Before moving further on, let us step back and first review an analogous reasoning in AdS3 . In the case of asymptotically AdS3 spacetimes, the asymptotic symmetry algebra contains two Virasoro algebras. Also, one can define a two-dimensional flat cylinder at the boundary of AdS3 using the Fefferman-Graham theorem . One is then led to identify quantum gravity in AdS3 spacetimes with a two-dimensional CFT defined on the cylinder. The known examples of AdS/CFT correspondences involving AdS3 factors can be understood as a correspondence between an ultraviolet completion of quantum gravity on AdS3 and a specific CFT. The vacuum AdS3 spacetime is more precisely identified with the invariant vacuum of the CFT, which is separated with a mass gap of from the zero-mass black holes. Extremal black holes with AdS3 asymptotics, the extremal BTZ black holes , are thermal states in the dual CFT with one chiral sector excited and the other sector set to zero temperature. It was further understood in  that taking the near-horizon limit of the extremal BTZ black hole corresponds to taking the DLCQ of the dual CFT (see Section 3.2 for a review of the DLCQ procedure and [31, 151] for further supportive studies). The resulting CFT is chiral and has a frozen right sector.
Given the close parallels between the near-horizon geometry of the extremal BTZ black hole (121) and the near-horizon geometries of four-dimensional extremal black holes (25), it has been suggested in  that extremal black holes are described by a chiral limit of two-dimensional CFT. This assumption nicely accounts for the fact that only one Virasoro algebra appears in the asymptotic symmetry algebra and it is consistent with the conjecture that no non-extremal excitations are allowed in the near-horizon limit as we discussed earlier. Moreover, the assumption that the chiral half of the CFT originates from a limiting DLCQ procedure is consistent with the fact that there is no natural invariant geometry in the boundary conditions (111), which would be dual to the vacuum state of the CFT. Indeed, even in the three-dimensional example, the geometric dual to the vacuum state (the AdS3 geometry) does not belong to the phase space defined in the near-horizon limit of extremal black holes. It remains an enigma why there is no natural invariant geometry in gravity at all that is dual to the vacuum state.
Let us now take as an assumption that the near-horizon geometry of the extremal Kerr black hole is described by the left-sector of a CFT that we will denote as . The details of this CFT will depend on the ultraviolet completion of gravity, but these details will be (fortunately) unimportant here. Instead, we will show that one can account for the entropy using the universal properties of that CFT. First, we can identify a non-trivial temperature for the excited states. We saw in Section 2.6 that scalar quantum fields in the analogue of the Frolov–Thorne vacuum restricted to extremal excitations have the temperature (67). Individual modes are co-rotating with the black hole along . Since we identify the left-sector of the CFT with excitations along and the right sector is frozen, the CFT left-moving states are described by a thermal density matrix with temperatures
It is remarkable that applying blindly Cardy’s formula (90) using the central charge given in (129) and using the temperatures (138), one reproduces the extremal Bekenstein–Hawking black-hole entropy. This matching is clearly not a numerical coincidence. For any spinning extremal black hole of the theory (1), one can associate a left-moving Virasoro algebra of central charge given in (128). The black-hole entropy (50) is then similarly reproduced by Cardy’s formula (141). As remarkably, taking any higher curvature correction to the gravitational Lagrangian into account, one also reproduces the Iyer–Wald entropy (52) using Cardy’s formula, while the central charge (131) is computed (apparently) completely independently from the entropy!
One can easily be puzzled by the incredible matching (141) valid for virtually any extremal black hole and outside the usual Cardy regime, as discussed in Section 3.1. Indeed, there are no arguments for unitarity and modular invariance of the dual CFT. It might suggest that Cardy’s formula has a larger range of applicability than what has been proven so far. Alternatively, this might suggest the existence of a long string CFT, as reviewed in Section 3.3. Note also that the central charge depends on the black-hole parameters, such as the angular momentum or the electric charge. This is not too surprising since, in known AdS/CFT correspondences where the black hole contains an AdS3 factor in the near-horizon geometry, the Brown–Henneaux central charge  also depends on the parameters of the black hole because the AdS length is a function of the black hole’s charge .
Let us now add an additional dimension to the scope of microscopic models. It turns out that when electromagnetic fields are present, another CFT description is available. Instead of assigning the left-moving temperature as (138), one might instead emphasize that electrically-charged particles are immersed in a thermal bath with temperature , as derived in (70) in Section 2.6. Identifying the left sector of the dual field theory with a density matrix at temperature and assuming again no right excitations at extremality, we make the following assignment. As one can easily check, the entropy of the general Kerr–Newman–AdS black hole can be similarily reproduced, as shown in [79, 71, 78, 76, 82]. We will refer to the class of CFTs with Virasoro algebra (120) by the acronym . Note that the entropy matching does not depend on the scale of the Kaluza–Klein dimension , which is arbitrary in our analysis.
Finally, when two symmetries are present, one can apply a modular transformation mixing the two and one obtains a different CFT description for each choice of element. Indeed, we argued that the set of generators (119) obeys the Virasoro algebra with central charge (136). After performing an change of basis in the Boltzman factor (66), we deduce the temperature of the CFT and Cardy’s formula is similarly reproduced. We will denote the corresponding class of CFTs by the acronym .
Living Rev. Relativity 15, (2012), 11
This work is licensed under a Creative Commons License.