Remarks on holographic models of the Kerr-AdS_{5} geometry
Julián Barragán Amado, Bruno Carneiro da Cunha, Elisabetta Pallante
RRemarks on holographic models of the
Kerr-AdS geometry Juli´an Barrag´an Amado, a,b
Bruno Carneiro da Cunha, a and Elisabetta Pallante b,ca Departamento de F´ısica, Universidade Federal de Pernambuco, 50670-901, Recife, Pernam-buco, Brazil b Van Swinderen Institute for Particle Physics and Gravity, University of Groningen, 9747Groningen, Netherlands c NIKHEF, Science Park 105, 1098 XG Amsterdam, Netherlands
E-mail: [email protected] , [email protected] , [email protected] Abstract:
We study the low-temperature limit of scalar perturbations of the Kerr-AdS black-hole for generic rotational parameters. We motivate the study by consid-ering real-time holography of small black hole backgrounds. Using the isomonodromictechnique, we show that corrections to the extremal limit can be encoded in the mon-odromy parameters of the Painlev´e V transcendent, whose expansion is given in termsof irregular chiral conformal blocks. After discussing the contribution of the intermedi-ate states to the quasi-normal modes, we perform a numerical analysis of the low-lyingfrequencies. We find not only that excited perturbations are superradiant, as expectedfrom thermodynamical considerations, but also that the fundamental mode presents adynamical phase transition, becoming unstable at specific small values of the asymp-totic charges. We close by considering the holographic interpretation of the unstablemodes and the decaying process. Keywords:
Black Hole Scattering, Gauge/Gravity Correspondence, Holographic Mod-els. a r X i v : . [ h e p - t h ] F e b ontents black hole 5 Holography has been considered a tool to study strongly-coupled phenomena in highenergy physics for some time now. In particular, the description of strongly-coupledplasma put forward by [1, 2] and [3–5] laid ground to a very precise procedure to studysystems such as finite temperature deformations of conformal field theories based on N = 4 SYM from gravitational physics. It is particularly successful in dealing with– 1 –he hydrodynamical limit of such systems, and quantities like the entropy to viscosityratio [1] and energy loss calculations [6, 7].More recently, it has been stressed the necessity of modeling rotation and vorticityinto the holographic picture [8, 9], which necessitates the holographic consideration ofthe generic rotating, vacuum black hole in anti-de Sitter (AdS) backgrounds – Kerr-AdS black hole for short [10, 11]. Usually, the study relies on thermodynamical argu-ments about the behavior of fluctuations in the black hole background. For instance,in [12] the authors analyzed the propagation of a string in this background to estimateenergy losses in a rotating fluid.For a given field theory, the holographic description is based on the Ward identitiesassociated to the conformal group currents [13]. The departure from a given ultravioletfixed point in the renormalization group flow is implemented by the (inwards) radialevolution on the currents using Einstein’s equations from the asymptotic boundary AdSstructure. In this paper, we study the validity of the description for generic rotationparameters at low temperatures. The motivation for this is threefold.First and foremost, the usefulness of holography to study the infrared (IR) be-havior of the associated field theories depends on the gravitational perturbations atlow temperatures. As we will revisit below, large black holes have a lower bound inthe temperature, so, this study will focus on small black holes in the low temperatureregime. At high temperatures, these black holes were studied by using the hydrody-namical approach in [4], which, among other things, recovers (flat space) Navier-Stokesequations by assuming that the mean free path of the components of the fluid is smallwhen compared to the AdS scale. This hints at the fact that the dual field theory is inthe strong coupling regime. This construction is at odds with holography, which startsfrom the assumption that the field theory has small corrections to the ultraviolet (UV)free or weakly interacting conformal point. More specifically, the available examples ofholography usually come from theories that are weakly coupled at the UV, and becomestrongly coupled at the IR. We will make an unwarranted – although not unprecedented– assumption that the description based on conformal currents is still valid in the latterregime. We also remark that it is an open question whether this generic view of holog-raphy can be applied to the particular case of QCD-like asymptotically free theorieswith a mass gap. In the best case scenario, the holographic interpretation in the restof this paper will refer to generic asymptotically conformal systems, not necessarily– 2 –ubject to the analysis in [14].Secondly, a more comprehensive study of real-time holography [15] for the Kerr-AdS black hole at low temperatures is desirable. Rotation mixes time and angular co-ordinates, and induces a “non-flat” asymptotic structure of the black hole background.The first fact poses problems for the analytic continuation of quantities computed inEuclidean signature and the second gives rise to trace anomalies which are heavilydependent on the particular field theory dual considered. We note however, that ina “classical limit” where the AdS radius is much larger than the Planck length, thequantities computed at different asymptotic structures, in our case essentially the onearising from global AdS R × S and the one arising from the Poincar´e patch R , arerelatable by a conformal transformation. Also, the repercussions of the AdS instabili-ties found in [16], of which the scalar field linear unstable modes found by the authorsin [17] are a particular example, are expected to be of fundamental relevance to theunderstanding of holography, and the study presented here seemed like a natural stepfurther.The third reason is somewhat technical. As argued in [18], black hole perturbationsshould be described by a particular limit of the four-dimensional conformal block [19].As argued in [20] (see also [21]), the pertinent perturbation can be computed using two-dimensional chiral conformal blocks. Semi-classical aspects of the latter whichare relevant to the discussion were anticipated in [22], and a full fledged solution waspresented in the seminal work of [23], also uncovering a relation between semiclassicaland c = 1 conformal blocks, both related to the Painlev´e VI tau function.The low temperature black hole considered here necessitates a careful considerationof the confluence limit, which is given in terms of the Painlev´e V tau function [24].Albeit in this paper we are focused on the study of the quasi-normal modes (QNMs)for small black holes r + (cid:28)
1, the resulting relation holds for generic black holes in thenear-extremal limit. On the one hand, this is to our knowledge a new application ofthe Painlev´e V transcendent – see also [25]. On the other hand, the appearance of thetwo-dimensional conformal blocks may be seen from one point of view as a calculationaltool, but, as we will argue, the connection between these and four-dimensional blocksmerits further investigation.While the writing of this work was in course, [26] came out with a similar study of(four-dimensional) quasi-normal modes of black holes using the properties of conformal– 3 –locks. The authors of [26] employ the usual relation between Fuchsian equationsand semiclassical conformal blocks as the starting point, and their relation to Seiberg-Witten theory [27] has been once more studied in [28]. We note, however, that there areadvantages and drawbacks to each approach. The expansion in terms of semiclassicalconformal blocks is more direct and leads directly to an asymptotic expansion of theaccessory parameter (called K below) of the associated differential equation (3.1).The c = 1 expansion used here implements an explicit solution to the Riemann-Hilbertproblem of finding both accessory parameters t and K in terms of monodromy data.As we saw in an earlier paper [17], an asymptotic expansion for the quasi-normal modesin the five-dimensional Kerr-AdS black hole is more effectively obtained using c = 1conformal blocks.With all this in mind, we have structured the paper as follows. In Sec. 2 we re-view the parameters of the Kerr-AdS black hole, as well as its asymptotic structureand conserved charges – mass and angular momenta. We spend some time analyzingthe holographic interpretation of the asymptotic structure, in particular the interplaybetween the R × S (global structure) and R , (Poincar´e patch), in a “near the ul-traviolet” description which allows us to identify the state in the putative field theorycorresponding to the black hole and to digress about the fate of perturbations withinthe scope of linear analysis.Sec. 3 comprises the bulk of the results. We consider scalar perturbations of theKerr-AdS black hole and show that the low temperature limit of the QNMs is givenby zeros of the Painlev´e V tau function. Using asymptotic expansions of the Painlev´eV transcendent for small parameter, we compute the QNMs frequencies for small blackholes. We then turn to discuss the CFT description of the perturbations, and find thatthe contributions to the fundamental QNM are indeed given by the vacuum conformalblock, at least for small black holes. The higher modes are correspondingly given byspecial blocks with internal dimensions given by degenerate “heavy” operators, in achiral version of the analysis in [29]. Finally, we discuss a dynamical phase transitionassociated to the fundamental QNM, not anticipated by the thermodynamical argu-ments of [30], as well as the numerical characterization of the higher eigenfrequenciesas functions of the global parameters. We find, as expected, unstable modes for allvalues of the angular momentum.In Sec. 4 we discuss the holographic consequences of the analysis, focussing on– 4 –he unstable modes. We see that the naive extrapolation of the near-ultraviolet result,pointing to the irrelevance of the perturbation, is misguided, and that the instabilitiesarising from the same perturbations as treated in Sec. 3 may change considerablythe fate of the state as one approaches the infrared. We also find that the spatialdependence of the decaying modes may be confined to small angles as the mass ofthe state increases. We close in Sec. 5 with some remarks on the results and theirinterpretation.There are two Appendices. Appendix A gives a self-contained derivation of theconserved charges for pure gravity in asymptotically AdS spaces and emphasizes therole of the conformal structure, as a special subcase of [31]. In Appendix B we reviewthe Fredholm determinant formulation of the Painlev´e VI transcendent derived fromthe generic isomonodromic tau function proposed in [32], and its dependence on therelevant monodromy parameters is made explicit. black hole The Kerr-AdS black hole is represented by the metric [10] ds = − ∆ r ρ (cid:18) dt − a sin θ − a dφ − a cos θ − a dφ (cid:19) + ∆ θ sin θρ (cid:18) a dt − ( r + a )1 − a dφ (cid:19) + 1 + r r ρ (cid:18) a a dt − a ( r + a ) sin θ − a dφ − a ( r + a ) cos θ − a dφ (cid:19) + ∆ θ cos θρ (cid:18) a dt − ( r + a )1 − a dφ (cid:19) + ρ ∆ r dr + ρ ∆ θ dθ , (2.1)where we set the AdS radius to one and∆ r = 1 r ( r + a )( r + a )(1 + r ) − M, ∆ θ = 1 − a cos θ − a sin θ, ρ = r + a cos θ + a sin θ, (2.2)and a and a are two independent rotation parameters. The mass relative to theAdS vacuum solution and the angular momenta were the subject of some discussion[31, 33–35]. To add to the confusion, we include our own derivation in Appendix A.– 5 –he values obtained using our procedure match those of [34] M = πM (2(1 − a ) + 2(1 − a ) − (1 − a )(1 − a ))4(1 − a ) (1 − a ) , (2.3) J = πM − a ) a (1 − a ) , J = πM − a ) a (1 − a ) . (2.4)In the following it will be useful to redefine the black hole parameters in terms ofthe roots of ∆ r ∆ r = 1 r ( r − r )( r − r − )( r − r ) (2.5)which, for a regular, finite charged, dressed horizon black hole satisfy r < − <
