A The Discrete Modes for the Graviton

Logarithmic Corrections to Extremal Black Hole Entropy in N=2,4 and 8 Supergravity

Abstract:

We compute the logarithmic correction to black hole entropy about exponentially suppressed saddle points of the Quantum Entropy Function corresponding to orbifolds of the near horizon geometry of the extremal black hole under study. By carefully accounting for zero mode contributions we show that the logarithmic contributions for quarter–BPS black holes in supergravity and one–eighth BPS black holes in supergravity perfectly match with the prediction from the microstate counting. We also find that the logarithmic contribution for half–BPS black holes in supergravity depends non-trivially on the orbifold. Our analysis draws heavily on the results we had previously obtained for heat kernel coefficients on orbifolds of spheres and hyperboloids in arXiv:1311.6286 and we also propose a generalization of the Plancherel formula to orbifolds of hyperboloids to an expression involving the Harish-Chandra character of , a result which is of possible mathematical interest.

Quantum Gravity, Black Holes in String Theory, AdS–CFT Correspondence
4

1 Introduction

Since the work of Bekenstein and Hawking, it has been known that black holes have entropy in a quantum theory of gravity. In particular, the entropy of a black hole is determined in terms of the area of its event horizon

 SBH=AH4GN, (1.1)

where we have set all fundamental constants except the Newton’s constant to one. However, it is expected that this formula will receive corrections in a complete theory of quantum gravity from two sources. Firstly, there may be classical corrections from higher derivative terms and secondly, there would be further quantum corrections to this formula, given that it has been obtained in the semi-classical approximation. Incorporation of higher-derivative corrections is achieved by means of the Wald formula [1], which is technically challenging but conceptually well-understood. The incorporation of quantum corrections is a more formidable task. In this situation, it is advisable to seek simpler settings in which the problem may potentially be solved as explicitly as possible, with the expectation that the lessons thus learnt would carry over to the more general cases.

Extremal black holes are an ideal setting to carry out this program. For one, since the near–horizon geometry of an extremal black hole always contains an factor [2, 3], their classical entropy formula already admits important simplifications, and can be computed by solving an algebraic set of equations [4, 5], a remarkable simplification over the general procedure for computing the Wald entropy. Remarkably, an expression for the full quantum entropy of these black holes has been proposed using the AdS/CFT correspondence, again exploiting the presence of the factor in the near–horizon geometry. In particular, the full quantum degeneracy associated with the event horizon for extremal black holes carrying charges is proposed to be given by the string path integral [6, 7]

where the subscript ‘finite’ means that the volume divergence of the path integral due to the factor has been regulated in accordance with the general principles of the AdS/CFT correspondence. We refer the reader to [8, 9, 10, 11, 12, 13, 14] for additional work on this proposal and the lectures [15] for a review.

The path integral receives contributions from all field configurations which asymptote to the near horizon geometry of the extremal black hole as is usual in AdS/CFT. This will be reviewed below in more detail. Since the work of Strominger and Vafa [16], in many cases the exact formula for entropy of supersymmetric extremal black holes has been computed in and string theory, see for example [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 14]. One can therefore try to explicitly check the validity of the proposal of [6, 7] in this context. More ambitiously, one can try to use (1.2) to compute the quantum entropy of black holes where the microscopic string answer is so far not available. This would be a non-trivial prediction arising from this proposal. Given this motivation, one would be interested in evaluating the path integral (1.2) either exactly using localisation [29, 30, 31, 32], or perturbatively in a loop expansion [33, 34, 35]. In this paper, we shall adopt the second approach.

In the semiclassical approximation, the dominant contribution to the path integral (1.2) comes from its saddle points. The leading saddle point corresponds to the attractor geometry itself, and it has been shown that the on-shell action evaluated on this saddle point correctly reproduces the Wald entropy of the black hole [6]. Further, loop corrections about this saddle point have also been computed for and supergravity to find that [33, 34]

 Misplaced & (1.3)

These expressions precisely match with the prediction from the microscopic answer from string theory [19, 22, 27, 14] and therefore provide a non-trivial quantum test of the validity of the quantum entropy function. Similar computations have also been carried out for BPS black holes to find [36]

 SBH=AH4GN+(2−χ24)lnAHGN+O(1), (1.4)

where is the Euler characterestic of the Calabi-Yau three-fold on which IIA string theory is compactified to obtain the string theory, and and are the number of hypermultiplets and vector multiplets present in the theory. In this case, the corresponding microscopic result is not known, though [36] observed an intriguing match with a version of the OSV formula computed in an a priori different scaling limit of black hole charges [37].

