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Kerr/CFT correspondence in a 4D extremal rotating regular black hole with a non-linear magnetic monopole

We carry out the Kerr/CFT correspondence in a four-dimensional extremal rotating regular black hole with a non-linear magnetic monopole (NLMM). One problem in this study would be whether our geometry can be a solution or not. We search for the way making our rotating geometry into a solution based on the fact that the Schwarzschild regular black hole geometry with a NLMM can be a solution. However, in the attempt to extend the Schwarzschild case that we can naturally consider, it turns out that it is impossible to construct a model in which our geometry can be a exact solution. We manage this problem by making use of the fact that our geometry can be a solution approximately in the whole space-time except for the black hole's core region. As a next problem, it turns out that the equation to obtain the horizon radii is given by a fifth-order equation due to the regularization effect. We overcome this problem by treating the regularization effect perturbatively. As a result, we can obtain the near-horizon extremal Kerr (NHEK) geometry with the correction of the regularization effect. Once obtaining the NHEK geometry, we can obtain the central charge and the Frolov-Thorne temperature in the dual CFT. Using these, we compute its entropy through the Cardy formula, which agrees with the one computed from the Bekenstein-Hawking entropy.


Introduction
It is known from an analogy with the thermodynamics [1,2,3] that black holes have entropy proportional to the horizon area. Such entropy is named the Bekenstein-Hawking entropy. If black holes are formed by collapsing of stars, intuitively considering what entropies of black holes is proportional to the volume would be natural. Hence the Bekenstein-Hawking entropy appears odd. Furthermore, black holes should not have characteristics more than mass, angular momentum and electric charge at the classical level according to the no-hair theorem [4]. Hence microscopic unavailing of the Bekenstein-Hawking entropy is very intriguing. Although at present it has not been completed yet, it could be considered that quantum gravity's degree of freedom would be closely related to its microscopic origin. Hence microscopic unavailing of the Bekenstein-Hawking entropy would be considered to lead to some sort of important understanding in the quantum gravity, which would be a very important outstanding problem.
Although many studies to understand the Bekenstein-Hawking entropy from the microscopic point of view have been performed, at 2008, in a four-dimensional Kerr black hole, Guica, Hartman, Song and Strominger could succeed in reading off the central charge and temperature in chiral half of 2D CFT dual for it, which can reproduce the entropy agreeing to the Bekenstein-Hawking entropy through the Cardy formula [5]. This is refereed to as Kerr/CFT correspondence. Now stars considered as Kerr black holes at mostly zero temperature are actually observed in the sky (e.g. GRS 1915+105 [6] and cygnus X-1 [7]). Hence it could be said that Kerr/CFT correspondence could be considered as a potential candidate leading to microscopic unavailing of the Bekenstein-Hawking entropy of realistic black holes.
Here we briefly state features in current Kerr/CFT correspondence: (i) Kerr/CFT can treat Kerr black holes in arbitrarily cosmological constants.
(ii) Entropy in extremal Kerr black holes can be computed exactly.
(iii) Underlying fundamental objects like D-branes which bridges between the gravity and CFT pictures are unclear.
(iv) Dual CFT in Kerr/CFT is considered as probably an effective description of the quantum gravity which would be obtained if we could succeed in quantizing the gravity on a given Kerr black hole background.
(v) Supersymmetries are not needed in Kerr/CFT.
Regarding (iii), although Kerr/CFT correspondence is realized using the Brown-Henneaux's method [8], the Brown-Henneaux's method is independent of the microscopic details of a given Kerr black hole background. Here in the Brown-Henneaux's method, some deviations (boundary conditions) are put on asymptotic region in a given space-time by hand. Correspondingly Kerr/CFT correspondence is also independent of microscopic details of a given Kerr black hole background. Rules to determine the deviations is not known at present, which is one of the open questions. Further (ii) would be a remarkable different points from AdS/CFT correspondence [9,10,11] between weekly coupling gravities and strongly coupled CFT.