12 + M ¯∆ θ (cid:19) Ω + O (Ω ) , (2.21)not to be confused with ρ in (2.2). We have the following structure for the metric interms of (cid:37) , following the procedure in [13], ds = d(cid:37) (cid:37) + 1 (cid:37) (cid:0) − d ¯ t + d Ω (cid:1) + 12 (cid:0) − d ¯ t − d Ω (cid:1) + (cid:37) (cid:18)(cid:18)
116 + M θ (cid:19) ( − d ¯ t + d Ω ) + 2 M ¯∆ θ ( d ¯ t − a sin ¯ θ d ¯ φ − a cos ¯ θ d ¯ φ ) (cid:19) + . . . , (2.22)– 8 –gain up to O ( (cid:37) ) terms. We recognize the induced metric ¯ g ab , corresponding to theinduced line element − d ¯ t + d Ω . The particular conformal structure of R × S is notinvariant under the transformation Ω → ω Ω. By a suitable choice of ω (see (2.30)),one could change the induced metric to the flat Minkowski one R , . In the following,we will refer to the R × S induced metric as ¯ g ab , or global AdS structure , and to theflat R , as ˆ g ab , or flat AdS structure . In global AdS, the holographic stress-energy tensor can be read directly from (2.22)above and has the expression [12] T hab = 14 πG N, (cid:18)
116 ¯ g ab + 2 M ¯∆ θ (cid:18) u ( a u b ) + 14 ¯ g ab (cid:19)(cid:19) (2.23)whose M -dependent term satisfies the conformally invariant, perfect fluid relation withenergy density ¯ ρ and pressure ¯ P πG N, ¯ ρ = 12 πG N, ¯ P = 3 M/ θ , (2.24)hence ¯ P = ¯ ρ/
3, and the holographic stress-energy tensor is traceless. The normalizedfluid velocity is u a = χ a / (cid:112) ¯∆ θ and it is divergence-free ¯ ∇ a u a = 0, where ¯ ∇ a is thecovariant derivative associated to the metric ¯ g ab . Despite depicting a classical energyprofile, and being dependent on the choice of conformal structure, (2.23) is by construc-tion a valid approximation to the stress-energy tensor of the state in the field theoryin the UV limit, which holographically corresponds to ¯ r → ∞ .The acceleration of u a is u a ¯ ∇ a u b = − a − a ¯∆ θ sin ¯ θ cos ¯ θ ∂∂ ¯ θ = − ρ ¯ ∇ b ρ = 12 ¯∆ θ ¯ ∇ b ¯∆ θ , (2.25)and it follows that T hab in (2.23) is conserved with respect to the metric ¯ g ab , i.e. ¯ ∇ a T hab = 0. The term independent of M in (2.23) corresponds to the four-dimensionalgravitational conformal anomaly [13], and it can be disregarded for large enough M .The total energy associated to the fluid is4 πG N, E = 4 πG N, (cid:90) S d Ω ¯ ρ = 3 π M (1 − a )(1 − a ) , (2.26)– 9 –hich can be compared with the total ADM mass M in (2.4).The perfect fluid form of the holographic stress-energy tensor for the Kerr-AdSblack hole in (2.23) mirrors the result for four-dimensional backgrounds from [37], andis in fact expected on general grounds. The conformally invariant nature of (2.23) hintsat an underlying conformally invariant field theory dual. Following the holographicrenormalization group flow towards the IR, the corrections to the metric will in generalbreak this conformal invariance, which is interpreted as the departure from a conformalfixed point in the field theory. If we think of the field theory as free or weakly coupledat the UV, the flow towards the IR may bring the theory to a strongly coupled regime,resulting in, among other things, corrections to the perfect fluid form of its stress-energytensor (2.23). In the spirit of holographic renormalization group flow, we will assumethat the radial evolution given by Einstein equations will give the proper picture forthe dual field theory as the energy scale is brought down.For high-temperatures, one can interpret the results of [38] and [4] as the equiva-lence between the general relativity equations and the Navier-Stokes equations for thefluid mechanics. In our study, we want to consider low temperatures, outside the scopeof their analysis. This necessarily leads us to consider small black holes, because largeblack holes in AdS are hot. To see this latter point, let us recall that, when written interms of r + , a and a , the temperature of the black hole outer horizon (2.6) is2 πT + = r + (cid:18) r + 1 r + a + r + 1 r + a − r (cid:19) , (2.27)which, for constant r + , is a monotonically decreasing function of a and a . However,for a , a < T + is always larger than T min = (2 r − / (2 πr + ). We can thereforeconclude that there cannot be a zero-temperature black hole for r > / T + . The other is to view the correspon-dence between the gravitational and the field theoretical system as dynamical, and giveup on the equilibrium view of the fluid, from which the equivalence between mean freepath and r + was deduced. Either choice will have interesting consequences.Dwelling in the second choice for a moment, if the fluid is no longer in equilibrium,there is no reason to restrict the analysis to time-independent configurations. The– 10 –onformal transformation to conformally flat boundary coordinates mixes time andspace and induces time-dependence for the dynamics of the state. Using the (classical)conformal symmetry of the fluid, let us then perform the transformation of ¯ g ab toconformally flat coordinates. The transformationˆ t = 12 tan ¯ t + ¯ χ t − ¯ χ , ˆ r = 12 tan ¯ t + ¯ χ −
12 tan ¯ t − ¯ χ , (2.28)where cos ¯ χ = sin ¯ θ cos ¯ φ , tan ˆ θ = cos ¯ θ sin ¯ θ sin ¯ φ (2.29)and ˆ φ = ¯ φ , turns the metric into d ˆ s = − d ˆ t + d ˆ r + ˆ r ( d ˆ θ + sin ˆ θ d ˆ φ )(1 + (ˆ t + ˆ r ) )(1 + (ˆ t − ˆ r ) ) . (2.30)We recall that we use bars to refer to coordinates in the R × S manifold ( { ¯ t, ¯ θ, ¯ φ , ¯ φ } )and hatted coordinates for Minkowski space R , ( { ˆ t, ˆ r, ˆ θ, ˆ φ } ). The flat metric will behenceforth referred to as ˆ g ab .We can use the conformal symmetry of the leading order perturbation h ab in (2.19)to derive the Poincar´e coordinates, flat AdS version of the holographic stress-energytensor, in the same approximation as (2.23). The perturbation is explicitly given bythe same formula as (2.18) h ab = 2 M z ˆ∆ ( n a n b + 1ˆ∆ ˆ χ a ˆ χ b ) (2.31)where n a = ( dz ) a , ˆ χ a = ˆ g ab χ b , andˆ∆ = − ˆ g ab χ a χ b = (1 + (ˆ t + ˆ r ) )(1 + (ˆ t − ˆ r ) ) − (1 + (ˆ t − ˆ r )) a − ˆ r ( a cos ˆ θ + a sin ˆ θ ) . (2.32)The flat-space version of the stress-energy tensor ˆ T hab = ω − T hab , with ω = (1 + (ˆ t + ˆ r ) )(1 + (ˆ t − ˆ r ) ) (2.33)– 11 –s conserved with respect to the flat metric ˆ g ab (see Appendix D in [39]), and it readsˆ T hab = ˆ ρ ˆ u a ˆ u b + ˆ P (ˆ g ab + ˆ u a ˆ u b ) , ˆ ρ = 3 ˆ P = 14 πG N, M , (2.34)and ˆ u a = χ a / (cid:112) ˆ∆ is the normalized velocity of the fluid with respect to ˆ g ab . − − − − − − t = − − − − − − t = − − − − − − t = Figure 1 . Three snapshots of the energy density profile (2.34), as sagittal projection. Thefree propagation shows two “blobs” of radiation traveling along the z axis and meeting closeto the origin as t →
0. The two parameters a and a control the spread of each blob awayfrom the symmetry axis and the astigmatism as the two blobs meet. In order to study the near ultraviolet behavior in the presence of the perturbation,we turn to the effect of the asymptotic form of the correction h ab . The pure AdS metricencodes the conformal structure – at least of its conformal part. One can then interpretthe effect of h ab as encoding the departure from the ultraviolet fixed point as one turnson the renormalization group flow towards the infrared.One should note at this point that there is an ambiguity in choosing ˆ φ = ¯ φ inpassing from the R × S conformal structure to Minkowski space R , . The Kerr-AdSblack hole background can be interpreted as the resulting CFT (thermal) state obtainedby turning expectation values for the operators comprising the generators of the Cartansubalgebra of the conformal group SO(4 , ∂∂ ¯ t ∼ M , ∂∂ ¯ φ ∼ M , ∂∂ ¯ φ ∼ M (2.35)– 12 –here the right-hand side makes use of an explicit representation of the conformal groupas six-dimensional “Lorentz” generators of R , . In choosing ˆ φ = ¯ φ , we explicitly chose M as the angular momentum generator in the four-dimensional picture. One couldas well choose M , which would amount to setting ˆ φ = ¯ φ . The effect of this choicewould be simply to interchange a with a in quantities such as (2.32) and (2.34). Wewill come back to this point in Sec. 4.A comparatively simple and preliminary calculation is to study the effect on theconformal dimension ∆, as read from the two-point function of two scalar primaryoperators as we depart from the conformal point in the ultraviolet. In general, oneexpects a number of these primary operators to share the same conformal dimension,and the presence of the background does not lift this degeneracy. As it is well-known,the primary operators satisfy a conformal Casimir Ward identity, which can be seenas the scalar wave equation in AdS. To give the calculation a purpose independent ofthe background conformal structure chosen at infinity, we will introduce the SO(4 , σ : σ = z i + z j + | x i − x j | z i z j , (2.36)which is related to the invariant AdS geodesic distance (cid:96) by σ = cosh (cid:96) . As it iswell-known, the pure AdS scalar propagator G ( σ ) = (cid:104) Φ( z i , x i )Φ( z j , x j ) (cid:105) = 2 ∆ Kσ − ∆2 F ( ∆ , ∆ + ; ∆ − σ − ) , (2.37)reproduces the conformal two-point function for two local primary operators with con-formal dimension ∆ sitting at the same value of the cut-off parameter z = z i = z j . Asone considers large values of z , however, the simple CFT expression | x i − x j | − nolonger holds, since keeping the coordinate z fixed (and large) is akin to keeping fixedthe length of the flux tube between the insertions [40], which is not usually achieved inQFT calculations. At any rate, as seen in [41] – see also [20], this dependence can betrivialized for pure AdS by a suitable choice of the z coordinate.The correction h ab to the pure AdS metric g ab at spatial infinity modifies thisdependence (indices are raised with the vacuum metric):¯ ∇ G = ∇ G − g ab g cd ( ∇ a h db − ∇ d h ab ) ∇ c G − h ab ∇ a ∇ b G = ∆(∆ − G, (2.38)– 13 –here g ab and ∇ a are the vacuum AdS metric and covariant derivative, and we areassuming non-coincident insertion points. The last term ∆(∆ −
