Logarithmic corrections to black hole entropy from Kerr/CFT

Journal of High Energy Physics, Apr 2017

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.

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Logarithmic corrections to black hole entropy from Kerr/CFT

Received: March Logarithmic corrections to black hole entropy from Kerr/CFT Oscar Varela 0 1 2 3 5 Open Access 0 1 3 c The Authors. 0 1 3 0 Am Muhlenberg 1 , D-14476 Potsdam , Germany 1 Santa Barbara , CA 93106 , U.S.A 2 Department of Physics, Utah State University 3 Cambridge , MA 02138 , U.S.A 4 Department of Physics , UCSB 5 Center for the Fundamental Laws of Nature, Harvard University 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 ve-dimensional near extremal Kerr-Newman black hole. Here we compute these corrections microscopically using a stringy embedding of the Kerr/CFT correspondence and nd perfect agreement. cMax-Planck-Institut fur Gravitationsphysik (Albert-Einstein-Institut) 1 Introduction 2 3 The A Computation of kQ B Change of ensemble Introduction ve-dimensional Kerr-Newman black hole Near horizon, near extremal limit Frolov-Thorne temperatures Logarithmic correction to entropy Microscopic computation Match of the macroscopic and microscopic computations The Bekenstein-Hawking area-entropy law universally applies to any self-consistent quantum theory of gravity. E orts to understand how the former constrains the latter have led to 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 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 e ects of a particle of mass m can be accounted for by integrating it out, which generates local higher derivative terms in the e ective action. These lead to corrections to the entropy which are suppressed by inverse powers of m2A 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 re ned than the test provided by the area law. Sen extensively analyzed a number of stringy examples all of which passed the test with ying 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{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 ve-dimensional Kerr-Newman black hole into a string compacti cation.1 The microscopic dual is a twodimensional eld theory de ned as the IR xed point of the worldvolume eld theory on a certain brane con guration with scaled uxes. This IR limit is certainly nontrivial but 1The near-extremal regime sidesteps subtleties with logarithmic corrections at extremality. is not a conventional 2D CFT. Its properties are incompletely understood and have been studied in a variety of approaches: see e.g. [14{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 avor: all logs vanish in a certain thermodynamic ensemble, with logs generated on both sides of the match by Legendre transforms to a di erent ensemble. However, later examples become quite intricate and provide compelling tests of a variety of stringy constructions. In the vedimensional 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 a necessary non-vanishing level of the current dual to the electric eld is provided by a Chern-Simons term which is crucially present in the e ective action obtained from string While it is reassuring that the simplest case works, perhaps more challenging matches such as the extremal 4D case may eventually provide more re ned 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 di erent enhancements of the global symmetries. We match the Bekenstein-Hawking entropy of the near extremal black hole to rst 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. Previous work on logarithmic corrections to black hole entropy includes [21{31]. ve-dimensional Kerr-Newman black hole We consider a charged and rotating black hole solution of ve-dimensional Einstein gravity minimally coupled to a gauge eld. The dynamics of the latter is speci ed by the YangMills-Chern-Simons Lagrangian, so that the complete action is,2 S5 = Speci cally, we are interested in the following Kerr-Newman black hole solution to (2.1) considered in [12], (a2 + r^2)(a2 + r^2 ds52 = A = M0 sinh 2 (d ^2 + d ^2 + 2 cos d ^ d ^) + 2This coincides with the bosonic sector of minimal supergravity in ve dimensions. where we have de ned the quantities B = a2 + r^2 M0s4(2s2 + 3) ; F = a(r^2 + a2)(c3 + s3) = r^2 + a2 + M0s2 ; f = r^2 + a2 ; cosh . The geometry depends on three independent parameters (a; M0; ) 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 M = JL = aM0 (c3 + s3) ; Q = M0sc : In ve dimensions, it is possible to have a second angular momentum, JR, but we set This black hole displays inner and outer horizons located at r2 = 12 (M0 At the (outer) horizon, the angular velocities are ^ = s3) + (c3 + s3)p1 4a2=M0 ^ = 0 ; SBH ext = 8 a3(c3 and the electric potential is Finally, the Hawking temperature is given by s2c + (c2s + s2c)p1 s3 + (c3 + s3)p1 4a2=M0 4a2=M0 TH = 4a2=M0 s3 + (c3 + s3)p1 4a2=M0 and the Bekenstein-Hawking entropy is SBH = 2 M0 (c6 + s6)M0 The black hole approaches extremality in the limit M0 ! 