Revisiting Logarithmic Correction to Five Dimensional BPS Black Hole Entropy

We compute logarithmic correction to five dimensional BPS black hole entropy using finite temperature black hole geometry and find perfect agreement with the microscopic results and macroscopic computations based on zero temperature near horizon geometry. We also reproduce the Bekenstein-Hawking term for zero temperature black hole entropy from the corresponding term for finite temperature black hole.


Introduction
Bekenstein-Hawking formula gives the leading contribution to the black hole entropy.It is expected to receive corrections that are suppressed by inverse powers of the black hole size and also corrections proportional to the logarithm of the black hole size.Logarithmic correction to the black hole entropy has proved a fruitful direction of research in the past since it can be computed purely from the infrared data of the theory -massless fields and their interaction.
Yet it puts strong constraint on any ultra-violet completion of the theory by requiring that explicit counting of microstates must reproduce the same correction.This has been tested for supersymmetric black holes for many four and five dimensional string compactifications [1][2][3][4][5][6].
Traditionally the computation of the logarithmic corrections to the entropy of non-extremal and extremal (BPS) black holes have used somewhat different routes, with the former employing gravitational path integral in the full asymptotically flat space-time geometry and the latter using the near horizon geometry of the black hole.This is necessitated by the fact that in the full geometry of extremal black holes there are two large length scales -the size of the black hole and the inverse temperature that is taken to infinity, and it is technically challenging to separate the logarithm of the size of the black hole from the logarithm of the temperature. 1ogarithmic corrections to BPS black hole entropy computed from the near horizon geometry could be tested against the results of explicit counting of microstates in supersymmetric string compactifications, while for non-supersymmetric black holes far from extremality, there is no independent microscopic computation in string theory.
This situation changed with the work of [9] which proposed a way to compute the index of supersymmetric states2 using gravitational path integral in the full asymptotically flat spacetime geometry.(Earlier work along the same line for AdS spaces can be found in [10] and for near horizon geometry can be found in [11]).The idea is to work at a finite temperature and introduce a chemical potential conjugate to the angular momentum that introduces a factor of (−1) F in the trace computed by the gravitational path integral.With some additional factor of angular momentum inserted into the path integral (discussed in section 3) this is precisely the index that counts supersymmetric ground states.Since this analysis can be done at any temperature, we do not encounter the problem we had with extremal black holes and in particular can choose the inverse temperature to be controlled by the same length scale that controls the black hole size.
In a previous paper we carried out this analysis for four dimensional black holes and found perfect agreement between the near horizon calculation and the full geometry calculation of the logarithmic correction to the entropy [12].This also means that the full geometry calculation gives result in agreement with the result of microscopic calculation whenever the latter results are available.In this paper we shall extend the analysis to five dimensional black holes and again find perfect agreement with the earlier results based on the near horizon geometry and microscopic counting.While in spirit the analysis is similar to that in four dimensions, we find that in five dimensions the change of ensemble produces non-vanishing logarithmic corrections and must be taken into account in order to get agreement with the earlier results.This also confirms the general procedure that was used earlier for computing logarithmic corrections to non-extremal black hole entropy.
As part of our analysis we also check that the classical entropy of the extremal black hole comes out correctly from the saddle point representing non-extremal black holes.The agreement is somewhat non-trivial since the agreement is not between the entropies computed on the two sides, but between the entropy of an extremal black hole and the entropy of a non-extremal black hole plus a term proportional to the angular momentum carried by the black hole [9].
The rest of the paper is organized as follows.In section 2 we review the scaling properties of various thermodynamic parameters of a black hole in arbitrary dimension.This determines how we should take the macroscopic limit of the various parameters so that the black hole metric scales by an overall constant factor in this limit.In section 3 we review the general formalism given in [9] for computing the supersymmetric index of the black hole from gravitational partition function.Particular emphasis is placed on the point that in order to go from the partition function to the index, we need to undergo a change of ensemble, and this produces some non-trivial logarithmic terms that must be taken into account.In section 4 we examine the classical limit of the relation between index and partition function.This leads to a nontrivial relation between the entropies of zero temperature and finite temperature black holes and we verify this relation using the known five dimensional solutions of [13,14].In section 5 we compute the logarithmic correction to the partition function for a class of supersymmetric black holes in five dimensions, and combine this result with the result of section 3 to compute the logarithmic correction to the index of these black holes.The result agrees perfectly with the earlier results based on the near horizon analysis and also the results of microscopic counting.