Thanks to Kerr/CFT correspondence, current situation could be said to have been reached a stage that quantum theories of gravities on Kerr black holes at the zero temperature would be possible to be effectively described by chiral half of 2D CFT, and all the Bekenstein-Hawking entropy of Kerr black holes at the zero temperature could be reproduced exactly in terms of the dual CFT † . Hence our next issue in unveiling the microscopic origin of the Bekenstein-Hawking entropy would be to extend Kerr/CFT correspondence to more realistic one. One way in these attempts would be extension to finite temperature (e.g. [27,28,29,30,31,32,33,34,35,36]). Another one would be to consider some more realistic black holes. If we could do this (and specify the actions of dual CFT), we might be able to give some sort of answers to open problems in black holes like the information paradox, cosmic censorship problem etc.
In such a situation, as an extension of Kerr/CFT correspondence to more realistic black holes, in this paper we apply Kerr/CFT correspondence to a four-dimensional regular rotating black hole at the zero temperature (extremal limit). We explain what is the regular black hole in what follows.
It is known that the cores of classical black holes are singular. However it is expected that quantum gravity effects working among matters in the cores would become strong repulsive forces as those matters close up to the plank scale length. As a result there are considerations that indeed the singularities in the core are not present. Those black holes are named regular black hole or non-singular black hole. Currently many stars expected as black holes are observed in the sky, and the regular black holes are considered to correspond to those actual black holes.
Regarding the formulations for the regular black holes until now are as follows: [37,38,39,40,41] for the Schwarzschild regular black hole and [42,43,44,45] for the rotating regular black hole.
In this study, if we treat the equation to obtain the horizon radii as it is, it turns out that it is given as a fifth-order equation due to the regularization effect for the core of the black holes, whereas it is a second-order equation in classical Kerr black holes. In order to handle this situation, we treat the regularization effect in perturbative way by expanding the regularized mass in terms of the parameter for the regularization effect up to the first cubic-order. We call this correction for the regularization effect expanded up to the first cubic-order as first cubic-order expansion. As a result, the equation becomes forth-order, and we can obtain an NHEK (near-horizon extremal Kerr) geometry of the rotating regular black hole at the extremal limit up to the first cubic-order expansion. † E.g. (i) For arbitrary dimensions and extension to gauged/ungauged SUGRA, [12,13], (ii) For general expression of conserved charges [14,15], (iii) For Kerr-Newman BH with arbitrary cosmological constant, [14], (iv) For 5D Kaluza-Klein BH, [16], (v) For 5D black rings, [17,18,19], (vi) For 5D D1-D5-P and BMPV BH's, [20,21], (vii) For higher order derivative corrections, [22,23,18,24], (viii) For Kerr/CFT based on M/superstring theories, [25,26] Then obtaining the Virasolo algebra with the central charge and the Frolov-Thorne temperature in the chiral half of of dual 2D CFT, we compute the entropy in the dual CFT using the Cardy formula. We can see that it agrees with the Bekenstein-Hawking entropy. Here the central charge, the CFT temperature and the entropy we compute involve the correction of the first cubic-order expansion.
As the significant point in this study, it is taken that this study would be a more realistic Kerr/CFT, since regular black holes could correspond to actual black holes.
The organization of this paper is as follows: In Sec.2, the rotating regular black hole we consider in this study is given. In Sec.3.1, it turns out that the equation used to obtain the horizon radii is given as a fifth-order. In order to handle that, the first cubic-order expansion which we mentioned above is proposed. Then in Sec.3.2 various fundamental quantities which are necessary to obtain the NHEK geometry are computed, and in Sec.3.3 the NHEK geometry is obtained. In Sec.4, the Virasolo algebra and the central charge in the dual CFT are obtained using the Brown-Henneaux's method. In Sec.5, the Frolov-Thorne temperature wchih can play the role of the temperature in the dual CFT is obtained. Then using the Cardy formula, the entropy in the dual CFT agreeing with the Bekenstein-Hawking entropy is compute. In Sec.6 is devoted to a problem in this study that there is a discrepancy in values of angular velocities obtained from different two ways are mentioned. Sec.7 is the summary of this paper.

