Quantum entropy and exact 4D/5D connection

Quantum entropy and exact 4D/5D connection

João Gomes
DAMTP, Center for Mathematical Sciences
University of Cambridge, Wilberforce road
Cambridge, CB3 0WA, UK
Emails : jmg84 at cam.ac.uk
Abstract:

We consider the holographic correspondence near the horizon of rotating five-dimensional black holes preserving four supersymmetries in supergravity. The bulk partition function is given by a functional integral over string fields in and is related to the quantum entropy via the Sen’s proposal. Under certain assumptions we use the idea of equivariant localization to non-rigid backgrounds and show that the path integral of off-shell supergravity on the near horizon background, which is a circle fibration over , reduces to a finite dimensional integral over parameters , where is the number of vector multiplets of the theory while the mode corresponds to a normalizable fluctuation of the metric. The localization solutions, which rely only on off-shell supersymmetry, become after a field redefinition, the solutions found for localization of supergravity on . We compute the renormalized action on the localization locus and show that, in the absence of higher derivative corrections, it agrees with the four dimensional counterpart computed on . These results together with possible one-loop contributions can be used to establish an exact connection between five and four dimensional quantum entropies.

black holes, superstrings, holography
preprint:

1 Introduction

In string theory or in any consistent quantum theory of gravity we should be able to describe a black hole as an ensemble of quantum states. The statistical entropy of the black hole or simply quantum entropy is given by the Boltzmann formula

(1)

with the number of states with charge . In the thermodynamic limit or large charge regime the expression above is well approximated by the famous Bekenstein-Hawking area formula111In units where the Newton’s constant [1, 2, 3, 4, 5, 6]

(2)

which then gives the semiclassical, leading contribution to the black hole’s quantum entropy. Hawking’s formula is in a sense very general and universal and therefore it does not tell much about the microscopic details of the theory. On the other hand, for extremal black holes, which have an near horizon geometry, finite charge corrections to the area formula can be used to test the holographic correspondence [7] beyond the thermodynamic limit and infer details of the dual quantum theory. In this sense it is of great interest to compute finite charge corrections to the entropy and compare them for example with known contributions from BPS state counting.

Sen’s proposal [8, 9] relates the quantum entropy of an extremal black hole to a path integral of string fields over with some Wilson line insertions at the boundary. Via the correspondence, it counts the number of ground states of the dual conformal quantum mechanics in a particular charge sector . The entropy is then given by the statistical formula . By putting the boundary of at finite radius we generate an IR cuttoff [10] which can be used to extract relevant information via holographic renormalization. This definition then respects all the symmetries of the theory and reduces to the Wald formula in the corresponding limit of low curvatures or large horizon radius.

The quantum entropy function constitutes a powerful tool to compute finite charge corrections to the area law which can then be compared to microscopic calculations 222By microscopics we mean the dual conformal quantum mechanics theory. In some contexts we count the BPS states by looking at the supersymmetric states of a 2d SCFT, like the D1-D5 low energy effective string.. For many examples of supersymmetric black holes in both four dimensional string theories there are microscopic formulas for an indexed number333By indexed number of BPS states we mean a helicity trace index , where is the helicity quantum number and is the number of complex fermion zero modes of BPS states [11, 12, 13, 14, 15, 16, 17, 18, 19] valid in a large region of the charge configuration space. Using either the Cardy formula or an asymptotic expansion in the large charge limit [20, 21], the microscopic index agrees with the exponential of the Wald’s entropy. Additional subleading corrections can then be compared with those obtained via the quantum entropy framework. For instance one-loop determinants of fluctuations of massless string fields over the attractor background give logarithmic corrections to the area formula that are in perfect agreement with the microscopic answers [22, 23, 24] .

