D0 brane gravity background
The gravity solution dual to the matrix model, in the regime (1) is where we have written the metric in string frame.The time t should be identified with the time in the matrix model.We see that the dimensionful coupling λ is only setting the units of time.
In other words, in terms of the dimensionless time τ the metric is completely independent of λ.
We have given the temperature both with respect to the dimensionless time τ , T τ , and the matrix model time t, T .The solution is valid in the following range of temperatures The upper limit comes from demanding that the curvature of the sphere is not too large at the horizon.This radius of curvature becomes of order one in string units when ρ 0 ∼ 1.The lower limit comes when the dilaton becomes of order one, or e ϕ | ρ=ρ 0 ∼ 1.Of course, for this range to be wide enough, we need N ≫ 1.In our conventions, 16πG N = (2π) 7 α ′4 , and α ′ disappears from the action.The metric can also be written as which shows that, for h = 1, the geometry is conformally equivalent to AdS 2 × S 8 [3,18,19].Strictly speaking, the thermodynamic state described by ( 2) is not quite stable, since it can decay by emission of D0 branes that go to infinity.In the large N limit this process is exponentially suppressed (see the rates computed in e.g.[20]) and we will ignore it in this paper.

Similarity transformations
An important property of this solution is that under the transformation the metric gets rescaled, and the dilaton gets shifted in such a way that the whole action is changed by This is in contrast to usual purely AdS solutions where the rescaling ( 5) is an actual symmetry of the metric that leaves the action invariant.This means that the transformation ( 5) is a symmetry of the equations of motion and of classical observables.But it is not a symmetry of the quantum theory.However, if we are interested in leading order in N results, the classical theory is enough, and this is as good as a symmetry.
Transformations that rescale the action are sometimes called "similarities" [21] rather than symmetries, and that is the name we will use here.This type of similarities have also been called "hyperscaling violation", with the hyperscaling violating exponent θ = − θ [7,8].
Similarities are common.For example, Einstein gravity with zero cosmological constant has a well known similarity under g µν → κ 2 g µν under which the action changes as S → κ D−2 S .This can be used to argue that the Euclidean action of a Schwarzschild black hole in D spacetime dimensions should scale like S ∝ r D−2 s .It is also the reason that the size of quantum gravity corrections is set by G N /r D−2 s .The classical type IIA gravity action has two similarities.The first corresponds to changing the string coupling and the RR field strengths In our case, since N is proportional to the flux of a RR field strength, this similarity implies the familiar result that the action scales like N 2 .This similarity is exact in α ′ and it extends to the weakly coupled regime (but not to the very strongly coupled regime, below the lower limit in (15)).
In the matrix model it follows from large N counting involving planar diagrams.
The second similarity of the type IIA supergravity action is which is simply the statement that we have a two derivative action.The rescaling (5) does not leave the metric or the gauge and scalar fields invariant.Instead, it changes them in the same way as a particular combination of the similarities ( 7) and ( 8), with where the last relation ensures that N is not changed under (5).This implies that the action changes as in (6).The scaling similarity (5) implies that the action and the entropy of the finite temperature solution behave as S ∝ N 2 T 9/5 ∝ N 2 T 9/5 λ −3/5 (10) where in the last step we restored the λ dependence by dimensional analysis.Here we used that the temperature changes as T → γ −1 T under (5) since it is proportional to 1/β, and β is the length of Euclidean time which changes as β → γβ according to (5).Note that the N 2 factor is fixed by (7).This similarity of the asymptotic form of the metric can also be used to classify the various perturbations of the theory.They can be characterized by the decay of normalizable perturbations at large ρ, or small z, χ ∝ z ∆ , as z → 0 (11) We can then think of ∆ as the dimension of the corresponding operator.We will explain this in more detail later.Let us stress that neither N nor λ are changed when we apply the transformation (5).A similar sounding scaling symmetry was discussed in [15,22] 3 where they changed λ → γ −3 λ (keeping N fixed).This is an exact "symmetry" of the full matrix model which is usually called dimensional analysis.We put "symmetry" in quotation marks because it involves changing a coupling constant.We emphasize that the scaling similarity (5) is not dimensional analysis.It is a non-trivial similarity that emerges at low enough energies and reflects a non-trivial critical behavior of the matrix model.In particular, (5) is definitely not a symmetry (or a similarity4 ) of the classical matrix model action.
This rescaling similarity is somewhat analogous to the conformal symmetry of the SYK model.Both emerge at relatively low temperatures.The parameter J in SYK is analogous to λ 1/3 here.Both set the scale at which the model becomes strongly coupled.And in both cases, the critical behavior is modified when we go to temperatures that are parametrically small in the 1/N expansion.
The rescaling similarity is not the symmetry associated to the 11 dimensional boost symmetry that should emerge at extremely low energies according to the BFSS conjecture [1].That is yet another emergent symmetry appearing at lower energies 5 .Nevertheless, the similarity does have a some connection to eleven dimensional boosts as we explain in section 3.1.