Rotating regular black hole we consider in this study
The rotating regular black hole we consider in this paper is ds 2 = g tt dt 2 + 2g tφ dtdφ + g φφ dφ 2 + g rr dr 2 + g θθ dθ 2 with where µ, ν = t, r, θ, φ, and m is the ADM mass and L is the angular momentum of the black hole, and l p is the Plank length which originates in the quantum gravity effect regularizing the core of black holes. The metric in eq.(1) is obtained [42] from the Hayward type regular black hole [40] through the Newman-Janis algorithm [46,47,48].
3 NHEK geometry in this study

Our first cubic-order expansion
Now we attempt to obtain the NHEK geometry in eq.(1). To this purpose, we first try to solve∆ = 0 to get the radii of the outer and inner horizons r ± analytically. Then we encounter a difficulty. Because it turns out that∆ = 0 is a fifth-order equation. To handle this, in this paper we consider the situation: and treat the regularized massm in eq.(5) perturbatively as In this paper we call this expansion as first cubic-order expansion. This expansion has been firstly considered in [49], and is the basic idea in this study. Corresponding to this first cubic-order expansion, our analysis in what follows is basically always performed up to the ρ 3 -correction with O(ρ 6 ). In all the following descriptions of equations we omit to write O(ρ 6 ) for the simplicity in description.

Fundamental quantities to get the NHEK under the first cubic-order expansion
Then solving∆ = 0, wherem in∆ is now gotten the first cubic-order expansion, we get four horizon radii as where r I is a multiple root of two solutions. We can see that r I is an unphysical horizon radius, and r + and r − are the outer and inner horizon radii, respectively. Actually we can see that the first two terms in eq.(9) are the typical one in the four-dimensional classical Kerr geometry. Further at this time, we can see that∆ can be written as Then by solving r + = r − , we obtain the a and m at the extremal limit. Writing these as a ext and m ext , these in terms of m and a are obtained as Here although we can find that there are four a ext 's and three m ext 's as solutions in r + = r − , in the above we have written only physically acceptable one. If we solve eq.(11) with regard to m, we can find that one of solutions is eq. (12). Contrary, if we solve eq.(12) with regard to a, we can find that one of solutions is eq.(11). Then substituting the a ext and m ext to r ± in eqs. (8) and (9), we can obtain the horizon radii at the extremal limit as Further, at the extremal limit∆ can also be written as We here define the angular velocity Ω in this study. In general angular velocities of black holes can be obtained from either one of the following two ways: (i) Read off from the Killing vector field, ξ = ∂ t + Ω ∂ φ , on the horizon.
(ii) Read off from the coefficient of dL in the variation of the horizon area A(m, L).
The Ω obtained from the (i) is On the other hand, if we obtain Ω in the (ii), although we do not write it here explicitly (see eq.(102)), it turns out that these two Ω's obtained from each way do not agree each other.
In this study, we adopt Ω obtained from the (i). Namely we consider Ω in eq.(16) in this paper. In this case, at the extremal limit, where Ω ext ≡ lim r→rext Ω. We take up the problem of this discrepancy in two Ω's obtained from the (i) and (ii) and state why we adopt the Ω obtained from the (i) in Sec.6.

NHEK geometry of our rotating regular black holes in the first cubic-order expansion
Now that we have obtained values of various quantities at the extremal limit, let us obtain the NHEK geometry of eq.(1) in the co-rotating frame. We write the coordinates in the co-rotating frame as (t,r,θ,φ), which are related with the normal coordinates (t, r, θ, φ) in eq.(1) aŝ where taking the λ → 0 limit means taking both extremal and near-horizon limits.
