Near Horizon of 5D Rotating Black Holes from 2D Perspective

We study the CFT dual to five dimensional extremal rotating black holes, by investigating the two dimensional perspective of their near horizon geometry. From two dimensional point of view, we show that both gauge fields, related to the two rotations, appear in the same manner in the asymptotic symmetry and in the associated central charge. We find that, our results are in perfect agreement with the generalization of Kerr/CFT approach to five dimensional extremal rotating black holes.


Introduction
Five dimensional black holes have been interested from the seminal work on computing the entropy of a 5D black hole by Strominger and Vafa in the context of string theory [1]. Recently by extending the Kerr/CFT [2] approach to 5D extremal black holes a wide class of such solutions are studied [3]- [11]. An important point of these studies is the appearance of two rotating coordinates in the near horizon geometry of the most of these solutions beside an AdS 2 part. It was proposed in some of those works that, corresponding to each of the rotating coordinate there is one CFT.
It was shown in [10] that these two CFTs are related to each other by the SL(2,Z) group which is the symmetry of the near horizon geometry of the 5D black holes with two rotating coordinates. For this propose the boundary conditions associated to the rotating coordinates are of the same order, and so the symmetry of the rotating coordinates is not destroyed by the fluctuations. This symmetry does not exist for the 5D black objets with only one.
The 2D approach to this problem, which is the subject of study in this paper, implies that the consistency of boundary conditions requires that the two gauge fields should be treated on the same footing in the study of asymptotic symmetries. This results in only one CFT (at least a unique central charge) corresponding to the near horizon geometry of the 5D extremal black holes.
From the 2D point of view, each of the rotating coordinates corresponds to one of the gauge field and we have a AdS 2 solution. Both of them should be considered to defining the consistent boundary conditions and boundary energy momentum tensor, and at the end both of them appear in computing the central charge of the chiral CF T 2 , which is conjectured as CFT, correspond to the AdS 2 solution. This approach to evaluating the contribution of gauge fields resembles the Sen's approach in quantum entropy function [12] where all of the gauge fields are considered in the same manner in the thermodynamics of the extremal black holes.
In this paper following the Castro and Larsen [13] we reduce the near horizon geometry of 5D extremal rotating black hole solution to a 2D theory and investigate the properties of the boundary energy momentum tensor of the AdS solution.
We show that the variation of the energy momentum tensor under diffeomorphisms which should be combined with gauge transformations [14] admits a central charge. We calculate the associated central charge For Myers-Perry black hole [15] and show the agreement with previous results.
The remaining of this paper is organized as follows. Since we intend this article will be self contained we briefly review the 5D extremal rotating black hole and its CFT dual from 5D point of view in §2. In §3 following [13] we study the reduction of 5D extremal rotating black hole to 2D and the AdS solution of this theory. Then we derive the boundary terms of the action and investigate the consistency of the boundary conditions which are allowed from 2D viewpoint. After that in §4 by using the notion of Peierls bracket [16] and counter-term subtraction charge [17] we define the associated charge and compute the central charge associated to the variation of the boundary energy momentum tensor. In §5 we study the Myers-Perry black hole. The near horizon geometry of this black hole has two rotations. We show the agreement of our results with the previous calculations. §6 is devoted to conclude our results and give some discussions.

2
Review of 5D ERBH/CFT In this section we give a brief review of the Kerr/CFT approach to 5D extremal rotating black holes [10]. The reader who is familiar with the Kerr/CFT approach can skip this section. We only review the main steps of calculations and will not discuss the details, which can be found in [10]. The near horizon geometry of 5D ERBH is given by [22] 2 A possible boundary condition for the fluctuations around the geometry (1) is, in the basis (t, r, θ, φ 1 , φ 2 ). This boundary conditions is consistent with the symmetry of the near horizon geometry which combine the φ i s coordinate with each other. The general diffeomorphism preserving the boundary conditions (2) is given by, 2 We absorb the AdS 2 radius in F (θ).
where ǫ(φ 1 , φ 2 ) and λ i (φ 1 , φ 2 ) are arbitrary smooth periodic functions of φ 1 and φ 2 . It was shown [10] that a class of diffeomorphism's generators has basis satisfying the Virasoro algebra [ζ m , ζ n ] Lie = −i(m − n)ζ m+n . These generators correspond to a chiral conformal group of CFT 2 . Using the definition of diffeomorphism's charges [18,19] and the Brown-Henneaux approach [20] it was shown that there is a Virasoro algebra between the associated charges with the central charge which is given by One should note that both of the k i s, which correspond to the angular momenta, contribute in the value of central charge (5).
In the next two sections we confirm this result from the 2D perspective. For this propose we reduce the near horizon geometry to 2D by integrating out the angular coordinates. The resulting solution has an AdS 2 metric and two gauge fields related to two angular momenta. We show that the combination of the diffeomorphisms and gauge transformations of both of the gauge fields is consistent for investigating the variation of the boundary energy momentum tensor. In §5 we show the agreement of 2D results with the 5D results for Myers-Perry black hole.