In this paper, we shall study loop corrections about saddle points of (1.2) which correspond to orbifolds of the near horizon geometry, where the quotient group is a subgroup of the isometry group . We shall define the action of the quotient group on the attractor geometry in Section 2. In the microscopic picture, these saddle points are expected to correspond to exponentially suppressed corrections to the asymptotic formula for the statistical degeneracy [11, 29, 14]. These corrections are certainly present in the microscopic formulae in and string theory [11, 14], and the leading contribution to has been computed from both the microscopic formula and the Quantum Entropy Function and matched with each other [29]. We shall focus on the next-to-leading contribution, the ‘log term’, in this paper. It is expected on general grounds [38] that this contribution arises only from one-loop contributions of massless fields, and only from the two-derivative sector of the theory. Therefore, though the path integral (1.2) is over all string fields, this particular contribution may be computed entirely from massless supergravity fields in the two-derivative approximation. We will find that the log term matches perfectly with the microscopic results from string theory in all cases where it is available.

Since the contribution to log terms appears only from the one-loop partition function it takes the form of certain determinants of operators of Laplace type, which may be computed efficiently using the heat kernel method [39]. This, however, requires us to compute the heat kernel of the kinetic operator corresponding to the graviphoton background over the orbifold of . This was initiated in [35] where an expression for the heat kernel of the Laplacian for vectors, scalars and spin– fields was obtained on these quotient spaces. Then, the heat kernel for a single vector multiplet was computed in the graviphoton background, and the log term was shown to vanish, in accordance with the expectation from the microscopic counting. This also required a careful counting of zero modes of the gauge field in the vector multiplet.

In this paper, we shall extend the analysis of [35] to include gravitons and spin– fields as well. This will enable us to compute the contribution of the gravity multiplet of and supergravity, as well as the gravity, vector and hypermultiplets of theories. As the computation is involved, we would like to present here a brief overview of the strategy and results obtained. Firstly, since the background we’re computing the determinants on is a quotient of the attractor geometry, the analysis of [33] and especially [34] will be of great utility. It was found there that incorporating the graviphoton flux into the heat kernel changes the eigenvalues of the kinetic operator from those of the Laplacian, but the degeneracies do not change5 The shift in eigenvalues could be systematically taken into account either exactly, or to the order relevant to extracting the log term. The analysis of this paper is exactly along the lines of [34]. In particular, the kinetic operator and the one-loop determinant being evaluated is exactly the same. The only modification is that we evaluate the determinant the subspace of fields which survive the orbifolding. The effect of the orbifolding is to change the degeneracies of the eigenvalues of the kinetic operator from those on the unquotiented space to new ones on . Again, using the same regularisation for the volume, the degeneracy can be made well-defined. We shall use the results of [35] to propose and motivate a very simple form for the degeneracy on these quotient spaces. In particular, the group theoretic form for the heat kernel will be especially useful for this purpose.

This leads to an important simplification in the computation. It was shown in [35] that the heat kernel on the orbifold background is a sum of times the heat kernel on the unorbifolded space plus contributions which come from the fixed points of the orbifold. Due to technical reasons which we shall outline, the contribution of the fixed points to logarithmic corrections is independent of the eigenvalues of the kinetic operator and depends only on the degeneracy of the eigenvalue. Hence, this contribution may be directly read off from the heat kernel of the Laplacian itself. An important distinction between the partition function on the unquotiented and the orbifold backgrounds is that of zero modes. Unlike the case of vector zero modes which were dealt with in [35], we find that the number of zero modes of the kinetic operator evaluated on gravitinos and gravitini changes non-trivially. On carefully analysing the contribution of these zero modes to the partition function, one obtains a match with the microscopic results for and string theory. We finally find the following results for the logarithmic corrections about the orbifold background ,

 ln(dhor,N)=AH4NGN+O(1),N=4,ln(dhor,N)=AH4NGN−4lnAHGN+O(1),N=8, (1.5)

where is the contribution to horizon degeneracy as computed via the path integral (1.2) evaluated about the saddle point corresponding to the orbifold of the attractor geometry of the black hole. These results are in perfect agreement with the microscopic computation [11, 14]. Additionally, the analysis may be extended to include half–BPS black holes in supergravity, where we find

 ln(dhor,N)=AH4NGN+(2−Nχ24)lnAHGN+O(1). (1.6)