Eleven dimensional uplift
When e ϕ becomes large, the metric (2) is no longer a good description.But we can use an eleven dimensional metric which is simply the Kaluza Klein uplift of (2) given by where φ ∼ φ + 2π.This is the metric of a plane wave in eleven dimensions.In this form of the metric the dimensionless time τ is given in terms of the energy scale g 2/3 .
The range of validity of this eleven dimensional solution is [2] The lower limit is due to the appearance of a Gregory Laflamme instability [24] 6 .Since the eleven dimensional metric is related to the ten dimensional metric by the simple transformation (12), it inherits the rescaling similarity (5), The meaning of the rescaling of φ requires some explanation because φ is a periodic variable, and we will not be changing its period.To explain it, let us note that this set up, (12) also has the two similarities (7) and (8) present in general IIA theories.The most obvious similarity of the eleven dimensional metric is a simple transformation in which the whole eleven dimensional metric is rescaled, We distinguish this from the one we had in ten dimensions (8).Now, a metric that is translation invariant along the φ direction also has a similarity under replacing φ → φ/κ in the metric ansatz (12).What we mean is that we keep φ periodic but we proportional to ds 2 ∝ −2dτ ′ dφ+ dφ 2 y ′ 7 +d ⃗ y ′ 2 .Then we expect that, once we go to finite temperature, the GL instability appears for β′ ∼ 1 which translates into the lower limit in (15).redefine ϕ, A, and g str so as to absorb κ.More explicitly, the transformation is e 4ϕ 3 (dφ − A) = e 4ϕ ′ 3 with The action would be invariant if we were to change the period of φ, but since we do not change it, the action changes as indicated.This is obviously a symmetry of the equations of motion because it can be viewed as a simple coordinate transformation for φ, once we forget about its period.
Of course, each of the two transformations (17) (18) corresponds to a combination of the two transformations (7) (8).
The rescaling similarity (5), which in the coordinates of (13) amounts to τ → γ τ , y → γ −2/5 y, does not leave the eleven dimensional metric invariant.Instead, it changes it in the same way as a particular combination of the two similarities (17) ( 18) So the rescaling of φ in ( 16) should be interpreted as implementing the M-theory similarity transformation (18).This eleven dimensional picture is particularly useful for determining the dimensions of the operators.The reason is that we can look at the behavior of the fields at large y, where they are simply perturbations around flat space.We can then expand the fields in powers of y and look at their eigenvalue with respect to the transformation (16).For this analysis, we note that ( 16) is a combination of a rescaling of all the coordinates τ → γ −2/5 τ , φ → γ −2/5 φ , ⃗ y → γ −2/5 ⃗ y (22) with a boost b that acts as At large y the perturbations can be expanded as χ ∼ y ℓ , where y ℓ denotes a symmetrized traceless homogeneous degree ℓ polynomial (it is traceless because it needs to obey the flat space Laplace equation).The rescaling of y in (22) will contribute to the transformation of the field.Let us consider, for example, a perturbation of the metric component δg ij ∼ y ℓ along the transverse dimensions.This will scale like γ −2ℓ/5 .When we compute the scaling of δg ij , we pull out an overall factor of γ −4/5 that comes from rescaling dy i dy j in δg ij dy i dy j .This is because the dimensions of the fields come from the scaling of the metric fluctuations relative to the scaling of the background.From this scaling we can read off the dimension as we explain below.
We are considering fields that are constant in time and growing at infinity.They can be viewed as the result of adding an operator to the action In a conformal theory the growth of the field in the bulk is related to dimensions of ζ.In other words, the operator insertion corresponds to a bulk field with a non-normalizable component going as χ → ζz s as z → 0 (25) We see that if we assign ζ a dimension s then we can keep χ invariant under scaling.In a d dimensional CFT, the analog of S pert in ( 24) should be invariant.This implies that where ∆ is the dimension of O.
In our case, the similarity rescales the action.Therefore its natural to require that S pert scales as S pert → γ − θS pert with θ = 9/5.This means then that we should assign to O the dimension ∆ such that s where we defined d in analogy with formulas we have in a CFT.In section 3.3 we will see another sense in which d analogous the dimension of the AdS boundary.This means that the δg ij metric components that go like y ℓ have s = −2ℓ/5, which, using ( 27), leads to In cases that the metric fluctuation has τ or φ indices, then there is another contribution to the scaling that comes from the boost eigenvalue of the fields.As noted above, the scaling similarity ( 16) can be viewed as a combination of ( 22) plus the boost (23).The boost leads to extra factors of γ depending on how many τ or φ indices the field has.We find that the possible eigenvalues under (16) are then where b is the boost eigenvalue of the field.For example δg τ ,i has b = −1.It might sound a little strange that the boost eigenvalue contributes to the scaling dimension, since the latter might seem to involve only the powers of y.To explain this more clearly, it is convenient to write both the background metric (13) and its fluctuations in terms of vielbeins as follows (for y 0 = 0) e τ = y 7/2 dτ , e φ = y −7/2 dφ , e î = dy i All the vielbeins transform as e μ → γ −2/5 e μ under (16).Here the μ and ν indices run over all eleven values.It is natural to think that the canonically normalized fields will be the metric fluctuations δh μν that multiply the vielbeins as in (30).Compared to the naively defined metric flucutations δh µν , they have extra powers of y 7/2 which depend on how many φ or τ indices it has.More precisely, they depend on the boost eigenvalue of the field.So, for example a fluctuation of δg φφ ∼ y ℓ leads to a δh φ φ ∼ y ℓ+7 and a transformation under scalings which leads to (29) with b = 2.
The fermionic fields have half integer values of b.We have also defined ∆, whose precise meaning is explained below.A full table of operators with the precise lower bounds on the possible values of ℓ for each case is given in [15] 8 .
The set of dimensions computed in (29) is the same as what we would obtain if instead we looked at the decaying part of the field, where we define the dimension via χ ∝ z ∆ .In this case, we look at fields in flat space going like y −7−ℓ .This also gives the set (29) after we take into account the boost eigenvalues of the fields.
We presented here a quick way to read off the dimensions.The answers agree with the detailed analysis discussed in [15].

