Exact holography and black hole entropy in \mathcal{N}=8 and \mathcal{N}=4 string theory

Exact holography and black hole entropy in and string theory

Abstract:

We compute the exact entropy of one-eighth and one-quarter BPS black holes in and string theory respectively. This includes all the CHL models in both and compactifications. The main result is a measure for the finite dimensional integral that one obtains after localization of supergravity on . This measure is determined entirely by an anomaly in supersymmetric Chern-Simons theory on local and takes into account the contribution from all the supergravity multiplets. In Chern-Simons theory on compact manifolds, this is the anomaly that computes a certain one-loop dependence on the volume of the manifold. For one-eighth BPS black holes, our results are a first principles derivation of a measure proposed in arXiv:1111.1161, while in the case of one-quarter BPS black holes our result computes exactly all the perturbative or area corrections. Moreover, we argue that instantonic contributions can be incorporated and give evidence by computing the measure, which matches precisely the microscopics. Along with this, we find a unitary condition that truncates the answer to a finite sum of instantons in perfect agreement with a microscopic formula. Our results therefore solve a number of puzzles related to localization in supergravity and constitute a larger number of examples where holography can be shown to hold exactly.

holography, supergravity, Localization
1

1 Introduction

In recent years there has been remarkable progress in computing quantum corrections to the entropy of extremal black holes. By relating the exact entropy to the partition function of string fields in the near horizon geometry, the proposal of [1] has led to a novel insight into the holographic nature of these black holes. Recently, in the context of supersymmetric black holes, localization techniques in supergravity [2, 3, 4] have opened the possibility of computing the partition function exactly for any value of the charges, thus constituting a remarkable step forward in our understanding of quantum corrections in gravity and more generally of finite corrections in holography.

In this work, we continue the study of the partition function using localization techniques. Our goal is to derive the exact measure for the finite dimensional integral that one obtains using localization [3]- it has been a long-standing problem determining the exact contribution from all the supergravity multiplets. In particular we are interested in giving a fundamental principles derivation of the measure proposed in [2] for the case of one-eighth BPS states in string theory and extend it also to one-quarter BPS states in string theory, that is, IIB on and CHL models including orbifolds.

The reason to study one-eighth and one-quarter BPS black holes is twofold. On one hand, from microscopics, the spectrum of BPS states, which is known exactly for a large class of charge configurations, includes rich information about non-perturbative physics. On the other hand, on the black hole side, we can use an equality between index and degeneracy [5, 6] to extract the exact degeneracy of the black hole that we can use to guide and test the bulk computations.

From the microscopic study, we find that perturbative corrections to the area formula are captured exactly by a modified Bessel function of the first kind2, which in the case of one-eighth BPS states has the form [2]

(2)

while for one-quarter BPS states it is

(3)

with the constant for the CHL models on and respectively and is a certain positive integer that depends on the CHL orbifold. Without loss of generality, we have set , where and are the T-duality invariants. The validity of the expressions (2) and (3) holds up to exponentially suppressed terms for sufficiently large charges.

Formula (3) was derived in [6] for , that is, for the case of one-quarter BPS states in IIB on , which agrees with a Rademacher expansion [7]. In section §2.2 we present a novel way to compute the leading (3) and subleading Bessel behaviour (4) of the one-quarter BPS degeneracy for the CHL models on and .

To next leading order, the formula (3) is corrected by terms that are also of Bessel type. Developing on a formula for the asymptotic behaviour of the degeneracy of dyons, first proposed in [8], we find a series of subleading Bessels of the form

(4)

with

(5)

and are some constant coefficients. For and fixed , the leading term in this sum, that is, the term with and , reproduces the Bessel (3), whereas the subleading terms are suggestive of a Rademacher type of expansion [7]. In contrast, for the case these corrections are not present, with (2) being corrected at a much subleading order by terms of the form , with an integer greater than one. These corrections are also present in the case but they will not be an object of study.

In this work, the focus is on the exact computation of the Bessel functions (2) and (3) including the precise coefficients. The emphasis will be on the one-quarter BPS case but we review at the same time the case of one-eighth BPS states, in particular, the measure proposed in [2]. Furthermore, we argue that the subleading corrections in (4) may originate from instanton contributions to the path integral. Under certain assumptions, we determine not only the exact Bessel functions but we also compute an instanton corrected measure, which reproduces the exact coefficients in the microscopic answers (4).