4) is equal to the AdSmass squared, which is equal to the SO(4 ,
2) quadratic Casimir. Continuing with thecalculation of (2.38), we have, due to the tracelessness of h ab , g ab g cd ( ∇ a h db − ∇ d h ab ) = ∇ a h ac , (2.39)where indices are raised with the pure AdS metric g ab . By means of the formulas (A.2)and (A.4), with the choice of ˜Ω = z so that ˜ g ab = ˆ g ab in Appendix A, we relate thisdivergence to ∇ a h ac = z (cid:18) ˆ ∇ a ˆ h ac − M z ˆ∆ ˆ n c (cid:19) = 0 , (2.40)which vanishes owing to the tracelessness of h ab and the relation between the accelera-tion of u a and the gradient of ˆ∆. The term h ab ∇ a ∇ b G in (2.38) gives less trouble, andthe resulting equation is (cid:16) σ − − z ˆ h ab ∇ a σ ∇ b σ (cid:17) d Gdσ + 5 σ dGdσ = ∆(∆ − G, (2.41)where ˆ h ab = ˆ g ac h cd ˆ g db has its indices raised by the flat metric ˆ g ab . Using the formulafor σ given in terms of the flat coordinates and z , z ˆ h ab ∇ a σ ∇ b σ = 2 M ˆ∆ ( x i ) | x ij | + O ( z , | x ij | ) , (2.42)the simple Ansatz G ∝ σ − ∆ (cid:48) then captures the asymptotic behavior of the solution,with ∆ (cid:48) = ∆ + ∆(∆ + 1)2(∆ − (cid:32) z | x ij | + 2 M z ˆ∆ ( x i ) (cid:33) + . . . = ∆ AdS (cid:18) πG N, −
2) ˆ ρz (cid:19) + . . . (2.43)where we can read the anomalous dimension as the correction to the pure AdS value∆ AdS , defined by the expansion of the vacuum propagator (2.37), that is, in the absenceof the black hole.Note that the effect of the background, at least perturbatively, is to increase theconformal dimension of the perturbation for ∆ >
2, which applies in particular to the– 14 –arginal case ∆ = 4. For z infinitesimally close to zero and ∆ sufficiently close to 4,(2.43) tells us that, as the correction to the AdS value ∆ AdS increases, ∆ (cid:48) crosses themarginal value of 4 and the corresponding perturbation becomes irrelevant, and as suchshould not alter substantially the fate of the flow as one approaches the infrared limit.On the other hand, the fact that corrections to the bare AdS value do become large at z ∼ M − can be seen as hint that the theory is becoming increasingly strongly coupledtowards the IR. As we will see in the following, the remark about the fluctuations notsubstantially altering the fate of the state as it flows will have to be revised. A complete holographic interpretation of a scalar perturbation requires the analysisof the field in the exact Kerr-AdS background, which we will consider in the lowtemperature regime. Let us start with generic considerations on the scalar field. TheKlein-Gordon equation in Kerr-AdS is separable in terms of two (Fuchsian) ordinarydifferential equations, which can be cast in the standard form: d ydz + p ( z ) dydz + q ( z ) y ( z ) = 0 ,p ( z ) = 1 − θ z + 1 − θ t z − t + 1 − θ z − ,q ( z ) = ( θ + θ t + θ + θ ∞ − θ + θ t + θ − θ ∞ − z ( z − − t ( t − K z ( z − z − t ) , (3.1)where the set { θ , θ t , θ , θ ∞ } are called the single monodromy parameters and { t , K } are called the accessory parameters . For the Klein-Gordon field mode Φ n,(cid:96),m ,m , withΦ n,(cid:96),m ,m ( t, r, θ, φ , φ ) = e − iωt + im φ + im φ R n,(cid:96),m ,m ( r ) S (cid:96),m ,m ( θ ) , (3.2)the radial system’s single monodromy parameters are θ Rad , = θ − , θ Rad ,t = θ + , θ Rad , = 2 − ∆ , θ Rad , ∞ = θ where θ k = i π (cid:18) ω − m Ω k, − m Ω k, T k (cid:19) , k = ± , , θ ∞ = 2 − ∆ , (3.3)– 15 –nd the respective accessory parameters are t Rad , = z = r − r − r − r , (3.4)4 z ( z − K Rad , = − C (cid:96) + r − ∆(∆ − − ω r − r − ( z − θ − + θ + − − θ − − z (cid:2) θ + − − ∆) + (2 − ∆) − (cid:3) . (3.5)The angular system’s single monodromy parameters are given by θ Ang , = m , θ Ang ,t = m , θ Ang , = 2 − ∆ , θ Ang , ∞ ≡ ς = ω + a m + a m , (3.6)and the accessory parameters are t Ang , = u = a − a a − , (3.7)4 u ( u − K Ang , = − C (cid:96) − ω − a ∆(∆ − − a − u (cid:2) ( m + ∆ − − m − (cid:3) − ( u − (cid:2) ( m + m + 1) − ς − (cid:3) . (3.8)In previous work [17, 20], the authors stressed how the eigenvalue problem forthe angular equation and the QNM problem can be solved in terms of the compositemonodromy parameters associated with the differential equation (3.1). In particular,the transformation between the accessory parameters { t , K } and the composite mon-odromy parameters { σ t , σ t } is defined in terms of the Painlev´e VI transcendent taufunction: τ ( θ , θ t , θ , θ ∞ ; σ t , σ t ; t ) = 0 , (3.9) ∂∂t log τ ( θ , θ t − , θ , θ ∞ + 1; σ t − , σ t − t ) − ( θ t − θ t − − ( θ t − θ t = K . (3.10)– 16 –he expansion of the tau function for small t is given in Appendix B, and the readeris referred to [32] for further details.As derived in [20], the quantization condition for the angular eigenfunctions canbe cast in terms of the monodromy parameters: σ Ang , t = − (cid:96), (cid:96) ∈ N , (3.11)from which we can compute the small u expansion of the separation constant C (cid:96) : C (cid:96) = ω + (cid:96) ( (cid:96) + 2) − ς − a + a (cid:0) (cid:96) ( (cid:96) + 2) − ς − ∆(∆ − (cid:1) − ( a − a ) ( m − m )2 (cid:96) ( (cid:96) + 2) (cid:0) (cid:96) ( (cid:96) + 2) − ς + (∆ − (cid:1) − ( a − a ) − a (cid:20) ( (cid:96) ( (cid:96) + 2) + m − m ) ( (cid:96) ( (cid:96) + 2) + (∆ − − ς )2 (cid:96) ( (cid:96) + 2) − (cid:96) ( (cid:96) + 2) + 132 (cid:0) m + ς ) − m + (∆ − ) (cid:1) − (( m + 1) − m ) ((1 − m ) − m ) ((∆ − − ς ) ((∆ − − ς )32( (cid:96) − (cid:96) + 3)+ (( m − m )((∆ − − ς ) + 8) − − m + m )((∆ − − ς ) (cid:96) ( (cid:96) + 2) − m − m ) ((∆ − + ς )32 (cid:96) ( (cid:96) + 2) − ( m − m ) ((∆ − − ς ) (cid:18) (cid:96) + 2) − (cid:96) (cid:19) (cid:21) + O (cid:32)(cid:18) a − a − a (cid:19) (cid:33) . (3.12)The radial QNMs can also be determined with similar techniques. The purely ingo-ing boundary condition at infinity requires that the composite monodromy parametersatisfies: cos πσ t = cos π ( θ t + θ ) . (3.13)This condition can be conveniently written in terms of the expansion parameter s that enters the tau function expansion (B.6). From the explicit representation for the– 17 –onodromy matrices (see [17] or [42]), we havesin πσ t cos πσ t = cos πθ cos πθ ∞ + cos πθ t cos πθ − cos πσ t (cos πθ cos πθ + cos πθ t cos πθ ∞ ) −
12 (cos πθ ∞ − cos π ( θ − σ t ))(cos πθ − cos π ( θ t − σ t )) s −
12 (cos πθ ∞ − cos π ( θ + σ t ))(cos πθ − cos π ( θ t + σ t )) s − . (3.14)Note that this equation is of the form A − Bs − Cs − = 0, and by direct calculation itcan be checked that A = B + C + sin πσ t (cos π ( θ t + θ ) − cos πσ t ) . Then, due to the quantization condition (3.13), the roots are readily seen to be s = 1and s = C/B , with the last one compatible with the parametrization of the monodromymatrices. The parameter s can be written explicitly in terms of the monodromy pa-rameters as s = sin π ( θ t + θ + σ ) sin π ( θ t − θ + σ )sin π ( θ t + θ − σ ) sin π ( θ t − θ − σ ) sin π ( θ + θ ∞ + σ ) sin π ( θ − θ ∞ + σ )sin π ( θ + θ ∞ − σ ) sin π ( θ − θ ∞ − σ ) . (3.15)This expression for s , when substituted in the tau function with the θ k parameters ofthe radial equation, will allow us through (3.10) to constrain the frequencies ω , andthus determine the QNMs of the scalar perturbation. At small r + , one can check from (2.27) that the requirement of positive temperaturelimits the range of available a and a , which will be taken to be of order r + . Bearingthis in mind, we define the parameters (cid:15) = r − r − r , α = a + a r , α − = a − a r , (3.16)with the understanding that α + will be of order 1. We will see below that the value of (cid:15) will be constrained by temperature requirements, whereas α − is completely determinedfrom (cid:15) , α + and r + . – 18 –he radii and mass parameters are in turn given in terms of (cid:15) , and α ± by r − = (1 − (cid:15) ) r , − r = 1 + 2(1 + α − (cid:15) ) r , (3.17)2 M = − (1 + r )(1 + r )(1 + r − ) = 2 r (1 + r )(1 + α − (cid:15) )(1 + (1 − (cid:15) ) r ) . (3.18)Given that the black hole charges are determined by three parameters, the followingrelation between r + , (cid:15) , α + and α − ( α − (1 − (cid:15) ) r ) − α − = (1 − (cid:15) )(1 + r )(1 + (1 − (cid:15) ) r ) (3.19)holds, so that, for fixed r + and (cid:15) , the set of available α and α − spans a hyperbola. Therequisite of positive temperature will truncate the hyperbola well before α < /r ,which corresponds to a , a <
1. We will assume that a > a , i.e. α − positive.Let us now use z = ( r − r − ) / ( r − r ) as extremality parameter. We can seethat it is proportional to the temperature by employing (2.6)2 πT + = r + ( r − r − )( r − r )( r + a )( r + a ) = z (1 + (3 + 2 α − (cid:15) ) r ) r (1 + α − (cid:15) )(1 + (1 − (cid:15) ) r ) , (3.20)and it vanishes with (cid:15) : z = r − r − r − r = 2 (cid:15)r α − (cid:15) ) r , (cid:15) = z α ) r z ) r . (3.21)The parametrization of T + in terms of z is relevant for the near-extremal limit of themonodromy parameters (3.3), to be discussed below. Dwelling on this a little further,consider the parametrization of T + in terms of (cid:15) from (3.20)2 πT + = (cid:15)r + α − (cid:15) ) r (1 + α − (cid:15) )(1 + (1 − (cid:15) ) r ) . (3.22)We find that the temperature is generically proportional to (cid:15) , but small temperaturesare defined relative to r + . This means that, for small black holes, it is not sufficient torequire small (cid:15) , but also that (cid:15) (cid:28) r + . In addition, we will take α ± to be of order one,following the argument at the beginning of the Section.With the parametrization above, the θ ± parameters defined by (3.3) have the– 19 –symptotic form: θ ± = ± i Λ z + ψ ± , (3.23)where Λ and ψ ± will have their expansions in r + given below. We will see that therelevant quantities will depend on the combination θ ∗ = ψ + + ψ − . Low temperature( (cid:15) (cid:28) r + expansions of the relevant quantities are θ = ω + ( m α + m α ) r + − α ) ωr + . . . (3.24)Λ = − (1 + α ) m α r − (1 + α ) m α r + 2(1 + α ) ωr +((6 + 3 α + 2 α )(1 + α ) m α − (6 + 2 α + 3 α )(1 + α ) m α ) r + . . . (3.25) θ ∗ = − i − α m α − i − α m α + i (1 + α ) ωr + + i α + α − α ) m α r + i α + α − α ) m α r + . . . (3.26)where α i = a i /r + , and the ellipsis stands for terms of higher order in r + and (cid:15) , whichwill not be relevant for the following analysis. Note that the leading behavior of Λ and θ ∗ with r + depends on whether m and m vanish or not. In particular, if m = m = 0,then Λ vanishes as r and θ ∗ vanishes as r + . Otherwise, it vanishes as r and has afinite limit, respectively. This behavior does not depend on whether (cid:15) is zero, whichimplies zero temperature, or not.The quantization conditions (3.10) can be applied, along with (3.15) to find theQNMs’ frequencies. For generic temperatures, the study was performed in [17]. Thelow-temperature limit is a confluence limit of the Heun equation, given that both θ + and θ − diverge as z →
