Before moving further on, let us step back and first review an analogous reasoning in AdS_{3} [251]. In the
case of asymptotically AdS_{3} spacetimes, the asymptotic symmetry algebra contains two Virasoro
algebras. Also, one can define a two-dimensional flat cylinder at the boundary of AdS_{3} using the
Fefferman-Graham theorem [137]. One is then led to identify quantum gravity in AdS_{3} spacetimes with a
two-dimensional CFT defined on the cylinder. The known examples of AdS/CFT correspondences
involving AdS_{3} factors can be understood as a correspondence between an ultraviolet completion of
quantum gravity on AdS_{3} and a specific CFT. The vacuum AdS_{3} 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 AdS_{3}
asymptotics, the extremal BTZ black holes [28], 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 [30]
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 [30]
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 AdS_{3} 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

where is given in (67). The other quantities and defined in (61) are then better interpreted as being proportional to auxiliary chemical potentials. One can indeed rewrite the Boltzman factor (66) as where the left chemical potentials are defined asIt 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

as first shown in [156]. 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 AdS_{3} factor in the
near-horizon geometry, the Brown–Henneaux central charge [58] also depends on
the parameters of the black hole because the AdS length is a function of the black hole’s
charge [206].

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

The other quantities and defined in (61) are then better interpreted as being proportional to auxiliary chemical potentials. One can indeed rewrite the Boltzman factor (66) as where is the probe electric charge in units of the Kaluza–Klein length and the left chemical potentials are defined as We argued above that in the near-horizon region, excitations along the gauge-field direction fall into representations of the Virasoro algebra defined in (120). As supported by non-extremal extensions of the correspondence discussed in Sections 5 and 6, the left sector of the dual field theory can be argued to be the chiral half of a CFT. Remarkably, Cardy’s formula (90) with temperatures (142) and central charge (134) also reproduces the entropy of the Kerr–Newman black hole. When the angular momentum is identically zero, the black-hole entropy of the Reissner–Nordström black hole is then reproduced from Cardy’s formula with left central charge given in (134) and left temperature as originally obtained in [159]. 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
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