Logarithmic corrections to black hole entropy from Kerr/CFT

It has been shown by A. Sen that logarithmic corrections to the black hole area-entropy law are entirely determined macroscopically from the massless particle spectrum. They therefore serve as powerful consistency checks on any proposed enumeration of quantum black hole microstates. Sen’s results include a macroscopic computation of the logarithmic corrections for a five-dimensional near extremal Kerr-Newman black hole. Here we compute these corrections microscopically using a stringy embedding of the Kerr/CFT correspondence and find perfect agreement.


Introduction
The Bekenstein-Hawking area-entropy law universally applies to any self-consistent quantum theory of gravity.Efforts to understand how the former constrains the latter have led to a wealth of insights.
Five years ago Sen et al. [1][2][3][4][5][6] pointed out that the leading corrections to this law, which are of order log A (where A is the area) are also universal in that they depend only on the massless spectrum of particles and are insensitive to the UV completion of the theory.The basic reason for this is that the effects of a particle of mass m can be accounted for by integrating it out, which generates local higher derivative terms in the effective action.These lead to corrections to the entropy which are suppressed by inverse powers of m 2 A and cannot give log A terms.
The macroscopically computed logarithms serve as a litmus test for any proposed enumeration of quantum black hole microstates which is more refined than the test provided by the area law.Sen extensively analyzed a number of stringy examples all of which passed the test with flying colors [7], and further noted that the macroscopic computation does not match the loop gravity result.He also posed a matching of the logarithms as a challenge for Kerr/CFT [8][9][10][11].
In this paper we show that the logarithms indeed match for one microscopic realization [12,13] of Kerr/CFT obtained by embedding a certain near-extremal five-dimensional Kerr-Newman black hole into a string compactification. 1 The microscopic dual is a twodimensional field theory defined as the IR fixed point of the worldvolume field theory on a certain brane configuration with scaled fluxes.This IR limit is certainly nontrivial but is not a conventional 2D CFT.Its properties are incompletely understood and have been studied in a variety of approaches: see e.g.[14][15][16][17][18][19][20].
The example of Kerr/CFT we chose is the simplest possible case.In the simplest examples considered by Sen, the macro-micro match has a somewhat trivial flavor: all logs vanish in a certain thermodynamic ensemble, with logs generated on both sides of the match by Legendre transforms to a different ensemble.However, later examples become quite intricate and provide compelling tests of a variety of stringy constructions.In the fivedimensional Kerr/CFT match herein, all logs vanish in a certain thermodynamic ensemble, and the character of the match is as in the simplest of Sen's examples.Non-trivial aspects of our match reside in its reliance on the values of the Kerr/CFT central charges and Kac-Moody levels which enter the microscopic computation.In particular, we find that a necessary non-vanishing level of the current dual to the electric field is provided by a Chern-Simons term which is crucially present in the effective action obtained from string theory.While it is reassuring that the simplest case works, perhaps more challenging matches such as the extremal 4D case may eventually provide more refined and compelling tests of Kerr/CFT.This paper is organized as follows.In section 2 we describe the black hole solution, take its near horizon limit, and determine the corresponding quantum mixed state in the CFT.In section 3 we begin by stating the result of [6] for the logarithmic corrections to the Bekenstein-Hawking entropy and then proceed to compute them microscopically in the dual theory for two different enhancements of the global symmetries.We match the Bekenstein-Hawking entropy of the near extremal black hole to first order in the Hawking temperature with the Cardy formula and, using the result of Appendix A for the gauge Kac-Moody level, we show that the logarithmic corrections also agree.
2 The five-dimensional Kerr-Newman black hole We consider a charged and rotating black hole solution of five-dimensional Einstein gravity minimally coupled to a gauge field.The dynamics of the latter is specified by the Yang-Mills-Chern-Simons Lagrangian, so that the complete action is,2 Specifically, we are interested in the following Kerr-Newman black hole solution to (2.1) considered in [12], where we have defined the quantities and s ≡ sinh δ , c ≡ cosh δ.The geometry depends on three independent parameters (a, M 0 , δ) and the physical quantities of the black hole, i.e., its mass, angular momentum and electric charge, are given in terms of those parameters by In five dimensions, it is possible to have a second angular momentum, J R , but we set J R = 0. Note that the SU (2) L angle is identified ψ ∼ ψ + 4π.This black hole displays inner and outer horizons located at At the (outer) horizon, the angular velocities are and the electric potential is Finally, the Hawking temperature is given by and the Bekenstein-Hawking entropy is The black hole approaches extremality in the limit M 0 → 4a 2 .In this limit, the two horizons (2.6) coalesce at r + = a and the Hawking temperature (2.9) vanishes.The charges (2.5) become and the angular velocity (2.7) and electric potential (2.8) become At extremality, the Bekenstein-Hawking entropy (2.10) reduces to In this paper we are interested in the near-extreme case so we introduce a small parameter κ that measures the deviation from extremality and write M 0 = 4a 2 + a 2 κ2 .Substituting this into (2.10) and keeping terms up to linear order in κ, the near extremal entropy is

