Confluent conformal blocks and the Teukolsky master equation

Quasinormal modes of usual, four dimensional, Kerr black holes are described by certain solutions of a confluent Heun differential equation. In this work, we express these solutions in terms of the connection matrices for a Riemann-Hilbert problem, which was recently solved in terms of the Painlev\'e V transcendent. We use this formulation to generate the small-frequency expansion for the angular spheroidal harmonic eigenvalue, and derive conditions on the monodromy properties for the radial modes. Using exponentiation, we relate the accessory parameter to a semi-classical conformal description and discuss the properties of the operators involved. For the radial equation, while the operators at the horizons have Liouville momenta proportional to the entropy intake, we find that spatial infinity is described by a Whittaker operator.


I. INTRODUCTION
The Kerr black hole is described by the metric, in Boyer-Lindquist coordinates [1]: where ∆ = r 2 − 2M r + a 2 = (r − r + )(r − r − ), Σ = r 2 + a 2 cos 2 θ, a = J M , referring to a solution of the four-dimensional, vacuum Einstein equations which is asymptotically flat and has mass M and angular momentum J = aM . It has two event horizons at r ± and the region r < r + cannot affect causally the region r > r + . The importance of this solution to the development of general relativity and all theories that generalize it can hardly be overestimated, since it is shown to be the most general vacuum metric with mass and angular momenta. Subsequent studies trying to reconcile its apparent simplicity with the multitude of processes which can in principle surround a black hole led to the concept of black hole entropy. The microscopic description of the latter for generic black holes remains an outstanding problem in theoretical physics. However, the significance of the Kerr metric goes beyond formal developments due to the various astrophysical applications of phenomena in the Kerr background, especially the experimental detection of the black hole ringdown after a black hole merging event [2], as well as the recent image of a supermassive black hole whose shadow region [3] gives strong evidence of the existence of an event horizon. As a matter of fact, both experiments are interpreted as a direct evidence of a Kerr black hole, and from the raw data the black hole parameters, such as M and a, are measured.
All of these phenomena underscore the importance of the study of fluctuations of the Kerr metric. Their evolution is described by the linearized Einstein equations, with the metric fluctuations decomposed into a linear (spin 0), vector (spin 1) and tensor (spin 2) parts. The resulting partial differential equations are linear, separable and the spin s solution can be written as a sum of solutions of two ordinary differential equations, 1 sin θ d dθ sin θ dS dθ + a 2 ω 2 cos 2 θ − 2aωs cos θ − (m + s cos θ) 2 sin 2 θ + s + λ S(θ) = 0, where K(r) = (r 2 + a 2 )ω − am, ∆ = r 2 − 2M r + a 2 = (r − r + )(r − r − ), Sec. V.

II. PREAMBLE: THE CONFLUENT HEUN EQUATION
Both equations (3) and (4) can be brought to the confluent Heun canonical form: The differential equation (6) has 3 singular points: two regular at z = 0 and z = t 0 and an irregular singular point of Poincaré rank 1 at z = ∞. Series expansions for the solutions y(z) at the regular points can be obtained from the Frobenius method. The point at infinity is trickier, because the solutions present the Stokes phenomenon: convergence is conditional to sectors of the complex plane, depending on the direction one takes the limit z → ∞. Near a regular singular point z i , the Frobenius method allows us, in general, to construct two solutions, whose local behavior is which we will call the local Frobenius solutions at z = z i . In general, one given Frobenius solution at z = z i will be expressed as a linear combination of the Frobenius solutions constructed at a different point z = z j . For a particular set of parameters in the differential equation (6), namely discrete values of the accessory parameter c t0 , there will be a solution which has definite behavior at both z = z i and z = z j , for instance: Finding the (discrete) values of c t0 for which such a y(z) exists will be referred to as the eigenvalue problem. The formulation of the eigenvalue problem at the irregular singular point z = ∞ is a bit more complicated and will be dealt with later. Let us briefly describe the solution to the eigenvalue problem proposed in [14]. The second order differential equation (6) can be cast as a first order matrix equation: where we introduced the fundamental matrix of solutions Φ(z): with y 1,2 (w) satisfying our original equation (6) and w 1,2 (z) related to y 1,2 (z) by differentiation and multiplication by a rational function: We note that any two solutions of (9) are related by right multiplication. We also note that one can change the value of α i at will by multiplication of the solution (10) by a factor i (z − z i ) 1 2 αi , with exception of the singular point at infinity.
The basis of the method is to see the parameter t in (9) as a gauge parameter in the space of flat holomorphic connections A(z, t), and to recover the differential equation (6) as we take t to t 0 . The usefulness of this deformation stems from the fact that we can translate conditions such as the quantization condition (68) in terms of gauge-invariant properties of (9), called monodromy data.