Matrix model computation of the critical exponent θ
From the matrix model point of view the scaling similarity is an emergent similarity.In other words, it is not obvious from the matrix model side.An important property of the similarity is the action scaling exponent (6).In this section, we discuss a computation of this scaling exponent assuming that we have a similarity with an unknown action scaling exponent θ, and then computing the exponent using the constraints of supersymmetry.
We start from the matrix model and assume that we have an emergent similarity transformation t → γt.Then we Higgs the gauge group to U (N/2) × U (N/2) by giving a diagonal vev to one of the nine bosonic matrices of the form X 1 = xσ 3 ⊗ 1 N/2 .This vacuum expectation value spontaneously breaks the scaling similarity.Since this similarity is a symmetry of the action, it should also be a symmetry of the effective action for this expectation value.To facilitate the analysis we define a new coordinate z which is a yet to be determined power of x, z = x a for some power a. z is chosen so that it transforms as z → γz under the scaling similarity.
We now imagine that z (or x) is changing slowly.The effective action for this motion is constrained by the scaling similarity to be of the form where f i are some constants and the overall factor of N 2 comes from the usual 't Hooft scaling.We write a hat in θ because it is still an unknown number, the action scaling exponent.Of course we want to argue that θ = θ in the end.Supersymmetry puts some constraints on these coefficients.First, supersymmetry implies that f 0 = 0. Then it says that the second term, the kinetic term, is not renormalized if we express it in terms of the diagonal components of the matrix This relates the coordinate x to the coordinate z.Then the quartic term in velocities becomes It was argued using supersymmetry that the power of x should be −7 [28] and, furthermore, the coefficient is one loop exact 10 .This then gives the equation This can be then viewed as a derivation, from the matrix model, of the action scaling exponent for the similarity transformation.This scaling exponent determines the temperature dependence of the action as we saw in (10).
This computation also implies that the low energy action on the moduli space also has a scaling similarity with the same exponent θ.This fact underlies the observation in [4,5], who proposed an explanation for the entropy of these black holes using the moduli space approximation 11 .From the gravity point of view, the full action for a brane probe S ∝ dtz −d [ √ 1 − ż2 − 1] also has the scaling similarity, with d as in (35) 12 .
A curiosity is that the dimension of x in (33) is such that it scales as a free field in d dimensions.We should also mention that the non-renormalization of the ⃗ x 2 term, including the angular components, implies that the ratio of the AdS and sphere radii should be as obtained from the gravity solution (4).
We can also use this non-renormalization argument to fix at least some of the operator dimensions.We can imagine giving vevs ⃗ x i to N − k of the diagonal components and the same vev ⃗ x to the other k components.If k ≪ N we expect that we can approximate the solution in terms of a brane probe on a background geometry.The brane probe action has terms of the form where the last term is just an expansion in scalar spherical harmonics.We also defined x = ⃗ x/|x|.Supersymmetry determines the form of the first expression for the action, which implies that the action has a rescaling similarity.The second term is the schematic form of the expansion for In the last term we summarized the average of x ℓ i as the expectation value of an operator, O a 1 •••a ℓ , in the theory of the N − k branes.Imposing that the first and last terms in (37) transform in the same way under the scaling similarity implies that the dimension of the operator is We can similarly compute the dimensions of other operators by looking at terms involving velocities of the N − k branes and so on.
Let us emphasize that the form of the low energy action ( 37) is constrained by supersymmetry even in the region where we do not have the scaling similarity, for example in the weakly coupled region, λ/x 3 ≪ 1.So we needed to make the non-trivial assumption that we had a similarity (with an unknown action scaling exponent) in order to fix θ.Of course, it is also desirable to have a first principles matrix model derivation for the emergence of the similarity itself.
The weakly coupled theory also has a scaling similarity, which is the similarlity of a quartic classical action of the schematic form On the other hand, the action ( 36) is also valid in the weakly coupled region, λ/x 3 ≪ 1.However, (36) does not have that similarity of the weakly coupled action.In the weakly coupled region, the ẋ4 /x 7 term in (36) should be viewed as a quantum correction which breaks the similarity of the classical theory.On the other hand, in the strong coupling regime we were assuming that we have a scaling similarity, whose ℏ is given by 1/N 2 .Therefore, in that regime it is reasonable to expect that both terms in (36) should be compatible with the similarity, since they are both part of the classical action when the ℏ expansion is the 1/N 2 expansion13 .