4a2. In this limit, the charges (2.5) become Mext = 6a2 cosh 2 ; JL ext = 4a3 (c3 + s3) ; Qext = 4a2sc ; and the angular velocity (2.7) and electric potential (2.8) become 2(c3 + s3)2a2 + (c4 + c2s2 + s4)pM0(M0 4a2) : (2.10) L ext = ext = 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 this into (2.10) and keeping terms up to linear order in ^, the near extremal entropy is SBH near ext = 8 a3(c3 s3) + 4 a3(c3 + s3) ^ + O(^2) (6JL ext) Near horizon, near extremal limit Consider the coordinate transformation t = 12 r = = ^ = ^ : Here, r+ is the location of the outer horizon given in (2.6) and L ext is the extremal angular velocity (2.12). Making this coordinate transformation in the 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 de ned by Then the metric (2.2) gives rise to M0 = 4a2 + a2 2 2 : ds52 = r(r + 2 ) r(r + 2 )dt2 + ! 0 limit. Here, we have de ned This notation will be clari ed in the next subsection. The location of the horizon in (2.17) temperature by When we identify with the parameter ^ 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) obtain (2.17) with TR The gauge eld corresponding to the near horizon, near extremal geometry is obtained by accompanying the coordinate transformation (2.15) with the gauge transformation Then the gauge eld (2.20) becomes ! 0 limit. Frolov-Thorne temperatures A = ae tanh 2 + cos d + e 2 (r + )dt TR = TL = TQ = 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 eld ' = e i!t^+im ^ R^(r^) S( ) T ( ^) on the the black hole background (2.2), with charge q under the gauge eld (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 ' = eiq e inRt+inL R(r) S( ) T ( ) m = nL ; ! = Now, the scalar eld is in a mixed quantum state whose density matrix has eigenvalues given by the Boltzmann factor e T1H (! m L+q ), where TH is the Hawking temperature (2.9). = e TL TR TQ and using (2.24) we nd the following Frolov-Thorne temperatures: TH = TR = TH = TH = s3 + (c3 + s3)p1 4a2=M0 4a2=M0 4a2=M0 Near extremality, M0 is given by (2.16) and (2.26){(2.28) become, in the ! 0 limit, Recall that both TR and TL 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. TL = TQ = 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 Smc M; J~; Q~ = SBH + log a Clocal where NC = D 1 is the number of Cartan generators of the spatial rotation group and nV is the number of vector elds in the theory. Clocal arises from one loop determinants of massless elds uctuating in the black hole background and vanishes in odd dimensions. The remaining contribution in (3.1) comes from zero modes and Legendre transforms. Smc = SBH with, for the case at hand, SBH 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 of a 1+1 eld theory living on the brane intersection. This xed point is a possibly nonlocal deformation of an ordinary 1+1 conformal eld theory which preserves at least one in nite-dimensional conformal symmetry. While the string theoretic construction implies the existence of the xed 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 isometry subgroup coming from the isometries of the AdS2 submanifold and the unbroken SU(2)L rotation isometry respectively. Various in nite-dimensional enhancements of this global isometry, involving di erent boundary conditions, have been extensively considered in the literature, and may be relevant in di erent circumstances or for di erent computations. See [20] for a recent discussion. We consider two of them which turn out to both give the same log corrections.3 In this subsection we consider a CFT in which the global symmetries are enhanced as SL(2; R)R U(1)L ! V irR where V irL and V irR are left and right moving Virasoro algebras with generators Ln and Ln respectively. L0 generates rotations and L0 generates AdS2 time translations. 3Had they been di erent, the matching of logarithmic corrections would have singled one out. (EL; ER; p~) ' be evaluated by saddle point methods. The integrand reaches an extremum at 0 = 0 = 0i = where the matrix kij is the inverse of kij and P pipj kij . The leading contribution to the entropy is obtained by evaluating (3.10) at the saddle (3.11). This gives S = log 0 = 2 R). Standard modular invariance of this partition function 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 Z( ; ; ~ ) = e Here i are left chemical potentials associated with the left moving conserved charges P i i j kij with kij 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, , at high temperatures by Z( ; ; ~ ) = dEL dER dnp (EL; ER; p~) e2 i EL 2 i ER+2 i ipi ; where EL; ER; pi are the eigenvalues of L0; L0; P i respectively. For small , (3.7) imThen, inverting (3.8), we obtain the following expression for the density of states: We put the CFT on a circle along t and consider the ensemble We assume that Z( ; ) = Tr e2 i L0 2 i L0 : we see that (3.14) matches the near-extremal Bekenstein-Hawking entropy (2.14) to linear order in . 