Logarithmic Corrections to Rotating Extremal Black Hole Entropy in Four and Five Dimensions
aa r X i v : . [ h e p - t h ] M a r Logarithmic Corrections to Rotating Extremal BlackHole Entropy in Four and Five Dimensions
Ashoke Sen
Harish-Chandra Research InstituteChhatnag Road, Jhusi, Allahabad 211019, India
E-mail: [email protected]
Abstract
We compute logarithmic corrections to the entropy of rotating extremal black holes usingquantum entropy function i.e. Euclidean quantum gravity approach. Our analysis includes fivedimensional supersymmetric BMPV black holes in type IIB string theory on T and K × S aswell as in the five dimensional CHL models, and also non-supersymmetric extremal Kerr blackhole and slowly rotating extremal Kerr-Newmann black holes in four dimensions. For BMPVblack holes our results are in perfect agreement with the microscopic results derived from stringtheory. In particular we reproduce correctly the dependence of the logarithmic corrections onthe number of U(1) gauge fields in the theory, and on the angular momentum carried by theblack hole in different scaling limits. We also explain the shortcomings of the Cardy limitin explaining the logarithmic corrections in the limit in which the (super)gravity descriptionof these black holes becomes a valid approximation. For non-supersymmetric extremal blackholes, e.g. for the extremal Kerr black hole in four dimensions, our result provides a stringenttesting ground for any microscopic explanation of the black hole entropy, e.g. Kerr/CFTcorrespondence. 1 ontents K ( y ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.5 Computation of β r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.6 Computation of R d d y G ( y ) ¯ K r ( y ) . . . . . . . . . . . . . . . . . . . . . . . . . 19 K × S / ZZ N . . . . . . . . . . . . . . 285.2 Evaluation of the index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.3 Removal of the additional hair contribution . . . . . . . . . . . . . . . . . . . 345.4 BMPV black hole in type IIB on T . . . . . . . . . . . . . . . . . . . . . . . . 355.5 Comparison with the Cardy limit . . . . . . . . . . . . . . . . . . . . . . . . . 37 A Counting of zero modes in
AdS Supersymmetric extremal black holes enjoy a certain set of non-renormalization properties,and this makes them a very useful testing ground for comparing the macroscopic predictions2or the entropy against the microscopic entropy [1, 2]. In particular for a class of N = 4supersymmetric string theories in four dimensions one can use Wald’s formula [3–6] adapted toBPS black holes [7–9] for computing higher derivative corrections to the black hole entropy, andthe results are in remarkable agreement with the microscopic results [10–20]. Nevertheless, asthere have been other attempts to explain extremal black hole entropy without making directuse of string theory [21, 22], it is useful to explore to what extent string theory can do betterthan these general methods, particularly in situations where quantum gravity corrections toblack hole entropy become important.A generalization of Wald’s formula, based on path integral over various fields in the nearhorizon geometry of the black hole, can be used to compute quantum corrections to the extremalblack hole entropy [23, 24]. Recently this method has been used to compute logarithmiccorrections to the entropy in a class of four dimensional
N ≥ e.g. in a different approach to quantizing gravity, the logarithmic corrections exist in anygeneric theory of gravity. Furthermore they are determined purely from the low energy data– spectrum of massless fields and their interactions – and are insensitive to the spectrum ofmassive fields and higher derivative corrections. At the same time, they are not universal sincethey are sensitive to what kind of massless fields the theory has, and also their interactions.Thus once we compute these logarithmic corrections from the low energy data, they will providea testing ground for any proposed ultraviolet completion of gravity and the description of blackhole microstates in this theory. A microscopic theory that does not reproduce the correctlogarithmic corrections must not be the correct microscopic theory of gravity.So far the computations have been done for spherically symmetric black hole solutionsand whenever microscopic results are available in string theory, e.g. in N = 4 and N = 8string theories in four dimensions [10–20, 29], the microscopic and the macroscopic results forthe logarithmic corrections are in perfect agreement. For N = 2 , As has already been emphasized in the past, extremal black holes refer to extremal limit of non-extremalblack holes. On the macroscopic side this is apparent from the form of the Euclidean
AdS metric (2.1),which is an analytic continuation of the Lorentzian near horizon metric − ( r − dt + dr / ( r −
1) ( r = cosh η , t = − iθ ). This near horizon geometry arises in the zero temperature limit of black holes and still has a bifurcateKilling horizon at r = 1. The gravity / string theory partition function in this geometry is then comparedwith the microscopic partition function T r ( e − H/T ) in the zero temperature limit, which, for a gapped system,approaches d e − E /T where E is the ground state energy and d is the ground state degeneracy. Q , Q , n ∼ Λ, J ∼ Λ / − ( n V −
3) ln Λ Q , Q , n ∼ Λ, J = 0 − ( n V + 3) ln ΛTable 1: Logarithmic corrections to the entropy of a BMPV black hole in type IIB stringtheory compactified on K × S / ZZ N for N = 1 , , , , n V ≡ N +1 + 3 denotes the totalnumber of U (1) gauge fields in the five dimensional theory. The entropy given in this tablecounts the number of states with fixed values of the total angular momentum as well as thethird component of the angular momentum and hence does not include the degeneracy factorof ( J + 1) from the multiplicity of SU (2) L spin J/ states. Each of the results in this tablehas been calculated independently in the macroscopic and microscopic description and in eachcase we find perfect agreement between the two sides.the macroscopic side [27, 28], but no microscopic results are available yet. Some results fornon-BPS black holes are also available [27] from the macroscopic side and yet others have beenproposed [28] but there are no known microscopic results to compare them with.In this paper we extend the computation to extremal rotating black holes. Before we de-scribe our results it is necessary to say a few words about the ensemble in which we computethe entropy. It follows from the general analysis of [23] that the extremal black hole computesthe entropy in the microcanonical ensemble, carrying fixed charges associated with all thegauge fields on
AdS . In this case the massless gauge fields on AdS include all the Maxwellfields of the original theory as well as the gauge fields arising out of the metric due to rota-tional isometries of the black hole solution. For a rotating black hole carrying SU(2) angularmomentum J = j , only the isometry associated with rotation about the third axis gives riseto massless gauge fields in AdS and hence the entropy computed by quantum entropy func-tion counts states with fixed J = j without any restriction on the total angular momentum.We shall denote by e d ( j ) the degeneracy of states with this restriction. We can also introduceanother ensemble in which we fix the total angular momentum ~J to j ( j + 1) besides fixing J to j . The corresponding degeneracy will be denoted by d ( j ). The latter is the relevant numberfor j = 0 since a black hole with j = 0 will have full SU (2) isometry and as a consequencequantum entropy function will count states for which all components of the angular momen-tum are fixed to 0. We note however that d ( j ) and e d ( j ) are related by the simple relation d ( j ) = e d ( j + 1) − e d ( j ); thus knowledge of one determines the other. We shall in fact see that4or generic j , e d ( j + 1) is related to e d ( j ) by a multiplicative factor of order unity (it followsfrom the fact that entropy scales in the same way as j ), and hence ln d ( j ) and ln e d ( j ) differby an additive term of order unity and have identical logarithmic corrections. The situationchanges when j takes value that is parametrically smaller than the generic value; this case willbe discussed separately later.Another remark that is relevant for the supersymmetric black holes is the relation betweenindex and entropy. Black holes compute degeneracies whereas the quantity that is robustagainst changes of parameters and hence can be compared between the macroscopic and themicroscopic results is an appropriate supersymmetric index. It has however been argued in[30,31] that for a supersymmetric black hole quantum entropy function also computes an indexand hence we can directly compare the results obtained from quantum entropy function withthe index computed in the microscopic theory. This argument will be reviewed in § K × S / ZZ N where N is a prime integer (1, 2, 3, 5 or 7), and the ZZ N actsas a shift along S and a transformation on K N = 1this reduces to the original BMPV black hole in type IIB in K × S [2] while other values of N correspond to BMPV black holes in five dimensional CHL models [33–36]. n V ≡ N +1 + 3denotes the total number of U (1) gauge fields in the five dimensional theory including anyvector field that can come from dualizing a 2-form field, e.g. for N = 1 we have n V = 27.The classical black hole solution under consideration carries Q units of D1-brane charge along S , Q units of D5-brane charge along K × S , − n/ N units of momentum along S and SO (4) = SU (2) L × SU (2) R angular momentum ~J R = 0, J L = J L = 0, J L = J . The tableshows the results for the logarithmic corrections to the entropy in the limit when Λ is large.Each of the results in this table has been calculated independently in the macroscopic andmicroscopic description and in each case we find perfect agreement between the two sides. Forthis comparison we need to ensure the correct choice of ensemble on the microscopic side, sothat the result can be compared with the microcanonical entropy that the macroscopic sidecomputes. There are two relevant ensembles: in the first one ~J R = 0 and J L is fixed at J/ ~J R = 0, J L is fixed at J/ ~J L is fixed at J ( J + 1). Both ensembles ofcourse have all the charges fixed. As discussed above, a direct macroscopic calculation gives the5ntropy in the first ensemble for J = 0 and in the second ensemble for J = 0. We have howeverused the known relation between the entropies in the two ensembles to express all the resultsin table 1 in the second ensemble. Also important for this comparison is the relation betweenthe index and entropy for black holes preserving four supersymmetries [30, 31] so that we cansensibly compare the entropy on the macroscopic side to the logarithm of the index computedon the microscopic side. Finally, this analysis can be easily generalized to other allowed valuesof N and with K3 replaced by T using the results of [19, 20], with the only difference that thesimple relation between N and n V will be lost, and some of the modular functions which willappear later in our microscopic formula will have a more complicated form.Our results also hold for BMPV black holes in type IIB string theory compactified on T .In fact on the macroscopic side there is no difference between the analysis in the CHL modelsor T compactifications, except that in the latter case n V has a specific value 27. Thus for thescaling Q , Q , n ∼ Λ, J ∼ Λ / the result for logarithmic correction to the entropy takes theform − . (1.1)On the other hand for Q , Q , n ∼ Λ, J = 0 we get a logarithmic correction of the form: −