AdS 2+ θ uplift as a mathematical trick
In this section, we discuss a mathematical trick which leads to a simple way to derive the wave equation for fluctuations around the near extremal geometry [14].In addition, we will get another perspective on the similarity transformation by relating it to a more familiar AdS situation.We do not claim that this trick has any physical meaning, it is just a mathematical trick.
The trick involves viewing the ten dimensional action involving the metric, dilaton, and gauge field as coming from a higher dimensional action involving only the metric and the gauge field, reinterpreting the dilaton as the volume of the extra dimensions.It is convenient to dualize the field strength in ten dimensions and view it as an eight form, F 8 = * 10 F 2 , whose flux on the 8-sphere is N , We now go to 10 + θ dimensions, keeping the 8-form an 8-form in the higher dimensional space.We will assume that this metric has the form dŝ 2 = e aσ ds 2  10 + e 2σ d⃗ x 2 θ (40 where d⃗ x 2 θ is the flat metric in θ dimensions and ds 2 10 is the ten dimensional string metric.Dimensionally reducing to ten dimensions we obtain the original string frame action (see appendix A) after choosing θ = 9/5 , a The solution becomes very simple in the higher dimensional space, it is just AdS 2+ θ × S 8 , where the ratio of the two radii is, as in (4), When we deal with this higher dimensional metric, we will take all metric components independent of the extra dimensions and we will only allow a metric perturbation by the field σ (40) in the θ extra dimensions.In particular, there will be no metric fluctuations with indices in the extra dimensions.
The near extremal solution is just the usual black brane Under this uplift, the operators of dimension ∆ are related to scalar fields with mass given by where the set of dimensions is given by ( 29) in our case.This also implies that the two point functions of operators in the original ten dimensional solution are given by since they have the form of correlators in d dimensions that have been integrated over θ of the dimensions in order to make them translation invariant along those θ dimensions.This agrees with the expressions found in [15].

Uplifting H 3 and F 4
The above discussion shows how to uplift the metric, dilaton and RR two form (dualized to an eightform).Here we just mention that we can also uplift the other forms in a rather formal way.
We need to take where the subscripts indicate the type of form we have in the higher dimensional space.This simply reflects how the volume appears in the kinetic term of these forms.

Massive string states
Here we comment on the effective equation for massive string states.In this case, we expect an effective action of the form where the metric is in the string frame, as in (2) 14 .This implies that in the semiclassical approximation of large mass we get an action of the form S = −m dτ g µν ẋµ ẋν (48) where g µν is the string frame metric in (2) and m is the flat space mass of the string state (m 2 = 4N/α ′ with integer N ).We will now consider the case with zero temperature, or ρ 0 = 0. We can write this action in terms of the z variables in (4) to obtain Let us comment that from the point of view of the higher dimensional AdS 2+ θ, this corresponds to the action of a massive particle with an effective mass The two point function of the operators that insert this string state does not display a power law behavior at long Euclidean times.Instead, the scaling similarity of the action (49) implies that the two point function goes as in the geodesic approximation, where C is a numerical constant that we can find by solving the classical problem associated to (49) So we get an exponential decay rather than the power law decay we had for supergravity fields (45).

Comments on relevant operators
In this section we make some comments on the operator spectrum of the model in the scaling regime (1).In the UV theory, or the weakly coupled matrix model, the operator X has dimension −1/2, so any operator of the form T r[X ℓ ] is relevant.However, as we go to the infrared and the coupling becomes strong, many of these operators acquire high anomalous dimensions.If the operator is dual to a massive string state this dimension grows with scale as in (51).For the smaller subset of operators that correspond to gravity modes, the dimensions are given in (29).If one is interested in a quantum or classical simulation of this matrix model, an interesting question is whether any of these operators are relevant.First one can wonder what we mean by a "relevant" operator.A relevant operator is one with ∆ < d, d = 14/5.This is the traditional condition for a relevant deformation in the AdS 2+ θ description.This is the correct condition since such an operator would give a growing deformation of the geometry, relative to the original geometry, as we go to the IR.It is also the condition that implies that the coefficient of the perturbation in (24) has positive mass dimension according to (27).We see that there are a few single trace operators in (29) which are relevant.

T r[X
where the indices are schematic [15].For example, T r[X ℓ ] really means T r[X (a 1 • • • X a ℓ ) ] where the indices are symmetrized and the traces are removed.We have also ignored fermion terms.For example the ℓ = 1 case of the last operator in (52) also contains a term A particular component of this operator is turned on by the mass deformation discussed in [30] 15 .
The matrix theory operators corresponding to various supergravity interactions and currents were identified in [31,32]; see there for a complete expression of the operators (52).
The significance of these operators is that they are the operators whose coefficients need to be fine tuned in order to get to the IR theory.The good news is that there are a small number of them.The bad news is that this number is nonzero, though this is not surprising when we want to get a quantum critical point.Notice that none of these operators is an SO(9) singlet.So if we could preserve the SO(9) symmetry, then there are no single trace relevant deformations.Since we have only a finite number, it is perhaps possible that a discrete subgroup of SO( 9) is enough to remove all single trace operators, but we did not investigate this in detail.On the other hand, there are double and triple trace relevant operators that are SO(9) singlets which are obtained by taking products of the operators in (52).One could also have products of fermionic operators that we have not listed explicitly here.
In addition to the SU (N ) singlet operators we discussed, in principle one could consider deformtations of the theory which are not SU (N ) singlets.It was argued in [33] that SU (N ) non-singlet states have higher energies and excite string states with endpoints near the boundary of the gravity region.This suggests that it is likely that these deformations are also irrelevant.However, this should be more carefully analyzed.
4 Quasinormal modes 4.1 Some generalities about QNM Quasinormal modes (QNM) characterize the response of the black hole to simple perturbations.They tell us how such perturbations decay at late times.Here, late means large compared to β but not compared to N 2 β.In the matrix theory, they describe how quickly perturbations of the thermal state relax back to equilibrium.In general we expect the QNM frequencies to be of order the temperature.In fact, the scaling similarity constrains them to be simple numbers times the temperature.This is because it implies that (2) is a solution to the vacuum Einstein equations for any ρ 0 .Then under (5), solutions to the equations of motion in one background are mapped to solutions in the rescaled background: e −iω(z 0 )t = e −iω(γz 0 )γt .So ω ∝ 1 z 0 ∝ T .We define where α n are dimensionless complex numbers.T = d 4πz 0 is the temperature with respect to t in (43).The α n can be computed by solving the wave equation with ingoing boundary conditions at the future horizon.In our case we also put Dirichlet boundary conditions at infinity, since the black hole is effectively in a box.