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 uctuations of the density of states (3.10) about the saddle (3.11): S = where A is the determinant of the matrix of second derivatives of the exponent in the integrand of (3.10) with respect to , i det A = 1 3 P2) n+23 ( ERv) 2 (4ER) 2 det kij : U(1) current algebra corresponding to SU(2) kQ is given in appendix A, the SU(2) level ELv = ERv = ER = S = rotations and the gauge eld. The SU(2) / Qext. The U(1) Kac-Moody level k22 following scalings, Bringing (3.17) to (3.15), (3.16), we obtain P2=4 ; ER S = In this subsection we consider a warped CFT, in which the global symmetries are enhanSL(2; R)R U(1)L ! U[(1)R Here U[(1)R is a left moving Kac-Moody algebra whose zero mode R~0 generates the right sector time translations in AdS2 and V irL is a left moving Virasoro algebra whose zero mode L~0 generates the left sector U(1)L rotational isometry. The symmetry algebra of our warped CFT is hL~m; L~ni = (m hR~m; R~ni = n)L~m+n + hL~m; R~ni = where L~m and R~m are the Virasoro and Kac-Moody generators respectively. Putting the theory on a circle along , the partition function at inverse temperature and angular is given by Z( ; ) = Tr e that by rede ning the charges as R~0+i L~0 . On the other hand, in [18] it was shown Ln = L~n 2 R~0R~n + Rn = 2 R~0R~n and putting the theory on the same circle but in the di erent ensemble4 the partition function obeys the usual CFT modular invariance: Z( ; ) = Tr e2 i L0 2 i R0 ; Z( ; ) = Z( 1= ; 1= ) : i( L + R) we may then proceed as in the previous section replacing L0 with R0 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 AdS3 [17]. A third more natural enhancement SL(2; R)R 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 identi cation in the bulk (along ) is precisely anti-aligned with the action of L0 (along t). It is an important outstanding problem in Kerr/CFT to generalize the arguments of [18] to accomodate U(1)L ! V irR 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 ve-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 4A change of ensemble may result in di erent logarithmic corrections to the entropy. explained in appendix B, the change of ensemble corresponding to the charge rede nitions (3.20) here does not imply any change in the logarithmic correction to the entropy. in (2.17). The transformation between the macroscopic and microscopic density of states requires a Jacobian factor (appendix B), 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 bulk = Sbulk = S + log a ; fQpm ; Qpn g = im 24 2TL2ae tanh 2 which indeed is satis ed by (3.2) and (3.18). Acknowledgments We thank Monica Guica, Thomas Hartman, Maria J. Rodriguez and Ashoke Sen for useful conversations. 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 Curie fellowship PIOF-GA-2012-328798. Computation of kQ In this appendix we compute the level kQ of the U(1) Kac-Moody algebra associated with the gauge eld A . We do not perform a full asymptotic symmetry group analysis here. We expect that with appropriate boundary conditions on the gauge eld this Kac-Moody is consistent with the rest of the asymptotic symmetries used in section 3. Here we are particularly interested in deriving the scaling of the level kQ with a. Thus we assume the U(1) current algebra is generated by TL . In modes, the generators satisfy the algebra = (y~) ; pn = 2 TL e iny~=(2 TL) ; [pm; pn] = 0 : Using the formulas in [33], one can compute the central extension in the corresponding Dirac bracket algebra. We nd: The central extension comes entirely from the Chern-Simons term in the action (2.1). Passing to the commutators f ; g ! i[ ; ] we obtain the current algebra, [Pm; Pn] = kQ = 12 (2 TL)2 ae tanh 2 : @(q1; q2; : : :) @(q10; q20; : : :) with level given by Change of ensemble appropriate Jacobian factor as However, kR / c change of ensemble. References [INSPIRE]. is therefore independent of the change of ensemble. 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Logarithmic corrections to black hole entropy from Kerr/CFT, Journal of High Energy Physics, 2017, DOI: 10.1007/JHEP04(2017)090