0. The tau function defined in (B.1) will have to be studied inthis limit.
The Plemelj operator D which composes the Fredholm determinant (B.1) has a fairlystraightforward confluence limit. We will assume that the σ and κ parameters, definedin (B.6), have finite limits as z →
0, which will be verified a posteriori . In this– 20 –ase, the hypergeometric functions entering the parametrices D can be expanded in anasymptotic series for small z yielding confluent hypergeometric functionsΨ( − σ, θ + , θ − ; z /z ) = Ψ (0) c ( − σ, θ ∗ ; i Λ /z ) + z Ψ (1) c ( − σ, ψ + , ψ − ; i Λ /z ) + . . . (3.27)where the confluent parametrix is, defining θ ∗ = ψ + + ψ − :Ψ (0) c ( − σ, θ ∗ ; i Λ /z ) = (cid:32) φ c ( − σ, θ ∗ ; i Λ /z ) χ c ( − σ, θ ∗ ; i Λ /z ) χ c ( σ, θ ∗ ; i Λ /z ) φ c ( σ, θ ∗ ; i Λ /z ) (cid:33) ,φ c ( ± σ, θ ∗ ; i Λ /z ) = F ( − θ ∗ ± σ ; ± σ ; − i Λ /z ) , (3.28) χ c ( θ ∗ , ± σ ; − i Λ /z ) = ± − θ ∗ ± σ σ (1 ± σ ) i Λ z F (1 + − θ ∗ ± σ , ± σ ; − i Λ /z ) . (3.29)It follows that the operator D is analytic in z and Λ D ( z , i Λ) = D ( i Λ) + z D ( i Λ) + z D ( i Λ) + . . . , (3.30)with each term computed recursively in terms of confluent hypergeometric functions.The first term D ( i Λ) defines the Fredholm determinant expression for the Painlev´e Vtranscendent defined in [24]. As we will see, we will not need the corrections D , D , . . . for our analysis.We will now assume that the s parameter that appears in (B.1) also has a finite z → a posteriori . The product of Gamma functionsrelating the s monodromy parameter to the κ parameter entering the Φ( t ) operatordefined in (B.4) has a comparatively peculiar limit. In the small z limit, terms of orderΛ − also survive. Following the analysis in [17], the first condition on the monodromyparameters (3.10) can be solved for κz σ , defining a series in z . The confluent limit canbe taken in that expression directly, resulting in κz σ = s Π( i Λ) σ (cid:18) − i σ (1 + ψ ∗ )2Λ z + O ( z ) (cid:19) (3.31)with Π defined as the confluent limit of the product of Gamma functions in (B.6)Π = Γ (1 − σ )Γ (1 + σ ) Γ(1 + ( θ ∗ − σ ))Γ(1 + ( θ ∗ + σ )) Γ(1 + ( θ − θ ∞ − σ ))Γ(1 + ( θ + θ ∞ − σ ))Γ(1 + ( θ − θ ∞ + σ ))Γ(1 + ( θ + θ ∞ + σ )) . (3.32)– 21 –et us now review the type of correction we have for the tau function away fromthe confluence point, where it coincides with the Painlev´e V transcendent. The relevantexpansions in Λ and z are:1. Expansion of the κ parameter, which is given in terms of powers of − iz / Λ.2. Expansion of the parametrices Ψ ( i ) c (3.27), which are given in terms of hypergeo-metric functions with monomials in z n ( i Λ) m .It is not difficult to see that, in the small Λ limit, the first type of expansion willdominate over the second. Disregarding the contributions of the second type, we haveexactly the same type of expansion as the Painlev´e V. This means that the first non-trivial correction to the accessory parameter problem of the confluent Heun equationcomes in the monodromy parameter, not in the tau function expansion. In short, thefirst non-trivial correction is obtained by expanding the κ parameter.The final step needed before the calculation can be performed is solving for thequantization condition for the radial equation, expressed in terms of the monodromyparameters in (3.15). It approaches, as z → s = e ∓ iπσ sin π ( θ ∗ + σ )sin π ( θ ∗ − σ ) sin π ( θ + θ ∞ + σ ) sin π ( θ − θ ∞ + σ )sin π ( θ + θ ∞ − σ ) sin π ( θ − θ ∞ − σ ) + . . . , (3.33)for (cid:60) Λ > (cid:60) Λ <
0, respectively. In this case the convergence is exponential, up toterms of order e − π | Λ | /z . Again, corrections are (exponentially) suppressed in terms of | Λ | . With these premises, the solution for the QNMs follows from the radial quantizationcondition, expressed in terms of s in (3.33), and the analysis is actually closely relatedto that of [17]. In the z → τ V ( θ ∗ , θ , θ ∞ ; σ, η ; i Λ) = 0 , (3.34) c = i Λ ddu log ˆ τ ( θ (cid:63) − , θ , θ ∞ + 1; σ − , η ; i Λ) + 14 (( σ − − ( θ (cid:63) − ) (3.35)with ˆ τ = det( − A Φ D Φ − ) and τ V ( i Λ) = ˜ C ( i Λ) − ( σ − θ (cid:63) ) e − i Λ θ ˆ τ is the Painlev´e Vtau function. The Plemelj operator A is defined in Appendix B and D is in (3.30).– 22 –he accessory parameter c is the limit as z → z K , given by (3.5) in our application.The procedure is now to tackle the condition in (3.34), which can be inverted forthe zero τ V = 0 in order to solve for κz σ as a series in Λ. The result is s ˜Π( i Λ) σ (cid:18) − i σ (1 + ψ ∗ )2Λ z + . . . (cid:19) = i ( σ + θ ∗ )(( σ + θ ) − θ ∞ )8 σ ( σ − Λ χ ( i Λ) (3.36a)where χ ( i Λ) is given as a series in Λ χ ( i Λ) = 1 + χ ( i Λ) + χ ( i Λ) + ..., (3.36b)with χ = ( σ − θ ∗ ( θ − θ ∞ ) σ ( σ − , (3.36c)and χ = θ ∗ ( θ − θ ∞ ) (cid:18) σ − σ − − σ − + 2 σ ( σ − (cid:19) − ( θ − θ ∞ ) + 2 θ ∗ ( θ + θ ∞ )64 (cid:18) σ − σ − (cid:19) + (1 − θ ∗ )( θ − ( θ ∞ − )( θ − ( θ ∞ + 1) )128 (cid:18) σ + 1) − σ − (cid:19) . (3.36d)The higher order terms can be computed recursively. For our application, we use thequantization condition (3.33) and (3.36a) now reads χ ( i Λ) = e − i πσ ΘΛ σ − (cid:18) − i σ (1 + ψ ∗ )2Λ z + . . . (cid:19) (3.37)whereΘ = i Γ (2 − σ )Γ( ( σ − θ ∗ ))Γ( ( σ − θ + θ ∞ ))Γ( ( σ − θ − θ ∞ ))Γ ( σ )Γ( (2 − σ − θ ∗ ))Γ( (2 − σ − θ + θ ∞ ))Γ( (2 − σ − θ − θ ∞ )) . (3.38)The treatment of the second condition (3.35) is analogous, substituting the value of χ ( i Λ) into the logarithmic derivative and solving for c as a function of Λ. The result,– 23 –ssuming (cid:60) σ >
0, is c = k + k ( i Λ) + k ( i Λ) + . . . + k n ( i Λ) n + . . . , (3.39a)with the first three terms in the expansion given by k = ( σ − − ( θ ∗ − , k = θ −
12 + θ ∗ θ ∗ ( θ − θ ∞ )4 σ ( σ − , (3.39b) k = 132 + θ ∗ ( θ − θ ∞ ) (cid:18) σ − σ − (cid:19) + (1 − θ ∗ )( θ ∞ − θ ) + 2 θ ∗ ( θ ∞ + θ )32 σ ( σ − − (1 − θ ∗ )(( θ ∞ − − θ )(( θ ∞ + 1) − θ )32( σ + 1)( σ − . (3.39c)Each term in the expansion of χ and c is a meromorphic function of σ , symmetricunder σ → − σ . There are single poles at σ = 3 , , , . . . , with the pole of order σ = n only seen at k n − term, and a pole of higher order near σ = 2, which is present at allorders save for k . The asymptotics of the χ n (3.36a) and k n (3.39) terms as σ (cid:39) σ − χ n (cid:39) k n = ( − n − C n − θ n ∞ ( θ − θ t ) n n ( σ − n − + . . . , n ≥ , (3.40)where the terms left out are of lower order in σ −
2. The relation k n = ( σ − χ n nolonger holds as one considers less divergent terms ( σ − − m , m < n −
1. Finally, C n is the n -th Catalan number: C n = 1 n + 1 (cid:32) nn (cid:33) = 1 , , , , , . . . (3.41)This remark will be useful when we discuss the CFT description and the fundamentalQNM in the following. Before delving into the QNMs, let us discuss the conformal aspects of the perturbations,both in the four-dimensional picture, and the “two-dimensional” point of view encoded– 24 –n the preceding analysis. If we take the massless scalar field (at ∆ = 4) to representthe scalar sector of gravitational perturbations in AdS, then the gravitational QNMscan be understood as resonances created by perturbations of the CFT state representedby the stress-energy tensor (2.23). The holographic view of the scattering process hasbeen considered in a great number of papers over the years, with the ideas listed herediscussed in [18] and [21]. The starting point is that scattering amplitudes are encodedin the 4-point function (cid:104)V
Heavy ( ∞ ) V Light ( x ) V Light ( x ) V Heavy (0) (cid:105) (3.42)with V Heavy vertex operators associated with the background charges – the mass andangular momenta of the black hole in our case – and V Light associated with the localinsertion of the scalar perturbations. In a Euclidean setting the positions of the heavyoperators are unambiguous, but in our Lorentz setting we will take 0 and ∞ to denotepast and future infinity respectively. The correlation functions such as in (3.42) havea rich history in the early development of CFTs, notably the seminal work of [41, 43–45]. The “partial wave” decomposition studied there can be thought of as a higherdimensional conformal block, in which the quantum numbers of the intermediate statesmirror the decomposition of the (scalar) wave equation in terms of SO(4 ,
2) quantumnumbers associated to the generators (2.35).For pure AdS, it has been known for some time that these higher dimensional con-formal blocks can be factorized in terms of “two-dimensional” classical conformal blocks(hypergeometric functions) [19, 46, 47]. Now, we want to view the conditions (3.10)relating the parameters of the differential equation to monodromy data as a deformedversion of this factorization, valid for the case with non-zero background charges (2.35).Given that the tau function expansion is given in terms of conformal blocks [48], thereare indeed some analogies, although in this case the two-dimensional CFT is chiral andhas unit central charge c = 1. Furthermore, z and u are independent, and in principlecomplex, variables.The latter point may be a consequence of Liouville exponentiation [49], and indeedthere is a semiclassical view also relating the accessory parameter of the Heun equationto the Painlev´e transcendent [22]. In the latter view, the accessory parameter also– 25 –ppears as the logarithmic derivative of the two-dimensional conformal block: (cid:104) V ∆ ( ∞ ) V ∆ (1)Π σ V ∆ ( t ) V ∆ (0) (cid:105) (3.43)where V ∆ i ( z i ) denotes each vertex operator, and Π σ the projection onto the Vermamodule constructed on the primary state, with Liouville momentum parametrized by σ . The conformal dimension ∆ i of each vertex operator V ∆ i ( z ) is given in terms of theLiouville momentum P i and the Liouville coupling constant b as∆ i = c −
124 + P i , c = 1 + 6 Q = 1 + 6( b + b − ) . (3.44)In this picture, all operators appearing in (3.10) are “heavy” in the sense that theyhave large Liouville momenta: P = θ − b , P t = θ + b , P = 2 − ∆ b , P ∞ = θ b , (3.45)which not only provide the Liouville interpretation for the single monodromy param-eters θ i , but also hint at a deeper connection. If the black hole absorbs a quantum ofenergy and angular momenta given by ω , m and m , respectively, then the increase inits entropy is given by δS = ω − m Ω , + − m Ω , + b T + = 2 π θ + b , (3.46)where we recovered Liouville theory’s (cid:126) = b . Thus, the entropy gained in the processis, up to the Liouville coupling, given by the Liouville momentum. This gives supportto the interpretation of V ∆ i , at least for the inner and outer horizon, as associated to a thermal state in the underlying dual theory. By assuming the latter to be unitary andmodular invariant, we can use Cardy’s formula [50] S π (cid:39) (cid:114) c (cid:16) L − c (cid:17) (3.47)for the entropy of such CFT at conformal dimension L = ∆ i . Assuming the classicallimit b →