152 ln Λ . (1.2)The computation on the microscopic side is quite different since we have a completely differentmicroscopic formula [37]. Nevertheless the final results agree precisely with (1.1) and (1.2).In computing the entropy of the black hole from the microscopic side one often invokes theCardy formula [38, 39]. Since the underlying CFT has central charge of order Q Q and theblack hole describes an ensemble of states with vanishing ¯ L eigenvalue and L eigenvalue oforder n , the central charge scales as Λ and the L eigenvalue scales as Λ in the scaling limitwe are considering. Thus the Cardy formula is not directly applicable. But one often uses theintuition that the CFT describing the D1-D5 system is a sigma model whose target space is thesymmetric product of Q Q copies of K T ) [40], and the twisted sector of this theory haslong string excitations whose dynamics is described by an effective CFT with central charge oforder one and L eigenvalue of order Q Q n ∼ Λ [41, 42]. This effectively amounts to keeping Q , Q fixed and scaling n as Λ , and Cardy formula can be applied. We have examined themicroscopic formula to examine the behaviour of the index in this limit with J ∼ Λ / − α for α ≥
0. We find that while for type IIB string theory on T the result agrees with the one shownin table 1 for n V = 27, for the CHL models (including type IIB string theory on K × S ) the6ogarithmic correction to the log of the index is given by − α ln Λ, which is quite different fromthe actual result − ( n V − α ) ln Λ (see table 1 for α = 0 and eq.(1.4) below for general α ). This mismatch for the CHL models shows that it is not always possible to extract thelogarithmic correction to black hole entropy by examining the corresponding formula in theCardy limit.For non-supersymmetric extreme Kerr black holes in four dimensions our result for loga-rithmic correction to the entropy takes the form:1180 ln A H (cid:18) n S − n V + 7 n F − n / + 64 (cid:19) , (1.3)if, besides gravity, the theory contains n S minimally coupled massless scalar, n V minimallycoupled massless U(1) gauge fields, n F minimally coupled massless Dirac fermions and n / minimally coupled massless real Rarita-Schwinger fields. Here A H is the area of the eventhorizon. It will be interesting to explore if (1.3) can be explained using Kerr/CFT correspon-dence [22], which is an attempt to explain the origin of extremel Kerr black hole entropy bypostulating the existence of a (1+1) dimensional conformal field theory dual to gravity in thenear horizon geometry of the black hole.Besides the scaling limits discussed above, we also consider the cases of slowly rotatingblack holes – black holes for which the angular momentum is parametrically smaller thanthe other charges so that the contribution of the angular momentum to the entropy becomesnegligible in the scaling limit. For example for BMPV black holes we can consider the scalinglimit Q , Q , n ∼ Λ and J ∼ Λ − α for some positive constant α . In this case the computationof the macroscopic entropy requires some additional assumptions about the spectrum of thekinetic operator in the background of such slowly rotating black holes. With this our resultfor the logarithmic correction to the entropy, in the ensemble in which ~J R = 0, J L is fixed tobe J/ ~J L is fixed to be J ( J + 1), is given by −
14 ( n V − α ) ln Λ . (1.4)This agrees with the microscopic results in the same ensemble. We also consider slowly rotatingextremal Kerr-Newmann black hole carrying charge q and angular momentum J , with rotationparameter γ ≡ J/q ∼ J/A H small. In this case we get a logarithmic correction of the form − n S + 62 n V + 11 n F ) ln A H + ln γ , (1.5)7f the theory contains, besides the metric and the Maxwell field under which the black hole ischarged, n S minimally coupled massess scalar field, n V minimally coupled additional masslessvector fields and n F minimally coupled massless Dirac fermions. (1.5) refers to the entropycomputed in an ensemble with fixed charges, J = J and ~J = J ( J + 1).Various other earlier approaches to computing logarithmic corrections to black hole entropycan be found in [39, 43–58]. Since in [27] we have given a detailed comparison between ourmethod and the method of [47, 58, 59] – which is closest to the method we use – we shall notrepeat the discussion here. However it will be prudent to point out that the naive applicationof the method of [47, 58, 59] would give zero result for the logarithmic correction to the BMPVblack hole entropy, since the trace anomaly on which their computation is based vanishes inodd dimensions. The reason that we get a non-zero result is due to the zero modes which mustbe treated separately, and is not correctly accounted for in the trace anomaly based approach.For extremal Kerr black hole the analysis of [47, 58, 59] would correctly give the dependence on n S , n V , n F and n / in (1.3) but would give the constant term to be 424 instead of 64. This isagain due to the additional corrections due to graviton zero modes which have been includedin (1.3).The rest of the paper is organized as follows. In § § § § § § e.g. logarithmic corrections to black hole entropy, are insensitive to theultraviolet completion of the theory and are determined solely by the infrared theory [59].Thus we can in principle use Euclidean quantum gravity to calculate even the entropy of non-extremal black holes, and any consistent ultraviolet completion of the theory must reproducethese results. This could provide a strong check on any proposed ultraviolet completion ofgravity, if the latter provides an independent derivation of the logarithmic correction to theblack hole entropy. In [25–27] we used the quantum entropy function [23] to compute logarithmic corrections tothe entropy of four dimensional spherically symmetric extremal black holes with
AdS × S near horizon geometry. In this section we shall generalize this analysis to rotating extremalblack holes in arbitrary dimensions. Rotating extremal black holes are expected to have an
AdS factor in their near horizongeometry [61–65], possibly fibered over some other directions. Besides the AdS factor thenear horizon geometry consists of various other compact directions, some of which could beinherited from the compact directions of the asymptotic geometry while others could be theangular coordinates of the non-compact part of the asymptotic space-time. Our goal will beto compute logarithmic corrections to the entropy in certain scaling limit in which the variouscharges grow in a certain way. In this limit some of these compact directions grow in sizewhile the others remain fixed in size. We shall label by K the space spanned by the compactdirections which grow in size, irrespective of whether they are inherited from the compactcoordinates or the angular coordinates of the asymptotic space-time. Let d be the dimensionof K . We shall Fourier decompose all string fields along the compact directions of fixed sizeand treat this as a ( d + 2) dimensional theory with the ( d + 2) coordinates containing thecoordinates of AdS and K . In what follows we shall denote the coordinates of AdS by x , the9oordinates along K by y and the coordinates ( x, y ) collectively by z . We shall for simplicityassume that the radius of curvature of AdS and the sizes of the other large dimensions labelledby y are of the same order a , but this assumption can be easily relaxed. The Euclidean nearhorizon ( d + 2) dimensional metric g µν can then be written as g µν dz µ dz ν = a f ( y )( dη + sinh η dθ ) + a ds K , (2.1)where ( η, θ ) are the coordinates along AdS , f ( y ) is some function of the coordinates y along K , and ds K is constructed out of differentials which are invariant under the SL (2 , R ) isometryof AdS . Note that we have explicitly factored out the large dimensional parameter a sothat f ( y ) and the metric ds K are of order unity. Some examples of such metrics can be foundin [62] (in a different coordinate system for labelling AdS metric). It follows from the SL (2 , R )isometry of AdS that p det g = sinh η G ( y ) , (2.2)for some function G ( y ) of order a d +2 . In particular G ( y ) has no dependence on the coordinatesof AdS .Let Z AdS denote the partition function of string theory in the near horizon geometry, eval-uated by carrying out functional integral over all the string fields weighted by the exponentialof the Euclidean action S , with boundary conditions such that asymptotically the field config-uration approaches the near horizon geometry of the black hole. Since in AdS the asymptoticboundary conditions fix the electric fields, or equivalently the charges carried by the blackhole, and let the constant modes of the gauge fields fluctuate, we need to include in the pathintegral a boundary term exp( − i H P k q k A ( k ) µ dx µ ) where A ( k ) µ are the gauge fields and q k arethe corresponding electric charges carried by the black hole [23]. The list of gauge fields includeall the gauge fields of the original theory, as well as any gauge field which might arise fromdimensional reduction of the metric along the compact directions of the near horizon geometry.Thus we have Z AdS = Z d Ψ exp(