Despite all this success, the techniques involved are quite limited which makes the computation of further perturbative corrections an extremely difficult problem. However, for supersymmetric theories we hope to use localization to compute all of them exactly. At least for rigid supersymmetric theories the principle is quite simple. We deform the original action by adding a -exact term of the form , where stands for some supersymmetry of the theory and the functional is chosen such that . Then using the fact that both the action and the deformation are invariant it can be shown that the path integral does not depend on . So in the limit the path integral collapses onto the saddle points of the deformation and the semiclassical approximation becomes exact. This explains the concept of localization. This technique has been used extensively and with great success to compute exactly many observables in non-abelian gauge theories defined on a sphere [25, 26] and recently many other cousins of these spaces. Recently the same technique was applied with great success to supergravity on [27] in the context of black hole entropy counting. A spectacular simplification was observed in that only a particular mode of the scalar fields was allowed to fluctuate, with the other fields fixed to their attractor values. We say that the path integral localized over a finite dimensional subspace of the phase configuration space. The renormalized action444This is the action of string fields on after removing IR cuttoff dependent terms. has a very simple dependence on the prepotential of the theory and is a function of parameters which have to be integrated. More recently in [28] these results were applied in the case of four dimensional big black holes in toroidally compactified IIB string theory. The microscopic degeneracy, given as a fourier coefficient of a Jacobi form, can be rewritten in the Rademacher expansion and then compared with the gravity computation. The leading term of this expansion was reproduced exactly from these considerations. The non-perturbative corrections to this result, possibly coming from additional orbifolds, are more subleading, rendering the agreement between microscopics and macroscopics almost exact.

It would be interesting in the view of correspondence to test these ideas in other examples. The study of higher dimensional black holes in this context is of particular interest for two main reasons. First, there is an interesting connection relating the microscopic partition functions of four and five dimensional black holes called lift [29, 30, 31, 32]. It would be very important to understand this connection from a bulk point of view at the quantum level. For instance the microscopic partition functions of four and five dimensional black holes in toroidally compactified string theory are the same 555We are skipping issues related to hair contributions [33, 34].. Since the quantum entropies have to agree one expects the five dimensional theory to ”reduce” to four dimensions exactly. Secondly, we want to understand how localization works in the presence of gravity, that is, in a non-rigid background. Since the four and five dimensional answers are related, it is expected that some mode of the five dimensional metric is left unfixed. As a matter of fact the near horizon geometry of a supersymmetric five dimensional black hole has the form of a circle fibered over [35] (which we denote as ). The fiber, which carries angular momentum, gives rise to a gauge field after dimensional reduction. Rigid supersymmetric localization is quite well understood. However localization in non-rigid backgrounds constitutes a new challenge and an interesting problem from a technical point of view.

At the level of two derivative supergravity action, the Bekenstein-Hawking entropy of the five-dimensional BMPV black hole [36] equals that of the four-dimensional supersymmetric black hole after identifying the five-dimensional angular momentum with four-dimensional electric charge. For black holes in toroidally compactified string theory this equality should hold even at quantum level since the microscopic answers in [11] and [17] are the same666The microscopic answers are the same except for a sign function , with the angular momentum.. However in the case of black holes the lift is non-trivial and the equality of quantum entropies is no longer true already at two derivative level777The logarithmic corrections computed using the two derivative supergravity action are different in four and five dimensional theories. [37] but always in agreement with the microscopic answers. In the view of the existing results for four-dimensional black holes in string theory [28] we give support for an exact bulk derivation of this property. We will find out that the renormalized action of both the five and four dimensional theories are the same.

This work has therefore two fundamental purposes: to compute the partition function of supergravity on using localization techniques and to establish a quantum version of the lift from a bulk perspective. Instead of reducing the theory down to , as in other perturbative computations [22, 23, 24], we consider the five dimensional theory and apply localization to the off-shell theory. Our work focus on the perturbative part of this computation, that is, in finding the saddle points of the localization action. We compute the renormalized action on the localization solutions in the case when higher derivative corrections are absent, which is appropriate for five dimensional black holes.

The use of localization in supergravity on , even though not fully understood, has remarkable results. As found in [27], the scalar in the compensating vector multiplet of four dimensional off-shell supergravity is left unfixed by the localization equations. This means that from a five dimensional perspective we expect some mode of the metric to be left unfixed, namely the dilaton that measures the size of the fiber. In other words we need to consider localization on a non-rigid background. While it is straightforward to show localization in rigid supersymmetric theories this is not the case when the background itself is dynamical like in supergravity. The problem resides on the fact that it is very difficult to construct an exact deformation, if such an object exists, that is both gauge invariant and background independent. For rigid theories we usually pick a Killing spinor whose associated Killing vector generates a compact symmetry of the background. This is then enough to show exactness of the deformation. In general this choice of Killing spinor breaks the symmetries of the background and therefore it cannot be a diffeomorphic invariant deformation. We explain this in more detail later in section §3. However, in supergravity this only makes sense in regions of the configuration space where we can use a partially fixed background. In these regions we can use a partially fixed Killing spinor that stills generates a compact symmetry. For instance we will show that by fixing the four dimensional metric to be while leaving the fiber off-shell it is still possible to construct an exact deformation that can be used to localize the theory. Even though our understanding of localization in supergravity is only partial, our belief is that supergravity, at least on spaces, localizes, that is, there are just a finite number of modes that capture all quantum corrections. This is a strong statement but it is quite clear from the microscopic answers that something similar might be happening.