2D View of 5D Extremal Rotating Black Holes
In this section, we study the 5D Extremal Rotating Black Holes (ERBH) from the 2D perspective. The next section is devoted to calculating the conserved charges and central charge following Castro and Larsen [13]. All of the steps and arguments are similar to [13] so we do not give the details. By using the reduction we will show both of the gauge fields in 2D, associated to two rotating coordinates, play the same role in studying the asymptotic symmetry and the AdS 2 /CFT 1 correspondence. This is the 2D evidence for the arguments in §2 where explained that one should consider the boundary conditions such that both of the rotating coordinates are of the same order.

5D ERBH
We start with the general form of the 5D ERBH with two angular momenta and by reduction on angular coordinates we obtain a 2D effective theory. As we mentioned in (1), the near horizon geometry of 5D ERBH with two nonzero rotations is given by where F (θ), σ(θ) and γ ij (θ) are the functions of only θ. x i ,i = 1, 2 correspond to the rotating coordinates. 2D theory is described by a general 2D metric and two gauge fields corresponding to the rotations as with µ, ν = t, r. We also couple the size of the angular coordinates to the scalar field ψ such that Raising and lowering of all indices and all of the operators are defined by g µν . The associated gauge field strengths are denoted by F i = dA i . The 5D Einstein Hilbert action is From the 2D viewpoint, using the ansatz (9) the 5D Ricci scalar simplifies as 3 in which and the determinant is where By integrating over the angular coordinates, the 2D effective action of ERBH can be derived as in which Action (15) treat in the manner of the generic dilaton gravity in 2D with gauge fields which was given in [21].

Solutions
Since we are interested in solutions which are corresponded to the geometry (6), we limit ourselves to the solutions with constant ψ and solve following equations of motion, The first and second equations can be simplified to Assuming β > 0, which is natural for reduction of extremal solution over angular coordinates, this solution is locally AdS 2 with radius As one can see from (17) l AdS is dimensionless. It is because we absorb the radius of the AdS 2 part of the near horizon geometry (6) in F (θ). Without losing generality we work in the gauge In this gauge, the general form of the solution of equations of motion are given by with the constraint Note that the constants λ i = 0 is inherited from k i = 0, which means the near horizon geometry of 5D ERBH (6) has two angular momenta. This solution can be described by the Fefferman-Graham expansion and the asymptotic behavior of the metric, scalar and gauge fields are given by The results of this section are similar to [13,23].

Boundary Terms
In this section, following the standard procedure for AdS/CFT we determine the normalized boundary action which is formally given by The first term is the Gibbons-Hawking-York term, where h is determinant of induced metric on ∂M, and K is the extrinsic curvature at the boundary, which for solution (27) is derived as As discussed in [13,23] the local form of the counter-term is given by in which the constants m 1 and m 2 will be determined by vanishing the variation of the action on-shell. The variation of the action is given by δS = ∂M π ab δh ab + π ψ δψ + π a i δA i a + Bulk terms, with Using the asymptotic behavior of the fields (29), and the extrinsic curvature (32), above expansions are reduced to Vanishing of these three boundary momenta is satisfied by two conditions, Note that, although there were two unknown constants m 1 and m 2 , we had three equations. Therefore finding a solution shows the consistency of our calculations. In this way the full action of reduced solution is given by