Note added
After submission, we learned that the first two QNM frequencies of the SO(9) scalar perturbation were first computed in [13] 16 .

Deriving the equation
Here we derive the equation for all modes.We have seen that the ten dimensional modes have the scaling of operators with dimensions ∆ given by (29).From the two dimensional point of view (after reducing on the S 8 ) these are all scalar fields (though they can have spin under SO( 9)).When we lift them to AdS 2+ θ = AdS d+1 we expect to get scalar fields of mass In our case, d = 14/5, but this discussion is valid for general d and ∆.We then need to solve the wave equation for such a scalar field in the black brane geometry at finite temperature (43).Writing the scalar field as Φ = e −iωt χ(z), we get The boundary conditions at infinity are that the field behaves as χ ∼ z ∆ for small z.And at the horizon we impose the boundary condition where ρ is the proper distance from the horizon.There is another solution near the horizon behaving as in (57) but with i → −i in the exponent.(57) corresponds to the solution which is regular at the future horizon.Including time dependence, (57) becomes Φ ∼ (X + ) − iω 2πT where X + = ρ exp(2πT t) is one of the Kruskal coordinates in the near horizon region.This expression is non-singular for finite X + and X − → 0 which is the future horizon.With these boundary conditions, there are solutions only for a discrete set of complex values of ω, which are the quasinormal mode frequencies.
Since our argument involved going to a fractional number of dimensions, as a sanity check, we also derived the equation for the SO(9) scalar mode by conventional methods.We expect a single SO(9) singlet mode with dimension ∆ = 28/5, obeying [15].We derived this by starting from the ten dimensional Schwarzschild black hole and lifting it to an eleven dimensional black string.In principle, we need to boost along the eleventh dimension x 11 in order to take the near horizon limit.Alternatively, we can consider only modes which in the final IIA picture will have no D0 brane charge.Those are the modes ϕ = e −iωt+ikx 11 whose frequency and momentum along x 11 are related by ω = k.We then expand the full metric in terms of SO(9) scalar fluctuations, fix the gauge, etc, in the standard way.With this method we indeed get the wave equation in [15] at zero temperature.For non-zero temperature we get the same wave equation obtained above (56).
In the following, we set z 0 = 1 and in the end restore the temperature dependence from the formula We now tackle the mathematical problem of determining these modes.

Numerical results for the quasinormal mode frequencies
QNM frequencies were computed using two independent numerical methods.In one method, we used a Mathematica package which implements a pseudospectral technique [34,35].The package replaces the radial variable of the wave equation by a grid z i of n points and writes the solution χ(z) as a linear combination of polynomials, the so-called cardinal functions, each of which has support on a single gridpoint.This turns the wave equation into an n × n generalized eigenvalue problem, which is solved using standard linear algebra.Boundary conditions are imposed implicitly by virtue of the cardinal functions used to approximate solutions.These functions are smooth and finite, and therefore cannot approximate solutions which are becoming arbitrarily large or oscillating arbitrarily quickly.Meanwhile, at the future horizon, the desired solution (57) is regular while the other solution is not.At the boundary, field redefinitions can be performed so that the solution which goes like z ∆ is normalizable while the solution we wish to set to zero (which goes like z d−∆ ) is not.With this setup, the desired boundary conditions are automatically imposed.
Notably, an n × n eigenvalue equation generally yields n eigenvalues.To see which eigenvalues correspond to QNM rather than numerical artifacts, we perform the same computation at various grid spacings and check for convergence.
In the second method, we construct Frobenius series solutions to the wave equation around the singular points at z = 0 and z = 1, restricting to solutions which obey the desired boundary conditions The coefficients a n and b n can be determined recursively using equation ( 56) and they depend on ω.
For general ω these define two independent solutions.We want to impose that they are proportional to each other.This can be ensured by setting their Wronskian to zero at some intermediate value of z.The Wronskian of two solutions χ 1 , χ 2 of (56) is W is the conserved inner product associated to (56).In other words, it is independent of the value of z at which we evaluate it.Demanding that W (χ hor , χ bdy ) = 0 (62) at some intermediate z 0 within the radius of convergence for the Frobenius series in (59) (60) yields an equation for ω.We can get a numerical approximation for ω by truncating the sums in (59) (60).The first eight QNM frequencies for a sampling of fields are included in Tables 1-3 below.An example of the typical spectrum for the lowest-lying modes is shown in Figure 1 for the SO(9) scalar.

