Remarks on holographic models of the Kerr-AdS$_{5}$ geometry

We study the low-temperature limit of scalar perturbations of the Kerr- AdS$_{5}$ black-hole for generic rotational parameters. We motivate the study by considering real-time holography of small black hole backgrounds. Using the isomonodromic technique, we show that corrections to the extremal limit can be encoded in the monodromy parameters of the Painlev\'e V transcendent, whose expansion is given in terms of irregular chiral conformal blocks. After discussing the contribution of the intermediate states to the quasi-normal modes, we perform a numerical analysis of the low-lying frequencies. We find not only that excited perturbations are superradiant, as expected from thermodynamical considerations, but also that the fundamental mode presents a dynamical phase transition, becoming unstable at specific small values of the asymptotic charges. We close by considering the holographic interpretation of the unstable modes and the decaying process.Comment: JHEP style, 51 pages, 8 figure