Coordinates appearing in what follows are always the ones in the co-rotating frame. Thus let us denote these as (t, r, θ, φ) withoutˆin what follows. Then performing the following manipulation to the geometry in eq.(1) straightforwardly in the following sequence: 1) Substitute eq. (15) 2) Substitute eq. (18) 3) Substitute eqs. (11), (13), and (14) 4) Take the terms up to the ρ 3 -order (the first cubic-order) 5) Take the leading order in the λ → 0 limit we can reach the NHEK geometry of eq.(1) as where and Here in the actual computation the parts now given as −f 1 r 2 and 2Λ 2 rf 5 in eq. (19) are given as O(l 12 p )/λ 2 + O(l 6 p )/λ − f 1 r 2 and O(l 6 p )/λ + 2Λ 2 rf 5 . Although there is a problem that the λ → 0 limit or disregarding the term more than O(l 3 p ) according to the first cubic-order expansion which one we consider first, we have disregarded the terms more than O(l 6 p ). Now writing eq.(19) as where µ ≡ f 1 + f 2 5 /f 4 . Then defining we can write the above into where Then defining it turns out that we can rewrite the NHEK in eq.(19) into the following form: where We can see from the form of the geometry in eq.(32) that our NHEK geometry with the ρ 3 -correction is also a θ-dependent S 1 (φ) -fibrated AdS 2 with an isometry SL(2,R) × U(1) (φ) as well as the NHEK geometry in [5].
The expression in eq.(32) has been obtained from straightforward computation through the manipulation above eq.(19) using mathematica. On the other hand, we will later obtain the same expression with eq.(32) from another computational way given in the appendix.B in [23] as in eq.(94). Finally we will see that those two results computed in such a different ways agree each other. The expression in eq.(32) would be one of the results in this study.
Although we move to the dual CFT side from the next section, lastly we evaluate the Bekenstein-Hawking entropy in our rotating regular black hole at the extremal limit. From the formula: We can obtain the above from any one of eqs. (1), (19) and (32).

Central charge in the dual CFT
In order to realize isometries on the NHEK geometry with a central charge with which we can compute the entropy matching with the Bekenstein-Hawking entropy, in this section using the Brown-Henneaux's method [8] we set deviations (boundary conditions) at the asymptotic region of the NHEK geometry as g µν =ḡ µν + O(1/r Pµν ) in the same way with [5], and look into the asymptotic symmetry group (ASG) of that. Herē g µν are the metrices of the NHEK geometry in eq. (19) and O(1/r Pµν ) are the deviations at the asymptotic region we set by hand, where P µν means the power characterizing the deviations.

Asymptotic Symmetry Group (ASG)
The deviation O(1/r Pµν ) we set on the NHEK geometryḡ µν in eq. (19) is where x µ = (t, φ, θ, r), and we write the metric in our NHEK geometry with the deviations as g µν =ḡ µν + O(1/r Pµν ). Then there appear diffeomorphisms in our NHEK geometry, and ASG (asymptotic symmetry group) is a set of diffeomorphisms defined as follows: Here 'Allowed diffeomorphisms' means a set of diffeomorphisms generated by Killing vectors ξ = ξ µ ∂ µ for x µ → x µ + ξ µ in g µν . On the other hand, 'Trivial diffeomorphisms' means a subset of 'Allowed diffeomorphisms' consisted of Killing vectors with the conserved charges Q ξ being always zero. Hence it can be also said that ASG is a subset of 'Allowed diffeomorphisms' consisted of Killing vectors with non-zero conserved charges.

Killing vector preserving the ASG
In the geometry in eq. (19) with the deviation in eq.(35), the most general Killing vectors belonging to the allowed diffeomorphisms can be written as [5] where C is some constant and ǫ(φ) is some function of φ. However since φ-direction in the NHEK geometry is periodic by 2π, ǫ(φ) is periodic as ǫ(φ) = ǫ(φ + 2π). Generic forms of conserved charges associated with diffemorphisms can be generally written as Q = boundary X + bulk Y , where X and Y are some conserved currents. In this expression, terms of bulk typically vanish due to constraint equations. Hence terms of boundary can give contribution to conserved charges (and diffemorphisms are generated by terms of boundary). Therefore since subleading terms in eq.(37) decay in the bulk and cannot reach the boundary, there is no contributions to conserved charges from subleading terms in eq.(37). Thus subleading terms in eq.(37) correspond to the trivial diffeomorphisms.
