# Near-horizon of 5D rotating black holes from 2D perspective

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## Abstract

We study the CFT dual to five-dimensional extremal rotating black holes, by investigating the two-dimensional perspective of their near-horizon geometry. From the 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 the Kerr/CFT approach to five-dimensional extremal rotating black holes.

### Keywords

Black Hole Central Charge Gauge Transformation Gauge Field Black Ring## 1 Introduction

Five-dimensional black holes have been interesting ever since 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 rotating black holes (ERBH) a wide class of such solutions have been studied [3, 4, 5, 6, 7, 8, 9, 10, 11]. A common feature of these studies is the appearance of two rotating coordinates in the near-horizon geometry of most of these solutions as well as an AdS\(_2\) part. It was proposed in some of these works that there are two dual CFTs, each of which corresponds to one of the rotations.

It was shown in [10] that these two CFTs are related to each other by the SL(2,\({\mathbb Z}\)) transformation which is a symmetry in the space of the moduli parameters of the near-horizon geometry of the 5D black holes with two rotating coordinates. For this propose the boundary conditions for the rotating coordinates are in the same order such that the symmetry of the rotating coordinates is preserved by boundary fluctuations. This is not the case for the 5D black holes with only one rotation, e.g. black ring.

In this note we show that the consistency of the boundary conditions from the 2D point of view requires that the two gauge fields should be treated on the same footing in the study of an asymptotic symmetry group. This leads to a chiral CFT, with a central special central charge, corresponding to the near-horizon geometry of 5D extremal double rotating black holes.

Each of the rotating coordinates in 5D reduces to one of the gauge fields from the 2D perspective. By introducing proper boundary conditions for the gauge fields and for the boundary energy-momentum tensor we calculate the corresponding central charge of the dual CFT. This approach resembles the quantum entropy function introduced by Ashoke Sen [12], where all of the gauge fields are considered in the same manner to study the thermodynamics of extremal black holes.

By using the approach introduced by Castro and Larsen [13], we reduce the near-horizon geometry of 5D extremal double rotating black holes to a 2D theory and investigate the properties of the boundary energy-momentum tensor of the AdS\(_2\) metric. We show that the variation of the energy-momentum tensor under diffeomorphism which should be combined with gauge transformations [14] admits one central charge. As an example we calculate the associated central charge for the Myers–Perry black hole [15] and show the agreement with known results in this case.

The remainder of this paper is organized as follows. In Sect. 2 we briefly review the 5D extremal rotating black hole and its CFT dual from the 5D point of view. In Sect. 3, following [13], we study the reduction of 5D extremal rotating black hole to the AdS solution of 2D theory. Then we derive the boundary terms of the 2D action and investigate the consistency of the boundary conditions which are allowed for 2D theory. By using the notion of the Peierls bracket [16] and the 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 Sect. 4. In Sect. 5 we study the Myers–Perry black hole with two rotations and we show the agreement of our results with the previous calculations. Finally, Sect. 6 contains our conclusions and a brief discussion.

## 2 Review of 5D ERBH/CFT

This section is devoted to a review of the generalization of Kerr/CFT approach for 5D extremal rotating black holes. The reader who is familiar with the Kerr/CFT approach can skip this section. We wil mention the main steps of the calculations and will not discuss the details, which can be found in [10].

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In the next two sections we confirm this result from the 2D perspective. For this propose we reduce the 5D near-horizon geometry (1) to 2D theory 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 diffeomorphism and gauge transformations of both of the gauge fields is consistent for investigating the variation of the boundary energy-momentum tensor. In Sect. 5 we show the agreement of 2D results with the 5D results for Myers–Perry black holes.

## 3 2D view of 5D extremal rotating black holes

In this section, we want to study the 5D ERBH from the 2D perspective. The next section is devoted to a calculation of the conserved charges and central charge following Castro and Larsen [13]. The steps and arguments are similar to [13], so we do not give all the details. By using the reduction we will show that both of the gauge fields in 2D, associated to two rotating coordinates, play the same role in studying the asymptotic symmetry and the AdS/CFT correspondence. This is 2D evidence for the arguments reviewed in Sect. 2.

### 3.1 5D ERBH

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### 3.2 Solutions

### 3.3 Boundary terms

### 3.4 Consistency of boundary conditions

As discussed in [14, 26], for the AdS solution with a gauge field the combination of diffeomorphism and gauge transformation should be consistent with the gauge conditions. In this section we show that for the solution (27), i.e. the 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 diffeomorphism. For this purpose we first determine the diffeomorphism of the metric and its consequences for the gauge fields. Then we find the compensating gauge transformations leaving the gauge fields in the gauge condition (25).

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From the 5D point of view this means that 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 the associated components of the metric, should be of the same order. This is in precise agreement with the result of [10], which is reviewed in Sect. 2.

## 4 Conserved charges and central charge

Now we want to investigate the asymptotic symmetries by employing the associated conserved charges. Since we are interested in the boundary energy-momentum tensor of a solution there are some subtleties we meet with as we define the associated conserved charges.

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### 4.1 Diffeomorphism charge

### 4.2 Gauge transformation charges

### 4.3 Central charge

Now we can explore the combination of the physical generators. Moreover, we can derive the central charge associated to the asymptotic transformations constructed in Sects. 4.1 and 4.2.

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### 4.4 Levels

## 5 Myers–Perry black hole

## 6 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 reducing the theory over the angular part of the 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 in 5D, appear in the same manner. By studying the variation of the boundary energy-momentum tensor we calculate the central charge of the CFT dual to the reduced solution which has a AdS\(_2\) geometry.

Although we did not trace the power of the fluctuations of the 5D metric due to the process of reduction, we showed that for consistent boundary conditions the results are in agreement with the calculations of 5D viewpoint [10]. The advantage of the consistency is that they compensate for the variations of both of the gauge fields with the diffeomorphism variation studied in Sect. 3.4. It is interesting to study the relation between consistency of boundary conditions from 5D and 2D point of views which are given in (2) and (42), respectively.

Finally, we conclude that from a 2D point of view that the consistency of the boundary conditions respects the symmetry of the near-horizon geometry discussed in [10] in the Kerr/CFT approach. Since the parameter \(\alpha \) appearing 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 the 5D point of view inherited by \(M_{ij}\), see (18), and its determinant. The result of this symmetry in the 5D perspective was investigated in [10], but it is not realized in the Myers–Perry black holes studied in Sect. 5.

Although we have not studied higher-dimensional extremal black holes we expect that one can apply this formalism in those cases and 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 [27]. 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 solutions 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 [28]. Microscopic description of this solution is studied using other methods in [3, 29, 30].

## Footnotes

- 1.
The AdS\(_2\) radius has been absorbed in \(F(\theta )\).

- 2.
We use indices, \(\mu , \nu \) for coordinates \(r, t\), and the indices \(i, j\) for the gauge fields.

- 3.
There is no summation over primed indices, in our notation.

- 4.
For details one can see [17].

- 5.
There are some subtleties in this relation which are discussed in [13].

## Notes

### Acknowledgments

I would like to thank Farhang Loran for useful discussions and comments on the draft and Department of Physics of Isfahan University of Technology where I did my Ph.D. and part of this work was done. This work is supported by Foundation for Polish Science MPD Programme co-financed by the European Regional Development Fund, agreement No. MPD/2009/6.

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