Near horizon OTT black hole asymptotic symmetries and soft hair

We study the near horizon geometry of both static and stationary extremal Oliva Tempo Troncoso (OTT) black holes. For each of these cases, a set of consistent asymptotic conditions is introduced. The canonical generator for the static configuration is shown to be regular. For the rotating OTT black hole, the asymptotic symmetry is described by the time reparametrization, the chiral Virasoro and centrally extended $u(1)$ Kac-Moody algebras.


Introduction
The longstanding problem of the origin of black hole entropy is one of the most important open questions in contemporary physics. There are many proposals how to interpret black hole entropy and corresponding microstates, let us mention some of them: entanglement entropy [1], as fuzzball [2] or as soft hair on the horizon [3,4]. It has, also, been the starting point of many ingenious discoveries, the most impressive of which is Holographic nature of gravity [5].
Holographic duality [6] states that gravitational theory in asymptotically locally anti de Sitter space-time is dual to non-gravitational theory defined on conformal boundary of the space-time. Although it, still, has a status of conjecture, there is vast number of results supporting it. Let us mention for the purpose of this paper that Holographic duality gave many insights into the black hole physics, including black hole information paradox and origin of black hole micro-states. Namely, Holography provided derivation of black hole entropy from the near horizon micro-states via Cardy formula [7], whose applicability crucially relies on the existence of 2D conformal symmetry as the subgroup of asymptotic group of symmetry. This, although invaluable, is not enough for the most general purposes, and we need generalization. Notable progress is derivation of Cardy like formula in Warped CFT (WCFT), see ref. [8].
Generalization of particular interest to us is given in [9], where authors hypothesized that extremal Kerr black hole is dual to the chiral 2D CFT. There are indications that this chiral CFT should arise as DLCQ [10]. More precise, extremal black hole, non necessarily Kerr-like, posses intriguing feature that its near horizon geometry is exact solution of the theory. This allows for the study of physics on the horizon by investigating properties of the near horizon geometry.
In this article we analyze the near horizon limit of hairy black hole known as the Oliva Tempo Troncoso (OTT) black hole [11], which is the solution of BHT gravity [12] as well as of the Poincaré gauge theory of gravity [13] for the special choice of action parameters. The leading idea of this analysis is study of the influence of hair parameter on micro-states of extremal black hole. Note that obtained near horizon geometries exist, without any reference to extremal black hole, as independent solutions and have importance in their own.
Firstly, we analyze the static OTT black hole, which becomes extremal for the specific value of the hair parameter and obtain the corresponding near horizon geometry. Then we study the asymptotic structure of the near horizon geometry and obtain the asymptotic symmetry group.
Secondly, we continue with the rotating OTT black hole which can be made extremal in two different ways. Extremality obtained by tuning of the hair parameter, surprisingly, leads to the same near horizon geometry as in the non-rotating case. The extremal OTT with maximal angular momentum leads to the geometry with richer structure. We conclude, in the end, that asymptotic symmetry is direct sum of time reparameterization, Virasoro algebra and centraly extended u(1) Kac-Moody algebra. The entropy of the extremal rotating OTT can be expressed in terms of central extension of Kac-Moody and on-shell value of the zero mode Virasoro generator Our conventions are the same as in Ref. [13]: the Latin indices (i, j, k, ...) refer to the local Lorentz frame, the Greek indices (µ, ν, ρ, ...) refer to the coordinate frame, e i is the orthonormal triad (coframe 1-form), ω ij is the Lorentz connection (1-form), the respective field strengths are the torsion T i = de i + ω i m ∧ e m and the curvature R ij = dω ij + ω i k ∧ ω kj (2-forms), the frame h i dual to e j is defined by h i e j = δ i j , the signature of the metric is (+, −, −), totally antisymmetric symbol ε ijk is normalized to ε 012 = 1, the Lie dual of an antisymmetric form X ij is X i := −ε ijk X jk /2, the Hodge dual of a form α is ⋆ α, and the exterior product of forms is implicit.

Conformally flat Riemannian solutions in PGT
The OTT black hole represents a vacuum solution of the BHT gravity in the sector with a unique AdS ground state [11]. In [13] it was proved that is also represents a Riemannian solution of PGT in vacuum, which parity preserving Lagrangian 3-form (mostly quadratic in field strengths) reads: where (a) T i and (a) R ij are irreducible components of the torsion and the RC curvature, see [14]. The above relation reveals a deep dynamical relation between the Riemannian sector of PGT and the BHT gravity. The content of this relation is expressed by a theorem stating that any conformally flat solution of the BHT gravity is also a Riemannian solution of PGT provided that: This is, in particular, true for the OTT black holes. In 3D, the Weyl curvature identically vanishes, and the Cotton 2-form C i is used to characterize conformal properties of spacetime [15]. It is defined by Re m is the Schouten 1-form. A spacetime is conformally flat when C i = 0.
An interesting interpretation of the identifications (2.2) is found by using the BHT condition that ensures the existence of the unique maximally symmetric background [13]:

Canonical generator and conserved charges
The standard construction of the canonical generator of the Poincaré gauge transformations -diffeomorphisms and Lorentz rotations [16], for the quadratic PGT makes use of the existence and classification of all constraints in the theory. The construction can be significantly simplified by going over to the first order Lagrangian (3-form) see [14]. Here, τ m and ρ ij are independent dynamical variables, the covariant field momenta conjugate to e i and ω ij , and the potential V ensures the on-shell relations τ i = T i , ρ ij = R ij , which transform L G into the standard quadratic form.
The first order formulation of L G simplifies the construction of the canonical generator G, the form of which can be found in [14]. Since G acts on the basic dynamical variables via the Poisson bracket operation, it must be a differentiable functional. To examine the differentiability of G, one starts from the form of its variation Here, Σ is the spatial section of spacetime, the variation is performed in the set of adopted asymptotic states, R stands for regular (differentiable) terms, and we use ρ i and ω i , the Lie duals of ρ mn and ω mn , to simplify the formulas. Diffeomorphisms are parametrized by ξ µ while parameters of local Lorentz rotations are θ i . In most cases asymptotic conditions ensure the regularity of the generator of Lorentz rotations, i.e. one finds δG 2 = R. In that case we get Thus, in general, δG = R and G is not differentiable. The problem can be corrected by going over to the improved generatorG := G + Γ, where the boundary term Γ is constructed so that δG = R. After making a partial integration in δG, one finds that Γ is defined by the variational equation 4 Static OTT black hole orbifold Extremal static OTT black hole. The metric of the static OTT black hole is given by: where N 2 = −µ + br + r 2 ℓ 2 . Black hole horizons are located at: The black hole is extremal iff horizons coincide r + = r − . This condition is satisfied if b 2 ℓ 2 + 4µ = 0. Let us note that existence of extremal black hole horizon implies b < 0.
Orbifold. Let us now consider the following coordinate transformation: Now metric becomes: In the limit ε → 0 the metric (with the prescription ρ → r) reads: It represents a perfectly regular solution an orbifold. We choose triad fields in the simple diagonal form: The Levi-Civita connection that corresponds to the triad fields reads Curvature has only one non-vanishing component: scalar curvature is constant R = 2 ℓ 2 , while Ricci and Shoutten one forms are given by: Solution is conformally flat (as OTT black hole) i.e. Cotton 2-form C i = ∇L i vanishes, and it solves equations of motion of BHT gravity and PGT in the sector b 4 + 2b 6 = 0.

Asymptotic conditions
Let us consider the following asymptotic conditions for the metric in the region r → ∞: where O n denotes term with asymptotic behavior r −n or faster. The asymptotics of the triad fields (in accordance with (4.6)) is given by Condition T i = 0 together with (4.7) gives the following asymptotic of the spin connection The diffeomorphisms that leave the metric (4.6) invariant are given by: Loretz transformations that leave asymptotic conditions invariant are In terms of Fourier modes ℓ n := δ 0 (S = e inϕ ) and j n := δ 0 (U = e imϕ ) algebra of residual gauge transformations takes a form of a semidirect sum of Virasoro and Kac-Moody algebra: (4.11)

Algebra of charges
Gauge generator is not a priori well-defined because with the given asymptotic behavior of fields its functional derivatives may be ill-defined, as we already mentioned in the section 3, and this problem can be resolved by construction of the improved generator by adding suitable surface terms [17]. By taking into account that τ i = 0, since our solution is Riemannian, relation (3.2) reduces to: For the particular case of generator and asymptotic conditions of this paper, after short calculation, we conclude that gauge generator is differentiable and there is no need for adding surface terms Γ = 0. (4.13) From this we immediately conclude that both the central charge of the Virasoro algebra and the level of U(1) Kac-Moody algebra are zero.