In appendix A we review the non-extremal five dimensional black hole solution of [13,14], and compute various quantities that are needed to test the relation between extremal and non-extremal black hole entropy, as discussed in section 4.

Scaling of the black hole parameters
In D dimensions the classical entropy S 0 of the black hole, given as a function of the charges ⃗ Q, mass M and angular momenta ⃗ J in the Cartan subalgebra of the rotation group, scales as [15] S 0 (λ The vector symbol on ⃗ J and ⃗ Ω labels different components of the angular momentum and angular velocity inside the Cartan subalgebra.This should not be confused with the vector symbol on ⃗ J L , ⃗ J R in later sections where it will denote all generators inside the SU (2) L and SU (2) R subgroups of the five dimensional rotation group.Throughout this paper we shall follow the convention that ⃗ J without a subscript denotes all generators of the Cartan subalgebra while ⃗ J L , ⃗ J R denote all generators of SU (2) L and SU (2) R respectively.The vector symbol on ⃗ Q is a reminder of the fact that the theory may have multiple U (1) gauge fields and the black hole may be charged under more than one of these gauge fields.⃗ Q is an n V dimensional vector if the theory has n V U(1) gauge fields.
(2.1) suggests that it is useful to introduce new parameters m, ⃗ q, ⃗ j via and define the macroscopic limit to be the large λ limit keeping m, ⃗ q, ⃗ j fixed at order one.Also in this limit the temperature β, chemical potential ⃗ µ and angular velocity ⃗ Ω scale as BPS black holes have infinite temperature and the mass and entropy are expressed as functions of the charges and the angular momenta.All quantities other than the temperature satisfy the same scaling relations, e.g. (2.4) For our analysis we shall also need the scaling properties of the black hole partition function, It follows from the results given above that ln Z 0 scales as (2.6)

Index from partition function
In five dimensions the rotation group is SO(4) = SU (2) L × SU (2) R where the subscripts L and R have been introduced for convenience.A generic black hole in five dimensions carry three U (1) charges, mass M and two angular momenta J ϕ and J ψ in orthogonal planes with azimuthal angles ϕ and ψ [13].The third components of SU (2) L and SU (2) R angular momenta are given in terms of J ϕ and J ψ as The conjugate variables are the inverse temperature β, the chemical potential ⃗ µ for the charges ⃗ Q and the chemical potentials (angular velocities) Ω L , Ω R conjugate to J 3L , J 3R .Classical supersymmetric black hole solutions have J 3R = 0.
The relevant index for a supersymmetric black hole that breaks 2n SU (2) L invariant supersymmetries is [16] with the trace taken over all states carrying fixed ( ⃗ Q, J 3L ) and zero momentum.This generalizes the helicity supertrace index in four dimensions [17,18].The sum is expected to pick up contribution only from BPS states that break 2n or less SU (2) L invariant supersymmetries.
Therefore N BP S = e S BP S counts BPS states with a fixed charge vector ⃗ Q, a fixed J 3L and zero momentum, but all values of ⃗ J 2 L , J 3R and ⃗ J 2 R , weighted by (−1) F (2J 3R ) n .We shall refer to this as the index.
For computing N BP S from the macroscopic side, we begin with the gravitational partition function Z with asymptotic boundary conditions appropriate to that of an Euclidean black hole with temperature β, chemical potentials µ and angular velocities Ω L,R , with a factor of (2J 3R ) n inserted into the path integral.The relevant boundary condition requires the Euclidean time τ and the azimuthal angles ⃗ ϕ conjugate to ⃗ J to be periodically identified as and the τ component of the gauge field at infinity to be fixed at Z defined this way computes: where the trace is taken over all the states.For computing the index N BP S = e S BP S we follow [9] and set the chemical potential Ω R dual to J 3R to −2πi/β.This computes Even though the trace runs over all states of the theory, it is expected to pick the contribution only from the BPS states.Let us denote by M BP S ( ⃗ Q, J 3L ) the mass of a BPS state carrying quantum numbers ( ⃗ Q, J 3L ).We can organize the trace in the expression for Z by first taking the trace over all quantum numbers other than ( ⃗ Q, J 3L , ⃗ k), and then sum over ⃗ Q, J 3L and integrate over ⃗ k.Since the first step produces N BP S ( ⃗ Q, J 3L ), this allows us to express Z as a weighted sum over N BP S ( ⃗ Q, J 3L ) = e S BP S ( ⃗ Q,J 3L ) .The ⃗ k integral can be easily done by noting that the ⃗ k dependence of the integrand comes only through the dependence of E on ⃗ k.We can replace E by M BP S + ⃗ k 2 /(2M BP S ) and express (3.6) as, where ⃗ k is an n T dimensional momentum vector that is invariant the rotation group element e −βΩ L J 3L and L is the physical size of the box in which we place the black hole.This restriction on ⃗ k stems from the fact that if ⃗ k is not invariant under e −βΩ L J 3L , then the action of this operator will produce a state with different ⃗ k and hence such a state will not contribute to the trace.We shall see that the final result for logarithmic correction is independent of n T .