Near horizon, near extremal limit
Consider the coordinate transformation Here, r + is the location of the outer horizon given in (2.6) and Ω L ext is the extremal angular velocity (2.12).Making this coordinate transformation in the five-dimensional geometry (2.2), (2.3), with M 0 fixed to its extremal value, M 0 = 4a 2 , and letting ǫ → 0, one obtains the extremal near horizon geometry given in [12].
Here, we are interested in reaching the near horizon geometry of the black hole close, but not exactly at, extremality.This is the analog of the so-called near-NHEK limit for 4D Kerr considered in [9].In order to do this, we still make the coordinate transformation (2.15), but now parametrize deviations from extremality with a parameter κ defined by (2.16) Then the metric (2.2) gives rise to in the ǫ → 0 limit.Here, we have defined This notation will be clarified in the next subsection.The location of the horizon in (2.17) is at r = 0 and the associated surface gravity is κ.We denote the corresponding Hawking temperature by When we identify κ with the parameter κ introduced in (2.14), the metric (2.17) corresponds to the near horizon geometry of the black hole (2.2) close to extremality in the following complementary sense as well.Making the coordinate transformation (2.15) with ǫ = 1 and expanding the metric components in (2.2) to leading order in r ∼ κ ≪ 1 we obtain (2.17) with κ = κ.In the rest of the paper we make this identification throughout.
The gauge field corresponding to the near horizon, near extremal geometry is obtained by accompanying the coordinate transformation (2.15) with the gauge transformation Then the gauge field (2.20) becomes in the ǫ → 0 limit.

Frolov-Thorne temperatures
We now move on to compute the Frolov-Thorne temperatures corresponding to the nearextremal Kerr-Newman black hole, by adapting the strategy of [9] to our present context.Consider a scalar field on the the black hole background (2.2), with charge q under the gauge field (2.3).Zooming into the near horizon region requires performing the coordinate transformation (2.15) combined with the gauge transformation (2.20).The charged scalar (2.22) thus becomes Now, the scalar field is in a mixed quantum state whose density matrix has eigenvalues given by the Boltzmann factor e , where T H is the Hawking temperature (2.9).Identifying and using (2.24) we find the following Frolov-Thorne temperatures: (2.28) Near extremality, M 0 is given by (2.16) and (2.26)-(2.28)become, in the ǫ → 0 limit, (2.29) Recall that both T R and T L have already appeared in our discussion: the former as the Hawking temperature (2.19) of the near-horizon, near-extremal metric (2.17) and the latter as a parameter, (2.18), in that metric.The present analysis elucidates the names given previously to those quantities.

Logarithmic correction to entropy
The logarithmic correction to the microcanonical entropy of a non-extremal, rotating charged black hole in general spacetime dimension D has been computed by Sen in [6].His result applies to the near extremal black hole considered in this paper, which has a small but non zero Hawking temperature.Equation (1.1) in [6] for the correction to the microcanonical entropy reads where is the number of Cartan generators of the spatial rotation group and n V is the number of vector fields in the theory.C local arises from one loop determinants of massless fields fluctuating in the black hole background and vanishes in odd dimensions.The remaining contribution in (3.1) comes from zero modes and Legendre transforms.Plugging D = 5, N C = 2 and n V = 1 into (3.1)we have with, for the case at hand, S BH given by (2.10).

