Vector perturbations of Kerr-AdS5 and the Painlevé VI transcendent

Abstract

We analyze the Ansatz of separability for Maxwell equations in generically spinning, five-dimensional Kerr-AdS black holes. We find that the parameter µ introduced in [1] can be interpreted as apparent singularities of the resulting radial and angular equations. Using isomonodromy deformations, we describe a non-linear symmetry of the system, under which µ is tied to the Painlevé VI transcendent. By translating the boundary conditions imposed on the solutions of the equations for quasinormal modes in terms of monodromy data, we find a procedure to fix µ and study the behavior of the quasinormal modes in the limit of fast spinning small black holes.

QNMs of scalar fields on small Reissner-Nordström-AdS$\mathbf{_5}$ black holes

“…It is also interesting to study the hydrodynamical modes in RN-AdS 5 black holes, since scalar perturbations in this background have been introduced as dual CFT states of strongly correlated holographic models with finite density, for which the bound on the speed of sound can be violated [35]. The method presented here and in [31] works for higher spin perturbations on rotating black holes [30] and can be used to study those perturbations in RN-AdS 5 .…”

“…s z σ 0 = χ(σ; z 0 ). (33) Equations ( 27) and (33) can be interpreted as an overdetermined system for σ in terms of the parameters of the differential equation, namely z 0 in (20a), K 0 in (20b) and θ in (30), which will only admit solutions for specific (discrete) values of ω. As we will see in the following, these equations can be solved perturbatively in the small r + limit to provide us with the QNMs frequencies.…”

“…Similar methods have been employed to study scalar and vector perturbations in Kerr-AdS 5 backgrounds [14,30], and more recently in the near-extremal limit [31]. In four dimensions, rotating black holes in asymptotically flat and de-Sitter spacetimes were analyzed in [28,29,32].…”

Remarks on holographic models of the Kerr-AdS5 geometry

“…Also, short representations of the CFT Virasoro algebra are relevant for the calculation of QNMs at higher levels. We defer the study of other interesting black hole limits, as well as higher spin perturbations [61] to future work.…”

More on the SW-QNM correspondence

“…Quite recently, attempting exact WKB quantization techniques [20][21][22][23][24][25], a new astonishing gauge-gravity connection between the QNM spectral problem and quantum Seiberg-Witten (SW) curves [26][27][28] for N = 2 SYM theories was suggested and tested in the case of Kerr BHs in 4-d [29]. It is worth mentioning that similar problems have been studied using the monodromy properties of differential equations yielding similar results [30][31][32][33][34][35][36][37][38][39][40][41][42][43].…”