After performing the integration over ⃗ k, (3.7) can be inverted as with appropriate choice of integration contours for ⃗ µ and Ω L .We have ignored overall numerical factors since they contribute constant terms in S BP S and are not of interest for the current paper.(3.7) and (3.8) holds for all β but we shall take β to scale as λ as in the case of non-extremal black holes given in (2.3) to facilitate our analysis.The leading classical result for ln Z is given by ln Z 0 given in (2.5).If we replace Z by Z 0 in (3.8) and carry out the integration over ⃗ k, ⃗ µ and Ω L using saddle point approximation, we get evaluated at the saddle point Due to (2.5) these equations are the same as the ones in (2.3).Using the various scaling properties described in section 2, and setting D = 5, we get where δ ln Z denotes further logarithmic corrections from integration over massless fields in the gravitational path integral and inclusion of the (2J 3 ) n factor in the definition of Z.Using (2.5) and βΩ R = −2πi, we get the sum of the leading contribution and logarithmic contribution to S BP S in D = 5 as4 Note that M and S 0 are the entropy of a classical black hole carrying temperature β, βΩ R = −2πi, charges ⃗ Q and SU (2) L charge J 3L , and are not a priori the same as M BP S and S BP S which are properties of a zero temperature black hole.
It should be understood that the equality in (3.12) holds for the Bekenstein-Hawking term and the term proportional to ln λ.In particular in five dimensions we also expect terms linear in λ that will not be kept track of.Even though these terms dominate over ln λ for large λ, they are sensitive to the details of the UV completion, e.g. higher derivative terms in the theory, and are not computable just from the low energy data.We can get rid of such polynomial terms by taking sufficient number of derivatives with respect to λ, so that at the end the dominant term comes from the coefficient of the ln λ term.
It is useful to compare (3.12) with similar result in D = 4.For this we can examine the result of the integrals in (3.9) for general D, and modify (3.12) accordingly.This would give, Here J 0 stands for whatever plays the role of J 3R in a general dimension, e.g. in four dimensions where the rotation group is SU (2), we take J 0 to be J 3 .n C is the dimension of the Cartan subalgebra of the rotation group, and the (n C − 1) factor gives the dimension of the analog of Ω L in general dimension.In D = 4 we have n C = 1 and hence all the logarithmic corrections in the second line of (3.13) vanish, leaving behind δ ln Z as the only source of logarithmic corrections.Therefore the logarithmic terms coming from change of ensemble, as discussed in this section, are absent in four dimensions.

Classical result
In this section we shall explore the consequences of the classical result S BP S for S BP S given by the first line of (3.12): In this equation M denotes the mass of the black hole solution corresponding to the saddle point that contributes to the index.On physical grounds we expect this to be equal to the BPS mass M BP S : However, this equality is in no way obvious since a priori M depends on β which can be chosen arbitrarily.If (4.2) holds then (4.1) may be expressed as which is also not an obvious relation from the point of view of black hole solutions since it relates the entropy of an extremal black hole to that of a non-extremal black hole.
