We investigate the retarded Greens function and the greybody factor in asymptotically AdS black holes. Using the connection coefficients of the Heun equation expressed in terms of the NekrasovShatashvili NS free energy of an SU2 supersymmetric gauge theory with four fundamental hypermultiplets we derive asymptotic expansions for both the retarded Greens function and the greybody factor in the small horizon limit. Furthermore we compute the corrections to these asymptotic expansions resulting from the resummation procedure of the instanton part of the NS function.
['Black Holes', 'AdS-CFT Correspondence', 'Supersymmetric Gauge Theory', 'High Energy Physics - Theory', 'General Relativity and Quantum Cosmology']
The effect of resummation on retarded Greens function and greybody factor in AdS black holes
{'Julián Barragán Amado a Shankhadeep Chakrabortty, b Arpit Maurya b': "- a Grupo de Física Matemática, Campo Grande Edifício C6, Lisboa 1749-016, Portugal\n- b Department of Physics, Indian Institute of Technology Ropar, Rupnagar, Punjab, India 140001.  \nE-mail:  \njjamado@fc.ul.pt , s.chakrabortty@iitrpr.ac.in ,  \narpit.20phz0009@iitrpr.ac.in  \nAbstract: We investigate the retarded Green's function and the greybody factor in asymptotically AdS black holes. Using the connection coefficients of the Heun equation, expressed in terms of the Nekrasov-Shatashvili (NS) free energy of an SU (2) supersymmetric gauge theory with four fundamental hypermultiplets, we derive asymptotic expansions for both the retarded Green's function and the greybody factor in the small horizon limit. Furthermore, we compute the corrections to these asymptotic expansions resulting from the resummation procedure of the instanton part of the NS function.", '1 Introduction': "A recent resurgence of black hole perturbation theory using the techniques originally developed for the Nekrasov Shatashvili phase of the Ω -background in 4D supersymmetric gauge theory [1-5] has opened up a significant possibility for achieving the exact analytical form of various physical observables including holographic thermal correlator, quasinormal modes, greybody factors and many more [6-16]. After the separation of variables, the equation of motion for the scalar mode in black hole perturbation theory results in linear second-order ordinary differential equations that can be represented in the form of Heun equations by appropriate redefinition of variables [17]. Heun's equation is known for its generality as a second-order linear differential equation with four regular singular points [18]. These singularities on the Riemann sphere characterize the equation as a generalization of the  \nhypergeometric equation [19]. The solution to Heun's equation, also known as the Heun function near a singular point, can be expanded in terms of the solutions around other singularities. The connection coefficients specify the explicit form of such expansion and are computed in terms of the Nekrasov-Shatashvili partition function [20].  \nThe connection coefficients and the underlying connection formula are best understood in the context of 2d Liouville CFT [21]. According to the 2D-4D duality of the Alday-Gaiotto-Tachikawa (AGT) correspondence, the BPS sector of four-dimensional N = 2 , SU (2) gauge theory in Ω is dual to the Liouville CFT on the Riemann sphere [4]. This duality implies that the four-point function of the Liouville CFT corresponds to the NekrasovShatashvili partition function in N = 2 , SU (2) gauge theory [22]. In particular, the duality offers a fascinating connection between the semiclassical limit (large central charge limit) on the parameters of Liouville CFT and the Nekrasov-Shatashvili limit on the parameters of the Ω background in the SU (2) gauge theory [5]. The BPZ equation governing the five-point correlator in Liouville theory provides an avenue for series solutions through the use of conformal blocks. The interplay of conformal blocks in different OPE channels (s, t, and u channels) establishes connections among the solutions to the BPZ equation. In the limit of a large central charge (semi-classical limit), the BPZ equation transforms into Heun's equation, a second-order linear differential equation with four regular singularities where these singular points signifies the insertion of primary operators in the theory. By the virtue of crossing symmetry, the relation among the solutions of the semi-classical BPZ equation leads to a connection formula among the solutions of Heun's equation [21].  \nVery recently, the authors of [23] established an exact analytical expression for the holographic retarded Green's function for AdS 5 Schwarzschild black hole, dual to the correlator in a thermal CFT living in the R 1 × S 3 boundary of the black hole spacetime. They have used a connection formula between the Heun functions associated with the incoming mode expansion of a scalar field near the horizon and those near the boundary of the black hole spacetime. This connection formula nicely captures the response function and source terms present in the boundary expansion of the scalar mode. Finally, by following the prescription of Lorentzian Green's function of black hole spacetime [24], they obtained the retarded Green's function by taking the ratio of source to response function. Further, exact retarded Green's function for a thermal CFT with chemical potential and angular momenta on R 1 × S 3 and also for a thermal CFT living in R 1 × H 3 are constructed in [25] and [26] respectively.  \nThe study of greybody factors in asymptotically flat spacetimes has been reviewed using the connection coefficients of the confluent Heun equation, resulting in asymptotic expansions proportional to the area of the black hole horizon in the low-energy limit [7, 9]. Nevertheless, in the case of asymptotically AdS spcetimes, the role of the connection coefficients becomes more elusive due to the nature of the boundary at spatial infinity. Harmark and collaborators [27] computed the greybody factors for static and spherically symmetric spacetime black holes in d -dimensions by splitting up the spacetime in three regions, such that the radial differential equation can be solved analytically, and then matched across the regions. Building on those considerations, [28] and [29] have investigated the greybody  \nfactors for rotating and non-asymptotically flat black holes.  \nIn this work, we study the retarded Green's function and the greybody factor for Kerr-AdS 5 black hole and Reissner-Nordström-AdS 5 black hole. In doing so, we analyze the uncharged and charged scalar mode perturbation in the respective black hole spacetimes. We show how the Klein-Gordon equation corresponding to the radial mode of scalar field takes the form of Heun's differential equation by successive applications of Möbius transformation followed by a s-homotopic transformation. We find the analytic form of two independent solutions of the Heun's equation at each of the two regular singular points corresponding to the black hole horizon and the boundary of the black hole spacetime. We explicitly show the emergence of the connection formula and the connection coefficients and describe the role they play in the subsequent study of the retarded Green's function and greybody factor. As expected, the Nekrasov partition function plays a crucial role in determining the connection coefficients. Most interestingly, we show how the singularity appearing from the pole structure of the Nekrasov partition function is reincarnated in to the singularity of the pole structure of a , the vacuum expectation value of the scalar in the vector hypermultiplets in the supersymmetric gauge theory, obtained by solving the Matone relation. Within the small black hole approximation, we use a resummation technique to cure the singularity structure in the a order by order in radius of black hole horizon. This exercise surprisingly accommodates a series correction terms in a that can be nicely presented in terms of the generating function of the Catalan numbers. Moreover, the presence of such correction terms in a closed form sigificantly simplify the final analytical results of Green's function and greybody factor.  \nThe plan of the paper is the following: In section 2 we discuss the explicit construction of the radial mode of the scalar field in Kerr-AdS 5 black hole spacetime. In section 3 we describe the radial mode of a charged scalar field in Reissner-Nordström-AdS 5 black hole spacetime. Section 4 is dedicated to Heun's equation, the solutions to Heun's equation near regular singular point and the connection formula. In section 5 and in section 6 we discuss the computtaions of retarded Green's function and Greybody factor for Kerr-AdS 5 and Reissner-Nordström-AdS 5 black holes respetively. Finally, we conclude in section 7.", '2 Scalar perturbations in Kerr-AdS 5': "In order to set the backdrop of a specific example of scalar perturbation theory in AdS black hole, let us review the five dimensional KerrAdS 5 black hole as presented in [30]. The explicit form of the corresponding metric takes the form,  \nds 2 = -∆ r ρ 2 ( dt -ˆ a 1 sin 2 θ Ξ 1 dϕ 1 -ˆ a 2 cos 2 θ Ξ 2 dϕ 2 ) 2 + ∆ θ sin 2 θ ρ 2 ( ˆ a 1 dt -r 2 +ˆ a 2 1 Ξ 1 dϕ 1 ) 2 + 1 + r 2 /L 2 r 2 ρ 2 ( ˆ a 1 ˆ a 2 dt -ˆ a 2 ( r 2 +ˆ a 2 1 ) sin 2 θ Ξ 1 dϕ 1 -ˆ a 1 ( r 2 +ˆ a 2 2 ) cos 2 θ Ξ 2 dϕ 2 ) 2 + ∆ θ cos 2 θ ρ 2 ( ˆ a 2 dt -r 2 +ˆ a 2 2 Ξ 2 dϕ 2 ) 2 + ρ 2 ∆ r dr 2 + ρ 2 ∆ θ dθ 2 , (2.1)  \nwhere  \n∆ r = 1 r 2 ( r 2 +ˆ a 2 1 ) ( r 2 +ˆ a 2 2 ) ( 1 + r 2 L 2 ) -2 M = 1 L 2 r 2 ( r 2 -r 2 0 ) ( r 2 -r 2 -) ( r 2 -r 2 + ) , ∆ θ = 1 -ˆ a 2 1 L 2 cos 2 θ -ˆ a 2 2 L 2 sin 2 θ, ρ 2 = r 2 +ˆ a 2 1 cos 2 θ +ˆ a 2 2 sin 2 θ, Ξ 1 = 1 -ˆ a 2 1 L 2 , Ξ 2 = 1 -ˆ a 2 2 L 2 (2.2)  \nHere L stands for AdS radius and M , ˆ a 1 and ˆ a 2 are the real parameters, related to the ADM mass ( M ) and angular momenta ( J 1 , J 2 ) respectively [31, 32]  \nM = πM (2Ξ 1 +2Ξ 2 -Ξ 1 Ξ 2 ) 4Ξ 2 1 Ξ 2 2 , J 1 = πM ˆ a 1 2Ξ 2 1 Ξ 2 , J 2 = πM ˆ a 2 2Ξ 1 Ξ 2 2 . (2.3)  \nWithin a physically sensible range of parameters described as M > 0 , and ˆ a 2 1 , ˆ a 2 2 < 1 , ∆ r = 0 allows two real roots r + and r -signifying the inner and outer horizons of the black hole respectively and one purely imaginary root r 0 that satisfies the following relation [31],  \n-r 2 0 = L 2 +ˆ a 2 1 +ˆ a 2 2 + r 2 -+ r 2 + . (2.4)  \nFor the purposes of this article, we will see the radial variable, or rather r 2 , as a generic complex number. It will be interesting for us to treat all three roots of ∆ r ,e.g., r 2 + , r 2 -and r 2 0 as Killing horizons. Actually, in the complexified version of the metric (2.1), in all three hypersurfaces defined by r = r 0 , r -and r + we have that each of the Killing fields  \nξ k = ∂ ∂t +Ω 1 ( r k ) ∂ ∂ϕ +Ω 2 ( r k ) ∂ ∂ψ , k = 0 , -, + , (2.5)  \nbecomes null [33]. The temperatures and angular velocities at each horizon are given by  \n̸  \n̸  \nT k = r 2 k ∆ ' r ( r k ) 4 π ( r 2 k +ˆ a 2 1 )( r 2 k +ˆ a 2 2 ) = r k 2 πL 2 ( r 2 k -r 2 i )( r 2 k -r 2 j ) ( r 2 k +ˆ a 2 1 )( r 2 k +ˆ a 2 2 ) , i = j = k, Ω k, 1 = ˆ a 1 Ξ 1 r 2 k +ˆ a 2 1 , Ω k, 2 = ˆ a 2 Ξ 2 r 2 k +ˆ a 2 2 . (2.6)  \nWithin the physically sensible range of parameters, T + is positive, T -is negative and T 0 is purely imaginary.", '2.1 The Klein-Gordon equation': "The Klein-Gordon (KG) equation for a scalar of mass µ , in the background (2.1), is determined by  \n1 √ -g ∂ µ ( √ -gg µν ∂ ν ) Φ -µ 2 Φ = 0 . (2.7)  \nWe consider the following ansatz  \nΦ( t, r, θ, ϕ 1 , ϕ 2 ) = e -iωt + im 1 ϕ 1 + im 2 ϕ 2 R ( r ) S ( θ ) , (2.8)  \nwhere ω is the frequency of the mode, and m 1 , m 2 ∈ Z are the azimuthal quantum numbers. With the suitable choice of ansatz (2.8),KG equation (2.7) decouples into two second-order ordinary differential equations of the form  \n[ 1 r d dr ( r ∆ r d dr ) + ( r 2 +ˆ a 2 1 ) 2 ( r 2 +ˆ a 2 2 ) 2 r 4 ∆ r ( ω -m 1 ˆ a 1 Ξ 1 r 2 +ˆ a 2 1 -m 2 ˆ a 2 Ξ 2 r 2 +ˆ a 2 2 ) 2 -1 r 2 (ˆ a 1 ˆ a 2 ω -ˆ a 2 Ξ 1 m 1 -ˆ a 1 Ξ 2 m 2 ) 2 -λ ℓ -µ 2 r 2 ] R ( r ) = 0 , (2.9a)  \n[ 1 sin θ cos θ d dθ ( sin θ cos θ ∆ θ d dθ ) -ω 2 L 2 -m 2 1 Ξ 1 sin 2 θ -m 2 2 Ξ 2 cos 2 θ + Ξ 1 Ξ 2 ∆ θ ( ωL + m 1 ˆ a 1 L + m 2 ˆ a 2 L ) 2 -µ 2 ( ˆ a 2 1 cos 2 θ +ˆ a 2 2 sin 2 θ ) + λ ℓ ] S ( θ ) = 0 , (2.9b)  \nwhere λ ℓ is the separation constant. We shall revisit the separation constant and describe its explicit form in section 5. For computational convenience, we introduce a dimensionless radial coordinate  \n˜ r = r L , ˜ r k = r k L , k = { 0 , -, + } (2.10)  \nand as a result ∆ ˜ r takes the form as ∆ ˜ r = ∆ r L 2 . If we scale the remaining parameters as  \n˜ ω = Lω, ˜ a i = ˆ a i L , ˜ µ = Lµ, (2.11)  \nwe obtain the dimensionless ODEs, which do not depend on the AdS radius #1 . In terms of the tilde notation, the radial equation (2.9a) takes the following form:  \n{ d 2 d ˜ r 2 + ( 1 ˜ r + ∆ ' ˜ r ∆ ˜ r ) d d ˜ r + (˜ r 2 +˜ a 2 1 ) 2 (˜ r 2 +˜ a 2 2 ) 2 ˜ r 4 ∆ 2 ˜ r ( ˜ ω -m 1 ˜ a 1 Ξ 1 ˜ r 2 +˜ a 2 1 -m 2 ˜ a 2 Ξ 2 ˜ r 2 +˜ a 2 2 ) 2 -1 ˜ r 2 ∆ ˜ r (˜ a 1 ˜ a 2 ˜ ω -˜ a 2 Ξ 1 m 1 -˜ a 1 Ξ 2 m 2 ) 2 -λ ℓ ∆ ˜ r -˜ µ 2 ˜ r 2 ∆ ˜ r } R (˜ r ) = 0 , (2.13a)  \nwhere  \n∆ ˜ r = ( ˜ r 2 -˜ r 2 0 ) ( ˜ r 2 -˜ r 2 -) ( ˜ r 2 -˜ r 2 + ) ˜ r 2 (2.13b)  \nEquation (2.13) possesses four regular singular points in ˜ r 2 variable, located at the roots of ∆ ˜ r and the point at infinity. The characteristic exponents of the Frobenius solutions near to each finite singularity are given by  \nβ ± k = ± 1 2 θ k , k = { + , -, 0 } (2.14)  \nT k = ˜ T k L , Ω k, 1 = ˜ Ω k, 1 L , Ω k, 2 = ˜ Ω k, 2 L . (2.12)  \nwhere  \nθ k = i 2 π ( ˜ ω -m 1 ˜ Ω k, 1 -m 2 ˜ Ω k, 2 ˜ T k ) , (2.15)  \nand for ˜ r = ∞ , we have  \nβ ± ∞ = 1 2 (2 ± θ ∞ ) , θ ∞ = √ 4 + ˜ µ 2 := ∆ -2 . (2.16)  \nIt turns out that θ + is the variation of the entropy δS of the black hole as it absorbs a quantum of frequency and angular momenta at the outer horizon. Furthermore, θ ∞ can be expressed in terms of the conformal dimension ∆ of the scalar operator dual to a scalar field in the bulk with mass ˜ µ , such that ∆(∆ -4) = ˜ µ 2 [34]. To bring the radial equation (2.13) to the canonical Heun form, we perform a change of variables followed by a s-homotopic transformation #2 ,  \nz = ˜ r 2 -˜ r 2 -˜ r 2 -˜ r 2 0 , R ( z ) = z β --( z 0 -z ) β -+ (1 -z ) β + ∞ f ( z ) , (2.17)  \nwhere  \nz 0 = ˜ r 2 + -˜ r 2 -˜ r 2 + -˜ r 2 0 . (2.18)  \nThe equation for f ( z ) is  \nd 2 f dz 2 + [ 1 -θ -z + 1 -θ + z -z 0 + ∆ -1 z -1 ] df dz + ( κ 1 κ 2 z ( z -1) -z 0 ( z 0 -1) K 0 z ( z -z 0 )( z -1) ) f ( z ) = 0 , (2.19)  \nwhere  \nκ 1 = 1 2 ( θ -+ θ + -∆ -θ 0 ) , κ 2 = 1 2 ( θ -+ θ + -∆+ θ 0 ) , (2.20a)  \n4 z 0 ( z 0 -1) K 0 = -( λ ℓ +∆(∆ -4)˜ r 2 --˜ ω 2 ) ˜ r 2 + -˜ r 2 0 -( z 0 -1) [ ( θ -+ θ + -1) 2 -θ 2 0 -1 ] -z 0 [ 2( θ + -1)(1 -∆) + (∆ -2) 2 -2 ] , (2.20b)  \nwhere K 0 is called the accessory parameter. We show the explicit form of angular differential equation in appendix B.", '3 Scalar perturbations in Reissner-Nordström-AdS 5': 'The line element of the Reissner-Nordström-AdS 5 (RN-AdS 5 ) black hole is  \nds 2 = -f ( r ) dt 2 + dr 2 f ( r ) + r 2 d Ω 2 3 (3.1)  \nwhere d Ω 2 3 is the metric of the unit three-sphere and the blackening function f ( r ) reads  \nf ( r ) = 1 -M r 2 + Q 2 r 4 + r 2 L 2 (3.2)  \nwhere M , Q and L are related to the black hole ADM mass, charge, and the radius of AdS respectively. The function f ( r ) is a polynomial of degree six with only even powers of r , such that the roots of f ( r ) can be expressed as  \nf ( r ) = ( r 2 -r 2 0 ) ( r 2 -r 2 1 ) ( r 2 -r 2 2 ) L 2 r 4 . (3.3)  \nSince we are interested in space-time configurations that possess black hole horizons, the outer horizon corresponds to the largest positive real root r 2 = r + , while the inner horizon is defined as r 1 = r -satisfying r -≤ r + , and r 0 is purely imaginary for generic values of the mass and charge. Furthermore, one can write mass parameter in terms of the the radius of the outer horizon as  \nM = r 2 + + Q 2 r 2 + + r 4 + L 2 , (3.4)  \nwhich implies  \nr 2 -= L 2 2   -1 -r 2 + L 2 + √ ( 1 + r 2 + L 2 ) 2 + 4 Q 2 L 2 r 2 +   r 2 0 = L 2 2   -1 -r 2 + L 2 -√ ( 1 + r 2 + L 2 ) 2 + 4 Q 2 L 2 r 2 +   (3.5)  \nThe temperature at each horizon is  \n̸  \n̸  \nT k = 1 4 π df ( r ) dr ∣ ∣ ∣ ∣ r = r k = 1 2 πL 2 ( r 2 k -r 2 i ) ( r 2 k -r 2 j ) r 3 k , i = j = k (3.6)  \nThe electromagnetic potential of the charged black hole can be written as  \nA µ dx µ = ( -√ 3 2 Q r 2 + C ) dt, (3.7)  \nand for a vanishing potential at spatial infinity, we set C = 0 .', '3.1 The Klein-Gordon equation': "In the charged black hole background (3.1), the Klein-Gordon equation for a massive charged scalar field perturbation reads as  \n1 √ -g D µ ( √ -gg µν D ν ) Φ -µ 2 Φ = 0 , (3.8)  \nwith D µ = ∂ µ -ieA µ , and e and µ being the charge and the mass of the field, respectively. Equation (3.8) can be decomposed into two second-order ODEs by implementing the following ansatz,  \nΦ( t, r, θ, ϕ, ψ ) = e -iωt Y m 1 ,m 2 ℓ ( θ, ϕ, ψ ) R ( r ) (3.9)  \nThe angular part reduces to the spherical harmonic function on the three-sphere Y m 1 ,m 2 ℓ ( θ, ϕ, ψ ) , which satisfies the eigenvalue equation  \n∆ Y m 1 ,m 2 ℓ ( θ, ϕ, ψ ) = -ℓ ( ℓ +2) Y m 1 ,m 2 ℓ ( θ, ϕ, ψ ) , (3.10)  \nwhere ℓ is the angular momentum quantum number, and m 1 and m 2 are integers associated with the magnetic quantum numbers. By means of (3.10), the differential equation for the radial function R ( r ) takes the following form  \n  1 r d dr ( r 3 f ( r ) d dr ) + r 2 f ( r ) ( ω -√ 3 2 eQ r 2 ) 2 -µ 2 r 2 -ℓ ( ℓ +2)   R ( r ) = 0 (3.11)  \nFor numerical convenience, we define  \n˜ r = r L , ˜ r k = r k L , f (˜ r ) = f ( r ) , k = { 0 , -, + } ˜ Q = Q L 2 , ˜ ω = Lω, ˜ µ = Lµ, ˜ e = √ 3 2 Le, (3.12)  \nso that equation (3.11) will take a dimensionless form:  \n   d 2 d ˜ r 2 + ( 3 ˜ r + f ' (˜ r ) f (˜ r ) ) d d ˜ r + 1 f (˜ r )   1 f (˜ r ) ( ˜ ω -˜ e ˜ Q ˜ r 2 ) 2 -ℓ ( ℓ +2) ˜ r 2 -˜ µ 2      R (˜ r ) = 0 , (3.13a)  \nwhere  \nf (˜ r ) = ( ˜ r 2 -˜ r 2 0 ) ( ˜ r 2 -˜ r 2 -) ( ˜ r 2 -˜ r 2 + ) ˜ r 4 . (3.13b)  \nEquation (3.13) possesses four regular singular points in ˜ r 2 variable, located at the roots of f (˜ r ) and the point at infinity. The characteristic exponents of the Frobenius solutions near to each finite singularity are given by  \nβ ± k = ± 1 2 θ k , k = { + , -, 0 } (3.14)  \nwhere  \nθ k = i 2 π ˜ T k ( ˜ ω -˜ e ˜ Q ˜ r 2 k ) , (3.15)  \nand for ˜ r = ∞ , we have  \nβ ± ∞ = 1 2 (2 ± θ ∞ ) , θ ∞ = √ 4 + ˜ µ 2 := ∆ -2 . (3.16)  \nSince our aim is to write the equation (3.13) in the canonical Heun form, we introduce a Möbius transformation followed by a s-homotopic transformation  \nz = ˜ r 2 -˜ r 2 -˜ r 2 -˜ r 2 0 , R ( z ) = z β --( z 0 -z ) β -+ (1 -z ) β + ∞ f ( z ) , (3.17)  \nwhere  \nz 0 = ˜ r 2 + -˜ r 2 -˜ r 2 + -˜ r 2 0 , (3.18)  \nthat leads to an equation for f ( z )  \nd 2 f dz 2 + [ 1 -θ -z + 1 -θ + z -z 0 + ∆ -1 z -1 ] df dz + ( κ 1 κ 2 z ( z -1) -z 0 ( z 0 -1) K 0 z ( z -z 0 )( z -1) ) f ( z ) = 0 , (3.19)  \nwith  \nκ 1 = 1 2 ( θ -+ θ + -∆ -θ 0 ) , κ 2 = 1 2 ( θ -+ θ + -∆+ θ 0 ) , (3.20a)  \n4 z 0 ( z 0 -1) K 0 = -ℓ ( ℓ +2) + ∆(∆ -4)˜ r 2 --˜ ω 2 ˜ r 2 + -˜ r 2 0 -( z 0 -1) [ ( θ -+ θ + -1) 2 -θ 2 0 -1 ] -z 0 [ 2( θ + -1)(1 -∆) + (∆ -2) 2 -2 ] (3.20b)  \nIt is important to note that while the radial equation (3.19) shares a similar formal structure with the radial equation (2.19), they are not identical. The characteristic exponents in each case depend on the details of the background. The accessory parameter K 0 includes the separation constant introduced to decouple the angular and radial components, which, in the case of RN-AdS 5 reduces to the eigenvalues associated to the scalar spherical harmonics on the three-sphere (3.10). Conversely, the angular eigenvalue problem in Kerr-AdS 5 is more complex and it is tipically solved perturbatively in specific limits, e.g., see [35, 36] for asymptotic expansions of λ in the near-equally rotating limit.", '4 Heun functions and connection coefficients': "In this section we show the explicit construction of the normal form of Heun's equation. We elaborate upon the solution of the Heun's equation near regular singular points and derive the connection formula which we frequently use in the computations of retarded Green's function and greybody factor in the subsequent sections.  \nThe canonical Heun's differential equation reads  \n( d 2 dz 2 + ( γ z + δ z -1 + ϵ z -t ) d dz + αβz -q z ( z -t )( z -1) ) f ( z ) = 0 , (4.1a)  \nwhere the coefficients satisfy the condition  \nα + β +1 = γ + δ + ϵ, (4.1b)  \nand the complex modulus of t is small, | t | ≪ 1 . Via the following transformation  \nf ( z ) = z -γ ( t -z ) -ϵ (1 -z ) -δ ψ ( z ) , (4.2)  \none can bring (4.1a) into its normal form  \n( ∂ 2 z + 1 4 -a 2 0 z 2 + 1 4 -a 2 t ( z -t ) 2 + 1 4 -a 2 1 ( z -1) 2 -1 2 -a 2 0 -a 2 t -a 2 1 + a 2 ∞ + u z ( z -1) + u z ( z -t ) ) ψ ( z ) = 0 , (4.3)  \nwith the identification  \na 0 = 1 -γ 2 , a t = 1 -ϵ 2 , a 1 = 1 -δ 2 , a ∞ = α -β 2 , u = γϵ -2 q +2 tαβ -t ( γ + δ ) ϵ 2( t -1) , (4.4)  \nwhich can be written in terms of a i 's and u as  \nα = 1 -a 0 -a 1 -a t + a ∞ , β = 1 -a 0 -a 1 -a t -a ∞ , γ = 1 -2 a 0 , δ = 1 -2 a 1 , ϵ = 1 -2 a t , q = 1 2 + ( a 2 0 + a 2 t + a 2 1 -a 2 ∞ ) t -a t -a 1 t + a 0 (2 a t -1 + t (2 a 1 -1)) + (1 -t ) u. (4.5)  \nBy comparing with the radial ODEs (2.19) and (3.19), one can recognize that  \nt = z 0 (4.6a)  \na 0 = ± θ -2 , a t = ± θ + 2 , a 1 = ± θ ∞ 2 , a ∞ = ± θ 0 2 , (4.6b)  \nu = -z 0 K 0 + 1 2 θ -(1 -θ + ) -1 2 ∆(1 -θ + ) -(1 -θ + ) (∆ -1) 2 ( z 0 -1) . (4.6c)  \nNote that the explicit form of the parameters, θ k , θ ∞ , z 0 and K 0 depends on the individual details of the corresponding black hole solutions such as the Kerr-AdS 5 and the ReissnerNordström-AdS 5 black holes.  \nIn [9, 20, 21], the authors provided explicit expressions for the local expansions of the Heun functions and their connection coefficients in terms of the Nekrasov partition functions. These concepts have been applied to study quasi-normal modes, tidal Love numbers and greybody factors in different space-times [10-14], post-Newtonian (PN) dynamics in the two-body problem [15, 37], as well as the computation of retarded Green's function in asymptotically AdS space-times [23, 25, 26].  \nFollowing [13, 20], two linearly independent solutions around z = t are given by  \nf ( t ) -( z ) =HeunG ( t t -1 , q -tαβ 1 -t , α, β, ϵ, δ, z -t 1 -t ) f ( t ) + ( z ) =( t -z ) 1 -ϵ HeunG ( t t -1 , q -αβt 1 -t -( ϵ -1) ( γ + δt t -1 ) , α -ϵ +1 , β -ϵ +1 , 2 -ϵ, δ, z -t 1 -t ) , (4.7a)  \nwhile the ones around z = 1 can be written as  \nf (1) -( z ) = ( z -t 1 -t ) -α HeunG ( t, q + α ( δ -β ) , α, δ + γ -β, δ, γ, t 1 -z t -z ) f (1) + ( z ) =(1 -z ) 1 -δ ( z -t 1 -t ) -α -1+ δ HeunG ( t, q -( δ -1) γt -( β -1)( α -δ +1) , -β + γ +1 , α -δ +1 , 2 -δ, γ, t 1 -z t -z ) . (4.7b)  \nIn addition, the connection formula between solutions near to z = t and near to z = 1 is given by  \nt -1 2 + a 0 ∓ a t (1 -t ) -1 2 + a 1 e ∓ 1 2 ∂ a t F ( t ) f ( t ) ( z ) =  \n∑ σ = ± M ± σ ( a t , a ; a 0 ) M ( -σ )+ ( a, a 1 ; a ∞ ) t σa e -σ 2 ∂ a F ( t ) ) (1 -t ) -1 2 + a t e iπ ( -a 1 + a t ) e -1 2 ∂ a 1 F ( t ) f (1) + ( z )  \n± ( ∑ σ = ± M ± σ ( a t , a ; a 0 ) M ( -σ ) -( a, a 1 ; a ∞ ) t σa e -σ 2 ∂ a F ( t ) ) (1 -t ) -1 2 + a t e iπ ( a 1 + a t ) e 1 2 ∂ a 1 F ( t ) f (1) -( z )+ ( ,  \n(4.8)  \nwhere F ( t ) is the instanton part of the NS free energy and the connection coefficients M 's are defined as  \nM θθ ' ( a 1 , a 2 ; a 3 ) = Γ( -2 θ ' a 2 ) Γ (1 + 2 θa 1 ) Γ ( 1 2 + θa 1 -θ ' a 2 + a 3 ) Γ ( 1 2 + θa 1 -θ ' a 2 -a 3 ) . (4.9)  \nBy combining (4.3) with (4.7) we construct the solution to the radial equations (2.19) and (3.19) at the regular singular point z = z 0 (or equivalently z = t ) as  \nf ( z ) = C z 0 -f ( z 0 ) -( z ) + C z 0 + f ( z 0 ) + ( z ) . (4.10)  \nConsequently, the asymptotic behavior of the radial solution R ( z ) at z = z 0 leads to  \nR ( z ) ≃ C z 0 -( z 0 -z ) -θ + / 2 + C z 0 + ( z 0 -z ) θ + / 2 (4.11)  \nfor which an incoming wave at the outer horizon z = z 0 (˜ r = ˜ r + ) requires that C z 0 + = 0 . Then, the remaining radial solution according to the boundary condition will be given by  \nR ( z ) = C z 0 -z -θ -/ 2 ( z 0 -z ) -θ + / 2 (1 -z ) ∆ / 2 f ( z 0 ) -( z ) . (4.12)  \nNow we can relate the solution around z = z 0 to the local solutions around z = 1 through the appropriate choice of the connection coefficients (4.8). Namely, we have  \nf ( z 0 ) -( z ) = ( M --( a t , a ; a 0 ) M + -( a, a 1 ; a ∞ ) z -a 0 e 1 2 ∂ a F ( z 0 ) + M -+ ( a t , a ; a 0 ) M --( a, a 1 ; a ∞ ) z a 0 e -1 2 ∂ a F ( z 0 ) ) z 1 2 -a 0 -a t 0 (1 -z 0 ) a t -a 1 × e iπ ( a 1 + a t ) e 1 2 ( ∂ a 1 F ( z 0 ) -∂ a t F ( z 0 ) ) f (1) -( z ) + ( M --( a t , a ; a 0 ) M ++ ( a, a 1 ; a ∞ ) z -a 0 e 1 2 ∂ a F ( z 0 ) + M -+ ( a t , a ; a 0 ) M -+ ( a, a 1 ; a ∞ ) z a 0 e -1 2 ∂ a F ( z 0 ) ) z 1 2 -a 0 -a t 0 (1 -z 0 ) a t -a 1 × e iπ ( a t -a 1 ) e -1 2 ( ∂ a 1 F ( z 0 )+ ∂ a t F ( z 0 ) ) f (1) + ( z ) . (4.13)  \nBy introducing (4.13) into (4.12), we obtain the radial solution at the horizon z = z 0 in terms of two local solutions at z = 1 as follows  \nR ( z ) = C 1 -C z 0 -z -θ -/ 2 ( z 0 -z ) -θ + / 2 (1 -z ) ∆ / 2 f (1) -( z ) + C 1+ C z 0 -z -θ -/ 2 ( z 0 -z ) -θ + / 2 (1 -z ) ∆ / 2 f (1) + ( z ) , (4.14)  \nwhere f (1) ± ( z ) are given in (4.7b) and C 1 , ± read  \nC 1 -= ( M --( a t , a ; a 0 ) M + -( a, a 1 ; a ∞ ) z -a 0 e 1 2 ∂ a F ( z 0 ) + M -+ ( a t , a ; a 0 ) M --( a, a 1 ; a ∞ ) z a 0 e -1 2 ∂ a F ( z 0 ) ) z 1 2 -a 0 -a t 0 (1 -z 0 ) a t -a 1 × e iπ ( a 1 + a t ) e 1 2 ( ∂ a 1 F ( z 0 ) -∂ a t F ( z 0 ) ) , (4.15a)  \nC 1+ = ( M --( a t , a ; a 0 ) M ++ ( a, a 1 ; a ∞ ) z -a 0 e 1 2 ∂ a F ( z 0 ) + M -+ ( a t , a ; a 0 ) M -+ ( a, a 1 ; a ∞ ) z a 0 e -1 2 ∂ a F ( z 0 ) ) z 1 2 -a 0 -a t 0 (1 -z 0 ) a t -a 1 × e iπ ( a t -a 1 ) e -1 2 ( ∂ a 1 F ( z 0 )+ ∂ a t F ( z 0 ) ) . (4.15b)  \nThe asymptotic behavior of the radial solution (4.14) corresponding to spatial infinity ˜ r → ∞ ( z → 1)  \nR ( r ) ≃ C 1 -C z 0 -( z 0 -1) -1 2 θ + ( ˜ r 2 --˜ r 2 0 ) 1 2 ∆ ˜ r -∆ + C 1+ C z 0 -( z 0 -1) -1 2 θ + ( ˜ r 2 --˜ r 2 0 ) 1 2 (4 -∆) ˜ r ∆ -4 (4.16)  \nwhere for ∆ ≥ 4 , the first term converges while the second one diverges at ˜ r → ∞ , and thus the asymptotic solutions correspond to normalizable and non-normalizable solutions, respectively. Furthermore, the retarded Green's function defined as the ratio of the response to the source