This answer has a slightly unexpected feature. In contrast to the results for and supergravity, the logarithmic correction depends on the orbifold under consideration, and surprisingly the parameter appears in the numerator. However, it does reduce to the unquotiented answer on setting , which on the face of it seems non-trivial as the number of zero modes changes when the orbifold is imposed. These results are summarised in Table 1.

We close this section with a brief overview of the rest of the paper. Section 2 contains a review of the saddle points of the Quantum Entropy Function that we compute the log term about. Section 3 is a review of the heat kernel method as it applies to extraction of the log term. Section 4 reviews the group theoretic form for the heat kernel on and contains a proposed expression for the degeneracy of eigenvalues of the Laplacian on , a result of possible mathematical interest. Sections 5, 6, 7 put these results together, along with the zero mode contributions to compute log corrections in , , and supergravity results, which have been tabulated above in Table 1. We then conclude. The Appendices contain some results useful in our analysis.

2 Exponentially Suppressed Contributions to Statistical Entropy and Saddle Points of the Quantum Entropy Function

This section is a review of asymptotic formulae for black hole entropy as obtained by microstate counting in string theory, along with the proposal of [6, 7] for the macroscopic origin of this formula. For definiteness, we focus on microscropic results in string theory [14], the case has already been reviewed in [35]. As our focus is just on the overall structure of the asymptotic expansion for microscopic degeneracy, we shall be heuristic here and refer the reader to the original papers [11, 14] for details and explanations. We shall consider, following [14], the string theory obtained by compactifying Type IIB string theory on . In this case the asymptotic formula for degeneracy of a dyon carrying charges and takes the form [14]

 dmicro(Q,P)≃(−1)Q⋅P+1∑ss(−1)Δs2+1(Δs2)−2eπ√Δs≡∑sdmicro,s(Q,P), (2.7)

where is an integer which obeys certain conditions explicitly enumerated in [14], and the Cremmer-Julia invariant, , is given by [43, 44]

 Δ(Q,P)=Q2P2−(Q⋅P)2. (2.8)

The leading contribution to (2.7) comes from the term in the sum. This corresponds to the usual Bekenstein-Hawking entropy on the macroscopic side. The terms with higher values of correspond to exponentially suppressed corrections to this leading answer. We will now review the proposal for the origin of these exponentially suppressed corrections within the framework of the quantum entropy function.

The quantum entropy function (1.2) is defined as the string path integral, with a Wilson line insertion, in asymptotically spacetimes. We now briefly review some saddle points of this path integral. We refer the reader to [29] for more details and a systematic classification of these saddle points. We consider, following [14], Type IIB string theory compactified on and six dimensional geometries that are asymptotically . The simplest saddle point of the string path integral is the attractor geometry itself, which is supported by fluxes.

 ds2=a(dη2+sinh2ηdθ2)+a(dψ2+sin2ψdϕ2)+R2τ2|dx4+τdx5|2,GI=18π2[QIsinψdx5∧dψ∧dϕ+PIsinψdx4∧dψ∧dϕ+% dual],ViI=constant,VrI=% constant. (2.9)

Here is the and radius, and is a real number which does not scale with . is a complex number which also does not scale with , is a matrix–valued scalar field. We refer the reader to [14] for more details regarding this geometry. The construction of the saddle point in supergravity is entirely analogous, the only difference is that IIB string theory is now compactified on K3 instead of [29].