0, and solving for L = ∆ i , we have again the interpretation of δS i / πb asthe Liouville momentum. – 26 –he construction outlined above is suitable for the generic case considered in [17],but it works in a completely analogous way for the confluence limit we consideredin Section 3.1.1. In the latter case, the two primary vertex operators associated tothe inner and outer horizons merge into a Whittaker operator [51]. The associatedconformal block is then an irregular one [52], instead of (3.43). The argument presentedhere about the nature of the intermediate states does have a direct parallel, though,and for further details we refer to the discussion in [25]. The calculation of the fundamental mode is technically very similar to the correspond-ing one in [17], where one solves for the first eigenfrequency ω , , , by applying theconditions (3.10) given the quantization of the angular and radial monodromy param-eters (3.11) and (3.13). From the discussion above, the corresponding task at lowtemperature involves solving the equations (3.37) and (3.39).As suggested by Fig. 2, for generic values of M , a and a a smooth zero-temperature limit for the fundamental QNM frequency exists. At small r + , we canwork out asymptotic formulas, with θ ∗ = iφ ∗ r + , i Λ = iλr , θ = ω − βr , σ = 2 − νr , (3.48)where the values of φ ∗ , λ and β can be read from (3.26), (3.25) and (3.24), setting m = m = 0, respectively: φ ∗ = (1 + α ) ω, λ = 2(1 + α ) ω, β = 3(2 + α ) ω (3.49)with ω (cid:39) ∆, as expected from the vacuum AdS limit. We will take the last equation in(3.48) to be the definition of ν , which then needs to be determined from the monodromyconditions (3.35).In the following we will focus on the case where ∆ is an integer, with the mostrelevant case for pure gravity perturbations being ∆ = 4. The Θ parameter in (3.37)has the small r + expansion for ∆ integer and (cid:96) = 0:Θ = φ ∗ ( ν − β )2 r ν ( ν + β ) (∆ − (cid:18) i νφ ∗ r + + O ( r ) (cid:19) . (3.50)– 27 – .0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 π T + R e ω π T + I m ω Figure 2 . The fundamental QNM frequency ω , , , as a function of the outer horizontemperature T + for a Kerr-AdS black hole with fixed r + = 0 . a = a − a = 0 . The accessory parameter c as in (3.39) and the χ function defined in (3.37) bothreceive contributions from all orders in z in the small r + limit. This is due to theproximity of σ to the critical value 2, as further clarified in the next subsection. Con-tributions to c and χ can be resummed using the generating formula for the Catalannumbers 1 + x + 2 x + 5 x + 14 x + . . . = ∞ (cid:88) n =0 C n x n = 1 − √ − x x . (3.51)The result for χ can then be written as χ = 12 + φ ∗ ( θ − θ )8 ν λ + 12 (cid:114) φ ∗ ( θ − θ )2 ν λ − ( θ − θ ) λ ν (cid:113) φ ∗ ( θ − θ )2 ν λ r + O ( r ) , (3.52)where θ = 2 − ∆. At this order in r + , we can substitute θ − θ = (2∆ + 2 n + (cid:96) − n + (cid:96) + 2) + O ( r ). The expansion for the accessory parameter c in the right-handside of (3.39a) can also be resummed in the same way: c = i φ ∗ r + + 14 φ ∗ r − νr (cid:114) φ ∗ ( θ ∞ − θ )2 ν λ + i θ − λr + O ( r ) . (3.53)– 28 –he same parameter c is defined as lim z → z K and can be computed from (3.5)and (3.12) by taking the appropriate r − → r + limit. In the (cid:15) (cid:28) r + (cid:28) (cid:96) = 0, this provides for the left-hand side of equation (3.39a), c = i φ ∗ r + + 14 φ ∗ r − α ω − ∆(∆ − r + i θ − λr + . . . , (3.54)which, equated to the right-hand side of (3.53), leads to an equation for ν as in (3.48) (cid:113) ν + (1 + α ) ω ( ω − (∆ − ) = 1 + α ω − ∆(∆ − . (3.55)The latter can be solved to yield, at zero T + and small r + , ν = 1 + α (cid:112) ( ω − ∆(∆ − + 4 ω ( ω + 4) + O ( (cid:15), r ) (3.56)and likewise for χχ = 14 + (1 + α )(3 ω − ∆(∆ − ν + (1 + α ) (3 ω − ∆(∆ − ν + O ( (cid:15), r ) (3.57)Finally, by substituting (3.56) and (3.57) into (3.37), we can compute the firstcorrection to the fundamental QNM frequency as a function of r + , including the firstnon-trivial O ( (cid:15) ) correction ω , , , = ∆ + (1 + α ) (cid:18) − X + 1 X − √ ∆ + 8 (cid:19) r − i (1 + α )∆ √ ∆ + 8 2 X ( X − (cid:18) √ ∆ + 8 r + − α ) ∆ (cid:15)r + (cid:19) r + . . . (3.58)where X = ∆ + 4∆ + 12 + 2(∆ + 4) √ ∆ + 82(∆ − . (3.59)The imaginary part of the frequency controls the relaxation times of the perturbation inthe dual CFT. Equation (3.58) shows that the temperature – proportional to (cid:15) by (3.22)– has a “reinforcing” effect on the fluctuations, increasing the (negative) imaginary partof the frequency, and thus increasing the decay time of the mode. In this sense, a phasetransition possibly occurs and the imaginary part becomes zero, when the system loses– 29 –his relaxation time scale. Perturbatively in T + , this happens at particular points ofthe parameter space, satisfying the relation r − r − (2 r + a + a ) (cid:39) ∆ √ ∆ + 82 = 16 √ , (3.60)where in the last equality we substituted ∆ = 4. Note that (3.60) comes from requiringthat (cid:15)/r + ∼ r + in (3.58), which respects the approximations taken. The condition(3.60) is actually a function of a + a and thus corresponds to a family of values of theasymptotic charges. Given that its left-hand side is proportional to the temperatureof the black hole T + , through the term r − r − , this transition is characterized by thedynamical critical relation t relax ∝ | T − T c | − , with critical exponent – called z/η in theliterature – equal to − T c (cid:39) √ π M (cid:113) M − π ( J + J ) , (3.61)where we employed the small r + expressions for the mass and angular momenta in(2.4). The dual character of this transition is rather mysterious because, unlike otherphase transitions of quantum nature, it happens at finite, albeit low T c ∝ r + (cid:28) m = m = 0 and it does notcouple to the “chemical potentials” Ω i, + associated to the angular velocities. At anyrate, these remarks pertain to the thermodynamics of the dual field theory, and not ofthe black hole, like argued in [53]. We hope to address this issue in future work. The two-dimensional version of (3.42), depicting the perturbation of the black holebackground, was studied in [29], where it was shown that the vacuum block, wherethe internal dimension is zero, was given by the generating function for the Catalannumbers in the heavy-light semiclassical limit, where two of the insertions have highconformal dimensions ∆ i ∝ c → ∞ , with c being the central charge.There is indeed a clear parallel to what is being counted in the fundamental modecalculations in the last subsection. The finite-temperature version of the calculationoutlined in [17] does have this hierarchy between the vertex operators, since the ones– 30 –ssociated with the inner and outer horizons do become “light” in the small black holeregime, and the residue at σ = 2 does indeed correspond to the identity operator asintermediate state in (3.42).Yet, there are subtleties. We have already remarked that (3.43) is a chiral conformalblock. A difference from [29] is that in principle all operators in (3.43) are “heavy”,in the sense that their conformal dimensions scaling with c . The parametric hierarchybetween operators arises from the parameter r + , which can be made small. It is acombined effect of the position of the insertion and the scaling of the dimensions ofthe “light” operators. The corresponding calculation using the Virasoro algebra seemsstraightforward, but lies outside the scope of this work.Another difference comes from the confluence limit taken as the temperature goesto zero. In this case the limiting form of the Liouville momenta for the inner and outerhorizon is given by (3.23), and the conformal dimensions of the operators associatedto θ ± are in fact much larger than c . The “light” operator is the resulting operatorcoming from the leading term of the OPE for the operators related to the inner andouter horizons: V P ( t ) V P (0) ∼ V P (0) + . . . (3.62)where we label the operators by the Liouville momentum, and P = P + P . As thefirst descendant of the resulting module [ V P ] one finds the Whittaker operator [51, 52] V θ ∗ /b,i Λ (0) ∼ : e θ ∗ /bφ L + i Λ ∂φ L : (3.63)which is labelled by the confluence parameters. We used the Feigin-Fuchs parametriza-tion of the Liouville field φ L (see [52] or [25]) in the right-hand side of (3.63) to compareour notation with the literature. In this case, the parameters in the regime investigatedhere do warrant the “light” requisite of [29] since they are proportional to r + and r ,respectively, as can be checked in (3.25) and (3.26). In light of this, the results obtainedin Sec. 3.3 point to an analogous simplification of the conformal block in terms of theCatalan number generating function when the two light operators in the analysis of[29] are substituted by an irregular light operator. For more on the relation betweenirregular conformal blocks and the Painlev´e V tau function we recommend [24].– 31 – .4 Higher quasi-normal modes Higher QNMs, with (cid:96) ≥
1, allow for non-zero values of m and m . In turn, thesemultiply the angular velocities Ω i, + , which are given in our parametrization byΩ i, + = a i (1 − a i ) r + a i = α i (1 − α i r ) r + (1 + α i ) , (3.64)with α i = a i /r + , i = 1 ,
2. As explained before, at small r + the range of allowed a i iscapped by r + , so it makes sense to consider fixed values of α i (cid:46)
1. Then in the small r + regime, the velocities Ω i, ± ∼ /r + become very large and the monodromy parameter(3.3) θ + = i π (cid:18) ω − m Ω , + − m Ω , + T + (cid:19) (3.65)varies so that its imaginary part can become negative. Note that r + can alternativelybe thought of as the ratio between the outer horizon and the AdS radius.As a matter of fact, it has been long established from heuristic arguments thatcombinations of ω, m and m such that (cid:61) θ + < (cid:61) θ + is negative,we have the superradiance window and unstable modes are bound to happen. Theinstability was anticipated by the thermodynamical analysis [11] and first observednumerically for four-dimensional black holes in [55]. The effect is, however, tiny andhard to study using conventional numerical methods, even at finite temperature [56].Our results in [17] provided supporting evidence for the positivity of the imaginarypart of the (cid:96) = 1 , m = 1 , m = 0 QNM frequency in the small r + limit, based on thequantization conditions (3.11) and (3.13).The low temperature version of the calculation for QNMs in [17] follows an analo-gous strategy, based on (3.37) and (3.39). Since the final analytical expression for theimaginary part of the (cid:96) ≥ r + for (cid:96) = 1. One sees that modes with either m or m positive are unstable for small values– 32 – .00 0.03 0.06 0.09 0.12 0.15 r + R e ω ‘ = 1 , m = − , m = 0 ‘ = 1 , m = 0 , m = − ‘ = 1 , m = 0 , m = 1 ‘ = 1 , m = 1 , m = 0 r + I m ω × Figure 3 . The (cid:96) = 1 modes for a = a − a = 0 .