In addition, writing conserved charges associated with t-translation as Q t , values of Q t correspond to deviation from the energy at the extremal limit. Since we study the extremal situation in which the energy should be zero, we impose a constraint: Q t = 0. Hence the leading term C in the ∂ t -term belongs to the trivial diffeomorphism in this paper.
Therefore the Killing vectors preserving the ASG in the NHEK geometry in eq.(32) with the deviations in eq.(35) is As mentioned above ǫ(φ) is periodic under φ → φ + 2π. So as well as [5] we expand Then ξ can be written as and we can find that the Lie bracket of ξ n form the one copy of the Virasoro algebra as

Central charge
where K[ξ, η] means the central charge. Since the system we are treating now has first class constraints, we impose a gauge fixing conditions and introduce the Dirac bracket.
It is known that δ η Q[ξ] gµν =ḡµν can be computed as [51] where G = 1 in the following, L n ≡ L ξn , ∂Σ is the boundary of the spatial surface at infinite r and where the covariant derivatives and raised indices are computed usingḡ µν and h ≡ g µν h µν , and ǫ tθφr = 1. Therefore from eq. (42), where Q m ≡ Q ξm , and we can see Q [ζm,ζn] L = −i(m−n)Q m+n from eq.(44) with eq.(40). Then let us compute the central charge.
In our geometry in eq.(32), k ξ [η,ḡ] can be computed as where ′ means ∂ φ , "+ · · · " in the first line means the terms not to contribute in the integration in eq.(44), which we disregard in the second line, and with d i (i = 1, · · · , 5) given in eq. (22). We shall note that although computational way to reach the above result is basically same with [5], in our computational process and the above result the ρ 3 -correction meaning the effect of the regularized black hole's core is involved. Now η µν in the above are L n g µν = ξ λ ∂ λ g µν + g αν ∂ µ ξ α + g µβ ∂ ν ξ β . Hence performing computation for each η µν appearing in the above as where J, Ω and Λ are given in eq.
where we have used In order to obtain quantum theory version of eq.(46) we perform replacements: The coefficients q 0 and q 3 are determined so that resultant algebra obtained by above replacement takes the form of the Virasolo algebra with a central charge c CFT in 2D CFT such as [L m , L n ] = (m − n)L m+n + 1 J c CFT 12 m(m 2 − 1)δ m+n . From the actual computation it turns out that q 0 and q 3 should be taken as After all, the quantum theory version of eq. (46) is where = 1 in the following. From this we can read off the central charge in the ASG in the NHEK geometry in eq.(32) with the deviations in eq.(35) as Now although (q 0 , q 3 ) could be determined as in eq.(57), we note that the result of q 3 at the ρ 3 -order is the one newly obtained in this study as the effect of the regularization treated up to the first cubic order expansion. Here, although we will see later that the result in eq.(59) is consistent one (see eq.(62)) and Kerr/CFT correspondence can hold (see eq.(98)), we would like to emphasize that when we determine (q 0 , q 3 ) we have had no intention to obtain the result in eq.(59), but we have just determined (q 0 , q 3 ) so that the constant part to be the central term can be bundled with m(m 2 − 1), and the result in eq.(59) is just a result of that.
