Logarithmic corrections to black hole entropy from Kerr/CFT
Abhishek Pathak, Achilleas P. Porfyriadis, Andrew Strominger, Oscar Varela
aa r X i v : . [ h e p - t h ] D ec Logarithmic corrections to black hole entropyfrom Kerr/CFT
Abhishek Pathak , Achilleas P. Porfyriadis , , Andrew Strominger and Oscar Varela , , Center for the Fundamental Laws of Nature, Harvard University, Cambridge, MA, USA Department of Physics, UCSB, Santa Barbara, CA, USA Max-Planck-Institut f¨ur Gravitationsphysik (Albert-Einstein-Institut), Potsdam, Germany. Department of Physics, Utah State University, Logan, UT, USA.
Abstract
It has been shown by A. Sen that logarithmic corrections to the black holearea-entropy law are entirely determined macroscopically from the masslessparticle spectrum. They therefore serve as powerful consistency checks on anyproposed enumeration of quantum black hole microstates. Sen’s results includea macroscopic computation of the logarithmic corrections for a five-dimensionalnear extremal Kerr-Newman black hole. Here we compute these correctionsmicroscopically using a stringy embedding of the Kerr/CFT correspondenceand find perfect agreement. ontents
V ir R × V ir L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.1.2 d U (1) R × V ir L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 Match of the macroscopic and microscopic computations . . . . . . . . . . . 9 A Computation of k Q
10B Change of ensemble 11
The Bekenstein-Hawking area-entropy law universally applies to any self-consistent quan-tum theory of gravity. Efforts to understand how the former constrains the latter have ledto a wealth of insights.Five years ago Sen et al. [1–6] 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 dependonly on the massless spectrum of particles and are insensitive to the UV completion ofthe theory. The basic reason for this is that the effects of a particle of mass m can beaccounted for by integrating it out, which generates local higher derivative terms in theeffective action. These lead to corrections to the entropy which are suppressed by inversepowers of m A and cannot give log A terms.The macroscopically computed logarithms serve as a litmus test for any proposed enu-meration of quantum black hole microstates which is more refined than the test providedby the area law. Sen extensively analyzed a number of stringy examples all of whichpassed the test with flying colors [7], and further noted that the macroscopic computationdoes not match the loop gravity result. He also posed a matching of the logarithms as achallenge for Kerr/CFT [8–11].In this paper we show that the logarithms indeed match for one microscopic realiza-tion [12, 13] of Kerr/CFT obtained by embedding a certain near-extremal five-dimensionalKerr-Newman black hole into a string compactification. The microscopic dual is a two-dimensional field theory defined as the IR fixed point of the worldvolume field theory ona certain brane configuration with scaled fluxes. This IR limit is certainly nontrivial butis not a conventional 2D CFT. Its properties are incompletely understood and have beenstudied in a variety of approaches: see e.g. [14–20]. The near-extremal regime sidesteps subtleties with logarithmic corrections at extremality.