yields  \nG ret (˜ ω, λ ) = ( ˜ r 2 --˜ r 2 0 ) ∆ -2 C 1 -C 1+ = e ∂ a 1 F ( z 0 )+ iπ 2 a 1 ( ˜ r 2 --˜ r 2 0 ) ∆ -2 Γ(2 a 1 ) Γ( -2 a 1 ) Σ 1 -Σ 1+ , (4.17a)  \nwhere  \n1  \nΣ 1 -= e -2 ∂ a F ( z 0 ) Γ(1 -2 a )Γ( -2 a ) Γ ( 1 2 -a -a 0 -a t ) Γ ( 1 2 -a + a 0 -a t ) Γ ( 1 2 -a + a 1 -a ∞ ) Γ ( 1 2 -a + a 1 + a ∞ ) × ( r 2 + -r 2 -r 2 + -r 2 0 ) a + e 1 2 ∂ a F ( z 0 ) Γ(1 + 2 a )Γ(2 a ) Γ ( 1 2 + a -a 0 -a t ) Γ ( 1 2 + a + a 0 -a t ) Γ ( 1 2 + a + a 1 -a ∞ ) Γ ( 1 2 + a + a 1 + a ∞ ) × ( r 2 + -r 2 -r 2 + -r 2 0 ) -a (4.17b)  \n-1  \nΣ 1+ = e 2 ∂ a F ( z 0 ) Γ(1 -2 a )Γ( -2 a ) Γ ( 1 2 -a -a 0 -a t ) Γ ( 1 2 -a + a 0 -a t ) Γ ( 1 2 -a -a 1 -a ∞ ) Γ ( 1 2 -a -a 1 + a ∞ ) × ( r 2 + -r 2 -r 2 + -r 2 0 ) a + e 1 2 ∂ a F ( z 0 ) Γ(1 + 2 a )Γ(2 a ) Γ ( 1 2 + a -a 0 -a t ) Γ ( 1 2 + a + a 0 -a t ) Γ ( 1 2 + a -a 1 -a ∞ ) Γ ( 1 2 + a -a 1 + a ∞ ) × ( r 2 + -r 2 -r 2 + -r 2 0 ) -a . (4.17c)  \nwhere the radial dictionary for a i , i = { 0 , t, 1 , ∞} is defined in (4.6b), and we will choose the one with the negative sign and we have verified that the final result is independent of this choice of sign. Expression (4.17a) is related to the propagator derived in [23, 25, 26] under the change a 1 →-a 1 , as a result of taking into account the asymptotic analysis of the radial function R ( z ) , instead of ψ ( z ) .", '5 Small Kerr-AdS 5 black holes': "In order to study the retarded Green's function and the greybody factor in asymptotically AdS 5 black holes, we will focus on the small radius limit, while considering equal rotation parameters ˜ a 1 = ˜ a 2 = ˜ a . For this special case the angular equation (2.9b) reduces to a hypergeometric differential equation with the angular eigenvalue given by  \nλ ℓ = ( 1 -˜ a 2 ) [ ℓ ( ℓ +2) -2˜ a ˜ ω ( m 1 + m 2 ) -˜ a 2 ( m 1 + m 2 ) 2 ] +˜ a 2 ( ˜ ω 2 +∆(∆ -4) ) . (5.1)  \nas it was investigated in [38]. Furthermore, we can define a critical rotation parameter as the maximal rotation parameter at extremality ( T + = 0)  \n˜ a c = ˜ r + √ 1 + 2˜ r 2 + , (5.2)  \nand we require ˜ a ≤ ˜ a c , in order to guarantee a regular outer horizon. We then parameterize the accessible values for the rotation parameter as ˜ a = α ˜ a c , where α ≤ 1 is a dimensionless extremality parameter for fixed ˜ r + > 0 . In the small ˜ r + limit, the indicial coefficients of the radial equation (2.15) and (2.16), as well as the conformal modulus (2.18) can be written as  \nθ -= -i ( m 1 + m 2 ) α 1 -α 2 + iα 2 ( 1 + α 2 ) ˜ ω ˜ r + 1 -α 2 -i ( m 1 + m 2 ) α ( 1 + α 2 ) ( 1 -3 α 2 ) ˜ r 2 + 2 (1 -α 2 ) + iα 2 ( 3 + 5 α 2 -7 α 4 -5 α 6 ) ˜ ω ˜ r 3 + 2 (1 -α 2 ) + i ( m 1 + m 2 ) α ( 5 + 36 α 2 +42 α 4 -52 α 6 -35 α 8 ) ˜ r 4 + 8 (1 -α 2 ) + O ( ˜ r 5 + ) ,  \n(5.3a)  \nθ + = -i ( m 1 + m 2 ) α 1 -α 2 + i ( 1 + α 2 ) ˜ ω ˜ r + 1 -α 2 + i ( m 1 + m 2 ) α ( 1 + α 2 ) ˜ r 2 + 1 -α 2 -2 i ˜ ω ˜ r 3 + 1 -α 2 -i ( m 1 + m 2 ) ( 3 -2 α 2 ) ˜ r 4 + 2 (1 -α 2 ) + O ( ˜ r 5 + ) , (5.3b)  \nθ 0 = ˜ ω +( m 1 + m 2 ) α ˜ r + -3 2 ( 1 + α 2 ) 2 ˜ ω ˜ r 2 + -1 2 ( m 1 + m 2 ) α ( 3 + 10 α 2 +5 α 4 ) ˜ r 3 + + 1 8 ( 23 + 28 α 2 +70 α 4 +100 α 6 +35 α 8 ) ˜ ω ˜ r 4 + + O ( ˜ r 5 + ) , (5.3c)  \nθ ∞ = ∆ -2 , (5.3d)  \nz 0 = ( 1 -α 2 ) ˜ r 2 + -2 ( 1 -α 2 ) ( 1 + 2 α 2 +3 α 4 + α 6 ) ˜ r 4 + + O ( ˜ r 6 + ) , (5.3e)  \nand the radial dictionary is defined as follows  \na 0 = ± θ -2 , a t = ± θ + 2 , a 1 = ± θ ∞ 2 , a ∞ = ± θ 0 2 , t = z 0 . (5.4)  \nIn addition, the Matone relation associates the accessory parameter of the Heun equation with the vacuum expectation value a of the gauge theory [39]  \nu = -1 4 -a 2 + a 2 0 + a 2 t + t∂ t F ( t ) , (5.5)  \nwhere F ( t ) is the instanton part of the NS free energy defined in (A16). Then, assuming an expansion for small ˜ r + of the form  \na = ∞ ∑ n =0 b n ˜ r n + (5.6)  \none can compute the coefficients of (5.6) recursively. The small ˜ r + expansion for a reads  \na = 1 2 ( ℓ +1) -( 1 + α 2 ) 2 ( 3 ℓ ( ℓ +2) -∆(∆ -4) + 3˜ ω 2 ) 8 ( ℓ +1) ˜ r 2 + -( m 1 + m 2 ) α ( 1 + α 2 ) 2 ( 6 ℓ ( ℓ +2) + (∆ -2) 2 -˜ ω 2 ) ˜ ω 4 ℓ ( ℓ +1)( ℓ +2) ˜ r 3 + + O ( ˜ r 4 + ) (5.7)  \nIn addition, the derivatives of the NS free energy in the case of the equally rotating KerrAdS 5 black hole are  \n∂ a F inst = -1 2 ( ℓ +1) ( 1 -α 4 ) ˜ r 2 + + ( m 1 + m 2 )( ℓ +1) α ( 1 + α 2 ) 2 ˜ ω ( (∆ -2) 2 -˜ ω 2 ) ℓ 2 ( ℓ +2) 2 ˜ r 3 + + O ( ˜ r 4 + ) (5.8a)  \n∂ a 1 F inst = 1 2 ( 1 -α 4 ) (∆ -2) ˜ r 2 + + ( m 1 + m 2 ) α ( 1 + α 2 ) 2 (∆ -2)˜ ω ℓ ( ℓ +2) ˜ r 3 + + O ( ˜ r 4 + ) (5.8b)  \n∂ a t F inst = -i ( m 1 + m 2 ) α ( 1 + α 2 ) ( ℓ ( ℓ +2) + (∆ -2) 2 -˜ ω 2 ) 2 ℓ ( ℓ +2) ˜ r 2 +  \n+ i (1 + α 2 ) ( ( 1 + α 2 ) ( ℓ ( ℓ +2) + (∆ -2) 2 -˜ ω 2 ) 2 ℓ ( ℓ +2) + ( m 1 + m 2 ) 2 α ℓ ( ℓ +2) ) ˜ ω ˜ r 3 + + O ( ˜ r 4 + ) (5.8c)  \nInterestingly, a and ∂ a F inst are independent of the sign choice in (5.4), while ∂ a 1 F inst and ∂ a t F inst will get a global minus sign depending on the choice. Moreover, the poles located at a = ± n/ 2 for n ∈ N in the analytic expansion of the NS free energy (A16) are  \ntranslated into poles for the values of the angular momentum quantum number of the form { ℓ, ℓ -1 , ℓ -2 , . . . } and will appear in the higher order terms of the asymptotic expansions for (5.7) and (5.8). It has been pointed out that the resummation procedure can eliminate the existing poles in the instanton partition function [40-42]. Bearing this in mind, one of the authors computed the low-energy absorption cross section of a Reissner-Nordström black hole in rainbow gravity by first applying the resummation procedure to the instanton contributions to the vacuum expectation value a , and then extending this method to the derivatives of the free energy [7].  \nBy inspecting the structure of a , one can recognize that up to a numerical factor, it coincides with the monodromy around two singular points in the Riemann-Hilbert map between the four-punctured Riemann sphere and Fuchsian systems [36]. Therefore, one can attempt an ansatz for the s-wave case ( ℓ = m 1 = m 2 = 0) of the form  \na ( ℓ = 0) = 1 2 -ν 0 ˜ r 2 + + O ( ˜ r 3 + ) (5.9)  \nand replace into the Matone relation (5.5). The left-hand side is given by equation (4.6c), with the radial accessory parameter K 0 taken from (2.20b), which for small ˜ r + yields  \nlhs = -1 2 -(1 + α 2 ) 2 [ (1 -α 2 ) + ∆(∆ -4) 2 -˜ ω 2 2(1 -α 2 ) 2 + α 2 (4 + α 2 )˜ ω 2 2(1 -α 2 ) 2 ] ˜ r 2 + + O ( ˜ r 4 + ) (5.10)  \nwhile the right hand side organizes as follows  \nrhs = -1 2 + ν 0 ˜ r 2 + -( 1 + α 2 ) 2 (1 + α 4 )˜ ω 2 4(1 -α 2 ) 2 ˜ r 2 + + 1 8 ( 1 -α 4 ) ( ˜ ω 2 -(∆ -2) 2 ) ˜ r 2 + -2 ν 0 x ( 1 + x +2 x +5 x 3 +14 x 4 + . . . ) ˜ r 2 + + O ( ˜ r 4 + ) , (5.11)  \nwhere  \n1 + x +2 x 2 +5 x 3 +14 x 4 + . . . = 1 -√ 1 -4 x 2 x , (5.12a)  \nx = ( 1 + α 2 ) 4 ˜ ω 2 ( (∆ -2) 2 -˜ ω 2 ) 2 6 ν 2 0 . (5.12b)  \nThe terms inside the parenthesis proportional to ν 0 in equation (5.11) are associated with the sequence of Catalan numbers. Consequently, one can introduce the generating function for these numbers to compute the first correction, ν 0 . Hence, equating (5.10) and (5.11) gives a quadratic equation for ν 0 up to the order O ( ˜ r 2 + ) :  \n1 8   ( 1 + α 2 ) 2 ( ∆(∆ -4) -3˜ ω 2 ) +8 ν 0 √ 1 + (1 + α 2 ) 4 ˜ ω 2 (˜ ω 2 -(∆ -2) 2 ) 2 4 ν 2 0   ˜ r 2 + + O ( ˜ r 4 + ) = 0 . (5.13)  \nIt turns out that ν 0 in terms of the black hole parameters has a surprisingly simple form  \nν 0 = ± 1 8 ( 1 + α 2 ) 2 √ (3˜ ω 2 -∆(∆ -4)) 2 -4˜ ω 2 (˜ ω 2 -(∆ -2) 2 ) , (5.14)  \nand x introduced in (5.12b) takes the following form,  \nx = ˜ ω 2 ( (∆ -2) 2 -˜ ω 2 ) (3˜ ω 2 -∆(∆ -4)) 2 -4˜ ω 2 (˜ ω 2 -(∆ -2) 2 ) . (5.15)  \nNotice that the series expansion for the vacuum expectation value (5.9), as well as the generating function of the Catalan numbers make sense if ˜ r 2 + < ν 0 ≤ 1 and | x | < 1 / 4 for given ˜ ω , α and ∆ .  \nBy the same token, one can apply the resummation procedure to the derivatives of the NS free energy as follows  \n∂ a F inst = -( 2 x +3 x 2 + 20 3 x 3 + 35 2 x 4 + 252 5 x 5 + . . . ) + O ( ˜ r 2 + ) (5.16a)  \n∂ a 1 F inst = 1 2 (∆ -2) [ ( 1 -α 4 ) + ( 1 + α 2 ) 4 ˜ ω 2 4 ν 0 ( 1 + x +2 x 2 +5 x 3 +14 x 4 + . . . ) ] ˜ r 2 + + O ( ˜ r 4 + ) (5.16b)  \n∂ a t F inst = -i ( 1 + α 2 ) 2 ( ˜ ω 2 -(∆ -2) 2 ) ˜ ω 8 ν 0 ( 1 + x +2 x 2 +5 x 3 +14 x 4 + . . . ) ˜ r + + O ( ˜ r 3 + ) (5.16c)  \nwhere ν 0 and x are defined in (5.14) and (5.15), respectively. We have observed that higher order terms in z 0 from the analytic expansion of the instanton part of the NS free energy contribute to the lower order terms and consequently we get a series in x as the correction terms nicely re-summed in the form of the corresponding generating functions #3  \n∂ a F inst := ∂ a F inst (0) + O ( ˜ r 2 + ) = log 4 -2 log ( 1 + √ 1 -4 x ) + O ( ˜ r 2 + ) (5.18a)  \n∂ a 1 F inst := ∂ a 1 F inst (2) ˜ r 2 + + O ( ˜ r 4 + ) √ (5.18b)  \n= 1 2 (∆ -2) [ ( 1 -α 4 ) + ( 1 + α 2 ) 4 ˜ ω 2 4 ν 0 1 -1 -4 x 2 x ] ˜ r 2 + + O ( ˜ r 4 + )  \n∂ a t F inst := ∂ a t F inst (1) ˜ r + + O ( ˜ r 3 + ) = -i ( 1 + α 2 ) 2 ( ˜ ω 2 -(∆ -2) 2 ) ˜ ω 8 ν 0 1 -√ 1 -4 x 2 x ˜ r + + O ( ˜ r 3 + ) (5.18c)", "5.1 Retarded Green's function": "In the previous section, we have computed the correction to a ( ℓ = 0) , as well as the derivatives of the instanton part of the NS free energy, which appear in the connection coefficients of the solutions of the Heun equation (4.8). Now we aim to derive an asymptotic  \n1 + x +2 x 2 +5 x 3 +14 x 4 + . . . = 1 -√ 1 -4 x 2 x , (5.17a)  \n2 x +3 x 2 + 20 3 x 3 + 35 2 x 4 + 252 5 x 5 + . . . = log 4 -2 log ( 1 + √ 1 -4 x ) (5.17b)  \nexpansion for the s-wave retarded Green's function in the equal angular momenta limit. This will be achieved by substituting (5.14), (5.15), and (5.18) into (4.17a) and expanding for small ˜ r +  \nG ret (˜ ω, ∆) = e -iπ ∆ Γ(2 -∆)Γ ( 1 2 (∆ -˜ ω -2) ) Γ ( 1 2 (∆ + ˜ ω -2) ) Γ(∆ -2) Γ ( 1 2 (2 -∆ -˜ ω ) ) Γ ( 1 2 (2 -∆+˜ ω ) ) { 1 + [ ∂ a 1 F inst (2) +(∆ -2) ( 1 + 2 α 2 (1 + α 2 ) ) -4(∆ -2) ν 0 ((∆ -2) 2 -˜ ω 2 ) + 2 π sin π ∆ cos π ˜ ω -cos π ∆ e ∂ a F inst (0) -x e ∂ a F inst (0) + x ν 0 + 3 4 ( 1 + α 2 ) 2 ˜ ω ( ψ (0) ( 1 2 (∆ -˜ ω -2) ) -ψ (0) ( 1 2 (2 -∆ -˜ ω ) ) -ψ (0) ( 1 2 (∆ + ˜ ω -2) ) + ψ (0) ( 1 2 (2 -∆+˜ ω ) ) )] ˜ r 2 + } + O ( ˜ r 3 + ) , (5.19)  \nwhere ψ (0) ( z ) corresponds to the digamma function, ∂ a F inst (0) and ∂ a 1 F inst (2) are given by (5.18a) and (5.18b), and refer to the coefficients of the series expansion in ˜ r + for the derivatives of the instanton part of the NS free energy with respect to a and a 1 , respectively. Note that expression (5.19) assumes that ∆ is not an integer (see [43] regarding the case when ∆ is an integer). Then, the asymptotic expansion for the retarded Green's function is given by  \nG ret (˜ ω, ∆) = e -iπ ∆ Γ(2 -∆)Γ ( 1 2 (∆ -˜ ω -2) ) Γ ( 1 2 (∆ + ˜ ω -2) ) Γ(∆ -2) Γ ( 1 2 (2 -∆ -˜ ω ) ) Γ ( 1 2 (2 -∆+˜ ω ) ) { 1 + 1 4 [ 3 ( 1 + α 2 ) 2 ˜ ω × ( ψ (0) ( 1 2 (∆ -˜ ω -2) ) -ψ (0) ( 1 2 (2 -∆ -˜ ω ) ) -ψ (0) ( 1 2 (∆ + ˜ ω -2) ) + ψ (0) ( 1 2 (2 -∆+˜ ω ) ) ) -( 1 + α 2 ) 2 ( 2(∆ -2) ((∆ -2) 2 -˜ ω 2 ) + π sin π ∆ cos π ∆ -cos π ˜ ω ) √ (∆(∆ -4) -3˜ ω 2 ) 2 +2(∆ -2) ( 3 + 4 α 2 +3 α 4 ) ] ˜ r 2 + } + O ( ˜ r 3 + ) . (5.20)", '5.2 Greybody factor': "The computation of the greybody factor in asymptotically AdS spacetimes is a bit more subtle than in asymptotically flat or dS spacetimes. Due to the nature of the boundary condition at spatial infinity, the radiation produced at the horizon can travel all the way to the spatial infinity to be reflected back towards the black hole. As a result, the thermal equilibrium of black holes in AdS is ensured by this infinite mechanism [27]. At the level of the scattering problem, the required boundary conditions cannot be satisfied since one cannot identify an outgoing wave solution at infinity. Nevertheless, using an approximation scheme the greybody factors of static and spherically symmetric, and rotating black holes in asymptotically AdS have been studied in [27, 28], respectively. By considering different regions of the spacetime, the radial equation simplifies and analytical solutions can be found. These solutions can be matched in an overlapping region, resulting in a consistent definition for the fluxes at the horizon and at spatial infinity.  \nIn addition, [29] has performed the numerical computation of the absorption cross section in a similar fashion way. The asymptotic analysis at infinity remains the same as [27, 28], while in the near-horizon region, the radial solutions are given in terms of the Heun functions instead of the Hypergeometric functions. Our approach follows both ideas: an approximate radial equation in the far-region and Heun functions at the horizon. However, we will introduce the exact connection coefficients rather than the ratio of the Wronskians as [29]. Finally, we will define the conserved fluxes as  \nF = 1 2 i ( R ∗ ˜ r ∆ ˜ r dR d ˜ r -R ˜ r ∆ ˜ r dR ∗ d ˜ r ) , (5.21)  \nso that the greybody factor is the ratio between the flux at the horizon and the flux coming in from infinity  \nγ ( ℓ ) = F hor F ( ∞ ) in (5.22)  \nWe consider radial equation in the far region approximation ˜ r ≫ 1 , such that the radial equation (2.13) reduces to  \nR '' + 5 ˜ r R ' -[ ∆(∆ -4) ˜ r 2 + ℓ ( ℓ +2) -˜ ω 2 ˜ r 4 ] R = 0 , (5.23)  \nthen, we introduce a new radial coordinate  \nu = ˜ ω ˜ r (5.24)  \nand consider the limit u ≪ ˜ ω . As a result, equation (5.23) yields  \nR '' -3 u R ' + ( 1 -ℓ ( ℓ +2) ˜ ω 2 -∆(∆ -4) u 2 ) R = 0 , (5.25)  \nwhose solution is given by the linear combination of the Bessel functions, J ν ( z ) and Y ν ( z ) , of the form  \nR ( u ) = C 1 u 2 J ∆ -2 ( √ 1 -ℓ ( ℓ +2) ˜ ω 2 u ) + C 2 u 2 Y ∆ -2 ( √ 1 -ℓ ( ℓ +2) ˜ ω 2 u ) , (5.26)  \nwhich can be written, more conveniently, in terms of the Hankel functions  \nR ( u ) = 1 2 ( C 1 -iC 2 ) u 2 H (1) ∆ -2 ( u ) + 1 2 ( C 1 + iC 2 ) u 2 H (2) ∆ -2 ( u ) , (5.27)  \nsince their asymptotic structures describe the incoming and outgoing part of the wave function. Namely, the H (1) ∆ -2 ( u ) is associated with the incoming part of the solution, while H (2) ∆ -2 ( u ) controls the outgoing part. In the limit ˜ r →∞ the radial solution for the s-wave case behaves like  \nR ( r ) ≃ ( C 1 A J + C 2 A Y ) ˜ r -∆ + C 2 B Y ˜ r ∆ -4 , (5.28)  \nwhere  \nA J = 4 Γ(∆ -1) ( ˜ ω 2 ) ∆ , A Y = -4 π cos π (∆ -2)Γ(2 -∆) ( ˜ ω 2 ) ∆ , B Y = -4 π Γ(∆ -2) ( ˜ ω 2 ) 4 -∆ . (5.29)  \nBy comparing the asymptotic behavior of the solutions (5.28) with (4.16), we obtain  \nC 1 = A 1 -A J C 1 --A 1+ A Y A J B Y C 1+ , C 2 = A 1+ B Y C 1+ , (5.30)  \nwhere C 1 ± are given in (4.15) and  \nA 1 -= C z 0 -( z 0 -1) -1 2 θ + ( ˜ r 2 --˜ r 2 0 ) 1 2 ∆ , A 1+ = C z 0 -( z 0 -1) -1 2 θ + ( ˜ r 2 --˜ r 2 0 ) 1 2 (4 -∆) . (5.31)  \nFurthermore, by inspecting the radial solution (5.27), we have  \nC in ≡ 1 2 ( C 1 -iC 2 ) , C out ≡ 1 2 ( C 1 + iC 2 ) , (5.32)  \nsuch that one can reproduce the (incoming) outcoming coefficients in [29]. Therefore, the asymptotic incoming radial solution with quantum numbers ( ℓ = m 1 = m 2 = 0) reads  \nR ( ∞ ) in (˜ r ) = C in ( ˜ ω ˜ r ) 2 H (1) ∆ -2 ( ˜ ω ˜ r ) , (5.33)  \nsuch that after substituting (5.33) into equation (5.21), we get the incoming flux at spatial infinity  \nF ( ∞ ) in = -2 | C in | 2 ˜ ω 4 π (5.34)  \nOn the other hand, the solution at the horizon is given by (4.12), and the associated flux at the horizon is  \nF hor = -2 ∆ | C z 0 -| 2 ˜ ω ˜ r 3 + ( 1 + ˜ r 2 + ) ∆ / 2 ( 1 + (1 + 2˜ r 2 + ) α 2 ) 2    √ 1 + ( 1 + 2(1 + 2˜ r 2 + ) α 2 ) 2 ˜ r 2 + 1 + ( 3 + 2(1 + ˜ r 2 + ) α 2 ) ˜ r 2 + + √ 1 + ˜ r 2 + √ 1 + ( 1 + 2(1 + 2˜ r 2 + ) α 2 ) 2 ˜ r 2 +    ∆ (5.35)  \nThe greybody factor given as the ratio of the flux at the horizon and the incoming flux at spatial infinity reads  \nγ ( ℓ =0) = F hor F ( ∞ ) in = | C z 0 -| 2 | C in | 2 2 ∆ -1 π ˜ r 3 + ˜ ω 3 ( 1 + ˜ r 2 + ) ∆ / 2 ( 1 + (1 + 2˜ r 2 + ) α 2 ) 2    √ 1 + ( 1 + 2(1 + 2˜ r 2 + ) α 2 ) 2 ˜ r 2 + 1 + ( 3 + 2(1 + ˜ r 2 + ) α 2 ) ˜ r 2 + + √ 1 + ˜ r 2 + √ 1 + ( 1 + 2(1 + 2˜ r 2 + ) α 2 ) 2 ˜ r 2 +    ∆ (5.36)  \nSince we are interested in the contribution of the resummation technique for the greybody factor in the small ˜ r + limit, we will derive asymptotic expansions without and with the resummation procedure. For the former we replace (5.7) and (5.8) into (5.36), while keeping  \nthe expansions up to second order O ( ˜ r 2 + ) to avoid the pole at ℓ = 0 . Bearing this in mind, the greybody factor reads  \nγ (0) = 2 π 2 ( 1 + α 2 ) 2 ˜ r 3 + 2 2∆+6 π ˜ ω 2∆+5 sin 2 π ∆ (cos π ∆ -cos π ˜ ω ) 2 ( 3˜ ω 2 -∆(∆ -4) ) 4 ((3˜ ω 2 -∆(∆ -4)) 2 -˜ ω 2 (˜ ω 2 -(∆ -2) 2 )) 2 1 χ + O ( ˜ r 4 + ) (5.37)  \nwhere  \nχ = 4 2∆ Γ ( 1 2 (∆ -˜ ω -2) ) 2 Γ ( 1 2 (∆ + ˜ ω -2) ) 2 ˜ ω 8 +4 4 Γ ( 1 2 (2 -∆ -˜ ω ) ) 2 Γ ( 1 2 (2 -∆+˜ ω ) ) 2 ˜ ω 4∆ + 4 ∆+4 π 2 ˜ ω 2∆+4 cos 2 π ∆ (cos π ∆ -cos π ˜ ω ) ((∆ -2) 2 -˜ ω 2 ) . (5.38)  \nOn the other hand, for the latter case we substitute (5.14), (5.15), and (5.16) into (5.36), which implies  \nγ (0) = 2 π 2 ( 1 + α 2 ) 2 ˜ r 3 + 2 2∆+6 π ˜ ω 2∆+5 sin 2 π ∆ (cos π ∆ -cos π ˜ ω ) 2 e ∂ a F inst (0) ( e ∂ a F inst (0) + x ) 2 1 χ + O ( ˜ r 4 + ) (5.39)  \nWe observe that the highlighted factor in (5.37) is corrected by a factor (highlighted in blue) containing all the contributions given by the derivative of the instanton part of the free energy at zeroth order, ∂ a F inst (0) . Furthermore, one can replace it by its generating function (5.18a), to obtain  \ne ∂ a F inst (0) ( e ∂ a F inst (0) + x ) 2 = 1 , (5.40)  \nwhich reduces the greybody factor (5.39) to  \nγ (0) = 2 2∆+6 π ˜ ω 2∆+5 ˜ A sin 2 π ∆ (cos π ∆ -cos π ˜ ω ) 2 1 χ + O ( ˜ r 4 + ) , (5.41)  \nwhere ∆ is not integer, and ˜ A = 2 π 2 ( 1 + α 2 ) 2 ˜ r 3 + is related to the area of the fivedimensional Kerr-AdS black hole with equal angular momenta in the small-radius limit. Note that in the non-rotating limit α → 0 , equation (5.41) gives the greybody factor for massive scalar fields in small Schwarzschild-AdS 5 black holes.  \nIn Fig. 1, we present the greybody factor for small Kerr-AdS 5 black holes with equal rotation parameters and ( ℓ = m 1 = m 2 = 0) modes, as calculated using formula (5.41). The analysis is performed for different values of the extremality parameter α = { 0 , 1 / 3 , 2 / 3 , 9 / 10 , 99 / 100 } , while keeping fixed ˜ r + = 1 / 1000 , and ∆ = 41 / 10 , and varying the frequency ˜ ω . It is worth mentioning that the greybody spectrum increases with α for fixed ˜ ω , which contrasts with previous results for massless scalar fields in rotating cohomogeneity -1 BH space-times [28]. As observed in [27], the large-amplitude oscillations in the greybody spectrum are consistent with the spacing between the normal modes frequencies in pure  \nfive-dimensional AdS spacetime. In addition, we observed an intriguing dynamics in the spectrum at small ˜ ω . As we increase the conformal dimension ∆ , two peaks appear and start to move closer, merging and then separating again, as illustrated in Fig. 2.  \nIn Fig. 3, we compare the greybody factor calculated with and without the resummation procedure. The solid curves, representing the asymptotic formula with resummation (5.41), display a compressed spectrum compared to the dashed curves, which correspond to the expression without resummation (5.37).  \nFigure 1 . Greybody factor (5.41) as a function of ˜ ω for different values of α , and fixed ˜ r + = 1 / 1000 , and ∆ = 41 / 10 .  \n<!-- image -->", '6 Small Reissner-Nordström-AdS 5 black holes': "We now turn to study the retarded Green's function and the greybody factor of a RN-AdS 5 black hole in the small-radius limit. To this matter, it is convenient to express the charge ˜ Q in terms of ˜ r + . Specifically, the temperature at the outer horizon defined as  \n˜ T + = 1 2 π [ 1 ˜ r + -˜ Q 2 ˜ r 5 + +2˜ r + ] , (6.1)  \nvanishes at extremality, indicating that the maximal charge is given by  \n˜ Q c = ˜ r 2 + √ 1 + 2˜ r 2 + , (6.2)  \nsuch that Q ≤ Q c must be satisfied to ensure a regular outer horizon. We then parametrize the accessible charge values as  \n˜ Q = q ˜ Q c = q ˜ r 2 + √ 1 + 2˜ r 2 + , 0 ≤ q ≤ 1 , (6.3)  \nFigure 2 . Greybody factor as a function of ˜ ω for fixed values of α and ˜ r + , while varying ∆ .  \n<!-- image -->  \nFigure 3 . Comparison between the greybody factors with (solid lines) and without resummation (dashed lines) for different values of α and fixed ˜ r + = 1 / 1000 , and ∆ = 41 / 10 .  \n<!-- image -->  \nwhere q is an extremality parameter, which describes the Schwarzschild-AdS 5 black-hole solution for q = 0 and the extremal RN-AdS 5 black hole for q = 1 . Plugging (6.3) into (3.15) and expanding for small ˜ r + yields  \nθ -= i ( q ˜ ω -˜ e ) q 2 ˜ r + 1 -q 2 -i ( ˜ e -3 q 2 ˜ e -q ( 1 -5 q 2 ) ˜ ω ) q 2 ˜ r 3 + 2 (1 -q 2 ) + O ( ˜ r 5 + ) , (6.4a)  \nθ + = i (˜ ω -q ˜ e ) ˜ r + 1 -q 2 -i (2˜ ω -q ˜ e ) ˜ r 3 + 1 -q 2 + O ( ˜ r 5 + ) , (6.4b)  \nθ 0 = ˜ ω -( 3 2 ( 1 + q 2 ) ˜ ω -q ˜ e ) ˜ r 2 + + 1 8 (( 23 + 26 q 2 +35 q 4 ) ˜ ω -4 ( 3 + 5 q 2 ) q ˜ e ) ˜ r 4 + + O ( ˜ r 6 + ) , (6.4c)  \nθ ∞ = ∆ -2 , (6.4d)  \nwhile the conformal modulus (3.18) reads  \nz 0 = ( 1 -q 2 ) ˜ r 2 + -2 ( 1 -q 4 ) ˜ r 4 + + O ( ˜ r 6 + ) (6.4e)  \nNext, one can solve Matone's relation (5.5) to obtain an expansion for the vacuum expectation value a in the form  \na = 1 2 ( ℓ +1) -( ( 1 + q 2 ) ( 3˜ ω 2 -∆(∆ -4) + 3 ℓ ( ℓ +2) ) 8( ℓ +1) -q ˜ e ˜ ω 2( ℓ +1) ) ˜ r 2 + + O ( ˜ r 4 + ) (6.5)  \nanalogous to (5.7). In addition, the derivatives of the NS free energy are  \n∂ a F inst = -1 2 ( 1 -q 2 ) ( ℓ +1) ˜ r 2 + + O ( ˜ r 4 + ) (6.6a)  \n∂ a 1 F inst = 1 2 ( 1 -q 2 ) (∆ -2) ˜ r 2 + + O ( ˜ r 4 + ) (6.6b)  \n∂ a t F inst = i (˜ ω -q ˜ e ) ( (∆ -2) 2 + ℓ ( ℓ +2) -˜ ω 2 ) ˜ r 3 + 2 ℓ ( ℓ +2) + O ( ˜ r 5 + ) (6.6c)  \nDue to the presence of a pole structure in the angular momentum quantum number ℓ within the expansions (6.5) and (6.6), it is necessary to sum the contributions from all orders in z 0 , as demonstrated in [44]. Specifically, we limit our analysis to the s-wave case ( ℓ = 0) , following the same procedure introduced for the equal angular momenta Kerr-AdS 5 black hole in section 5. In other words, we consider an ansatz of the form  \na ( ℓ = 0) = 1 2 -ν 0 ˜ r 2 + + O ( ˜ r 4 + ) (6.7)  \nand replace into the Matone's relation (5.5). The left-hand side is given by equation (4.6c), with the radial accessory parameter K 0 taken from (3.20b), which for small ˜ r + yields  \nlhs = -1 2 -1 4 [ ∆(∆ -4) -6 q 2 (1 -q 2 ) -˜ ω 2 -(3˜ ω -4 q ˜ e ) q 2 ˜ ω -( 1 + q 2 ) q 2 ˜ e 2 -2 ( 1 -q 6 ) (1 -q 2 ) 2 ] ˜ r 2 + + O ( ˜ r 4 + ) (6.8)  \nwhile the right hand side organizes as follows  \nrhs = -1 2 + ν 0 ˜ r 2 + -1 8 ( 1 -q 2 ) ( (∆ -2) 2 -˜ ω 2 ) ˜ r 2 + -( (1 + q 6 )˜ ω 2 -2(1 + q 4 ) q ˜ e ˜ ω +(1 + q 2 ) q 2 ˜ e 2 ) ˜ r 2 + 4 (1 -q 2 ) 2 -2 ν 0 x ( 1 + x +2 x +5 x 3 +14 x 4 + . . . ) ˜ r 2 + + O ( ˜ r 4 + ) , (6.9)  \nwhere  \nx = ( (∆ -2) 2 -˜ ω 2 ) ( ( 1 + q 2 ) (˜ ω -q ˜ e ) 2 + q 4 ( ˜ ω 2 -˜ e 2 ) ) 2 6 ν 2 0 , (6.10)  \nand the terms inside the parenthesis in equation (6.9) are associated with the sequence of Catalan numbers (5.12a), as presented in the previous section. Equating (6.8) and (6.9) leads to an equation for ν 0 that can be solved to yield  \nν 0 = ± 1 8 √ √ √ √ √ 4 ( (∆ -2) 2 -˜ ω 2 ) ( ( 1 + q 2 ) (˜ ω -q ˜ e ) 2 + q 4 ( ˜ ω 2 -˜ e 2 ) ) + (( 1 + q 2 ) ( 3˜ ω 2 -∆(∆ -4) ) -4 q ˜ e ˜ ω ) 2 , (6.11)  \nwhich in the limit q → 0 gives the correction to Schwarzschild-AdS 5 black-hole solution. Furthermore, x reduces to  \nx = 1 ( 4 + ((1+ q 2 )(3˜ ω 2 -∆(∆ -4)) -4 q ˜ e ˜ ω ) 2 ((∆ -2) 2 -˜ ω 2 ) ( (1+ q 2 )(˜ ω -q ˜ e ) 2 + q 4 (˜ ω 2 -˜ e 2 ) ) ) . (6.12)  \nAnalogously, the derivatives of the NS free energy are  \n∂ a F inst = -( 2 x +3 x 2 + 20 3 x 3 + 35 2 x 4 + 252 5 x 5 + . . . ) + O ( ˜ r 2 + ) (6.13a)  \n∂ a 1 F inst = 1 2 (∆ -2) [ ( 1 -q 2 ) + ( ( 1 + q 2 ) (˜ ω -q ˜ e ) 2 + q 4 ( ˜ ω 2 -˜ e 2 ) ) 4 ν 0 (6.13b)  \n× ( 1 + x +2 x 2 +5 x 3 +14 x 4 + . . . ) ] ˜ r 2 + + O ( ˜ r 4 + )  \n∂ a t F inst = -i (˜ ω -q ˜ e ) ( (∆ -2) 2 -˜ ω 2 ) 8 ν 0 ( 1 + x +2 x 2 +5 x 3 +14 x 4 + . . . ) ˜ r + + O ( ˜ r 3 + ) (6.13c)  \nwhere ν 0 and x are defined in (6.11) and (6.12), respectively. By means of (5.17), one can substitute the series expansions in (6.13) by their generating functions as follows  \n∂ a F inst := ∂ a F inst (0) + O ( ˜ r 2 + ) = log 4 -2 log ( 1 + √ 1 -4 x ) + O ( ˜ r 2 + ) (6.14a)  \n∂ a 1 F inst := ∂ a 1 F inst (2) ˜ r 2 + + O ( ˜ r 4 + )  \n= 1 2 (∆ -2) [ ( 1 -q 2 ) + ( ( 1 + q 2 ) (˜ ω -q ˜ e ) 2 + q 4 ( ˜ ω 2 -˜ e 2 ) ) 4 ν 0 1 -√ 1 -4 x 2 x ] ˜ r 2 + + O ( ˜ r 4 + ) (6.14b)  \n∂ a t F inst := ∂ a t F inst (1) ˜ r + + O ( ˜ r 3 + ) = -i (˜ ω -q ˜ e ) ( (∆ -2) 2 -˜ ω 2 ) 8 ν 0 1 -√ 1 -4 x 2 x ˜ r + + O ( ˜ r 3 + ) (6.14c)  \nInterestingly, the corrections to the derivatives of F inst in the case of small RN-AdS 5 BHs match the order of ˜ r + in the corrections found in the case of small Kerr-AdS 5 BHs with equal rotation (5.18). Furthermore, the most singular terms at each given order of ∂ a F inst are responsible for the appearance of the branch cut after the resummation, as we have seen in the coefficient ∂ a F inst (0) , which contains a logarithm.", "6.1 Retarded Green's function": "The retarded Green's function, defined as the ratio of the response to the source, is given by (4.17), where the radial dictionary (4.6b) is realized by (6.4). The new elements in the connection coefficients, such as a ( ℓ = 0) and the derivatives of F inst are obtained from (6.7) and (6.14). Expanding for small ˜ r + , we derive  \nG  \nret (˜ ω, ∆) = e -iπ ∆ Γ(2 -∆)Γ ( 1 2 (∆ -˜ ω -2) ) Γ ( 1 2 (∆ + ˜ ω -2) ) Γ(∆ -2) Γ ( 1 2 (2 -∆ -˜ ω ) ) Γ ( 1 2 (2 -∆+˜ ω ) ) { 1 + [ ∂ a 1 F inst (2) -( e ∂ a F inst (0) -x e ∂ a F inst (0) + x ν 0 -1 4 ( 3(1 + q 2 )˜ ω -2 q ˜ e ) ) ( ψ (0) ( 1 2 (∆ -˜ ω -2) ) -ψ (0) ( 1 2 (2 -∆ -˜ ω ) ) ) -( e ∂ a F inst (0) -x e ∂ a F inst (0) + x ν 0 + 1 4 ( 3(1 + q 2 )˜ ω -2 q ˜ e ) ) ( ψ (0) ( 1 2 (∆ + ˜ ω -2) ) -ψ (0) ( 1 2 (2 -∆+˜ ω ) ) ) +(∆ -2) ( 1 + 2 q 2 ) -8 ν 0 (∆ -2) ((∆ -2) 2 -˜ ω 2 ) e ∂ a F inst (0) ( e ∂ a F inst (0) + x ) ] ˜ r 2 + } + O ( ˜ r 3 + ) , (6.15)  \nwhich can be further simplified, yielding  \nG ret (˜ ω, ∆) = e -iπ ∆ Γ(2 -∆)Γ ( 1 2 (∆ -˜ ω -2) ) Γ ( 1 2 (∆ + ˜ ω -2) ) Γ(∆ -2) Γ ( 1 2 (2 -∆ -˜ ω ) ) Γ ( 1 2 (2 -∆+˜ ω ) ) { 1 + 1 4 [ 6 ( 1 + q 2 ) (∆ -2) -4(∆ -2) √ ((1 + q 2 )(3˜ ω 2 -∆(∆ -4) -4 q ˜ e ˜ ω ) 2 ((∆ -2) 2 -˜ ω 2 ) -1 2 ( 4 q ˜ e -6(1 + q 2 )˜ ω + √ ((1 + q 2 )(3˜ ω 2 -∆(∆ -4) -4 q ˜ e ˜ ω ) 2 ) × ( ψ (0) ( 1 2 (∆ -˜ ω -2) ) -ψ (0) ( 1 2 (2 -∆ -˜ ω ) ) ) + 1 2 ( 4 q ˜ e -6(1 + q 2 )˜ ω -√ ((1 + q 2 )(3˜ ω 2 -∆(∆ -4) -4 q ˜ e ˜ ω ) 2 ) × ( ψ (0) ( 1 2 (∆ + ˜ ω -2) ) -ψ (0) ( 1 2 (2 -∆+˜ ω ) ) ) ] ˜ r 2 + } + O ( ˜ r 3 + ) . (6.16)", '6.2 Greybody factor': 'The computation of the greybody factor for the RN-AdS 5 black hole follows section 5.2. In the far-region approximation, the radial equation (3.13) simplifies to (5.23), then one can consider that the incoming flux at spatial infinity is  \nF ( ∞ ) in = -2 | C in | 2 ˜ ω 4 π , (6.17)  \nwhile the flux at the horizon corresponds to the incoming radial solution (4.12), expressed in terms of the RN-AdS 5 black hole parameters:  \nF hor = -2 ∆ | C z 0 -| 2 ( ˜ ω -q ˜ e √ 1 + 2˜ r 2 + ) ˜ r 3 +   √ 1 + ( 2 + ˜ r 2 + + q 2 ( 4 + 8˜ r 2 + )) ˜ r 2 + 1 + 3˜ r 2 + + √ 1 + ( 2 + ˜ r 2 + + q 2 ( 4 + 8˜ r 2 + )) ˜ r 2 +   ∆ (6.18)  \nTherefore, the greybody factor for the s-wave reads  \nγ ( ℓ =0) = F hor F ( ∞ ) in = | C z 0 -| 2 | C in | 2 2 ∆ -1 π ˜ r 3 + ˜ ω 4 ( ˜ ω -q ˜ e √ 1 + 2˜ r 2 + )   √ 1 + ( 2 + ˜ r 2 + + q 2 ( 4 + 8˜ r 2 + )) ˜ r 2 + 1 + 3˜ r 2 + + √ 1 + ( 2 + ˜ r 2 + + q 2 ( 4 + 8˜ r 2 + )) ˜ r 2 +   ∆ , (6.19)  \nwhere C in is defined in (5.32). By replacing (6.5) and (6.6) into (6.19), while retaining terms up to second order O ( ˜ r 2 + ) to avoid the pole at ℓ = 0 , the greybody factor in the small ˜ r + limit becomes:  \nγ (0) = 2 2∆+7 π 3 ˜ r 3 + ˜ ω 2∆+4 (˜ ω -q ˜ e ) sin 2 π ∆ (cos π ∆ -cos π ˜ ω ) 2 1 ( 1 + ((∆ -2) 2 -˜ ω 2 ) ( (1+ q 2 )(˜ ω -q ˜ e ) 2 + q 4 (˜ ω 2 -˜ e 2 ) ) ((1+ q 2 )(3˜ ω 2 -∆(∆ -4)) -4 q ˜ e ˜ ω ) 2 ) 1 χ + O ( ˜ r 4 + ) (6.20)  \nwhere  \nχ = 4 2∆ Γ ( 1 2 (∆ -˜ ω -2) ) 2 Γ ( 1 2 (∆ + ˜ ω -2) ) 2 ˜ ω 8 +4 4 Γ ( 1 2 (2 -∆ -˜ ω ) ) 2 Γ ( 1 2 (2 -∆+˜ ω ) ) 2 ˜ ω 4∆ + 4 ∆+4 π 2 ˜ ω 2∆+4 cos 2 π ∆ (cos π ∆ -cos π ˜ ω ) ((∆ -2) 2 -˜ ω 2 ) , (6.21)  \nNevertheless, the resummation technique gives a contribution to the asymptotic expansion, determined by (6.11) and (6.14),  \nγ (0) = 2 2∆+7 π 3 ˜ r 3 + ˜ ω 2∆+4 (˜ ω -q ˜ e ) sin 2 π ∆ (cos π ∆ -cos π ˜ ω ) 2 e ∂ a F inst (0) ( e ∂ a F inst (0) + x ) 2 1 χ + O ( ˜ r 4 + ) (6.22)  \nγ (0) = 2 2∆+7 π 3 ˜ r 3 + ˜ ω 2∆+4 (˜ ω -q ˜ e ) sin 2 π ∆ (cos π ∆ -cos π ˜ ω ) 2 1 χ + O ( ˜ r 4 + ) (6.23)  \nIn Fig. 4, we present the greybody factor for small RN-AdS 5 black holes with ℓ = 0 modes, as calculated using formula (6.23). The analysis is performed for different values of  \nthe extremality parameter q = { 0 , 1 / 3 , 2 / 3 , 9 / 10 , 99 / 100 } , while keeping fixed ˜ r + = 1 / 100 , ˜ e = 2 √ 3 , and ∆ = 46 / 10 , and varying the frequency ˜ ω . Interestingly, our results show that the greybody spectrum decreases as q increases, which contrasts our previous results in Fig. 1. Although α and q play the role of extremality parameters, their physical nature is different: the former is associated with the angular momentum of the black hole, while the latter corresponds to the charge of the black hole. It is interesting to note that in the γ (0) vs ˜ ω plot for Kerr-AdS 5 , γ (0) increases with α for a fixed ˜ ω . On the other hand, for RN-AdS 5 black hole due the presence of a factor (˜ ω -q ˜ e ) , the greybody factor γ (0) decreases with q for a fixed value of ˜ ω . We see this distinctive behavior by comparing the plot 1 with plot 4. In addition, for 0 < ˜ ω < q ˜ e , the greybody factor is negative, as shown in the panel on the upper-left corner of figure 4, pointing out a superradiant frequency window. The first negative peak increases (in absolute value) as we increase q , contrary to the behavior seen for ˜ ω > q ˜ e , where the spectrum shrinks.  \nIn Fig. 5, we compare the greybody factor calculated with and without the resummation procedure. The solid curves, representing the asymptotic formula with resummation (6.23), show a compressed spectrum compared to the dashed curves, which describe the expression without the resummation (6.20).  \nFigure 4 . Greybody factor (6.23) as a function of ˜ ω for different values of q , and fixed ˜ r + = 1 / 100 , ˜ e = 2 √ 3 , and ∆ = 46 / 10 .  \n<!-- image -->', '7 Discussion': "In this paper, we have investigated scalar perturbations in asymptotically AdS sapcetimes, focusing on Kerr-AdS 5 and Reissner-Nordström-AdS 5 black holes. In both cases, the radial ODE can be transformed into a Heun equation -a second-order ODE with four regular singular points. Recently, Heun solutions and their connection coefficients have been computed in terms of Virasoro conformal blocks and, via AGT correspondence, can be expressed in term of the Nekrasov-Shatashvili partition function of an SU (2) supersymmetric gauge theory with four fundamental hypermultiplets [21]. By means of these tools, we have computed the retarded Green's functions and the greybody factor.  \nFigure 5 . Comparison between the greybody factors with resummation (solid lines) and without resummation (dashed lines) for different values of q , and fixed ˜ r + = 1 / 100 , ˜ e = 2 √ 3 , and ∆ = 46 / 10 .  \n<!-- image -->  \nInterestingly, these functions introduce new elements, such as the vacuum expectation value of the scalar in the vector hypermultiplet a , and the derivatives of the NS function with respect to a and the masses of the gauge theory, ∂ a F inst and ∂ a i F inst , as seen in (4.15). We observed that by solving Matone's relation for a at small ˜ r + , the poles in the analytic expansion of the NS function (A16), located at a = ± n/ 2 for n ∈ N , correspond to poles in the angular momentum quantum number ℓ for the series expansion of a . Consequently, these poles appear in the expansions of the derivatives of F inst . Bearing this in mind, we perform the computation of the retarded Green's function and the greybody factor for the pole at ℓ = 0 using two strategies: one based on keeping the expansions up to order O ( ˜ r 2 + ) to avoid the pole and one based on curing the pole by considering contributions from all orders in the analytic expansion of the NS function [7]. It is worth mentioning that these features are present in both backgrounds, making the computation analogous, and we restrict our analysis for small ˜ r + black holes, including an extremality parameter.  \nAt order O ( ˜ r 2 + ) , the quantities a , ∂ a F inst , and ∂ a i F inst remain finite in the case ℓ = 0 , as shown in (5.8), such that by substituting these into (4.17) and expanding for small ˜ r + , we derive an asymptotic expression for G ret in this limit. For the Kerr-AdS 5 with equal rotation, the result is given by (C1), while for RN-AdS 5 , G ret takes the form (C2).  \nIn asymptotically AdS spacetimes, the computation of the greybody factor as the ratio of the flux at the horizon to the incoming flux from infinity is more subtle, due to the asymptotic behavior of the solutions at infinity (4.16), which prevents us identifying the ingoing and outgoing wave behavior. However, using a far-region approximation introduced in [27], we can transform the radial equation into a Bessel's diferrential equation (5.23), where the asymptotic form of the solutions reproduces the ingoing and outgoing waves. For the flux at the horizon, we will use the Heun functions and their connections coefficients. Regarding the computation without the resummation procedure, the greybody factors for small Kerr-AdS 5 and RN-AdS 5 are given in (5.37) and (6.20), respectively.  \nOn the other hand, the pole structure in the analytic expansion of the instanton part of  \nthe NS free energy suggests that at each order, the most singular term provides a contribution that can be resummed. For instance, we have shown that at ℓ = 0 , the first correction ν 0 to a ( ℓ = 0) is determined by the generating function of the Catalan numbers [45]. It turns out that the derivatives of F inst also receives corrections from the most singular terms at each order. For instance, the leading singularities from ∂ a F inst sum up into a logarithm at order O ( ˜ r 0 + ) . Then, we use them into the formulae (4.17) and (5.22).  \nThe asymptotic expansions for the retarded Green's functions in both spacetimes are corrected in a non-trivial manner, involving terms like ν 0 , ∂ a 1 F inst (2) and e ∂aF inst (0) -x e ∂aF inst (0) + x . Surprisingly, these corrections simplify our results (5.20) and (6.16) in comparison with the asymptotic expansions in Appendix C.  \nFor the s-wave greybody factor, the effect of resummation simplifies the asymptotic expansions, as well as it leads to a greybody spectrum γ (0) (˜ ω ) that is compressed in comparison with the spectrum without the resummation procedure, as shown in Fig. 3 and 5. Furthermore, we observe in Fig. 1 that the greybody factor of Kerr-AdS 5 with equal rotation increases when the rotation parameter α increases, contrary to previous results [28]. In Fig. 2 an interesting dynamics show up at small frequencies, as we vary the conformal dimension, two peaks emerge and move closer possibly merge and the separate again.  \nFor RN-AdS 5 black holes, the spectrum decreases as we increase the extremality parameter q . For 0 < ˜ ω < q ˜ e , the greybody factor is negative, corresponding to a superradiant instability (see Fig. 4). Finally, it will be interesting to incorporate the higher order corrections in a and see the effect in physical observables. It is also worth to explore the effect of resummation in AdS hyperbolic black hole.", 'Acknowledgments': 'We thank Atanu Bhatta for useful discussion and collaboration at the initial stage of the work. JBA acknowledges financial support from the Fundação para a Ciência e a Tecnologia (FCT) - Portugal through the research project 10.54499/2022.03702.PTDC (GENIDE). SC acknowledges ANRF grant CRG/2022/006165. AM would like to thank the Council of Scientific and Industrial Research (CSIR), Government of India, for the financial support through a research fellowship (File No.: 09/1005(0034)/2020-EMR-I).', 'A Nekrasov-Shatashvili function with N f = 4 fundamental hypermultiplets': "Here, we review the SU (2) Nekrasov-Shatashvili function with N f = 4 fundamental hypermultiplets. Nekrasov considered a deformed lagrangian of 4 d , N = 4 gauge theory by introducing two deformation parameters ϵ 1 , 2 [3, 46]. These parameters parameterize the SO (4) rotation of the spacetime R 4 . As a result, the translation symmetry is broken. Nekrasov partition function depends on the coupling parameter τ , VEV ' a ' of the adjoint scalars in the vector multiplets, and hypermultiplet masses m . It consists of three parts,  \nnamely, the classical, the one-loop and the instanton part:  \nZ ( τ, a, m ; ϵ 1 , 2 ) = Z classical Z 1-loop Z inst (A1)  \nIts key characteristic is that it provides the prepotential of the theory in the limit ϵ 1 , 2 → 0 and coincides with the prepotential as determined by the Seiberg-Witten curve. Here, we focus on instanton part of Nekrasov partition function for SU (2) gauge theory which is obtained from the U (2) partition function by dividing it with U (1) -factor [4, 9]. The U (2) partition function is expressed in terms of the combinatorial formula which we review below. Let us denote a partition (Young Tableau) by  \nY = ( y 1 , y 2 , ... ) (A2)  \nwhere y i is the height of the i -th column and y i = 0 , when i is larger than the width of the tableau. Its transpose is given by  \nY t = ( y t 1 , y t 2 , ... ) (A3)  \nWe write a vector of Young Tableau as  \nY = ( Y 1 , Y 2 ) (A4)  \nFor a given Young diagram Y , we denote the arm length and the leg length of a box ' s ' with respect to the diagram Y as  \nA Y ( i, j ) = y j -i, L Y ( i, j ) = y t i -j (A5)  \nwhere ( i, j ) denotes the coordinates of the box ' s '. We do not restrict ' s ' to be in Y . A Y and L Y can be negative if the box ' s ' lies outside Y [20].  \nThe instanton part of Nekrasov partition function is given by the summation over the Young tableaux, whose summand is the product of factors corresponding to the field content of the Lagrangian.  \nZ U (2) inst ( t,⃗a, m ; ϵ 1 , 2 ) = ∑ ⃗ Y t | ⃗ Y | z vec ( ⃗a, ⃗ Y ) z hyp ( ⃗a, ⃗ Y , m ) , (A6)  \nwhere ⃗a = ( a 1 , a 2 ) is the VEV of the scalar in the vector multiplet, | ⃗ Y | = | Y 1 | + | Y 2 | denotes the total number of boxes (i.e. instanton number) in both Y 1 and Y 2 . The instanton counting parameter t is given by  \nt = e 2 πiτ (A7)  \nwhere τ is related with the gauge coupling constant by  \nτ = θ 2 π + i 4 π g 2 Y M (A8)  \nThe hypermultiplet and vector contributions are given by [47, 48]  \nz hyp ( ⃗a, ⃗ Y , m ) = ∏ I =1 , 2 ∏ s ∈ Y I [ a I + m + ϵ 1 ( i -1 2 ) + ϵ 2 ( j -1 2 )] , z vec ( ⃗a, ⃗ Y ) = 2 ∏ I,J =1 ∏ s ∈ Y I 1 a I -a J -ϵ 1 L Y I ( s ) + ϵ 2 ( A Y I ( s ) + 1) × ∏ t ∈ Y J 1 a I -a J + ϵ 1 ( L Y I ( t ) + 1) -ϵ 2 A Y J ( t ) (A9)  \nIn our work, we always consider a 1 = -a 2 = a . Furthermore, let us denote with m 1 , m 2 , m 3 and m 4 the masses of the four hypermultiplets. Additionally, we introduce the gauge parameters a 0 , a t , a 1 and a ∞ which are related to the masses m i of the hypermultiplets via  \nm 1 = a t + a 0 , m = a a , m = a + a , m = a a .  \n2 t -0 3 1 ∞ 4 1 -∞ (A10)  \nOn the other hand, the U (1) -partition function for N f = 4 is given by  \nZ N f =4 U (1) = (1 -t ) 2( a 1 + ϵ 2 )( a t + ϵ 2 ) /ϵ 1 ϵ 2 (A11)  \nwhere ϵ = ϵ 1 + ϵ 2 . Now, we define the SU (2) partition function which is given by  \nZ SU (2) inst ( t,⃗a, m 1 , m 2 , m 3 , m 4 ; ϵ 1 , 2 ) = Z -1 U (1) ( t, m 1 , m 2 , m 3 ; ϵ 1 , 2 ) Z U (2) inst ( t,⃗a, m 1 , m 2 , m 3 , m 4 ; ϵ 1 , 2 ) (A12)  \nMoreover, to work in Nekrasov-Shatashvili (NS) limit, we consider ϵ 2 → 0 while keeping ϵ 1 fixed (we set ϵ 1 = 1 ). Then, the instanton part of NS free energy is defined as  \nF ( N f =4) inst ( a, m i , t ) = lim ϵ 2 → 0 ( ϵ 2 log Z SU (2) inst ( a, m i , t ; ϵ 2 ) ) (A13)  \nThis can be rewritten as [9, 20, 23]  \nF (4) inst ( a, m i , t ) = lim ϵ 2 → 0 ϵ 2 log [ (1 -t ) -2 ϵ -1 2 ( a 1 + 1 2 )( a t + 1 2 ) ∑ ⃗ Y t | ⃗ Y | z vec ( ⃗a, ⃗ Y ) z hyp ( ⃗a, ⃗ Y , m i ) ] (A14)  \nAn interesting property of NS-free energy is that it is a convergent series in t . Hence, it can be written as  \nF (4) inst ( a, m i , t ) = ∞ ∑ n ≥ 1 c n ( a, m 1 , m 2 , m 3 , m 4 ) t n (A15)  \nThe expansion up to the order t 2 is given below;  \nF (4) inst ( a ; m i ; t ) = [ 1 8 ( 1 -4 a 2 ) -1 2 ( m 1 m 2 + m 3 m 4 ) -m 1 m 2 m 3 m 4 2 ( a -1 2 ) + m 1 m 2 m 3 m 4 2 ( a + 1 2 ) ] t + [ 1 128 ( 9 -26 a 2 ) -( m 1 + m 2 ) 2 64 -( m 3 + m 4 ) 2 64 -7 32 ( m 1 m 2 + m 3 m 4 ) -( 1 -4 m 2 1 ) ( 1 -4 m 2 2 ) ( 1 -4 m 2 3 ) ( 1 -4 m 2 4 ) 2048 ( a -1) + ( 1 -4 m 2 1 ) ( 1 -4 m 2 2 ) ( 1 -4 m 2 3 ) ( 1 -4 m 2 4 ) 2048 ( a +1) -m 2 1 m 2 2 ( m 2 3 + m 2 4 ) + m 2 3 m 2 4 ( m 2 1 + m 2 2 ) +4 m 1 m 2 m 3 m 4 (1 -m 1 m 2 m 3 m 4 ) 16 ( a -1 2 ) + m 2 1 m 2 2 ( m 2 3 + m 2 4 ) + m 2 3 m 2 4 ( m 2 1 + m 2 2 ) +4 m 1 m 2 m 3 m 4 (1 -m 1 m 2 m 3 m 4 ) 16 ( a + 1 2 ) + m 2 1 m 2 2 m 2 3 m 2 4 16 ( a -1 2 ) 3 -m 2 1 m 2 2 m 2 3 m 2 4 16 ( a + 1 2 ) 3 ] t 2 + O ( t 3 )  \n(A16)", 'B Angular Equation': 'By two consecutive transformations χ = sin 2 θ , and u = χ/ ( χ -χ 0 ) , with χ 0 = (1 -˜ a 2 1 ) / (˜ a 2 2 -˜ a 2 1 ) , we can take the four singular points of (2.9b) to be located at  \nu = 0 , u = 1 , u = u 0 = ˜ a 2 1 -˜ a 2 2 1 -˜ a 2 2 , u = ∞ , (B1)  \nand the indicial exponents are  \nα ± 0 = ± m 1 2 , α ± 1 = 1 2 (2 ± θ ∞ ) , α ± u 0 = ± m 2 2 , α ± ∞ = ± 1 2 ς, (B2)  \nwith  \nθ ∞ = ∆ -2 , ς = (˜ ω + m 1 ˜ a 1 + m 2 ˜ a 2 ) . (B3)  \nBy introducing the following transformation  \nS ( u ) = u α + 0 ( u 0 -u ) α + u 0 (1 -u ) α + 1 y ( u ) , (B4)  \nwe bring the angular equation to the canonical Heun form  \nd 2 y du 2 + ( 1 + m 1 u + 1 + m 2 u -u 0 + ∆ -1 u -1 ) dy du + ( q 1 q 2 u ( u -1) -u 0 ( u 0 -1) Q 0 u ( u -1)( u -u 0 ) ) y ( u ) = 0 , (B5)  \nwhere q 1 , q 2 and the accessory parameter Q 0 are definend as  \nq 1 = 1 2 ( m 1 + m 2 +∆ -ς ) , q 2 = 1 2 ( m 1 + m 2 +∆+ ς ) , (B6a)  \n4 u 0 ( u 0 -1) Q 0 = ( ˜ ω 2 +˜ a 2 1 ∆(∆ -4) -λ ) 1 -˜ a 2 2 -u 0 [ ( m 2 + θ ∞ +1) 2 -m 2 2 -1 ] -( u 0 -1) [ ( m 1 + m 2 +1) 2 -ς 2 -1 ] . (B6b)', "C Retarded Green's function": "For completeness, we provide the asymptotic expansions of the s-wave retarded Green's functions for both the Kerr-AdS 5 black hole with equal rotation and the RN-AdS 5 black hole in the small ˜ r + limit, without applying the resummation procedure.", 'Kerr-AdS 5 :': 'G  \nret (˜ ω, ∆) = e -iπ ∆ Γ(2 -∆)Γ ( 1 2 (∆ -˜ ω -2) ) Γ ( 1 2 (∆ + ˜ ω -2) ) Γ(∆ -2) Γ ( 1 2 (2 -∆ -˜ ω ) ) Γ ( 1 2 (2 -∆+˜ ω ) ) { 1 + [ 3 4 ˜ ω ( 1 + α 2 ) 2 × ( ψ (0) ( 1 2 (∆ -˜ ω -2) ) -ψ (0) ( 1 2 (2 -∆ -˜ ω ) ) + ψ (0) ( 1 2 (2 -∆+˜ ω ) ) -ψ (0) ( 1 2 (∆ + ˜ ω -2) ) ) -( 1 + α 2 ) 2 ( 3˜ ω 2 -∆(∆ -4) ) ( ( 3˜ ω 2 -∆(∆ -4) ) 2 -˜ ω 2 ( (∆ -2) 2 -˜ ω 2 ) ) 8((3˜ ω 2 -∆(∆ -4)) 2 + ˜ ω 2 ((∆ -2) 2 -˜ ω 2 )) 2 π sin π ∆ (cos π ∆ -cos π ˜ ω ) + (∆ -2) ( 2(1 + α 4 )(∆ -3)(∆ -1) -α 2 ( 3˜ ω 2 -∆(∆ -4) ) -3(1 + α 4 )˜ ω 2 ) ((∆ -2) 2 -˜ ω 2 ) +2 α 2 (∆ -2) ] ˜ r 2 + } + O ( ˜ r 3 + )  \n(C1)', 'RN-AdS 5 :': 'G ret (˜ ω, ∆) = e -iπ ∆ Γ(2 -∆)Γ ( 1 2 (∆ -˜ ω -2) ) Γ ( 1 2 (∆ + ˜ ω -2) ) Γ(∆ -2) Γ ( 1 2 (2 -∆ -˜ ω ) ) Γ ( 1 2 (2 -∆+˜ ω ) ) { 1 + [ ( 3 ( 1 + q 2 ) ˜ ω -2 q ˜ e ) 2 ( ψ (0) ( 1 2 (2 -∆+˜ ω ) ) -ψ (0) ( 1 2 (∆ + ˜ ω -2) ) ) + (∆ -2) ( (1 + q 2 )(2(∆ -2) 2 -3˜ ω 2 ) -(1 + q 2 )(3˜ ω +2) + 2 q ˜ e (˜ ω +1) ) ((∆ -2) 2 -˜ ω 2 ) + ( ( (1 + q 2 )(3˜ ω 2 -∆(∆ -4)) -4 q ˜ e ˜ ω ) 8 -( 3 ( 1 + q 2 ) ˜ ω -2 q ˜ e ) 4 + 1 4 ( ((1+ q 2 )(3˜ ω 2 -∆(∆ -4)) -4 q ˜ e ˜ ω ) (˜ ω 2 -(∆ -2) 2 )((1+ q 2 )(˜ ω -q ˜ e ) 2 + q 4 (˜ ω 2 -˜ e 2 )) -1 ((1+ q 2 )(3˜ ω 2 -∆(∆ -4)) -4 q ˜ e ˜ ω ) ) ) ( ψ (0) ( 1 2 (2 -∆ -˜ ω ) ) + ψ (0) ( 1 2 (2 -∆+˜ ω ) ) -ψ (0) ( 1 2 (∆ -˜ ω -2) ) -ψ (0) ( 1 2 (∆ + ˜ ω -2) ) -4(∆ -2) ((∆ -2) 2 -˜ ω 2 ) ) ] ˜ r 2 + } + O ( ˜ r 3 + )  \n(C2)', 'References': "- [1] N. 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