WKB approximations
For large mass we expect that the QNM will be given by a geodesic in the black brane background that is sitting at some value of Extremizing with respect to z we get  In principle we have several roots.We pick the ones with least imaginary part, namely (−1) 1/d = e iπ/d .We can also expand the action for small fluctuations around this solution.Inserting (64) into the action (63) then gives us the QNM frequencies [36] 2πT where n = 0, 1, 2, • • • is an integer.In the second expression we have inserted a more accurate expression derived in [36].Note that (65) has an overall factor of e −iπ/d which explains why the modes tend to lie on a straight line, see Figure 1.They do not lie perfectly along such a line because (65) is just an approximation.
In principle, (65) is valid only for large ∆, but the formula works pretty well for the low lying values obtained above when ∆ ≳ d.Writing the fractional error between a mode α and the WKBpredicted value in (65) as we find that ϵ 0 < 0.09 when ∆ ≥ d.The error decreases with n, or as we move away from the fundamental mode toward higher frequencies.For the ∆ = 28/5 SO(9) scalar, ϵ 0 ∼ 0.03.Values of ∆ below d yield greater disagreement.For the lightest field in the spectrum, ∆ = 4/5, ϵ 0 ∼ 0.9.
As n → ∞, we expect [36] the QNM spacing to asymptote to We find that (67) converges more quickly to the numerical results for larger ∆.However, we expect (67) also to hold for small ∆, as long as n is taken sufficiently large.For example, the percent error 67) decreases to 0.0008 by n = 8 for the ∆ = 28/5 SO(9) scalar, while for the ∆ = 4/5 field, (67) is still off by 18% at n = 8.

Sensitivity to the UV boundary conditions
The computation of quasinormal modes discussed in this section was done using gravity.Since the gravity description breaks down near the boundary, one could wonder about the sensitivity of the quasinormal modes to the precise boundary conditions 17 .We can explore this as follows.Instead of putting the boundary condition at z = 0, we put it at z = ϵ, where ϵ ∝ T λ −1/3 in units where the horizon is at z = 1.Then the solution that obeys the modified boundary condition is where χ bdy is given in (60), and χ ′ bdy is the other solution, with ∆ → d − ∆ in (60).The equation determining the quasinormal modes, analogous to (62), can now be expanded as where c is a constant proportional to W (χ hor , χ ′ bdy ) which is order one in ϵ.We have also used that W (χ hor , χ bdy ) is proportional to δω = ω − ω 0 n , where ω 0 n is the quasinormal mode frequency when we select the original boundary conditions, see (62).
(69) describes the dependence of δω on ϵ.To determine the ω dependence, we consider a simpler problem, that of a massive scalar in empty AdS d+1 with a hard wall at z = 1.Although the bulk boundary condition is different from the one we are interested in, we note that moving the UV boundary condition to z = ϵ only affects the form of the wavefunction at small values of z.Therefore we expect the AdS hard wall problem to correctly capture the scaling of δω even for the case where a black hole is present.
In empty AdS the wave equation is solvable and the solutions are Bessel functions, z d/2 J ±(∆−d/2) (ωz).We first solve for the normal mode frequencies when Dirichlet boundary conditions are imposed at z = 1 and z = 0. We then shift the UV boundary condition to z = ϵ and solve for the resulting change δω in the frequencies, using the asymptotic form of the Bessel functions This calculation suggests that δω n of the black hole QNM should scale like where C ω 0 n T , ∆, d is an order one factor that depends on the details of the bulk solution and boundary condition.In restoring temperature dependence, ϵω 0 n has been replaced by ω 0 n /λ 1/3 .The scaling (70) is supported by numerical analysis.
When 2∆ − d > 0 the quasinormal modes are not very sensitive to the precise boundary conditions as ϵ → 0. In fact, this change in boundary conditions can be interpreted as the insertion of a double trace operator O 2 ∆ whose dimension is approximately 2∆ [37].When this is integrated over d dimensions, this leads to effects that scale as ϵ 2∆−d .In other words, when the double trace operator is irrelevant, the effects are small.
On the other hand, if the double trace operator is relevant, 2∆ − d < 0, then the effects of a modified boundary condition is large.This is the case, for example, for the ∆ = 4/5 mode with QNM frequencies listed in Table 1.For such modes, we need a more sophisticated argument for selecting the boundary condition.For example, we could use the fact that these double trace operators typically break supersymmetry, so that the right boundary condition for the supersymmetry preserving model is such that supersymmetry is also preserved in the gravity approximation.