Consistency of Boundary Conditions
As discussed in [23,14], for the AdS solution with a gauge field the combination of diffeomorphisms and gauge transformation is consistent with the gauge conditions. In this section we show that for solution (27), i.e. AdS metric with two gauge fields, the consistency requires that the gauge transformations of both of the gauge fields should be included in addition to the diffeomorphisms. For this propose we first determine the diffeomorphisms of the metric and its induction on the gauge fields. Next, we find the compensating gauge transformations leaving the gauge fields in the gauge condition (25).
General diffeomorphisms transform the metric as The gauge condition (25) has fixed the g ρρ and g tρ components of the metric to zero and by using the Fefferman-Graham form we have fixed the asymptotic value of the g tt (29). Thus one can find the associated diffeomorphisms that preserve these conditions by requiring that, δ ǫ g ρρ = 0, δ ǫ g tρ = 0, δ ǫ g tt = 0.O(e 2ρ/l ).
One can show that these conditions are satisfied if where ζ(t) is an arbitrary function of coordinate t. It is easy to show that under this diffeomorphisms the boundary metric transforms as The transformation of the gauge fields A i µ , which is defined by under the diffeomorphisms (43) are given by To restore the gauge condition A (i) ρ = 0 (25) one should compensate the diffeomorphisms with a gauge transformations for each of the gauge fields as with gauge functions Therefore, the combination of the allowed diffeomorphisms (43) and the gauge transformations (47) satisfy the gauge condition (25), 4 One can easily show that under the combination of the transformations the variation of the gauge fields are Noting that, for satisfying the gauge condition (25) the gauge transformations of both of the gauge fields (50) are required to compensate with the the diffeomorphisms (43) and one can not turn off one of them consistently.
From the 5D point of view this means the rotating coordinates must play the same role in the asymptotic behavior of the metric. In other words, this implies that the boundary conditions of rotating coordinates, which determine the fluctuations of associated components of the metric, should be in the same order. This is in precise agreement with the results of §2.

Conserved Charges and Central Charge
Now we can construct the combination of the associated conserved charges for studying the asymptotic symmetries. Since we are interested in the boundary energy momentum tensor of a solution there are some subtleties in defining the associated conserved charge.
As shown in [17], the generators of the asymptotic symmetries should be defined via the counter-term subtraction method (CTSM). These charges can differ from those defined usually. This method is based on the Peierls bracket [16] which has a covariant construction and is equivalent to the Poisson bracket on the space of observables. The charge defined in this way is called counter-term subtraction charge (CTSC) which is given by where ξ is an infinitesimal transformation parameter and G is a regular function such that near the past boundary G = 0 and near the future boundary G = 1.
As mentioned above, we focus on the boundary fields and the boundary energy momentum tensor. So, we need to determine the associated transformations. The induced transformation of an arbitrary boundary filed Φ is defined by For details one can see [17].

Diffeomorphism Charge
Under a general diffeomorphisms transformation x µ → x µ + ǫ µ the variation of the the full action is given by Using the induced transformation (52) the variation of the full action gets simplified as In the last step we used the definition of Neother charge associated to the Peierls bracket. Using the definition CTSC (51) the associated charge is given by Both of the gauge fields combined with the energy momentum tensor appear in the generator of the diffeomorphisms. One should note that, although the energy momentum tensor T tt and the U(1) currents J i,t diverge as ρ → ∞, the above combination asymptotically is finite.
Note that for the extremal solution where f (t) = 0 all non-extremal excitations are vanishing. This is a consistency condition since it implies that the excitations considered above keeps the solution in the extremal limit [2].

Gauge Transformation Charges
Again one can explore the variation of the action under a gauge transformation δ Λ A i a = ∂ a Λ i by using the CTSM given by where The 2nd term in (61) vanishes, as like as (58), and by definition the charges of the gauge transformations are given by Using the asymptotic behavior of the gauge fields and metric (29) one finds For the near horizon extremal solution the gauge transformation charges are given by For the 4D extremal Kerr solution it was shown that this charge equals to the angular momentum in 4D point of view [13].

Central charge
Now we can explore the combination of the true generators to see if there is a central charge associated to the combined asymptotic transformations constructed in §4.1 and §4.2. The combined generator is given by For studying the transformations of this charge, it is natural to relate the transformation parameters, ǫ and Λ i , to each other to treat the combined charge as a charge with one transformation parameter. The asymptotic behavior of the transformation parameters are given by and up to leading order one can write 5 Thus, the gauge transformations part of the combined charge (66) are simplified as Because of the relation (69), the first order of ǫ t where appear in (70) is e −2ρ/l . Now we are able to study the transformations of the combined charge (66). At the first we calculate transformation of the diffeomorphisms part of the combined charge, T tt +J t,i A i t , which asymptotically is given by Asymptotically, the variation of the gauge transformations part of the combined charge (66) is given by It seems the weight of this part is zero. But as discussed in [13], we do not worry about this. As mentioned above for this part of the combined charge asymptotically ǫ t ∼ e −2ρ/l and it has the weight two effectively.
Since the AdS 2 radius l AdS is dimensionless in the standard normalization of the transformation of the energy momentum tensor the central charge is defined by Thus one can read the associated central charge from (24), (71), (72) and (73) as

Levels
The level k of the U(1) gauge transformation is defined by For the currents (55) associated to the gauge transformation of the U(1) charges one can derive the level by and by using (24) Thus, both of the gauge fields have the same level as This relation between the central charge and the level of the R-currents is as like as what obtained in [13,23].