Near-horizon geometry of rotating OTT
Rotating OTT black hole. The rotating OTT black hole is defined by the metric The roots of N = 0 are The metric (5.1) depends on three free parameters, µ, b and η. For η = 1, it represents the static OTT black hole, and for b = 0, it reduces to the rotating BTZ black hole with parameters (m, j), such that 4Gm := µ and 4Gj := µℓ 1 − η 2 . The conserved charges of the rotating black hole solution take the following form: Rotating OTT black hole is a three parameter solution, so that extremal limit can be achieved in two different ways. The first way is the same as in the non-rotating case 4µ + b 2 ℓ 2 = 0. However, it is easily checked that we obtain the same geometry as if black hole was non-rotating. This is not a surprising result if we note that in this case energy and angular momentum vanish.
The second way to obtain an extremal black hole is to take η = 0 which means that angular momentum takes maximal possible value. This corresponds to the usual procedure for the Kerr black hole.
The horizon is located at Coordinate change is given as An interesting departure from the usual redefinition of coordinates in the literature is that in order to obtain non-singular metric we have to scale time coordinate with the same parameter used in the rescaling of the radial coordinate to the power of the minus two, instead of standard minus one.
After the change of coordinates and taking ǫ = 0 we obtain near-horizon metric It is convenient to further rescale time coordinate and obtain aesthetically more pleasing form of the metric We chose triads to be of the form The Levi-Civita connection is given by: The solution is maximally symetric and therefore we have: Rotating OTT black hole for b = 0 reduces to rotating BTZ black hole, what about the corresponding near-horizon geometries? If we introduce ρ = r 2 we obtain near-horizon BTZ with two times smaller ℓ and different r 0 [22]. The only trace of the hair parameter is hidden in r 0 and will lead to the different values of central charges. So we are able to recover results for near-horizon BTZ from the corresponding results for OTT, but not in a naive way by simply taking b = 0.

Asymptotic conditions
We consider the following asymptotic form of the metric The asymptotics of the triad fields is chosen in accordance with the asymtotic behavior of the metric (5.11) Asymptotic form of the spin connection reads The condition of vanishing torsion T i = 0 together with (5.12) and (5.13) leads to the following constraints The diffeomorphisms that leave the metric (5.11) invariant are given by Loretz transformations that leave asymptotic form of triads and spin connection invariant are

Algebra of charges
The improved generator is given byG After a shorter calculation we obtain that surface term is The charge is finite due to the conditions following from the constraint T i = 0. Using the composition law for the local Poincaré transformations we derive the Poisson bracket algebra between improved canonical generators (which is well-defined generator, too [18]). The Virasoro algebra is not centrally extended For related works see [19,20].
The entropy of the extremal OTT black hole S = πr 2 0 G can be reproduced in terms of purely algebraic quantities via peculiar formula The entropy formula has striking resemblance to the entropy formula of [8]. In our case J on−shell 0 = 0 so our formulas is the consequence of the general formula for entropy in WCFT iff

Sugawara-Sommerfeld construction
It is well known that it is possible to construct Virasoro algebra as a bilinear combination of the elements of the Kac-Moody algebra. The procedure is known as the Sugawara-Somerfeld construction [21], which we now apply to the algebra of the previous section. First we introduce auxiliary operators The value of parameter α is fixed by demanding that Virasoro algebra satisfies some canonical relation. We shall fix it by requiring that Cardy formula reproduces entropy correctly. For the orbifold values of Virasoro zero modes are

Thermodynamics at extremality
There is an equivalent Cardy formula which instead of using background values of Virasoro zero modes uses temperature. Consequently, the required addition piece of information is temperature of dual CFT which we will derive from black hole thermodynamics. Starting point is the first law of black hole thermodynamics where J is angular momentum, Ω is angular velocity and q i are additional conserved charges while Φ i are potentials conjugate to q i . In the case of extremal black hole, for which Hawking temperature is zero T H = 0 the first law implies that energy in extremal case is function of conserved charges The corresponding generalized temperatures are defined as which is called left moving temperature, and Now we need thermodynamical quantities of extremal OTT, which read: entropy while the angular momentum is From the variation of the entropy of extremal OTT we determine the left moving temperature T L = r 0 πℓ . (7.9) In extremal case the right moving temperature is zero T R = 0. (7.10) From the demand that, alternative form of the Cardy formula S C = π 2 e T L c L + π 2 e T R c R , (7.11) reproduces the entropy of the extremal OTT we conclude that the value of the c R is undetermined and the left central charge is twice the value of the Brown-Henneaux central charge c L = 3ℓ G . (7.12) This can be used to fix constant α that appeared in Sugawara-Sommerfeld construction. From equation (6.13) and previous formula we derive α = 1 2 . (7.13)

Concluding remarks
We investigated near horizon symmetry of static and stationary OTT black hole in quadratic PGT. In the static case corresponding asymptotic symmetry is trivial, while in the stationary case set of consistent asymptotic conditions leads to a symmetry consisting of time reparametrization and semi-direct sum of centrally extended u(1) Kac-Moody and chiral Virasoro algebra. Improved asymptotic conditions that follow from the vanishing of torsion (5.14c) can be further strengthened thus making time reparametrization pure gauge. Near horizon limit corresponds to deep infrared sector of the theory, which implies that only soft part of the charge survives. This means that corresponding charges represent soft hair on the black hole horizon. Formula shows that there is an intimate relation between black hole entropy and soft hairs on the horizon, while more precise relation calls for further inspection. Using Sugawara-Sommerfeld construction we build Virasoro algebra as bilinear combination of u(1) Kac-Moody and chiral Virasoro algebra. Presence of conformal symmetry enables us to use Cardy formula for entropy which correctly reproduces black hole entropy.