A. Monodromy data
Let us first describe the latter. The monodromy data associated to the matrix of solution Φ(z) of (9) is its behavior under analytical continuation around the singular points: which defines the monodromy matrix as the decomposition of the analytic continuation of each of the solutions in terms of themselves. As defined above, the matrices M i are independent of the homotopy class of the curve we choose for analytic continuation. The matrices M i are also independent on the sum of the indicial exponents at each singular point, the α i in (8), due to the fact that these can be changed by multiplication of a scalar function. For the irregular singular point z = ∞ there is a subtlety, due to the Stokes phenomenon. Let us follow [35] (see also [36] -and define sectors S k as in each the asymptotic solution for (9) is where θ i are defined asθ The analytic continuation of the solution Φ k (z) can be now described as the connection between Φ k (z) in different sectors: where S k are the Stokes matrices. By (14), so only two of the Stokes matrices are independent. It can be checked from the discussion that they have the structure where the parameters s 2k , s 2k+1 are called Stokes multipliers. It is customary to define the monodromy matrix at z = ∞ in the sector k = 2: with the corresponding matrices for generic k defined through the recursion M ∞ (k + 1) = S −1 k M ∞ (k)S k . The monodromy matrix M ∞ can be obtained from the Stokes matrices by and satisfies the relation With these definitions, we define the monodromy data ρ associated to the matrix equation (9) as the basis independent data in the matrices M i : It will be convenient to define the trace of M ∞ as an independent parameter 2 cos πσ = Tr M ∞ = 2 cos πθ ∞ + s 1 s 2 e −iπθ∞ .

B. Connection matrix and the quantization condition
We can now phrase the eigenvalue problem (8) in terms of monodromy data. Let us choose the fundamental solution at z = 0, Φ(z; z 0 = 0) with y 1 (z) and y 2 (z) in (10) constructed using the Frobenius method at z = 0.
where 1 2 (α 0 ±θ 0 ) are the eigenvalues of A 0 . It is clear that the monodromy matrix around z 0 = 0 for this basis is diagonal: The α 0 , abelian part of the monodromy can be removed by a "s-homotopic transformation" like (70) and can be taken to be zero. We therefore have that, in this basis of solutions M 0 = e iπθ0σ3 . The monodromy around z = t is likewise diagonal with the fundamental solution Φ(z; z 0 = t), but in terms of Φ(z, z 0 = 0) above is called the connection matrix between the singular points at z = 0 and z = t. Now, we can see that if the parameters in the matrix system (9) are such that the conditions (68) are satisfied, then the connection matrix C t0 is either lower triangular or upper triangular. Simple algebra shows that, if this is the case, then It can be checked that the converse is also true: if this trace property is satisfied, then C t0 is either lower or upper triangular. This is a condition to be satisfied when λ corresponds to the angular eigenvalue. Using the property (21), we have Tr M 0 M t = Tr M −1 ∞ , and, by the definition of σ above (23), we arrive at where we underscored the dependence of theσ parameter on λ, but in fact it depends on all parameters in (6). The condition (28) does not provide a full solution of the system, however, because it may involve non-normalizable solutions of the differential equation (6). In our applications below, it will be clear from the context which values of θ i lead to the proper modes.

