Extremal Rotating Black Holes in the Near-Horizon Limit: Phase Space and Symmetry Algebra

We construct the NHEG phase space, the classical phase space of Near-Horizon Extremal Geometries with fixed angular momenta and entropy, and with the largest symmetry algebra. We focus on vacuum solutions to $d$ dimensional Einstein gravity. Each element in the phase space is a geometry with $SL(2,\mathbb R)\times U(1)^{d-3}$ isometries which has vanishing $SL(2,\mathbb R)$ and constant $U(1)$ charges. We construct an on-shell vanishing symplectic structure, which leads to an infinite set of symplectic symmetries. In four spacetime dimensions, the phase space is unique and the symmetry algebra consists of the familiar Virasoro algebra, while in $d>4$ dimensions the symmetry algebra, the NHEG algebra, contains infinitely many Virasoro subalgebras. The nontrivial central term of the algebra is proportional to the black hole entropy. This phase space and in particular its symmetries might serve as a basis for a semiclassical description of extremal rotating black hole microstates.

Questions regarding black holes have been at the frontiers of astrophysics and high energy physics. On the theoretical side the possible microscopic origin of thermodynamical aspects of black holes [1], the information loss problem and its the recent developments [2], have been active research areas in the last forty years. These questions are usually regarded as test grounds for, and windows to, models of quantum gravity. On the observational side, and with the advance in X-ray astronomy (see e.g. [3]), we now have several approved candidates of black holes in a wide range of masses and spins. Extremal spinning black holes, namely black holes with maximum possible spin for a given mass, are an important special class of black holes to study. Remarkably, several near-extremal Kerr black holes have been observationally identified [4]. In the extremal limit, the Hawking temperature vanishes and very close to the horizon one finds a Near-Horizon Extremal Geometry (NHEG) with enhanced SL(2, R) × U (1) isometry where the dynamics is decoupled from the region far from the black hole horizon [5]. The Kerr NHEG is therefore an appealing starting point for modeling accretion disks, jets and gravitational waves around astrophysical near-extreme rotating black holes [6].
Earlier analyses have established uniqueness of the Kerr NHEG as the 4d Einstein vacuum solution with SL(2, R)× U (1) isometry [7]. This uniqueness has been extended to more general NHEG's which are solutions to pure Einstein vacuum gravity (with or without cosmological constant) in d dimensions with SL(2, R) × U (1) d−3 isometry [8]. The latter is the class of solutions we focus on in this work. The metric has the general form where ds 2 2 = −r 2 dt 2 + dr 2 r 2 and i, j = 1, 2, · · · , d − 3. Requiring the geometry to be smooth and Lorentzian implies Γ > 0 and the eigenvalues of γ ij to be real and nonnegative.
Moreover, around locations θ = θ 0 where one eigenvalue of γ ij vanishes it should scale like (θ − θ 0 ) 2 and the other eigenvalues should remain finite.
The solution (1) is specified by d−3 constant parameters k = (k 1 , . . . , k d−3 ) which are thermodynamically conjugate to angular momenta J. One can associate an entropy S to this geometry which is a Noether-Wald [9] conserved charge [10] and obeys the entropy law [10] S 2π Here, H denotes codimension two, constant arbitrary t, r surfaces. Such H's form infinitely many bifurcation surfaces of the geometry (1), as detailed in [10,11].
Carrying out an asymptotic symmetry group analysis, it was shown that the U (1) isometry of the four dimensional Kerr NHEG enhances to an infinite dimensional Virasoro algebra [12]. This observation led to the Kerr/CFT proposal stating that there is a chiral two dimensional CFT dual to the dynamics around the extremal Kerr horizon [12]. Soon afterwards, it was realized that any solution asymptotic to the Kerr NHEG must have vanishing SL(2, R) charges and should be diffeomorphic to the Kerr NHEG itself [13]. As a consequence, no dynamical correspondenceà la AdS/CFT [14] exists for extremal black holes.
Nonetheless, the presence of nontrivial charges signals the existence of states which represent the symmetry algebra. After quantization, the corresponding quantum microstates will be physically independent since they will be labelled by different eigenvalues of the charge operators. A semiclassical description of these microstates requires the existence of a well-defined phase space, which has not yet been identified. One of the main results of this letter is to provide a consistent phase space for the Kerr NHEG and prove its uniqueness under a set of well-motivated as-sumptions. Our construction also resolves technical difficulties (such as ambiguities, defects at the poles and lack of SL(2, R) invariance) encountered in [15].