Now it may be shown that the leading contribution to the quantum entropy function from this saddle point is precisely where is the Wald entropy of the extremal black hole. Further, by expanding in fluctuations about this saddle point, the contribution to the quantum entropy function which scales as log() may be matched from the microscopic result [33, 34]. Our focus in this paper will be field configurations which correspond to orbifolds of the attractor geometry (2.9), where the quotient group acts via

 (θ,ϕ,x5)↦(θ+2πN,ϕ−2πN,x5+2πkN),k,N∈Z,gcd(k,N)=1. (2.10)

This is a freely acting orbifold of (2.9). It is possible to show by a change of coordinates that this orbifold obeys the same asymptotic boundary conditions as (2.9). It is therefore an admissible saddle point of the quantum entropy function. Additionally, the classical contribution to the quantum entropy function is . Further, though these orbifolds break supersymmetry partly, the integration over the zero modes corresponding to broken symmetries does not make the path integral vanish automatically [29]. Therefore, this orbifold is a promising candidate to correspond to the exponentially suppressed terms in (2.7) if and are identified to each other.

In this paper we shall perform a further test of this conjecture along the lines of [33] and [34]. In particular, note that for any value of ,

 Extra open brace or missing close brace (2.11)

In particular, we shall reproduce the second term in this expansion from quadratic fluctuations about the quotient (2.10) of (2.9). We carry out first the same computation in supergravity, where the term proportional to is known to vanish from the microscopic side, and then extend our computation to supergravity. We find that our macroscopic results match perfectly with the predications from the microscopic theory. We then carry out the corresponding computation for supergravity.

3 The Heat Kernel Method

We have reviewed how the saddle points of the Quantum Entropy Function corresponding to orbifolds of the black hole attractor geometry are promising candidates to reproduce the exponentially suppressed contributions to the statistical dyon degeneracy computed from and string theory in the large charge limit. In particular, they correctly reproduce the leading growth of the exponentially suppressed saddle points of the microscopic degeneracy formula. The goal of this paper is to show that the next to leading growth in charges is also correctly reproduced by these orbifolds. On the microscopic side, the subleading term corresponds to a contribution to which goes as . To compare this from the macroscopic side, we need to extract the term from the contribution of the saddle point (2.9), (2.10) to the partition function (1.2). For this purpose, we will think of the string theory that (2.9) is embedded in as a supergravity theory on coupled to an infinite number of massive fields which may be stringy or Kaluza-Klein modes. This leads to some remarkable simplifications on very general grounds [38]. In particular, it may be shown that the term which scales as in the partition function can arise only from one–loop fluctuations of massless fields in the four-dimensional geometry [33, 34, 38]. The heat kernel method is therefore ideally suited to analyse this problem, and we review it briefly here in order to set up notation and remind the reader of a few salient facts, mostly following the discussion in [45]. We refer the reader to [39] for a general overview of the heat kernel method, and to [45, 35] for more details regarding the discussion below. The one-loop partition function for a dimensional theory with overall length scale takes the form

 Z1−ℓ=(det′(M))−12⋅Zzero(a), (3.12)

where is an operator which in our case is a positive semi-definite operator whose eigenvalues scale as . The prime indicates that the determinant is evaluated on fields which are not zero modes of the operator , is the zero mode contribution to the partition function, and the argument in reminds us that the zero mode contribution also scales non-trivially with . Now the determinant of may be related to its trace via

 lndet′(M)=−∫∞0dttTr′(e−tM). (3.13)

Now the trace can be conveniently computed by means of the integrated heat kernel

 K(t)=Tr(e−tM)=∑n~mn∑m=1∫Mdd+1x√gψ∗n,m(x)ψn,me−ta2En, (3.14)

where are eigenfunctions of the operator belonging to the -fold degenerate eigenvalue . Note that this expression is perfectly well-defined for compact manifolds like , but is naively divergent for non-compact spaces like hyperboloids. However, it is by now well-understood how this divergence may be regulated in accordance with the general principles of the AdS/CFT correspondence and a sensible answer extracted for [33, 34, 46]. This involves putting a cutoff on the radial coordinate of global and extracting the term which is of order 1 in the large expansion. This procedure also extends nicely to quotients of hyperboloids both with and without fixed points [47, 35]. We also note that the heat kernel in (3.14) is evaluated over all the eigenfunctions of , including zero modes. To obtain the determinant over non-zero modes, one has to subtract out the zero mode contribution

 lndet′(M)=−∫∞0dtt(K(t)−n0M), (3.15)

where is the number of zero modes of the operator , defined via

 n0M=n0M∑m=1∫Mdd+1x√gψ∗0,m(x)ψ0,m,M|ψ0,m⟩=0. (3.16)

