Abstract
We present a firstprinciples derivation of the main results of the Kerr/CFT correspondence and its extensions using only tools from gravity and quantum field theory, filling a few gaps in the literature when necessary. Firstly, we review properties of extremal black holes that imply, according to semiclassical quantization rules, that their nearhorizon quantum states form a centrallyextended representation of the onedimensional conformal group. This motivates the conjecture that the extremal Kerr and ReissnerNordström black holes are dual to the chiral limit of a twodimensional CFT. We also motivate the existence of an SL(2, ℤ) family of twodimensional CFTs, which describe in their chiral limit the extremal KerrNewman black hole. We present generalizations in antide Sitter spacetime and discuss other mattercoupling and higherderivative corrections. Secondly, we show how a nearchiral limit of these CFTs reproduces the dynamics of nearsuperradiant probes around nearextremal black holes in the semiclassical limit. Thirdly, we review how the hidden conformal symmetries of asymptoticallyflat black holes away from extremality, combined with their properties at extremality, allow for a microscopic accounting of the entropy of nonextremal asymptoticallyflat rotating or charged black holes. We conclude with a list of open problems.
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1 Introduction
It is known since the work of Bardeen, Bekenstein, Carter and Hawking [42, 32, 162] that black holes are thermodynamical systems equipped with a temperature and an entropy. In analogy to Bolzmann’s statistical theory of gases, one expects that the entropy of black holes counts microscopic degrees of freedom. Understanding what these degrees of freedom actually are is one of the main challenges that a theory of quantum gravity should address.
Since the advent of string theory, many black holes enjoying supersymmetry have been understood microscopically. In many cases, supersymmetry and its nonrenormalization theorems allow one to map the blackhole states to dual states in a weaklycoupled description, which also provides a method to microscopically reproduce Hawking radiation; see [253, 60] and subsequent work. For all supersymmetric black holes that contain in their nearhorizon limit a factor of threedimensional antide Sitter spacetime AdS_{3} or a quotient thereof, a simpler microscopic model is available. Since quantum gravity in asymptotically AdS_{3} geometries is described by a twodimensional conformal field theory (2d CFT) [58, 251], one can account for the entropy and the Hawking radiation of these supersymmetric or nearly supersymmetric black holes using only the universal properties of a dual CFT description defined in the nearhorizon region [209, 104] (for reviews, see [155, 113]). Ultraviolet completions of these AdS/CFT correspondences can be constructed using string theory [205, 265].
These results can be contrasted with the challenge of describing astrophysical black holes that are nonsupersymmetric and nonextremal, for which these methods cannot be directly applied. Astrophysical black holes are generically rotating and have approximately zero electromagnetic charge. Therefore, the main physical focus should be to understand the microstates of the Kerr black hole and to a smaller extent the microstates of the Schwarzschild, the KerrNewman and the ReissnerNordström black hole.
Recently, considerable progress has been made in reproducing the entropy of the Kerr black hole as well as reproducing part of its gravitational dynamics using dual field theories that share many properties with twodimensional CFTs [156, 53, 68] (see also [104]).^{Footnote 1} The Kerr/CFT correspondence will be the main focus of this review. Its context is not limited to the sole Kerr black hole. Indeed, it turns out that the ideas underlying the correspondence apply as well to a large class of black holes in supergravity (in four and higher dimensions) independently of the asymptotic region (asymptoticallyflat, antide Sitter…) far from the black hole. These extensions of the Kerr/CFT correspondence only essentially require the presence of a U(1) axial symmetry associated with angular momentum. It is important to state that at present the Kerr/CFT correspondence and its extensions are most understood for extremal and nearextremal black holes. Only sparse but nontrivial clues point to a CFT description of black holes away from extremality [104, 68, 107].
Before jumping into the theory of black holes, it is important to note at the outset that rotating extremal black holes might be of astrophysical relevance. Assuming exactly zero electromagnetic charge, the bound on the Kerr angular momentum derived from the cosmiccensorship hypothesis is J ≤ GM^{2}. No physical process exists that would turn a nonextremal black hole into an extremal one. Using details of the accretion disk around the Kerr black hole, Thorne derived the bound J ≤ 0.998 GM^{2} assuming that only reasonable matter can fall into the black hole [258]. Quite surprisingly, it has been claimed that several known astrophysical black holes, such as the black holes in the Xray binary GRS 1905+105 [218] and Cygnus X1 [152], are more than 95% close to the extremality bound. Also, the spintomasssquare ratio of the supermassive black holes in the active galactic nuclei MCG63015 [57] and 1H 0707495 [134] has been claimed to be around 98%. However, these measurements are subject to controversy since independent data analyses based on different assumptions led to opposite results as reviewed in [138]: the spintomasssquare ratio of the very same black hole in the Xray binary GRS 1905+105 has been evaluated as J/(GM^{2}) = 0.15 [182], while the spin of the black hole in Cygnus X1 has been evaluated as J/(GM^{2}) = 0.05 [219]. If the measurements of high angular momenta are confirmed or if precise measurements of other nearby highlyspinning black holes can be performed, it would promote extremal black holes as “nearly physical” objects of nature.
In this review, we will present a derivation of the arguments underlying the Kerr/CFT correspondence and its extensions starting from firstprinciples. For that purpose, it will be sufficient to follow an effective field theory approach based solely on gravity and quantum field theory. In particular, we will not need any detail of the ultraviolet completions of quantum gravity except for one assumption (see Section 1.1 for a description of the precise classes of gravitational theories under study). We will assume that the U(1) electromagnetic field can be promoted to be a KaluzaKlein vector of a higherdimensional spacetime (see Section 1.2 for some elementary justifications and elaborations on this assumption). If this assumption is correct, it turns out that the Kerr/CFT correspondence can be further generalized using the U(1) electric charge as a key quantity instead of the U(1) angular momentum [159]. We will use this assumption as a guiding principle to draw parallels between the physics of static charged black holes and rotating black holes. Our point of view is that a proper understanding of the concepts behind the Kerr/CFT correspondence is facilitated by studying in parallel staticcharged black holes and rotating black holes.
Since extremal black holes are the key objects of study, we will spend a large amount of time describing their properties in Section 2. We will contrast the properties of static extremal black holes and of rotating extremal black holes. We will discuss how one can decouple the nearhorizon region from the exterior region. We will then show that one can associate thermodynamical properties with any extremal black hole and we will argue that nearhorizon geometries contain no local bulk dynamics. Since we aim at drawing parallels between black holes and twodimensional CFTs, we will quickly review some of their most relevant properties for our concerns in Section 3.
After this introductory material, we will discuss the core of the Kerr/CFT correspondence starting from the microscopic counting of the entropy of extremal black holes in Section 4. There, we will show how the nearhorizon region admits a set of symmetries at its boundary, which form a Virasoro algebra. Several choices of boundary conditions exist, where the algebra extends a different compact U(1) symmetry of the black hole. Following semiclassical quantization rules, the operators, which define quantum gravity in the nearhorizon region, form a representation of the Virasoro algebra. We will then argue that nearhorizon quantum states can be identified with those of a chiral half of a twodimensional CFT. This thesis will turn out to be consistent with the description of nonextremal black holes. The thermodynamical potential associated with the U(1) symmetry will then be interpreted as the temperature of the density matrix dual to the black hole. The entropy of the black hole will finally be reproduced from the asymptotic growth of states in each chiral half of these CFTs via Cardy’s formula.
In Section 5 we will move to the description of nonextremal black holes, and we will concentrate our analysis on asymptoticallyflat black holes for simplicity. We will describe how part of the dynamics of probe fields in the nearextremal KerrNewman black hole can be reproduced by correlators in a family of dual CFTs with both a left and a rightmoving sector. The leftmoving sector of the CFTs will match with the corresponding chiral limit of the CFTs derived at extremality. In Section 6 we will review the hidden local conformal symmetry that is present in some probes around the generic KerrNewman black hole. We will also infer from the breaking of this conformal symmetry that the KerrNewman black hole entropy can be mapped to states of these CFTs at specific left and rightmoving temperatures. Finally, we will summarize the key results of the Kerr/CFT correspondence in Section 7 and provide a list of open problems. This review complements the lectures on the Kerr black hole presented in [54] by providing an overview of the Kerr/CFT correspondence and its extensions for general rotating or charged black holes in gravity coupled to matter fields in a larger context. Since we follow an effective fieldtheory approach, we will cover stringtheory models of black holes only marginally. We refer the interested reader to the complementary string theoryoriented review of extremal black holes [244].^{Footnote 2}
1.1 Classes of effective field theories
The Kerr/CFT correspondence is an effective description of rotating black holes with an “infrared” CFT. Embedding this correspondence in string theory has the potential to give important clues on the nature of the dual field theory. Efforts in that direction include [225, 22, 157, 98, 23, 116, 31, 243, 246, 115, 130]. However, the details of particular CFTs are irrelevant for the description of astrophysical black holes, as long as we don’t have a reasonable control of all realistic embeddings of the standard model of particle physics and cosmology in string theory. Despite active research in this area, see, e.g., [153, 117, 217], a precise description of how our universe fits in to the landscape of string theory is currently outofreach.
In an effective field theory approach, one concentrates on longrange interactions, which are described by the physical EinsteinMaxwell theory. However, it is instructive in testing ideas about quantum gravity models of black holes to embed our familiar EinsteinMaxwell theory into the larger framework of supergravity and study the generic properties of rotating black holes as toy models for a physical string embedding of the KerrNewman black hole.
Another independent motivation comes from the AdS/CFT correspondence [205, 265]. Black holes in antide Sitter (AdS) spacetime in d + 1 dimensions can be mapped to thermal states in a dual CFT or CFT in d dimensions. Studying AdS black holes then amounts to describing the dynamics of the dual stronglycoupled CFT in the thermal regime. Since this is an important topic, we will discuss in this review the AdS generalizations of the Kerr/CFT correspondence as well. How the Kerr/CFT correspondence fits precisely in the AdS/CFT correspondence in an important open question that will be discussed briefly in Section 7.2.
In this review we will consider the following class of fourdimensional theories,
possibly supplemented with Plancksuppressed higherderivative corrections. We focus on the case where f_{AB}(χ) and k_{IJ}(χ) are positive definite and the scalar potential V(χ) is nonpositive in (1). This ensures that matter obeys the usual energy conditions and it covers the case of zero and negative cosmological constant. Some theories of interest contained in the general class (1) are the EinsteinMaxwell gravity with negative or zero cosmological constant and the bosonic sector of \({\mathcal N} = 8\) supergravity. Note that the phenomenology described by the action (1) is limited by the absence of charged scalars, massive vectors, nonabelian gauge fields and fermions.
The explicit form of the most general singlecenter spinningblackhole solution of the theory (1) is not known;however, see [270, 221] for general ansätze. For Einstein and EinsteinMaxwell theory, the solutions are, of course, the Kerr and KerrNewman geometries that were derived about 45 years after the birth of general relativity. For many theories of theoretical interest, e.g., \({\mathcal N} = 8\) supergravity, the explicit form of the spinningblackhole solution is not known, even in a specific Uduality frame (see, e.g., [49] and references therein). However, as we will discuss in Section 2.3, the solution at extremality greatly simplifies in the nearhorizon limit due to additional symmetries and takes a universal form for any theory in the class (1). It is for this reason mainly that we find convenient to discuss theory (1) in one swoop.
1.2 Gauge fields as KaluzaKlein vectors
Since the work of Kaluza and Klein, one can conceive that our U(1) electromagnetic gauge field could originate from a KaluzaKlein vector of a higherdimensional spacetime of the form \({{\mathcal M}_4} \times X\), where \({{\mathcal M}_4}\) is our spacetime, X is compact and contains at least a U(1) cycle (the total manifold might not necessarily be a direct product). Experimental constraints on such scenarios can be set from bounds on the deviation of Newton’s law at small scales [197, 2].
If our U(1) electromagnetic gauge field can be understood as a KaluzaKlein vector, it turns out that it is possible to account for the entropy of the ReissnerNordström black hole in essentially the same way as for the Kerr black hole [159]. This mainly follows from the fact that the electric charge becomes an angular momentum J_{2} = Q in the higherdimensional spacetime, which is on the same footing as the fourdimensional angular momentum J_{1} = J lifted in the higherdimensional spacetime.
Assumption We will assume throughout this review that the U(1) electromagnetic gauge field can be promoted as a KaluzaKlein vector.
As far as the logic goes, this assumption will not be required for any reasoning in Section 2, even though it will help to understand striking similarities between the effects of rotation and electric charge. The assumption will be a crucial input in order to formulate the ReissnerNordström/CFT correspondence and its generalizations in Section 4 and further on. This assumption is not required for the Kerr/CFT correspondence and its (extremal or nonextremal) extensions, which are exclusively based on the U(1) axial symmetry of spinning black holes.
In order to make this idea more precise, it is important to study simple embeddings of the U(1) gauge field in higherdimensional spacetimes as toy models for a realistic embedding. In asymptoticallyflat spacetimes, let us introduce a fifth compact dimension χ ∼ χ + 2πR_{χ}, where 2πR_{χ} is the length of the U(1) KaluzaKlein circle and let us define
The metric (2) does not obey fivedimensional Einstein’s equations unless the metric is complemented by matter fields. One simple choice consists of adding a U(1) gauge field A_{(5)}, whose field strength is defined as
where ⋆_{(4)} is the fourdimensional Hodge dual. The fivedimensional metric and gauge field are then solutions to the fivedimensional EinsteinMaxwellChernSimons theory, as reviewed, e.g., in [185].
These considerations can also be applied to black holes in antide Sitter spacetimes. However, the situation is more intricate because no consistent KaluzaKlein reduction from five dimensions can give rise to the fourdimensional EinsteinMaxwell theory with cosmological constant [204]. As a consequence, the fourdimensional KerrNewmanAdS black hole cannot be lifted to a solution of any fivedimensional theory. Rather, embeddings in elevendimensional supergravity exist, which are obtained by adding a compact sevensphere [69, 109].
Therefore, in order to review the arguments for the ReissnerNordström/CFT correspondence and its generalizations, it is necessary to discuss fivedimensional gravity coupled to matter fields. We will limit our arguments to the action (1) possibly supplemented by the ChernSimons terms
where C_{IJK} = C_{(IJK)} are constants. This theory will suffice to discuss in detail the embedding (2)–(3) since the fivedimensional EinsteinMaxwellChernSimons theory falls into that class of theories. We will not discuss the supergravities required to embed AdSEinsteinMaxwell theory.
Let us finally emphasize that even though the scale R_{χ} of the KaluzaKlein direction is arbitrary as far as it allows one to perform the uplift (2), it is constrained by matter field couplings. For example, let us consider the toy model of a probe charged massive scalar field ϕ(x) of charge q_{e} in four dimensions, which is minimally coupled to the gauge field. The wave equation reads as
where the derivative is defined as D_{α} = ∂_{α} − iq_{e}A_{α}. This wave equation is reproduced from a fivedimensional scalar field ϕ_{(5d)} (x, χ) probing the fivedimensional metric (2), if one takes
and if the fivedimensional mass is equal to \(\mu _{(5d)}^2 = {\mu ^2} + q_e^2\). However, the fivedimensional scalar is multivalued on the circle χ unless
This toy model illustrates that the scale R_{χ} can be constrained from consistent couplings with matter. We will use this quantization condition in Section 6.4.
2 Extremal Black Holes as Isolated Systems
In this section, we review some key properties of extremal black holes in the context of fourdimensional theories of gravity coupled to matter. In one glance, we show that the nearhorizon regions of extremal black holes are isolated geometries, isolated thermodynamical systems and, more generally, isolated dynamical systems. We first contrast how to decouple from the asymptotic region the nearhorizon region of static and rotating black holes. We then derive the thermodynamic properties of black holes at extremality. We finally discuss uniqueness of nearhorizon geometries and their lack of local bulk dynamics.
2.1 Properties of extremal black holes
For simplicity, we will strictly concentrate our analysis on stationary black holes. Since the Kerr/CFT correspondence and its extensions are only concerned with the region close to the horizon, one could only require that the nearhorizon region is stationary, while radiation would be allowed far enough from the horizon. Such a situation could be treated in the framework of isolated horizons [11, 10] (see [14] for a review). However, for our purposes, it will be sufficient and much simpler to assume stationarity everywhere. We expect that all results derived in this review could be generalized for isolated horizons (see [268] for results along these lines).
Many theorems have been derived that characterize the generic properties of fourdimensional stationary black holes that admit an asymptoticallytimelike Killing vector. First, they have one additional axial Killing vector — they are axisymmetric^{Footnote 3} — and their event horizon is a Killing horizon^{Footnote 4}. In asymptoticallyflat spacetimes, black holes have spherical topology [163].
Extremal black holes are defined as stationary black holes with vanishing Hawking temperature,
Equivalently, extremal black holes are defined as stationary black holes whose inner and outer horizons coincide.
No physical process is known that would make an extremal black hole out of a nonextremal black hole.^{Footnote 5} If one attempts to send finelytuned particles or waves into a nearextremal black hole in order to further approach extremality, one realizes that there is a smaller and smaller window of parameters that allows one to do so when approaching extremality. In effect, a nearextremal black hole has a potential barrier close to the horizon, which prevents it from reaching extremality. Also, if one artificially continues the parameters of the black holes beyond the extremality bound in a given solution, one typically obtains a naked singularity instead of a black hole. Such naked singularities are thought not to be reachable, which is known as the cosmic censorship hypothesis. Extremal black holes can then be thought of as asymptotic or limiting black holes of physical black holes. The other way around, if one starts with an extremal black hole, one can simply throw in a massive particle to make the black hole nonextremal. Therefore, extremal black holes are finely tuned black holes. Nevertheless, as we will discuss, studying the extremal limit is very interesting because many simplifications occur and powerful specialized methods can be used.
Extremal spinning or charged rotating black holes enjoy several interesting properties that we will summarize below. In order to be selfcontained, we will also first provide some properties of generic (extremal or nonextremal) black holes. We refer the reader to the excellent lecture notes [259] for the derivation of most of these properties.

Angular velocity. Spinning black holes are characterized by a chemical potential — the angular velocity Ω_{J} — conjugate to the angular momentum. The angular velocity can be defined in geometrical terms as the coefficient of the blackholehorizon generator proportional to the axial Killing vector
$$\xi = {\partial _t} + {\Omega _J}{\partial _\phi}.$$(9)The net effect of the angular velocity is a framedragging effect around the black hole. This gravitational kinematics might be the clue of an underlying microscopic dynamics. Part of the intuition behind the extremal spinning black hole/CFT correspondence is that the degrees of freedom responsible for the black hole entropy are rotating at the speed of light at the horizon.

Electrostatic potential. Electricallycharged black holes are characterized by a chemical potential — the electrostatic potential Φ_{e} — conjugated to the electric charge. It is defined on the horizon r = r_{+} as
$$\Phi _e^I =  {\xi ^\mu}A_\mu ^I{\vert _{r = {r_ +}}},$$(10)where ξ is the horizon generator defined in (9). Similarly, one can associate a magnetic potential \(\Phi _m^I\) to the magnetic monopole charge. The form of the magnetic potential can be obtained by electromagnetic duality, or reads as the explicit formula derived in [99] (see also [91] for a covariant expression). Part of the intuition behind the extremal charged black hole/CFT correspondence is that this kinematics is the sign of microscopic degrees of freedom “moving along the gauge direction”. We will make that statement more precise in Section 4.1.

