4.4 Microscopic counting of the entropy

In Section 4.3 we have shown the existence of an asymptotic Virasoro algebra at the boundary r = ∞ of the near-horizon geometry. We also discussed that the SL (2,ℝ ) symmetry is associated with zero charges. Following semi-classical quantization rules, the operators that define quantum gravity with the boundary conditions (111), (116View Equation), (115View Equation) form a representation of the Virasoro algebra and are in a ground state with respect to the representation of the SL (2,ℝ ) symmetry [251Jump To The Next Citation Point, 156Jump To The Next Citation Point]. A consistent theory of quantum gravity in the near-horizon region, if it can be defined at all, is therefore either a chiral CFT or a chiral half of a two-dimensional CFT. A chiral CFT is defined as a holomorphically-factorized CFT with zero central charge in one sector, while a chiral half of a 2d CFT can be obtained, e.g., after a chiral limit of a 2d CFT, see Section 3.2. We will see in Sections 5 and 6 that the description of non-extremal black holes favors the interpretation of quantum gravity in extremal black holes as the chiral half of a full-fledged two-dimensional CFT. Moreover, the applicability of Cardy’s formula as detailed later on also favors the existence of a two-dimensional CFT. Since the near-horizon geometry is obtained as a strict near-horizon limit of the original geometry, the CFT might be thought of as describing the degrees of freedom of the black-hole horizon.

Before moving further on, let us step back and first review an analogous reasoning in AdS3 [251]. 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 [137]. 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 SL (2,ℝ ) × SL (2,ℝ ) invariant vacuum of the CFT, which is separated with a mass gap of − c∕24 from the zero-mass black holes. Extremal black holes with AdS3 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 [30Jump To The Next Citation Point] 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 SL (2,ℝ ) right sector.

Given the close parallels between the near-horizon geometry of the extremal BTZ black hole (121View Equation) and the near-horizon geometries of four-dimensional extremal black holes (25View Equation), 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 SL (2,ℝ ) × SL (2,ℝ ) 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 SL (2,ℝ ) × SL (2,ℝ ) 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 2d CFT that we will denote as CFTJ. 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 (67View Equation). Individual modes are co-rotating with the black hole along ∂ϕ. Since we identify the left-sector of the CFT with excitations along ∂ ϕ and the right SL (2,ℝ) R sector is frozen, the CFT left-moving states are described by a thermal density matrix with temperatures

T = T , T = 0, (138 ) L ϕ R
where Tϕ is given in (67View Equation). The other quantities Te and Tm defined in (61View Equation) are then better interpreted as being proportional to auxiliary chemical potentials. One can indeed rewrite the Boltzman factor (66View Equation) as
( ) m − q μJ,e − q μJ,m exp − ℏ------e-L-----m--L-- , (139 ) TL
where the left chemical potentials are defined as
J,e Tϕ J,m T ϕ μ L ≡ − --, μL ≡ − ---. (140 ) Te Tm

It is remarkable that applying blindly Cardy’s formula (90View Equation) using the central charge cL = cJ given in (129View Equation) and using the temperatures (138View Equation), one reproduces the extremal Bekenstein–Hawking black-hole entropy

! 𝒮CFT = 𝒮ext, (141 )
as first shown in [156Jump To The Next Citation Point]. This matching is clearly not a numerical coincidence. For any spinning extremal black hole of the theory (1View Equation), one can associate a left-moving Virasoro algebra of central charge cL = cJ given in (128View Equation). The black-hole entropy (50View Equation) is then similarly reproduced by Cardy’s formula (141View Equation). As remarkably, taking any higher curvature correction to the gravitational Lagrangian into account, one also reproduces the Iyer–Wald entropy (52View Equation) using Cardy’s formula, while the central charge (131View Equation) is computed (apparently) completely independently from the entropy!

One can easily be puzzled by the incredible matching (141View Equation) 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 c = 3l∕2G3 [58] also depends on the parameters of the black hole because the AdS length l 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 (138View Equation), one might instead emphasize that electrically-charged particles are immersed in a thermal bath with temperature T χ = RχTe, as derived in (70View Equation) in Section 2.6. Identifying the left sector of the dual field theory with a density matrix at temperature T χ and assuming again no right excitations at extremality, we make the following assignment

T = T = R T , T = 0. (142 ) L χ χ e R
The other quantities Tϕ and Tm defined in (61View Equation) are then better interpreted as being proportional to auxiliary chemical potentials. One can indeed rewrite the Boltzman factor (66View Equation) as
( ) q − m μQ,ϕ − q μQ,m exp − ℏ-χ------L------m--L-- , (143 ) Tχ
where qχ = R χqe is the probe electric charge in units of the Kaluza–Klein length and the left chemical potentials are defined as
T T μQL,ϕ≡ − -χ-, μQL,m ≡ − -χ-. (144 ) Tϕ Tm
We argued above that in the near-horizon region, excitations along the gauge-field direction fall into representations of the Virasoro algebra defined in (120View Equation). 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 2d CFT. Remarkably, Cardy’s formula (90View Equation) with temperatures (142View Equation) and central charge (134View Equation) 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 𝒮 = πQ2 ext is then reproduced from Cardy’s formula with left central charge cL = cQ given in (134View Equation) and left temperature TL = Rχ ∕(2πQ ) 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 [79Jump To The Next Citation Point, 71Jump To The Next Citation Point, 78Jump To The Next Citation Point, 76, 82]. We will refer to the class of CFTs with Virasoro algebra (120View Equation) by the acronym CFTQ. Note that the entropy matching does not depend on the scale of the Kaluza–Klein dimension R χ, which is arbitrary in our analysis.

Finally, when two U(1) symmetries are present, one can apply a modular transformation mixing the two U (1) and one obtains a different CFT description for each choice of SL (2,ℤ ) element. Indeed, we argued that the set of generators (119View Equation) obeys the Virasoro algebra with central charge (136View Equation). After performing an SL (2,ℤ) change of basis in the Boltzman factor (66View Equation), 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 CFT (p1,p2,p3).

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