C. The τ function and Painlevé V system
To calculate σ as a function of the differential equation parameters is a version of the Riemann-Hilbert problem, whose solution we will make use of. The idea goes back to the theory of isomonodromic deformations as introduced by [21][22][23], and is based on interpreting t as a gauge parameter. If we accompany (9) by its Lax pair: the existence of the mixed derivative ∂ z ∂ t Φ = ∂ t ∂ z Φ requires that A 0 and A t satisfy the Schlesinger equations: whose solution gives a one-parameter family of matrix systems with different values of t but the same monodromy data. Since A 0 and A t are now arbitrary, let us consider the generic differential equation satisfied by the first row of Φ(z) in (9) where λ is the root of A 12 (z) and µ = A 11 (z = λ). c t is related to λ and µ by The algebraic condition (33) tells us that the singularity at z = λ in (32) is an apparent one: the indicial equation gives integer exponents 0 and 2, and there is no logarithmic behavior due to (33). The monodromy matrix around z = λ is then trivial. The Schlesinger equations induce a flow to λ and µ, and the corresponding differential equation for λ is equivalent to the Painlevé V transcendent. The family of isomonodromic connections will include our original equation (6) if and note that, per (23),σ = σ − 1. These conditions are more conveniently written in terms of the Jimbo-Miwa-Ueno (JMU) τ function where we left explicitly the dependence of the JMU τ function on the monodromy data ρ due to its expansions [35], [25]. Therefore, (34) is The second condition (36) stems from the second derivative of the τ function, calculated using the Schlesinger equations and imposing (34). The left hand side can be related through the Toda equation [37] to a product of τ functions: where K V is independent of t and the ρ ± are related to ρ by simple shifts: Miwa's theorem [24] tells us that τ defined by (35) is analytic in t except at the critical points t = 0 and t = ∞. Therefore either factor of the numerator in (37) has to vanish. The proof of (37) is straightforward, from a fundamental solution Φ(z) one defines the derived solutions where σ + = 0 1 0 0 and σ − = 0 0 1 0 are nilpotent combinations of Pauli matrices. Given Φ ± (z), one can establish the Toda equation (37) by comparing the corresponding expressions for each τ function (35), and choosing p ± and q ± in order to keep the form of the new connection, defined through (9), maintain the partial fraction form at z = t and z = ∞. It is clear that the monodromy data of Φ ± (z) are related to that of Φ(z) by (38). Further algebraic manipulation shows that Given that the first line has a divergent limit λ → t, we conclude that we can substitute the second condition in (36) by the simpler one where the monodromy data is that of (6): whereas, in terms of ρ, the first condition in (36) is given by with the shift in ρ − analogous to that ofρ above.

D. The Nekrasov expansion
In this section we are going to drop the "hatted" notation in order not to overburden the formulas. The Nekrasov expansion of the Painlevé V τ -function is given by [25] Here ρ = {θ 0 , θ t , θ ∞ ; σ, s V } is the monodromy data. The definition of the parameter σ in terms of the Stokes parameters is given by (23), and we will discuss the parameter s V below. The function B V is analytic near t = 0 and closely related to the irregular conformal blocks of the first kind [32,34]. It is based on the Nekrasov expansion, which a scalar function associated to a pair of Young diagrams λ, µ, a complex parameter b, as well as a complex number α: where a λ (i, j) and l λ (i, j) are respectively the arm-length and the leg-length of the box (i, j) in the diagram λ. The parameter b is related to the central charge of the Virasoro algebra by c = 1 + 6Q 2 = 1 + 6(b + b −1 ) 2 (79). As it can be checked in [38], the expansion of irregular conformal block of the first kind is given by where ∆ i = Q 2 4 + P 2 i and B λ,µ is given by ratios of Nekrasov functions As stated in [25], the expansion of the τ -function for the Painlevé V near t = 0 is given in terms of c = 1 irregular conformal blocks. These are obtained taking b = √ −1 -and therefore Q = 0 -in the expressions above, as well as setting the parameters P i to the monodromy parameters: Coming back to (45), one can recognize in the B V expansion the terms of the same functions B λ,µ appearing in the expansion of the irregular conformal blocks (48): where, again, the sum runs over all pairs of Young diagrams (λ, µ), with each coefficient in the series given by the appropriate reduction of (48): and the the hook lenght is defined by h λ (i, j) = a λ (i, j) + l λ (i, j) + 1. The structure constants C V in (45) are rational products of Barnes functions where the Barnes function G(z) is defined by functional equation G(1 + z) = Γ(z)G(z) plus some convexity requirements. The functional equation is its only property required to recover the results in this paper.

