The working assumption of the microscopic model is that the near-horizon region of any near-extremal spinning black hole can be described and therefore effectively replaced by a dual two-dimensional CFT. In the dual CFT picture, the near-horizon region is removed from the spacetime and replaced by a CFT glued along the boundary. Therefore, it is the near-horizon region contribution alone that we expect to be reproduced by the CFT. The normalization defined in (164) will then be dictated by the explicit coupling between the CFT and the asymptotically-flat region.
Remember from the asymptotic symmetry group analysis in Section 4.1 and 4.3 that boundary conditions were found where the exact symmetry of the near-horizon extremal geometry can be extended to a Virasoro algebra as
We will now assume that quantum gravity states form a representation of both a left and a right-moving Virasoro algebra with generators and . The value of the right-moving central charge will be irrelevant for our present considerations. At near-extremality, the left sector is thermally excited at the extremal left-moving temperature (67). We take as an assumption that the right-moving temperature is on the order of the infinitesimal reduced Hawking temperature. As discussed in Sections 2.9 and 4.2, the presence of right-movers destabilize the near-horizon geometry. For the Kerr–Newman black hole, we have
In order to match the bulk scattering amplitude for near-extremal Kerr–Newman black holes, the presence of an additional left-moving current algebra is required [106, 160]. This current algebra is expected from the thermodynamic analysis of charged rotating extremal black holes. We indeed obtained in Section 2.6 and in Section 4.4 that such black holes are characterized by the chemical potential defined in (140) associated with the electric current. Using the expressions (74), we find for the Kerr–Newman black hole the value
As done in , we also assume the presence of a right-moving current algebra, whose zero eigenmode is constrained by the level matching condition). This right-moving current algebra and the matching condition (182) will turn out to be adequate to match the gravitational result, as detailed below. Note that three-dimensional analogues of this level matching condition appeared in logically independent analyses [95, 94].
Therefore, under these assumptions, the symmetry group of the CFT dual to the near-extremal Kerr–Newman black hole is given by the product of a current and a Virasoro algebra in both sectors,
In the description where the near-horizon region of the black hole is replaced by a CFT, the emission of quanta is due to couplings
The conformal weight can be deduced from the transformation of the probe field under the scaling (24) in the overlap region . The scalar field in the overlap region is . Using the rules of the AdS/CFT dictionary , this behavior is related to the conformal weight as . One then infers that 
In general, the weight (185) will be complex and real weight will not be integers. However, a curious fact, described in [129, 224], is that for any axisymmetric perturbation () of any integer spin of the Kerr black hole, the conformal weight (185) is an integer. Counter-examples exist in higher dimensions and for black holes in AdS . There is no microscopic accounting of this feature at present.
Throwing the scalar at the black hole is dual to exciting the CFT by acting with the operator . Reemission is represented by the action of the Hermitian conjugate operator. Therefore, the absorption probability is related to the thermal CFT two-point function [53, 106, 160] 15
In order to compare the bulk computations to the CFT result (191), we must match the conformal weights and the reduced momenta . The gravity result (178) agrees with the CFT result (191) if and only if we choose.) This justifies why a right-moving current algebra was assumed in the CFT. The dependence of the conformal weights in is similarly made possible thanks to the existence of the left-moving current with . The matching is finally complete.
Now, let us notice that the matching conditions (193) – (194) are “democratic” in that the roles of angular momentum and electric charge are put on an equal footing, as noted in [79, 71]. One can then also obtain the conformal weights and reduced left and right frequencies using alternative CFT descriptions such as the with Virasoro algebra along the gauge field direction, and the mixed family of CFTs. We can indeed rewrite (192) in the alternative form
In summary, near-superradiant absorption probabilities of probes in the near-horizon region of near-extremal black holes are exactly reproduced by conformal field theory two-point functions. This shows the consistency of a CFT description (or multiple CFT descriptions in the case where several symmetries are present) of part of the dynamics of near-extremal black holes. We expect that a general scattering theory around any near-extremal black-hole solution of (1) will also be consistent with a CFT description, as supported by all cases studied beyond the Kerr–Newman black hole [106, 73, 242, 71, 80, 46].
Finally, let us note finally that the dynamics of the CFTs dual to the Kerr–Newman geometry close to extremality can be further investigated by computing three-point correlation functions in the near-horizon geometry, as initiated in [40, 39].
Living Rev. Relativity 15, (2012), 11
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