The derivation for both one-eighth and one-quarter BPS black holes is similar. This follows from a truncation of supergravity, which allows us to see the one-eighth BPS black hole effectively as a one-quarter BPS black hole of supergravity. The truncation consists in projecting onto even states in which we set the RR and RNS fields to zero, with the world-sheet fermion number 3. On the microscopic side, this is also consistent as the counting is valid only for charges vectors that are purely NSNS or that can be brought to such a configuration by a duality transformation [9]. Equivalently the counting is performed in a region of the moduli space invariant under a right subalgebra, the one that is invariant; the one-eighth BPS states are effectively one-quarter BPS states of that subalgebra. For this reason, we can treat both supersymmetric examples in a similar way with the difference that in the one-eighth BPS case we need to take into account the contribution from the odd fields, that is, the fields which are odd under .

To carry out this task, our starting point is a formula for that one obtains using localization in supergravity [10, 3]. It was argued in [3] that the path integral of off-shell supergravity on reduces to the finite dimensional integral

(6)

where are respectively the electric and magnetic charges of the black hole. Here is the prepotential and is the number of vector-multiplets. The variables parametrize a normalizable mode of the scalars that is left unfixed by the localization regulator. In a saddle point approximation of (6) one obtains precisely the area formula for the entropy.

Despite this success, formula (6) lacks the correct measure that reproduces the microscopic answers (2) and (3). In a way, this is partially understood because the localization computation of [3] only takes into account the vector-multiplets. Instead, the full answer must take into account not only the contribution from the other matter multiplets but also the contribution from the gravity multiplet, which in this context can be problematic. That is, if we want to apply localization in supergravity we need to deal with local supersymmetry and thus with a proper gauge fixed path integral, which is by itself a very difficult problem. Furthermore, the localization technique relies on a certain cohomological structure of the underlying space- equivariant cohomology to be more precise, and hence it is not clear how this can be translated to gravity, which can raise questions of background independence. Despite these issues, the problem of computing the localization one-loop determinants for the matter multiplets has been addressed recently in [11, 12] using off-shell supergravity on . Here we give a qualitatively different approach based mostly on three dimensional supergravity but it includes the contribution from all the massless fields. In particular, we argue that the Kahler dependence of the one-loop determinants and the induced measure claimed in [11, 12] for the and examples do not hold. We explain, nevertheless, why the approach for followed in [11] correctly reproduces the exact microscopic answer and how it can be correctly extended for the case.

To achieve our goal, we use well-known properties of and its relation to holography to derive the measure for the integral (6). Partially justified by the computation of [13] which fixes the background to be exactly , our derivation is based on the assumption that the full path integral has the form (6), that is, the contribution from other multiplets enters only through the measure. After all, the integral (6) reproduces correctly the area formula in all known examples and so it is not expected that other fields contribute already at the exponential level.

The key idea is as follows. We map the problem of computing the measure to a certain anomaly in the path integral of super Chern-Simons theory on . From a bulk point of view, this anomaly is related to a dependence on a metric choice for the Chern-Simons path integral. Using holography, we can relate this dependence to the modular weight of the dual partition function- for the supersymmetric black holes this partition function is an elliptic genus. It is well known that in general the partition function on the torus is not invariant under global diffeomorphisms but transforms covariantly with a certain weight under modular transformations. Equivalently, we can say that the partition function is anomalous under diffeomorphisms. In turn, the low/high temperature modular transformation can be used to show that the asymptotic growth of the Fourier coefficients of the partition function [14], and thus the partition function, are of Bessel type. Therefore, by understanding the anomalous modular transformation in the bulk we have immediate access to the structure of perturbative corrections to the entropy, which are determined by the leading Bessel in (3). Nevertheless, this is not the full answer. In going from the gravity picture to the Chern-Simons formulation we need to keep track of an anomalous field redefinition- it is anomalous due to zero modes. This contribution, in turn, can be identified with a certain degeneracy that accompanies the Bessel function, which, in the theory of Jacobi forms, can be identified with a “polar” coefficient.