Microscopic computation
We now change gears and compute the logarithmic correction to the entropy of the microscopic theory dual to the Kerr-Newman black hole.In [12] this solution was embedded into string theory and the microscopic dual thereby shown to be the infrared fixed point of a 1+1 field theory living on the brane intersection.This fixed point is a possibly nonlocal deformation of an ordinary 1+1 conformal field theory which preserves at least one infinite-dimensional conformal symmetry.While the string theoretic construction implies the existence of the fixed point theory, it exhibits a new kind of 1+1 D critical behavior and is only partially understood.The near horizon geometry (2.17) has a SL(2, R) R × U (1) L isometry subgroup coming from the isometries of the AdS 2 submanifold and the unbroken U (1) L ⊂ SU (2) L rotation isometry respectively.Various infinite-dimensional enhancements of this global isometry, involving different boundary conditions, have been extensively considered in the literature, and may be relevant in different circumstances or for different computations.See [20] for a recent discussion.We consider two of them which turn out to both give the same log corrections.
In this subsection we consider a CFT in which the global symmetries are enhanced as where V ir L and V ir R are left and right moving Virasoro algebras with generators L n and Ln respectively.L 0 generates ψ rotations and L0 generates AdS 2 time translations.
We put the CFT on a circle along ψ − t and consider the ensemble Standard modular invariance of this partition function is Z(τ, τ ) = Z(−1/τ, −1/τ ).The microscopic dual to the Kerr-Newman black hole we are considering in this paper has an additional SU (2) × U (1) global symmetry, corresponding to the SU (2) rotation isometry and the U (1) gauge symmetry.Turning on the associated chemical potentials, the partition function becomes and it obeys the modular transformation rule Here µ i are left chemical potentials associated with the left moving conserved charges P i and µ 2 ≡ µ i µ j k ij with k ij the matrix of Kac-Moody levels of the left moving currents.In our case i, j run from 1 to 2 but, for the sake of generality, we temporarily assume they run from 1 to n.This partition function is related to the density of states, ρ, at high temperatures by where E L , E R , p i are the eigenvalues of L 0 , L0 , P i respectively.For small τ , (3.7) implies that Then, inverting (3.8), we obtain the following expression for the density of states: where we have assumed that the vacuum is electrically neutral, p i v = 0.This integral may be evaluated by saddle point methods.The integrand reaches an extremum at where the matrix k ij is the inverse of k ij and P 2 ≡ p i p j k ij .The leading contribution to the entropy is obtained by evaluating (3.10) at the saddle (3.11).This gives Putting we have The analysis of [12,20] yields c = 6J L ext and using the values for T L , T R obtained in (2.29), we see that (3.14) matches the near-extremal Bekenstein-Hawking entropy (2.14) to linear order in κ.This extends the match of [12] from the extremal to the near-extremal regime.
The logarithmic correction ∆S to the leading entropy (3.12) is generated by Gaussian fluctuations of the density of states (3.10) about the saddle (3.11): where A is the determinant of the matrix of second derivatives of the exponent in the integrand of (3.10) with respect to τ , µ i , τ .We find We now fix n = 2 for the left moving SU (2)×U (1) current algebra corresponding to SU (2) rotations and the gauge field.The SU (2) × U (1) charges are p 1 = 0 (because J R = 0) and , and k 12 = k 21 = 0. Taking into account (2.11), we thus have the following scalings, (3.17) Bringing (3.17) to (3.15, 3.16), we obtain ∆S = −5 log a . (3.18) In this subsection we consider a warped CFT, in which the global symmetries are enhanced as Here U (1) R is a left moving Kac-Moody algebra whose zero mode R0 generates the right sector time translations in AdS 2 and V ir L is a left moving Virasoro algebra whose zero mode L0 generates the left sector U (1) L rotational isometry.The symmetry algebra of our warped CFT is where Lm and Rm are the Virasoro and Kac-Moody generators respectively.Putting the theory on a circle along ψ, the partition function at inverse temperature β and angular potential θ is given by Z(β, θ) = Tr e −β R0 +iθ L0 .On the other hand, in [18] it was shown that by redefining the charges as and putting the theory on the same circle but in the different ensemble4 the partition function obeys the usual CFT modular invariance: we may then proceed as in the previous section replacing L0 with R 0 everywhere starting from equation (3.6) onwards.We thus arrive at the same results for the leading entropy and its logarithmic correction.
It should be noted that the enhancement (3.19) is somewhat unusual in the context of warped AdS 3 [17].A third more natural enhancement SL(2, R) R × U (1) L → V ir R × U (1) L in that context is also possible [20].However, this case may not be treated as (3.19) above because the arguments of [18] do not apply to the case when the identification in the bulk (along ψ) is precisely anti-aligned with the action of L 0 (along t).It is an important outstanding problem in Kerr/CFT to generalize the arguments of [18] to accomodate this case.

Match of the macroscopic and microscopic computations
We have already exhibited the match, in the near-extremal regime, of the bulk and microscopic results for the leading term of the entropy of the five-dimensional Kerr-Newman black hole under consideration: the Cardy formula (3.14) reproduces the near-extremal Bekenstein-Hawking entropy (2.14).
We will now show that the logarithmic corrections also agree.In order to furnish a sensible comparison, one must ensure that both results are given in the same ensemble.This is not the case for the macroscopic, (3.2), and microscopic, (3.18), results given above.The former assumes the entropy to be a function of the energy Q[∂ t] conjugate to the asymptotic time which features in the full black hole solution (2.2), while the latter is instead a function of the energy Q[∂ t ] conjugate to the near horizon time which appears in (2.17).The transformation between the macroscopic and microscopic density of states requires a Jacobian factor (Appendix B),

B Change of ensemble
Under a charge redefinition, q = q ( q ′ ), the density of states, ρ( q), transforms with the appropriate Jacobian factor as ρ ′ ( q ′ ) = ∂(q 1 , q 2 , . ..) ∂(q ′ 1 , q ′ 2 , . ..) ρ( q) .(B.1) The leading piece of the entropy S = log ρ typically scales like a D−2 for large q ∼ a and is therefore independent of the change of ensemble.However, the logarithmic correction, which scales like log a, often picks up contributions from the Jacobian factor above.We have seen this explicitly in section 3.2 where the Jacobian (3.25) scales with a.
Another instance of a change of ensemble was mentioned in relation to the charge redefinitions in (3.20).In this case the Jacobian is However, k R ∝ c ∼ a 3 [20] and R 0 ∼ a 3 so in this instance the Jacobian does not scale with a and therefore the logarithmic correction to the entropy is left intact by this particular change of ensemble.
24))Now, from the change of coordinates (2.15) and the expression for the extremal angular velocity in (2.12), we see that this Jacobian scales like