001 as a function of r + and fixed, verysmall temperature T + ∼ − . The modes m = 1 , m = 0 (dark red) and m = 0 , m = 1(light red) display a small positive imaginary part for values where (cid:61) θ + < of r + and moderate values of a and a , of which the example shown is characteristic.From the superradiant condition (cid:61) θ + <
0, one sees that there are two competingfactors determining whether the mode will be unstable or not: we have the positivecontribution from the real part of the frequency and the negative contribution fromthe angular velocity. Now, as one turns on the rotation parameters a and a , the realpart of the frequency gets negative corrections that overwhelm the growth in angularvelocity, effectively reducing the superradiant window. The “kink” in the zoomed ininset as (cid:61) θ + crosses to positive values comes from the change of sign of Λ in (3.23).In the zero temperature limit, this change of sign will induce a discontinuity in the s value obtained from the quantization condition (3.33).From the conformal block point of view, the calculation of the QNMs is trulyperturbative in the CFT levels, with the internal momentum centered at σ = 2+ (cid:96) − ν (cid:96) r .It is interesting to note that these should correspond in the r + → σ (cid:39) (cid:96) . As a matter of fact, the latter valuesfor σ do indeed correspond to poles in the conformal block expansion at level t (cid:96) +10 . It– 33 – .00 0.03 0.06 0.09 0.12 0.15 r + R e ω ‘ = 2 , m = − , m = 0 ‘ = 2 , m = − , m = − ‘ = 2 , m = − , m = 1 ‘ = 2 , m = 0 , m = − ‘ = 2 , m = 0 , m = 0 ‘ = 2 , m = 1 , m = − r + I m ω r + R e ω ‘ = 2 , m = 0 , m = 2 ‘ = 2 , m = 1 , m = 1 ‘ = 2 , m = 2 , m = 0 r + I m ω × Figure 4 . Stable (top, blue) and unstable (bottom, red) modes for (cid:96) = 2 and a − a = 0 . r + . Like in the (cid:96) = 1 case, the positive imaginary part is very small andrestricted to the condition (cid:61) θ + < would be interesting to investigate whether this alternative approach could lead to asimplified analytic study of these modes.In Fig. 4 we display the (cid:96) = 2 case. Again, the unstable modes correspond topositive m and/or m , and the region of instability is determined by the superradiantwindow (cid:61) θ + <
0. The comparison to (cid:96) = 1 – see also Fig. 5 – shows that the decay rate– 34 – .02 0.04 0.06 0.08 0.10 0.12 0.14 r + R e ω r + I m ω × a = 0 . a = 0 . a = 0 . a = 0 . a = 0 . a = 0 . a = 0 . a = 0 . Figure 5 . The frequencies of the unstable modes for (cid:96) = 1 as a function of r + for variousvalues of a = a − a . Note that the superradiant window induced by the condition (cid:61) θ + < a due to the larger reduction of the real part of theeigenfrequency. of the most unstable mode [11], i.e. , the one with largest m Ω , + + m Ω , + decreaseswith (cid:96) . Indeed, Fig. 6 shows that the maximum of the decay rate as a function of r + does decrease from (cid:96) = 1 to (cid:96) = 4, making those higher (cid:96) states longer-lived. Ifthis trend continues for larger (cid:96) , it would mean that the decaying profile of the scalaremissions from the black hole is a combined effect of the angular dependence of thecoupling between the black hole and the scalar perturbations on the one hand and thehalf-life of the perturbation as a function of (cid:96) , and the radial eigenvalue n on the other.One should also note from Fig. 5 that the superradiant window appears to de-crease as a = a − a increases, indicating that, as far as the contributions to θ + are concerned, the negative correction to the real part of the eigenfrequency in (3.65)dominates over the effect of the increased angular rotation. So, as a increases, thereal part of (cid:61) θ + turns positive. In fact, around the value a (cid:39) .
045 there is nosuperradiance for (cid:96) = 1, as can be inferred from Fig. 5. This is in contrast with– 35 – .02 0.04 0.06 0.08 0.10 0.12 0.14 r + R e ω r + I m ω × ‘ = 1 , m = 1 , m = 0 ‘ = 2 , m = 2 , m = 0 ‘ = 3 , m = 3 , m = 0 ‘ = 4 , m = 4 , m = 0 Figure 6 . The most unstable mode m = (cid:96) as a function of r + for (cid:96) = 1 , , ,
4. Notethat the smaller decay rate for larger values of (cid:96) is accompanied by a larger width of thesuperradiant window. The black hole is expected to decay through different values of (cid:96) , m and m depending on the values of its charges. the fragmentation phenomenon, which is expected at sufficiently distorted – i.e, highlyrotating – five-dimensional black holes as argued in [57]. Given the results of Sec. 3 for the unstable QNMs, we may reflect on the fate of thecorresponding state in the putative dual CFT. The instabilities signal that the stateassociated to the black hole will decay. The particular features of the decay, such asits rate and final products, will of course depend on the coupling between the blackhole and the perturbation fields, which in holography can be read from the stress-energy tensor. For the scalar type of perturbations considered here we can deduce aninteraction Hamiltonian of the sort H int = λh ab T ab . As we learned from the discussionabove, the imaginary part of the QNMs frequencies remains fairly constant with (cid:96) , withthe most unstable mode at given (cid:96) being the one with the largest m Ω , + + m Ω , + .Let us first consider the case where the field theory lives in the global boundary– 36 – × S . The energy density profile can be read from (2.23)4 πG N, ¯ ρ = 3 M θ = 3 M (cid:0) a sin ¯ θ + a cos ¯ θ ) + . . . (cid:1) . (4.1)We interpret the expansion in the right-hand side to be the contribution of the higher S spherical harmonics. From (4.1) we conclude that the generation of the (cid:96) -th modewill be dampened by a factor of a (cid:96)i . Since a and a are parametrically small, of order r + , the corresponding five-dimensional spheroidal harmonics can be approximated bytheir zero rotation counterparts, i.e , the three-dimensional spherical harmonics [58] Y m ,m (cid:96) (¯ θ, ¯ φ , ¯ φ ) = (cid:114) (cid:96) + 12 π (cid:115) (( (cid:96) + m + m ) / (cid:96) − m − m ) / (cid:96) + m − m ) / (cid:96) − m + m ) / × (sin ¯ θ ) m (cos ¯ θ ) m P ( m ,m ) ( (cid:96) − m − m ) (cos 2¯ θ ) e im ¯ φ + im ¯ φ , (4.2)where P ( a,b ) n ( z ) are the Jacobi polynomials. At given (cid:96) , the most unstable mode has theenergy dependence proportional to the tt -component of the scalar stress-energy tensor,in turn roughly proportional to the absolute value squared of the field itself¯ ρ fluc . ∝ | Φ (cid:96),m = (cid:96),m =0 | = (cid:12)(cid:12)(cid:12) e − iω ( (cid:96) )¯ t Y (cid:96), (cid:96) (cid:12)(cid:12)(cid:12) ∝ e (cid:61) ω ( (cid:96) )¯ t sin (cid:96) ¯ θ, (4.3)which favors localization at the ¯ θ (cid:39) π/ (cid:61) ω ( (cid:96) ) correspondsto the imaginary part of the eigenfrequency ω ,(cid:96),(cid:96), , which is assumed positive for theunstable mode.In the case where the dual theory lives in flat coordinates, we can in principle justapply the transformation (2.29) to translate from the global results above. However,our analysis in Sec. 3 was made by assuming a > a , which in global coordinates isnot a restriction because one can interchange the azimuthal variables ¯ φ and ¯ φ . Whenwe transform to flat coordinates, we choose one of them to map to the coordinate ˆ φ ,and we have to consider the two choices separately.For the case a > a , we have ˆ φ = ¯ φ as in Sec. 3, and the energy profile for thefluctuation reads ¯ ρ fluc . ∝ e (cid:61) ω ( (cid:96) )¯ t (cid:32) (1 + (ˆ t − ˆ r )) + ˆ r cos ˆ θ (1 + (ˆ t + ˆ r ) )(1 + (ˆ t − ˆ r ) ) (cid:33) (cid:96)/ (4.4)– 37 –here ¯ t = arctan ˆ t + ˆ r t − ˆ r . (4.5)The case a < a , can be realized by setting ˆ φ = ¯ φ , and the coordinate transformationchanges from (2.29) to sin ¯ θ = sin ˆ χ sin ˆ θ, (4.6)which corresponds to interchanging a ↔ a in the profile (2.32). The energy dissipatedin the perturbation mode is now¯ ρ fluc . ∝ e (cid:61) ω ( (cid:96) )¯ t (cid:32) ˆ r sin ˆ θ (1 + (ˆ t + ˆ r ) )(1 + (ˆ t − ˆ r ) ) (cid:33) (cid:96)/ , (4.7)with ¯ t given by (4.5).We note that, with the second choice, there is a tendency for the decay to occur inthe ˆ x − ˆ y plane (sin ˆ θ (cid:39) ρ fluc . . As we discussed above,the specific value of (cid:96) which will be preferred by the decay will depend on the detailsof the state and the coupling. In Fig. 6 we see that, even though the maximum ofthe imaginary part of the eigenfrequency decreases with (cid:96) , larger values of r + tend tofavor larger values of (cid:96) for the decay. This leads us to infer that, at low temperatures,the localization effect of the decay products resulting from (4.7) increases with themass M . We also point out the difference between the time dependence of the decaymodes depending whether we consider global coordinates R × S (bar coordinates) orconformally flat coordinates R , (hatted coordinates). Whereas in global coordinatesthe fluctuations grow exponentially without bound, at least in the linear analysis wedo here, the growth in conformally flat coordinates is capped as ˆ t → ∞ due to (4.5). In this paper we analyzed the holographic aspects of the generically rotating Kerr-AdS black hole in Lorentzian signature, focussing on the low-temperature limit T + (cid:39) r + (cid:28)
1. We reviewed the definition of asymptotic charges, discussedthe geometrical meaning of the regularization involved in the definition of mass, anddetermined the holographic stress-energy tensor associated to the black hole in thedual field theory – whose thermal state is characterized by non-zero vacuum expectation– 38 –alues for all generators of the Weyl subgroup of the conformal group SO(4 , (cid:96) .For (cid:96) >
0, the existence of instabilities in the QNMs was anticipated by [30], usingthermodynamical arguments. For (cid:96) = 0, the first order correction in the temperaturehas an enhancement effect which decreases the decay time of the perturbation. Theeffect is large enough that, even at parametrically low temperature, the imaginary partof the QNM frequency turns positive, and the mode becomes unstable. We have alsostudied numerically the dependence of the (cid:96) > (cid:96) . These instabilities of AdS space are expected from general arguments[16], and we attempted a holographic interpretation by studying the instabilities of thethermal field theory state corresponding to the black hole and the spatial dependenceof its decay products. We found qualitative hints that the ejecta tend to collimate atthe axis of highest rotation as the black hole mass increases.We have also shown that the general procedure of [17], which solved for the non-local boundary conditions related to QNMs in terms of monodromy data of the Heundifferential equations involved, can be used effectively to find QNMs in the zero temper-ature limit. This limit led us to consider the monodromy parameters to be determinedby the Painlev´e V transcendent, obtained from the confluence limit of the Painlev´eVI. The importance of the expansion of the Painlev´e transcendents in terms of c = 1Virasoro conformal blocks, as studied in [24], provides us with the same interestingfactorization of the four-dimensional conformal blocks in terms of (semi-classical) two-dimensional ones, as first anticipated by [19]. The new ingredient here is that thetwo-dimensional conformal blocks are of the irregular type, as also appeared in theblack hole perturbation context in [25]. The semiclassical conformal blocks associatedto the radial perturbations seem to arise from a unitary theory, and the Vertex oper-ators associated to the inner and outer horizons can be interpreted as thermal states,– 39 –ith their degeneracy equating the entropy of the scalar perturbation with quantumnumbers given by ω, m and m as it is absorbed by the black hole. Finally, the studyof QNMs at small values of r + showed the relevance of short Virasoro representationsfor the intermediate states of the conformal blocks, again anticipated in a differentcontext by [29].The motivation for considering low-temperature black holes stemmed from thenecessity for pushing the hydrodynamical analogy of [38], if we want to understandthe holographic interpretation of instabilities in AdS space. We have seen that thephase diagram of these states in the dual theory has a richer set of associated phenom-ena than what is expected from the “naive” near-UV considerations, with instabilitiesin all modes and collimation of decay products. Tools usually used in holographicrenormalization-group flow are not really suitable to determine the fate of the state asone flows to the infrared. We can use the same methods proposed here to study whathappens at larger values of r + , but the appearance of instabilities as well as fragmen-tation issues [57] – the latter absent here – can further contribute to complicate theissue of determining the fate of the decay process.Finally, we stress that the isomonodromic method used here for the numericalanalysis overcomes a major hurdle arising in usual Frobenius matching. At low tem-peratures, the latter method suffers from large spatial oscillations of the modes nearthe outer horizon r + . The isomonodromic method has no such hindrance. Moreover,our investigation also shows that the mode expansion used in Frobenius matching isnot suitable to study the fundamental mode. Indeed, in the finite temperature case, wehave shown that the monodromy parameters, and thus the actual value of the QNMfrequency, receive contributions from all levels in the Frobenius expansion. These con-tributions can be interpreted, in the two-dimensional conformal block attribution, ascoming from all descendants of the CFT primaries associated to the intermediate states.We verified that these contributions can be resummed in terms of the generating func-tion of the Catalan numbers. Also, short representations of the CFT Virasoro algebraare relevant for the calculation of QNMs at higher levels. We defer the study of otherinteresting black hole limits, as well as higher spin perturbations [59] to future work.– 40 – cknowledgements We apologize for any omission in this important and long standing field of research.The authors would like to thank Dmitry Melnikov, Monica Guica, Oleg Lisovyy forcomments and suggestions along the way.
