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

The right sector was taken to be frozen at extremality. The resulting chiral limit of the CFT with central charge sufficed to account for the extremal black-hole entropy.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 [53], we also assume the presence of a right-moving current algebra, whose zero eigenmode is constrained by the level matching condition

The level matching condition is consistent with the fact that the excitations are labeled by three instead of four conserved quantities. The CFT state is then assumed to be at a fixed chemical potential . This right-moving current algebra cannot be detected in the extremal near-horizon geometry in the same way that the right-moving Virasoro algebra cannot be detected, so its existence is conjectural (see, however, [67]). 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

between bulk modes and operators in the CFT. The structure of the scattering cross section depends on the conformal weights and charges of the operator. The normalization of the coupling is also important for the normalization of the cross section.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 [265], this behavior is related to the conformal weight as . One then infers that [160]

The values of the charges are simply the charges of the probe, where the charge follows from the matching condition (182). We don’t know any first-principle argument leading to the values of the right-moving chemical potential , the right-moving temperature and the left-moving conformal weight . We will deduce those values from matching the CFT absorption probability with the gravitational result.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

where . One can generalize this result to any axisymmetric perturbation of any vacuum five-dimensional near-horizon geometry [224]. Counter-examples exist in higher dimensions and for black holes in AdS [129]. 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 [209]

where are the coordinates of the left and right moving sectors of the CFT. At left and right temperatures and at chemical potentials an operator with conformal dimensions and charges has the two-point function which is determined by conformal invariance. From Fermi’s golden rule, the absorption cross section is [53, 106, 160] Performing the integral in (190), we obtainIn 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

The right conformal weight matches with (185), consistent with conformal invariance. The left conformal weight is natural for a spin field since . The value for is consistent with the temperature (180) and chemical potential (181). Indeed, since the left-movers span the direction of the black hole, we have . We then obtain after using the value (175). The value of is fixed by the matching. It determines one constraint between , and . However, there is a subtlety in the above matching procedure. The conformal weights and depend on through . This dependence cannot originate from since is introduced after the Fourier transform (190), while are already defined in (189). One way to introduce this dependence is to assume that there is a right-moving current algebra and that the dual operator has the zero-mode charge , which amounts to imposing the condition (182). (It is then also natural to assume that the chemical potential is , but the matching does not depend on any particular value for [53].) 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

where is the left-moving temperature of the , is the chemical potential defined in (144) and is the probe electric charge in units of the Kaluza–Klein circle length. The identification of the right-moving sector is unchanged except that now . One can trivially extend the matching with the family of CFTs conjectured to describe the (near-)extremal Kerr–Newman black hole.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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