In this sense we adopt a different approach in this work. We start from an ansatz for the metric and do the same for the Killing spinor. To be able to use localization we need a fermionic symmetry that generates a compact bosonic symmetry. More specifically we need

(3)

where is the Lie derivative along a Killing vector and are gauge transformations. The operator has the name of twisted de Rham operator in differential geometry. As we will see later, in order for to act equivariantly, that is, in the sense of (3), we need to impose certain conditions on the fields.

In order to use localization we need an off-shell realization of supersymmetry in five dimensions. A beautiful construction is given by the five dimensional superconformal formalism developed recently in [38, 39, 40, 41]. Although our interest is on BPS black holes in theories, for which we have microscopic answers, we will use the formalism where these black holes can be embedded.

The localization solutions, presented in section §5, look very complicated from a five dimensional perspective. There are a great number of fields both from the weyl multiplet and vector multiplets left unfixed by the localization equations. The hypermultiplet fields however remain fixed to their background values. This is only an assumption since we do not have an off-shell representation of their supersymmetric variations. After a field redefinition, these solutions are recognized to be the solutions found in [27, 42] for localization of supergravity on . They are parametrized by parameters , with the number of vector multiplets, and label normalizable fluctuations of the four dimensional scalar fields. These four dimensional scalar fields result from a combination of the five dimensional scalars together with the fifth component of the gauge fields and a mode coming from the auxiliary antisymmetric field , in the Weyl multiplet, that we denote as , in the form

At asymptotic infinity is related to the five dimensional angular momentum . The reduced euclidean theory has R-symmetry which explains the use of paracomplex scalars 888Euclidean supersymmetry in four dimensions has R-symmetry [43]. The vector multiplet scalars are ”charged” under this non-compact R-symmetry.. The localization equations leave unfixed the ”real” part of the paracomplex scalar fields which in hyperbolic coordinates999In hyperbolic coordinates metric is written as . have the following spacetime dependence

where denotes the on-shell value. On the other hand the dilaton which measures the size of the five dimensional circle combines also with to give an additional paracomplex scalar

(4)

and gives an extra mode that has to be integrated out

with an arbitrary constant. As in [27] these modes can fluctuate if certain auxiliary fields are also allowed to fluctuate above their attractor values. The localization equations show that any dependence on the five dimensional coordinate drops out, rendering the reduction exact. However we need to be cautious about possible contributions from Kaluza Klein modes in the one-loop determinants but we do not consider this problem here.

Note that the localization analysis uses off-shell supersymmetry and therefore it is independent of the particular details of the action. As explained later in section §5.1, to guarantee that the supergravity action is invariant under the fermionic symmetry we need to add appropriate boundary terms, Wilson lines in this case. These boundary terms are important to guarantee a consistent variational principle. These Wilson lines are different from the electric Wilson lines used in [8, 9, 27] as they do not carry explicit information about the five dimensional charges as we could have expected. Nonetheless, after some algebra, the renormalized action shows dependence on the five dimensional charges in a way that is consistent with the four dimensional results of [27]. The final answer for the quantum degeneracy , in the absence of higher derivative terms, is a finite dimensional integral over variables,

(5)

where denotes an effective measure on the space of , and are the charges and angular momentum respectively, and the renormalized action has the form

(6)

where is a completely symmetric constant matrix. After a suitable analytic continuation of the renormalized action matches the four dimensional counterpart as expected from the lift. The measure should in principle be computed from the one-loop determinants. However since the localization equations are valid only in a region of the full configuration space we do not have access to all fluctuations orthogonal to the localization locus but only a subset.

The paper is organized as follows. In section §2 we review the quantum entropy function formalism. In section §3 we explain the technique of equivariant localization starting from finite dimensional integrals and then introducing the case of infinite dimensional integrals. In section §4 we review the five dimensional superconformal formalism. We introduce the supersymmetric variations of the various supermultiplets, the lagrangian and also the full BPS attractor equations. Finally in section §5 we do localization of the supergravity theory and compute the renormalized action on the localization locus.