Ergoregion. Although the Killing generator associated with the mass of the black hole, ∂_{t}, is timelike at infinity, it does not need to be timelike everywhere outside the horizon. The region where ∂_{t} is spacelike is called the ergoregion and the boundary of that region where ∂_{t} is lightlike is the ergosphere. If there is no ergoregion, ∂_{t} is a global timelike Killing vector outside the horizon. However, it should be noted that the presence of an ergoregion does not preclude the existence of a global timelike Killing vector. For example, the extremal spinning KerrAdS black hole has an ergoregion. When the horizon radius is smaller than the AdS length, the horizon generator becomes spacelike at large enough distances and there is no global timelike Killing vector, as for the Kerr black hole. On the contrary, when the horizon radius is larger than the AdS length, the horizon generator is timelike everywhere outside the horizon and is therefore a global timelike Killing vector.

Superradiance. One of the most fascinating properties of some rotating black holes is that neutral particles or waves sent towards the black hole with a frequency ω and angular momentum m inside a specific band
$$0 < \omega < m{\Omega _J}$$(11)come back to the exterior region with a higher amplitude. This amplification effect or Penrose effect allows the extraction of energy very efficiently from the black hole. Superradiance occurs for the Kerr and KerrNewman black hole and is related to the presence of the ergoregion and the lack of a global timelike Killing vector. Because of the presence of a global timelike Killing vector, there is no superradiance for large KerrAdS black holes (when reflective boundary conditions for incident massless waves are imposed) [165, 264].

Electromagnetic analogue to superradiance. Charged black holes contain electrostatic energy that can also be extracted by sending charged particles or waves with frequency ω and charge q_{e} inside a specific band [84] (see [177] for a review)
$$0 < \omega < {q_e}{\Phi _e}.$$(12)There is no ergoregion in the fourdimensional spacetime. However, for asymptoticallyflat black holes, there is a fivedimensional ergoregion when considering the uplift (2). For the ReissnerNordström black hole, the fivedimensional ergoregion lies in the range r_{+} < r < 2M, where M is the mass and r the standard BoyerLindquist radius.
The combined effect of rotation and charge allows one to extract energy in the range
$$0 < \omega < m{\Omega _J} + {q_e}{\Phi _e}.$$(13)When considering a wave scattering off a black hole, one can define the absorption probability σ_{abs} or macroscopic greybody factor as the ratio between the absorbed flux of energy at the horizon and the incoming flux of energy from infinity,
$${\sigma _{{\rm{abs}}}} = {{d{E_{{\rm{abs}}}}/dt} \over {d{E_{{\rm{in}}}}/dt}}.$$(14)In the superradiant range (13), the absorption probability is negative because the outgoing flux of energy is higher than the incoming flux.

No thermal radiation but spontaneous emission. Taking quantum mechanical effects into account, nonextremal black holes radiate with a perfect blackbody spectrum at the horizon at the Hawking temperature T_{h} [162]. The decay rate of a black hole as observed from the asymptotic region is the product of the blackbody spectrum decay rate with the greybody factor σ_{abs},
$$\Gamma = {1 \over {{e^{{{\omega  m{\Omega _J}  {q_e}{\Phi _e}} \over {{T_H}}}}}  1}}{\sigma _{{\rm{abs}}}}.$$(15)The greybody factor accounts for the fact that waves (of frequency ω, angular momentum m and electric charge q_{e}) need to travel from the horizon to the asymptotic region in the curved geometry. In the extremal limit, the thermal factor becomes a step function. The decay rate then becomes
$${\Gamma _{{\rm{ext}}}} =  \Theta ( \omega + m{\Omega _J} + {q_e}{\Phi _e}){\sigma _{{\rm{abs}}}}.$$(16)As a consequence, ordinary Hawking emission with σ_{abs} > 0 and ω > mΩ_{J} + q_{e}Φ_{e} vanishes while quantum superradiant emission persists. Therefore, extremal black holes that exhibit superradiance, spontaneously decay to nonextremal black holes by emitting superradiant waves.

Innermost stable orbit approaching the horizon in the extremal limit. Nearextremal black holes have an innermost stable circular orbit (ISCO) very close to the horizon. (In BoyerLindquist coordinates, the radius of such an orbit coincides with the radius of the horizon. However, since the horizon is a null surface, while the ISCO is timelike, the orbit necessarily lies outside the horizon, which can be seen explicitly in more appropriate coordinates. See Figure 2 of [34]^{Footnote 6}). As a consequence, the region of the black hole close to the horizon can support accretion disks of matter and, therefore, measurements of electromagnetic waves originating from the accretion disk of nearextremal rotating black holes contain (at least some marginal) information from the nearhorizon region. For a careful analysis of the physical processes around rotating black holes, see [34]. See also [154] for a recent discussion.