We consider a charged and rotating black hole solution of five-dimensional Einstein gravityminimally 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, S = 14 π Z d x (cid:18) √− g (cid:18) R − F (cid:19) + 14 ǫ abcde A a F bc F de (cid:19) . (2.1)Specifically, we are interested in the following Kerr-Newman black hole solution to (2.1)considered in [12], ds = − ( a + ˆ r )( a + ˆ r − M )Σ d ˆ t + Σ (cid:18) ˆ r d ˆ r f − M ˆ r + dθ (cid:19) − M F Σ ( d ˆ ψ + cos θ d ˆ φ ) d ˆ t + Σ4 ( d ˆ ψ + d ˆ φ + 2 cos θ d ˆ ψ d ˆ φ ) + a M B ( d ˆ ψ + cos θ d ˆ φ ) , (2.2) A = M sinh 2 δ (cid:18) d ˆ t − ae δ ( d ˆ ψ + cos θ d ˆ φ ) (cid:19) , (2.3) This coincides with the bosonic sector of minimal supergravity in five dimensions. B = a + ˆ r − M s c − M s (2 s + 3) , F = a (ˆ r + a )( c + s ) − aM s , Σ = ˆ r + a + M s , f = ˆ r + a , (2.4)and s ≡ sinh δ , c ≡ cosh δ . The geometry depends on three independent parameters( a, M , δ ) and the physical quantities of the black hole, i.e. , its mass, angular momentumand electric charge, are given in terms of those parameters by M = 3 M δ , J L = aM ( c + s ) , Q = M sc . (2.5)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 ˆ ψ ∼ ˆ ψ + 4 π .This black hole displays inner and outer horizons located at r ± = ( M − a ) ± p M ( M − a ) . (2.6)At the (outer) horizon, the angular velocities areΩ L ≡ Ω ˆ ψ = 4 aM c − s ) + ( c + s ) p − a /M , Ω R ≡ Ω ˆ φ = 0 , (2.7)and the electric potential isΦ = c s − s c + ( c s + s c ) p − a /M c − s + ( c + s ) p − a /M . (2.8)Finally, the Hawking temperature is given by T H = 1 π √ M p − a /M c − s + ( c + s ) p − a /M , (2.9)and the Bekenstein-Hawking entropy is S BH = π √ M q ( c + s ) M − c + s ) a + ( c + c s + s ) p M ( M − a ) . (2.10)The black hole approaches extremality in the limit M → a . In this limit, the twohorizons (2.6) coalesce at r + = a and the Hawking temperature (2.9) vanishes. The charges(2.5) become M ext = 6 a cosh 2 δ , J L ext = 4 a ( c + s ) , Q ext = 4 a sc , (2.11)and the angular velocity (2.7) and electric potential (2.8) becomeΩ L ext = 1 a ( c − s ) , Φ ext = c s − s cc − s . (2.12)At extremality, the Bekenstein-Hawking entropy (2.10) reduces to S BH ext = 8 πa ( c − s ) . (2.13)3n this paper we are interested in the near-extreme case so we introduce a small parameterˆ κ that measures the deviation from extremality and write M = 4 a + a ˆ κ . Substitutingthis into (2.10) and keeping terms up to linear order in ˆ κ , the near extremal entropy is S BH near ext = 8 πa ( c − s ) + 4 πa ( c + s ) ˆ κ + O (ˆ κ )= π J L ext ) (cid:18) π c − s c + s + ˆ κ π (cid:19) + O (ˆ κ ) . (2.14) Consider the coordinate transformation t = ǫ Ω L ext ˆ t , r = ˆ r − r ǫ r , ψ = ˆ ψ − Ω L ext ˆ t , φ = ˆ φ . (2.15)Here, r + is the location of the outer horizon given in (2.6) and Ω L ext is the extremalangular velocity (2.12). Making this coordinate transformation in the five-dimensionalgeometry (2.2), (2.3), with M fixed to its extremal value, M = 4 a , and letting ǫ → κ defined by M = 4 a + a ǫ κ . (2.16)Then the metric (2.2) gives rise to ds = M ext (cid:20) − r ( r + 2 κ ) dt + dr r ( r + 2 κ ) + dθ + sin θdφ + 27 J L ext M (cid:0) πT L ( dψ + cos θ dφ ) + ( r + κ ) dt (cid:1) (cid:21) (2.17)in the ǫ → T L ≡ π c − s c + s . (2.18)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 Hawkingtemperature by T R ≡ κ π . (2.19)When we identify κ with the parameter ˆ κ introduced in (2.14), the metric (2.17)corresponds to the near horizon geometry of the black hole (2.2) close to extremality inthe 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 ∼ ˆ κ ≪ κ = ˆ κ . In the rest of the paper we make this identification throughout.4he gauge field corresponding to the near horizon, near extremal geometry is obtainedby accompanying the coordinate transformation (2.15) with the gauge transformation A → A − d Λ , with Λ ≡ Φ ext