Lastly, it is known [14,15] that when a NHEK geometry is written as the central charge we have obtained above can be written as without exploiting the Brown-Henneaux's method like eq. (35). The form of eq.(60) is the same with eq. (32). Hence now let us perform computation for the central charge based on the expression of eq.(32) according to eq.(61). Then it turns out that central charge same with eq.(59) can be obtained again as Before closing this section, we would like to give a comment. The computational process to reach c CFT in eq.(59) would be completely different from the one for c CFT in eq.(62). Actually c CFT in eq.(59) has been reached from analysis starting with modifying the fundamental background geometry according to the Brown-Henneaux's method, and then looking into the central charge in the algebra for isometries, and performing the replacement in eq.(56) with eq.(57). The remarkable point in this analysis is that it has been carried out with the ρ 3 -correction. In such a case there would be no guarantee that c CFT in eq.(59) can agree with c CFT in eq.(62). Despite of this, two c CFT 's in eqs.(59) and (62) could agree each other finally. Here we would like to emphasize that the determination of (q 0 , q 3 ) in eq.(57) has not been performed with aiming at this agreement, but performed just for realization of the Virasolo algebra with a central term, in other words, so that the constant part to be the central term can be bundled with m(m 2 − 1). This agreement could be considered as a non-trivial check for the expression of eq.(61) given in [14,15], and the value of the central charge we could obtained as in eq.(59) would be one of the results in this study.

Entropy in the dual CFT
In this section we compute the entropy in the dual CFT considered as an effective description of the quantum gravity on the NHEK geometry in eq.(1) through the Cardy formula: Here c CFT is the central charge in the Virasolo algebra of the dual CFT which has been already obtained in eq.(59). Hence in this section we obtain the temperature in the dual CFT.

Temperature in the dual CFT
For the reason written in [5], the temperature in the NHEK geometry of the classical Kerr geometries is given by the Frolov-Thorne temperature [50]. The definition of the Frolov-Thorne temperature T φ is given as where T H is the Hawking temperature. It is turned out that the temperature in the NHEK geometry of the regular rotating black hole in this study up to the first cubicorder expansion can be also given by the above Frolov-Thorne temperature. Below we show this by the same discussion in [5].
We consider a Fourier expanded scalar filed as Φ = w,m,l f l (r, θ) e −i(wt−mφ) . Here the quantities withoutˆmean the quantities in the normal coordinate, and the quantities withˆmean the quantities in the co-rotating frame, though those do not appear in the following. This notation is the same with eq. (18). Then when the coordinate is changed to the co-rotating frame, it can be seen that the shoulder in the exponential in the above weight transforms as where the coordinate transformation: (t, φ) → (t,φ) has been performed along with eq. (18). Therefore respectively defining the charges associated with ∂ t and ∂ φ as the Boltzmann weight in the co-rotating frame can be written from the Boltzmann weight in the normal coordinate at a hawking temperature T H as where from the left to the middle the λ → 0 limit is taken, which means both nearhorizon and extremal limits have been taken, and from the middle to the right we have rewritten using T φ in eq.(64).
Now we have confirmed that the temperature in our NHEK geometry is also given by the Frolov-Thorne temperature defined in eq.(64). Since it is generally known that the temperature in the dual CFT in Kerr/CFT correspondence is given by the Frolov-Thorne temperature, the CFT temperature in this study can also be considered as Hence in what follows, in order to obtain the CFT temperature we compute the Frolov-Thorne temperature based on the definition in eq.(64). The computation below is gone ahead along essentially the same way in [23].
First we write our original geometry in eq.(1) at the neighborhood of the extremal limit without the near-horizon limit in the following form: Here, and Then the Hawking temperature T H at the extremal limit can be written as where and In the above computation, ′ means ∂ φ and we have used∆ = 0. Using this expression, we can obtain T CFT in eq.(64) as where we have used r ext in eq. (14) with substituting a ext in eq. (11).
Here let us mention that indeed the actual computation the author did in eq.(80) to compute the Hawking temperature was the one using the metric in eq. (19). Hence eqs.(82)-(85) and a result in r.h.s. of eq.(86) are indeed the results in the metric in eq. (19), and the actual equation author has used in the computation to reach eq.(86) is not eq.(80) but (79). How to identify the computational result to the form in eq.(80) is comparing the computational result obtained using the metric in eq. (19) with the computational result obtained by following eq.(B.5) in [23].
