The Kerr/CFT Correspondence and its Extensions
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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.
Keywords
Black Hole Extremal Black Hole Kerr Black Hole Hide Conformal Symmetry Asymptotic Symmetry Algebra1 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]).^{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].^{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.
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.
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].
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^{3} — and their event horizon is a Killing horizon^{4}. In asymptoticallyflat spacetimes, black holes have spherical topology [163].
No physical process is known that would make an extremal black hole out of a nonextremal black hole.^{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.
 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 vectorThe 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.$$\xi = {\partial _t} + {\Omega _J}{\partial _\phi}.$$(9)
 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_{+} aswhere 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.$$\Phi _e^I =  {\xi ^\mu}A_\mu ^I{\vert _{r = {r_ +}}},$$(10)

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 bandcome 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].$$0 < \omega < m{\Omega _J}$$(11)
 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)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.$$0 < \omega < {q_e}{\Phi _e}.$$(12)The combined effect of rotation and charge allows one to extract energy in the rangeWhen 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,$$0 < \omega < m{\Omega _J} + {q_e}{\Phi _e}.$$(13)In the superradiant range (13), the absorption probability is negative because the outgoing flux of energy is higher than the incoming flux.$${\sigma _{{\rm{abs}}}} = {{d{E_{{\rm{abs}}}}/dt} \over {d{E_{{\rm{in}}}}/dt}}.$$(14)
 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},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$$\Gamma = {1 \over {{e^{{{\omega  m{\Omega _J}  {q_e}{\Phi _e}} \over {{T_H}}}}}  1}}{\sigma _{{\rm{abs}}}}.$$(15)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.$${\Gamma _{{\rm{ext}}}} =  \Theta ( \omega + m{\Omega _J} + {q_e}{\Phi _e}){\sigma _{{\rm{abs}}}}.$$(16)

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]^{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
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.^{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].
2.3 Nearhorizon of extremal spinning geometries
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].^{9}
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].
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
2.4.2 Nearhorizon geometry of extremal ReissnerNordström
2.4.3 Nearhorizon geometry of extremal KerrNewman
2.4.4 Nearhorizon geometry of extremal KerrNewmanAdS
2.5 Entropy
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.
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.
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.)
2.6.1 Temperatures and entropies of specific extremal black holes
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.
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 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
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.
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
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].
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.
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 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.
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.
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.
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 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 }.
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.
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].
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.
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.
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.
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.
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.
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.
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.

Nearhorizon region: x ≪ 1,

Far region: x ≫ τ_{ H },

Overlap region: τ_{ H } ≪ x ≪ 1.
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.
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.
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
6.2 Scalar wave equation in KerrNewmanAdS
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.^{17} Since much remains to be understood, we will not discuss AdS black holes further.
6.3 Nearregion scalarwave equation
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.
6.4 Local SL(2, ℝ) × SL(2, ℝ) symmetries
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 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].
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.
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
 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.
Footnotes
 1.
 2.
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].
 3.
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].
 4.
 5.
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].
 6.
We thank the anonymous referee for pointing out this reference.
 7.
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].
 8.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 polesSimilar regularity requirements apply for the scalar and gauge fields.$${{\gamma {{(\theta)}^2}} \over {\alpha {{(\theta)}^2}}}\sim{\theta ^2} + O({\theta ^3})\sim{(\pi  \theta)^2} + O({(\pi  \theta)^3}).$$(26)
 9.
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)).
 10.
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 \({T_H}\delta {\mathcal S} = \delta {\mathcal M}  {\Omega _J}\delta {\mathcal J}  {\Phi _e}\delta {{\mathcal Q}_e}\).
 11.
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.
 12.
Compère, in preparation, (2012).
 13.
We thank Tom Hartman for helping deriving this central charge during a private communication.
 14.
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.
 15.
 16.
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.
 17.
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.
Notes
Acknowledgments
This review originates from lectures given at Iberian Strings 2012 in Bilbao. I am very grateful to the organizers I. Bandos, I. Egusquiza, J.L. Mañes, M.A. Valle and C. Meliveo for the invitation to lecture in this outstanding and agreeable conference. I gratefully thank V. Balasubramanian, J. de Boer, B. Chen, C.M. Chen, B. Chowdury, A. Castro, S. Detournay, J. Jottar, F. Larsen, S. Markoff, K. Murata, M. Rangamani, H. Reall, S. SheikhJabbari, K. Skenderis, A. Strominger, A. Virmani and especially M. Guica and T. Hartman for interesting exchanges during the writing of this review. This work has been financially supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO) via an NWO Vici grant. It is also currently supported by the FNRS, Belgium. I finally thank the organizers D. Berman, J. Conlon, N. Lambert, S. Mukhi and F. Quevedo of the program “Mathematics and Applications of Branes in String and Mtheory” at the Isaac Newton Institute, Cambridge for support and hospitality during the final stages of this work.
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