S − i I X k q k A ( k ) µ dx µ ) , (2.3)where Ψ stands for all the string fields. AdS /CF T correspondence tells us that the fullquantum corrected entropy S macro is related to Z AdS via [23]: e S macro − E L = Z AdS , (2.4)10here E is the energy of the ground state of the black hole carrying a given set of charges,and L denotes the length of the boundary of AdS in a regularization scheme that renders thevolume of AdS finite by putting an infrared cut-off η ≤ η .Let ∆ L eff ( y ) denote the quantum correction to the ( d + 2) dimensional effective lagrangiandensity evaluated in the background geometry (2.1). Due to the SL (2 , R ) invariance, ∆ L eff is independent of the AdS coordinates x but could depend on the coordinates y of the largecompact dimensions. Then the quantum correction to Z AdS is given byexp (cid:20)Z p det g dη dθ d d y ∆ L eff (cid:21) = exp (cid:20) π (cosh η − Z d d y G ( y ) ∆ L eff ( y ) (cid:21) . (2.5)The term proportional to cosh η in the exponent has the interpretation of − L ∆ E + O ( L − )where L = 2 π sinh η is the length of the boundary of AdS parametrized by θ and ∆ E = − R d d y G ( y ) ∆ L eff ( y ) is the shift in the ground state energy. Alternatively, this can becancelled by a boundary counterterm which gives a contribution proportional to the length ofthe boundary ı.e. sinh η . The rest of the contribution in the exponent can be interpreted asthe quantum correction to the black hole entropy [23]. Thus we have∆ S macro = − π Z d d y G ( y ) ∆ L eff ( y ) . (2.6)While the term in the exponent proportional to L and hence ∆ E can get further correctionsfrom boundary terms in the action, the finite part ∆ S macro is defined unambiguously. Thisreduces the problem of computing quantum correction to the black hole entropy to that ofcomputing quantum correction to L eff . If on the other hand we replace ∆ L eff in (2.6) by theclassical Lagrangian density then we get back the Wald entropy [66] of the classical black hole.So far our analysis has been completely general without making any approximation. How-ever if we are interested in computing the logarithmic correction to the entropy then we canfocus on the contribution to ∆ L eff from the massless fields only [25,26]. We shall now describethe general procedure for calculating one loop contribution to ∆ L eff ( y ) from massless fields. Let us suppose that the string theory under consideration contains a set of massless fields { φ i } (or fields of mass ∼ a − ) on AdS × K . Here the index i could run over several scalar In order to fix the normalization in the definition of L and E we need to fix the normalization of the metricon AdS after dimensional reduction on K . We can for example take ( dη + sinh ηdθ ) as the definition of themetric on AdS in which case the boundary has length 2 π sinh η . Any other normalization will simply rescale E and L in opposite directions without affecting the rest of the analysis. f ( i ) n ( x, y ) denote an orthonormal basisof eigenfunctions of the kinetic operator expanded around the near horizon geometry, witheigenvalue κ n : Z d x d d y p det g G ij f ( i ) n ( x, y ) f ( j ) m ( x, y ) = δ mn , (2.7)where G ij is a metric in the space of fields induced by the metric on the near horizon geometry, e.g. for a vector field A µ , G µν = g µν . Then the heat kernel K ij ( x, y ; x ′ , y ′ ) is defined as K ij ( x, y ; x ′ , y ′ ; s ) = X n e − κ n s f ( i ) n ( x, y ) f ( j ) n ( x ′ , y ′ ) . (2.8)In (2.7), (2.8) we have assumed that we are working in a basis in which the eigenfunctions arereal; if this is not the case then we need to replace one of the f ( i ) n ’s by f ( i ) ∗ n . Among the f ( i ) n ’sthere may be a special set of modes for which κ n vanishes. We shall denote these zero modesby the special symbol g ( i ) ℓ ( x, y ), normalized as Z d x d d y p det g G ij g ( i ) ℓ ( x, y ) g ( j ) ℓ ′ ( x, y ) = δ ℓℓ ′ , (2.9)and define ¯ K ij ( x, y ; x ′ , y ′ ) = X ℓ g ( i ) ℓ ( x, y ) g ( j ) ℓ ( x ′ , y ′ ) , (2.10) K ( y ; s ) = G ij K ij ( x, y ; x, y ; s ) , ¯ K ( y ) = G ij ¯ K ij ( x, y ; x, y ) . (2.11)Note that due to the SL (2 , R ) symmetry, G ij K ij ( x, y ; x, y ; s ) and G ij ¯ K ij ( x, y ; x, y ) dependonly on y but not on x . Using orthonormality of the wave-functions we get Z d x d d y p det g (cid:0) K ( y ; s ) − ¯ K ( y ) (cid:1) = X n ′ e − κ n s , (2.12)where P ′ n denotes sum over the non-zero modes only. Since the one loop contribution to Z AdS from the non-zero modes is given by Q ′ n κ − / n = exp[ − ln κ n ], the contribution to the one loopeffective action can now be expressed as∆ S = − X n ′ ln κ n = 12 Z ∞ ǫ dss X n ′ e − κ n s = 12 Z ∞ ǫ dss Z d x d d y p det g (cid:0) K ( y ; s ) − ¯ K ( y ) (cid:1) , (2.13)where ǫ is an ultraviolet cut-off which we shall take to be of order unity, ı.e. string scale.Identifying (2.13) as the contribution to R d x d d y √ det g ∆ L eff ( y ) we get the contribution to∆ L eff ( y ) from the non-zero modes:∆ L ( nz ) eff ( y ) = 12 Z ∞ ǫ dss (cid:0) K ( y ; s ) − ¯ K ( y ) (cid:1) . (2.14)12ubstituting this into (2.6) we get the one loop contribution to ∆ S macro due to the non-zeromodes: ∆ S macro = − π Z ∞ ǫ dss Z d d y G ( y ) (cid:0) K ( y ; s ) − ¯ K ( y ) (cid:1) . (2.15)Since we have labelled by a the overall scale factor in the metric, the non-zero eigenvalues κ n of the kinetic operator scale as 1 /a and hence K ( y, s ) − ¯ K ( y ) is a function of ¯ s ≡ s/a .Thus it is more natural to express (2.15) as∆ S macro = − π Z ∞ ǫ/a d ¯ s ¯ s Z d d y G ( y ) (cid:0) K ( y ; s ) − ¯ K ( y ) (cid:1) . (2.16)In this case the logarithmic contribution to the entropy – term proportional to ln a – arisesfrom the ǫ/a << ¯ s <<
1, ı.e. ǫ << s << a region in the s integral. If we expand K ( y ; s ) in aLaurent series expansion in ¯ s = s/a in the region ¯ s <<
1, and if K ( y ) denotes the coefficientof the constant mode in this expansion, then using (2.16) we see that the net logarithmiccorrection to the entropy from the non-zero modes will be given by − π ln a Z d d y G ( y ) (cid:0) K ( y ) − ¯ K ( y ) (cid:1) . (2.17) The contribution to Z AdS from integration over the zero modes can be evaluated as follows.First note that we can use (2.9), (2.10), (2.11) to define the number N zm of zero modes: Z d x d d y p det g ¯ K ( y ) = X ℓ N zm . (2.18)In fact often the matrix ¯ K ij takes a block diagonal form in the field space, with different blocksrepresenting zero modes of different sets of fields. In that case we can use the analog of (2.18)to define the number of zero modes of each block. If these different blocks are labelled by { A r } and we define ¯ K r ( y ) ≡ X ℓ ∈ A r G ij g ( i ) ℓ ( x, y ) g ( j ) ℓ ( x, y ) , (2.19)then the number of zero modes belonging to the r -th block will be given by N ( r ) zm = Z d x d d y p det g ¯ K r ( y ) = 2 π (cosh η − Z d d y G ( y ) ¯ K r ( y ) . (2.20) If on the other hand the theory has a cosmological constant, e.g. for BTZ black holes, then the scale ofthe eigenvalues of the kinetic operator are set by the cosmological constant instead of the parameter a . In thiscase the entire logarithmic corrections would come only from integration over the zero modes. a -independent parametrization of the asymptotic symmetry group so that the groupcomposition laws and the range over which the parameters take value is a -independent. Sup-pose for the zero modes in the r ’th block the Jacobian for the change of variables from the fieldsto supergroup parameters gives a factor of a β r for each zero mode. Then the net a dependentcontribution to Z AdS from the zero mode integration will be given by a P r β r N ( r ) zm = exp " π ln a (cosh η − Z d d y G ( y ) X r β r ¯ K r ( y ) . (2.21)As before we can interpret the term in the exponent proportional to cosh η as a contributionto the ground state energy and the finite term as a contribution to ∆ S macro . Adding this to(2.17) we get ∆ S macro = − π ln a Z d d y G ( y ) K ( y ) + X r ( β r −
1) ¯ K r ( y ) ! . (2.22)We shall refer to the term proportional to P r ( β r − R d d y G ( y ) ¯ K r ( y ) as the zero modecontribution although it should be kept in mind that only the term proportional to β r arisesfrom integration over the zero modes, and the − { ψ i } denote the set of fermion fields in the theory. Here i labels the internal indices orspace-time vector index (for the gravitino fields) but the spinor indices are suppressed. Withoutany loss of generality we can take the ψ i ’s to be Majorana spinors satisfying ¯ ψ i = ( ψ i ) T e C where e C is the charge conjugation operator. Then the kinetic term for the fermions have the form −
12 ¯ ψ i D ij ψ j = −
12 ( ψ i ) T e C D ij ψ j , (2.23)for some appropriate operator D . We can now proceed to define the heat kernel of the fermionsin terms of eigenvalues of D in the usual manner, but with the following simple changes. Sincethe integration over the fermions produce (det D ) / instead of (det D ) − / , we need to include14n extra minus sign in the definition of the heat kernel. Also since the fermionic kinetic operatoris linear in derivative, it will be convenient to first compute the determinant of D and thentake an additional square root of the determinant. This is implemented by including an extrafactor of 1 / K f ( y ) the s -independentpart of the trace of the fermionic heat kernel in the small s expansion after taking into accountthis factor of − /
2. To identify the zero modes however we need to work with the kineticoperator and not its square since the zero mode structure may get modified upon taking thesquare e.g. the kinetic operator may have blocks in the Jordan canonical form which squaresto zero, but the matrix itself may be non-zero. Let us denote by ¯ K f ( y ) the total fermionzero mode contribution to K f ( y ). Then we arrive at an expression similar to (2.17) for thefermionic non-zero mode contribution to the entropy: − π ln a Z d d y G ( y ) (cid:16) K f ( y ) − ¯ K f ( y ) (cid:17) . (2.24)Next we need to carry out the integration over the zero modes. Taking into account the extrafactor of − / N ( f ) zm now takes the form N ( f ) zm = − π (cosh η − Z d d y G ( y ) ¯ K f ( y ) . (2.25)Let us further assume that integration over each fermion zero modes gives a factor of a − β f / forsome constant β f . Then the total a -dependent contribution from integration over the fermionzero modes is given by a − β f N ( f ) zm / = exp (cid:20) π ln a (cosh η − Z d d y G ( y ) β f ¯ K f ( y ) (cid:21) . (2.26)As usual the coefficient of cosh η can be interpreted as due to a shift in the energy E ,whereas the η independent term has the interpretation of a contribution to the black holeentropy. Combining this with the contribution (2.24) from the non-zero modes we arrive atthe following expression for the logarithmic correction to the entropy from the fermion zeromodes: ∆ S macro = − π ln a Z d d y G ( y ) (cid:16) K f ( y ) + ( β f −
1) ¯ K f ( y ) (cid:17) . (2.27) This problem would not arise if we work with e C D instead of D since e C D is represented by an anti-symmetricmatrix. However it is easier to work with D instead of ( e C D ) .