To compute this anomaly we use three dimensional supergravity. By convenience, this can be written as a super Chern-Simons action based on the gauge group . On top of this, we will consider additional abelian Chern-Simons terms. We put this theory on , which is the same as the quotient of global by an additive group , and use microcanonical boundary conditions consistent with the path integral. We explore the orbifold construction to argue that the one-loop contribution in Chern-Simons theory must hold for any value of the charges. After all, this leads to a correction of the form to the effective action, with the order of the group, and therefore cannot be renormalized by a local counterterm 4. In particular we show that the one-loop approximation to the partition function has the form

(7)

where is a flat connection and is the Chern-Simons action on properly renormalized. In particular, we find that the one-loop correction is

(8)

where is the size of in the physical theory, and are respectively the and levels of the non-supersymmetric Chern-Simons terms, and is the abelian Chern-Simons level. By identifying the parameter with the variable in the integral (6), we argue that the component uniquely determines the measure in the integral (6).

In essence, the main result is an exact formula for , which in the case takes the form

(9)

with

(10)

where is the tree level prepotential, is the contribution from the odd fields and is the number of vector-multiplets of the truncation. This formula correctly reproduces the Bessel answer (2). Furthermore, it gives a fundamental principles derivation of the measure proposed in [2], as we wanted to show.

In the case, the partition function has a similar expression except that the contribution from the odd fields is absent and the prepotential is modified by instanton corrections. We compute the exact zero instanton path integral,

(11)

with

(12)

Here is the zero instanton prepotential , with taking values for the and CHL models respectively, and is the number of vectormultiplets. Note that for large the zero instanton measure (12) reproduces the one-eighth BPS measure (10). This is as expected since the measure comes entirely from the supergravity multiplet and therefore for large charges it should become the same in all examples.

The formula (11) leads precisely to the microscopic answer (3), including the precise coefficients that multiply the Bessel function. Furthermore, we argue that corrections due to instantons can be incorporated by integrals similar to (11) and we compute the exact measure in agreement with microscopics. The idea relies on the observation that the integral (6) with the instanton corrected prepotential suggests a similar Chern-Simons computation but with renormalized levels. From this, it follows an unitarity condition that truncates the instanton sum and leads to the same tail of Bessel functions (4) that we find from microscopics.

The plan of the paper is as follows. In section §2 we study the exact microscopic answers for both one-eighth and one-quarter BPS states. We derive the exact Bessel function that captures all perturbative corrections to the area formula and we discuss the role of the subleading contributions. In section §3 we first describe the on-shell background and then we review the application of localization techniques in the computation of the path integral. Finally, in section §4 we describe the computation of the measure. We divide this section into three main parts. First, we consider the one-eighth BPS case and the contribution coming from the odd fields and then we determine both the vector and supergravity multiplet measures using the Chern-Simons formulation. We conclude with a discussion about open problems and other future directions.

2 Microscopic degeneracy

In the following sections we describe the microscopic partition functions that capture the spectrum of one-eighth and one-quarter BPS states. We are mainly interested in the behaviour of the degeneracy for sufficiently large charges. For the case we review a formula derived originally in [2]. In the case, we present a novel way to compute the leading Bessel function and subleading corrections by developing on the formula for the asymptotic degeneracy of dyons [8, 15].

2.1 string theory and one-eighth BPS states

We consider IIB string theory compactified on . An important difference with the case is that here we consider only a subspace of all the one-eighth BPS configurations. In particular, we consider those that carry only NS-NS charges or that can be mapped to this case after an U-duality transformation [16]. The microscopic formula that we present is valid in a region of moduli space where the RR moduli are turned off. Effectively we are counting one-quarter BPS states of a right subalgebra of the supersymmetry algebra 5. In this region of the moduli space, the U-duality group is broken to .

The one-eighth BPS configuration that was considered in [17, 9] consists of a D5-brane wrapped on , D1-branes wrapped along , a Kaluza-Klein monopole associated with the circle , units of momentum along the circle and units of momentum along the circle . This configuration can be mapped to a purely NS-NS configuration in IIA after a set of U-duality transformations as described in [16].

The index that captures one-eighth BPS states has the expression

(13)

with the Fourier coefficients of the Jacobi form 6

(14)

Here with , and are the T-duality invariants.