The Kerr/CFT analysis and appearance of a Virasoro algebra has been extended to many near horizon extremal geometries of the form (1) for d > 4. In five dimensions, an SL(2, Z) family of boundary conditions has been proposed [16] where a specific circle on the 2-torus spanned by ϕ 1 , ϕ 2 is selected and a Virasoro algebra along that circle is constructed (see also [17]). In all cases analyzed so far, the asymptotic symmetry algebra is given by a single copy of the Virasoro algebra (for a review, see [18]).
In this letter, we will treat the whole family of NHEG in (1) on an equal footing and we will prove that a phase space constructed through a family of finite diffeomorphisms exists, which is specified by a single arbitrary function of all ϕ i coordinates. All the dynamics will then be expressed in terms of the diffeomorphism and the associated physical conserved charges. The symmetry algebra, defined as the Dirac bracket of conserved charges, admits a central extension given by the entropy.

Summary of the results:
The NHEG phase space. Given the statements about absence of dynamical perturbations on the NHEG [10,13], we are naturally led to construct the classical phase space of an NHEG with given angular momenta by the action of diffeomorphisms on (1). The vector field which, as we will outline, is appropriate for this purpose is within the family where ǫ( ϕ) is an arbitrary periodic function of and With the above we construct the phase space G[F ] as the family of metrics obtained through (4), viewed as an active transformation. Explicitly, after dropping the primes, G[F ] is set of metrics where τ = t + b r and The "background NHEG" (1) is the F = 0 element in G[F ]. Obtained from diffeomorphisms (4), G[F ] contains metrics which are smooth everywhere. We will be defining the conserved charges through integration of (d − 2)-forms on the constant t, r surfaces H whose metric is Requiring these surfaces to be smooth fixes b = 1.
Given our construction above, one clearly sees that the SL(2, R) × U (1) d−3 isometries of the background extend to each metric of the form (6) in the NHEG phase space G[F ]. Notice that the angular momenta are not associated with ∂ ϕi but rather with the background U (1) vector fields transformed by the diffeomorphism (4). This implies that the angular momenta, defined as Komar integrals, are constant over the phase space. Also, after the choice b = 1 each bifurcate Killing horizon has a bifurcation surface with the same area as the background. In that sense, the NHEG phase space has constant entropy.
The most important property of the NHEG phase space is the existence of a finite and conserved symplectic structure, allowing one to define the classical and semiclassical dynamics. The standard Lee-Wald symplectic structure [19] built from the Einstein action diverges, as was noted in [15]. Nonetheless, as we will discuss below, there exist boundary terms which once added remove the divergences. The resulting symplectic form vanishes everywhere on-shell. In the analogous case of vacuum Einstein gravity in three dimensions, there is also no bulk dynamics while boundary conditions exist which enjoy two copies of the Virasoro algebra as symmetry algebra [20]. In that setting, it has been recently shown in [21] that the symplectic form vanishes on-shell on the phase space [22], which implies that the symmetries act everywhere in the bulk spacetime. The situation is analogous here. Since the symplectic form is zero on-shell instead of at infinity only, the asymptotics is not a special place and symmetries act everywhere. We will hence refer to them as symplectic symmetries in contrast with asymptotic symmetries.
The NHEG symplectic symmetry algebra. Since the symplectic structure is nontrivial off-shell, one can define physical surface charges associated with the symplectic symmetries χ 1 [ǫ n ], where ǫ n = e i n· ϕ , n i ∈ Z. The generators of these charges denoted by L n satisfy the NHEG algebra In the semi-classical limit, we can quantize the algebra by promoting L n as operators and replacing the classical bracket by −i times the commutator. The angular momenta J i and the entropy S obeying (2) commute with L n and are therefore central elements of the NHEG algebra. For the four dimensional Kerr case, k = 1 and one obtains the familiar Virasoro algebra [12] [L m , L n ] = (m − n)L m+n + c 12 m 3 δ m+n,0 with central charge c = 12 S 2π = 12J. In higher dimensions, the NHEG algebra (8) is a new infinite-dimensional algebra in which the entropy appears as the central extension.