In what follows, we show that the expression for T CFT in the middle of eq.(86) appears in the expression of the geometry in the co-rotating frame at the extremal and near-horizon limits, and its value being read off from the geometry can agree with the value in eq.(86).
We rewrite the geometry in eq.(69) into the co-rotating frame in eq.(18) and take the λ → 0 limit which means to take the extremal and near-horizon limits. Then each part in eq.(69) can be written as Here, a a φ 2 = 8m 2 sin 2 θ 3 + cos(2θ) + 4m 2 (1 + 2 cos(2θ)) sin 2 θ (1 + cos 2 θ) 2 ρ 3 , and Ω ext is given in eq. (17). Therefore we can rewrite the geometry in eq.(69) into the co-rotating frame at the near-horizon and extremal limits as We perform the following transformation: Then from eq.(92), where we can compute as Here JΩf 2 , f 3 /f 2 , f 4 /f 2 and κ mean the quantities in eq.(33). Indeed we have already written down our geometry into the form of eq.(94) at eq.(33), and can see that eq.(94) is consistent with eq.(33). Further, from the fact (86)) appears in the expression in eq.(94), we can see that we can read off the CFT temperature from the value of that part as We can see that this result is consistent with the result obtained from the definition of the Frolov-Thorne temperature in eq.(64).
The agreement between the CFT temperature in eqs.(86) and (97) would not be trivial things. Because the actual computational processes we have taken to reach eqs.(86) and (97) are different.
Actually, the result of the CFT temperature in r.h.s. of eq.(86) has been reached by going through computation for the Hawking temperature in eq.(80) based on the original geometry in eq.(1) as mentioned under eq.(86). On the other hand, although the CFT temperature in eq.(97) has been obtained from reading off of the part represented as κ in eq.(32), in reaching eq.(32) we have just performed the manipulation written above eq.(19) toward the original geometry in eq.(1) without any particular settings. The result in the middle of eq.(96) is the one just taken from eq.(33), although of course we can get it from direct computation of the l.h.s. in eq.(96). The remarkable point here is that these analysis have been carried out with the ρ 3 -correction, and we could consider that what the CFT temperature in eqs.(86) and (97) can agree each other from these analysis with the ρ 3 -correction would not be trivial things, as well as the agreement in the central charges stated in the last of Sec.4.3.
Despite of this finally we could reach a fact that eqs.(86) and (97) agree each other. This agreement could be considered as a non-trivial check for the way to read off the CFT temperature from the part written as κ in eq. (32), and would be also one of the results in this study.
5.2 Entropy in the dual CFT for our rotating regular black holes in the first cubic-order expansion Substituting eqs. (59) and (97) into the Cardy formula in eq.(63), we can obtain the entropy in the dual CFT as We can see that this agrees with the Bekenstain-Hawking entropy obtained from the horizon area in eq.(34). In what follows we write Ω's obtained from each way as Ω (i) and Ω (ii) , respectively. Then it turns out that these two Ω's obtained from each way are different each other. Actually, where from the first to second lines, we have substituted r + and expanded it up to the first cubic-order expansion. On the other hand, if we try to obtain Ω (ii) , we first have to obtain the horizon area, which can be computed as Toward this A, substituting r + in eq.(9) and a = L/m, we take variation in terms of m and L. We write it as dA = ∂ m A dm + ∂ L A dL. Then rewriting it into the as In the computation to reach the above, we have backed ρ and a to l p /m and L/m respectively when taking the variation with regard to m and L. ‡ Then we can find that Ω (i) and Ω (ii) do not agree each other: Ω (i) = Ω (ii) at the ρ 3order. Basically it seems that those should agree each other, and this discrepancy might suggest some breakdown of the black hole thermodynamics in the rotating regular black hole treated by the first cubic-order expansion. Although author does not know certain reason for this discrepancy at the moment, we can confirm that our NHEK geometry in eq. (19) does not satisfy the Einstein equation. Author guesses that when geometries satisfy the Einstein equation, Ω (i) and Ω (ii) agree each other, but if geometries do not satisfy the Einstein equation like this study, Ω (i) and Ω (ii) may not need to agree each other. Returning to our subject, the current situation is that there are two Ω's, and we have to select either of Ω's. In this study we would like to select Ω by which S CFT can agree with S BH as we state below.