15n other words, we can use (2.22) to represent contributions from both the bosonic and thefermionic modes provided we include the extra factors of − / β r for the fermions.At the end of this process we are still left with an a -independent contribution from in-tegration over the supergroup which contains both bosonic and fermonic zero modes. Usingsupersymmetric localization we can get finite result for this integral by canceling the infinitiesfrom the bosonic zero mode integration against the zeroes from the fermion zero mode integra-tion [67, 68]. However since there is no a -dependence in this contribution we shall not discussthis any further. K ( y ) In [25–27] K ( y ) was evaluated for various fields by finding the eigenfunctions and eigenvaluesof the kinetic operator in the AdS × S near horizon geometry. Since this is a difficulttask in absence of rotational symmetry, we shall now describe an indirect method [69–77]for computing K ( y ) which works under special circumstances. It can be argued on generalgrounds that the small s expansion of K ( y ; s ) contains even (odd) powers of s / in even (odd)dimensions (see e.g. [76] for a recent review). As a result K ( y ), which is the coefficient ofthe s term in the small s expansion of K ( y ; s ), vanishes in odd dimensions. Thus in the restof this subsection we shall restrict our analysis to the four dimensional (ı.e. d = 2) case. Inthis case one can show that [69–77] if we have purely gravitational background, e.g. as in thecase of extremal Kerr black holes, then in a theory with n S minimally coupled massless scalarfields, n V minimally coupled massless vector fields, n F minimally coupled massless Dirac fields, n / minimally coupled massless spin 3/2 field and n minimally coupled massless spin 2 fields, K ( y ) is given by K ( y ) = − π ( n S + 62 n V + 11 n F ) E − π ( n S + 12 n V + 6 n F − n / + 4243 n ) I , (2.28)where E = 164 (cid:0) R µνρσ R µνρσ − R µν R µν + R (cid:1) I = − (cid:18) R µνρσ R µνρσ − R µν R µν + 13 R (cid:19) . (2.29)In pure gravity R µν and R vanish for classical solutions, and (2.28) can be written as K ( y ) = − π ( − n S + 26 n V − n F + 2332 n / − n ) E . (2.30)16n the presence of background electromagnetic field strength eq.(2.30) gets additional contri-bution involving powers of the background field strength. In principle one could write downthe most general four derivative terms on the right hand side of (2.28) and calculate the co-efficients of these terms by perturbative computation of the trace anomaly. Alternatively insupersymmetric theories one could invoke supersymmetry to constrain the terms on the righthand side of (2.28). In particular for theories with
N ≥ K ( y ) [27, 73]. In our analysis we shallmainly focus on solutions without any background flux and hence use (2.30). β r We shall now describe the procedure for computing β r for various fields following [27]. Let usbegin with the contribution from an U (1) gauge field A µ . The path integral measure over A µ is normalized via Z [ DA µ ] exp (cid:20) − Z d d +2 z p det g g µν A µ A ν (cid:21) = 1 , (2.31)where, as mentioned earlier, z µ stand for both the coordinates x along AdS and the coordinates y along the large compact dimensions. In the scaling limit we consider g µν can be written as a g (0) µν where a scales as some power of Λ and g (0) µν is an a independent constant metric. Thuswe can express (2.31) as Z [ DA µ ] exp (cid:20) − a d Z d d +2 z p det g (0) g (0) µν A µ A ν (cid:21) = 1 . (2.32)From this we see that up to an a independent normalization constant, [ DA µ ] actually cor-responds to integration with measure Q µ,z d ( a d/ A µ ( z )). On the other hand the gauge fieldzero modes are associated with deformations produced by the gauge transformations withnon-normalizable parameters: δA µ ∝ ∂ µ Λ( z ) for some functions Λ( z ) with a -independent inte-gration range. Thus the result of integration over the gauge field zero modes can be found byfirst changing the integration over the zero modes of ( a d/ A µ ) to integration over Λ and thenpicking up the contribution from the Jacobian in this change of variables. This gives a factorof a d/ from integration over each zero mode of A µ . Comparing this with the definition of β r given above (2.21) we see that for gauge fields we have β v = d/ h µν of the metric (includingthose of the gauge fields arising from the dimensional reduction of the metric along the large17ompact dimensions) can be found in a similar way, with (2.31), (2.32) replaced by Z [ Dh µν ] exp (cid:20) − Z d d +2 z p det g g µν g ρσ h µρ h νσ (cid:21) = 1 , (2.33)ı.e. Z [ Dh µν ] exp (cid:20) − a d − Z d d +2 z p det g (0) g (0) µν g (0) ρσ h µρ h νσ (cid:21) = 1 . (2.34)Thus the correctly normalized integration measure, up to an a independent constant, is Q z, ( µν ) d ( a ( d − / h µν ( z )). Now the zero modes of the metric are associated with diffeomor-phisms with non-normalizable parameters: h µν ∝ D µ ξ ν + D ν ξ µ , with the diffeomorphism pa-rameter ξ µ ( z ) having a independent integration range. Thus the a dependence of the integralover the metric zero modes can be found by finding the Jacobian from the change of variablesfrom a ( d − / h µν to ξ µ . Lowering of the index of ξ µ gives a factor of a , leading to a factor of a ( d +2) / per zero mode. Thus for the metric we have β m = ( d + 2) / ψ µ . In thiscase eqs.(2.31), (2.32) are replaced by: Z [ Dψ µ ] exp (cid:20) − Z d d +2 z p det g g µν ¯ ψ µ ¯ ψ ν (cid:21) = 1 , (2.35)ı.e. Z [ Dψ µ ] exp (cid:20) − a d Z d d +2 z p det g (0) g (0) µν ¯ ψ µ ¯ ψ ν (cid:21) = 1 . (2.36)Thus up to an a independent normalization constant [ Dψ µ ] stands for Q d ( a d/ ψ µ ). On theother hand these zero modes are associated with the deformations corresponding to localsupersymmetry transformation ( δψ µ ∝ D µ ǫ ) with supersymmetry transformation parameters ǫ which do not vanish at infinity. Now since the anti-commutator of two supersymmetrytransformations correspond to a general coordinate transformation with parameter ξ µ = ¯ ǫγ µ ǫ ,and since γ µ ∼ a − , we conclude that ǫ = a − / ǫ provides a parametrization of the asymptoticsupergroup in which the group composition laws become a -independent. Writing δ ( a d/ ψ µ ) ∝ a ( d +1) / D µ ǫ , using the fact that the integration over the supergroup parameter ǫ produces an a independent result and that d ( λǫ ) = λ − dǫ for a grassmann variable ǫ , we now see thateach fermion zero mode integration produces a factor of a − ( d +1) / . Comparing this with thedefinition of β f given below (2.25) we see that β f = d + 1.The results of this subsection can be summarized in the relations β v = d , β m = d + 22 , β f = d + 1 . (2.37)For d = 2 this reproduces the results of [27]. 18 .6 Computation of R d d y G ( y ) ¯ K r ( y ) Finally we shall discuss the computation of R d d y G ( y ) ¯ K r ( y ) for various fields. This is bestdone with the help of (2.20), (2.25) and the results in appendix A. For each massless gaugefield on AdS the number of zero modes is given by eq.(A.5) and for each massless symmetrictensor field on AdS the number of zero modes is given by (A.8). Comparing these with (2.20)we see that each gauge field on AdS contributes a factor of 1 to 2 π R d d y G ( y ) ¯ K r ( y ) andeach symmetric rank two tensor field in AdS contributes a factor of 3 to 2 π R d d y G ( y ) ¯ K r ( y ).Thus for example if we consider the extremal Kerr black hole in four dimensions whose nearhorizon geometry has a squashed sphere with U(1) symmetry besides the AdS factor, weget a U(1) gauge field and a symmetric rank two tensor field on AdS after we dimensionallyreduce the metric on the squashed sphere. This gives a net contribution of 1 + 3 = 4 to2 π R d d y G ( y ) ¯ K r ( y ). Finally for computing R d d y G ( y ) ¯ K f ( y ) for BMPV black holes we canuse the result (A.20) for the total number of fermion zero modes. Identifying this with (2.25)we get − π R d d y G ( y ) ¯ K f ( y ) = 8. We can test this for quarter BPS black holes in fourdimensional N = 4 supersymmetric string theories or half BPS black holes in four dimensional N = 2 supersymmetric string theories which have identical number of fermion zero modes inthe near horizon geometry. For these cases we have d = 2, β f = 3 and the net contribution ofthe fermion zero modes to ∆ S macro computed from (2.27) will be 8 ln a . This agrees with theresults of [26, 27]. We now turn to the analysis of logarithmic corrections to the entropy of an extremal Kerrblack hole solution in Einstein gravity. We take the gravitational part of the action to be S = Z d x p − det g L , L = R . (3.1)We shall assume that the theory has, besides the metric, n S minimally coupled massless scalars, n V minimally coupled massless vector fields, n F minimally coupled massless Dirac fermions and n / minimally coupled massless Majorana Rarita-Schwinger fields. The near horizon geometryof an extremal Kerr black hole in this theory is given by (see e.g. [61,62] where the near horizon19olution is written down in somewhat different coordinate systems) ds = a (1 + cos ψ ) (cid:8) − ( r − dt + dr / ( r −
1) + dψ (cid:9) + 4 a sin ψ ψ ( dφ − ( r − dt ) (3.2)where ( φ, ψ ) label the azimuthal and the polar coordinates, ( r, t ) denote the radial and thetime coordinates and a is a constant related to the angular momentum J via the relation J = 16 πa . (3.3)The classical Bekenstein-Hawking entropy of the black hole, obtained as 1 / G N = 4 π timesthe area of the event horizon, is given by S BH = 8 πa Z π dφ Z π sin ψdψ = 32 π a = 2 πJ . (3.4)In order to compute the logarithmic correction to the entropy we first write down theEuclidean near horizon geometry by replacing t by − iθ . We also introduce the new radialcoordinate η = cosh − r for convenience. This gives ds = a (1 + cos ψ )( dη + sinh ηdθ + dψ ) + 4 a sin ψ ψ ( dφ + i (cosh η − dθ ) . (3.5)Substituting this into eq.(2.2) we get G ( y ) = 2 a sin ψ (1 + cos ψ ) . (3.6)Using (2.22), (2.27) and carrying out the φ integral we now get∆ S macro = − π a ln a Z dψ sin ψ (1 + cos ψ ) K ( ψ ) + X r ( β r −
1) ¯ K r ( ψ ) ! , (3.7)where the sum over r runs over the bosonic as well as the fermionic fields.We can now use (2.30) for computing K ( ψ ). In this case R µν = 0, R = 0 and [78, 79] R µνρσ R µνρσ = 48 a ψ ) (1 −
15 cos ψ + 15 cos ψ − cos ψ ) . (3.8)This gives K ( ψ ) = 1120 π a (cid:18) n S − n V + 7 n F − n / + 424 (cid:19) × ψ ) (1 −
15 cos ψ + 15 cos ψ − cos ψ ) . (3.9)20ubstituting this into (3.7) and using the result Z π dψ sin ψ (1 + cos ψ ) − (1 −