Again, this procedure is well-defined for compact manifolds and, using the regularisation procedure mentioned above, can be made well-defined for AdS and its quotients as well. Now the heat kernel has a small expansion of the form

 K(t)=1(4π)d+12∞∑n=0tn−d+12∫Mdd+1x√gan(x), (3.17)

where a few leading coefficients are known explicitly for the Laplacian on both smooth and conical spaces [39, 48]. Now we may extract the contribution from that scales as . It turns out that the term that scales as in this determinant receives contributions only from the term in the heat kernel expansion. This will be important in our subsequent analysis. That is

 Missing \left or extra \right (3.18)

where we have been careful to subtract out the contribution of the zero mode from the heat kernel. The dots denote extra terms which do not scale as .

We conclude the section with a discussion of the contribution of zero modes to the term to . This has been analysed on very general grounds in [38] and we quote the final result. The zero mode contribution from a field to the path integral can be shown to scale as [33, 34, 38]

 Zzero=(anϕβϕ)Z0, (3.19)

where does not scale with , is the number of zero modes of the operator evaluated over the field and is a number which can be computed for every field on a case–by–case basis. For a vector field [33], for spin– fields, and for the graviton [34]. In that case, using equations (3.18) and (3.19), we can show that

 ln(Z1−ℓ)=(K(0;t)−n0M)ln(a)+∑ϕn0ϕβϕln(a)+…, (3.20)

where is the term in the heat kernel expansion when the heat kernel is evaluated over all the eigenmodes of the operator , including zero modes. Again ’’ denote terms that do not scale as . Using the fact that , the total number of zero modes of , we find that the total contribution to the partition function proportional to when expanded about the given saddle point is

 ln(Z)log=⎛⎝K(0;t)+∑ϕn0ϕ(βϕ−1)⎞⎠ln(a). (3.21)

We also note here that the number of zero modes (3.16) can be read off from the coefficient of the term in the full heat kernel expansion (3.14).

4 The Heat Kernel on (AdS2⊗S2)/ZN

In this section we will review the results we previously obtained in [35] for the scalar and spin heat kernels in . It will turn out that this analysis will be enough to extract the log term corresponding to the determinant of the kinetic operator evaluated over the orbifold (2.10) of the graviphoton background (2.9). We will find an interesting simplification in the behaviour of the heat kernel which allows us to relate the heat kernel of the rediagonalised fields on this orbifold of the black hole near horizon geometry to the unorbifolded heat kernel plus some extra ‘conical’ terms which can be very simply computed. We now define the geometry and the orbifold projection that we shall impose. In global coordinates, the metric on is given by

 ds2=a2(dη2+sinhη2dθ2)+a2(dρ2+sin2ρdϕ2), (4.22)

where are coordinates on and are coordinates on . We shall compute the heat kernel on the space obtained by imposing the following orbifold on

 (θ,ϕ)↦(θ+2πN,ϕ−2πN). (4.23)

We refer to the space thus obtained as . This space is related by analytic continuation to the orbifold (4.23) of the space

 ds2=a21(dχ2+sinχ2dθ2)+a22(dρ2+sin2ρdϕ2), (4.24)

where we have to continue and to reach . Our strategy in this paper will be the same as that adopted in [33, 34, 35]. We will express vectors, gravitini and gravitons in a basis of states obtained by acting derivatives and gamma matrices on scalars and Dirac fermions, as applicable. This is outlined partially in Appendices A and B, and we further refer the reader to [33, 34] for complete details. For this reason, we shall now concentrate on the heat kernels over over scalars and Dirac fermions on , reviewing the expressions obtained in [35].

4.1 The Scalar Heat Kernel

Our strategy for computing the heat kernel of the Laplacian over scalar fields on was to evaluate the heat kernel on and analytically continue the result to . This was done both explicitly and by observing certain group theoretic properties of the heat kernel on these quotient spaces and proposing an appropriate analytic continuation. In this section we shall briefly review the latter approach and build slightly on it. We begin with the heat kernel of the scalar on , which is given by [35]