A Asymptotic charges of AdS spaces
Let the asymptotic metric be given by g ab = 1˜Ω ˜ g ab + h ab (A.1)where h ab is understood to be small with respect to the first term in the ˜Ω → g ab is either the flat metric (Poincar´e patch),called ˆ g ab in the main text, or the R × S metric (global coordinates), referred to by ¯ g ab in the main text. Let us introduce the covariant derivates ∇ a , associated with g ab and˜ ∇ a , associated with ˜ g ab . If χ a is a vector field, we can use Leibniz rule to show thatthe difference ∇ a χ b − ˜ ∇ a χ b is a linear operator on χ a , therefore, for any two covariantderivatives, we have ∇ a χ b = ˜ ∇ a χ b + C bac χ c , (A.2)with the difference between the connections given by the conditions that ∇ a is com-patible with g ab and ˜ ∇ a with ˜ g ab . To first order in h ab : C bac = 12 g bd ( ˜ ∇ a g dc + ˜ ∇ c g ad − ˜ ∇ d g ac ) (A.3)= 12 ˜Ω (˜ g bd − ˜Ω h bd )( ˜ ∇ a ( 1˜Ω ˜ g dc + h dc ) + ˜ ∇ c ( 1˜Ω ˜ g ad + h ad ) − ˜ ∇ d ( 1˜Ω ˜ g ac + h ac )) , (A.4)– 41 –ith indices in the right hand side now and hereafter raised with ˜ g ab , so that h ab ≡ ˜ g ac h cd ˜ g db . Expanding up to quadratic terms in h ab , C bac = − δ bc ˜ ∇ a ˜Ω + δ ba ˜ ∇ c ˜Ω − ˜ g ac ˜ ∇ b ˜Ω) + ˜Ω( h bc ˜ ∇ a ˜Ω + h ba ˜ ∇ c ˜Ω − ˜ g ac h bd ˜ ∇ d ˜Ω)+ ˜Ω ( ˜ ∇ a h bc + ˜ ∇ c h ba − ˜ ∇ b h ac ) . (A.5)Conserved quantities associated with vector fields ξ a are defined as the integral ofthe ( d − Q [ ξ ] a ...a d − = − κ (cid:15) [ g ] aba ...a d − g ac ∇ c ξ b , (A.6)with (cid:15) [ g ] the volume form associated with g ab . The contravariant derivative can alsobe computed to first order in h ab : g ac ∇ c ξ b = ˜Ω (˜ g ac − ˜Ω h ac )( ˜ ∇ c ξ b + C bcd ξ d ) (A.7)or g ac C bcd = − ˜Ω( δ bd ˜ ∇ a ˜Ω + ˜ g ab ˜ ∇ d ˜Ω − δ ad ˜ ∇ b ˜Ω) + ˜Ω ( h bd ˜ ∇ a ˜Ω + h ab ˜ ∇ d ˜Ω − h ad ˜ ∇ b ˜Ω)+ ˜Ω ( δ bd h ac ˜ ∇ c ˜Ω + h ab ˜ ∇ d ˜Ω − δ ad h bc ˜ ∇ c ˜Ω) + ˜Ω ( ˜ ∇ a h bd + ˜ ∇ d h ab − ˜ ∇ b h ad ) , (A.8)Now, the vector fields we are interested in are Killing vector fields of the inducedmetric at the conformal boundary ˜Ω = 0. These will satisfy: ξ d ˜ ∇ d ˜Ω = 0 , ˜ ∇ a ξ b + ˜ ∇ b ξ a = 0 , (A.9)denoting the vanishing Lie derivatives with respect to ξ a of ˜Ω and ˜ g ab , respectively.Note that we are assuming that ξ a and ˜ ∇ a ˜Ω are orthogonal, which is suitable for theasymptotically AdS case.Wrapping it all up, g ac ∇ c ξ b = 2 ˜Ω ξ [ a ˜ ∇ b ] ˜Ω + ˜Ω ˜ ∇ [ a ξ b ] − ( ξ d h [ ad ˜ ∇ b ] ˜Ω + ξ [ a h b ] c ˜ ∇ c ˜Ω)+ ˜Ω ( ξ d ˜ ∇ [ a h b ] d − h d [ a ˜ ∇ d ξ b ] + £ ξ h ab ) + O ( h ) (A.10) For convenience, a term equal to the volume of the d − Q [ ξ ] is omitted. – 42 –here £ ξ h ab is the Lie derivative of h ab with respect to ξ a . The volume form is readilycomputed: (cid:15) [ g ] = 1˜Ω d (cid:15) [˜ g ](1 + ˜Ω h aa + O ( h )) (A.11)and so the charge density can be expanded Q [ ξ ] a ...a d − = − κ (cid:15) [˜ g ] aba ...a d − d − (2 ξ [ a ˜ ∇ b ] ˜Ω + ˜Ω ˜ ∇ [ a ξ b ] − ( ξ d h [ ad ˜ ∇ b ] ˜Ω + ξ [ a h b ] c ˜ ∇ c ˜Ω − h cc ξ [ a ˜ ∇ b ] ˜Ω)+ ˜Ω ( ξ d ˜ ∇ [ a h b ] d − h d [ a ˜ ∇ d ξ b ] + h cc ˜ ∇ [ a ξ b ] ) + O ( h )) (A.12)By integrating Q [ ξ ] in the conformal boundary ˜Ω → (cid:96) AdS = 1) near infinity is written as: R ac = − ( d − g ac , (A.13)which, when written in terms of ˜ g ab , yields, up to terms of order h ab :˜ R ac + ( d −
2) ˜ ∇ a ˜ ∇ c ˜Ω˜Ω − ( d − g ac ˜ g bd ˜ ∇ b ˜Ω ˜ ∇ d ˜Ω˜Ω + ˜ g ac ˜ g bd ˜ ∇ b ˜ ∇ d ˜Ω˜Ω = − ( d −
1) ˜ g ac ˜Ω . (A.14)Requiring that the term proportional to ˜Ω − vanishes at the boundary implies that n a = ˜ g ab ˜ ∇ b ˜Ω is normalized space-like according to ˜ g ab in the ˜Ω → O ( ˜Ω − ) term will set the (conformal, Ricci) geometry at infinitythrough ˜ R ac − d −
1) ˜ g ac ˜ R = − ( d −
2) 1˜Ω ˜ ∇ a ˜ ∇ c ˜Ω . (A.15)Therefore the conformal geometry chosen for the four-dimensional manifold at infinitywill influence the asymptotics of ˜Ω, and thus the actual value of the asymptotic charges.We may consider now two cases for Q [ ξ ], depending on ξ a :1. ξ a is the time translation operator, and therefore orthogonal to the surface ofintegration used to define the conserved quantity;2. ξ a generates rotations, so it is tangent to the surface of integration;– 43 –n the first case, the term ξ [ a ˜ ∇ b ] ˜Ω is a bi-vector normal to the surface of integrationand Q [ ξ ] diverges in the ˜Ω → ∇ a ˜Ω approaches the unit vector at spatial infinity n a by: n a = (1 + N ) − / ˜ ∇ a ˜Ω , N = ˜Ω h ab ˜ ∇ a ˜Ω ˜ ∇ b ˜Ω . (A.16)The second term in brackets in the first line of (A.12) is ˜Ω ˜ ∇ [ a ξ b ] which we willalso take to vanish, either because ξ a is covariantly constant with respect to ˜ g ab , asin the case of time translation, or because the derivative is tangent to the surface ofintegration, as in the case of rotations. The next terms will converge in the ˜Ω → h ab = ˜Ω d − γ ab . Substituting in (A.12) and taking the limit we have: Q [ ξ ] a ...a d − = 1 κ (cid:15) [˜ g ] aba ...a d − (( d − n [ a γ b ] c ξ c − ξ [ a γ b ] c n c + ( γ cc + N ) ξ [ a n b ] ) + O ( ˜Ω) . (A.17)One can then conclude that the value for generic charges in asymptotically AdS spacesstems not only from the conformal structure encoded in ˜ g , but also from the notion of“normalized direction in RG flow”, encoded by the normalization of n a . B The Fredholm determinant formulation of the Painlev´e VItranscendent
This is a review of the Fredholm determinant formulation of the Painlev´e VI, borrowingheavily from [32] τ ( t ) = const · t ( σ − θ − θ t ) (1 − t ) − θ t θ det( − A Φ( t ) D Φ( t ) − ) , (B.1)where the Plemelj operators A , D act on the space of pairs of square-integrable functionsdefined on C , a circle on the complex plane with radius R < A g )( z ) = (cid:73) C dz (cid:48) πi A ( z, z (cid:48) ) g ( z (cid:48) ) , ( D g )( z ) = (cid:73) C dz (cid:48) πi D ( z, z (cid:48) ) g ( z (cid:48) ) , g ( z ) = (cid:32) f + ( z ) f − ( z ) (cid:33) , (B.2)– 44 –ith kernels given, for | t | < R , explicitly by A ( z, z (cid:48) ) = Ψ( σ, θ , θ ∞ ; z )Ψ − ( σ, θ , θ ∞ ; z (cid:48) ) − z − z (cid:48) ,D ( z, z (cid:48) ) = − Ψ( − σ, θ t , θ ; t/z )Ψ − ( − σ, θ t , θ ; t/z (cid:48) ) z − z (cid:48) . (B.3)The operators A and D can be thought of as projecting the analytic and principal partsof a generic function defined on the circle C into the space generated by functions withdefinite monodromy. The parametrix Ψ and the “gluing” matrix Φ are Ψ( α , α , α ; z ) = (cid:32) φ ( α , α , α ; z ) χ ( α , α , α ; z ) χ ( − α , α , α ; z ) φ ( − α , α , α ; z ) (cid:33) , Φ( κ, σ ; t ) = (cid:32) t − σ/ κ − / t σ/ κ / (cid:33) , (B.4)with φ and χ given in terms of Gauss’ hypergeometric function – note the overall minussign with respect to the conventions in [32]: φ ( α , α , α ; z ) = F ( ( α − α + α ) , ( α − α − α ); α ; z ) χ ( α , α , α ; z ) = α − ( α − α ) α (1 + α ) z F (1 + ( α − α + α ) , ( α − α − α ); 2 + α ; z ) . (B.5)Finally, κ is a known function of the monodromy parameters: κ = s Γ (1 − σ )Γ (1 + σ ) Γ(1 + ( θ t + θ + σ ))Γ(1 + ( θ t − θ + σ ))Γ(1 + ( θ t + θ − σ ))Γ(1 + ( θ t − θ − σ )) × Γ(1 + ( θ + θ ∞ + σ ))Γ(1 + ( θ − θ ∞ + σ ))Γ(1 + ( θ + θ ∞ − σ ))Γ(1 + ( θ − θ ∞ − σ )) , (B.6)and, finally, the parameter s is given in terms of the monodromy parameters { σ t , σ t } : s = ( w t − p t − p t p ) − ( w − p − p t p t ) exp( πiσ t )(2 cos π ( θ t − σ t ) − p )(2 cos π ( θ − σ t ) − p ∞ ) , (B.7)where p i = 2 cos πθ i , p ij = 2 cos πσ ij ,w t = p p t + p p ∞ , w t = p p t + p p ∞ , w = p p + p t p ∞ . (B.8)The expansion for the tau function at small t can be computed directly from its– 45 –efinition: τ ( t ) = Ct ( σ − θ − θ t ) (1 − t ) θ θ t (cid:18) (cid:18) θ θ t θ − θ t − σ )( θ ∞ − θ − σ )8 σ (cid:19) t − ( θ − ( θ t − σ ) )( θ ∞ − ( θ − σ ) )16 σ (1 + σ ) κt σ − ( θ − ( θ t + σ ) )( θ ∞ − ( θ + σ ) )16 σ (1 − σ ) κ − t − σ + . . . (cid:19) . (B.9)The accessory parameter K , defined by (3.10) as essentially the logarithm deriva-tive of τ , has an analogous expansion for small t and can be seen in [17]. References [1] G. Policastro, D. T. Son, and A. O. Starinets,
The Shear viscosity of strongly coupledN=4 supersymmetric Yang-Mills plasma , Phys. Rev. Lett. (2001) 081601,[ hep-th/0104066 ].[2] R. Baier, P. Romatschke, D. T. Son, A. O. Starinets, and M. A. Stephanov, Relativisticviscous hydrodynamics, conformal invariance, and holography , JHEP (2008) 100,[ arXiv:0712.2451 ].[3] S. Bhattacharyya, V. E. Hubeny, S. Minwalla, and M. Rangamani, Nonlinear FluidDynamics from Gravity , JHEP (2008) 045, [ arXiv:0712.2456 ].[4] S. Bhattacharyya, R. Loganayagam, S. Minwalla, S. Nampuri, S. P. Trivedi, and S. R.Wadia, Forced Fluid Dynamics from Gravity , JHEP (2009) 018, [ arXiv:0806.0006 ].[5] S. Bhattacharyya, R. Loganayagam, I. Mandal, S. Minwalla, and A. Sharma, Conformal Nonlinear Fluid Dynamics from Gravity in Arbitrary Dimensions , JHEP (2008) 116, [ arXiv:0809.4272 ].[6] S. S. Gubser, Drag force in AdS/CFT , Phys. Rev. D (2006) 126005,[ hep-th/0605182 ].[7] H. Liu, K. Rajagopal, and U. A. Wiedemann, Calculating the jet quenching parameterfrom AdS/CFT , Phys. Rev. Lett. (2006) 182301, [ hep-ph/0605178 ].[8] B. Mcinnes, Applied holography of the AdS –Kerr space–time , Int. J. Mod. Phys. A (2019), no. 24 1950138, [ arXiv:1803.02528 ]. – 46 –
9] B. McInnes,
Fragmentation of AdS -Kerr Black Holes, With Two Applications , arXiv:2005.03869 .[10] S. W. Hawking, C. J. Hunter, and M. Taylor, Rotation and the AdS / CFTcorrespondence , Phys. Rev.