E. Monodromy matrices
The parameter s V in (45) has a geometrical interpretation in terms of the monodromy data. Following [35,36], we will introduce an explicit representation for the monodromy matrices. Let The connection matrices C 0 and C t allow the following parametrization: with D t , D 0 and D diagonal matrices and The κ parameter in the monodromy matrix is related to the s V parameter in the Nekrasov expansion (45) by a string of gamma functions As a comment, the diagonal matrices D 0 and D t represent the ambiguity in diagonalizing M t and M 0 , which is in turn tied to the choice of normalization of the Frobenius basis y i (z; z 0 ) at each point. Likewise, C ∞ diagonalizes M ∞ and D represents the ambiguity in the basis normalization at ∞. The parameter κ (or s V ) then has the interpretation of the relative normalization between the system at ∞ and the system at 0, t, which is an isomonodromy invariant as can be checked from the asymptotic analysis like that in [35] or [36]. Alternatively, one can relate the s V = e iη to the relative twist between the "gluing" of the 3-point Riemann-Hilbert problem with monodromies {θ 0 , θ t , σ} -which is solved by hypergeometric functions -to the 2-point irregular Riemann-Hilbert problem {−σ, θ ∞ , s 1 , s 2 } -solved by confluent hypergometrics -as was defined in [38].

F. The accessory parameter for the confluent Heun equation
Solving (44) involves finding the root of the JMU τ function and then using the value of this root to find c t0 as the derivative of the logarithm of the shifted function. Given the structure of (45), it is interesting to write whereτ involves only the combinatorial expansion of the irregular conformal blocks (50) and ratios of Barnes functions which can be written in terms of Euler's gamma functions. The asymptotics ofτ is given by [35]: where parameter κ is as above. Theσ appearing in (60) is related to the monodromy parameter by the addition of an even integerσ = σ − 2p, p ∈ Z. This indeterminacy stems from the quasi-periodicity of the Nekrasov expansion (45) with respect to σ: This quasi-periodicity will impose a multi-valuedness in the monodromy parameters found by solving (42). The nontrivial zeros of τ are those ofτ , but, to work the asymptotics we have to make sure that the terms in the expansion (60) are indeed dominant. To that end, it is useful to define the variableκ = κt σ . Seen as a function ofκ and t,τ is meromorphic inκ and soτ (κ, t 0 ) = 0 can be inverted to giveκ( θ, σ; t 0 ). The quasi-periodicity means that, from one such solution, we can create a series labelled by the integer p: where Y ( θ, σ) is related to the string of gamma functions in (58), and X( θ, σ; t 0 ) is analytic, obtained by inverting (60). We quote the first three terms, valid if ℜσ > 0: X( θ, σ; t 0 ) = 1 + χ 1 t 0 + χ 2 t 2 0 + . . . + χ n t n 0 + . . . with and The value of p in (62) will be determined, later, by the requirement that the quantities have a sensible limit as t 0 → 0. For the accessory parameter (44), this ambiguity is just the shift on σ by an even integer, which will play no further role. In order to use (44) and find the accessory parameter, we must shift the monodromy parameters by one unit. A simple calculation using (58) yields:κ Now, using (44) and expanding theτ term, we find the asymptotic formula for the accessory parameter with the three first terms in the expansion given by , (67b) where we assumed ℜσ > 0. The corresponding expression for ℜσ < 0 can be obtained by sending σ → −σ. Higher order terms can be consistently computed using (45). Although the terms become increasingly complicated, we have the structure where the term k n is a rational function of the monodromy parameters, and analytic in the single monodromy parameters θ. As a function of σ it is meromorphic, with poles at integer values. The structure of the poles at order n is • poles of order 2n − 1 and below at σ = 0 and σ = ±2; • single poles at σ = ±3, . . . , ±(n + 1) -note that the structure of (67a) for negative σ is illusory, since it is only valid for ℜσ > 0; • analytic at σ = 1.
This structure mirrors that of the accessory parameter for the (non-confluent) Heun equation found in [15]. There, the structure was inherited from the corresponding structure of conformal blocks [39]. It seems that irregular conformal blocks display the same traits. It should be stressed that (42) and (44) are exact relations, even though their usefulness stems from our ability to compute the τ function for Painlevé V efficiently. Miwa's theorem [24] shows that the τ function is analytic in the whole complex plane except at t = 0 and t = ∞. Thus, the expansion (45) has infinite radius of convergence, even if it becomes exponentially hard to compute the higher order coefficients in t, due to their combinatorial nature. These limitations should be overcome by the Fredholm determinant formulation of the τ function proposed recently [38], which would be of great help for numerical studies.
At t = ∞, the expansion of the Painlevé V τ function is substantially more complicated. No general expansion exists, but formulas for t → ∞ along specific rays, such as arg t = 0, π/2, π, 3π/2 have been proposed, see [38] for a review as well as the relation between these expansions and the different types of irregular conformal blocks at c = 1.
In the application of interest in this work, however, the parameter t 0 depends on ω, which will be complex for the general case, therefore straying from these rays. We hope to study the large frequency asymptotic of the quasi-normal modes in the context presented here in future work.