 Ks=1NKsS2⊗S2+1NN−1∑m=1∞∑ℓ,~ℓ=0χℓ,~ℓ(πmN)e−¯s1Eℓe−¯s2E~ℓ, (4.25)

where

 χℓ,~ℓ(πmN)=sin((2ℓ+1)πmN)sin(πmN)sin((2~ℓ+1)πmN)sin(πmN) (4.26)

is the Weyl character in the representation of , , , and and are the radii of the two spheres. is the heat kernel on the unquotiented space . Now a simplification that will be important for the rest of the analysis is the following. The sum over in (4.25) is finite as approach zero (i.e. approaches zero). In particular, the order term from the fixed points is given by

 1N∞∑m=1∞∑ℓ,~ℓ=0χℓ,~ℓ(πmN)=(N2−1)(N2+11)180N. (4.27)

We refer the reader to Appendix D for details. We therefore obtain

 Ks(S2⊗S2)/ZN(¯s1,¯s2)=1NKsS2⊗S2+(N2−1)(N2+11)180N+O(t). (4.28)

Importantly, we observe that the fixed point contributions to the heat kernel in (4.25), associated with the terms there, are completely finite in the goes to zero limit. All the behaviour comes from the global contribution. Therefore, firstly it is justified to do a naive power-series expansion in for the terms. Secondly the term in the expansion of these terms, which is the only term that contributes to the log term, is completely independent of the eigenvalues. Hence if scalars on mix among each other due to a background flux, leading to a shift in the eigenvalues, the contribution of the fixed point terms to the term of the heat kernel is always just

 Klogconical=mNN−1∑s=1∞∑ℓ,~ℓ=0χℓ,~ℓ(πsN)=mKlogconical,1scalar. (4.29)

All the effects of rediagonalisation of the quadratic operator manifest themselves in the ‘global’ term only. This is the main simplification we will use.

The second feature of the expression (4.25) which we would like to highlight is the following. The heat kernel is essentially the trace of the exponential of the Laplacian. Therefore, on general grounds it should be expressible as

 K(t)=Tr(e−tΔ)=∑ndne−tEn, (4.30)

where is the degeneracy of the eigenvalue , which we define via

 dn=∑m∫Mψ∗n,mψn,m, (4.31)

where denotes integration over the spacetime manifold that the fields live on, and span a normalized basis of states in the vector space consisting of eigenstates of the Laplacian belonging to the eigenvalue 6. Comparing (4.30) with (4.25), we find that the degeneracy of the eigenvalue is given by

 dℓ,~ℓ=1N∞∑ℓ,~ℓ=0(2ℓ+1)(2~ℓ+1)+1NN−1∑m=1∞∑ℓ,~ℓ=0χℓ,~ℓ(πmN). (4.32)

We now outline how the expressions (4.25) and (4.28) may be analytically continued to the AdS case. This is again a review of the arguments of [35]. Firstly we note that the term in the heat kernel expression is just times the heat kernel on the unquotiented space , therefore, it can be replaced by the scalar heat kernel , which was explicitly evaluated in [33] by relating it to the coincident heat kernel. In particular, it was found that [33]

Here the volume of is divergent as is non-compact. It may be regulated by putting a cutoff on the radial coordinate at a large value and retaining the term which is of order 1 in the large expansion. Physically, this is because the term divergent in the large limit represents a shift in the ground state energy, and the one-loop correction to black hole entropy is contained in the finite part [6, 7]. Then the regularised volume of is found to be [33]

and the (regulated) heat kernel becomes

 Missing or unrecognized delimiter for \right (4.35)

Comparing this to (4.30) we see that on regulating the volume divergence in the heat kernel, we have obtained a well defined measure of degeneracy of the eigenvalue of the scalar Laplacian on . It is given by

 ~dλℓ=−λtanh(πλ)(2ℓ+1), (4.36)

which is essentially the Plancherel measure. Now the rest of the terms in (4.28) may be continued to by continuing the radii to and multiplying by an overall half [35]. This is because under the orbifold that we’re imposing, has half the number of fixed points as . In that case, (4.28) gets analytically continued to [35]

Additionally, motivated by the analytic continuation carried out in [47] for thermal AdS and other freely acting orbifold groups, we also proposed that an equivalent way of carrying out the analytic continuation would be to replace the Weyl character corresponding to the being analytically continued to by the Harish-Chandra (global) character7 [49]

 χbλ(πsN)=cosh(π−2πsN)λcosh(πλ)sin(πsN), (4.38)

and multiplying the conical terms by half. We therefore obtain

where we further defined

 χbλℓ(πsN)=χbλ(πsN)χℓ(πsN). (4.40)

By regulating the volume divergence in the first term of (4.39) as was done for the unquotiented example, we find that a measure of the degeneracy of eigenvalues is given by

 dλℓ=−1Nλtanh(πλ)+12NN−1∑s=1χbλℓ(πsN). (4.41)

Therefore, as in the case of the compact manifold , the effect of the orbifold is to change the degeneracy of the eigenvalue from to . Both and are perfectly well-defined once the volume divergence has been regulated as above.