15 cos ψ + 15 cos ψ − cos ψ ) = − , (3.10)we get the non-zero mode contribution to the black hole entropy to be: − π a ln a Z dψ sin ψ (1 + cos ψ ) K ( ψ ) = 190 ln a (cid:18) n S − n V + 7 n F − n / + 424 (cid:19) . (3.11)This coincides with the results in [47, 58, 59].The contribution from the zero modes can be computed as follows. Possible zero modes inthis case arise from the gauge fields and the metric, – since the black hole is non-supersymmetricthere are no gravitino zero modes. Since here d = 2, (2.37) gives β v = 1 and β m = 2 [27].Eq.(2.22) now shows that the contribution from the gauge field zero modes, being proportionalto ( β v − ψ, φ ) direction gives a U (1) gauge field and a massless symmetric rank 2 tensorfield on AdS . According to the discussion in § π R d d y G ( y ) ¯ K m ( y ). The second term in eq.(2.22) now gives a net contribution of − a to S macro . Adding this to the non-zero mode contribution (3.11) we get the net contribution to∆ S macro to be 190 ln a (cid:18) n S − n V + 7 n F − n / + 64 (cid:19) = 1180 ln A H (cid:18) n S − n V + 7 n F − n / + 64 (cid:19) , (3.12)where A H ∝ a is the area of the event horizon.Let us denote by C tot the net coefficient of the ln A H term appearing in (3.12). In orderto seek a microscopic explanation of this result we need to specify for which ensemble (3.12)gives the logarithmic correction to the entropy. As discussed in the introduction, this giveslogarithmic correction to ln e d ( J ), e d ( J ) being number that counts all states with fixed J = J and all gauge charges set to 0, but no restriction on ~J . We can extract from this the number d ( J ) where ~J is also fixed to be J ( J + 1) as follows. First note that while e d ( J ) counts allstates with fixed J = J and ~J ≥ J ( J + 1), e d ( J + 1) counts all states with fixed J and ~J ≥ ( J + 1) ( J + 2). Furthermore due to SU (2) symmetry, once we fix ~J , the index isindependent of the chosen value of J , and hence in both cases we can take J = J . Thus21 d ( J ) − e d ( J + 1) will count all states with ~J = J ( J + 1) and J = J . This is the desiredmicroscopic index d ( J ). Thus we have d ( J ) = e d ( J + 1) − e d ( J ) = e S BH ( J +1)+ C tot ln A H + ··· − e S BH ( J )+ C tot ln A H + ··· , (3.13)where S BH ( J ) = 2 πJ is the classical Bekenstein-Hawking entropy given in (3.4) and · · · denoteterms of order unity. It is easy to see using (3.4) that the right hand side is given by ( e π − e d ( J ). Thus the logarithmic correction to ln d ( J ) is the same as that for ln e d ( J ), and isgiven by (3.12).It will be interesting to explore if Kerr/CFT correspondence [22] or any other approachcould give us a microscopic explanation of (3.12). In this section we shall analyze logarithmic corrections to the entropy of a five dimensionalBMPV black hole in type IIB string theory on K × S , as well as in a class of CHL models[33–36] obtained by taking ZZ N orbifolds of type IIB string theory on K × S . The ZZ N transformation acts by 2 π/ N shift along S and an appropriate action on K T and its various orbifolds discussed in [19, 20] with the only change that the metric b g mn in (4.2) represents metric on T . This macroscopic result for logarithmic correction to theentropy will then be compared with the microscopic results derived in § / ) and the first subleading order(charge / ). The logarithmic corrections to be studied here constitute the next leading correc-tion to the entropy. In the α ′ = 1 unit the ten dimensional action of type IIB string theory takes the form S = Z d x √− det G L , L = 1(2 π ) (cid:20) e − (cid:0) R + 4 G MN ∂ M Φ ∂ N Φ (cid:1) − F MNP F MNP (cid:21) , (4.1)22here G MN is the ten dimensional string metric, Φ is the dilaton and F MNP is the RR 3-formfield strength under which the D1 and D5 branes are electrically and magnetically charged. Weshall choose the coordinate along S such that S / ZZ N has period 2 πR , and as we move 2 πR along this direction we come back to the same point on the circle but a ZZ N transformed pointon K
3. A unit momentum will be defined as 1 /R . Since there are four non-compact spacedirections, this theory has an SO (4) = SU (2) L × SU (2) R rotational symmetry. We shall denoteby J iL and J iR for i = 1 , , SU (2) L and SU (2) R rotation groups. In thistheory we consider a classical BPS black hole solution carrying Q units of D5-brane chargealong K × S , Q units of D1-brane charge along S (including the − Q units of D1-branecharge that is induced by wrapping Q D5-branes on K − n/ N = − e n units of momentumalong S / ZZ N , J/ J L charge and zero J L , J L and J iR charge [2]. The Lorentzianten dimensional near horizon geometry takes the form [90, 91] dS = r dρ ρ + dχ + r ( dx + cos ψdφ ) + e J r dχ ( dx + cos ψdφ ) − ρdχdτ + r (cid:0) dψ + sin ψdφ (cid:1) + b g mn du m du n ,e Φ = λ ,F = r λ " ǫ + ∗ ǫ + e J r dχ ∧ (cid:18) ρ dρ ∧ ( d x + cos ψ dφ ) + sin ψ dψ ∧ dφ (cid:19) , ( ψ, φ, x ) ≡ (2 π − ψ, φ + π, x + π ) ≡ ( ψ, φ + 2 π, x + 2 π ) ≡ ( ψ, φ, x + 4 π ) , ( χ, ~u ) ≡ ( χ + 2 πR , h ~u ) . (4.2)Here u m are coordinates along K b g mn is the metric along K h represents the action of theZZ N generator on the coordinates of K λ is an arbitrary constant, ǫ = sin ψ dx ∧ dψ ∧ dφ denotes the volume form on a 3-sphere of coordinate radius 2 labelled by the coordinates( x , ψ, φ ) and ∗ denotes Hodge dual in six dimensions spanned by the coordinates t , χ , ρ , x , ψ and φ . The constants r , R , e J and the volume V of K r = λQ , R = s λ e nQ , e J = J Q p Q e n λ / , V ≡ Z d u p det b g = (2 π ) Q Q . (4.3)The Bekenstein-Hawking entropy of this black hole can be computed by dividing the area of theevent horizon, spanned by the coordinates ( χ, x , θ, φ, ~u ), by four times the effective Newton’s23onstant at the horizon, read out from (4.1)-(4.3). The result is: S BH = 2 π r Q Q e n − J . (4.4)In order to make the SL (2 , R ) symmetry of AdS manifest, we define A = √ r − e J r ! − / , B = − e J r A , (4.5)and change coordinates to e τ = Aτ /r . (4.6)In these coordinates the near horizon metric takes the form dS = r dρ ρ − r ρ d e τ + ( dχ − Aρd e τ ) + r ( dx + cos ψdφ − Bρ d e τ ) + r (cid:0) dψ + sin ψdφ (cid:1) + e J r ( dχ − Aρd e τ )( dx + cos ψdφ − Bρd e τ ) + b g mn du m du n . (4.7)This metric has an SL (2 , R ) isometry generated by [61] L = ∂ e τ , L = e τ ∂ e τ − ρ∂ ρ , L − = (1 / /ρ + e τ ) ∂ e τ − ( e τ ρ ) ∂ ρ + ( A/ρ ) ∂ χ + ( B/ρ ) ∂ x . (4.8)In order to make connection with the coordinate systems used in (2.1) we need to make acoordinate change cosh η = 12 ( ρ + ρ − − ρ e τ ) , e − t = (1 − e τ ) − ρ − (1 + e τ ) − ρ − , (4.9)together with some ( ρ, e τ ) dependent shifts on the coordinates χ and x and then make theanalytic continuation t → − iθ . The classical entropy function method [66], applied to thenear horizon geometry with the action (4.1), give the same result for the entropy as (4.4) asexpected [90].The scaling limit we shall consider is Q ∼ Λ , Q ∼ Λ , e n ∼ Λ , J ∼ Λ / , (4.10)with Λ large. Also we shall keep the undetermined constant λ fixed as we take the large chargelimit. In this limit r grows as Λ, e J grows as Λ / , R and the volume V of K a of AdS spanned by ( ρ, e τ ) as well as the the size of the squashed 3-sphere labelledby ( x , ψ, φ ) grow as a ≡ √ r ∼ Λ / . (4.11)Thus in this situation we can apply the formalism developed in § ~u, χ ) and regard this as a five dimensional theory living on thespace spanned by the coordinates ( x , ψ, φ, ρ, e τ ). We also dualize the 3-form field strength F MNP as well as all other 3-form field strengths to 2-form field strengths so that we can applythe formalism of § In this subsection we shall make some comments on the ensemble in which we compute theentropy. As argued in [23], the quantum entropy function computes the entropy in the micro-canonical ensemble in which all charges and angular momenta, which have the interpretationof charges associated with gauge fields in
AdS , are fixed. This means in particular that forthe BMPV black hole, J iR = 0 and J L = J/
2. This is exactly analogous to the situation forextremal Kerr black hole as discussed at the end of §
3. On the other hand for the StromingerVafa black hole carrying zero angular momentum all components of ~J L are associated withgauge charges on AdS and hence we have J iR = 0 and J iL = 0.Now while comparing our result with the microscopic results we need to use a protectedindex instead of the degeneracy [31, 32, 37, 91, 92]. In the present situation we can consider twodifferent indices. One of them will be defined as d ( n, Q , Q , J ) ≡ − p ! T r (cid:2) ( − J R (2 J R ) p (cid:3) , (4.12)where p takes the value 2 for type IIB string theory on K × S and the CHL models, but is6 for type IIB string theory on T . The trace is taken over all states carrying fixed Q , Q , n and J L = J/ ~J L = J (cid:0) J + 1 (cid:1) but different values of J R and ~J R . The second index will be This agrees with that used in [31, 91] but apparently differs from those in the earlier papers e.g. [32, 37].For example for type IIB on K × S , [32] would have p = 0 whereas for type IIB on T , [37] would have p = 2. This difference can be attributed to the fact that the trace in (4.12) is taken over all the modes of thesystem, whereas in the definition of the index given in [32, 37] the trace over the D1-D5 center of mass modeswas factored out. The definition we are using is in the same spirit as the helicity trace index used in [93, 94]for four dimensional black holes. e d ( n, Q , Q , J ) ≡ − p ! f T r (cid:2) ( − J R (2 J R ) p (cid:3) , (4.13)where f T r denotes that while taking the trace we sum over all states with fixed Q , Q , n and J L = J/
2, but different values of J R , ~J R and ~J L . The difference between the two indices isthat the ~J L value is not fixed in the second index. The two indices are related by a formulasimilar to (3.13): d ( n, Q , Q , J ) = e d ( n, Q , Q , J ) − e d ( n, Q , Q , J + 2) . (4.14)It has been argued in detail in [23, 31] that on the black hole side the exponential of theentropy in fact computes an index. This essentially follows from factoring the trace in (4.12)or (4.13) into a trace over the horizon degrees of freedom and the trace over the hair modes –modes living outside the horizon. In particular the hair modes include a set of 2 p fermion zeromodes associated with broken supersymmetry which are charged under ~J R but not under ~J L . T r [( − J R (2 J R ) p ] appearing in (4.12) receives a non-zero constant contribution from thesemodes, but each of the p factors of 2 J R are needed to prevent the contribution from vanishing.After factoring out trace over these zero modes, we are left with T r ( − J R with the tracetaken over the rest of the modes. Since BPS black holes with four unbroken supersymmetriesare forced to have ~J R = 0 [31], all the states represented by the black hole horizon have( − J R = 1, and hence the contribution to the index from the degrees of freedom associatedwith the horizon will be given by T r (1) = e S macro . If there are no additional hair modes besidesthe zero modes mentioned above then this allows us to compare exp[ S macro ] directly with themicroscopic index. A similar argument holds for the index (4.13). Whether one also fixes ~J L ornot depends on the situation; for non-zero J , the black hole counts states with fixed J L = J/ ~J L . Thus e d is the relevant index. On the other hand for J = 0, all componentsof ~J L are gauge charges and fixed to be zero. Thus the index d is the correct choice.If there are additional hair modes then we also need to compute their contribution to theindex and combine this with the contribution exp[ S macro ] from the horizon modes. It is howevermore convenient to identify the macroscopic contribution to the index as the contributionexp[ S macro ( n, Q , Q , J )] , (4.15)from the horizon modes only and remove the contribution of the (macroscopic) hair modesfrom the microscopic index before comparing the macroscopic and the microscopic results . It26ill be argued in § We now describe the macroscopic computation of logarithmic corrections to the BMPV blackhole entropy. According to (2.22) this involves three parts: computation of K ( y ), compu-tation of β r and computation of ¯ K r ( y ). As already discussed in § K ( y ) vanishes in odddimensions. Thus we are left with only the contribution from the second term of (2.22).Let us begin with the contribution from a U (1) gauge field A µ . Eq.(2.37) gives β v = 3 / d = 3. On the other hand since each five dimensional gauge field upon dimensional reductionon the squashed S in the near horizon geometry gives a gauge field on AdS , we have, from § π R d d y G ( y ) ¯ K v ( y ) = 1. Eq.(2.22) now shows that for each gauge field we havea contribution of − ln a to ∆ S macro . Since a ∼ Λ / we see that n V vector fields will give alogarithmic correction of the form − ( n V /