The coefficients admit an exact Rademacher expansion [2]. This is an exact formula for the Fourier coefficients of Jacobi forms of non-positive weight. In essence, it consists of an infinite but convergent sum of Bessel functions. In this case, the leading behaviour is controlled by

(15)

where the refer to terms that are exponentially supressed. Furthermore, it is convenient to write the integral in form

(16)
7

where we defined

(17)

The contour takes over the imaginary axis and over the axis with . This form of the integral will be useful later on, as a way to physically check the bulk computation of the answer.

2.2 string theory and one-quarter BPS states

In this section we consider one-quarter BPS states in CHL compactifications. These are particular orbifolds of IIB string theory on either or . We present a summarized description of the spectrum and BPS-state counting for both cases. The discussion presented is completely systematic in . This is advantageous for a comparison with the gravitational computation and it will help us highlight the key points of our derivation. For a more detailed description of CHL compactifications and BPS-state counting we point the reader to [18] and references therein.

One of the goals of this section is to rewrite the microscopic degeneracies in a manner suitable for a gravitational comparison. We will find that the microscopic answer can be written as a finite sum of Bessel functions, up to much subleading terms.

We consider IIB string theory on , with modded out by a symmetry group. The orbifold identification involves a shift along the circle and an order transformation on . The element commutes with the supersymmetry generators and therefore the orbifold preserves all the supersymmetry of the parent theory. By convention, we take the radius of the circle to have size in the parent theory. Here runs over in the case and in the case. Under the orbifold only a subgroup of the U-duality group survives, with the total number of gauge fields. The integer is given by

(18)
(19)

Let us consider a configuration with a single D5-brane wrapping , D1-branes wrapping , a single Kaluza-Klein monopole associated with the circle , units of momentum along and units of momentum along . At low energies this system is described by a two dimensional SCFT on . At the orbifold point in the moduli space, this theory is described by a symmetric product sigma model and we can compute the supersymmetric index that counts one-quarter BPS states. The index, which we denote by , has the form

(20)

where is a three dimensional contour in the complexified space with

(21)
(22)

and , and are the T-duality invariants. The function is a modular form. In particular, for it becomes the Igusa cusp form: the unique weight ten Siegel modular form, while for other we obtain Siegel modular forms of congruence subgroups.

We now study the degeneracy in the regime of large charges. In a saddle point approximation of the integral (20) we deform the contour and pick poles of . The leading contribution comes from the residue at a quadratic divisor of and has final expression [8, 19, 15, 18],

(23)

with and . The function is determined by the pole structure of . In the case of it is given by

(24)

while for it has the form

(25)

with given as in (18).

The formula (23) is not in a form that is suitable for a comparison with the localization computation. The reason is that the measure in (23) depends on the electric charges, while from a gravitational point of view they appear only at level of the Wilson lines. Besides, the contour in (23) has to be chosen appropriately. The only requirement at this point is that it passes near the leading physical saddle 8. We show that there is a choice for which we recover not only the leading Bessel function (3) [6] but also an additional tail of subleading Bessel type corrections. Namely, we choose a contour with complex defined as

(26)

Here is small but positive (we will make precise what we mean by small in due course). This contour ensures that we always have and positive- here and are not necessarily complex conjugate. This in turn leads to the exact Bessel function determined in [7] for the leading asymptotics of one-quarter BPS states in IIB on .

We proceed with an integration by parts. First we rewrite the expression (23) in the convenient way

(27)

with

(28)

The exponential is just the entropy function of Sen (65)

(29)

Using the identity

(30)

we can write the measure in (27) as

(31)

which leads to

(32)

The first two terms on the R.H.S. can be written as a total derivative

(33)

which vanishes for the contour (26). Hence, the final expression for the degeneracy is

(34)

with

(35)

Note the similarities between the one-quarter and the one-eighth BPS formulas, respectively (34) and (16). In particular, if we neglect the factor of in the one-eighth BPS formula (16), then both integrands have the form of the exponential of the entropy function times the quantum corrected Kähler potential [19] and a factor of , where is the total number of vector fields of the supergravity (truncation in the one-eigth BPS case).

We can proceed further and expand the modular functions in as a Fourier series in powers of and . Note that we have always for the contour (26) and thus we expand as

(36)

with the Fourier coefficients

(37)

Here for and orbifolds respectively.