Classical singularities approaching the horizon in the extremal limit. Stationary axisymmetric nonextremal black holes admit a smooth inner and outer horizon, where curvatures are small. However, numerical results [52, 50, 51, 112] and the identification of unstable linear modes using perturbation theory [220, 125, 124] showed that the inner horizon is unstable and develops a curvature singularity when the black hole is slightly perturbed. The instability is triggered by tiny bits of gravitational radiation that are blueshifted at the inner Cauchy horizon and which create a null singularity. In the nearextremality limit, the inner horizon approaches the outer horizon and it can be argued that test particles encounter a curvature singularity immediately after they enter the horizon of a nearextremal black hole [212].
2.2 Nearhorizon geometries of static extremal black holes
As a warmup, let us first review the nearhorizon limit of static extremal black holes. In that case, the generator of the horizon (located at r = r_{+}) is the generator of time translations ∂_{t} and the geometry has SO(3) rotational symmetry. Since the horizon generator is null at the horizon, the coordinate t diverges there. The nearhorizon limit is then defined as
with λ → 0. The scale r_{0} is introduced for convenience in order to factor out the overall scale of the nearhorizon geometry. In the presence of electrostatic potentials, a change of gauge is required when taking the nearhorizon limit (17). Indeed, in the nearhorizon coordinates (17) the gauge fields take the following form,
where \(\Phi _e^I\) is the static electric potential of the gauge field A^{I}. Upon taking the nearhorizon limit one should, therefore, perform a gauge transformation A^{I} → A^{I} + dΛ^{I} of parameter
where \(\Phi _e^{I,{\rm{ext}}}\) is the static electric potential at extremality.
It is important to note that one is free to redefine the nearhorizon limit parameter λ as λ → αλ for any α > 0. This transformation scales r inversely proportionally to t. Therefore, the nearhorizon geometry admits the enhanced symmetry generator
in addition to ζ_{−1} = ∂_{t} and the SO(3) symmetry generators. Using the properties of static horizons, one can further derive an additional symmetry generator at the horizon ζ_{1}, which together with ζ_{−1} and ζ_{0} forms a SL(2, ℝ) algebra. This argument is purely kinematical and does not involve the field equations; see, e.g., [194] for a detailed derivation. The general nearhorizon solution compatible with an SL(2, ℝ) × SO(3) symmetry is then given by
where v_{1}, v_{2}, \(\chi _\ast^A\), e_{I}, p^{I} are parameters, which are constrained by the equations of motion. The geometry consists of the direct product AdS_{2} × S^{2}.
For some supersymmetric theories, the values v_{1}, v_{2}, \(\chi _\ast^A\), e_{I} are generically completely fixed by the electric (q^{I}) and magnetic (p^{I}) charges of the black hole and do not depend continuously on the asymptotic value of the scalar fields in the asymptotic region — the scalar moduli. This is the attractor mechanism [141, 250, 140]. It was then realized that it still applies in the presence of certain higherderivative corrections [199, 200, 198]. The attractor mechanism was also extended to nonsupersymmetric extremal static black holes [139, 240, 150, 179]. As a consequence of this mechanism, the entropy of these extremal black hole does not depend continuously on any moduli of the theory.^{Footnote 7} The index that captures the entropy can still have discrete jumps when crossing walls of marginal stability in the scalar moduli space [227, 118]. This allows one to account for their blackhole entropy by varying the moduli to a weaklycoupled description of the system without gravity, where states with fixed conserved charges can be counted. Therefore, the attractor mechanism led to an explanation [18, 111] of the success of previous string theory calculations of the entropy of certain nonsupersymmetric extremal black holes [181, 172, 110, 260, 131, 132].
As will turn out to be useful in the development of the ReissnerNordström correspondence, let us discuss the nearhorizon geometry (21) under the assumption that one gauge field A can be lifted as a KaluzaKlein vector to a higherdimensional spacetime, as discussed in Section 1.2. In the simple model (2), the change of gauge A → A + dΛ is implemented as the change of coordinates χ → χ + Λ. Using the definition of the electrostatic potential \(\Phi _e^{{\rm{ext}}}\) (10) at extremality, it is straightforward to obtain that in the geometry (2) the horizon is generated by the vector field \({\xi _{{\rm{tot}}}} = {\partial _t} + \Phi _e^{{\rm{ext}}}{\partial _\chi}\). The change of coordinates (17) combined with χ → χ + Λ with Λ defined in (19) then maps this vector to
2.3 Nearhorizon of extremal spinning geometries
Let us now consider extremal spinning black holes. Let us denote the axis of rotation to be ∂_{ϕ}, where ϕ ∼ ϕ + 2π and let r = r_{+} be the blackhole horizon. The generator of the horizon is \(\xi \equiv {\partial _t} + \Omega _J^{{\rm{ext}}}{\partial _\phi}\) where \(\Omega _J^{{\rm{ext}}}\) is the extremal angular velocity. We choose a coordinate system such that the coordinate t diverges at the horizon, which is equivalent to the fact that g^{tt} diverges at the horizon. As in the static case, one needs to perform a gauge transformation of parameter (19), when electrostatic fields are present. One can again interpret this change of gauge parameter as a change of coordinates in a higherdimensional auxiliary spacetime (2). The nearhorizon limit is then defined as
with λ → 0. The scale r_{0} is again introduced in order to factor out the overall scale of the nearhorizon geometry. The additional effect with respect to the static nearhorizon limit is the shift in the angle in order to reach the frame comoving with the horizon. The horizon generator becomes ζ = λ/r_{0}∂_{t} in the new coordinates. Including the gauge field, one has precisely the relation (22). As in the static case, any finite energy excitation of the nearhorizon geometry is confined and amounts to no net charges in the original (asymptotically flat of AdS) geometry.
One is free to redefine λ as λ → αλ for any α > 0 and, therefore, the nearhorizon geometry admits the enhanced symmetry generator
in addition to ζ_{−1} = ∂_{t} and L_{0} = ∂_{ϕ}. Together ζ_{0} and ζ_{−1} form a noncommutative algebra under the Lie bracket.
Now, contrary to the static case, the existence of a third Killing vector is not guaranteed by geometric considerations. Nevertheless, it turns out that Einstein’s equations derived from the action (1) imply that there is an additional Killing vector ζ_{1} in the nearhorizon geometry [194, 19] (see also [64] for a geometrical derivation). The vectors ζ_{−1}, ζ_{0}, ζ_{1} turn out to obey the SL(2, ℝ) ∼ SO(2, 1) algebra. This dynamical enhancement is at the origin of many simplifications in the nearhorizon limit. More precisely, one can prove [194] that any stationary and axisymmetric asymptoticallyflat or antide Sitter extremal blackhole solution of the theory described by the Lagrangian (1) admits a nearhorizon geometry with SL(2, ℝ) × U(1) isometry. The result also holds in the presence of higherderivative corrections in the Lagrangian provided that the black hole is big, in the technical sense that the curvature at the horizon remains finite in the limit where the higherderivative corrections vanish. The general nearhorizon geometry of extremal spinning black holes consistent with these symmetries is given by
where Γ(θ) > 0, γ(θ) ≥ 0, χ^{A}(θ), f^{I}(θ) and k, e_{I} ∈ ℝ are fixed by the equations of motion. By inverting t and redefining A^{I} → −A^{I}, we can always set k ≥ 0, e_{I} ≥ 0. The function α(θ) ≥ 0 can be removed by redefining θ but it is left for convenience because some nearhorizon geometries are then more easily described.^{Footnote 8}
The term \( {{{e_I}} \over k}d\phi\) in (25) is physical since it cannot be gauged away by an allowed gauge transformation. For example, one can check that the nearhorizon energy \({{\mathcal Q}_{{\partial _t}}}\) would be infinite in the KerrNewman nearhorizon geometry if this term would be omitted. One can alternatively redefine f^{I}(θ) = b^{I}(θ) + e_{I}/k and the gauge field takes the form
The static nearhorizon geometry (21) is recovered upon choosing only SO(3) covariant quantities with a welldefined static limit. This requires k → 0 and it requires the form
where p^{I} are some pure numbers, which are the magnetic charges.
Going back to the spinning case, the SL(2, ℝ) × U(1) symmetry is generated by
In addition, the generator ζ_{1} should be accompanied by the gauge transformation of parameter Λ^{1} = −e_{I}/r so that \({{\mathcal L}_{{\zeta _1}}}A_\mu ^I + {\partial _\mu}{\Lambda ^I} = 0\). Note that all of these symmetries act within a threedimensional slice of fixed polar angle θ. The metric is also invariant under discrete symmetry, which maps
This is often called the tϕ reflection symmetry in blackhole literature. The parity/time reversal transformation (30) reverses the electromagnetic charges of the solution.
The geometry (25) is a warped and twisted product of AdS_{2} × S^{2}. The (r, t) coordinates are analogous to Poincaré coordinates on AdS_{2} with an horizon at r = 0. One can find global coordinates in the same way that the global coordinates of AdS_{2} are related to the Poincaré coordinates [33]. Let
The new axial angle coordinate φ is chosen so that dϕ + krdt = dφ + kydτ, with the result
In these new coordinates, the nearhorizon geometry becomes
after performing an allowed gauge transformation (as the change of gauge falls into the boundary conditions (115) derived in Section 4.1). Note that the τ= 0 hypersurface coincides with the t=0 hypersurface, and that ϕ = φ on this hypersurface. The geometry has two boundaries at y = −∞ and y = +∞.
Geodesic completeness of these geometries has not been shown in general, even though it is expected that they are geodesically complete. For the case of the nearhorizon geometry of Kerr, geodesic completeness has been proven explicitly in [33] after working out the geodesic equations.
At fixed polar angle θ, the geometry can be described in terms of 3d warped antide Sitter geometries; see [8] for a relevant description and [226, 158, 238, 127, 223, 175, 174, 6, 119, 43, 26, 93, 222] for earlier work on these threedimensional geometries. Warped antide Sitter spacetimes are deformations of AdS_{3}, where the S^{1} fiber is twisted around the AdS_{2} base. Because of the identification ϕ ∼ ϕ + 2π, the geometries at fixed θ are quotients of the warped AdS geometries, which are characterized by the presence of a Killing vector of constant norm (namely ∂_{ϕ}). These quotients are often called selfdual orbifolds by analogy to similar quotients in AdS_{3} [100].^{Footnote 9}
The geometries enjoy a global timelike Killing vector (which can be identified as ∂_{τ}) if and only if
if there is no global timelike Killing vector, there is at least one special value of the polar angle θ_{⋆}, where kγ(θ_{⋆}) = 1. At that special value, the slice θ = θ_{⋆} is locally an ordinary AdS_{3} spacetime and acquires a local SL(2, ℝ) × SL(2, ℝ) isometry. At all other values of θ, one SL(2, ℝ) is broken to U(1). Note that there is still a global time function for each nearhorizon geometry. Constant global time τ in the global coordinates (33) are spacelike surfaces because their normal is timelike,
Hence, there are no closed timelike curves.
One can show the existence of an attractor mechanism for extremal spinning black holes, which are solutions of the action (1) [17]. According to [17], the complete nearhorizon solution is generically independent of the asymptotic data and depends only on the electric charges \({\mathcal Q}_e^I\), magnetic charges \({\mathcal Q}_m^I\) and angular momentum \({\mathcal J}\) carried by the black hole, but in special cases there may be some dependence of the near horizon background on this asymptotic data. In all cases, the entropy only depends on the conserved electromagnetic charges and the angular momentum of the black hole and might only jump discontinuously upon changing the asymptotic values of the scalar fields, as it does for static charged black holes [227, 118].
One can generalize the construction of nearhorizon extremal geometries to higher dimensions. In five dimensions, there are two independent planes of rotation since the rotation group is a direct product SO(4) ∼ SO(3) × SO(3). Assuming the presence of two axial U(1) symmetries \({\partial _{{\phi _i}}}\), i = 1, 2 (with fixed points at the poles), one can prove [194] that the nearhorizon geometry of a stationary, extremal blackhole solution of the fivedimensional action (1) possibly supplemented by ChernSimons terms (4) is given by
In particular, the solutions obtained from the uplift (2)–(3) fall into this class. In general, these solutions can be obtained starting from both black holes (with S^{3} horizon topology) and black rings (with S^{2} × S^{2} horizon topology) [133].
2.4 Explicit nearhorizon geometries
Let us now present explicit examples of nearhorizon geometries of interest. We will discuss the cases of the extremal Kerr and ReissnerNordström black holes as well as the extremal KerrNewman and KerrNewmanAdS black holes. Other nearhorizon geometries of interest can be found, e.g., in [88, 121, 203].
2.4.1 Nearhorizon geometry of extremal Kerr
The nearhorizon geometry of extremal Kerr with angular momentum \({\mathcal J} = J\) can be obtained by the above procedure, starting from the extremal Kerr metric written in usual BoyerLindquist coordinates; see the original derivation in [33] as well as in [156, 54]. The result is the NHEK geometry, which is written as (25) without matter fields and with
The angular momentum only affects the overall scale of the geometry. There is a value \({\theta _\ast} = \arcsin (\sqrt 3  1) \sim 47\) degrees for which ∂_{t} becomes null. For θ_{⋆} < θ < π − θ_{⋆}, ∂_{t} is spacelike. This feature is a consequence of the presence of the ergoregion in the original Kerr geometry. Near the equator we have a “stretched” AdS_{3} selfdual orbifold (as the S^{1} fiber is streched), while near the poles we have a “squashed” AdS_{3} selfdual orbifold (as the S^{1} fiber is squashed).
2.4.2 Nearhorizon geometry of extremal ReissnerNordström
The extremal ReissnerNordström black hole is determined by only one parameter: the electric charge Q. The mass is \({\mathcal M} = Q\) and the horizon radius is r_{+} = r_{−} = Q. This black hole is static and, therefore, its nearhorizon geometry takes the form (21). We have explicitly
2.4.3 Nearhorizon geometry of extremal KerrNewman
It is useful to collect the different functions characterizing the nearhorizon limit of the extremal KerrNewman black hole. We use the normalization of the gauge field such that the Lagrangian is proportional to R − F_{ab}F^{ab}. The black hole has mass \({\mathcal M} = \sqrt {{a^2} + {Q^2}}\). The horizon radius is given by \({r_ +} = {r_ } = \sqrt {{a^2} + {Q^2}}\). One finds
In the limit Q → 0, the NHEK functions (37) are recovered. The nearhorizon geometry of extremal KerrNewman is therefore smoothly connected to the nearhorizon geometry of Kerr. In the limit a → 0 one finds the nearhorizon geometry of the ReissnerNordström black hole (38). The limiting procedure is again smooth.
2.4.4 Nearhorizon geometry of extremal KerrNewmanAdS
As a last example of nearhorizon geometry, let us discuss the extremal spinning charged black hole in AdS or KerrNewmanAdS black hole in short. The Lagrangian is given by L ∼ R + 6/l^{2} − F^{2} where l^{2} > 0. It is useful for the following to start by describing a few properties of the nonextremal KerrNewmanAdS black hole. The physical mass, angular momentum, electric and magnetic charges at extremality are expressed in terms of the parameters (M, a, Q_{e}, Q_{m}) of the solution as
where Ξ = 1 − a^{2}/l^{2} and \({Q^2} = Q_e^2 + Q_m^2\) The horizon radius r_{+}(r_{−}) is defined as the largest (smallest) root, respectively, of
Hence, one can trade the parameter M for r_{+}. If one expands Δ_{r} up to quadratic order around r_{+}, one finds
where Δ_{0} and r_{⋆} are defined by
In AdS, the parameter r_{⋆} obeys r_{−} ≤ r_{⋆} ≤ r_{+}, and coincides with r_{−} and r_{+} only at extremality. In the flat limit l → ∞, we have Δ_{0} → 1 and r_{⋆} → r_{−}. The Hawking temperature is given by
The extremality condition is then r_{+} = r_{⋆} = r_{−} or, more explicitly, the following constraint on the four parameters (r_{+}, a, Q_{e}, Q_{m}),
The nearhorizon geometry was obtained in [159, 71] (except the coefficient e given here). The result is
where we defined
The nearhorizon geometry of the extremal KerrNewman black hole is recovered in the limit l → ∞.
2.5 Entropy
The classical entropy of any black hole in Einstein gravity coupled to matter fields such as (1) is given by
where Σ is a crosssection of the blackhole horizon and G_{n} is the fourdimensional Newton’s constant. In the nearhorizon geometry, the horizon is formally located at any value of as a consequence of the definition (23). Nevertheless, we can move the surface Σ to any finite value of r without changing the integral, thanks to the scaling symmetry ζ_{0} of (29). Evaluating the expression (49), we obtain
In particular, the entropy of the extremal Kerr black hole is given by
In units of ℏ the angular momentum \({\mathcal J}\) is a dimensionless halfinteger. The main result [156, 203, 21, 159, 225, 83, 173, 22, 204, 97] of the extremal spinning black hole/CFT correspondence that we will review below is the derivation of the entropy (50) using Cardy’s formula (90).
When higher derivative corrections are considered, the entropy does not scale any more like the horizon area. The blackhole entropy at equilibrium can still be defined as the quantity that obeys the first law of blackhole mechanics, where the mass, angular momenta and other extensive quantities are defined with all higherderivative corrections included. More precisely, the entropy is first defined for nonextremal black holes by integrating the first law, and using properties of nonextremal black holes, such as the existence of a bifurcation surface [262, 176]. The resulting entropy formula is unique and given by
where ϵ_{ab} is the binormal to the horizon, i.e., the volume element of the normal bundle to Σ. One can define it simply as ϵ_{ab}=n_{a}ϵ_{b}− ξ_{a}n_{b}, where ξ is the generator of the horizon and n is an outgoing null normal to the horizon defined by n^{2} = 0 and n^{a}ξ_{a} = −1. Since the Lagrangian is diffeomorphism invariant (possibly up to a boundary term), it can be expressed in terms of the metric, the matter fields and their covariant derivatives, and the Riemann tensor and its derivatives. This operator δ^{cov}/δR_{abcd} acts on the Lagrangian while treating the Riemann tensor as if it were an independent field. It is defined as a covariant EulerLagrange derivative as
Moreover, the entropy formula is conserved away from the bifurcation surface along the future horizon as a consequence of the zeroth law of blackhole mechanics [178]. Therefore, one can take the extremal limit of the entropy formula evaluated on the future horizon in order to define entropy at extremality. Quite remarkably, the IyerWald entropy (52) can also be reproduced [20] using Cardy’s formula as we will detail below.
In fivedimensional Einstein gravity coupled to matter, the entropy of extremal black holes can be expressed as
where Γ(θ) and α(θ) have been defined in (36) and γ(θ)^{2} = det(γ_{ij}(θ)^{2}).
From the attractor mechanism for fourdimensional extremal spinning black holes [17], the entropy at extremality can be expressed as an extremum of the functional
where \({\mathcal L}\) is the Lagrangian. The entropy then only depends on the angular momentum \({\mathcal J}\) and the conserved charges \({\mathcal Q}_{e,m}^I\),
and depend in a discontinuous fashion on the scalar moduli [240]. The result holds for any Lagrangian in the class (1), including higherderivative corrections, and the result can be generalized straightforwardly to five dimensions.
When quantum effects are taken into account, the entropy formula also gets modified in a nonuniversal way, which depends on the matter present in quantum loops. In Einstein gravity, the main correction to the area law is a logarithmic correction term. The logarithmic corrections to the entropy of extremal rotating black holes can be obtained using the quantum entropy function formalism [241].
2.6 Temperature and chemical potentials
Even though the Hawking temperature is zero at extremality, quantum states just outside the horizon are not pure states when one defines the vacuum using the generator of the horizon. Let us review these arguments following [156, 159, 83]. We will drop the index I distinguishing different gauge fields since this detail is irrelevant to the present arguments.
From the expression of the entropy in terms of the charges \({{\mathcal S}_{{\rm{ext}}}}({\mathcal J},{{\mathcal Q}_e},{{\mathcal Q}_m})\), one can define the chemical potentials
. Note that electromagnetic charges are quantized, but when the charges are large one can use the continuous thermodynamic limit. These potentials obey the balance equation
Another way to obtain these potentials is as follows. At extremality, any fluctuation obeys
where \(\Omega _J^{{\rm{ext}}}\) is the angular potential at extremality and \(\Phi _{e,m}^{{\rm{ext}}}\) are electric and magnetic potentials at extremality; see Section 2.1 for a review of these concepts.
One can express the first law at extremality (58) as follows: any variation in \({\mathcal J}\) or \({{\mathcal Q}_{m,e}}\) is accompanied by an energy variation. One can then solve for \({\mathcal M} = {{\mathcal M}_{{\rm{ext}}}}({\mathcal J},{{\mathcal Q}_e},{{\mathcal Q}_m})\). The first law for a nonextremal black hole can be written as
Let us now take the extremal limit using the following ordering. We first take extremal variations with \(\delta {\mathcal M} = \delta {{\mathcal M}_{{\rm{ext}}}}({\mathcal J},{{\mathcal Q}_e},{{\mathcal Q}_m})\). Then, we take the extremal limit of the background configuration. We obtain (57) with
where the extremal limit can be practically implemented by taking the limit of the horizon radius r_{+} to the extremal horizon radius r_{ext}.
The interpretation of these chemical potentials can be made in the context of quantum field theories in curved spacetimes; see [47] for an introduction. The HartleHawking vacuum for a Schwarzschild black hole, restricted to the region outside the horizon, is a density matrix \(\rho = {e^{ \omega/{T_H}}}\) at the Hawking temperature T_{h}. For spacetimes that do not admit a global timelike Killing vector, such as the Kerr geometry, the HartleHawking vacuum does not exist, but one can use the generator of the horizon to define positive frequency modes and, therefore, define the vacuum in the region where the generator is timelike (close enough to the horizon). This is known as the FrolovThorne vacuum [144] (see also [128]). One can take a suitable limit of the definition of the FrolovThorne vacuum to provide a definition of the vacuum state for any spinning or charged extremal black hole.