D59 (1999) 064005, [ hep-th/9811056 ].[11] S. W. Hawking and H. S. Reall,
Charged and rotating AdS black holes and their CFTduals , Phys. Rev.
D61 (2000) 024014, [ hep-th/9908109 ].[12] I. Y. Aref’eva, A. A. Golubtsova, and E. Gourgoulhon,
Holographic drag force in 5dKerr-AdS black hole , arXiv:2004.12984 .[13] S. de Haro, K. Skenderis, and S. N. Solodukhin, Holographic reconstruction ofspacetime and renormalization in the ads/cft correspondence , Communications inMathematical Physics (Mar, 2001) 595–622.[14] M. Bochicchio,
Renormalization in large- N QCD is incompatible with open/closedstring duality , Phys. Lett. B (2018) 341–349, [ arXiv:1703.10176 ].[15] K. Skenderis and B. C. van Rees,
Real-time gauge/gravity duality: Prescription,Renormalization and Examples , JHEP (2009) 085, [ arXiv:0812.2909 ].[16] S. R. Green, S. Hollands, A. Ishibashi, and R. M. Wald, Superradiant instabilities ofasymptotically anti-de Sitter black holes , Class. Quant. Grav. (2016), no. 12 125022,[ arXiv:1512.02644 ].[17] J. Barrag´an Amado, B. Carneiro da Cunha, and E. Pallante, Scalar quasinormal modesof Kerr-AdS5 , Phys. Rev.
D99 (2019), no. 10 105006, [ arXiv:1812.08921 ].[18] E. Hijano, P. Kraus, E. Perlmutter, and R. Snively,
Witten diagrams revisited: The adsgeometry of conformal blocks , arXiv:1508.00501 .[19] F. Dolan and H. Osborn, Conformal partial waves and the operator product expansion , Nucl. Phys. B (2004) 491–507, [ hep-th/0309180 ].[20] J. B. Amado, B. Carneiro da Cunha, and E. Pallante,
On the Kerr-AdS/CFTcorrespondence , JHEP (2017) 094, [ arXiv:1702.01016 ].[21] B. Carneiro da Cunha and M. Guica, Exploring the BTZ bulk with boundary conformalblocks , arXiv:1604.07383 .[22] A. Litvinov, S. Lukyanov, N. Nekrasov, and A. Zamolodchikov, Classical ConformalBlocks and Painleve VI , arXiv:1309.4700 .[23] O. Gamayun, N. Iorgov, and O. Lisovyy, How instanton combinatorics solves Painlev´eVI, V and IIIs , J. Phys.
A46 (2013) 335203, [ arXiv:1302.1832 ]. – 47 –
24] O. Lisovyy, H. Nagoya, and J. Roussillon,
Irregular conformal blocks and connectionformulae for Painlev´e V functions , J. Math. Phys. (2018), no. 9 091409,[ arXiv:1806.08344 ].[25] B. Carneiro da Cunha and J. P. Cavalcante, Confluent conformal blocks and theTeukolsky master equation , arXiv:1906.10638 .[26] G. Aminov, A. Grassi, and Y. Hatsuda, Black Hole Quasinormal Modes andSeiberg-Witten Theory , arXiv:2006.06111 .[27] L. F. Alday, D. Gaiotto, and Y. Tachikawa, Liouville Correlation Functions fromFour-dimensional Gauge Theories , Lett. Math. Phys. (2010) 167–197,[ arXiv:0906.3219 ].[28] N. Nekrasov, Blowups in BPS/CFT correspondence, and Painlev´e VI , arXiv:2007.03646 .[29] A. L. Fitzpatrick, J. Kaplan, M. T. Walters, and J. Wang, Hawking from catalan , Journal of High Energy Physics (May, 2016).[30] S. W. Hawking,
Stability of AdS and phase transitions , Class. Quant. Grav. (2000)1093–1099.[31] S. Hollands, A. Ishibashi, and D. Marolf, Comparison between various notions ofconserved charges in asymptotically AdS-spacetimes , Class. Quant. Grav. (2005)2881–2920, [ hep-th/0503045 ].[32] P. Gavrylenko and O. Lisovyy, Fredholm Determinant and Nekrasov SumRepresentations of Isomonodromic Tau Functions , Commun. Math. Phys. (2018),no. 7 1–58, [ arXiv:1608.00958 ].[33] A. M. Awad and C. V. Johnson,
Higher dimensional Kerr - AdS black holes and theAdS / CFT correspondence , Phys. Rev.
D63 (2001) 124023, [ hep-th/0008211 ].[34] G. W. Gibbons, M. J. Perry, and C. N. Pope,
The First law of thermodynamics forKerr-anti-de Sitter black holes , Class. Quant. Grav. (2005) 1503–1526,[ hep-th/0408217 ].[35] R. Olea, Regularization of odd-dimensional AdS gravity: Kounterterms , JHEP (2007) 073, [ hep-th/0610230 ].[36] C. Fefferman and C. R. Graham, Conformal invariants , Ast´erisque (1985) 95–116.[37] V. Cardoso, ´O. J. C. Dias, G. S. Hartnett, L. Lehner, and J. E. Santos,
Holographic – 48 – hermalization, quasinormal modes and superradiance in Kerr-AdS , JHEP (2014)183, [ arXiv:1312.5323 ].[38] S. Bhattacharyya, S. Lahiri, R. Loganayagam, and S. Minwalla, Large rotating AdSblack holes from fluid mechanics , JHEP (2008) 054, [ arXiv:0708.1770 ].[39] R. M. Wald, General Relativity . The University of Chicago Press, 1984.[40] A. M. Polyakov and V. S. Rychkov,
Loop dynamics and AdS / CFT correspondence , Nucl. Phys. B (2001) 272–286, [ hep-th/0005173 ].[41] S. Ferrara, A. F. Grillo, G. Parisi, and R. Gatto,
Covariant expansion of the conformalfour-point function , Nucl. Phys.
B49 (1972) 77–98. [Erratum: Nucl.Phys.B53,643(1973)].[42] A. Its, O. Lisovyy, and A. Prokhorov,
Monodromy dependence and connection formulaefor isomonodromic tau functions , Duke Math. J. (2018), no. 7 1347–1432,[ arXiv:1604.03082 ].[43] S. Ferrara, A. F. Grillo, and R. Gatto,
Manifestly conformal covariant operator-productexpansion , Lett. Nuovo Cim. (1971) 1363–1369. [Lett. Nuovo Cim.2,1363(1971)].[44] S. Ferrara, A. F. Grillo, G. Parisi, and R. Gatto,
Canonical scaling and conformalinvariance , Phys. Lett.
B38 (1972) 333–334.[45] A. Polyakov,
Nonhamiltonian approach to conformal quantum field theory , Zh. Eksp.Teor. Fiz. (1974) 23–42.[46] F. Dolan and H. Osborn, Conformal four point functions and the operator productexpansion , Nucl. Phys. B (2001) 459–496, [ hep-th/0011040 ].[47] F. Dolan and H. Osborn,
Conformal Partial Waves: Further Mathematical Results , arXiv:1108.6194 .[48] O. Gamayun, N. Iorgov, and O. Lisovyy, Conformal field theory of Painlev´e VI , JHEP (2012) 038, [ arXiv:1207.0787 ]. [Erratum: JHEP10,183(2012)].[49] N. Iorgov, O. Lisovyy, and J. Teschner, Isomonodromic tau-functions from Liouvilleconformal blocks , Commun. Math. Phys. (2015), no. 2 671–694,[ arXiv:1401.6104 ].[50] J. L. Cardy,
Operator Content of Two-Dimensional Conformally Invariant Theories , Nucl. Phys. B (1986) 186–204.[51] D. Gaiotto,
Asymptotically free N = 2 theories and irregular conformal blocks , J. Phys.Conf. Ser. (2013), no. 1 012014, [ arXiv:0908.0307 ]. – 49 –
52] H. Nagoya,
Irregular conformal blocks, with an application to the fifth and fourthPainlev equations , J. Math. Phys. (2015), no. 12 123505, [ arXiv:1505.02398 ].[53] E. Berti and V. Cardoso, Quasinormal modes and thermodynamic phase transitions , Phys. Rev. D (2008) 087501, [ arXiv:0802.1889 ].[54] R. Brito, V. Cardoso, and P. Pani, Superradiance: Energy Extraction, Black-HoleBombs and Implications for Astrophysics and Particle Physics , vol. 906. Springer, 2015.[55] V. Cardoso and O. J. C. Dias,
Small Kerr-anti-de Sitter black holes are unstable , Phys.Rev.
D70 (2004) 084011, [ hep-th/0405006 ].[56] V. Cardoso, O. J. C. Dias, and S. Yoshida,
Classical instability of Kerr-AdS black holesand the issue of final state , Phys. Rev.
D74 (2006) 044008, [ hep-th/0607162 ].[57] R. Emparan and R. C. Myers,
Instability of ultra-spinning black holes , JHEP (2003)025, [ hep-th/0308056 ].[58] L. Lindblom, N. W. Taylor, and F. Zhang, Scalar, Vector and Tensor Harmonics onthe Three-Sphere , Gen. Rel. Grav. (2017), no. 11 139, [ arXiv:1709.08020 ].[59] J. Barrag´an-Amado, B. Carneiro da Cunha, and E. Pallante, Vector perturbations ofKerr-AdS and the Painlev´e VI transcendent , arXiv:2002.06108 .[60] E. Witten, A Note On Boundary Conditions In Euclidean Gravity , arXiv:1805.11559 ..