4) ln Λ.The effect of integration over the zero modes of the fluctuations h µν of the metric (includingthose of the gauge fields arising from the dimensional reduction of the metric on squashed S )can be found in a similar way. Eq.(2.37) gives β m = 5 / d = 3. On the other hand sincethe five dimensional metric leads to SU (2) × U (1) gauge fields and a metric on AdS afterdimensional reduction on the squashed sphere, we see, after using the results in § π R d d yG ( y ) K m ( y ) = 4 + 3 = 7. Eq.(2.22) now shows that the logarithmic correction to S macro from the five dimensional metric is given by − (21 /
2) ln a = − (21 /
4) ln Λ.Finally we turn to the contribution due to the zero modes of the gravitino field ψ µ . Inthis case eq.(2.37) gives β f = 4 for d = 3. On the other hand the analysis in § − π R d d y G ( y ) ¯ K f ( y ) = 4. Eq.(2.27) now shows that the net logarithmic correction to theentropy from the gravitino zero modes is given by 3 × a = 6 ln Λ.Combining the contributions from the vector, metric and the gravitino zero modes we geta net logarithmic correction of (cid:20) − n V −
214 + 6 (cid:21) ln Λ = −
14 ( n V −
3) ln Λ . (4.16)As already discussed, this logarithmic correction refers to the logarithm of the index e d ( n, Q , Q , J )from the macroscopic side. We shall denote this index by e d macro ( n, Q , Q , J ). It now follows27rom (4.4) and (4.14) that logarithmic correction to d macro ( n, Q , Q , J ) takes the form d macro ( n, Q , Q , J ) = e S BH ( J ) − ( n V −
3) ln Λ+ ··· − e S BH ( J +2) − ( n V −
3) ln Λ+ ··· = e S BH ( J ) − ( n V −
3) ln Λ+ ··· (4.17)when the charges and the angular momentum scale as in (4.10). Here · · · denote terms oforder 1. In the term on the right hand side we have included an additional factor of orderunity, involving the ratio J/ p Q Q e n ∼
1, in the · · · . (4.17) is in perfect agreement with themicroscopic result (5.22). It is also in agreement with (5.36) for α = 0 and n V = 27, thelatter being the number of vector fields in type IIB string theory compactified on T . Bycomparing (4.16) and (4.17) we also see that in this limit the logarithmic corrections to d and e d are identical.If we take J = 0 instead of J ∼ Λ / then we get two extra massless gauge fields on AdS from the reduction of the metric on S , giving the total contribution to 2 π R d d y G ( y ) ¯ K m ( y )from the metric to be 6 + 3 = 9. This changes the metric contribution of − / − /
4, and we get the following result for the logarithmic correction to the entropy: −
14 ( n V + 3) ln Λ . (4.18)Furthermore this now directly computes the logarithmic correction to the index d ( n, Q , Q , J ).(4.18) is in perfect agreement with the microscopic result (5.26). This is also in agreementwith the microscopic result (5.38) for type IIB string theory on T if we set n V = 27. In this section we shall derive the microscopic formulæ for the index of a black hole in typeIIB string theory compactified on K × S / ZZ N for N = 1 , , , , N symmetry acts by 2 π/ N units of translation along S and by a geometric transformationon K T . K × S / ZZ N As in § Q D5-branes wrapped on K × S / ZZ N , carrying Q unitsof D1-brane charge wrapped on S , − n/ N units of momentum along S and SU (2) L angular28omentum J L = J/
2. We first consider the microscopic index e d micro ( n, Q , Q , J ) ≡ − f T r (cid:2) ( − J R (2 J R ) (cid:3) , (5.1)where the trace is taken over states carrying fixed Q , Q , n and J L = J/
2, but different valuesof ~J L , J R and ~J R . The expression for e d micro ( n, Q , Q , J ) can be obtained from the knownexpression for the elliptic genus of the D1-D5 conformal field theory [32]. It will however beconvenient for us to begin with the expression for the index of quarter BPS states in the fourdimensional theory obtained by compactifying type IIB string theory on K × S × e S / ZZ N [10, 15, 18], and then use the fact that the latter is given by placing the five dimensional systemwe want in the background of a Kaluza-Klein monopole associated with the e S compactification[95]. Thus we simply need to remove [84, 96] from the index of the four dimensional black holecomputed in [10, 15, 18] the contribution of the Kaluza-Klein monopole, and the contributionfrom the supersymmetric quantum mechanics that binds the D1-D5 system to the Kaluza-Klein monopole, and then multiply this by the contribution from some additional fermion zeromodes which are present in the five dimensional system [91]. This gives the microscopic indexof such states to be (the N = 1 result was written down explicitly in [31])( − J e d micro ( n, Q , Q , J ) = − N Z d e ρ Z d e σ Z d e v e − πi ( e ρn + e σQ/ N + e vJ ) ( e πi e v − e − πi e v ) f ( N e ρ ) e Φ( e ρ, e σ, e v ) , (5.2)where e Φ is a known function of its arguments [15–18] and can be found e.g. in eqs.(C.18) ofthe review [97], f ( N e ρ ) = η ( e ρ ) k +2 η ( N e ρ ) k +2 , k + 2 ≡ N + 1 = 12 ( n V − , (5.3) Q ≡ Q Q . (5.4)The contour integration over the complex variables ( e ρ, e σ, e v ) in (5.2) runs along the real axes inthe range (0 , , N ) and (0 ,
1) respectively at fixed values of Im( e ρ, e σ, e v ). The ( − J factoron the left hand side of (5.2) can be traced to the fact that the index is normally definedwith a ( − J L +2 J R factor inserted into the trace whereas in the definition of e d micro we havejust inserted ( − J R into the trace. − / e Φ is the partition function for the four dimensionalindex. On the other hand 1 /f ( N e ρ ) is the partition function of the index associated with theKaluza-Klein monopole, and a factor of − ( e πi e v − e − πi e v ) − represents the partition functionassociated with the supersymmetric quantum mechanics that describes the D1-D5 center of29ass motion in the KK monopole background. Both these factors must be removed fromthe four dimensional partition function − / e Φ [84, 96], accounting for a multiplicative factorof − ( e πi e v − e − πi e v ) f ( N e ρ ) in the integrand. Another multiplicative factor of − ( e πi e v − e − πi e v ) in (5.2) represents the index associated with the fermion zero modes of the D1-D5 systemcarrying non-trivial SU (2) L quantum numbers [31, 91].Using (4.14) for the microscopic index we can compute the index d micro ( n, Q , Q , J ) wherewe also fix ~J L = J (cid:0) J + 1 (cid:1) . This gives d micro ( n, Q , Q , J ) = e d micro ( n, Q , Q , J ) − e d micro ( n, Q , Q , J + 2)= ( − J +1 N Z d e ρ Z d e σ Z d e v e − πi ( e ρn + e σQ/ N + e vJ ) ( e πi e v − e − πi e v ) (1 − e − πi e v ) f ( N e ρ ) e Φ( e ρ, e σ, e v ) . (5.5) (5.5) may be evaluated by deforming the contours of integration of ( e ρ, e σ, e v ) and picking upcontributions from the residues at various poles. The leading contribution comes from theresidue at [10, 11, 15, 18] e ρ e σ − e v + e v = 0 , (5.6)where e Φ has a zero. We now make a change of variables e ρ = 1 N v − ρ − σ , e σ = N v − ρσ v − ρ − σ , e v = v − ρ v − ρ − σ , (5.7)or equivalently ρ = e ρ e σ − e v N e ρ , σ = e ρ e σ − ( e v − N e ρ , v = e ρ e σ − e v + e v N e ρ . (5.8)In these variables we have d e ρ ∧ d e σ ∧ d e v = − (2 v − ρ − σ ) − dρ ∧ dσ ∧ dv , (5.9)and the pole at (5.6) is situated at v = 0 . (5.10) This was originally derived for the limit in which Q Q and n scale in the same way, and J scales at eitherthe same rate or slower than Q Q and n . But a careful analysis shows that this pole also gives the dominantcontribution in other scaling limits [31]. e Φ satisfies (see e.g. eq.(C.21) of the review [97] where also all the other propertiesof e Φ and b Φ discussed here can be found) e Φ( e ρ, e σ, e v ) = − ( i ) k C (2 v − ρ − σ ) k b Φ( ρ, σ, v ) (5.11)where C is a real positive constant, b Φ( ρ, σ, v ) is a new function defined e.g in eq.(C.19) of [97],and k has been defined in (5.3). Using these relations in (5.5) we get the leading contributionto d micro to be d micro ( n, Q , Q , J ) ≃ ( − J +1 ( i ) − k N C Z C ′ dρ ∧ dσ ∧ dv (2 v − ρ − σ ) − k − b Φ( ρ, σ, v ) e − πi ( e ρn + e σQ/ N + e vJ ) ( e πi e v − e − πi e v ) (1 − e − πi e v ) f ( N e ρ ) (5.12)where C ′ denotes a contour around v = 0.Now near v = 0 b Φ( ρ, σ, v ) behaves as [15, 18, 97] b Φ( ρ, σ, v ) = − π v g ( ρ ) g ( σ ) + O ( v ) , (5.13)where g ( ρ ) = η ( ρ ) k +2 η ( N ρ ) k +2 . (5.14)This allows us to evaluate the integration over v in (5.12) using the residue theorem. Makinga further change of variables ρ = τ + iτ , σ = − τ + iτ , (5.15)so that near the pole v = 0 we have e ρ = i N τ (cid:18) − i vτ + O ( v ) (cid:19) , e σ = i N τ + τ τ (cid:18) − i vτ + O ( v ) (cid:19) , e v = (cid:18) − i τ τ (cid:19) (cid:18) − i vτ − vτ + iτ + O ( v ) (cid:19) , (5.16)we can express (5.12) as d micro ( n, Q , Q , J ) ≃ Z d ττ G ( τ , τ ) , (5.17)where G ( τ , τ ) = exp (cid:20) πτ n n N + Q ( τ + τ ) − τ J o(cid:21) { g ( τ + iτ ) g ( − τ + iτ ) } − (2 τ ) − ( n V − / f (cid:18) i τ (cid:19) (cid:26) (cid:18) πτ τ (cid:19)(cid:27) (cid:0) − e − πτ /τ (cid:1) × (cid:26) n V − πτ (cid:16) n N + Q ( τ + τ ) − τ J (cid:17) + i τ f ′ ( i/ τ ) f ( i/ τ ) + 4 π τ τ tanh πτ τ +4 π τ τ e − πτ /τ − e − πτ /τ (cid:27) × constant . (5.18) ≃ in (5.17) implies equality up to exponentially suppressed contributions. If we had consideredthe index e d micro ( n, Q , Q , J ) then the factor of (cid:0) − e − πτ /τ (cid:1) from the second line and the4 π τ τ e − πτ /τ − e − πτ /τ factor from the last line of (5.18) will be absent.We now consider the limit n ∼ Q ∼ Q ∼ Λ, J ∼ Λ , with large Λ. In this case Q ≡ Q Q ∼ Λ and the leading contribution to (5.17) as well as systematic corrections tothis formula can be found using saddle point method. The saddle point values of τ , τ atthe leading order, obtained by extremizing the exponent of the first exponential in (5.18) withrespect to τ , τ , are given by τ = J Q ∼ Λ − / , τ = s n N − J Q Q ∼ Λ − / . (5.19)Substituting this into the exponential of the first exponent in (5.18) we get the result2 π q Q Q n N − J which is the leading contribution to the entropy given in (4.4).In order to calculate the logarithmic corrections to the entropy we first note that theintegration over the τ , τ coordinates run along the imaginary τ , τ directions [97]. In thesedirections the first exponential term in (5.18) is sharpely peaked around the saddle point witha width of order ∆ τ ∼ Λ − / , ∆ τ ∼ Λ − / , (5.20)which can be found by studying the second derivative of the term in the exponent with respectto ( τ , τ ). The logarithmic corrections to the log of the index come from the following factorsin (5.17), (5.18) containing powers of Λ: ( τ ) − : Λ∆ τ : Λ − / ∆ τ : Λ − / { g ( τ ) g ( − ¯ τ ) } − (2 τ ) − ( n V − / : Λ − ( n V − / (cid:26) n V − πτ (cid:16) n N + Q ( τ + τ ) − τ J (cid:17) + i τ f ′ ( i/ τ ) f ( i/ τ ) + 4 π τ τ tanh πτ τ π τ τ e − πτ /τ − e − πτ /τ (cid:27) : Λ / . (5.21)Note that in order to extract the small τ behaviour of g ( τ ), given in the fourth line of (5.21), weneed to make use of the modular properties of the η -functions appearing in (5.14). Multiplyingthe various factors in (5.21) we get a net contribution ofΛ − ( n V − / = exp (cid:20) −