Despite common features, our four dimensional construction contrasts from the one originally performed in [12] in several respects: (i) The symmetry generator is χ 1 , not χ 0 ; this resolves the pathologies at the north and south pole of the sphere (cf. [12,15]). (ii) The symmetries are symplectic rather than asymptotic and all radial dependences are exact. (iii) The phase space is built explicitly and it furnishes a representation of the Virasoro algebra. (iv) The Virasoro algebra is not an enhancement of the U (1) isometry of the background. All the points in the phase space are SL(2, R) × U (1) invariant and the angular momentum is constant over the phase space.
For d > 4 the NHEG algebra contains infinitely many Virasoro subalgebras. To see the latter, one may focus on the generators L n where n = n e for any given vector e, e · k = 0. It is then readily seen that ℓ n ≡ 1 k· e L n form a Virasoro algebra of the form (9) with central extension c = 12S 2π k · e. The entropy might then be written in the suggestive form S = π 2 3 c T F.T. where T −1 F.T. = 2π k · e is the inverse Frolov-Thorne temperature, as reviewed in [18]. The NHEG algebra also contains many affine U (1) Kač-Moody symmetries spanned by generators of the form L n where n = n v and v · k = 0, under the condition that v is on the lattice.
On the choice of symmetry generator. The NHEG background (1) enjoys SL(2, R) × U (1) d−3 isometry. Let us denote the SL(2, R) generators by ξ 1 , ξ 2 , ξ 3 We also define the two vectors and denote by ξ 1 , ξ 2 , ξ 3 , η 1 , η 2 the push-forward of these vectors on a generic element of the phase space after acting with the diffeomorphism (4). Starting with the most general diffeomorphism generator χ, we highlight conditions singling out χ b , which is the basic object both in construction of the NHEG phase space G[F ] and the NHEG algebra (8). The following six requirements uniquely fix χ b=1 in the family (3).
An arbitrary t, r can be mapped onto any given constant t 0 , r 0 under a ξ 1 , ξ 2 transformation. ξ 1 , ξ 2 invariance implies that the charges associated with geometries in the NHEG phase space are independent of the codimension two surface H (bifurcation horizons of the NHEG) over which the charges are defined.
4. We fix ǫ t = −b ∂ ϕ · ǫ and impose b = 1. The diffeomorphism then preserves one of two expansion-free rotation-free and shear-free null geodesic congruences which is labelled by the normal to constant v = t + 1 r surfaces (The other congruence is related to u = t− 1 r ) [23]. 5. We impose ǫ to be θ independent. This condition guarantee smoothness of the t, r constant surfaces H and it also guarantees that the angular momenta are constant. It also makes the volume of H be invariant under χ-diffeomorphisms, which leads to a conserved entropy. This latter is explicitly seen from (7).
The first choice leads to a Kerr/CFT type diffeomorphism and phase space, that we will describe in [11]. The second choice leads to the NHEG phase space that we describe here.
The symplectic structure. The solution space G[F ] can be promoted to a phase space only when the symplectic structure is defined. It is well-known that the Lee-Wald (d−1) symplectic form ω LW [δ 1 Φ, δ 2 Φ; Φ] for a generic theory with fields Φ and field variations δΦ is ambiguous up to the addition of boundary terms [19]. According to the holographic renormalization framework, the total symplectic form takes the form (13) where Y [δΦ, Φ] is the (d−2)-form boundary pre-symplectic potential [24]. The symplectic structure is then defined for a codimension one surface Σ as Σ ω. Since we only consider diffeomorphisms, metric variations are Lie derivatives, δ χ g µν = L χ g µν . We fix the ansatz for Y [δΦ, Φ] by requiring the following. (i) Since the bulk action has two derivatives, we require Y to have at most one derivative. (ii) We allow Y to depend on the metric and on η 1 , η 2 . This fixes the fields to be Φ = {g µν , η 1 , η 2 }. With the above structure we find a large set of different terms that might linearly contribute to Y .