Quantum fields for nonextremal black holes can be expanded in eigenstates with asymptotic energy \({\hat \omega}\) and angular momentum \({\hat m}\) with \({\hat t}\) and \({\hat \phi}\) dependence as \({e^{ i\hat \omega \hat t + i\hat m\hat \phi}}\). When approaching extremality, one can perform the change of coordinates (23) in order to zoom close to the horizon. By definition, the scalar field ϕ in the new coordinate system x^{a} = (t, ϕ, θ, r) reads in terms of the scalar field \({\hat \phi}\) in the asymptotic coordinate system \({{\hat x}^a} = (\hat t,\hat \phi ,\theta ,\hat r)\) as \(\phi ({x^a}) = \hat \phi ({{\hat x}^a})\). We can then express
and the nearhorizon parameters are
When no electromagnetic field is present, any finite energy ω in the nearhorizon limit at extremality λ → 0 corresponds to eigenstates with \(\hat \omega = \hat m\Omega _J^{{\rm{ext}}}\). When electric fields are present, zooming in on the nearhorizon geometry from a nearextremal solution requires one to perform the gauge transformation A(x) → A(x) + dΛ(x) with gauge parameter given in (19), which will transform the minimallycoupled charged scalar wavefunction by multiplying it by \({e^{i{q_e}\Lambda (x)}}\). Finite energy excitations in the nearhorizon region then require \(\hat \omega = m\Omega _e^{{\rm{ext}}} + {q_e}\Phi _e^{{\rm{ext}}}\). Invoking (classical) electromagnetic duality, the magnetic contribution has the same form as the electric contribution. In summary, the general finiteenergy extremal excitation has the form
Following Frolov and Thorne, we assume that quantum fields in the nonextremal geometry are populated with the Boltzmann factor
where \({{\hat q}_{e,m}}\) are the electric and magnetic charge operators. We also assume that modes obey (64) at extremality. Using the definitions (60)–(61), we obtain the nontrivial extremal Boltzmann factor in the extremal and nearhorizon limit
where the mode number m and charges q_{e,m} in the nearhorizon region are equal to the original mode number and charges \(\hat m,{{\hat q}_{e,m}}\). This completes the argument that the FrolovThorne vacuum is nontrivially populated in the extremal limit.
Now, as noted in [4], there is a caveat in the previous argument for the Kerr black hole and, as a trivial generalization, for all black holes that do not possess a global timelike Killing vector. For any nonextremal black hole, the horizongenerating Killing field is timelike just outside the horizon. If there is no global timelike Killing vector, this vector field should become null on some surface at some distance away from the horizon. This surface is called the velocity of light surface. For positiveenergy matter, this timelike Killing field defines a positive conserved quantity for excitations in the nearhorizon region, ruling out instabilities. However, when approaching extremality, it might turn out that the velocity of light surface approaches asymptotically the horizon. In that case, the horizongenerating Killing field of the extreme black hole may not be everywhere timelike. This causes serious difficulties in defining quantum fields directly in the nearhorizon geometry [183, 229, 228]. However, (at least classically) dynamical instabilities might appear only if there are actual bulk degrees of freedom in the nearhorizon geometries. We will argue that this is not the case in Section 2.9. As a conclusion, extremal FrolovThorne temperatures can be formally and uniquely defined as the extremal limit of nonextremal temperatures and chemical potentials. However, the physical interpretation of these quantities is better understood finitely away from extremality.
The condition for having a global timelike Killing vector was spelled out in (34). This condition is violated for the extremal Kerr black hole or for any extremal KerrNewman black hole with \(a \geq Q/\sqrt 3\), as can be shown by using the explicit values defined in (2.4). (The extremal KerrNewman nearhorizon geometry does possess a global timelike Killing vector when \(a < Q/\sqrt 3\) and the KerrNewmanAdS black holes do as well when \(4{a^2}/({\Delta _0}r_ + ^2) < 1\), which is true for large black holes with r_{+} ≫ l. Nevertheless, there might be other instabilities due to the electric superradiant effect.)
The extremal FrolovThorne temperatures should also be directly encoded in the metric (25). More precisely, these quantities should only depend on the metric and matter fields and not on their equations of motion. Indeed, from the derivation (60)–(61), one can derive these quantities from the angular velocity, electromagnetic potentials and surface gravity, which are kinematical quantities. More physically, the Hawking temperature arises from the analysis of free fields on the curved background, and thus depends on the metric but not on the equations of motion that the metric solves. It should also be the case for the extremal FrolovThorne temperatures. Using a reasonable ansatz for the general blackhole solution of (1), including possible higherorder corrections, one can derive [83, 20] the very simple formula
From similar considerations, it should also be possible to derive a formula for T_{e} in terms of the functions appearing in (25). We propose simply that
while we do not have a direct proof of the equivalence between (68) and (61), the formula is consistent with the thermodynamics of (AdS)KerrNewman black holes as one can check from the formulae in Section 2.4. It would be interesting to generalize the arguments of [83, 20] to prove the equivalence.
Similarly, one can work out the thermodynamics of fivedimensional rotating black holes. Since there are two independent angular momenta \({{\mathcal J}_1},{{\mathcal J}_2}\), there are also two independent chemical potentials \({T_{{\phi _1}}},{T_{{\phi _2}}}\) associated with the angular momenta. The same arguments lead to
where k_{1} and k_{2} are defined in the nearhorizon solution (36).
When considering the uplift (2) of the gauge field along a compact direction of length 2πR_{χ}, one can use the definition (69) to define the chemical potential associated with the direction ∂_{χ}. Since the circle has a length 2πR_{χ}, the extremal FrolovThorne temperature is expressed in units of R_{χ},
where T_{e} is defined in (68).
2.6.1 Temperatures and entropies of specific extremal black holes
The entropy of the extremal Kerr black hole is \({{\mathcal S}_{{\rm{ext}}}} = 2\pi J\). Integrating (57) or using the explicit nearhorizon geometry and using (67), we find
and T_{e} is not defined.
The entropy of the extremal ReissnerNordström black hole is \({{\mathcal S}_{{\rm{ext}}}} = \pi {Q^2}\). Integrating (57), we obtain
while T_{ϕ} is not defined.
For the electricallycharged KerrNewman black hole, the extremal entropy reads as \({{\mathcal S}_{{\rm{ext}}}} = \pi ({a^2} + {r^2})\). Expressing the entropy in terms of the physical charges \(Q = \sqrt {r_ + ^2  {a^2}}\) and J = ar_{+}, we obtain
Using (57) and reexpressing in terms of the parameters (a, r_{+}) we find
We can also derive T_{ϕ} from (67) and the explicit nearhorizon geometry (39). T_{e} is consistent with (68).
For the extremal KerrNewmanAdS black hole, the simplest way to obtain the thermodynamics at extremality is to compute (60)–(61). Using the extremality constraint (46), we obtain
where we used the definitions (48). The magnetic potential T_{m} can then be obtained by electromagnetic duality. The expressions coincide with (67)–(68). These quantities reduce to (74) in the limit of no cosmological constant when there is no magnetic charge, q_{m} = 0. The extremal entropy is given by \({{\mathcal S}_{{\rm{ext}}}} = \pi (r_ + ^2 + {a^2})/\Xi\).
2.7 Nearextremal nearhorizon geometries
An important question about nearhorizon geometries is the following: how much dynamics of gravity coupled to matter fields is left in a nearhorizon limit such as (23)? We will explore in the following Sections 2.8 and 2.9 several aspects of the dynamics in the nearhorizon limit. In this section, we will discuss the existence of nearextremal solutions obtained from a combined nearhorizon limit and zero temperature limit. We will discuss in Section 2.8 the absence of nonperturbative solutions in the nearhorizon geometries, such as black holes. In Section 2.9, we will argue for the absence of local bulk degrees of freedom, and finally in Section 4.4 we will discuss nontrivial boundary dynamics generated by large diffeomorphisms.
Let us first study infinitesimal perturbations of the nearhorizon geometry (25). As a consequence of the change of coordinates and the necessary shift of the gauge field (23), the nearhorizon energy \(\delta {{\mathcal Q}_{{\partial _t}}}\) of an infinitesimal perturbation is related to the charge associated with the generator of the horizon \({\xi _{{\rm{tot}}}} \equiv (\xi ,\Lambda) = ({\partial _t} + \Omega _J^{{\rm{ext}}}{\partial _\phi},\Phi _e^{{\rm{ext}}})\) as follows,
as derived in Sections 2.2 and 2.3. Assuming no magnetic charges for simplicity, the conserved charge \(\delta {{\mathcal Q}_{{\xi _{{\rm{tot}}}}}}\) is given by \(\delta {\mathcal M}  \Omega _J^{{\rm{ext}}}\delta {\mathcal J}  \Phi _e^{{\rm{ext}}}\delta {{\mathcal Q}_e}\).^{Footnote 10} Using the first law of thermodynamics valid for arbitrary (not necessarily stationary) perturbations, the lefthand side of (76) can be expressed as
Any geometry that asymptotes to (25) will have finite nearhorizon energy \({{\mathcal Q}_{{\partial _t}}}\). Indeed, an infinite nearhorizon energy would be the sign of infrared divergences in the nearhorizon geometry and it would destabilize the geometry. It then follows from (76)–(77) that any infinitesimal perturbation of the nearhorizon geometry (25) will correspond to an extremal blackhole solution with vanishing Hawking temperature, at least such that T_{H} = O(λ). Common usage refers to blackhole solutions, where T_{H} ∼ λ as nearextremal black holes. Nevertheless, it should be emphasized that after the exact limit λ → 0 is taken the Hawking temperature of such a solution is exactly zero.
We can obtain a nearextremal nearhorizon geometry as follows. Starting from a stationary nonextremal black hole of mass M in BoyerLindquist coordinates, we perform the nearhorizon scaling limit (23) together with the scaling of the temperature
while the form of the general nonextremal solution would be required to perform that limit in detail, all examples so far in the class of theories (1), such as the KerrNewmanAdS black hole, lead to the following metric
The nearextremal nearhorizon solution (79) is diffeomorphic to the nearhorizon geometry in Poincaré coordinates (25). Denoting the finite temperature coordinates by a subscript T and the Poincaré coordinates by a subscript P, the change of coordinates reads as [207, 247, 4, 53]
where
Therefore, the classical geometries are equivalent. However, since the diffeomorphism is singular at the boundary r_{F} → ∞, there is a distinction at the quantum level. Since the asymptotic time in nearextremal geometries (79) is different than in extremal geometries (25), fields will be quantized in a different manner in the two geometries.
Let us now compute the energy of these geometries. Multiplying Eq. (76) by r_{0}/λ and using (77) and (78), we get that the energy variation around the nearextremal geometry is given by
where the extremal entropy \({{\mathcal S}_{{\rm{ext}}}}\) can be expressed in terms of the nearhorizon quantities as (50). We denote the variation by \(\not \delta\) to emphasize that the energy is not the exact variation of a quantity unless T^{near−ext} is constant or \({{\mathcal S}_{{\rm{ext}}}}\) is fixed (which would then lead to zero energy). Therefore, the charge \(\not \delta{{\mathcal Q}_{{\partial _t}}}\) is a heat term, which does not define a conserved energy. Since our derivation of the formula (84) was rather indirect, we check that it is correct for the KerrNewmanAdS family of black holes by computing the energy variation directly using the Lagrangian charges defined in [36, 90, 97].
2.8 Uniqueness of stationary nearhorizon geometries
We reviewed in Section 2.3 that for any stationary extremal spinning black hole one can isolate a geometry in the vicinity of the horizon, which has enhanced symmetry and universal properties. We discussed in Section 2.7 that another class of stationary nearhorizon geometries can be defined, which are, however, related to the extremal nearhorizon geometries via a diffeomorphism. It is natural to ask how unique the stationary nearhorizon geometries are.
In the case of Einstein gravity, one can prove that the NHEK (nearhorizon extremal Kerr geometry) is the unique (up to diffeomorphisms) regular stationary and axisymmetric solution asymptotic to the NHEK geometry with a smooth horizon [4]. This can be understood as a Birkoff theorem for the NHEK geometry. This can be paraphrased by the statement that there are no black holes “inside” of the NHEK geometry. One can also prove that there is a nearhorizon geometry in the class (25), which is the unique (up to diffeomorphisms) nearhorizon stationary and axisymmetric solution of AdSEinsteinMaxwell theory [192, 193, 191]. The assumption of axisymmetry can be further relaxed since stationarity implies axisymmetry [170]. It is then natural to conjecture that any stationary solution of the more general action (1), which asymptotes to a nearhorizon geometry of the form (25) is diffeomorphic to it. This conjecture remains to be proven.
2.9 Absence of bulk dynamics in nearhorizon geometries
In this section, we will review arguments pointing to the absence of local degrees of freedom in the nearhorizon geometries (25), following the arguments of [4, 122] for Einstein gravity in the NHEK geometry. The only nontrivial dynamics can be argued to appear at the boundary of the nearhorizon geometries due to the action of nontrivial diffeomorphisms. The analysis of these diffeomorphisms will be deferred until Section 4.1.
One usually expects that conserved charges are captured by highlysymmetric solutions. From the theorems presented in Section 2.8, we infer that in (AdS)EinsteinMaxwell theory there is no candidate nontrivial nearhorizon solution charged under the SL(2, ℝ) × U(1) symmetry (× U(1) symmetry when electric charge is present), except for a solution related via a diffeomorphism to the nearhorizon geometry. If the conjecture presented in Section 2.8 is correct, there is no nontrivial candidate in the whole theory (1). One can then argue that there will be no solution — even nonstationary — with nonzero mass or angular momentum (or electric charge when a Maxwell field is present) above the background nearhorizon geometry, except solutions related via a diffeomorphism.
In order to test whether or not there exist any local bulk dynamics in the class of geometries, which asymptote to the nearhorizon geometries (25), one can perform a linear analysis and study which modes survive at the nonlinear level after backreaction is taken into account. This analysis has been performed with care for the spin 2 field around the NHEK geometry in [4, 122] under the assumption that all nonlinear solutions have vanishing SL(2, ℝ) × U(1) charges (which is justified by the existence of a Birkoff theorem as mentioned in Section 2.8). The conclusion is that there is no linear mode that is the linearization of a nonlinear solution. In other words, there is no local spin 2 bulk degree of freedom around the NHEK solution. It would be interesting to investigate if these arguments could be generalized to scalars, gauge fields and gravitons propagating on the general class of nearhorizon solutions (25) of the action (1), but such an analysis has not been done at that level of generality.
This lack of dynamics is familiar from the AdS_{2} × S^{2} geometry [207], which, as we have seen in Sections 2.2–2.3, is the static limit of the spinning nearhorizon geometries. In the above arguments, the presence of the compact S^{2} was crucial. Conversely, in the case of noncompact horizons, such as the extremal planar AdSReissnerNordström black hole, flux can leak out the ℝ^{2} boundary and the arguments do not generalize straightforwardly. There are indeed interesting quantum critical dynamics around AdS_{2} × ℝ^{2} nearhorizon geometries [136], but we will not touch upon this topic here since we concentrate exclusively on compact black holes.
3 TwoDimensional Conformal Field Theories
Since we aim at drawing parallels between black holes and twodimensional CFTs (2d CFTs), it is useful to describe some key properties of 2d CFTs. Background material can be found, e.g., in [120, 149, 234]. An important caveat to keep in mind is that there are only sparse results in gravity that can be interpreted in terms of a 2d CFT. Only future research will tell if 2d CFTs are the right theories to be considered (if a holographic correspondence can be precisely formulated at all) or if generalized field theories with conformal invariance are needed. For progress in this direction, see [130, 169].
A 2d CFT is defined as a local quantum field theory with local conformal invariance. In twodimensions, the local conformal group is an infinitedimensional extension of the globallydefined conformal group SL(2, ℝ) × SL(2, ℝ) on the plane or on the cylinder. It is generated by two sets of vector fields L_{n}, \({L_n},{{\bar L}_n},n \in {\mathbb Z}\), n ∊ ℤ obeying the Lie bracket algebra
From Noether’s theorem, each symmetry is associated to a quantum operator. The local conformal symmetry is associated with the conserved and traceless stressenergy tensor operator, which can be decomposed into left and right moving modes \({{\mathcal L}_n}\) and \({{\bar {\mathcal L}}_n}\), n ∊ ℤ. The operators \({{\mathcal L}_n},{{\bar {\mathcal L}}_n}\) form two copies of the Virasoro algebra
where \({{\mathcal L}_{ 1}},{{\mathcal L}_0},{{\mathcal L}_1}\) (and \({{\bar {\mathcal L}}_{ 1}},{{\bar {\mathcal L}}_0},{{\bar {\mathcal L}}_1}\)) span a SL(2, ℝ) subalgebra. The pure numbers c_{L} and c_{r} are the left and rightmoving central charges of the CFT. The auxiliary parameters A_{L}, A_{r} depend if the CFT is defined on the plane or on the cylinder. They correspond to shifts of the background value of the zero eigenmodes \({{\mathcal L}_0},{{\bar {\mathcal L}}_0}\). In many examples of CFTs, additional symmetries are present in addition to the two sets of Virasoro algebras.
A 2d CFT can be uniquely characterized by a list of (primary) operators \({\mathcal O}\), the conformal dimensions of these operators (their eigenvalue under \({{\mathcal L}_0}\) and \({{\bar {\mathcal L}}_0}\)) and the operator product expansions between all operators. Since we will only be concerned with universal properties of CFTs here, such detailed data of individual CFTs will not be important for our considerations.
We will describe in the next short Sections 3.1, 3.2 and 3.3 some properties of CFTs that are conjectured to be relevant to the Kerr/CFT correspondence and its extensions: the Cardy formula, some properties of the discrete lightcone quantization (DLCQ) and some properties of symmetric product orbifold CFTs.
3.1 Cardy’s formula
In any unitary and modular invariant CFT, the asymptotic growth of states in the microcanonical ensemble is determined only by the left and right central charge and the left and right eigenvalues \({{\mathcal L}_0},{{\bar {\mathcal L}}_0}\) as
when \({{\mathcal L}_0} \gg {c_L},{{\bar {\mathcal L}}_0} \gg {c_R}\). This is known as Cardy’s formula derived originally in [61, 48] using modular invariance of the CFT. A review can be found, e.g., in [62]. Transforming to the canonical ensemble using the definition of the left and right temperatures,
we get
and, therefore, we obtain an equivalent form of Cardy’s formula,
valid when T_{l} ≫ 1, T_{r} ≫ 1.
3.2 DLCQ and chiral limit of CFTs
The role of the DLCQ of CFTs in the context of the Kerr/CFT correspondence was suggested in [30] (for closely related work see [252]). Here, we will review how a DLCQ is performed and how it leads to a chiral half of a CFT. A chiral half of a CFT is here defined as a sector of a 2d CFT defined on the cylinder, where the rightmovers are set to the ground state after the limiting DLCQ procedure. We will use these considerations in Section 4.4.
Let us start with a CFT defined on a cylinder of radius R,
Here the coordinates are identified as (t, ϕ) ∼ (t, ϕ + 2πR), which amounts to
The momentum operators P^{v} and P^{u} along the u and v directions are L_{0} and \({{\bar L}_0}\), respectively. They have a spectrum
where the conformal dimensions obey h, \(\bar h \geq 0\) and n, \(\bar n \neq 0\) are quantized left and right momenta.
Following Seiberg [239], consider a boost with rapidity γ
The boost leaves the flat metric invariant. The discrete lightcone quantization of the CFT is then defined as the limit γ → ∞ with R′ ≡ Re^{γ} fixed. In that limit, the identification (92) becomes
Therefore, the resulting theory is defined on a null cylinder. Because of the boosted kinematics, we have
Keeping P^{u′} (the momentum along v′) finite in the γ → ∞ limit requires \(\bar h = 0\) and \(\bar n = 0\).
Therefore, the DLCQ limit requires one to freeze the rightmoving sector to the vacuum state. The resulting theory admits an infinite energy gap in that sector. The leftmoving sector still admits nontrivial states. All physical finiteenergy states in this limit only carry momentum along the compact null direction u′. Therefore, the DLCQ limit defines a Hilbert space \({\mathcal H}\),
with left chiral excitations around the SL(2, ℝ) × SL(2, ℝ) invariant vacuum of the CFT 0⟩_{L} ⊗ 0⟩_{r}. As a consequence, the rightmoving Virasoro algebra does not act on that Hilbert space. This is by definition a chiral half of a CFT.
In summary, the DLCQ of a 2d CFT leads to a chiral half of the CFT with central charge c = c_{L}. The limiting procedure certainly removes most of the dynamics of the original CFT. How much dynamics is left in a chiral half of a CFT is an important question that is left to be examined in detail in the future.
3.3 Long strings and symmetric orbifolds
Given a set of Virasoro generators \({{\mathcal L}_n}\) and a nonzero integer N ∈ ℤ_{0}, one can always redefine a subset or an extension of the generators, which results in a different central charge (see, e.g., [25]). One can easily check that the generators
obey the Virasoro algebra with a larger central charge c^{short} = N c. Conversely, one might define
In general, the generators \({\mathcal L}_n^{{\rm{long}}}\) with n ≠ N k, k ∈ ℤ do not make sense because there are no fractionalized Virasoro generators in the CFT. Such generators would be associated with multivalued modes e^{inϕ/N} on the cylinder (t, ϕ) ∼ (t, ϕ + 2π). However, in some cases, as we review below, the Virasoro algebra (101) can be defined. The resulting central charge is smaller and given by c^{long} = c/N.
If a CFT with generators (101) can be defined such that it still captures the entropy of the original CFT, the Cardy formula (90) applied in the original CFT could then be used outside of the usual Cardy regime T_{L} ≫ 1. Indeed, using the CFT with leftmoving generators (101) and their rightmoving analogue, one has
which is valid when NT_{L} ≫ 1, NT_{r} ≫ 1. If N is very large, Cardy’s formula (90) would then always apply. We will use the assumption of the existence of such a “long string CFT” in Section 4.4 to justify the validity of Cardy’s formula outside the usual Cardy regime as done originally in [157].