Also, from the above expressions, it is manifest that the conical terms in the heat kernel (4.39) are finite in the limit. Their contribution to the term, which is the only term in the heat kernel expansion relevant for computing the log term, in the heat kernel on is therefore insensitive to the precise form of the eigenvalues . This is what we will use when we consider the full kinetic operator of the gravity multiplet fields of supergravity and the fields in the graviphoton background.

4.2 The Fermion Heat Kernel

We now review the computation of the heat kernel of the Dirac operator on carried out in [35]. As the analysis is entirely analogous to the scalar, we will mostly enumerate main steps and the final results. The heat kernel for the Dirac operator is computed by evaluating the heat kernel of its square8. We will denote this heat kernel as . Now since the square of the eigenvalues of the Dirac operator on or is the sum of the squares of the eigenvalues of the Dirac operator on (or ) and , we thus obtain

and similarly for . The overall minus sign is on account of the Grassmann–odd nature of fermions. was explicitly evaluated on the quotient space in [35]. It is given by

 Kf(S2⊗S2)/ZN(a1,a2)=1NKfS2⊗S2−4NN−1∑s=1∞∑ℓ,ℓ′=0χℓ+12,ℓ′+12(πsN)e−tEℓ,ℓ′, (4.43)

where , and is the product of characters in the spin- and spin- representations. As for the scalar, we analytically continue this expression by replacing one of the Weyl characters by the Harish-Chandra character for and multiplying the conical terms by half. This character is given by

 Missing or unrecognized delimiter for \right (4.44)

Then the heat kernel for the fermion on is given by

We may further check that the above expression for reduces to

We thus see that the contribution from the fixed points is independent of the actual form of the eigenvalues as for the scalar case. Again, this leads us to define the following degeneracy of eigenvalues

 dλℓ=−1Nλcoth(πλ)+2NN−1∑s=1sinh(π−2πsN)λsinh(πλ)sin(πsN)χℓ+12(πsN) (4.47)

for the Dirac fermion. We note again that for Majorana fermions, all the expressions for and would need to be divided by half.

5 14–BPS Black Holes in N=4 Supergravity

Having reviewed the essential ingredients for the computations that we will perform, we now turn to the main task at hand: computations of logarithmic corrections to exponentially suppressed saddle points of the quantum entropy function. We do this first for the case of –BPS black holes in supergravity, for which we had studied the contribution of a vector multiplet in [35] and found that the answer vanishes. As there is an entire class of string theories containing an arbitrary number of vector multiplets for which the corresponding microscopic answer vanishes, it was natural to conclude that the contribution of a single vector multiplet to the log term should vanish. We found that this was indeed the case in the macroscopic computation. In this section we will compute the contribution of the gravity multiplet to the log correction to black hole entropy. We find that this answer also vanishes once the zero modes of the graviton and gravitini are carefully accounted for. This result completes the matching of the macroscopic and microscopic results for black holes at the level of log terms.