14 ( n V −
3) ln Λ (cid:21) . (5.22)The term in the exponent on the right hand side of (5.22) is the net logarithmic correctionto ln d micro . If instead of considering the index d micro we had analyzed e d micro we would havegotten an identical result since the extra factors mentioned below (5.18), which distinguish d micro from e d micro , do not contribute to the logarithmic corrections. The result (5.22) for d micro is in perfect agreement with the macroscopic result (4.17).We can also consider the case when J ∼ Λ (3 / − α for some positive constant α . In thiscase it follows from (5.19) that τ ∼ Λ − (1 / − α , τ ∼ Λ − / and hence an additional logarithmiccorrection comes from the following factor in (5.18) (cid:0) − e − πτ /τ (cid:1) : Λ − α (5.23)Thus we get the net power of Λ in the expression for d ( n, Q , Q , J ) to beΛ − α − ( n V − / = exp (cid:20) −
14 ( n V − α ) ln Λ (cid:21) . (5.24)This agrees with the macroscopic result (6.2). If we had considered the index e d micro we wouldget the result (5.22) since the factor given in (5.23) is absent from the expression for e d micro .The above analysis needs some modification when J vanishes exactly since the factor (cid:0) − e − πτ /τ (cid:1) vanishes at the saddle point. In this case we proceed by expanding this ina power series in τ : (cid:0) − e − πτ /τ (cid:1) = 2 π τ τ − π (cid:18) τ τ (cid:19) + · · · (5.25)Substituting this into (5.17), (5.18) we see that the contribution from the term linear in τ vanishes by τ → − τ symmetry of the rest of the integrand. The term proportional to ( τ ) For this we need to ignore the last term inside the curly bracket in (5.18) which is in any case subdominant. τ integral to be approximately a gaussian with a width oforder ∆ τ ∼ Λ − / around the saddle point. Thus ( τ ) factor can be replaced by a term oforder Λ − / . On the other hand since the saddle point value of τ is of order Λ − / which islarger than the width of the gaussian Λ − / , we can replace the ( τ ) in the denominator byits saddle point value of order Λ − . This gives a net factor of Λ − / / Λ − ∼ Λ − / , and we getthe net logarithmic correction to d micro for J = 0 to beΛ − / − ( n V − / = exp (cid:20) −
14 ( n V + 3) ln Λ (cid:21) , (5.26)in agreement with the macroscopic result (4.18). We now see that if we had used the index e d micro we would get the result (5.22) which will disagree with the macroscopic result. But asdiscussed in § d micro is the correct index to compare.Before concluding this section we would like to mention that the results for the microscopicindex exist for a more general class of N = 4 supersymmetric theories obtained by taking ZZ N orbifolds of type IIB string theory on K × S with non-prime N , as well as orbifolds of typeIIB string theory on T . For these models the formula for the index takes a form similar to(5.2) and the relation between k and n V given in (5.3) still holds although the relation between N and n V given in (5.3) is lost. Also the functions g ( τ ) and f ( τ ) have more complicated form,but their large and small τ behavior are identical to what we have discussed. Thus the results(5.22), (5.24) and (5.26) hold for these models as well. It has been argued in [91, 98] that some hair degrees of freedom may live outside the horizonof the black hole and hence their contribution must be removed from the microscopic partitionfunction before we can compare the results to the macroscopic index associated with thehorizon degrees of freedom. The hair modes for a BMPV black hole in type IIB string theoryon K × S were analyzed in [98]. A similar analysis is also possible for the orbifold models.It was found in [98] that the hair mode contribution to the partition function includes a factorof ( e πi e v − e − πi e v ) associated with the fermion zero modes carrying J L charge and also a e ρ dependent factor h ( e ρ ) associated with the modes of the gravitino field. Thus we must multiplythe integrand in (5.5) by a factor of ( e πi e v − e − πi e v ) − ( h ( e ρ )) − . Now it follows from (5.16) thatat the saddle point ( e πi e v − e − πi e v ) = i cosh πτ τ , e ρ = i N τ . (5.27)34sing (5.19) we see that ( e πi e v − e − πi e v ) remains finite at the saddle point. On the other handsince e ρ ∼ i Λ / , and since h ( e ρ ) is made of products of (1 − e πiℓ e ρ ) [98], it does not give anyfactor involving powers of Λ. Thus removal of the hair contribution from the partition functiondoes not introduce any new logarithmic correction to the entropy. T We shall now briefly discuss the microscopic computation of the index in type IIB string theoryon T . In this case the known microscopic index is [31, 37] e d micro ( n, Q , Q , J ) = − f T r (cid:2) ( − J R (2 J R ) (cid:3) , (5.28)where, as before, we take the trace over states with fixed Q , Q , n and J L = J/
2, but all J R , ~J R and ~J L . Explicit computation gives this index to be [37] e d micro ( n, Q , Q , J ) ≃ ( − J Z dτ Z dv e − πiQ Q nτ − πiJv ( e πiv − e − πiv ) ϑ ( v | τ ) η ( τ ) , (5.29)up to exponentially suppressed terms. As in (5.5) we can find the index d micro for fixed Q , Q , n , J R = J/ ~J R = J (cid:0) J + 1 (cid:1) by inserting in the integrand a factor of (1 − e − πiv ): d micro ( n, Q , Q , J ) = e d micro ( n, Q , Q , J ) − e d micro ( n, Q , Q , J + 2) ≃ ( − J Z dτ Z dv e − πiQ Q nτ − πiJv ( e πiv − e − πiv ) ϑ ( v | τ ) η ( τ ) (1 − e − πiv ) . (5.30)We now consider the scaling limit: Q , Q , n ∼ Λ , J ∼ Λ − α , (5.31)and try to evaluate the integral using saddle point method. Anticipating that at the saddlepoint τ is small and v ∼
1, we can approximate the integral by d micro ( n, Q , Q , J ) ≃ ( − J Z dτ Z dv e − πiQ Q nτ − πiJv ( e πiv − e − πiv ) e − πiv /τ e πiv/τ (1 − e − iπv/τ ) ( − iτ ) (1 − e − πiv ) . (5.32)Extremizing the integrand with respect to v and τ we find the approximate saddle point inthe range 0 ≤ Re ( v ) < v = 12 − J τ + · · · , τ = i/ p nQ Q − J + · · · ∼ Λ − / , (5.33) For type IIB on K × S we have h ( e ρ ) = Q ∞ ℓ =1 (1 − e πiℓ e ρ ) . · · · denote subleading terms. The value of the integrand at this saddle point isexp[ π p nQ Q − J + · · · ] . (5.34)This gives the leading contribution to d micro ( n, Q , Q , J ). To examine the logarithmic correc-tions we determine the various powers of Λ coming from different terms in (5.32):∆ v : Λ − / ∆ τ : Λ − / ( − i τ ) : Λ − (1 − e − πiv ) : Λ − α (5.35)where ∆ v and ∆ τ denotes the range of v and τ integration beyond which the integrand fallsoff sharply. Taking the product of these factors we get the net power of Λ in the expressionfor the index: Λ − − α = exp[ − (6 + α ) ln Λ] . (5.36)For α = 0 this agrees with the macroscopic result (4.17) for n V = 27. On the other hand for α >
0, (5.36) agrees with the macroscopic result (6.2). For computing the index e d micro weneed to drop the (1 − e − πiv ) from the integrand in (5.32) with no other change. According to(5.35), we now get the result Λ − = exp[ − . (5.37)If we instead set J = 0 then the relevant index for comparison with the macroscopic result d micro . In this case special care is needed to deal with the factor of (1 − e − πiv ). Since thisvanishes at the saddle point, we expand it to second order in fluctuations about the saddlepoint and replace it by a term of order ∆ v ∼ Λ − / . This gives a net contribution ofΛ − − / = exp[ −
152 ln Λ] , (5.38)to d micro . Again this agrees with the macroscopic result (4.18) for n V = 27. However if wehad considered the index e d micro we would have gotten the result (5.37) which would disagreewith the macroscopic result. This again illustrates the necessity for making the correct choiceof ensembles while comparing the microscpic and the macroscopic results.36 .5 Comparison with the Cardy limit Since the success of the black hole microstate counting is often associated with the Cardyformula – applicable for n → ∞ limit at fixed Q , Q – we shall analyze in this subsection thelogarithmic correction to the entropy in the Cardy limit. For this we consider the scaling n ∼ Λ , J ∼ Λ / − α , (5.39)with Q , Q fixed. Eq.(5.19) shows that in this limit τ ∼ Λ / , τ ∼ Λ / − α . (5.40)Thus the logarithmic correction to ln d micro comes from the terms: ( τ ) − : Λ − ∆ τ : Λ / ∆ τ : Λ / g ( τ ) g ( − ¯ τ )(2 τ ) − ( n V − / : Λ − n V − / f ( i/ τ ) : Λ n V − / (cid:26) n V − πτ (cid:16) n N + Q ( τ + τ ) − τ J (cid:17) + i τ f ′ ( i/ τ ) f ( i/ τ ) + 4 π τ τ tanh πτ τ +4 π τ τ e − πτ /τ − e − πτ /τ (cid:27) : Λ / (1 − e − πτ /τ ) : Λ − α (5.41)Note that we now need to use the modular property of the η -functions appearing in (5.3) tofind the behavior of f for small value of its argument. Taking the product of all the factorswe get Λ − α = exp[ − α ln Λ] , (5.42)which is quite different from (5.24). In particular there is no dependence on n V . This showsthat the Cardy limit is not always reliable for extracting the black hole entropy in the limit If we had been considering the elliptic genus of symmetric products of K3 – which is a weak Jacobi formof weight zero – then for α = 0 we should have gotten a power correction of order n − ∼ Λ − [38]. Howeverthe contribution from the center of mass oscillation modes turns the elliptic genus into a (meromorphic) weakJacobi form of weight two, and kills the logarithmic term in the asymptotic expansion of the Fourier coefficients. Q Q >> n is related to theCardy limit of a different CFT that appears in the dual type IIA description.We can also consider a similar limit for the formula (5.30) for type IIB string theory on T .From (5.30) we see that the index depends only on the combination Q Q n and not individuallyon Q , Q and n . Thus in this case scaling n by Λ keeping Q and Q fixed is equivalent toscaling Q , Q and n by Λ, and we get a result identical to (5.36):exp[ − (6 + α ) ln Λ] . (5.43) In the macroscopic analysis of § i ) when the angularmomentum is large so that its contribution to the classical Bekenstein-Hawking entropy scalesin the same way as that from the charges, and ( ii ) when the angular momentum vanishes.In this section we shall consider some cases when the angular momentum is non-zero butparametrically small so that it gives negligible contribution to the classical Bekenstein-Hawkingentropy. This analyis will however require us to make some assumptions about the spectrumof the kinetic operator in the near horizon geometry of a slowly rotating black hole. For thisreason the results of this section should be regarded as somewhat tentative. The microscopic analysis for BMPV black holes yields the result for the index in the scalinglimit Q , Q , n ∼ Λ, J ∼ Λ − α for some positive number α – as given in eqs.