The “long string CFT” can be made more explicit in the context of symmetric product orbifold CFTs [186], which appear in the AdS_{3}/CFT_{2} correspondence [206, 114, 123] (see also [230] and references therein). These orbifold CFTs can be argued to be relevant in the present context, since the Kerr/CFT correspondence might be understood as a deformation of the AdS_{3}/CFT_{2} correspondence, as argued in [157, 98, 23, 116, 31, 243, 246, 115, 130].
Let us then briefly review the construction of symmetric product orbifold CFTs. Given a conformallyinvariant sigmamodel with target space manifold \({\mathcal M}\), one can construct the symmetric product orbifold by considering the sigmamodel with identical copies of the target space manifold \({\mathcal M}\), identified up to permutations,
where S_{N} is the permutation group on N objects. The low energy (infrared) dynamics is a CFT with central charge c_{Sym} = Nc if the central charge of the low energy CFT of the original sigma model is c. The Virasoro generators of the resulting infrared CFT can then be formally constructed from the generators \({{\mathcal L}_m}\) of the original infrared CFT as (100). Conversely, if one starts with a symmetric product orbifold, one can isolate the “long string” sector, which contains the “long” twisted operators. One can argue that such a sector can be effectively described in the infrared by a CFT, which has a Virasoro algebra expressed as (101) in terms of the Virasoro algebra of the low energy CFT of the symmetric product orbifold [211]. The role of these constructions for the Kerr/CFT correspondence remains to be fully understood.
4 Microscopic Entropy of Extremal Black Holes
We discussed that nearhorizon geometries of compact extremal black holes are isolated systems with universal properties and we reviewed that in all analyzed cases they have no local bulk dynamics. Given the nontrivial thermodynamic properties of these systems even at extremality, one can suspect that some nontrivial dynamics are left. It turns out that such nontrivial dynamics appears at the boundary of the nearhorizon geometry. We now show that nearhorizon geometries can be extended to a large class describing extremal boundary excitations. The set of all nearhorizon geometries will admit additional symmetries at their boundary — asymptotic symmetries — which will turn out to be given by one copy of the Virasoro algebra. We will then argue that these nearhorizon geometries are described by chiral limits of twodimensional CFTs, which we will use to microscopically derive the entropy of any charged or spinning extremal black hole.
4.1 Boundary conditions and asymptotic symmetry algebra
Let us discuss the existence and the construction of a consistent set of boundary conditions that would define “the set of solutions in the nearhorizon region of extremal black holes”. Since the nearhorizon region is not asymptotically flat or asymptotically antide Sitter, one cannot use previous results in those spacetimes to derive the boundary conditions in the nearhorizon region. Rather, one has to derive the relevant boundary conditions from first principles. A large literature on the theory of boundary conditions and asymptotic charges exists, see [9, 237, 59, 196, 35, 36] (see also [90] for a review). We will use the Lagrangian methods [35, 36] to address the current problem.
A set of boundary conditions always comes equipped with an asymptotic symmetry algebra. Restricting our discussion to the fields appearing in (1), the boundary conditions are preserved by a set of allowed diffeomorphisms and U(1) gauge transformations (ζ^{μ}, Λ), which act on the fields as
Asymptotic symmetries are the set of all these allowed transformations that are associated with nontrivial conserved charges. The set of allowed transformations that are associated with zero charges are “pure gauge” or “trivial” transformations. The set of asymptotic symmetries inherits a Lie algebra structure from the Lie commutator of diffeomorphisms and U(1) gauge transformations. Therefore, the asymptotic symmetries form an algebra,
where [ζ_{m}, ζ_{n}] is the Lie commutator and
Consistency requires that the charge associated with each element of the asymptotic symmetry algebra be finite and well defined. Moreover, as we are dealing with a spatial boundary, the charges are required to be conserved in time. By construction, one always first defines the “infinitesimal variation of the charge” \(\delta {\mathcal Q}\) from infinitesimal variations of the fields around a solution. If \(\delta {\mathcal Q}\) is the exact variation of a quantity \({\mathcal Q}\), the quantity \({\mathcal Q}\) is the welldefined charge and the charges are said to be integrable.
Imposing consistent boundary conditions and obtaining the associated asymptotic symmetry algebra requires a careful analysis of the asymptotic dynamics of the theory. If the boundary conditions are too strong, all interesting excitations are ruled out and the asymptotic symmetry algebra is trivial. If they are too weak, the boundary conditions are inconsistent because transformations preserving the boundary conditions are associated to infinite or illdefined charges. In general, there is a narrow window of consistent and interesting boundary conditions. There is not necessarily a unique set of consistent boundary conditions.
There is no universal algorithm to define the boundary conditions and the set of asymptotic symmetries. One standard algorithm used, for example, in [168, 167] consists in first promoting all exact symmetries of the background solution as asymptotic symmetries and second acting on solutions of interest with the asymptotic symmetries in order to generate tentative boundary conditions. The boundary conditions are then restricted in order to admit consistent finite, well defined and conserved charges. Finally, the set of asymptotic diffeomorphisms and gauge transformations, which preserve the boundary conditions are computed and one deduces the full asymptotic symmetry algebra after computing the associated conserved charges.
As an illustration, asymptotically antide Sitter spacetimes in spacetime dimensions d + 1 admit the SO(2, d) asymptotic symmetry algebra for d ≥ 3 [1, 15, 168, 167] and two copies of the Virasoro algebra for d = 2 [58]. Asymptoticallyflat spacetimes admit as asymptotic symmetry algebra the Poincaré algebra or an extension thereof depending on the precise choice of boundary conditions [9, 231, 147, 237, 13, 12, 16, 37, 38, 92, 261]. From these examples, we learn that the asymptotic symmetry algebra can be larger than the exact symmetry algebra of the background spacetime and it might in some cases contain an infinite number of generators. We also notice that several choices of boundary conditions, motivated from different physical considerations, might lead to different asymptotic symmetry algebras.
Let us now motivate boundary conditions for the nearhorizon geometry of extremal black holes. There are two boundaries at r = ∞ and r = −∞. It was proposed in [156, 159] to build boundary conditions on the boundary r = ∞ such that the asymptotic symmetry algebra contains one copy of the Virasoro algebra generated by
Part of the physical motivation behind this ansatz is the existence of a nonzero temperature T_{ϕ} associated with modes corotating with the black hole, as detailed in Section 2.6. This temperature suggests the existence of excitations along ∂_{ϕ}. The ansatz for Λ_{ϵ} will be motivated in (117). The subleading terms might be chosen such that the generator ζ_{ϵ} is regular at the poles θ = 0, π. This ansatz has to be validated by checking if boundary conditions preserved by this algebra exist such that all charges are finite, well defined and conserved. We will discuss such boundary conditions below. Expanding in modes as^{Footnote 11}
the generators L_{n} ≡ (ζ_{n}, Λ_{n}) obey the Virasoro algebra with no central extension
where the bracket has been defined in (105)
Finding consistent boundary conditions that admit finite, conserved and integrable Virasoro charges and that are preserved by the action of the Virasoro generators is a nontrivial task. The details of these boundary conditions depend on the specific theory at hand because the expression for the conserved charges depend on the theory. (For the action (1), the conserved charges can be found in [97]). Specializing in the case of the extremal Kerr black hole in Einstein gravity, the problem of finding consistent boundary conditions becomes more manageable but is still intricate (see discussions in [5]). In [156], the following falloff conditions
were proposed as a part of the definition of boundary conditions. The zero energy excitation condition
was imposed as a supplementary condition. We will discuss in Section 4.2 the relaxation of this condition. A nontrivial feature of the boundary conditions (111)–(112) is that they are preserved precisely by the Virasoro algebra (107), by ∂_{t} and the generator (24) (as pointed out in [5]) and subleading generators. (Note that these boundary conditions are not preserved by the action of the third SL(2, ℝ) generator (29).) It was shown in [156] that the Virasoro generators are finite given the falloff conditions and well defined around the background NHEK geometry. It was shown in [5] that the Virasoro generators are conserved and well defined around any asymptotic solution given that one additionally regularizes the charges using counterterm methods [96]. Therefore, up to some technical details that remain to be fully understood, it can be claimed that consistent boundary conditions admitting (at least) a Virasoro algebra as asymptotic symmetry algebra exist. The set of trivial asymptotic symmetries comprise two of the SL(2, ℝ) generators. It is not clear if the boundary conditions could be enhanced in order to admit all SL(2, ℝ) generators as trivial asymptotic symmetries.
Let us now generalize these arguments to the electricallycharged KerrNewman black hole in EinsteinMaxwell theory. First, the presence of the chemical potential T_{e} suggests that some dynamics are also present along the gauge field. The associated conserved electric charge \({{\mathcal Q}_e}\) can be shown to be canonically associated with the zeromode generator J_{0}= (0, −1) with gauge parameter Λ = −1. It is then natural to define the current ansatz
which obeys the commutation relations
The nontrivial step consists in establishing the existence of boundary conditions such that the Virasoro and the current charges are well defined and conserved. Ongoing work is in progress in that direction.^{Footnote 12} One can simplify the problem of constructing boundary conditions by imposing the following additional constraints
which discard the current algebra. Such a simplification was used in [159] and the following boundary conditions were proposed (up to the term e/k, which was omitted in [159])
which are preserved upon acting with the Virasoro generator (107)–(108). In particular, the choice of the compensating gauge transformation Λ_{ϵ} (108) is made such that
It can be shown that the Virasoro generators are finite under these boundary conditions.
Let us also discuss what happens in higher dimensions (d > 4). The presence of several independent planes of rotation allows for the construction of one Virasoro ansatz and an associated FrolovThorne temperature for each plane of rotation [203, 173, 21, 225, 83]. More precisely, given n compact commuting Killing vectors, one can consider an SL(n, ℤ) family of Virasoro ansätze by considering all modular transformations on the U(1)^{n} torus [201, 76]. However, preliminary results show that there is no boundary condition that allows simultaneously two different Virasoro algebras in the asymptotic symmetry algebra [21]. Rather, there are mutuallyincompatible boundary conditions for each choice of Virasoro ansatz.
Since two U(1) circles form a torus invariant under SL(2, ℤ) modular transformations, one can then form an ansatz for a Virasoro algebra for any circle defined by a modular transformation of the ϕ_{1} and ϕ_{2}circles. More precisely, we define
where p_{1}p_{4} − p_{2}p_{3} = 1 and we consider the vector fields
The resulting boundary conditions have not been thoroughly constructed, but evidence points to their existence [21, 201].
The occurrence of multiple choices of boundary conditions in the presence of multiple U(1) symmetries raises the question of whether or not the (AdS)ReissnerNöordstrom black hole admits interesting boundary conditions where the U(1) gauge symmetry (which is canonically associated to the conserved electric charge Q) plays the prominent role. One can also ask these questions for the general class of (AdS)KerrNewman black holes.
It was argued in [159, 204] that such boundary conditions indeed exist when the U(1) gauge field can be promoted to be a KaluzaKlein direction of a higherdimensional spacetime, or at least when such an effective description captures the physics. Denoting the additional direction by ∂_{χ} with χ + 2πR_{χ}, the problem amounts to constructing boundary conditions in five dimensions. As mentioned earlier, evidence points to the existence of such boundary conditions [21, 201]. The Virasoro asymptoticsymmetry algebra is then defined using the ansatz
along the gauge KaluzaKlein direction. The same reasoning leading to the SL(2, ℤ) family of Virasoro generators (119) would then apply as well. The existence of such a Virasoro symmetry around the KerrNewman black holes is corroborated by nearextremal scattering amplitudes as we will discuss in Section 5, and by the hidden conformal symmetry of probes, as we will discuss in Section 6.
4.2 Absence of SL(2, ℤ) asymptotic symmetries
The boundary conditions discussed so far do not admit solutions with nontrivial charges under the SL(2, ℤ) exact symmetry group of the background geometry generated by ζ_{0,±1} (29). In fact, the boundary conditions are not even invariant under the action of the generator ζ_{1}. One could ask the question if such an enlargement of boundary conditions is possible, which would open the possibility of enlarging the asymptoticsymmetry group to include the SL(2, ℝ) group and even a Virasoro extension thereof. We will now argue that such enlargement would result in trivial charges, which would not belong to the asymptoticsymmetry group.
First, we saw in Section 2.7 that there is a class of nearextremal solutions (79) obeying the boundary conditions (111)–(116) with nearhorizon energy \(\not \delta{{\mathcal Q}_{{\partial _t}}} = {T^{{\rm{near  ext}}}}\delta {{\mathcal S}_{{\rm{ext}}}}\). However, the charge \(\not \delta{{\mathcal Q}_{{\partial _t}}}\) is a heat term, which is not integrable when both T^{near−ext} and \({{\mathcal S}_{{\rm{ext}}}}\) can be varied. Moreover, upon scaling the coordinates as t → t/α and r → αr using the SL(2, ℝ) generator (24), one obtains the same metric as (79) with T^{near−ext} → T^{near−ext}/α. If one would allow the class of nearextremal solutions (79) and the presence of SL(2, ℝ) symmetries in a consistent set of boundary conditions, one would be forced to fix the entropy \({{\mathcal S}_{{\rm{ext}}}}\) to a constant, in order to define integrable charges. The resulting vanishing charges would not belong to the asymptoticsymmetry algebra. Since there is no other obvious candidate for a solution with nonzero nearhorizon energy, we argued in Section 2.9 that there is no such solution at all. If that assumption is correct, the SL(2, ℝ) algebra would always be associated with zero charges and would not belong to the asymptotic symmetry group. Hence, no additional nonvanishing Virasoro algebra could be derived in a consistent set of boundary conditions. For alternative points of view, see [215, 216, 236, 214].
Second, as far as extremal geometries are concerned, there is no need for a nontrivial SL(2, ℝ) or second Virasoro algebra. As we will see in Section 4.4, the entropy of extremal black holes will be matched using a single copy of the Virasoro algebra, using the assumption that Cardy’s formula applies. Matching the entropy of nonextremal black holes and justifying Cardy’s formula requires two Virasoro algebras, as we will discuss in Section 6.6. However, nonextremal black holes do not admit a nearhorizon limit and, therefore, are not dynamical objects described by a consistent class of nearhorizon boundary conditions. At most, one could construct the near horizon region of nonextremal black holes in perturbation theory as a large deformation of the extremal nearhorizon geometry. This line of thought was explored in [67]. In the context of the nearextremal Kerr black hole, it was obtained using a dimensionallyreduced model such that the algebra of diffeomorphisms, which extends the SL(2, ℝ) algebra, is represented on the renormalized stressenergy tensor as a Virasoro algebra. It would be interesting to further define and extend these arguments (which go beyond a standard asymptoticsymmetry analysis) to nondimensionallyreduced models and to other nearextremal black holes.
Finally, let us also note that the current discussion closely parallels the lower dimensional example of the nearhorizon limit of the extremal BTZ black hole discussed in [30]. There it was shown that the nearhorizon geometry of the extremal BTZ black hole of angular momentum J is a geometry with SL(2, ℝ) × U(1) isometry
with ϕ ∼ ϕ + 2π, which is known as the selfdual AdS_{3} orbifold [100]. It was found in [30] that the asymptotic symmetry group consists of one chiral Virasoro algebra extending the U(1) symmetry along ∂_{ϕ}, while the charges associated with the SL(2, ℝ) symmetry group are identically zero. These observations are consistent with the analysis of fourdimensional nearhorizon geometries (25), whose constant θ sections share similar qualitative features with the threedimensional geometries (121). It was also shown that an extension of the boundary conditions exists that is preserved by a second Virasoro algebra extending the SL(2, ℝ) exactsymmetry algebra [24]. All associated charges can be argued to be zero, but a nontrivial central extension still appears as a background charge when a suitable regularization is introduced. However, the regularization procedure does not generalize to the fourdimensional geometries mainly because two features of the threedimensional geometry (121) are not true in general (∂_{t} is null in (121) and ∂_{t} − ϵ∂_{ϕ} with ϵ ≪ 1 is a global timelike Killing vector).
4.3 Virasoro algebra and central charge
Let us now assume in the context of the general theory (1) that a consistent set of boundary conditions exists that admit the Virasoro algebra generated by (107)–(108) as asymptoticsymmetry algebra. Current results are consistent with that assumption but, as emphasized earlier, boundary conditions have been checked only partially [156, 5, 21].
Let us define the Dirac bracket between two charges as
Here, the operator \({\delta _{({\zeta _m},{\Lambda _m})}}\) is a derivative in phase space that acts on the fields \({g_{\mu \nu}},A_\mu ^I,{\chi ^A}\) appearing in the charge \({\mathcal Q}\) as (104). From general theorems in the theory of asymptoticsymmetry algebras [59, 35, 36], the Dirac bracket represents the asymptotic symmetry algebra up to a central term, which commutes with each element of the algebra. Namely, one has
where the bracket between two generators has been defined in (105) and \({\mathcal K}\) is the central term, which is antisymmetric in its arguments. Furthermore, using the correspondence principle in semiclassical quantization, Dirac brackets between generators translate into commutators of quantum operators as \(\{\ldots \} \rightarrow  {i \over \hbar}[ \ldots ]\). Note that, according to this rule, the central terms in the algebra aquire a factor of 1/ℏ when operator eigenvalues are expressed in units of ℏ (or equivalently, when one performs \({\mathcal Q} \rightarrow \hbar {\mathcal Q}\) and divide both sides of (123) by ℏ.).
For the case of the Virasoro algebra (110), it is well known that possible central extensions are classified by two numbers c and A. The general result has the form
where A is a trivial central extension that can be set to 1 by shifting the background value of the charge \({{\mathcal L}_0}\). The nontrivial central extension c is a number that is called the central charge of the Virasoro algebra. From the theorems [59, 35, 36], the central term in (123) can be expressed as a specific and known functional of the Lagrangian \({\mathcal L}\) (or equivalently of the Hamiltonian), the background solution \(\bar \phi = ({{\bar g}_{\mu \nu}},\bar A_\mu ^I,{{\bar \chi}^A})\) (the nearhorizon geometry in this case) and the Virasoro generator (ζ, Λ) around the background
In particular, the central charge does not depend on the choice of boundary conditions. The representation theorem leading to (124) only requires that such boundary conditions exist. The representation theorem for asymptotic Hamiltonian charges [59] was famously first applied [58] to Einstein’s gravity in three dimensions around AdS, where the two copies of the Virasoro asymptoticsymmetry algebra were shown to be centrally extended with central charge \(c = {{3l} \over {2{G_N}\hbar}}\), where l is the AdS radius and G_{N} Newton’s constant.
For the general nearhorizon solution (25) of the Lagrangian (1) and the Virasoro ansatz (107)–(108), one can prove [159, 97] that the matter part of the Lagrangian (including the cosmological constant) does not contribute directly to the central charge, but only influences the value of the central charge through the functions Γ(θ), α(θ), γ(θ) and k, which solve the equations of motion. The central charge (125) is then given as the m^{3} factor of the following expression defined in terms of the fundamental charge formula of Einstein gravity as [35]
where \({{\mathcal L}_{{L_{ m}}}}\bar g\) is the Lie derivative of the metric along L_{−m} and
Here, \(d{S_{\mu \nu}} = {1 \over {2(d  2)!}}\sqrt { g} {\varepsilon _{\mu \nu {\alpha _{1 \cdots {\alpha _{d  2}}}}}}d{x^{{\alpha _1}}} \wedge \cdots \wedge d{x^{{\alpha _{d  2}}}}\) is the integration measure in d dimensions and indices are raised with the metric g^{μν}, h ≡ g−m^{μν}h_{μν} and S is a surface at fixed time and radius r. Physically, \(Q_\xi ^{{\rm{Einstein}}}[h;g]\) is defined as the charge of the linearized metric h_{μν} around the background g_{μν} associated with the Killing vector ξ, obtained from Einstein’s equations [1]. Substituting the general nearhorizon solution (25) and the Virasoro ansatz (107)–(108), one obtains
We will drop the factors of G_{N} and ℏ from now on. In the case of the NHEK geometry in Einstein gravity, substituting (37), one finds the simple result [156]
The central charge of the Virasoro ansatz (107)–(108) around the KerrNewman black hole turns out to be identical to (129). We note in passing that the central charge c_{J} of extremal Kerr or KerrNewman is a multiple of six, since the angular momentum is quantized as a halfinteger multiple of ℏ. The central charge can be obtained for the KerrNewmanAdS solution as well [159] and the result is
where Δ_{0} has been defined in (44).
