Extremal RN/CFT in Both Hands Revisited

We study RN/CFT correspondence for four dimensional extremal Reissner-Nordstrom black hole. We uplift the 4d RN black hole to a 5d rotating black hole and make a geometric regularization of the 5d space-time. Both hands central charges are obtained correctly at the same time by Brown-Henneaux technique.


I. INTRODUCTION
To fully understand Bekenstein-Hawking entropy of a black hole in a microscopic point of view is still a challenge. An important progress has been made in [1] by using the Brown-Henneaux technique [2], i.e. the Kerr/CFT correspondence. The central charge of the dual CFT reproduces the exact Bekenstein-Hawking entropy of the 4-dimensional extremal Kerr black hole by Cardy's formula. This method has been extensively studied for Kerr black hole and other more general rotating black holes [3][4][5][6][7][8][9][10][11][12][13][14][15][16]. However, in the original Kerr/CFT method, the Virasoro algebra was realized only from an enhancement of the rotational U (1) isometry, which corresponds to left hand central charge, not from the SL (2, R). Later, it was found that the right hand central charge of rotating black holes can be obtained by delicately choosing appropriate boundary conditions, [17][18][19][20][21][22][23][24][25].
Nevertheless, the method in Kerr/CFT correspondence cannot be directly applied to non-rotating charge black holes, such as Reissner-Nordstrom (RN) black hole. One found that the central charge for 4d RN black hole vanishes by directly using the Brown-Henneaux technique. One possible way to solve the problem is to uplift the space-time to higher dimensions. The left hand central charge c L = 6Q 2 of 4d extremal RN black hole was obtained by uplifting it to 5 dimension [4,9,21,[26][27][28]. The other possible way is to consider a 2-dimensional effective theory of 4d extremal RN black hole via a dimensional reduction following the idea in [19], by which the right hand central charge c R = 6Q 2 of 4d extremal RN way obtained in [28,29]. Although both hands central charges can be obtained, but one has to use difference method. It is an interesting question that whether we are able to calculate both hands central charges for extremal RN black hole at the same time by choosing appropriate boundary conditions as in the case of Kerr/CFT? The answer is yes, but with a geometric regularization.
In this paper, we calculate both hands central charges for 4d extremal RN black hole by using the Brown-Henneaux technique. We first uplift the 4d RN black hole to a 5d black hole, following the idea in [24], we then deform the 5d black hole by a geometric regularization.
Both hands central charges c R = c L = 6Q 2 are obtained in this way at the same time.
The paper is organized as follows. In section II, we briefly review the near horizon geometry of 4d extremal RN black hole. We then uplift 4d RN black hole to 5d and calculate both hands asymptotic Killing vectors in section III. In section IV, we deform the 5d black hold by the geometric regularization to obtain both hands central charges. Our conclusion and discussion are included in section V.

II. NEAR HORIZON GEOMETRY OF 4D EXTREMAL RN BLACK HOLE
In four dimension, the Einstein-Maxwell theory admits a unique spherical electro-vacuum solution, the RN black hole. The metric of 4d non-extremal RN black hole is where dΩ 2 2 = dθ 2 + sin 2 θdφ 2 and Two horizons are obtained by solving f (r) = 0, The Hawking temperature and black hole entropy can be calculated as follows, In the extremal limit of RN black hole, i.e. r + = r − = m = q, we have and with To consider the near horizon limit, r → r + , we make the following coordinates transformation, and take the limit ǫ → 0. Finally, the metric of 4d near horizon extremal RN black hole is obtained as  [4,28]. The metric of the uplifted 5-dimensional black hole can be expressed as, where A (2) is a gauge 2-form and the new coordinate y is compactifed on a circle with a proper period [27] y = y + 2πq. The metric (11) with the 2-form (12) is a solution of 5d Einstein-Maxwell theory, where G 5 = 2πqG 4 and F (3) = dA (2) . We should notice that the 2-form field A (2) has no contribution to the central charge based on the argument in [6,26].
By choosing the following boundary condition, the asymptotic Killing vector (AKV) is obtained as, where ǫ (τ ) and η (y) are arbitrary functions of τ and y, respectively. The left hand and right hand AKVs reads Since y = y + 2πq is periodic, we can express the function η (y) as its Fourier bases e −iny/q .
Similarly, if we assign a period for τ with τ = τ + 2πβ with β an arbitrary real number, the function ǫ (τ ) can be expressed by e −inτ /β . With appropriate normalizations, we can write The corresponding AKVs composes two copies of Virasoro algebra without central extension as expected, To obtain the central charges, we define the asymptotic charge Q ζ associated to the AKV ζ as, where the 2-form k ζ is defined for a perturbation h µν around the background metric g µν [3], with * denotes the Hodge dual, and Σ is the 4-dimensional equal-time hypersurface at τ = const.
The Dirac bracket of the asymptotic charge is given by We define the generators L n of Virasoro algebra as with α an arbitrary c-number. With the usual rule of Dirac brackets {Q ζm , Q ζn } DB → −i [Q ζm , Q ζn ], the commutation relations of the generators L n can be calculated as where L is Lie derivative. Plugging AKVs (18) into the 2-form (21), we obtain Comparing to the standard commutation relations of Virasoro algebra, the central charges are read off as 1 , We see that the above calculation only gives us the correct left hand central charge with the right hand central charge vanishes. However, because there is no gravitational anomaly, we expect that c L = c R . In next section, we will make a geometric regularization of the 5d black hole and obtain the correct both hands central charges together.
In the above calculation, we took the "equal-time hypersurface" Σ with τ = const when we defined the asymptotic charge Q ζ in Eq. (20). However, it is easy to see that ds 2 → 0 for τ = const from Eq. (27), so that Σ is not a space-like surface but a light-like one on the boundary. Therefore our naive choice of the "equal-time hypersurface" Σ causes problem when we consider the asymptotic behavior near the boundary. To define a spacelike surface at the boundary, we follows [24] to regularize the 5d geometry by a coordinates transformation, where a is the regularization parameter. We should remark that the resulted central charges should be independent of the regularization parameter a.
With the coordinates transformation (28), the 5d black hole metric (11) becomes which, at the boundary ρ = const → ∞, reduces to Now the shifted "equal-time hypersurface" Σ ′ with τ ′ = const becomes a space-like surface We also note that, to respect the periodicity y = y + 2πq,. we should fix the period of τ by β = aq in Eq. (17). and the AKVs can be rewritten by the new coordinates (τ ′ , y ′ ) as which also satisfy Virasoro algebra without central charge as in Eq. (19). Similar calculations can be carried on with the results, Comparing to the standard commutation relations of Virasoro algebra 25, the above results lead to the correct both hands central charges of 4d extremal RN black hole, We see that the central charges are indeed independent of the regularization parameter a as we promised.
Together with the CFT temperature of both hands [28,29], we obtain the microscopic entropy by Cardy's formula from the central charges (33), which correctly reproduces the Bekenstein-Hawking entropy of the extremal RN black hole (7).

V. CONCLUSION
In this paper, we studied near horizon geometry of 4d extremal RN black hole. We uplifted the 4d extremal RN black hole to a 5d black hole, and deformed the 5d black hole by a geometric regularization. Both hands central charges of 4d extremal RN black hole c L = c R = 6Q 2 were correctly obtained by using the Brown-Henneaux technique. The crucial step to get the result is the geometric regularization by which the shifted "equaltime hypersurface" becomes space-like. The resulted central charges are independent of the regularization as we expected.