ˆ t . (2.20)Then the gauge field (2.20) becomes A = − ae δ tanh 2 δ (cid:16) dψ + cos θdφ + e − δ ( r + κ ) dt (cid:17) (2.21)in the ǫ → We now move on to compute the Frolov-Thorne temperatures corresponding to the near-extremal Kerr-Newman black hole, by adapting the strategy of [9] to our present context.Consider a scalar field ϕ = e − iω ˆ t + im ˆ ψ ˆ R (ˆ r ) S ( θ ) T ( ˆ φ ) (2.22)on the the black hole background (2.2), with charge q under the gauge field (2.3). Zoom-ing 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 ϕ = e iq Λ e − in R t + in L ψ R ( r ) S ( θ ) T ( φ ) (2.23)with Λ = 2Φ ext ǫ Ω L ext t , m = n L , ω = 12 ǫ Ω L ext (cid:16) n R + 2 ǫ n L − q Φ ext ǫ Ω L ext (cid:17) . (2.24)Now, the scalar field is in a mixed quantum state whose density matrix has eigenvaluesgiven by the Boltzmann factor e − TH ( ω − m Ω L + q Φ) , where T H is the Hawking temperature(2.9). Identifying e − TH ( ω − m Ω L + q Φ) = e − nLTL − nRTR − qTQ (2.25)and using (2.24) we find the following Frolov-Thorne temperatures: T R = 2 ǫ Ω L ext T H = 2 aǫπ √ M ( c − s ) p − a /M c − s + ( c + s ) p − a /M , (2.26) T L = − L − Ω L ext T H = 2 aπ √ M ( c − s ) c + s + ( c − s ) p − a /M , (2.27) T Q = 1Φ − Φ ext T H = 12 π √ M c − s s c . (2.28)Near extremality, M is given by (2.16) and (2.26)–(2.28) become, in the ǫ → T R = κ π , T L = 1 π c − s c + s , T Q = 14 πa c − s s c . (2.29)Recall that both T R and T L have already appeared in our discussion: the former as theHawking temperature (2.19) of the near-horizon, near-extremal metric (2.17) and the latteras a parameter, (2.18), in that metric. The present analysis elucidates the names givenpreviously to those quantities. 5 Logarithmic correction to entropy
The logarithmic correction to the microcanonical entropy of a non-extremal, rotatingcharged 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 asmall but non zero Hawking temperature. Equation (1.1) in [6] for the correction to themicrocanonical entropy reads S mc (cid:16) M, ~J , ~Q (cid:17) = S BH (cid:16) M, ~J , ~Q (cid:17) +log a (cid:18) C local − D − − D − N C − D − n V (cid:19) (3.1)where N C = (cid:2) D − (cid:3) 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 ofmassless 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 S mc = S BH − a , (3.2)with, for the case at hand, S BH given by (2.10). We now change gears and compute the logarithmic correction to the entropy of the mi-croscopic theory dual to the Kerr-Newman black hole. In [12] this solution was embeddedinto string theory and the microscopic dual thereby shown to be the infrared fixed pointof a 1+1 field theory living on the brane intersection. This fixed point is a possibly non-local deformation of an ordinary 1+1 conformal field theory which preserves at least oneinfinite-dimensional conformal symmetry. While the string theoretic construction impliesthe existence of the fixed point theory, it exhibits a new kind of 1+1 D critical behavior andis 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 submanifold and the unbroken U (1) L ⊂ SU (2) L rotation isometry respectively. Various infinite-dimensional enhance-ments of this global isometry, involving different boundary conditions, have been exten-sively considered in the literature, and may be relevant in different circumstances or fordifferent computations. See [20] for a recent discussion. We consider two of them whichturn out to both give the same log corrections. V ir R × V ir L In this subsection we consider a CFT in which the global symmetries are enhanced as SL (2 , R ) R × U (1) L → V ir R × V ir L , (3.3)where V ir L and V ir R are left and right moving Virasoro algebras with generators L n and¯ L n respectively. L generates ψ rotations and ¯ L generates AdS time translations. Had they been different, the matching of logarithmic corrections would have singled one out.