We then restrict the corresponding coefficients through the following requirements: (i) The symplectic structure should be finite and conserved. Given the ξ 1 , ξ 2 invariance, one has ω t ∼ 1/r, ω r ∼ r. This leads to a logarithmically divergent symplectic structure with infinite flux unless ω t = 0 = ω r on-shell, which we therefore require. (ii) We require that ω θ = 0 = ω ϕ i on-shell. It implies that any smooth deformation of the surface H will lead to the same conserved charges; (iii) 1 r ∂ t should be a pure gauge and the central charge should be independent on b. We find that a boundary term which guarantees these requirements is where η b = η 2 +bη 1 , Θ[δg µν , g µν ] is the d−1 form appearing in the on-shell variation of the Einstein action δL ≈ dΘ [9] and ǫ ⊥ is the binormal to the two shear-free expansionfree and rotation-free null congruences, normalized as ǫ ⊥ = dt ∧ dr on the background. No boundary term in the class exists when ǫ = Kǫ(ϕ i ), with ǫ an arbitrary function of all angles ϕ i and K = k, which justifies the 6th requirement in the choice of symmetry generator.
The conserved charges. Given the symplectic form ω, we can define variations of surface charges around any NHEG element (6). One consistency requirement is to be able to integrate these charge variations into finite charges. The latter is known as the integrability conditions which read as [25] H χ·ω[δ 1 Φ, δ 2 Φ; Φ] = 0 for any field variations δ 1 Φ, δ 2 Φ and fields Φ and any symmetry generator χ. It turns out that the integrability conditions are obeyed as a consequence of χ t ω r = χ r ω t which holds off-shell.
Given the symplectic structure one can compute the charges Q ǫ [19]. It is easier in this case to first compute the Poisson bracket δ ǫ ′ Q ǫ use the algebra to deduce the charges Q ǫ [11]. It is straightforward to check that acting on the phase space with the symmetry generator χ b [ǫ( ϕ)], keeps the metric in the same functional form as (6) but with F shifted as δ ǫ F = (1 + ∂F )ǫ = e Ψ ǫ where ∂ denotes the "directional derivative" ∂ ≡ k · ∂. One can translate this transformation law in terms of Ψ defined in (5) as Therefore Ψ transforms like a Liouville field, which we dub as the NHEG boson and transforms as The charges associated with χ b=1 [ǫ( ϕ)] then turn out to be where dH = Γ d−2 2 √ det γdθd ϕ. If Q ǫ for ǫ = e i m· ϕ is denoted by L m , the charge algebra {Q ǫ , Q ǫ ′ } ≡ δ ǫ ′ Q ǫ exactly reproduces the NHEG algebra (8).

Discussion and outlook.
In this work we put forward a proposal for the semiclassical phase space of near-horizon extremal geometries which are solutions to vacuum Einstein gravity with SL(2, R) × U (1) d−3 isometry. We gave an explicit construction for the NHEG phase space G[F ] through diffeomorphisms χ b=1 defined in (3). We showed that each geometry in G[F ] is labeled by a periodic function of d − 3 variables Ψ( ϕ) which determines the charges L n . These charges are generators of the NHEG algebra (8) which has entropy S as its central extension. Here we did not show the details of the construction which will be presented in our upcoming publication [11].
Our preliminary analysis indicates that this construction for NHEG phase space and the NHEG algebra are not limited to the class of geometries given in (1) and is true for all Near-Horizon Extremal Geometries [8] including solutions in theories involving gauge and scalar fields.
Our analysis shows that besides χ b defined in (3), the "Kerr/CFT-type" choice for χ leads to an alternative consistent phase space with a single Virasoro symmetry algebra (cf. discussions in item 6.). The comparison between our proposed NHEG phase space G[F ] and the latter will be studied in [11].
We also gave the expression for charges associated with each geometry in (6) as a function of the NHEG boson Ψ, and discussed a realization of the NHEG algebra (8) in terms of this field. This construction is in a sense a (d−2) dimensional version of the Liouville field theory. We expect that in a fully quantized phase space, the algebra (8) appears as the fundamental symmetry and the field theory based on Ψ may appear as an effective description. It is of course very exciting to explore this direction which may be useful for a semiclassical microstate counting.
MMShJ would like to thank Hossein Yavartanoo for discussions at the early stages of this work. MMSHJ, AS and KH would like to thank Allameh Tabatabaii Prize Grant of Boniad Melli Nokhbegan of Iran. MMSHJ, KH and AS