(5.24) and(5.36). Since J/ Λ / is the parameter that controls the deviation of the black hole solutionfrom the rotationally invariant configuration, our first guess will be that the partition functionon the macroscopic side can be computed using perturbation expansion in J/ Λ / around therotationally invariant configuration J = 0. Furthermore, one would naively expect that thecorrections to the non-zero eigenvalues would have negligible effect for small J/ Λ and we onlyneed to take into account the shift in the zero eigenvalues using perturbation theory. This wouldproduce small but non-zero eigenvalues, and, as a result, compared to the case of J ∼ Λ / , theone loop determinant will get some extra power of J/ Λ from the small eigenvalues. Howeverthis naive expectation is not quite correct due to the following reason. We recall that the zero38odes which are lifted by switching on J are the ones associated with the W ± L gauge fields on AdS arising from SU (2) L . Now a non-zero J corresponds to switching on a constant U (1) L electric field along AdS . Since the W ± L fields are charged under the U (1) L , any quantum of W ± L will be subject to a constant force in the presence of a U (1) L electric field. Hence wecannot use perturbation theory to study the effect of the switching on a U (1) L electric field onthe zero modes of W ± L – we expect these zero modes to be lifted altogether in the presence ofnon-zero J . While we do not have a concrete analysis of the eigenvalue equations in the newbackground, it is natural to assume that all the modes, except the exact zero modes which arein any case neutral under U (1) L , will have eigenvalues of order a − . In other words the effectof switching on even a small J takes us to back to the case of generic J/ Λ / ∼
1, ı.e. gives alogarithmic correction to ln e d macro ( n, Q , Q , J ) given in (4.16). Using the fact that S BH givenin (4.4) satisfies S BH ( J ) − S BH ( J + 2) ≃ π J p Q Q e n ∼ Λ − α (6.1)for J ∼ Λ − α , we see from (4.17) that the logarithmic correction to d macro ( n, Q , Q , J ) nowtakes the form: d macro ( n, Q , Q , J ) = e S BH ( J ) − ( n V −
3) ln Λ+ ··· − e S BH ( J +2) − ( n V −
3) ln Λ+ ··· = e S BH ( J ) − ( n V − α ) ln Λ+ ··· . (6.2)This agrees with the microscopic result (5.24) and also with (5.36) for n V = 27. In this subsection we shall briefly discuss the case of slowly rotating extremal Kerr-Newmannblack hole in four dimensional Einstein-Maxwell theory. For zero angular momentum, ı.e. forReissner-Nordstrom black hole, the result for the logarithmic correction was calculated in [27]with the result: − n S + 62 n V + 11 n F ) ln A H (6.3)if the theory contains, besides the metric and the Maxwell field under which the black hole ischarged, n S minimally coupled massless scalar fields, n V minimally coupled additional massless This is corroborated by the fact that if in (A.2) we replace d Φ ( ℓ ) by D Φ ( ℓ ) where D denotes the gaugecovariant derivative for the W ± L fields in the background of U (1) L electric field, then the modes cease to besquare integrable. n F minimally coupled massless Dirac fermions. Here A H ∼ a is the area ofthe event horizon and we have set the Newton’s constant G N to unity.Let us now consider the effect of deforming the solution so that the black hole carries notonly electric charge q but also angular momentum J while remaining extremal. The nearhorizon geometry still contains an AdS factor [61, 62], but the SO (3) isometry of the nearhorizon geometry is now broken to U (1) and as a result from the dimensional reduction of themetric we get only one massless gauge field on AdS instead of three. The departure of themetric from that of the Reissner-Nordstrom black hole is given by the rotation parameter γ ≡ Jq . (6.4)We shall work with γ small (but still J large) so that the geometry is almost that of theextremal Reissner-Nordstrom black hole. Following the discussion in § γ will be to get rid of the zero modes of two of thethree gauge fields coming from the unbroken SO (3) isometry of the non-rotating black holesand make all the eigenvalues, except those associated with the exact zero modes, of order a − . In the language of this paper, this translates to the fact that 2 π R d yG ( y ) ¯ K m ( y ), insteadof taking the value 3 + 3 = 6 will now take value 1 + 3 = 4. Since here β m = 2, this meansaccording to (2.22) that we lose an additive contribution of − a = − ln A H from the entropy.Adding ln A H to (6.3) we get e d macro ( q, J ) = exp (cid:20) S BH ( q, J ) − n S + 62 n V + 11 n F ) ln A H + · · · (cid:21) . (6.5)Note that we have used the fact that for non-zero angular momentum the macroscopic compu-tation yields the degeneracy e d macro ( q, J ) in the ensemble of fixed J = J , instead of d macro ( q, J )which will be the degeneracy in the ensemble of fixed J = J and ~J = J ( J + 1). However wecan use (6.5) and the result for the classical Bekenstein-Hawking entropy: S BH ( q, J ) = 2 π p q + J = 2 π q (cid:18) γ + O ( γ ) (cid:19) , (6.6)to compute d macro ( q, J ): d macro ( q, J ) = e d macro ( q, J ) − e d macro ( q, J + 1) ≃ exp (cid:20) S BH ( q, J ) − n S + 62 n V + 11 n F ) ln A H + ln γ (cid:21) . (6.7)40 cknowledgement: I wish to thank Monica Guica and Rajesh Gupta for useful commu-nications. This work was supported in part by the J. C. Bose fellowship of the Department ofScience and Technology, India and the project 11-R&D-HRI-5.02-0304.
A Counting of zero modes in
AdS
AdS andcount their numbers. For definiteness we shall take the AdS metric to be( g AdS ) µν dx µ dx ν = a ( dη + sinh ηdθ ) , (A.1)although the result for the number of zero modes will be independent of what we take to bethe AdS size a .First consider the case of a U (1) gauge field A µ . The normalized basis of zero modes ofsuch a field on AdS is given by [99] A = d Φ ( ℓ ) , Φ ( ℓ ) = 1 p π | ℓ | (cid:20) sinh η η (cid:21) | ℓ | e iℓθ , ℓ = ± , ± , ± , · · · , (A.2)satisfying Z AdS d x √ g AdS g mnAdS ∂ m Φ ( ℓ ) ∗ ∂ n Φ ( ℓ ′ ) = δ ℓℓ ′ . (A.3)The basis states (A.2) also satisfy X ℓ g mnAdS ∂ m Φ ( ℓ ) ∗ ( x ) ∂ n Φ ( ℓ ) ( x ) = 12 πa . (A.4)This can be derived using the fact that due to homogeneity of AdS this sum is independentof x and hence can be evaluated at η = 0, and that at η = 0 only the ℓ = ± AdS is given by N = Z AdS d x √ g AdS X ℓ g mnAdS ∂ m Φ ( ℓ ) ∗ ( x ) ∂ n Φ ( ℓ ) ( x ) = 12 π Z η sinh η dη Z dθ = cosh η − . (A.5)A similar analysis can be done for a symmetric rank two tensor representing the gravitonfluctuation on AdS . The normalized basis of zero mode deformations is given by [99] h mn = w ( ℓ ) mn ,w ( ℓ ) mn dx m dx n = a √ π (cid:20) | ℓ | ( ℓ − (cid:21) / (sinh η ) | ℓ |− (1 + cosh η ) | ℓ | e iℓθ ( dη + 2 i sinh η dηdθ − sinh η dθ ) ℓ ∈ ZZ , | ℓ | ≥ . (A.6)41ocally these can be regarded as deformations generated by a diffeomorphism on AdS , butthese diffeomorphisms themselves are not square integrable. The basis states (A.6) satisfy X ℓ g mnAdS g pqAdS w ( ℓ ) ∗ mp ( x ) w ( ℓ ) nq ( x ) = 32 πa . (A.7)We have derived this using the fact that due to homogeneity of AdS this sum is independentof x , and that at η = 0 only the ℓ = ± N = 3 cosh η − . (A.8)Finally we turn to the zero modes of the gravitino fields. We use the following conventionsfor the zweibeins and the gamma matrices e = a sinh η dθ, e = a dη , (A.9) γ = − τ , γ = τ , (A.10)where τ i are two dimensional Pauli matrices and a is the AdS size parameter. In this conven-tion the Dirac operator on AdS can be written as D AdS = a − (cid:20) − τ η ∂ θ + τ ∂ η + 12 τ coth η (cid:21) . (A.11)The eigenstates of D AdS are given by [100] χ ± k ( λ ) = 1 √ πa (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Γ (1 + k + iλ )Γ( k + 1)Γ (cid:0) + iλ (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e i ( k + ) θ (cid:18) i λk +1 cosh k η sinh k +1 η F (cid:0) k + 1 + iλ, k + 1 − iλ ; k + 2; − sinh η (cid:1) ± cosh k +1 η sinh k η F (cid:0) k + 1 + iλ, k + 1 − iλ ; k + 1; − sinh η (cid:1) (cid:19) ,η ± k ( λ ) = 1 √ πa (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Γ (1 + k + iλ )Γ( k + 1)Γ (cid:0) + iλ (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e − i ( k + ) θ (cid:18) cosh k +1 η sinh k η F (cid:0) k + 1 + iλ, k + 1 − iλ ; k + 1; − sinh η (cid:1) ± i λk +1 cosh k η sinh k +1 η F (cid:0) k + 1 + iλ, k + 1 − iλ ; k + 2; − sinh η (cid:1) (cid:19) ,k ∈ ZZ , ≤ k < ∞ , < λ < ∞ , (A.12)42atisfying D AdS χ ± k ( λ ) = ± i a − λ χ ± k ( λ ) , D AdS η ± k ( λ ) = ± i a − λ η ± k ( λ ) . (A.13)The zero modes of the gravitino fields ψ m can be expressed in terms of the spinors (A.12)via the relations ξ ( k ) ± m ≡ N ± k (cid:18) D m ± a γ m (cid:19) χ ± k ( i ) , b ξ ( k ) ± m ≡ b N ± k (cid:18) D m ± a γ m (cid:19) η ± k ( i ) , k = 1 , · · · ∞ , (A.14)where N ± k and b N ± k are appropriate normalization constants such that a Z sinh η dη dθ g mnAdS ξ ( k )+ † m ( η, θ ) ξ ( k ′ )+ n ( η, θ ) = δ kk ′ (A.15)etc. Although χ ± k ( i ) and η ± k ( i ) are not square integrable, the modes described in (A.14)are square integrable and hence they must be included among the eigenstates of the Rarita-Schwinger operator. These modes can be shown to satisfy the chirality projection condition τ ξ ( k ) ± m = − ξ ( k ) ± m , τ b ξ ( k ) ± m = b ξ ( k ) ± m . (A.16)Furthermore with the help of (A.13) and that χ − k = τ χ + k , η − k = τ η + k one can show that ξ ( k ) ± m are proportional to each other and b ξ ( k ) ± m are proportional to each other.Now suppose we have a set of gravitino zero modes given by ξ ( k )+ m with k ranging over allpositive integers. Then the total number of zero modes may be expressed as ∞ X k =1 a Z sinh η dη dθ ∞ X k =1 g mn ξ ( k )+ † m ( η, θ ) ξ ( k )+ n ( η, θ ) . (A.17)We now use the fact that after taking the sum over k the integrand must become independentof ( η, θ ) and hence we can evaluate it at η = 0. In this case the only contribution comes fromthe k = 1 term. Substituting k = 1, λ = i in (A.12) and choosing the normalization constant N ± so that ξ (1)+ m defined in (A.14) is normalized, we get ξ (1)+ θ = 1 √ π e iθ/ (cid:18) η / cosh η (cid:19) , ξ (1)+ η = − i sinh η ξ (1)+ θ . (A.18)This gives ∞ X k =1 g mn ξ ( k )+ † m ( η, θ ) ξ ( k )+ n ( η, θ ) = g mn ξ (1)+ † m ( η, θ ) ξ (1)+ n ( η, θ ) | η =0 = 12 πa . (A.19)43ubstituting this into (A.17) we get the total number of zero modes to be (cosh η − b ξ ( k )+ m .We should however remember that the spinors ξ ( k )+ m and b ξ ( k )+ m are tensored with spinorsassociated with the tangent space spinors of other directions transverse to AdS and hence eachzero mode associated with ξ ( k )+ m and b ξ ( k )+ m may actually represent multiple zero modes. In orderto determine this multiplicity we shall use the fact that the gravitino zero modes are associatedwith the deformations generated by the fermionic generators of the N = 4 superconformalalgebra labelled as G αβn where α and β each takes value ± n ∈ ZZ + [30, 67]. G αβ ± / are the global symmetry generators which, together with the bosonic generators, form the SU (1 , |
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