On the Kerr-AdS/CFT correspondence

Abstract

We review the relation between four-dimensional global conformal blocks and field propagation in AdS 5 . Following the standard argument that marginal perturbations should backreact in the geometry, we turn to the study of scalar fields in the generic Kerr-AdS 5 geometry. On one hand, the result for scattering coefficients can be obtained exactly using the isomonodromy technique, giving exact expressions in terms of c = 1 chiral conformal blocks. On the other hand, one can use the analogy between the scalar field equations to the Level 2 null field Ward identity in two dimensional Liouville field theory to write approximate expressions for the same coefficients in terms of semi-classical chiral Liouville conformal blocks. Surprisingly, the conformal block thus constructed has a well-behaved interpretation in terms of Liouville vertex operators.

Scalar quasinormal modes of Kerr−AdS5

“…As derived in [1], the converse is also true: if the composite monodromy is given by (2.38), then the monodromy matrices M i and M j are either both lower or upper triangular. We note that this formulation views the problem of finding eigenvalues for the angular equation similar in spirit to finding the quasinormal frequencies for the radial equation.…”

“…The functions described in this Section will be the main ingredient to compute the separation constant C j and the quasinormal modes, which will be the focus of next Section. A more extensive discussion of the strategy can be found in [1]. Let us begin by rewriting the Heun equation in the standard form as a first order differential equation.…”

“…The usefulness of the PVI τ function for the solution of the scattering and quasinormal modes for the scalar AdS perturbations is based on the relation between the scattering coefficients and the monodromy data [10,12]. For the quasinormal modes, the relationship was shown in [1]. Succintly, it states that conditions like (2.23) and (2.24) require that the relative connection matrix between the Frobenius solutions constructed at the singular points to be upper or lower triangular.…”

Spin-2 quasinormal modes in generalized Nariai spacetimes

“…In order to obtain the spectrum of the even part of the gravitational perturbation it was not necessary to use all field equations displayed in Eq. (24). More precisely, we have not solved E + tt = 0, E − xx = 0, and E I φnφn = 0 for n > 2.…”

“…Rotating black holes in four dimensions were tackled in the seminal works of Teukolsky [22,23]. Recently, some techniques based on monodromy calculations have been put forward to obtain analytical expressions for the quasinormal spectrum of perturbations in five-dimensional Kerr background [24]. However, the latter spectrum is written in terms of transcendental equations whose solutions must be found numerically [25].…”