6e put the CFT on a circle along ψ − t and consider the ensemble Z ( τ, ¯ τ ) = Tr e πiτL − πi ¯ τ ¯ L . (3.4)We assume that 4 πτ = β L − β R + i ( β L + β R ) (3.5)and 4 π ¯ τ = β L − β R − i ( β L + β R ). Standard modular invariance of this partition functionis Z ( τ, ¯ τ ) = Z ( − /τ, − / ¯ τ ). The microscopic dual to the Kerr-Newman black hole we areconsidering in this paper has an additional SU (2) × U (1) global symmetry, correspondingto the SU (2) rotation isometry and the U (1) gauge symmetry. Turning on the associatedchemical potentials, the partition function becomes Z ( τ, ¯ τ , ~µ ) = Tr e πiτL − πi ¯ τ ¯ L +2 πiµ i P i (3.6)and it obeys the modular transformation rule Z ( τ, ¯ τ , ~µ ) = e − πiµ τ Z (cid:18) − τ , − τ , ~µτ (cid:19) . (3.7)Here µ i are left chemical potentials associated with the left moving conserved charges P i and µ ≡ µ i µ j k ij with k ij the matrix of Kac-Moody levels of the left moving currents. Inour case i, j run from 1 to 2 but, for the sake of generality, we temporarily assume theyrun from 1 to n . This partition function is related to the density of states, ρ , at hightemperatures by Z ( τ, ¯ τ , ~µ ) = Z dE L dE R d n p ρ ( E L , E R , ~p ) e πiτE L − πi ¯ τE R +2 πiµ i p i , (3.8)where E L , E R , p i are the eigenvalues of L , ¯ L , P i respectively. For small τ , (3.7) impliesthat Z ( τ, ¯ τ , ~µ ) ≈ e − πiµ τ e − πiEvLτ + πiEvR ¯ τ + πiµipivτ . (3.9)Then, inverting (3.8), we obtain the following expression for the density of states: ρ ( E L , E R , ~p ) ≃ Z dτ d ¯ τ d n µ e πi (cid:18) − µ τ − EvLτ + EvR ¯ τ − E L τ + E R ¯ τ − µ i p i (cid:19) , (3.10)where we have assumed that the vacuum is electrically neutral, p iv = 0. This integral maybe evaluated by saddle point methods. The integrand reaches an extremum at τ = s E vL E L − P , ¯ τ = − s E vR E R , µ i = − k ij p j s E vL E L − P , (3.11)where the matrix k ij is the inverse of k ij and P ≡ p i p j k ij . The leading contribution tothe entropy is obtained by evaluating (3.10) at the saddle (3.11). This gives S = log ρ = 2 π q − E vL (4 E L − P ) + 2 π q − E vR (4 E R ) . (3.12)7utting E vL = E vR = − c , E L − P π c T L , E R = π c T R , (3.13)we have S = π c T L + π c T R . (3.14)The analysis of [12,20] yields c = 6 J 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 linearorder 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 Gaussianfluctuations of the density of states (3.10) about the saddle (3.11):∆ S = −
12 log det A (2 π ) n +2 , (3.15)where A is the determinant of the matrix of second derivatives of the exponent in theintegrand of (3.10) with respect to τ , µ i , ¯ τ . We finddet A = (2 π ) n +2
16 ( − E vL ) − n +12 (4 E L − P ) n +32 ( − E vR ) − (4 E R ) det k ij . (3.16)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 = 0 (because J R = 0) and p ∝ Q ext . The U (1) Kac-Moody level k ≡ k Q is given in Appendix A, the SU (2) levelis k ≡ k J ∝ c [32], and k = k = 0. Taking into account (2.11), we thus have thefollowing scalings, E vL , E vR , E L − P / , E R ∼ a , k Q ∼ a , k J ∼ a . (3.17)Bringing (3.17) to (3.15, 3.16), we obtain∆ S = − a . (3.18) d U (1) R × V ir L In this subsection we consider a warped CFT, in which the global symmetries are enhancedas SL (2 , R ) R × U (1) L → [ U (1) R × V ir L . (3.19)Here [ U (1) R is a left moving Kac-Moody algebra whose zero mode ˜ R generates the rightsector time translations in AdS and V ir L is a left moving Virasoro algebra whose zeromode ˜ L generates the left sector U (1) L rotational isometry. The symmetry algebra of ourwarped CFT is h ˜ L m , ˜ L n i = ( m − n ) ˜ L m + n + c