When higherderivative corrections are considered, the central charge can still be computed exactly, using as crucial ingredients the SL(2, ℝ) × U(1) symmetry and the (t, ϕ) reversal symmetry of the nearhorizon solution. The result is given by [20]
where the covariant variational derivative δ^{cov}/δR_{abcd} has been defined in (53) in Section 2.5. One caveat should be noted. The result [20] is obtained after auxiliary fields are introduced in order to rewrite the arbitrary diffeomorphisminvariant action in a form involving at most two derivatives of the fields. It was independently observed in [190] that the formalism of [35, 36] applied to the GaussBonnet theory formulated using the metric only cannot reproduce the central charge (131) and, therefore, the blackhole entropy as will be developed in Section 4.4. One consequence of these two computations is that the formalism of [35, 36, 90] is not invariant under field redefinitions. In view of the cohomological results of [35], this ambiguity can appear only in the asymptotic context and when certain asymptotic linearity constraints are not obeyed. Nevertheless, it has been acknowledged that boundary terms in the action should be taken into account [237, 164]. Adding supplementary terms to a welldefined variational principle amount to deforming the boundary conditions [56, 266, 213] and modifying the symplectic structure of the theory through its coupling to the boundary dynamics [96]. Therefore, it remains to be checked if the prescription of [96] to include boundary effects would allow one to reconcile the work of [190] with that of [20].
In fivedimensional Einstein gravity coupled to U(1) gauge fields and scalars, the central charge associated with the Virasoro generators along the direction \({\partial _{{\phi _i}}},i = 1,2\) can be obtained as a straightforward extension of (128) [159, 97]. One has
where the extra factor of 2π with respect to (128) originates from integration around the extra circle (see also [151, 166] for some higher derivative corrections). Since the entropy (54) is invariant under a SL(2, ℤ) change of basis of the torus coordinates (ϕ_{1}, ϕ_{2}) as (118), \({c_{{\phi _i}}}\) transforms under a modular transformation as k_{i}. Now, k_{i} transforms in the same fashion as the coordinate ϕ_{i}, as can be deduced from the form of the nearhorizon geometry (36). Then, the central charge for the Virasoro ansatz (119) is given by
Let us now discuss the central extension of the alternative Virasoro ansatz (120) for the extremal ReissnerNordström black hole of electric charge Q and mass Q. First, the central charge is inversely proportional to the scale R_{χ} set by the KaluzaKlein direction that geometrizes the gauge field. One can see this as follows. The central charge is bilinear in the Virasoro generator and, therefore, it gets a factor of (R_{χ})^{2}. Also, the central charge consists of the n^{3} term of the formula (127), it then contains terms admitting three derivatives along χ of e^{−inχ/R} and, therefore, it contains a factor of \(R_\chi ^{ 3}\). Also, the central charge is defined as an integration along χ and, therefore, it should contain one factor R_{χ} from the integration measure. Finally, the charge is inversely proportional to the fivedimensional Newton’s constant G_{5} = (2πR_{χ})G_{4}. Multiplying this complete set of scalings, one obtains that the central charge is inversely proportional to the scale R_{χ}.
Using the simple embedding of the metric and the gauge fields in a higherdimensional spacetime (2), as discussed in Section 1.2, and using the Virasoro ansatz (120), it was shown [159, 146, 77] that the central charge formula (126) gives
One might object that (2) is not a consistent higherdimensional supergravity uplift. Indeed, as we discussed in Section 1.2, one should supplement matter fields such as (3). However, since matter fields such as scalars and gauge fields do not contribute to the central charge (125) [97], the result (134) holds for any such consistent embedding.
Similarly, we can uplift the KerrNewman black hole to fivedimensions, using the uplift (2)–(3) and the fourdimensional fields (25)–(39). Computing the central charge (132) for the Virasoro ansatz (120), we find again^{Footnote 13}
Under the assumption that the U(1) gauge field can be uplifted to a KaluzaKlein direction, we can also formulate the Virasoro algebra (119) and associated boundary conditions for any circle related by an SL(2, ℤ) transformation of the torus U(1)^{2}. Applying the relation (133) we obtain the central charge
Let us discuss the generalization to AdS black holes. As discussed in Section 1.2, one cannot use the ansatz (2) to uplift the U(1) gauge field. Rather, one can uplift to eleven dimensions along a sevensphere. One can then argue, as in [204], that the only contribution to the central charge comes from the gravitational action. Even though no formal proof is available, it is expected that it will be the case given the results for scalar and gauge fields in four and five dimensions [97]. Applying the charge formula (126) accounting for the gravitational contribution of the complete higherdimensional spacetime, one obtains the central charge for the Virasoro algebra (120) as [204]
where parameters have been defined in Section 2.4.4 and 2πR_{χ} is the length of the U(1) circle in the sevensphere.
The values of the central charges (129), (130), (131), (132), (133), (135), (136), (137) are the main results of this section.
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 nearhorizon geometry. We also discussed that the SL(2, ℝ) symmetry is associated with zero charges. Following semiclassical quantization rules, the operators that define quantum gravity with the boundary conditions (111), (116), (115) form a representation of the Virasoro algebra and are in a ground state with respect to the representation of the SL(2, ℝ) symmetry [251, 156]. A consistent theory of quantum gravity in the nearhorizon region, if it can be defined at all, is therefore either a chiral CFT or a chiral half of a twodimensional CFT. A chiral CFT is defined as a holomorphicallyfactorized 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 nonextremal black holes favors the interpretation of quantum gravity in extremal black holes as the chiral half of a fullfledged twodimensional CFT. Moreover, the applicability of Cardy’s formula as detailed later on also favors the existence of a twodimensional CFT. Since the nearhorizon geometry is obtained as a strict nearhorizon limit of the original geometry, the CFT might be thought of as describing the degrees of freedom of the blackhole horizon.
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 twodimensional flat cylinder at the boundary of AdS_{3} using the FeffermanGraham theorem [137]. One is then led to identify quantum gravity in AdS_{3} spacetimes with a twodimensional 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 SL(2, ℝ) × SL(2, ℝ) invariant vacuum of the CFT, which is separated with a mass gap of −c/24 from the zeromass 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 nearhorizon 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 nearhorizon geometry of the extremal BTZ black hole (121) and the nearhorizon geometries of fourdimensional extremal black holes (25), it has been suggested in [30] that extremal black holes are described by a chiral limit of twodimensional 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 nonextremal excitations are allowed in the nearhorizon 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 threedimensional example, the geometric dual to the vacuum state (the AdS_{3} geometry) does not belong to the phase space defined in the nearhorizon 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 nearhorizon geometry of the extremal Kerr black hole is described by the leftsector of a 2d CFT that we will denote as CFT_{J}. 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 nontrivial temperature for the excited states. We saw in Section 2.6 that scalar quantum fields in the analogue of the FrolovThorne vacuum restricted to extremal excitations have the temperature (67). Individual modes are corotating with the black hole along ∂_{ϕ}. Since we identify the leftsector of the CFT with excitations along ∂_{ϕ} and the right SL(2, ℝ)_{R} sector is frozen, the CFT leftmoving states are described by a thermal density matrix with temperatures
where T_{ϕ} is given in (67). The other quantities T_{e} and T_{m} 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 as
It is remarkable that applying blindly Cardy’s formula (90) using the central charge c_{L} = c_{J} given in (129) and using the temperatures (138), one reproduces the extremal BekensteinHawking blackhole 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 leftmoving Virasoro algebra of central charge c_{L} = c_{J} given in (128). The blackhole 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 IyerWald 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 blackhole 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 nearhorizon geometry, the BrownHenneaux central charge c = 3l/2G_{3} [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 leftmoving temperature as (138), one might instead emphasize that electricallycharged particles are immersed in a thermal bath with temperature T_{χ} = R_{χ}T_{e}, as derived in (70) 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
The other quantities T_{ϕ} and T_{m} defined in (61) are then better interpreted as being proportional to auxiliary chemical potentials. One can indeed rewrite the Boltzman factor (66) as
where q_{χ} = R_{χ}q_{e} is the probe electric charge in units of the KaluzaKlein length and the left chemical potentials are defined as
We argued above that in the nearhorizon region, excitations along the gaugefield direction fall into representations of the Virasoro algebra defined in (120). As supported by nonextremal 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 (90) with temperatures (142) and central charge (134) also reproduces the entropy of the KerrNewman black hole. When the angular momentum is identically zero, the blackhole entropy of the ReissnerNordström black hole \({{\mathcal S}_{{\rm{ext}}}} = \pi {Q^2}\) is then reproduced from Cardy’s formula with left central charge c_{L} = c_{Q} given in (134) and left temperature T_{L} = Rχ/(2πQ) as originally obtained in [159]. As one can easily check, the entropy of the general KerrNewmanAdS 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 CFT_{Q}. Note that the entropy matching does not depend on the scale of the KaluzaKlein 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 (119) obeys the Virasoro algebra with central charge (136). After performing an SL(2, ℤ) 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 \({\rm{CF}}{{\rm{T}}_{({p_1},{p_2},{p_{3)}}}}\).
5 Scattering from NearExtremal Black Holes
In Section 4, we presented how the entropy of any extremal black hole can be reproduced microscopically from one chiral half of one (or several) twodimensional CFT(s). In this section, we will present arguments supporting the conjecture that this duality can be extended to nearextremal black holes dual to a CFT with a second sector slightly excited, following [53, 106, 160]. We will show that the derivation of [53, 106, 160] is supporting evidence for all CFTs presented in Section 4, as noted in [79, 71]. In the case of the CFT_{J} dual to nearextremal spinning black holes, one can think intuitively that the second CFT sector is excited for the following reason: lights cones do not quite coalesce at the horizon, so microscopic degrees of freedom do not rotate at the speed of light along the single axial direction. The intuition for the other CFTs (CFT_{Q}, \({\rm{(CF}}{{\rm{T}}_Q}{\rm{,CF}}{{\rm{T}}_{({p_1},{p_2},{p_{3)}}}})\)) is less immediate.
Nearextremal black holes are defined as black holes with a Hawking temperature that is very small compared with their inverse mass
At finite energy away from extremality, one cannot isolate a decoupled nearextremal nearhorizon geometry. As we discussed in Section 4, the extremal nearhorizon geometry then suffers from infrared divergences, which destabilize the nearhorizon geometry. This prevents one to formulate boundary conditions à la BrownHenneaux to describe nonchiral excitations. Therefore, another approach is needed.
If nearextremal black holes are described by a dual field theory, it means that all properties of these black holes — classical or quantum — can be derived from a computation in the dual theory, after it has been properly coupled to the surrounding spacetime. We now turn our attention to the study of one of the simplest dynamical processes around black holes: the scattering of a probe field. This route was originally followed for static extremal black holes in [208, 209]. In this approach, no explicit metric boundary conditions are needed. Moreover, since gravitational backreaction is a higherorder effect, it can be neglected. One simply computes the blackholescattering amplitudes on the blackhole background. In order to test the nearextremal black hole/CFT correspondence, one then has to determine whether or not the black hole reacts like a twodimensional CFT to external perturbations originating from the asymptotic region far from the black hole.
We will only consider fields that probe the nearhorizon region of nearextremal black holes. These probe fields have energy ω and angular momentum m close to the superradiant bound \(\omega \sim m\Omega _J^{{\rm{ext}}} + {q_e}\Phi _e^{{\rm{ext}}}\),
In order to simplify the notation, in this section we will drop all hats on quantities defined in the asymptotic region far from the black hole.
Since no general scattering theory around nearextremal blackhole solutions of (1) has been proposed so far, we will concentrate our discussion on nearextremal asymptoticallyflat KerrNewman black holes, as discussed in [53, 160] (see also [79, 72, 81, 74, 77, 3]). Extensions to the KerrNewmanAdS black hole or other specific black holes in four and higher dimensions in gauged or ungauged supergravity can be found in [53, 106, 73, 242, 46] (see also [71, 80, 129, 224]).
5.1 Nearextremal KerrNewman black holes
Nearextremal KerrNewman black holes are characterized by their mass M, angular momentum J = Ma and electric charge Q. (We take a, Q ≥ 0 without loss of generality.) They contain nearextremal Kerr and ReissnerNordström black holes as particular instances. The metric and thermodynamic quantities can be found in many references and will not be reproduced here.
The nearextremality condition (145) is equivalent to the condition that the reduced Hawking temperature is small,
Indeed, one has τ_{H} = M T_{H}[4π((r_{+}/M)^{2} + (a/M)^{2})/(r_{+}/M)] and the term in between the brackets is of order one since 0 ≤ a/M ≤ 1, 0 ≤ Q/M ≤ 1 and 1 ≤ r_{+}/M ≤ 2. Therefore, we can use interchangeably the conditions (145) and (147).
Since there is both angular momentum and electric charge, extremality can be reached both in the regime of vanishing angular momentum and vanishing electric charge Q. When angular momentum is present, we expect that the dynamics could be described by the CFT_{J}, while when electric charge is present the dynamics could be described by the CFT_{Q}. It is interesting to remark that the condition
implies (145)–(147) since \({\tau _H} = {{{T_H}} \over {{\Omega _J}}}(4\pi a/M) \ll 1\) but it also implies a > 0. Similarly, the condition
implies (145)–(147), since τ_{H} = 4πQT_{h}/Φ_{e}, but it also implies Q > 0. In the following, we will need only the nearextremality condition (145), and not the more stringent conditions (148) or (149). This is the first clue that the nearextremal scattering will be describable by both the CFT_{J} and the CFT_{Q}.
Nearextremal black holes are characterized by an approximative nearhorizon geometry, which controls the behavior of probe fields in the window (146). Upon taking T_{H} = O(λ) and taking the limit λ → 0 the nearhorizon geometry decouples, as we saw in Section 2.7.
Probes will penetrate the nearhorizon region close to the superradiant bound (146). When T_{H} = O(λ) we need
Indeed, repeating the reasoning of Section 2.6, we find that the Boltzman factor defined in the nearhorizon vacuum (defined using the horizon generator) takes the following form
where ω, m and q_{e} are the quantum numbers defined in the exterior asymptotic region and
is finite upon choosing (150). The conclusion of this section is that the geometries (79) control the behavior of probes in the nearextremal regime (145)–(146). We identified the quantity n as a natural coefficient defined near extremality. It will have a role to play in later Sections 5.3 and 5.4. We will now turn our attention to how to solve the equations of motion of probes close to extremality.
5.2 Macroscopic greybody factors
The problem of scattering of a general spin field from a Kerr black hole was solved in a series of classic papers by Starobinsky [248], Starobinsky and Churilov [249] and Press and Teukolsky [255, 256, 235, 257] in the early 1970s (see also [145, 4, 122]). The scattering of a spin 0 and 1/2 field from a KerrNewman black hole has also been solved [257], while the scattering of spins 1 and 2 from the KerrNewman black hole has not been solved to date.
Let us review how to solve this classic scattering problem. First, one has to realize that the KerrNewman black hole enjoys a remarkable property: it admits a KillingYano tensor [269, 232, 142]. (For a review and some surprising connections between KillingYano tensors and fermionic symmetries, see [148].) A KillingYano tensor is an antisymmetric tensor fh_{μν} = −f_{νμ}, which obeys
This tensor can be used to construct a symmetric Killing tensor
which is a natural generalization of the concept of Killing vector K_{μ} (obeying ∇_{(μ}K_{ν}) = 0). This Killing tensor was first used by Carter in order to define an additional conserved charge for geodesics [65]
and thereby reduce the geodesic equations in Kerr to firstorder equations. More importantly for our purposes, the Killing tensor allows one to construct a secondorder differential operator K^{μν}∇_{μ}∇_{ν}, which commutes with the Laplacian ∇^{2}. This allows one to separate the solutions of the scalar wave equation ∇^{2}Ψ^{s=0} = 0 as [65]
where is the real separation constant present in both equations for S(θ) and R(r). The underlying KillingYano tensor structure also leads to the separability of the Dirac equation for a probe fermionic field. For simplicity, we will not discuss further fermionic fields here and we refer the interested reader to the original reference [160] (see also [41]). The equations for spin 1 and 2 probes in Kerr can also be shown to be separable after one has conveniently reduced the dynamics to a master equation for a master scalar Ψ^{s}, which governs the entire probe dynamics. As a result, one has
The master scalar is constructed from the field strength and from the Weyl tensor for spin 1 (s = ±1) and spin 2 (s = ±2) fields, respectively, using the NewmanPenrose formalism. For the KerrNewman black hole, all attempts to separate the equations for spin 1 and spin 2 probes have failed. Hence, there is no known analytic method to solve those equations (for details, see [70]). Going back to Kerr, given a solution to the master scalar field equation, one can then in principle reconstruct the gauge field and the metric from the Teukolsky functions. This nontrivial problem was778ikm solved right after Teukolsky’s work [89, 87]; see Appendix C of [122] for a modern review (with further details and original typos corrected).
In summary, for all separable cases, the dynamics of probes in the KerrNewman geometry can be reduced to a secondorder equation for the angular part of the master scalar \(S_{\omega ,A,m}^{\mathcal S}(\theta)\) and a secondorder equation for the radial part of the master scalar \(R_{\omega ,A,m}^{\mathcal S}(r)\). Let us now discuss their solutions after imposing regularity as boundary conditions, which include ingoing boundary conditions at the horizon. We will limit our discussion to the nonnegative integer spins s = 0, 1, 2 in what follows.
The angular functions \(S_{\omega ,A,m}^{\mathcal S}(\theta)\) obey the spinweighted spheroidal harmonic equation
(The Kronecker δ_{s0} is introduced so that the multiplicative term only appears for a massive scalar field of mass μ.) All harmonics that are regular at the poles can be obtained numerically and can be classified by the usual integer number l with l ≥ m and l ≥ s. In general, the separation constant \(A = A_{a\omega ,l,m}^{\mathcal S}\) depends on the product aω, on the integer l, on the angular momentum of the probe and on the spin s. At zero energy (ω = 0), the equation reduces to the standard spinweighted sphericalharmonic equation and one simply has \(A_{0,l,m}^{\mathcal S} = l(l + 1)  {{\mathcal S}^2}\). For a summary of analytic and numerical results, see [44].
Let us now take the values \(A_{a\omega ,l,m}^{\mathcal S}\) as granted and turn to the radial equation. The radial equation reduces to the following SturmLiouville equation
where Δ(r) = (r − r_{+})(r − r_{−}) = r^{2} − 2Mr + a^{2} + Q^{2} in a potential V^{s}(r). The form of the potential is pretty intricate. For a scalar field of mass μ, the potential V^{0}(r) is real and is given by
where H(r) = ω(r^{2} + a^{2}) − q_{e}Qr − am. For a field of general spin on the Kerr geometry, the potential is, in general, complex and reads as
where H(r) = ω(r^{2} + a^{2}) − am. This radial equation obeys the following physical boundary condition: we require that the radial wave has an ingoing group velocity — or, in other words, is purely ingoing — at the horizon. This is simply the physical requirement that the horizon cannot emit classical waves. This also follows from a regularity requirement. The solution is then unique up to an overall normalization. For generic parameters, the SturmLiouville equation (159) cannot be solved analytically and one has to use numerical methods.
For each frequency ω and spheroidal harmonic (l, m), the scalar field can be extended at infinity into an incoming wave and an outgoing wave. The absorption probability σ_{abs} or macroscopic greybody factor is then defined as the ratio between the absorbed flux of energy at the horizon and the incoming flux of energy from infinity,
An important feature is that in the superradiant range (13) the absorption probability turns out to be negative, which results in stimulated as well as spontaneous emission of energy, as we reviewed in Section 2.1.
5.3 Macroscopic greybody factors close to extremality
The SturmLiouville problem (159) cannot be solved analytically. However, in the regime of nearextremal excitations (145)–(146) an approximative solution can be obtained analytically using asymptotic matched expansions: the wave equation is solved in the nearhorizon region and in the far asymptoticallyflat region and then matched along their common overlap region.
For that purpose, it is useful to define the dimensionless horizon radius x = (r − r_{+})/r_{+} such that the outer horizon is at x = 0. The two other singular points of the radial equation (159) are the inner horizon x = −τ_{H} and spatial infinity x = ∞. One then simply partitions the radial axis into two regions with a large overlap as