Myers-Perry Black Hole
To further study our results we investigate one example of 5D solutions in this section. We study the Myers-Perry black hole, which is a simple 5D solution with two angular momenta in the near horizon geometry. The near horizon geometry of the Myers-Perry black hole are of the form (6). Without lose of generality we assume that 0 < a < b, where a and b are the two parameters of the Myers-Perry black hole which are related to the two angular momenta. The parameters and functions of the metric (6) for this solution are given by [15] F (θ) = σ(θ) 4 , σ(θ) = ab + a 2 cos 2 θ + b 2 sin 2 θ, Considering these expansions one can derive the parameters (16)-(17) appeared in the central charge (74) and the levels (77) as One can see from (16) that α is proportional to the area of the horizon which gives the Wald entropy of the black hole. So it is natural to study the symmetries of the rotating coordinates which does not affect on the entropy. For general study of this argument one can see [10]. The geometry of Myers-Perry black hole (83) has a symmetry under a ↔ b compensated with θ → π/2 − θ. Thus after integrating over angular coordinates we expect that the central charge (74) has this symmetry. Using (74), (78) and (84) the central charge and levels are given by This is a sum of two central charges derived in [4]. One can show that this is also equal to the result obtained from Eq.(5) for the Myers-Perry black hole (83).

Conclusion and Discussion
In this paper we studied the near horizon geometry of the 5D extremal rotating black holes from the 2D point of view by reduction over the angular coordinates following [13]. We showed that the consistency of the boundary conditions implies that both of the gauge fields, which correspond to two angular momenta, should appear in the same manner. By studying the variation of the boundary energy momentum tensor we calculated the central charge of the CFT 1 dual to the reduced solution which is AdS 2 .
Although we did not trace the power of the fluctuations of the metric in 5D point of view (2) due to the process of reduction, we showed that by following consistency of the boundary conditions the results are in agreement with the calculations of 5D viewpoint [10]. The advantage of the consistency is, compensating of the variations of both of the gauge fields with the diffeomorphisms variation §3.4. It is interesting to study the relation between consistency of boundary conditions from 5D (2) and 2D (42) viewpoints.
Thus we conclude that from the 2D viewpoint the consistency of boundary conditions respect the symmetry of the near horizon geometry discussed in [10] in the Kerr/CFT approach systematically. Since α appeared in the central charge (74) is proportional to the Wald entropy (16), it is natural to study the effect of the symmetry of the near horizon geometry. The symmetry of the rotating coordinates in 5D point of view inherited to M ij (18) and its determinant. The result of this symmetry in 5D perspective was investigated in [10] but, it is not realized in our two examples studied in §5.
The reduction of extremal black holes in higher dimensions to 2D is also the basic argument of Sen's quantum entropy function [12]. This reduction produce an AdS 2 metric, some scalar fields and a number of gauge fields associated to the angular momenta. In [12] Sen showed that the thermodynamics of such solution can be derived from quantum entropy function which is defined by in the euclidean frame. dθA i θ denotes the integral of i-th gauge field along the boundary of AdS 2 and q i is the i-th electric charge. For complete discussion one can see [12]. It is clear that all of the gauge fields play the same role in the thermodynamics of the reduced extremal solution. Thus, it is natural to expect that there is one CFT corresponded to the near horizon geometry. This was discussed for 5D ERBH in [10] by Kerr/CFT approach and in this article by reduction to 2D. It is worth to study the relation between these approaches and Sen's quantum entropy function.
Although we have not studied higher dimensional extremal black holes we anticipate that after the reduction on angular coordinates, all of the gauge fields play the same role from 2D perspective and there is only one CFT corresponding to the near horizon geometry of extremal black holes.
Recently it was proposed a systematic method for deriving the order of the boundary conditions of the metric for topologically massive gravity [24]. Another interesting question is the extension of this method to higher dimensions and compare our results with this extension.
In this paper we limited ourself to the solution of 5D Einstein gravity but one can generalize this method to the solutions of other gravity theory e.g. supergravity. As a simple example with only one rotating coordinate one can study supersymmetric black ring [25]. Microscopic description of this solution is studied from other points of view [26,27,3].