12 ( m − m ) δ m + n , h ˜ R m , ˜ R n i = k R mδ m + n , h ˜ L m , ˜ R n i = − n ˜ R m + n , where ˜ L m and ˜ R m are the Virasoro and Kac-Moody generators respectively. Putting thetheory on a circle along ψ , the partition function at inverse temperature β and angular8otential θ is given by Z ( β, θ ) = Tr e − β ˜ R + iθ ˜ L . On the other hand, in [18] it was shownthat by redefining the charges as L n = ˜ L n − k R ˜ R ˜ R n + 1 k R ˜ R δ n , R n = 2 k R ˜ R ˜ R n − k R ˜ R δ n , (3.20)and putting the theory on the same circle but in the different ensemble Z ( τ, ¯ τ ) = Tr e πiτL − πi ¯ τR , (3.21)the partition function obeys the usual CFT modular invariance: Z ( τ, ¯ τ ) = Z ( − /τ, − / ¯ τ ) . (3.22)Assuming 4 πτ = β L − β R + i ( β L + β R ) (3.23)and 4 π ¯ τ = β L − β R − i ( β L + β R ) we may then proceed as in the previous section replacing¯ L with R everywhere starting from equation (3.6) onwards. We thus arrive at the sameresults for the leading entropy and its logarithmic correction.It should be noted that the enhancement (3.19) is somewhat unusual in the context ofwarped AdS [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) abovebecause 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 (along t ). It is an importantoutstanding problem in Kerr/CFT to generalize the arguments of [18] to accomodate thiscase. We have already exhibited the match, in the near-extremal regime, of the bulk and mi-croscopic results for the leading term of the entropy of the five-dimensional Kerr-Newmanblack hole under consideration: the Cardy formula (3.14) reproduces the near-extremalBekenstein-Hawking entropy (2.14).We will now show that the logarithmic corrections also agree. In order to furnish asensible 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 givenabove. The former assumes the entropy to be a function of the energy Q [ ∂ ˆ t ] conjugate tothe asymptotic time which features in the full black hole solution (2.2), while the latter isinstead a function of the energy Q [ ∂ t ] conjugate to the near horizon time which appearsin (2.17). The transformation between the macroscopic and microscopic density of statesrequires a Jacobian factor (Appendix B), ρ bulk = δQ [ ∂ t ] δQ [ ∂ ˆ t ] ρ . (3.24) A change of ensemble may result in different logarithmic corrections to the entropy. However, asexplained in Appendix B, the change of ensemble corresponding to the charge redefinitions (3.20) here doesnot imply any change in the logarithmic correction to the entropy. δQ [ ∂ t ] δQ [ ∂ ˆ t ] ∼ a . (3.25)Thus ∆ S bulk = ∆ S + log a , (3.26)which indeed is satisfied by (3.2) and (3.18). Acknowledgements
We thank Monica Guica, Thomas Hartman, Maria J. Rodriguez and Ashoke Sen for usefulconversations. AP and AS are supported in part by DOE grant DE-FG02-91ER40654.APP is supported by NSF grant PHY-1504541. OV is supported by the Marie Curiefellowship PIOF-GA-2012-328798.