Since c CFT in eq.(61) can be written as c CFT = κ 2π 2π 0 dφ π 0 dθ Γ(θ)α(θ)γ(θ) = κ 2π A, the Cardy formula can be written as S CFT = πκ 2 A T CFT . Hence, in order to reproduce the Bekenstein-Hawking entropy S BH = A/4, it can be seen that T CFT should be T CFT = 1/2πκ. In the previous section, we could succeed in deriving T CFT = 1/2πκ from the computation starting with the geometry in eq.(69). In that computation process the expression of Ω has taken the form of Ω (i) as in eq.(99). This means that if we performed computation for T CFT with Ω (ii) in eq.(100), we could not reach T CFT = 1/2πκ. Hence we select the angular velocity given by Ω (i) in this paper. ‡ At this time, the Hawking temperature can be obtained as

Summary
In the question for what the microscopic origin of the Bekenstein-Hawking entropy is, Kerr/CFT correspondence would be one of the very important directions. The current issue in Kerr/CFT correspondence is to enlarge its applicable range. In such a situation, in this study taking a regular rotating black hole which would be generally considered to be able to correspond to realistic black holes, we have carried out Kerr/CFT correspondence at the extremal limit.
As a specific problem, in this study the equation to obtain the horizon radii has been given as a fifth-order equation due to the regularization effect for the core of black holes. In order to handle this situation, we have treated the regularization effect by a perturbative way up to the first cubic-order expansion. As a result the equation has become forth-order, and we could reach a NHEK geometry up to the first cubic-order expansion as in eq. (32).
Once the NHEK geometry could be obtained, we could obtain the Virasolo algebra with the central charge as in eq.(59) and the Frolov-Thorne temperature in the chiral half of of dual 2D CFT through the Brown-Henneaux's method as in eq.(97). Then using these we could compute the entropy in the dual CFT through the Cardy formula as in eq.(98). The result could agree with the Bekenstein-Hawking entropy as mentioned in Sec.5.2. Here we would like to emphasize that our central charge, the CFT temperature and the entropy have involved the correction of the first cubic-order expansion.
As the significant point in this study, it is taken that this study would be a more realistic Kerr/CFT, since regular black holes could correspond to actual black holes.
In addition, the value of the central charge is known to be computed easily from eq.(61), and the temperature in the dual CFT is known to be easily read off from the part written as κ in the geometry in eq.(32) as shown in eq.(96). However it would be no guarantee that these ways can work actually and exactly in the situation that there is the ρ 3 -correction (the first cubic-order expansion). In this study we have straightforwardly computed the value of the central charge as in eq.(59), and the dual CFT temperature based on the definition of the Frolov-Thorne temperature as in eq.(64). Then we could see that those eqs.(59) and (64) can agree with the ones obtained from the another easy ways mentioned above as in eqs.(62) and (97). Those agreements could be considered as non-trivial checks for the formula of eq.(61) and the way to read off the Frolov-Thorne from the κ in the geometry in eq. (32).
However there has been a problem in the angular velocity used in the above computational process, which is that there is a discrepancy in the angular velocities computed from the black hole thermodynamics and the geometry in the first cubic-order expansion, as described in Sec.6. Regarding the cause, although author could not specify, it has been guessed that what our NHEK does not satisfy the Einstein equation would have some relation with this discrepancy. In the expression of the angular velocity in this study, we have chosen the one obtained from the geometry as the one by which we can reach the CFT entropy matching with the Bekenstein-Hawking entropy.