Other Dp brane geometries
In this section we generalize the discussion of the scaling similarity transformation for the near horizon geometries describing all Dp branes [2,3], see also [9].
The general extremal geometry is This metric, and the corresponding action, have a scaling similarity We can also formally add θ extra dimensions and uplift the solution to a solution which is AdS 2+p+ θ × S 8−p [14], where the quotients of the radii are Note that once we assume that the gauge theory has a scaling similarity, then it is possible to compute the scaling exponent θ by a procedure similar to the one discussed in section 3.2.Namely, we consider configurations describing separated branes that are slow-moving in the separation.We find that the vacuum expectation value of the matrix model field X scales like r ∝ z − d−2 2 in the above solution.Using that the (velocity) 4 term has a protected form we can fix the scaling exponent θ to (72).This constitutes an explanation for the peculiar powers of temperature that appear in the entropies of these solutions, where we reintroduced the 't Hooft coupling by dimensional analysis.Note that probe actions also have this scaling similarity, which explains the agreement with the thermodynamics in the moduli space approximation [5,6].The logic here is different and it consists of deriving the temperature dependence in (74) from the gauge theory, specifically by making the assumption that we have a scaling similarity and then deriving the scaling exponent θ from first principles by using supersymmetry constraints on the low energy dynamics of multicenter solutions.This argument also fixes the scaling dimensions of the simplest operators, T r[X ℓ ], to  49) and ( 51).The finite temperature solution uplifts to the usual AdS 1+d black brane.The equation for quasinormal modes can then be easily written as a perturbation of those black brane solutions, using the spectrum of fields in (75), and has the form (56) with d and ∆ given by (73) (75).
There is an interesting suggestion in [38,39] for how to go from the strong coupling dimensions of the operators T r[X ℓ ] in (75) to the weak coupling dimensions corresponding to ℓ free fields in p + 1 dimensions by taking a suitable large ℓ limit similar to the one in [30].
Let us discuss some special cases now.
• p = −1: In this case θ = 8 3 .One can wonder about the meaning of the exponent in this case.One observation is the following.Consider the fermion mass operators as in (53), which for general p have a dimension such that its coefficient ζ has mass dimension one, call it ζ = µ.This is a relevant deformation around the critical point.Then if we perturb the action by such operators, we predict that its action should scale as S ∝ µ 8/3 .This is a non-trivial prediction of the scaling similarity 18 .Note that we are saying that the matrix integral develops a nontrivial critical point which can be seen by looking at the values of the integral upon adding some deformations 19 .
• p = 1: Here θ = 1.we can think of the solution as S-dual to that of the fundamental string in type IIB, and this solution is similar in type IIA.In type IIA we can uplift the solution to the usual M2 brane solution or AdS 4 × S 7 which are the dimensions that also appear here for that case.
• p = 2: We get θ = 1/3, which is a non-trivial similarity exponent.Notice that in the far IR the boundary quantum field theory develops an actual conformal fixed point which is dual to AdS 4 × S 7 .This is different that the one we get in the intermediate energy regime described here.
• p = 3: We get θ = 0 and the usual CFT.In this case the two scaling similarities we discussed in (7) (8) combine to imply that the gravity answers are independent of the string coupling and have an action proportional to N 2 , which is of course a familiar result.
• p = 4: We get θ = 1 and the uplift is AdS 7 × S 4 .This is just saying that the metric is simply a dimensional reduction of the familiar solution for M5 branes.
• p = 5: Here θ = ∞ and the above formulas become singular.The proper interpretation is that the coordinates are not really rescaled but the radial direction, r, is rescaled, which is what happens when we go to the NS 5brane geometry.The similarity just corresponds to a shift of the dilaton as we shift the radial dimension.
• p = 6: Here θ = −9.The D6 brane solution can be uplifted to eleven dimensions as an A N −1 singularity and the rescaling is just the usual rescaling of all coordinates in eleven dimensions.This gives the usual scaling of the action as (length) 9 , which is the standard similarity in eleven dimensional flat space.

Note added 20
In the interesting paper [42], the sphere partition function was computed for the gauge theories living on the worldvolume of Dp branes.Using supersymmetric localization they computed the logarithm of the partition function on S p+1 .For p ≥ 1 they found that it scales as ( λ) p−3 5−p in terms of the dimensionless coupling λ = λR 3−p , where R is the radius of the sphere (see eqn. (2.19) in [42]).This is precisely the scaling that we expect on the basis of the scaling similarity, namely the total power of size of the sphere is then R − θ.In fact, this observation can be turned around and viewed as an argument for a computation of θ from the matrix model, logically similar to what we discussed in section 3.2.Namely, we assume that the theory has a scaling similarity and we used the localization results to compute the exponent.Of course, the power law dependence on λ observed in [42] is also evidence that the matrix model has this similarity.
In [42], the Wilson loop was also computed.This is also constrained by the scaling similarity, but with a different action scaling exponent than the one we have for the bulk gravity action.The relevant exponent is the one that appears in the action of the string, which is related to the scaling of the string frame metric, which goes as ds 2 str ∝ z . This implies that the scaling exponent for the fundamental string action is θ = p−3 5−p .Therefore the Wilson loop action is expected to scale as ( λ) 1/(5−p) , which is indeed what [42]