A Computation of k Q In this appendix we compute the level k Q of the U (1) Kac-Moody algebra associated withthe gauge field A µ . We do not perform a full asymptotic symmetry group analysis here.We expect that with appropriate boundary conditions on the gauge field this Kac-Moodyis consistent with the rest of the asymptotic symmetries used in Section 3. Here we areparticularly interested in deriving the scaling of the level k Q with a .Thus we assume the U (1) current algebra is generated byΛ η = η (˜ y ) , (A.1)where ˜ y = πT L ψ . In modes, the generators p n = − πT L e − in ˜ y/ (2 πT L ) , (A.2)satisfy the algebra [ p m , p n ] = 0 . (A.3)Using the formulas in [33], one can compute the central extension in the correspondingDirac bracket algebra. We find: { Q p m , Q p n } = − im π T L ae δ tanh 2 δ δ m + n . (A.4)The central extension comes entirely from the Chern-Simons term in the action (2.1).Passing to the commutators { , } → − i [ , ] we obtain the current algebra,[ P m , P n ] = k Q mδ m + n , (A.5)with level given by k Q = 12 (2 πT L ) ae δ tanh 2 δ . (A.6)10 Change of ensemble
Under a charge redefinition, ~q = ~q ( ~q ′ ), the density of states, ρ ( ~q ), transforms with theappropriate Jacobian factor as ρ ′ ( ~q ′ ) = ∂ ( q , q , . . . ) ∂ ( q ′ , q ′ , . . . ) ρ ( ~q ) . (B.1)The leading piece of the entropy S = log ρ typically scales like a D − for large q ∼ a andis 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. Wehave 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 chargeredefinitions in (3.20). In this case the Jacobian is ∂ ( L , R ) ∂ ( ˜ L , ˜ R ) = 2 ˜ R k R = 2 r R k R . (B.2)However, k R ∝ c ∼ a [20] and R ∼ a so in this instance the Jacobian does not scale with a and therefore the logarithmic correction to the entropy is left intact by this particularchange of ensemble. References [1] S. Banerjee, R. K. Gupta and A. Sen, “Logarithmic Corrections to ExtremalBlack Hole Entropy from Quantum Entropy Function,” JHEP , 147 (2011)[arXiv:1005.3044 [hep-th]].[2] S. Banerjee, R. K. Gupta, I. Mandal and A. Sen, “Logarithmic Corrections to N=4and N=8 Black Hole Entropy: A One Loop Test of Quantum Gravity,” JHEP ,143 (2011) [arXiv:1106.0080 [hep-th]].[3] A. Sen, “Logarithmic Corrections to N=2 Black Hole Entropy: An Infrared Windowinto the Microstates,” Gen. Rel. Grav. , no. 5, 1207 (2012) [arXiv:1108.3842 [hep-th]].[4] A. Sen, “Logarithmic Corrections to Rotating Extremal Black Hole Entropy in Fourand Five Dimensions,” Gen. Rel. Grav. (2012) 1947 [arXiv:1109.3706 [hep-th]].[5] S. Bhattacharyya, B. Panda and A. Sen, “Heat Kernel Expansion and Extremal Kerr-Newmann Black Hole Entropy in Einstein-Maxwell Theory,” JHEP , 084 (2012)[arXiv:1204.4061 [hep-th]].[6] A. Sen, “Logarithmic Corrections to Schwarzschild and Other Non-extremal BlackHole Entropy in Different Dimensions,” JHEP (2013) 156 [arXiv:1205.0971 [hep-th]]. 117] A. Sen, “Microscopic and Macroscopic Entropy of Extremal Black Holes in StringTheory,” Gen. Rel. Grav. , 1711 (2014) [arXiv:1402.0109 [hep-th]].[8] M. Guica, T. Hartman, W. Song and A. Strominger, “The Kerr/CFT Correspon-dence,” Phys. Rev. D (2009) 124008 [arXiv:0809.4266 [hep-th]].[9] I. Bredberg, T. Hartman, W. Song and A. Strominger, “Black Hole SuperradianceFrom Kerr/CFT,” JHEP , 019 (2010) [arXiv:0907.3477 [hep-th]].