List of Footnotes

1 For independent arguments pointing to the presence of conformal symmetry around arbitrary black holes, see [63, 245, 180]. We will not discuss these approaches here.
2 Nevertheless, let us mention that some classes of black holes admit a vanishing horizon area Ah and zero temperature T limit such that the ratio Ah∕T is finite. Such extremal vanishing horizon (EVH) black holes admit near-horizon limits, which contain (singular) identifications of AdS3 that can be used for string model building [157Jump To The Next Citation Point, 98Jump To The Next Citation Point, 116Jump To The Next Citation Point, 130Jump To The Next Citation Point, 115Jump To The Next Citation Point]. Most of the ideas developed for the Kerr/CFT correspondence and its extensions can be developed similarly for EVH black holes [243Jump To The Next Citation Point].
3 That has been proven for any non-extremal black hole in d = 4 Einstein gravity coupled to any matter obeying the weak energy condition with hyperbolic equations of motion and asymptotically-flat boundary conditions [161, 163Jump To The Next Citation Point, 254, 86, 143]. The proof has been extended to extremal black holes, to higher dimensions and to anti-de Sitter asymptotics in [171, 170Jump To The Next Citation Point, 85].
4 The original proofs were limited to non-extremal black holes, which have a bifurcation surface [66, 163Jump To The Next Citation Point]. The proof for extremal black holes can now be found in [170Jump To The Next Citation Point].
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 near-horizon parameters on the scalar moduli, but the entropy is constant under such continuous changes [17Jump To The Next Citation Point].
8 We fix the range of 𝜃 as 𝜃 ∈ [0,π]. Since the original black hole has S2 topology and no conical singularities, the functions γ(𝜃), α(𝜃) also obey regularity conditions at the north and south poles
2 γ(𝜃)--∼ 𝜃2 + O(𝜃3) ∼ (π − 𝜃)2 +O ((π − 𝜃)3). (26) α(𝜃)2
Similar regularity requirements apply for the scalar and gauge fields.
9 In singular limits where both the temperature and horizon area of black holes can be tuned to zero, while keeping the area-over-temperature–ratio fixed, singular near-horizon geometries can be constructed. Such singular near-horizon geometries contain a local AdS3 factor, which can be either a null self-dual orbifold or a pinching orbifold, as noted in [33Jump To The Next Citation Point, 29, 135, 23Jump To The Next Citation Point] (see [116Jump To The Next Citation Point] for a comprehensive study of the simplest three-dimensional model and [243Jump To The Next Citation Point] for a partial classification of four-dimensional vanishing area near-horizon solutions of (1View Equation)).
10 Our conventions for the infinitesimal charges associated with symmetries is as follows: the energy is δℳ = δ𝒬 ∂t, the angular momentum is δ𝒥 = δ𝒬 − ∂ϕ and the electric charge is δ𝒬 = δ𝒬 e −∂χ. In other words, the electric charge is associated with the gauge parameter Λ = − 1. The first law then reads T δ𝒮 = δℳ − Ω δ𝒥 − Φ δ𝒬 H J e 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 ℤ2 ambiguity in the definition of parameters since Eq. (171View Equation) 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 The two-point function (189View Equation) has a branch cut, and as a result, one must find a way to fix the choice of relative sign between the two exponentials in (191View Equation). The sign is fixed by matching the gravitational computation to be − (− 1)2s, where s is the spin of the corresponding field.
16 Note that at extremality 2 J = M, so the central charge at extremality (129View Equation) could as well be written as 2 cL = 12M. However, away from extremality, matching the black hole entropy requires that the central charge be expressed in terms of the quantized charge cL = 12J.
17 Alternatively, it was suggested in [73, 71] that one can describe the dynamics of the scalar field in the near-horizon region using the truncated expansion of Δr (r) around r+ at second order. However, the resulting function trunc Δ r 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.