We shall test both (4.2) and (4.3) for a class of three charge five dimensional black holes constructed in [13,14] (see also [19] for earlier construction for special charges).The solution has been reviewed in appendix A where we also examine the consequence of the βΩ R = −2πi equation.Here we just quote some of the relevant results from that appendix.For BPS black holes we have [20] and where Q (i) for 1 ≤ i ≤ 3 are the three U(1) charges carried by the black hole, all taken to be positive.Note that the number of gauge fields n V may still be arbitrary, but for the particular solution under consideration, the black hole does not carry charges under these other U(1) gauge fields.For the finite temperature solution with βΩ R = −2πi, we find in (A.13), This is the same as (4.4), verifying (4.2).On the other hand (A.14) shows that the entropy of this black hole is given by This, together with (4.5), confirms (4.3).Finally, (A.16) shows that J 3R and the inverse temperature β are related as This shows that β remains finite as long as J 3R is finite.Furthermore (4.8) is consistent with the scaling relation (2.3).We recover the extremal black hole by taking J 3R → 0 limit.

Logarithmic correction to the index
Using (4.1) we can express (3.12) as, Our goal in this section is to compute δ ln Z by computing the logarithmic contribution to ln Z from path integral over various fields following the procedure described in [15].As argued in [15], only one loop contribution from the massless fields are relevant for this.
Since the metric carries an overall factor of λ 2 , it follows that the eigenvalues of the kinetic operator for the bosons, involving two derivatives, scale as λ −2 and the eigenvalues of the kinetic operator for the fermions, involving single derivatives, scale as λ −1 .Therefore integration over each bosonic mode produces a factor of λ and integration over each fermionic mode produces a factor of λ −1/2 .The number of modes, although infinite, can be regulated using heat kernel and we get a finite power of λ from integration over these modes.However, in odd dimensions this power vanishes!Thus it would seem that ln Z does not receive corrections of order ln λ.
However the analysis described above ignores the presence of zero eigenvalues of the kinetic operator.If such zero modes are present, their effect needs to be taken into account.There are two effects: first for each bosonic zero mode we must remove a factor of λ from Z and for each fermionic zero mode we must remove a factor of λ −1/2 from Z since in the expression based on the heat kernel these modes are treated as if they contribute in the same way as the non-zero modes.Second, we need to explicitly evaluate the factors of λ that may arise from integration over the zero modes.We shall now describe how this is done for different zero modes.

Translation zero modes:
The Lorentzian signature black hole in five dimensions will have four translational zero modes.However for Euclidean signature black hole carrying angular momenta, the azimuthal angles conjugate to J 3L and J 3R are shifted by −iβΩ L and 2π respectively as we go around the Euclidean time circle.Therefore only those translational zero modes that remain invariant under this twist will be genuine zero modes.The number of such zero modes will depend on Ω L (in particular whether or not it vanishes), but it is the same number n T that determined the dimension of the momentum vector in our earlier analysis.We shall now compute the effect of integration over these translation zero modes.
Let h µν denote the fluctuation mode of the graviton around the Euclidean black hole solution.We take the path integral measure over h µν to be d(λ α h µν ) where α is determined by the condition: g µν being the background metric.We shall work in the coordinate system in which g µν 's are given by λ independent functions multiplied by λ 2 .Since g µν ∼ λ 2 , the coefficient of the h µν h ρσ term in the exponent is proportional to λ.This determines α to be 1/2.Therefore the integration measure over h µν is d(λ 1/2 h µν ).
Now consider the deformation of the metric generated by translation of the solution in some particular direction, and label the parameter by c.Then the zero mode deformation is generated by a diffeomorphism x µ → x µ + cf µ (x) where f µ (x)'s will be taken to be λ independent functions which approach some constant vector at infinity lying along the direction of translation.Thus the translational zero mode deformations of the metric have the form where the λ2 factor comes from the lowering of the index of f µ by the metric g µν .This gives d(λ 1/2 h µν ) ∼ λ 5/2 dc . ( Now, if we confine the black hole inside a box of physical size L as before, then the range of c is of order L/λ due to the λ 2 factor in the metric.This gives the result of a translational zero mode integral to be λ 5/2 × L/λ ∼ Lλ the symmetry generators that generate the zero modes do not commute with e −βΩ L J 3L .However as discussed in [12] in the context of four dimensional black holes, for the special choice βΩ R = −2πi, the twist in the azimuthal angle conjugate to J 3R becomes 2π and now the two rotation generators in SU (2) R are invariant under this twist.The contribution from these modes can be analyzed in the same way as for translational modes, with the only difference that the range of integration over the diffeomorphism parameters (analog of c for the translation modes) is a constant instead of order L/λ.