[10] I. Bredberg, C. Keeler, V. Lysov and A. Strominger, “Cargese Lectures onthe Kerr/CFT Correspondence,” Nucl. Phys. Proc. Suppl. (2011) 194[arXiv:1103.2355 [hep-th]].[11] G. Compere, “The Kerr/CFT correspondence and its extensions: a comprehensivereview,” Living Rev. Rel. , 11 (2012) [arXiv:1203.3561 [hep-th]].[12] M. Guica and A. Strominger, “Microscopic Realization of the Kerr/CFT Correspon-dence,” JHEP (2011) 010 [arXiv:1009.5039 [hep-th]].[13] W. Song and A. Strominger, “Warped AdS3/Dipole-CFT Duality,” JHEP , 120(2012) [arXiv:1109.0544 [hep-th]].[14] D. M. Hofman and A. Strominger, “Chiral Scale and Conformal Invariance in 2DQuantum Field Theory,” Phys. Rev. Lett. , 161601 (2011) [arXiv:1107.2917 [hep-th]].[15] S. El-Showk and M. Guica, “Kerr/CFT, dipole theories and nonrelativistic CFTs,”JHEP , 009 (2012) [arXiv:1108.6091 [hep-th]].[16] I. Bena, M. Guica and W. Song, “Un-twisting the NHEK with spectral flows,” JHEP , 028 (2013) [arXiv:1203.4227 [hep-th]].[17] T. Azeyanagi, D. M. Hofman, W. Song and A. Strominger, “The Spectrum of Stringson Warped AdS × S ,” JHEP , 078 (2013) [arXiv:1207.5050 [hep-th]].[18] S. Detournay, T. Hartman and D. M. Hofman, “Warped Conformal Field Theory,”Phys. Rev. D , 124018 (2012) [arXiv:1210.0539 [hep-th]].[19] S. Detournay and M. Guica, “Stringy Schroedinger truncations,” JHEP , 121(2013) [arXiv:1212.6792 [hep-th]].[20] G. Compere, M. Guica and M. J. Rodriguez, “Two Virasoro symmetries in stringywarped AdS ,” JHEP , 012 (2014) [arXiv:1407.7871 [hep-th]].[21] S. N. Solodukhin, “The Conical singularity and quantum corrections to entropy ofblack hole,” Phys. Rev. D , 609 (1995) [hep-th/9407001].[22] S. N. Solodukhin, “On ’Nongeometric’ contribution to the entropy of black hole dueto quantum corrections,” Phys. Rev. D , 618 (1995) [hep-th/9408068].1223] D. V. Fursaev, “Temperature and entropy of a quantum black hole and conformalanomaly,” Phys. Rev. D , 5352 (1995) [hep-th/9412161].[24] R. B. Mann and S. N. Solodukhin, “Universality of quantum entropy for extremeblack holes,” Nucl. Phys. B , 293 (1998) [hep-th/9709064].[25] S. Carlip, “Logarithmic corrections to black hole entropy from the Cardy formula,”Class. Quant. Grav. , 4175 (2000) [gr-qc/0005017].[26] S. Ferrara and A. Marrani, “Generalized Mirror Symmetry and Quantum Black HoleEntropy,” Phys. Lett. B , 173 (2012) [arXiv:1109.0444 [hep-th]].[27] C. Keeler, F. Larsen and P. Lisbao, “Logarithmic Corrections to N ≥ (2014) no.4, 043011 [arXiv:1404.1379 [hep-th]].[28] A. Chowdhury, R. K. Gupta, S. Lal, M. Shyani and S. Thakur, “Logarithmic Correc-tions to Twisted Indices from the Quantum Entropy Function,” JHEP (2014)002 [arXiv:1404.6363 [hep-th]].[29] F. Larsen and P. Lisbao, “Quantum Corrections to Supergravity on AdS × S ,” Phys.Rev. D , no. 8, 084056 (2015) [arXiv:1411.7423 [hep-th]].[30] A. M. Charles and F. Larsen, “Universal corrections to non-extremal black hole en-tropy in N ≥ (2015) 200 [arXiv:1505.01156 [hep-th]].[31] A. Belin, A. Castro, J. Gomes and C. A. Keller, “Siegel Modular Forms and BlackHole Entropy,” arXiv:1611.04588 [hep-th].[32] J. M. Maldacena and A. Strominger, “Universal low-energy dynamics for rotatingblack holes,” Phys. Rev. D , 4975 (1997) [hep-th/9702015].[33] G. Compere, K. Murata and T. Nishioka, “Central Charges in Extreme BlackHole/CFT Correspondence,” JHEP0905