III. SPHEROIDAL HARMONICS
We are interested in solutions of (6) which are regular at both the South and the North poles: which will place a restriction on the value of λ, allowing only a discrete set as possible values λ ℓ (s, m), ℓ ∈ N. Finding these correspond to the eigenvalue problem for the angular equation.
We are going to define the single monodromy parameters Upon the change of variables y(z) = (1 + cos θ) θt 0 /2 (1 − cos θ) θ0/2 S(θ), we bring the differential equation to a canonical confluent Heun form (6), with θ as above and Given the expansion (67a), it is a matter of direct substitution of the parameters of the spheroidal harmonic equation (69) and (71), using the quantization condition (28): The result is: which can be checked to agree with the literature [40] -see [8] for a thorough review. In order to recover the asymptotics, we chose j = ℓ + s + 1 in (28). As anticipated in [4], the minimum eigenvalue of ℓ is |s| and the azimuthal momentum is constrained so |m| ≤ ℓ.

IV. CONFORMAL BLOCKS AND THE RADIAL EQUATION
The Nekrasov expansion for some of the Painlevé τ functions has been interpreted in terms of c = 1 conformal blocks in [25,41]. The details of the structure stems from the AGT conjecture [18,20] and can be checked in the references. For Painlevé VI, the structure of the corresponding τ function is similar to (45), with "instanton sectors" labelled by n, and regular conformal blocks, defined as where V ∆i (z i ) are primary vertex operators, acting on the primary state |∆ j and its descendants (the Verma module built on the primary state) with an operator of dimension ∆ i and Π ∆ a projector onto the Verma module generated from |∆ (see [42] for details and notation). The conformal blocks are dependent on the Virasoro Algebra central charge c -which enters through the Kac-Shapovalov matrix of inner products of descendant states of |∆ . F can be seen to have the asymptotic expansion where again c = 1 + 6Q 2 . The higher order terms in t can be computed either recursively [39] or via the Nekrasov functions [19,20]. The correspondence between c = 1 conformal blocks and accessory parameters to Fuchsian ordinary differential equations have been stablished in [17]. On the other hand, in [31] it was outlined a method to compute the same accessory parameters using semi-classical blocks. These are obtained in the c → ∞ limit of the conformal blocks defined above. The analysis at the semi-classical limit is based on the property of exponentiation: where δ k , δ are obtained from a scaling procedure from ∆ k , ∆. With the parametrization Q = b + 1/b, we have It can be checked by applying the Virasoro algebra that the Verma module constructed from the "light" operator V (2,1) (z), with ∆ (2,1) = − 1 2 − 3b 2 4 has a null vector at level 2. Requiring that this vector decouples from correlation functions imply the condition When this condition is applied to correlation functions involving primary operators, we find Fuchsian differential equations, essentially due to the fact that the OPE between T (z) and primary operators at z i have no terms diverging faster than (z − z i ) −2 .
To describe irregular singular points we need to take confluent limits of two primary operators [32], which are associated to Whittaker modules of the Virasoro algebra, see [34] for a review and [38] for the relation between these conformal blocks to the asymptotics of Painlevé V. The confluent limit of the two colliding primary operators generates an non-primary operator, and in order to derive the corresponding Ward identity related to the null condition we will work with the Feigin-Fuchs representation of Liouville field theory, which can be seen to generate a central charge of c = 1 + 6Q 2 . The confluent primary vertex operator of rank 1, as defined in [33], is given by and by analogy with primary operators and Verma modules, V α,β (0) is associated to an Whittaker module of states. We find for singular terms of the OPE with the stress-energy tensor and then where ∆ = α(Q − α). The global conformal Ward identities on the correlation functions of N Whittaker operators follow from the commutation relations [L n , V α,β (z i )] for n = −1, 0, 1: Note that these expressions reduce to the well-known formulas involving primary operators if we take β i = 0. For the confluent Heun equation (6), the relevant conformal block has 3 insertions of primary operators and one with a non-trivial Whittaker operator: using the global conformal Ward identities to solve for ∂ zi G b and setting z 3 = 0, z 2 = 1 and z 1 = ∞, we find that the null vector condition is The semiclassical limit is obtained through the scaling: In this limit, the three insertions at 0, 1, ∞ become "heavy", and set the background over which the "light" operator V (2,1) (z 0 ) will induce fluctuations. Assuming exponentiation, the four-point function (84) should factorize as for the solution of (9) near z = ∞. The existence of this limit requires that the matrix B 0 is diagonal. The subleading term gives Since the diagonal terms of [B 1 , σ 3 ] vanish, the off-diagonal elements of A 0 + A t only alter the subleading O(1/z) terms of Φ k (z), and then preserve the asymptotic form of the wavefunction. The condition that the connection matrix between z = +i∞ and z = z 0 is lower triangular can be read from the explicit representation (57), Comparing to the analogue problem of finding quasinormal modes in Kerr-AdS 5 black holes [15], the usefulness of this expression is somewhat wanting. The lack of natural small parameters makes it difficult to study radial eigenmodes analytically. They can, however, be studied numerically. The isomonodromy method will most likely not be as fast as Leaver's method [43], but on the other hand we have more control on the analyticity of the functions involved. The investigation is under way and will be reported elsewhere.