Discussion
In this paper we discussed in some detail the scaling similarity of the D0 brane solution.We emphasized that it should be viewed as a non-trivial scaling transformation which is not obvious from the matrix model point of view, and it would be interesting to explain its emergence purely from the matrix model.On the other hand, the assumption that the matrix model develops a similarity, plus the constraints of supersymmetry, determine the important action scaling exponent θ.In turn, this determines how the finite temperature entropy depends on the temperature.Similar similarities exist for other Dp brane solutions (of course for p = 3 this is actually an ordinary scaling symmetry).
Here we noted that a particular many-body quantum system, the matrix model, develops a scaling similarity in the large N limit.It would be interesting to find other examples of this phenomenon.In particular one could wonder whether there is an SYK-like system that exhibits a scaling similarity with a non-zero action scaling exponent θ.In statistical mechanics, an example is the random field Ising model [7].In [20] some supersymmetric quantum mechanics models were analyzed by solving the one loop truncated 21 Dyson-Schwinger equations and they found a scaling similarity.
We have also used the eleven dimensional uplift of the D0 brane geometry to get the scaling dimensions of all gravity fields.A similar analysis can be done for Dp branes by starting with a plane wave geometry smeared along p dimensions and then performing U-dualities.This also leads to (75).
We then explored the quasinormal modes.At first sight, this is a difficult task because we need to expand all fields around the finite temperature solution.This task is simplified by a mathematical trick, uplifting the solution to 10 + θ dimensions, so that we have an AdS 1+d × S 8−p solution with d = 1 + p + θ.From the AdS 1+d solution point of view we have fields whose dimensions are given by (75) and we can simply write down the wave equation.
We analyzed this equation numerically for the D0 brane case, p = 0.The quasinormal modes are a unique "fingerprint" of the black hole and it would be interesting to reproduce them from the matrix model by either a classical numerical simulation or by a quantum simulation.An interesting feature of these modes is that the single trace operators come in special low spin representation of SO(9), essentially spin less than two.It is not obvious from the matrix model that this is the case, but it is obvious from the bulk, since spins are upper bounded by two in gravity.The fact that we only get these light modes is analogous to the spin gap condition that was discussed in [43] as a necessary (and probably sufficient [44]) condition for a gravity description.is no σ dependence in front of F 2  8 .This sets a = θ/3.We also use the formula for the form of the curvature under a rescaling of the metric R = e −aσ R − 9a∇ 2 σ − 18a 2 (∇σ) 2  (77) Inserting this into the action we get that where we used that a = θ/3.Setting this to be equal to the usual ten dimensional action, . We can follow the same procedure for Dp branes.In that case one has an F 8−p , so we get that a = θ/(3 − p), ϕ = −(7 − p)σ/2, θ = (3 − p) 2 /(5 − p), and d = 1 + p + θ = 2(7 − p)/(5 − p).
As a side comment, note that a gravity solution of the form AdS D × S D ′ with an F D ′ field on the S D ′ (or its dual) will have radii in the following ratio B Derivation of the one loop eight derivative correction In this appendix we provide a short derivation of the one loop R 4 correction that was obtained originally in [16,17], and was matched to numerical results in [45,46,47].
The ten dimensional effective action has higher derivative corrections which lead to a modification of the thermodynamic properties of black holes.These are particularly interesting because they can be compared with the numerical results in [46,47].The first corrections are eight derivative, or α ′3 , corrections that arise from two sources.There is a tree level correction and a loop level correction of the schematic form where the first term arises at string tree level and the second at one loop.The dots include terms involving the RR fields and gradients of the dilaton.The full form of the tree level term is not known.However, the one loop term is believed to be given simply by a Kaluza Klein reduction from a similar R 4 term in eleven dimensions.The eleven dimensional correction to the Euclidean action is [48,49] − where the invariants t 8 t 8 R 4 and E 8 can be expressed in terms of the Riemann tensor as in [50] (with E 8 = 8!Z, with Z in [50]).
Since the eleven dimensional solution ( 13) is simply a boosted version of a ten dimensional Schwarzschild black hole, we can simply evaluate the invariants on the ten dimensional Euclidean Schwarzschild black hole of the form We can compute the various invariants for this black hole: where we found the formulas in [50] useful.We have given the separate values of other invariants just for completeness.
We can now evaluate the correction to the free energy by integrating J M over the whole space.Note that we do not need to find the correction to the black hole metric, since that would only change the answer at higher order.We obtain where β t is the physical temperature with respect to the time in the matrix model, and β τ is the temperature, properly computed using (13).
Note that the one loop correction (89) is larger than naively expected from a scaling point of view.Namely, from the gravity side, one might naively expect that a one loop correction is the logarithm of a determinant which has at most a logarithmic scaling with the scale or the temperature, log λ 1/3 /T .On the other hand, (89) is a correction that grows as a power when we reduce the temperature.The reason is that this correction comes from a one loop counterterm to (80), which scales as (size) 2 /α ′ ∝ T −3/5 , where the size refers to the radius of the S 8 in string units at the horizon, see (2).This makes the correction larger than what we would expect if the gravity theory were one loop finite.Of course, the gravity theory is not finite, but the string theory is indeed finite, however the final answer is larger than naively expected and it depends explicitly on the string theory length α ′ .

T
These come from the C τ φi , C τ ij , C φij and C ijk components and the dimensions are again given by(29).

2 δ
trivial components of the Riemman tensor areR 1212 = 28µ , R 1i 1j = R 2i 2j = − 7 ij µ , R îĵîĵ = µ (no sum, i ̸ = j) ,µ ≡ the values in local Lorentz indices, with 1 = τ , 2 = ŷ, and i, j run over the eight values corresponding to the sphere directions.All other components not related to those above by the symmetries of the Riemann tensor are zero.