1 Introduction

It is known since the work of Bardeen, Bekenstein, Carter and Hawking [42, 32, 162Jump To The Next Citation Point] 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 non-renormalization theorems allow one to map the black-hole states to dual states in a weakly-coupled 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 near-horizon limit a factor of three-dimensional anti-de Sitter spacetime AdS3 or a quotient thereof, a simpler microscopic model is available. Since quantum gravity in asymptotically AdS3 geometries is described by a two-dimensional conformal field theory (2d CFT) [58Jump To The Next Citation Point, 251Jump To The Next Citation Point], 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 near-horizon region [209Jump To The Next Citation Point, 104Jump To The Next Citation Point] (for reviews, see [155, 113]). Ultraviolet completions of these AdS/CFT correspondences can be constructed using string theory [205Jump To The Next Citation Point, 265Jump To The Next Citation Point].

These results can be contrasted with the challenge of describing astrophysical black holes that are non-supersymmetric and non-extremal, 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 Kerr–Newman and the Reissner–Nordström 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 two-dimensional CFTs [156Jump To The Next Citation Point, 53Jump To The Next Citation Point, 68Jump To The Next Citation Point] (see also [104Jump To The Next Citation Point]).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 (asymptotically-flat, anti-de 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 near-extremal black holes. Only sparse but non-trivial clues point to a CFT description of black holes away from extremality [104, 68Jump To The Next Citation Point, 107Jump To The Next Citation Point].

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 cosmic-censorship hypothesis is J ≤ GM 2. No physical process exists that would turn a non-extremal black hole into an extremal one. Using details of the accretion disk around the Kerr black hole, Thorne derived the bound 2 J ≤ 0.998GM 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 X-ray binary GRS 1905+105 [218] and Cygnus X-1 [152], are more than 95% close to the extremality bound. Also, the spin-to-mass–square ratio of the supermassive black holes in the active galactic nuclei MCG-6-30-15 [57] and 1H 0707-495 [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 spin-to-mass–square ratio of the very same black hole in the X-ray binary GRS 1905+105 has been evaluated as J∕(GM 2) = 0.15 [182], while the spin of the black hole in Cygnus X-1 has been evaluated as 2 J∕(GM ) = 0.05 [219]. If the measurements of high angular momenta are confirmed or if precise measurements of other nearby highly-spinning 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 first-principles. 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 Kaluza–Klein vector of a higher-dimensional 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 [159Jump To The Next Citation Point]. 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 static-charged 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 near-horizon 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 near-horizon geometries contain no local bulk dynamics. Since we aim at drawing parallels between black holes and two-dimensional 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 near-horizon 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 semi-classical quantization rules, the operators, which define quantum gravity in the near-horizon region, form a representation of the Virasoro algebra. We will then argue that near-horizon quantum states can be identified with those of a chiral half of a two-dimensional CFT. This thesis will turn out to be consistent with the description of non-extremal 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 non-extremal black holes, and we will concentrate our analysis on asymptotically-flat black holes for simplicity. We will describe how part of the dynamics of probe fields in the near-extremal Kerr–Newman black hole can be reproduced by correlators in a family of dual CFTs with both a left and a right-moving sector. The left-moving 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 Kerr–Newman black hole. We will also infer from the breaking of this conformal symmetry that the Kerr–Newman black hole entropy can be mapped to states of these CFTs at specific left and right-moving 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 [54Jump To The Next Citation Point] 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 field-theory approach, we will cover string-theory models of black holes only marginally. We refer the interested reader to the complementary string theory-oriented review of extremal black holes [244].2

 1.1 Classes of effective field theories
 1.2 Gauge fields as Kaluza–Klein vectors

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