V. DISCUSSION
In this work we considered the Teukolsky master equation eigenvalue problem tackled by the isomonodromy method. We have seen from a more general perspective the relationship between the ensuing confluent Heun equations and confluent conformal blocks, built on Whittaker modules. The mixture of classical complex analysis, integrable systems (through Riemann-Hilbert problems) and conformal blocks has been drawn some attention of late [45,46], and we have found in this paper that the eigenvalue problems for both the angular and radial equation can be cast in terms of monodromy data and solved by expansions of the Painlevé V τ function.
Using the Painlevé V small isomonodromic time expansion [25,38], we derived expansions for the spheroidal harmonic angular eigenvalue in terms of the frequency. We have verified heuristically the exponentiation property for semiclassical confluent conformal blocks (of the first type as defined in [32]) and used then to rederive the small t expansion of the composite monodromy parameter σ. In turn, this allowed us to interpret the radial equation as the null condition of a composition of two primary operators at the radial positions of the inner and outer horizon and a Whittaker operator seated at radial infinity. Curiously, the primary operators can be seen to have real Liouville momenta, in an "unitary" description of sorts.
Using the relation between the accessory parameter and the zero of the isomonodromic tau function, we have an effective way to compute the monodromy parameters -and thus the connection matrix -of the solutions of the confluent differential equation (6). This gives an effective algorithm to compute scattering data and solving the eigenvalue problem which is procedural. We are currently investigating methods to efficiently compute the quasinormal modes in the notoriously hard quasi-extremal regime (r − → r + ). While in all probability the method will not be as fast as existing numerical methods for computing quasinormal modes -see [7], the analytical properties of monodromy parameters make a precision study amenable, as well as enhancements in precision.
The basic ingredients involved in the analysis are the monodromy parameters and their relation to primary/Whittaker operators of a CFT. We have found for both the radial and angular equations that these monodromy parameters are associated to CFTs which can be considered unitary, and in the radial equation the Liouville momentum of the operator at the outer horizon is proportional to the entropy intake. The angular eigenvalue condition has again the interpretation of an equilibrium condition between the "angular" and "radial" systems, just like the lore in [15]. It is tempting to try to interpret (105) as small perturbations on a "macroscopic" black hole state -the CFT vacuum in this case, and by composing these perturbations arrive at some macroscopic state corresponding to a different black hole. Given the c = 1 interpretation of the process, we can perhaps count the difference in the number of states using known facts about the representation of Virasoro algebra. We leave these as enticing prospects for future work.