Nearhorizon region: x ≪ 1,

Far region: x ≫ τ_{H},

Overlap region: τ_{H} ≪ x ≪ 1.
The overlap region is guaranteed to exist thanks to (147).
In the nearextremal regime, the absorption probability σ_{abs} gets a contribution from each region as
where ψ(x = x_{B})^{2} is the norm of the scalar field in the overlap region with τ_{H} ≪ x_{B} ≪ 1. One can conveniently normalize the scalar field such that it has unit incoming flux dE_{in}/dt = 1. The contribution \(\sigma _{{\rm{abs}}}^{{\rm{match}}}\) is then simply a normalization that depends on the coupling of the nearhorizon region to the far region.
In the nearhorizon region, the radial equation reduces to a much simpler hypergeometric equation. One can in fact directly obtain the same equation from solving for a probe in a nearextremal nearhorizon geometry of the type (79), which is, as detailed in Section 2.3, a warped and twisted product of AdS_{2} × S^{2}. The presence of poles in the hypergeometric equation at x = 0 and x = −τ_{H} requires one to choose the AdS_{2} base of the nearhorizon geometry to be
One can consider the nondiagonal term 2Γ(θ)γ(θ)kr dtdϕ appearing in the geometry (79) as a U(1) electric field twisted along the fiber spanned by dϕ over the AdS_{2} base space. It may then not be surprising that the dynamics of a probe scalar on that geometry can be expressed equivalently as a charged massive scalar on AdS_{2} with two electric fields: one coming from the U(1) twist in the fourdimensional geometry, and one coming from the original U(1) gauge field. By SL(2, ℝ) invariance, these two gauge fields are given by
The coupling between the gauge fields and the charged scalar is dictated by the covariant derivative
where ∇ is the LeviCivita connection on AdS_{2} and q_{1} and q_{2} are the electric charge couplings. One can rewrite more simply the connection as q_{eff}A, where q_{eff} = q_{1}α_{1} + q_{2}α_{2} is the effective total charge coupling and A = xdt is a canonicallynormalized effective gauge field. The equation for a charged scalar field Φ(t, x) with mass μ_{eff} is then
Taking \(\Phi (t,r) = {e^{ i{\omega _{{\rm{ef}}{{\rm{f}}^\tau}{H^t}}}}}\Phi (x)\), we then obtain the following equation for Φ(x),
Using the field redefinition
we obtain the equivalent equation,
where the potential is
Here, the parameters a, b, c are related to μ_{eff}, q_{eff} and ω_{eff} as^{Footnote 14}
Finally, comparing Eq. (171) with (159), where the potential V^{s}(r) is approximated by the nearhorizon potential, we obtain that these equations are identical, as previously announced, after identifying the parameters as
Moreover, using the expression of the frequency (150) near extremality, one can write the effective charge in the convenient form
where the extremal FrolovThorne temperatures T_{e} and T_{ϕ} are defined in (74).
We can now understand that there are two qualitatively distinct solutions for the radial field R^{s}(x). Uncharged fields in AdS_{2} below a critical mass are unstable or tachyonic, as shown by Breitenlohner and Freedman [55]. Charged particles in an electric field on AdS_{2} have a modified BreitenlohnerFreedman bound
in which the square mass is lifted up by the square charge. Below the critical mass, charged scalars will be unstable to Schwinger pair production [233, 184]. Let us define
Stable modes will be characterized by a real β ≥ 0, while unstable modes will be characterized by an imaginary β. This distinction between modes is distinct from superradiant and nonsuperradiant modes. Indeed, from the definition of n (152), superradiance happens at nearextremality when n < 0.
We can now solve the equation, impose the boundary conditions, compute the flux at the horizon and finally obtain the nearhorizon absorption probability. The computation can be found in [53, 106, 160]. The net result is as follows. A massive, charge e, spin s = 0, ½ field with energy ω and angular momentum m and real β > 0 scattered against a KerrNewman black hole with mass and charge has nearregion absorption probability
for a massless spin s = 1, 2 field scattered against a Kerr black hole, exactly the same formula applies, but with e = Q = 0. The absorption probability in the case where β is imaginary can be found in the original papers [235, 257].
We will now show that the formulae (178) are Fourier transforms of CFT correlation functions. We will not consider the scattering of unstable fields with β imaginary in this review. We refer the reader to [53] for arguments on how the scattering absorption probability of unstable spin 0 modes around the Kerr black hole match with dual CFT expectations.
5.4 Microscopic greybody factors
In this section we model the emission amplitudes from a microscopic point of view. We will first discuss nearextremal spinning black holes and we will extend our discussion to general charged and/or spinning black holes at the end of this section.
The working assumption of the microscopic model is that the nearhorizon region of any nearextremal spinning black hole can be described and therefore effectively replaced by a dual twodimensional CFT. In the dual CFT picture, the nearhorizon region is removed from the spacetime and replaced by a CFT glued along the boundary. Therefore, it is the nearhorizon region contribution alone that we expect to be reproduced by the CFT. The normalization \(\sigma _{{\rm{abs}}}^{{\rm{match}}}\) defined in (164) will then be dictated by the explicit coupling between the CFT and the asymptoticallyflat 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 nearhorizon 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 c_{J} = 12J sufficed to account for the extremal blackhole entropy.
We will now assume that quantum gravity states form a representation of both a left and a rightmoving Virasoro algebra with generators L_{n} and \({{\bar L}_n}\). The value of the rightmoving central charge will be irrelevant for our present considerations. At nearextremality, the left sector is thermally excited at the extremal leftmoving temperature (67). We take as an assumption that the rightmoving temperature is on the order of the infinitesimal reduced Hawking temperature. As discussed in Sections 2.9 and 4.2, the presence of rightmovers destabilize the nearhorizon geometry. For the KerrNewman black hole, we have
In order to match the bulk scattering amplitude for nearextremal KerrNewman black holes, the presence of an additional leftmoving 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 \(\mu _L^{J,e}\) defined in (140) associated with the U(1)_{e} electric current. Using the expressions (74), we find for the KerrNewman black hole the value
As done in [53], we also assume the presence of a rightmoving U(1) current algebra, whose zero eigenmode \({{\bar J}_0}\) is constrained by the level matching condition
The level matching condition is consistent with the fact that the excitations are labeled by three (ω, m, q_{e}) instead of four conserved quantities. The CFT state is then assumed to be at a fixed chemical potential μ_{R}. This rightmoving current algebra cannot be detected in the extremal nearhorizon geometry in the same way that the rightmoving Virasoro algebra cannot be detected, so its existence is conjectural (see, however, [67]). This rightmoving current algebra and the matching condition (182) will turn out to be adequate to match the gravitational result, as detailed below. Note that threedimensional 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 nearextremal KerrNewman black hole is given by the product of a U(1) current and a Virasoro algebra in both sectors,
In the description where the nearhorizon region of the black hole is replaced by a CFT, the emission of quanta is due to couplings
between bulk modes Φ_{bulk} and operators \({\mathcal O}\) in the CFT. The structure of the scattering cross section depends on the conformal weights (h_{L}, h_{R}) and charges (q_{L}, q_{R}) of the operator. The normalization of the coupling is also important for the normalization of the cross section.
The conformal weight h_{R} can be deduced from the transformation of the probe field under the scaling \({{\bar L}_0} = t{\partial _t}  r{\partial _r}\) (24) in the overlap region τ_{H} ≪ x ≪ 1. The scalar field in the overlap region is \(\Phi \sim {\Phi _0}(t,\theta ,\phi){r^{ {1 \over 2} + \beta}} + {\Phi _1}(t,\theta ,\phi){r^{ {1 \over 2}  \beta}}\). Using the rules of the AdS/CFT dictionary [265], this behavior is related to the conformal weight as \(\Phi \sim {r^{{h_{R  1}}}},{r^{ {h_R}}}\). One then infers that [160]
The values of the charges (q_{L}, q_{R}) are simply the U(1) charges of the probe,
where the charge q_{R} = m follows from the matching condition (182). We don’t know any firstprinciple argument leading to the values of the rightmoving chemical potential μ_{R}, the rightmoving temperature T_{R} and the leftmoving conformal weight h_{L}. 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 (m = 0) of any integer spin of the Kerr black hole, the conformal weight (185) is an integer
where l = 0, 1, …. One can generalize this result to any axisymmetric perturbation of any vacuum fivedimensional nearhorizon geometry [224]. Counterexamples exist in higher dimensions and for black holes in AdS [129]. There is no microscopic accounting of this feature at present.
Throwing the scalar Φ_{bulk} at the black hole is dual to exciting the CFT by acting with the operator \({\mathcal O}\). Reemission is represented by the action of the Hermitian conjugate operator. Therefore, the absorption probability is related to the thermal CFT twopoint function [209]
where t^{±} are the coordinates of the left and right moving sectors of the CFT. At left and right temperatures (T_{L}, T_{r}) and at chemical potentials (μ_{L}, μ_{r}) an operator with conformal dimensions (h_{L}, h_{r}) and charges (q_{L}, q_{R}) has the twopoint 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 obtain^{Footnote 15}
where
In order to compare the bulk computations to the CFT result (191), we must match the conformal weights and the reduced momenta \(({{\tilde \omega}_L},{{\tilde \omega}_R})\). 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 SL(2, ℝ)_{R} conformal invariance. The left conformal weight is natural for a spin s field since h_{L} − h_{R} = s. The value for \({{\tilde \omega}_L}\) is consistent with the temperature (180) and chemical potential (181). Indeed, since the leftmovers span the ϕ direction of the black hole, we have ω_{L} = m. We then obtain
after using the value (175). The value of \({{\tilde \omega}_R}\) is fixed by the matching. It determines one constraint between ω_{R}, μ_{R} and T_{R}. However, there is a subtlety in the above matching procedure. The conformal weights h_{L} and h_{R} depend on m through β. This m dependence cannot originate from ω_{l} = m since ω_{l} is introduced after the Fourier transform (190), while h_{L}, h_{R} are already defined in (189). One way to introduce this m dependence is to assume that there is a rightmoving current algebra and that the dual operator \({\mathcal O}\) has the zeromode charge q_{R} = m, which amounts to imposing the condition (182). (It is then also natural to assume that the chemical potential is μ_{R} ∼ Ω_{J}, but the matching does not depend on any particular value for μ_{R} [53].) This justifies why a rightmoving current algebra was assumed in the CFT. The dependence of the conformal weights in q_{e} is similarly made possible thanks to the existence of the leftmoving current with q_{L} = q_{e}. 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 \({{\tilde \omega}_L},{{\tilde \omega}_R}\) using alternative CFT descriptions such as the CFT_{Q} with Virasoro algebra along the gauge field direction, and the mixed SL(2, ℤ) family of CFTs. We can indeed rewrite (192) in the alternative form
where \(T_L^Q = {R_\chi}{T_e}\) is the leftmoving temperature of the CFT_{Q}, \(\mu _L^{\phi ,Q}\) is the chemical potential defined in (144) and q_{χ} = R_{χ}q_{e} is the probe electric charge in units of the KaluzaKlein circle length. The identification of the rightmoving sector is unchanged except that now q_{R} = q_{e}. One can trivially extend the matching with the SL(2, ℤ) family of CFTs conjectured to describe the (near)extremal KerrNewman black hole.
In summary, nearsuperradiant absorption probabilities of probes in the nearhorizon region of nearextremal black holes are exactly reproduced by conformal field theory twopoint functions. This shows the consistency of a CFT description (or multiple CFT descriptions in the case where several U(1) symmetries are present) of part of the dynamics of nearextremal black holes. We expect that a general scattering theory around any nearextremal blackhole solution of (1) will also be consistent with a CFT description, as supported by all cases studied beyond the KerrNewman black hole [106, 73, 242, 71, 80, 46].
Finally, let us note finally that the dynamics of the CFTs dual to the KerrNewman geometry close to extremality can be further investigated by computing threepoint correlation functions in the nearhorizon geometry, as initiated in [40, 39].
5.5 Microscopic accounting of superradiance
We mentioned in Section 2.1 that extremal spinning black holes that do not admit a globallydefined timelike Killing vector spontaneously emit quanta in the range of frequencies (11). This quantum effect is related by detailed balance to the classical effect of superradiant wave emission, which occur in the same range of frequencies.
It has been argued that the bound (11) essentially follows from FermiDirac statistics of the fermionic spincarrying degrees of freedom in the dual twodimensional CFT [121] (see also [132]). These arguments were made for specific black holes in string theory but one expects that they can be applied to generic extremal spinning black holes, at least qualitatively. Let us review these arguments here.
One starts with the assumption that extremal spinning black holes are modeled by a 2d CFT, where the left and right sectors are coupled only very weakly. Therefore, the total energy and entropy are approximately the sum of the left and right energies and entropies. The state corresponding to an extremal spinning black hole is modeled as a filled Fermi sea on the right sector with zero entropy and a thermal state on the left sector, which accounts for the blackhole entropy. The rightmoving fermions form a condensate of aligned spins s = +1/2, which accounts for the macroscopic angular momentum. It is expected from details of emission rates in several parametric regimes that fermions are only present on the right sector, while bosons are present in both sectors [105, 106].
Superradiant spontaneous emission is then modeled as the emission of quanta resulting from interaction of a left and a rightmoving mode. Using details of the model such as the fact that the Fermi energy should be proportional to the angular velocity Ω_{J}, one can derive the bound (11). We refer the reader to [132] for further details. It would be interesting to better compare these arguments to the present setup, and to see how these arguments could be generalized to the description of the bound (12) for static extremal rotating black holes. Let us finally argue that the existence of a qualitative process of superradiant emission in these models further supports the conjecture that the dual theory to extremal black holes is a chiral limit of a 2d CFT instead of a chiral CFT with no rightmoving sector.
6 Hidden Symmetries of NonExtremal Black Holes
In Section 4 we described evidence showing that the asymptotic growth of states of extremal rotating or charged black holes is controlled by a chiral half of a twodimensional CFT, at least in the semiclassical limit. We also reviewed in Section 5 how the nearhorizon dynamics of probes can be reproduced by manipulating nearchiral CFT twopoint functions in the nearextremal limit. These analyses strongly rely on the existence of a decoupled nearhorizon geometry for all extremal or nearextremal black holes. Away from extremality, one cannot decouple the horizon from the surrounding geometry. Therefore, it is unclear whether any of the previous considerations will be useful in describing nonextremal geometries.
It might then come as a surprise that even away from extremality, conformal invariance is present in the dynamics of probe scalar fields around the Kerr black hole in a specific regime (at low energy and close enough to the black hole as we will make more precise below) [68]. In that regime, the probe scalar field equation can be written in a SL(2, ℝ) × SL(2, ℝ) invariant fashion in a region close enough to the horizon. Such a local hidden symmetry is nongeometric but appears in the probe dynamics. The 2π periodic identification of the azimuthal angle ϕ breaks globallyconformal symmetry. Using the properties of this representation of conformal invariance, one can then argue that the Kerr black hole is described by a CFT with specific left and rightmoving temperatures [68]
The wellknown lowenergy scattering amplitudes coincide with correlators of a twodimensional CFT with these temperatures. Finally, quite remarkably, the entropy of the Kerr black hole is then reproduced by Cardy’s formula if one assumes that the CFT has left and rightmoving central charges equal to the value c_{L} = c_{R} = 12J, which matches with the value for the leftmoving central charge (129) derived at extremality.^{Footnote 16}
These observations are consistent with the interpretation of a 2d CFT dual to the Kerr black hole, but the existence of such a CFT is conjectural. For example, there is no known derivation of two Virasoro algebras with central charges c_{L} = c_{R} = 12J from the nonextremal Kerr geometry. Asymptotic symmetry group methods are not directly applicable here because the horizon is not an isolated system. Therefore, it is unclear how these Virasoro algebras could be derived in Kerr. However, as argued in [68], the resulting picture shows a remarkable cohesiveness and only future research can prove or disprove such a CFT interpretation.
Given the successful generalization of the extremal Kerr/CFT correspondence to several independent extremal black hole/CFT correspondences in gravity coupled to matter, as we reviewed above, it is natural to test the ideas proposed in [68] to more general black holes than the Kerr geometry. First, hidden symmetry can be found around the nonextremal ReissnerNordström black hole [81, 77] under the assumption that the gauge field can be understood as a KaluzaKlein gauge field, as done in the extremal case [160]. One can also generalize the analysis to the KerrNewman black hole [263, 74, 78]. In complete parallel with the existence of an SL(2, ℤ) family of CFT descriptions, there is a class of hidden SL(2, ℝ) × SL(2, ℝ) symmetries of the KerrNewman black hole related with SL(2, ℤ) transformations [75]. What has not been noted in the literature so far is that each member of the SL(2, ℤ) family of CFTs describes only probes with a fixed ratio of probe angular momentum to probe charge as we will discuss in detail in Section 6.4. Therefore, one needs a family of CFTs to fully describe the dynamics of low energy, low charge and low mass probes. Remarkably, for all cases where a hidden local conformal invariance can be described, the nonextremal blackhole entropy matches with Cardy’s formula using the central charges c_{R} = c_{L} and using the value c_{L} in terms of the quantized conserved charges derived at extremality. Fivedimensional asymptoticallyflat black holes were also discussed in [189, 80].
In attempting to generalize the hidden symmetry arguments to fourdimensional black holes in AdS one encounters an apparent obstruction, as we will discuss in Section 6.2. It is expected that hidden symmetries are present at least close to extremality, as illustrated by fivedimensional analogues [46]. However, the structure of the wave equation is more intricate far from extremality because of the presence of complex poles, which might have a role to play in microscopic models [102].
Quite surprisingly, one can also find a single copy of hidden SL(2, ℝ) symmetry around the Schwarzschild black hole [45], which turns out to be globally defined. As a consequence, no dual temperature can be naturally defined in that case. This hidden symmetry can be understood as a special case of a generalized notion of hidden conformal symmetry around the Kerr geometry [202]. At present, it is unclear how these hidden symmetries fit in the general picture of the Kerr/CFT correspondence since the derivations of the central charges of the CFT dual to Kerr, ReissnerNordström or KerrNewman black holes are done at extremality, which clearly cannot be done in the Schwarzschild case.
All arguments presented in the literature so far have been derived for a probe scalar field. It is not clear if any of these arguments can be generalized to higherspin fields, and, if such, a generalization would give the same values for the left and rightmoving CFT temperatures. It would certainly be interesting to understand whether this is a technical obstruction that can be overcome or whether it is a fundamental limitation in the CFT descriptions.
Hidden symmetries in asymptoticallyflat spacetimes only appear in a region close enough to the black hole. It has been suggested that one deform the geometry far from the black hole such that hidden symmetries appear in the entire resulting geometry [108, 107]. The resulting “subtracted” geometries are not asymptotically flat and are supported by additional matter fields [108, 107, 101]. The nature of these geometries and their role in the Kerr/CFT correspondence remains to be clarified. We will therefore not cover these constructions in this review.
In what follows, we present a summary of the derivation of the hidden symmetries of the KerrNewman black hole and we discuss their CFT interpretation. We will limit our presentation to the approach of [68] but we will generalize the discussion to the KerrNewman black hole, which contains several new interesting features. In particular, we will show that each member of the conjectured SL(2, ℤ) family of CFTs controls part of the dynamics of low energy, low charge and low mass probes. We do not review the matching of absorption probabilities with CFT correlation functions. This matching is very similar to the analysis already performed in Section 5 at nearextremality and it follows from local conformal invariance. As noted in [68], the only difference is that in the present context the region close enough to the horizon is not geometrically a nearhorizon region, but it does not affect the discussion.
6.1 Scalar wave equation in KerrNewman
Let us first discuss probe scalar fields on the KerrNewman black hole. The KleinGordon equation for a charged massive spin 0 field of mass μ and charge q_{e} was analyzed in Section 5.2. Expanding in eigenmodes and using the fact that the equation is separable, we have
The equations for the functions S(θ) and R(r) were written in (158) and (159). Substituting \(A_{a\omega ,l,m}^0 = {K_l} + {a^2}({\omega ^2}  {\mu ^2})\), the equations can be written in a convenient way as
where Δ(r) = (r − r_{+})(r − r_{−}). The function α(r) is defined as
and is evaluated either at r_{+} or r_{−} and
These equations can be solved by Heun functions, which are not among the usual special functions. A solution can be found only numerically.
6.2 Scalar wave equation in KerrNewmanAdS
The equations for probe scalars fields on the KerrNewmanAdS black hole can be obtained straightforwardly. We consider only massless probes for simplicity. Using again the decomposition (197), the KleinGordon equation is decoupled into an angular equation
and a radial equation
where C_{l} is a separation constant and the various functions and parameters in the equations have been defined in Section 2.4.4. In the flat limit, Eqs. (198)–(199) are recovered with K_{l} = C_{l} + 2maω − a^{2}ω^{2}.
The radial equation has a more involved form than the corresponding flat equation (199) due to the fact that Δ_{r} is a quartic instead of a quadratic polynomial in r; see (42). More precisely, the quartic polynomial Δ_{r} can be written as
where r_{c} is a complex root. The radial equation is a general Heun’s equation due to the presence of two conjugate complex poles in (203) in addition to the two real poles corresponding to the inner and outer horizons and the pole at infinity.
It has been suggested that all these poles have a role to play in the microscopic description of the AdS black hole [102]. It is an open problem to unravel the structure of the hidden symmetries, if any, of the full nonextremal radial equation (203). It has been shown that in the context of fivedimensional black holes, one can find hidden conformal symmetry in the nearhorizon region close to extremality [46]. It is expected that one could similarly neglect the two complex poles in the nearhorizon region of nearextremal black holes, but this remains to be checked in detail.^{Footnote 17} Since much remains to be understood, we will not discuss AdS black holes further.
6.3 Nearregion scalarwave equation
Let us go back to the scalar wave equation around the KerrNewman black hole. We will now study a particular range of parameters, where the wave equations simplify. We will assume that the wave has low energy and low mass as compared to the black hole mass and low electric charge as compared to the black hole charge,
where ϵ ≪ 1. From these approximations, we deduce that ωa, ωr_{+}, ωQ and μa = O(ϵ) as well.
We will only look at a specific region of the spacetime — the “near region” — defined by
Note that the near region is a distinct concept from the nearhorizon region r − r_{+} ≪ M. Indeed, for sufficiently small ω and μ, the value of r defined by the near region can be arbitrarily large.
Using the approximations (205), the wave equation greatly simplifies. It can be solved both in the near region and in the far region r ≫ M in terms of special functions. A complete solution can then be obtained by matching near and far solutions together along a surface in the matching region M ≪ r ≪ ω^{−1}. As noted in [68], conformal invariance results from the freedom to locally choose the radius of the matching surface within the matching region.
More precisely, using (205), the angular equation (198) reduces to the standard Laplacian on the twosphere
The solutions e^{imϕ}S(θ) are spherical harmonics and the separation constants are
In the near region, the function V(r) defined in (201) is very small, V(r) = O(ϵ^{2}). The near region scalarwave equation can then be written as
where α(r) has been defined in (200).
6.4 Local SL(2, ℝ) × SL(2, ℝ) symmetries
We will now make explicit the local SL(2, ℝ) × SL(2, ℝ) symmetries of the nearhorizon scalar field equations (209). For this purpose it is convenient to define the “conformal” coordinates (ω^{±}, y) defined in terms of coordinates (t, r, ϕ′) by (see [68] and [210] for earlier relevant work)
The change of coordinates is locally invertible if ΔΩ = Ω_{L} − Ω_{R} ≠ 0. We choose the chirality ΔΩ > 0, as it will turn out to match the chirality convention in the description of extremal black holes in Section 4.4.
Several choices of coordinate ϕ′ ∼ ϕ′ + 2π will lead to independent SL(2, ℝ) × SL(2, ℝ) symmetries. For the Kerr black hole, there is only one meaningful choice: ϕ′= ϕ. For the ReissnerNordström black hole, we identify ϕ′ = χ/Rχ, where χ is the KaluzaKlein coordinate that allows one to lift the gauge field to higher dimensions, as done in Section 4.3. For the KerrNewman black hole, we use, in general, a coordinate system (ϕ′, χ′) ∼ (ϕ′, χ′ + 2π) ∼ (ϕ′ + 2π, χ′) parameterized by a SL(2, ℤ) transformation
with p_{1}p_{4} − p_{2}p_{3} = 1 so that
Let us define locally the vector fields
and
These vector fields obey the SL(2, ℝ) Lie bracket algebra,
and similarly for \(({{\bar H}_0},{{\bar H}_{\pm 1}})\). Note that
The SL(2, ℝ) quadratic Casimir is
In terms of the coordinates (r, t, ϕ′), the Casimir becomes
where Δ(r) = (r − r_{+})(r − r_{−}).
We will now match the radial wave equation around the KerrNewman black hole in the near region (209) with the eigenvalue equation
The scalar field has the following eigenvalues ∂_{t}Φ = −iωΦ and ∂_{ϕ}Φ = imΦ. In the case where an electromagnetic field is present, one can perform the uplift (2) and consider the fivedimensional gauge field (6). In that case, the eigenvalue of the fivedimensional gauge field under ∂_{χ} is the electric charge ∂_{χ}Φ = iq_{e}Φ. Let us denote the eigenvalue along ∂_{ϕ′} as im′ ≡ i(p_{4}m − p_{3}q_{e}R_{χ}). Eqs. (209) and (220) will match if and only if the two following equations are obeyed
where α(r) has been defined in (200).
For simplicity, let us first discuss the case of zero probe charge q_{e} = 0 and nonzero probe angular momentum m ≠ 0. The matching equations then admit a unique solution
upon choosing ϕ′ = ϕ (and χ′ = χ/R_{χ}). This shows in particular that the Kerr black hole has a hidden symmetry, as derived originally in [68].
For probes with zero angular momentum m = 0, but electric charge q_{e} ≠ 0, there is also a unique solution,
upon choosing ϕ′ = χ/R_{χ} (and χ′ = −ϕ). This shows, in particular, that the ReissnerNordström black hole admits a hidden symmetry, as pointed out in [81, 77].
Finally, one can more generally solve the matching equation for any probe scalar field whose probe angular momentum and probe charge are related by
In that case, one chooses the coordinate system (211) and the unique solution is then
When p_{1} = 0 and Q ≠ 0 or p_{2} = 0 and J ≠ 0, one recovers the two previous particular cases. The condition (224) is equivalent to the fact that the scalar field has zero eigenvalue along ∂_{χ′}. Since m and q_{e}R_{χ} are quantized, as derived in (7), there is always (at least) one solution to (224) with integers p_{1} and p_{2}.
In conclusion, any low energy and low mass scalar probe in the near region (206) of the Kerr black hole admits a local hidden SL(2, ℝ) × SL(2, ℝ) symmetry. Similarly, any low energy, low mass and low charge scalar probe in the near region (206) of the ReissnerNordström black hole admits a local hidden SL(2, ℝ) × SL(2, ℝ) symmetry. In the case of the KerrNewman black hole, we noticed that probes obeying (205) also admit an SL(2, ℝ) × SL(2, ℝ) hidden symmetry, whose precise realization depends on the ratio between the angular momentum and the electric charge of the probe. For a given ratio (224), hidden symmetries can be constructed using the coordinate ϕ′ = p_{1}ϕ + p_{2}χ/R_{χ}. Different choices of coordinate ϕ′ are relevant to describe different sectors of the low energy, low mass and low charge dynamics of scalar probes in the near region of the KerrNewman black hole. The union of these descriptions cover the entire dynamical phase space in the near region under the approximations (205)–(206).
6.5 Symmetry breaking to U(1)_{L} × U(1)_{R}
The vector fields that generate the SL(2, ℝ) × SL(2, ℝ) symmetries are not globally defined. They are not periodic under the angular identification
Therefore, the SL(2, ℝ) symmetries cannot be used to generate new global solutions from old ones. In other words, solutions to the wave equation in the near region do not form SL(2, ℝ) × SL(2, ℝ) representations. In the (ω^{+}, ω^{−}) plane defined in (210), the identification (226) is generated by the SL(2, ℝ)_{L} × SL(2, ℝ)_{R} group element
as can be deduced from (217). This can be interpreted as the statement that the SL(2, ℝ)_{L} × SL(2, ℝ)_{R} symmetry is spontaneously broken to the U(1)_{L} × U(1)_{R} symmetry generated by \(({{\bar H}_0},{{\bar H}_0})\).
The situation is similar to the BTZ black hole in 2+1 gravity that has a SL(2, ℝ)_{L} × SL(2, ℝ)_{R} symmetry, which is spontaneously broken by the identification of the angular coordinate. This breaking of symmetry can be interpreted in that case as placing the dual CFT to the BTZ black hole in a density matrix with left and rightmoving temperatures dictated by the SL(2, ℝ)_{L} × SL(2, ℝ)_{R} group element generating the 2π identification of the geometry [210].
In the case of nonextremal blackhole geometries, one can similarly interpret the symmetry breaking using a CFT as follows [68]. First, we need to assume that before the identification, the near region dynamics is described by a dual twodimensional CFT, which possesses a ground state that is invariant under the full SL(2, ℝ)_{L} × SL(2, ℝ)_{R} symmetry. This is a strong assumption, since there are several (apparent) obstacles to the existence of a ground state, as we already discussed in the case of extremal black holes; see Section 4.4. Nevertheless, assuming the existence of this vacuum state, the two conformal coordinates (ω^{+}, ω^{−}) can be interpreted as the two null coordinates on the plane where the CFT vacuum state can be defined. At fixed r, the relation between conformal coordinates (ω^{+}, ω^{−}) and BoyerLindquist (ϕ, t) coordinates is, up to an rdependent scaling,
where
This is precisely the relation between Minkowski (ω^{±}) and Rindler (t^{±}) coordinates. The periodic identification (226) then requires that the Rindler domain be restricted to a fundamental domain under the identification
generated by the group element (227).
The quantum state describing this accelerating strip of Minkowski spacetime is obtained from the SL(2, ℝ)_{L} × SL(2, ℝ)_{R} invariant Minkowski vacuum by tracing over the quantum state in the region outside the strip. The result is a thermal density matrix at temperatures (T_{L}, T_{r}). Hence, under the assumption of the existence of a CFT with a vacuum state, nonextremal black holes can be described as a finite temperature (T_{L}, T_{r}) mixed state in a dual CFT.
It is familiar from the threedimensional BTZ black hole that the identifications required to obtain extremal black holes are different than the ones required to obtain nonextremal black holes [27, 210]. Here as well, the vector fields (214)–(215) are not defined in the extremal limit because the change of coordinates (210) breaks down. Nevertheless, the extremal limit of the temperatures T_{L} and T_{r} match with the temperatures defined at extremality in Section 5.4. More precisely, the temperatures T_{L} and T_{r} defined in (222), (223) and (225) match with the temperatures defined at extremality T_{ϕ}, R_{χ}T_{e} and \({({p_1}T_\phi ^{ 1} + {p_2}{({R_\chi}{T_e})^{ 1}})^{ 1}}\), respectively, where T_{ϕ} and T_{e} are defined in (74). This is consistent with the interpretation that states corresponding to extremal black holes in the CFT can be defined as a limit of states corresponding to nonextremal black holes.
6.6 Entropy matching
We will now argue that the temperatures T_{L} and T_{r} obtained in Section 6 combined with the analysis at extremality in Section 4 lead to a (several) microscopic counting(s) of the black hole entropy of the Kerr, ReissnerNordström and KerrNewman black holes.
Let us assume that there is a twodimensional CFT (CFT_{J}) describing the Kerr black hole, a twodimensional CFT (CFT_{Q}) describing the ReissnerNordström black hole and a SL(2, ℤ) family of twodimensional CFTs \(({\rm{CF}}{{\rm{T}}_{({p_1},{p_2},{p_3})}},{p_1},{p_2},{p_2} \in {\mathbb Z})\) describing the KerrNewman black hole. If these CFTs are dual to the black hole, the entropy is reproduced by Cardy’s formula
which is valid when T_{L} ≫ 1, T_{R} ≫ 1. As already mentioned in Section 4.4 and argued in [156], the regime T_{L} ≫ 1, T_{R} ≫ 1 is not a necessary condition for Cardy’s formula to be valid if these CFTs have special properties such as admitting a long string picture, as reviewed in Section 3.3.
Let us discuss the values of the central charges. In a CFT, the difference c_{R}− c_{L} is proportional to the diffeomorphism anomaly of the CFT [188, 187]. One can then argue from diffeomorphism invariance that the two left and right sectors should have the same value for the central charge,
We obtained the value c_{L} at extremality in Section 4.3 and checked that Cardy’s formula reproduces the extremal blackhole entropy. One way to uniquely fix the value c_{L} away from extremality would consist in matching Cardy’s formula (232) with the KerrNewman blackhole entropy
using (233) and the values for the temperatures derived in Section 6.4. Therefore, the matching of blackhole entropy is true by construction, which is clearly unsatisfactory. It would be more satisfactory to have an independent computation of c_{L} away from extremality, but such a computation is currently not available.
However, the resulting central charge c_{L} is, however, nontrivial. For the CFT_{J}, we obtain c_{L} = 12 J. For CFT_{Q}, we have c_{Q} = 6Q^{3}/R_{χ} and for the \({\rm{CF}}{{\rm{T}}_{({p_1},{p_2},{p_3})}}\), we find \({c_{({p_1},{p_2})}} = 6({p_1}(2J) + {p_2}{Q^3}/{R_\chi})\). Quite remarkably, these central charges are expressed solely in terms of quantized charges. They do not depend on the mass of the black hole. This is a nontrivial feature that has no explanation so far.
The presence of several CFTs dual to the KerrNewman black hole is curious but not inconsistent. Each CFT describes part of the lowenergy dynamics of probe scalar fields and multiple CFTs are needed in order to reproduce the full dynamics for arbitrary ratios of the probe angular momentum to probe electric charge. Therefore, each CFT description has therefore a range of applicability away from extremality.
7 Summary and Open Problems
7.1 Summary
Let us summarize the key results that have been derived so far. Any extremal black hole containing a compact U(1) axial symmetry admits a Virasoro algebra in its nearhorizon geometry with a nontrivial central charge. The blackhole entropy is reproduced by a chiral half of Cardy’s formula. This result is robust for any diffeomorphisminvariant theory and holds even including scalar and gauge field couplings and higherderivative corrections. Moreover, if a U(1) gauge field can be geometrized into a KaluzaKlein vector in a higherdimensional spacetime, a Virasoro algebra can be defined along the KaluzaKlein compact U(1) direction and all analysis goes through in a similar fashion as for the axial U(1) symmetry. The deep similarity between the effects of rotation and electric charge can be understood from the fact that these charges are on a similar footing in the higherdimensional geometry. When two U(1) symmetries are present, one can mix up the compact directions using a modular transformation and the construction of Virasoro algebras can still be made.
Independent of these constructions, the scattering probabilities of probes around the nearextremal KerrNewman black hole can be reproduced near the superradiant bound by manipulating nearchiral thermal twopoint functions of a twodimensional CFT. The result extends straightforwardly to other asymptoticallyflat or AdS black holes in various gravity theories. Finally away from extremality, hidden SL(2, ℝ) × SL(2, ℝ) symmetries are present in some scalar probes around the KerrNewman black hole close enough to the horizon. We showed that several CFTs are required to account for the entire probe dynamics in the near region in the regime of small mass, small energy and small charge. This analysis does not extend straightforwardly to AdS black holes.
These results — obtained in gravity coupled to matter — are naturally accounted for by assuming that the microstates of asymptoticallyflat black holes, at extremality and away from extremality, can be described by 2d CFTs and that the microstates of asymptoticallyAdS black holes at extremality can be described by chiral halves of 2d CFTs. Scattering amplitudes and hidden symmetries are also accounted for by assuming that part of the dynamics of black holes can be mapped to the dynamics of these CFTs once they are suitably coupled to the exterior blackhole region. By consistency with the gravitational analysis, several CFT descriptions are available when several compact U(1) symmetries are present. The existence of such CFTs is conjectural and only future research will tell how far these Kerr/CFT correspondences and their extensions can be made more precise.
A fair concluding remark would be that our understanding of the Kerr, ReissnerNordström and KerrNewman black hole has increased over the last four years, but there is still a long road ahead of us to comprehend what these CFTs really are and what they are telling us about the nature of quantum black holes.
7.2 Set of open problems
We close this review with a list of open problems. We hope that the interested reader will tackle them with the aim of shedding more light on the Kerr/CFT correspondence. We tried to order the problems with increasing difficulty but the evaluation is rough and highly subjective.