Using the expansion (36) we can write formula (34) as the sum

(38)

We can massage further the exponential and write the degeneracy in the form

(39)

with the integral defined as

(40)

with

(41)

and

(42)

Lets focus on the integral for the moment. After the change of variables it becomes

(43)

where we have defined . For large values of , we can perform the integral by a saddle point approximation. In this case, given the contour , it is enough to take . Moreover, in this limit we can neglect the term in the square and thus we can write the integral (43) as

(44)

In computing this integral by saddle point approximation, there are two cases to consider. The most relevant is when

(45)

In this case, the saddle is inside the contour of integration and thus the saddle point approximation of (44) is simply

(46)

up to terms that are exponentially decaying. In the other case, that is, when we can use the asymptotics of the complemetanty error function

(47)

to estimate

(48)

Putting these results back in the integral (40) we find two types of asymptotic behaviour. In the first, when , we obtain

(49)

which is of Bessel type. However, if we can close the contour at infinity and since there is no pole inside we obtain zero. In turn, this leads to the condition

(50)

At this point it is convenient to impose the condition that such that the formula (49) is valid exactly for . We assume this for now on.

In the case when the asymptotics are governed instead by

(51)

which is still of Bessel type but has different index. By the same argument that gives the condition (50), this integral will be non-zero only when the term proportional to in the exponential is positive. In this case we have to truncate further to the terms with for , and for , with the condition that , for a maximum that solves the equation . Note that, when , that is, for orbifolds, the exponential is always negative and thus these terms are not present. Under these conditions we find

(52)

With this analysis we conclude that the integral has two kinds of behaviour. For it behaves as a modified Bessel function with index while for and the Bessel has index . As we will see later, the ones which are of interest for us are the Bessels with index . Moreover, these are the ones that do not depend on the regularization, that is, on the choice of for .

Given the conditions that and it is easy to show that is bounded by

(53)

and thus the sum over the terms with is finite. This leads to an answer for the degeneracy which is a finite sum of Bessel functions, that is,

(54)

with the Kronecker delta function. The terms of order are Bessels of index . Some of these can compete asymptotically with the other Bessels but since they depend explicitly on , parameter for which we have some freedom to choose, we assume that they are not relevant for the physics we want to study.

In this work we are mainly interested in the zero instanton term which is the leading term in the tail (2.2). We find

(55)

In particular, the two following examples are instructive. In the case of and we obtain the Bessel function

(56)

where we have taken large. This in perfect agreement with the results in [6, 7]. Also, in the case of and we obtain, in the same charge limit,

(57)

Up to a factor of , this is precisely the same Bessel function one obtains from the one-eighth BPS formula (16). This will be important to understand the role of the odd fields in a truncation of supergravity.

3 Black hole entropy and supersymmetric localization

In the first part of this section we review the quantum entropy formalism introduced by Sen [1]. Later we describe recent developments on the computation of the path integral using supersymmetric localization.

The quantum entropy function is a proposal based on the correspondence that relates the quantum degeneracy of an extremal black hole with charges to a string theory path integral on , that is,

(58)

The path integral is performed in euclidean 9 and the Wilson line insertions are required to assure that the correct boundary conditions are preserved. On we fix the electric fields and integrate instead over the chemical potentials which are the normalizable modes. We denote the Wilson line path integral (58) simply by .

This formalism reduces to Wald’s formalism in the limit of low curvatures or large horizon radius. That is, in a saddle point approximation we can write as the contribution of the on-shell configuration,

(59)

where is an IR cuttoff [5] and Ren denotes renormalization by appropriate boundary counter terms. The most R.H.S expression is the exponential of Wald’s entropy or equivalently Sen’s entropy function [20], which in this context is interpreted as the on-shell renormalized action on .

Additional quantum corrections can be systematically computed in perturbation theory. For example, in [21, 22, 23, 24] logarithmic corrections to the entropy are computed by integrating out the massless fields 10. These are found to be in agreement with the microscopic answers described in sections §2.1 and §2.2.

3.1 supergravity and attractor background

In this section we describe the on-shell attractor background of one-quarter BPS black holes in four dimensional supergravity. This includes the CHL compactifications in both and compactifications. Along with this, we describe how the one-eighth BPS black hole can be embedded in supergravity. Additional details can be found in [18] and references therein.

As explained in section §2.2, CHL compactifications are orbifolds of IIB string theory on , where is either or , that preserve supersymmetry. After the orbifold the U-duality group is reduced to an