Therefore compared to the translational zero mode, the rotational zero modes have an additive contribution of ln(λ/L) to ln Z per zero mode.Since (5.6) gives a contribution of ln(Lλ 1/2 ) per translation zero mode, we see that we have a net extra contribution of 3. Since there are 2n broken supersymmetries, we also have 2n gravitino zero modes.To find their contribution we proceed as in [12].If we take the integration measure over the gravitino fields ψ µ to be d(λ γ ψ µ ) then we require d(λ γ ψ µ ) exp − d5 x det gg µν ψµ ψ ν = 1 . (5.8) Since g µν ∼ λ 2 , the coefficient of the ψµ ψ ν term in the exponent is of order λ 3 .Therefore we must choose γ = 3/2.On the other hand, the action for the gravitino field takes the form where E ρ a is the inverse vierbein.Therefore J 3R , constructed from the action using Noether prescription, also carries a factor of λ 2 and the part of J 3R involving the fermion zero modes ψ p has the form λ 2 c pq ψ p ψ q for some λ-independent constants c pq .The integration over the gravitino zero modes now takes the form (5.10) We need to multiply this by λ 1/2 for each of the 2n zero modes since the heat kernel counts a factor of λ −1/2 for each zero mode.Thus cancels the λ −n factor in (5.10).Therefore the net extra logarithmic correction to ln Z from the gravitino zero mode integration vanishes. 5  Combining these results we get the logarithmic correction to ln Z: Substituting this into (5.1)we get This is in agreement with the microscopic results as well as the result of near horizon analysis described in [3].
N BP S = e S BP S computed above counts states with fixed value of J 3L but all ⃗ J 2 L .The microscopic analysis also gives this directly [3].For J 3L = 0, [3] considered another quantity that counts all BPS states with fixed charge and ⃗ J 2 L = 0, i.e.only SU (2) L singlet states, since the macroscopic analysis based on near horizon AdS 2 × S 2 geometry gave this result directly.
We do not consider it here since in our analysis the ensemble with fixed J 3L but all ⃗ J 2 L is the natural ensemble in which the macroscopic results are found even for vanishing J 3L eigenvalue.However, it is possible to extract from our result the result for fixed charge and ⃗ J 2 L = 0 following the analysis described in [3,15].

A Classical black holes
The goal of this appendix is to verify the relation between BPS and non-BPS entropy and mass given in (4.2) and (4.3) for classical black hole solutions.
We use the solution given in [13], labelled by six parameters m, δ e3 , δ e1 , δ e2 , l 1 and l 2 .In G N = π/4 units, the metric is given by From this we see that the metric expressed in the ρ, t, θ, ϕ, ψ coordinate system is finite in the m → 0 limit.We should note however that in this limit (r 2 + + r 2 − ) diverges and hence ρ 2 and r 2 are related by an infinite shift.Therefore the metric expressed in the r, t, θ, ϕ, ψ coordinate system would not be finite.In particular the horizon at r = r + will be at infinite value of r.
Even though we have written down the Lorentzian version of the black hole metric, it is the Euclidean version of this solution that provides the saddle point for the gravitational path integral for the index.It should be possible to check that this metric (and other fields whose form can be found in [13]) admit Killing spinors and therefore describe a supersymmetric configuration.We have not checked this.However the fact that the solution saturates the BPS mass bound is a strong indication that this is indeed the case.
A Mathematica notebook containing various computations in this appendix is being submitted to the arXiv with this paper.
Acknowledgement: C.C. would like to acknowledge the organizers of the ST4 workshop and IIT Mandi for hospitality during the course of the work.Research at ICTS-TIFR is supported by the Department of Atomic Energy Government of India, under Project Identification No. RTI4001.The work of A.S. is supported by ICTS-Infosys Madhava Chair Professorship and the J. C. Bose fellowship of the Department of Science and Technology.
(2)m n T translation zero modes.However, as pointed out earlier, the heat kernel includes the contribution from each bosonic zero mode to ln Z as ln λ.Removing this contribution, we see that the net extra contribution to ln Z that we have from n T Since the black hole carries third components of angular momenta under SU (2) L and SU (2) R , we can rotate the Lorentzian solution about the 1 and 2 axes of each SU(2)groups to generate new solutions.This gives four rotational zero modes.However, in the Euclidean theory the azimuthal angles are twisted by −iβΩ L and −iβΩ R = 2π as we go around the time circle, and only those modes that remain invariant under this twist are genuine zero modes.This eliminates the rotational zero modes of SU (2) L since