1.
Hidden symmetries have been discussed so far for spin 0 probes. Discuss hidden symmetries for a probe gauge field or a probe graviton on Kerr or KerrNewman. Does one obtain the same temperatures T_{L} and T_{R} as in the scalar probe case?

2.
A black hole in de Sitter spacetime can be extremal in the sense that its outer radius coincides with the cosmological horizon. The resulting geometry, called the rotating Narirai geometry, has many similarities with the nearhorizon geometries of extremal black holes in flat spacetime or in AdS spacetime. The main difference is that the nearhorizon geometry is a warped product of dS_{2} with S^{2} instead of AdS_{2} with S^{2}. It has been conjectured that these extremal black holes are dual to the chiral half of a Euclidean CFT [7]. Test the conjecture by generalizing all arguments of the Kerr/CFT correspondence to this cosmological setting.

3.
Away from extremality, it is curious that the rightmoving temperature is given by R = T_{H}/Ω_{J} for the KerrNewman black hole. Account for this fact. Also, for all known asymptoticallyflat extremal black holes in Einstein gravity coupled to matter, the product of the horizon areas of the inner and outer horizon can be expressed in terms of quantized charges (J, Q, …) and fundamental constants only [195, 103, 106]. Explain this feature from a fundamental perspective.

4.
In the analysis of nearextremal superradiant scattering for any spin, we discarded the unstable modes that are below the BreitenlohnerFreedman bound. Such modes have imaginary β; see (177). Clarify the match between these modes and CFT expectations for the KerrNewman black hole.

5.
The probe scalar wave equation in KerrNewmanAdS has two complex poles in addition to poles corresponding to the inner and outer horizon and infinity. This prevented a straightforward generalization of the hidden SL(2, ℝ) × SL(2, ℝ) symmetry. Clarify the role of these additional poles. Also explain why the product of all horizon areas (inner, outer and complex horizons) seems in general not to depend on the mass of the black hole [102].

6.
Nearhorizon geometries of blackhole solutions of (1) have been classified. Classify the fourdimensional nearhorizon geometries of extremal black holes for gravity coupled to charged scalars, massive vectors, pforms and nonabelian gauge fields.

7.
Compute the central charges c_{L} and c_{R} away from extremality. Also, compute the quantum corrections to the central charge c_{L} and investigate the matching between the quantumcorrected entropy of extremal black holes derived in [241] and the asymptotic growth of states in the dual CFT.

8.
Understand how the extension of the Kerr/CFT correspondence to extremal AdS black holes fits within the AdS/CFT correspondence. As discussed in [204], the extremal AdSKerr/CFT correspondence suggests that one can identify a nontrivial Virasoro algebra acting on the lowenergy states of strongly coupled large N superYangMills theory in an extremal thermal ensemble. Try to make this picture more precise.

9.
From the point of view of 2d CFTs, study if a SL(2, ℤ) action exists that transforms a CFT into another CFT. This would clarify the existence of an SL(2, ℤ) family of CFTs dual to the KerrNewman black hole. Note that this can be done for threedimensional CFTs with a U(1) current [267].

10.
Compute the superradiant scattering amplitude of probe scalar fields on the KerrNewman geometry with firstorder backreaction. Compare the result with the scattering amplitude defined in the CFT at one loop order (using two and threepoint correlation functions).

11.
Formulate a general scattering theory around nearextremal blackhole solutions of (1). This would require one to classify the geometries admiting a KillingYano tensor so that the wave equation could be separated. A longstanding problem already consists in separating and decoupling the wave e quation of a probe spin 1 or spin 2 field in the KerrNewman geometry.

12.
Construct one example in string theory of an exact quantum field theory dual to (an embedding in string theory of) the Kerr black hole. Characterize whether that field theory is a CFT, a limit of a CFT, or a deformation thereof.
Notes
Nevertheless, let us mention that some classes of black holes admit a vanishing horizon area A_{h} and zero temperature T limit such that the ratio A_{h}/T is finite. Such extremal vanishing horizon (EVH) black holes admit nearhorizon limits, which contain (singular) identifications of AdS_{3} that can be used for string model building [157, 98, 116, 130, 115]. Most of the ideas developed for the Kerr/CFT correspondence and its extensions can be developed similarly for EVH black holes [243].
That has been proven for any nonextremal black hole in d = 4 Einstein gravity coupled to any matter obeying the weak energy condition with hyperbolic equations of motion and asymptoticallyflat boundary conditions [161, 163, 254, 86, 143]. The proof has been extended to extremal black holes, to higher dimensions and to antide Sitter asymptotics in [171, 170, 85].
Nevertheless, one can describe the process of spontaneous creation of extremal black holes in an electromagnetic field as an analogue to the Schwinger process of particle creation [126].
We thank the anonymous referee for pointing out this reference.
In some special cases, there may be some continuous dependence of the nearhorizon parameters on the scalar moduli, but the entropy is constant under such continuous changes [17].
We fix the range of θ as θ ∈ [0, π]. Since the original black hole has S^{2} topology and no conical singularities, the functions γ(θ), α(θ) also obey regularity conditions at the north and south poles
$${{\gamma {{(\theta)}^2}} \over {\alpha {{(\theta)}^2}}}\sim{\theta ^2} + O({\theta ^3})\sim{(\pi  \theta)^2} + O({(\pi  \theta)^3}).$$(26)Similar regularity requirements apply for the scalar and gauge fields.
In singular limits where both the temperature and horizon area of black holes can be tuned to zero, while keeping the areaovertemperatureratio fixed, singular nearhorizon geometries can be constructed. Such singular nearhorizon geometries contain a local AdS_{3} factor, which can be either a null selfdual orbifold or a pinching orbifold, as noted in [33, 29, 135, 23] (see [116] for a comprehensive study of the simplest threedimensional model and [243] for a partial classification of fourdimensional vanishing area nearhorizon solutions of (1)).
Our conventions for the infinitesimal charges associated with symmetries is as follows: the energy is \(\delta {\mathcal M} = \delta {{\mathcal Q}_{{\partial _t}}}\) the angular momentum is \(\delta {\mathcal J} = \delta {{\mathcal Q}_{ {\partial _\phi}}}\) and the electric charge is \(\delta {{\mathcal Q}_e} = \delta {{\mathcal Q}_{ {\partial _\chi}}}\). In other words, the electric charge is associated with the gauge parameter Λ = −1. The first law then reads \({T_H}\delta {\mathcal S} = \delta {\mathcal M}  {\Omega _J}\delta {\mathcal J}  {\Phi _e}\delta {{\mathcal Q}_e}\).
The sign choice in this expansion is motivated by the fact that the central charge to be derived in Section 4.3 will be positive with this choice. Also, the zero mode ϵ = −1 is canonically associated with the angular momentum in our conventions.
Compère, in preparation, (2012).
We thank Tom Hartman for helping deriving this central charge during a private communication.
There is a ℤ_{2} ambiguity in the definition of parameters since Eq. (171) is invariant upon replacing (a, b, c) by (is + 2b − a, b, c + (2b − is)(is + 2b − 2a)). We simply chose one of the two identifications.
Note that at extremality J = M^{2}, so the central charge at extremality (129) could as well be written as c_{L} = 12M^{2}. However, away from extremality, matching the black hole entropy requires that the central charge be expressed in terms of the quantized charge c_{L} = 12J.
Alternatively, it was suggested in [73, 71] that one can describe the dynamics of the scalar field in the nearhorizon region using the truncated expansion of Δ_{r} (r) around r_{+} at second order. However, the resulting function \(\Delta _r^{{\rm{trunc}}}\) has, in addition to the pole r_{+}, a fake pole r_{*}, which is not associated with any geometric or thermodynamic feature of the solution. Therefore, the physical meaning of this truncation is unclear.
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