Before describing the actual computation, we begin with an overview of the strategy and simplifications that will help us solve the problem. These remarks also carry over to the and results that we obtain in later sections. Firstly, the spectrum of quadratic fluctuations about the quarter–BPS black hole attractor geometry was completely analysed in [33, 34]. Imposing the orbifold (4.23) on the spectrum projects onto a subset of the modes. In particular, it does not change the eigenvalues of the kinetic operator, it only changes the degeneracy of the eigenvalue. Further, the overall strategy of [33, 34] was to study the heat kernel of fields minimally coupled to background gravity, include a coupling to the graviphoton flux and see how the system changes. It was found that while the eigenvalues shifted from those of fields minimally coupled to gravity, the degeneracy of the eigenvalues did not change. In particular, the degrees of freedom in quadratic fluctuations always organised themselves into scalars and Dirac fermions with shifted eigenvalues. In our computations we will therefore adopt the overall strategy of [33, 34]. In particular, we will take the spectrum obtained on the full attractor geometry and impose the orbifold by replacing the unquotiented degeneracies by the degeneracy (4.41) for the scalar and the degeneracy (4.47) for the Dirac fermion. This leads to a further simplification in the problem. We had pointed out that the conical terms in the heat kernels (4.39) and (4.46) are finite in the limit. Therefore, their contribution to the term in the heat kernel expansion is independent of the eigenvalues and remains the same even when the fields are coupled to the graviphoton flux. All the effects of the flux manifest themselves in the term contained in the unquotiented part of the heat kernels. This term has already been computed in [34]. To account for the fixed point contributions, we need to compute the finite terms, which are independent of the eigenvalues, and sum them up. We do this in the sections below.

Additionally, there is an additional discrete mode contribution from vectors, gravitini and the graviton, which will be computed explicitly as was done for the vector discrete modes in [35]. The discrete modes also furnish zero modes to the kinetic operator, and these will also be separately accounted for. We finally find that the contribution to the log term from the gravity multiplet in supergravity also vanishes.

5.1 Integer Spin Fields

We begin with the contribution of the integer spin fields in the gravity multiplet to the one–loop determinant on with the graviphoton flux. The physical spectrum consists of one graviton, 6 gauge bosons , , and two scalars which correspond to the axion-dilaton field. In addition, there are 12 scalar ghosts that arise during gauge-fixing the gauge symmetries associated with and two vector ghosts which correspond to gauge fixing the group of linearised diffeomorphisms about the background. The vector modes also have a discrete series contribution which we shall compute. We begin with the contribution of 4 vector fields , which do not couple to the background graviphoton flux. Their contribution on was evaluated in [35]. It is given by

 Kv=K(vT,s)+K(vL,s)+K(vd,s)+K(s,vT)+K(s,vL), (5.48)

where denote the transverse, longitudinal and discrete modes, and is the contribution of the transverse vector along the direction and the scalar along the direction and so on. Now on the orbifolded space, we had evaluated in [35] to be

 K(vd,s)=−1NN−1∑s=0∞∑ℓ=0χℓ(πsN)e−ta2ℓ(ℓ+1). (5.49)

Note that is a zero mode of the kinetic operator. We will account for the presence of these modes in our final results. We now turn to the contribution of the longitudinal and transverse modes of the 4 vector fields. Though the answer is already known from [35] we will explicitly evaluate it again using the approach we have outlined above, i.e. by replacing the ‘unquotiented’ degeneracy by the degeneracy on . The contribution of the longitudinal and transverse modes of the 4 vector fields on is given by [33, 34]9

 KvU=4[−∫∞0dλλtanh(πλ)∞∑ℓ=0(2ℓ+1)e−tEλℓa2(4−2δℓ,0)], (5.50)

where . In what follows, we shall denote the heat kernel on the unquotiented space by , and the heat kernel on the orbifolded space as . Thus we see that the entire heat kernel arranges itself into the scalar heat kernel where some modes are missing as they don’t give rise to non-trivial gauge fields. The degeneracy of eigenvalues is given by

 dλℓ=−λtanh(πλ)(2ℓ+1). (5.51)

To obtain the expression for the heat kernel on the quotient space, we will replace this degeneracy with the new degeneracy (4.41). We then obtain

 ~KvN(t)=1NKvU(t)+2NN−1∑s=1∞∑ℓ=0∫∞0dλχbλ(πsN)χℓ(πsN)e−Eλℓa2(4−2δl,0)−4NN−1∑s=1∞∑ℓ=0χℓ(πsN)e−tℓ(ℓ+1)a2. (5.52)

As we are interested in the short- expansion of this heat kernel, in particular in the term, we will retain just the leading term in the short time expansion of the conical contributions. We therefore obtain

 KvN=1NKvU+2NN−1∑s=1∞∑ℓ=0∫∞0dλχbλ(πsN)χℓ(πsN)(4−2δl,0)−4NN−1∑s=1∞∑ℓ=0χℓ(πsN). (5.53)

We can explicitly do the sums and integrals using the results listed in Appendix D to obtain

 KvN=1N