2 Quantum entropy function and correspondence

The quantum entropy function [9, 8, 44, 10], based on the correspondence, is a proposal for the quantum entropy of an extremal black hole. By quantum entropy we mean a generalization of the Wald’s formula that captures the entropy of a black hole described as an ensemble of quantum states. At least for BPS black holes for which there are precise microscopic answers it is believed that such a formula might exist. We have to stress the fact that notwithstanding the microscopic answer being an index, it has been argued in [10, 20] that for black holes that preserve at least four supercharges index equals degeneracy for the near horizon degrees of freedom.

The quantum degeneracy , where labels the charges of the black hole, counts the number of ground states of the conformal quantum mechanics and is related, via correspondence, to the partition function of supergravity on with Wilson lines inserted at the boundary:

(7)

where the geometry has euclidean signature101010As usual we perform a Wick rotation . In other instances we learned to take such that the path integrand becomes with positive providing a convergent integral. Here however the euclidean action is already divergent due to the infinite volume of and it is the renormalized action that provides the correct damping exponential. In short, .

The Wilson line insertions can be understood from two different but equivalent ways. From an holographic point view, the electric part of the gauge fields carries a non-normalizable component at asymptotic infinity, that is, in coordinates where the boundary is at the gauge field goes as , and therefore via the usual bulk/boundary correspondence dictionary [45] these modes have to be fixed while the normalizable component, the chemical potentials, have to be integrated out. This is in contrast with higher dimensional examples like in . The microcanonical ensemble is natural from this point of view. But we can also see these Wilson lines as a requirement of a consistent formulation of the path integral. Without the Wilson lines the equations of motion for the gauge fields are not obeyed at the boundary because they carry a non-normalizable component 111111In other words the boundary terms that arise from varying the action do not vanish at asymptotic infinity. Much like the Gibbons-Hawking terms, the path integral on requires appropriate boundary terms, Wilson lines in this case, that restore the validity of the equations of motion throughout all the space. We develop this idea further in section §5.1.

This formalism surpasses in many ways other attempts to compute quantum corrections to the entropy. The success comes essentially from two basic facts. Firstly, there is a natural UV cutoff, the string scale . Secondly, it introduces via holographic renormalization an IR cutoff which is essential for extracting relevant information even at the classical level. Besides, this formalism respects all the symmetries of string theory and reduces to Wald’s formalism in the limit of low curvatures or large horizon. To see this consider the following simple example. The relevant near horizon data of an extremal black hole is given by the metric

(8)

with conformal boundary at , and gauge fields and scalar fields

(9)

respectively. The leading contribution to (7) comes from evaluating the action on the configuration described above. Since has infinite volume we introduce a cutoff at and discard terms that are linearly divergent, that is,

(10)

where Ren denotes renormalization by appropriate boundary counter terms that remove the dependence. The most RHS expression is just the exponential of the Wald’s entropy. If we want to compute quantum contributions we look at normalizable fluctuations order by order in perturbation theory. This is essentially the work done in [22, 23, 24, 37]. The authors consider the reduced theory on and look at normalizable fluctuations of the background (8) and (9). They compute, using the heat kernel method, one-loop determinants in the two derivative supergravity action. These terms give corrections of order to the entropy, where is the horizon area in appropriate units.

When performing localization we will consider the path integral defined on the five-dimensional space instead of reducing all the fields down to . This method is obviously favorable for a number of reasons. However in the context of localization it requires the addition of appropriate boundary terms. Some of them arise by demanding that the equations of motion be obeyed also at the boundary, as explained before, others are necessary to restore gauge invariance. As we will explain later in the section §3 these terms are necessary for invariance of the action under supersymmetry, an essential ingredient for using localization. This condition will allow us to define in five dimensions an entropy function à la Sen [46]. Current available attempts [47] circumvent this problem by reducing to four dimensions which is clearly unsatisfactory in the view of our main goal.

3 Localization Principle

For illustrative purposes consider the example of a finite dimensional integral over a compact manifold ,

(11)

If has a finite number of non-degenerate critical points , a saddle point approximation gives an asymptotic expansion in

(12)

where the coefficients can be computed in terms of .

For a certain class of integrals an extraordinary simplification occurs. If is a symplectic manifold and is the hamiltonian of an action on then the ”higher loop” corrections, that is, the terms vanish and the saddle point approximation becomes exact. This is the simplified version of the Duistermaat-Heckman theorem [48]. In such a case, the integral localizes exactly over the critical points of which are also the fixed points of the action on

(13)

where is the dimension of and is the vector associated with the action. The fact that the integral only depends on the neighborhood data of the fixed points is commonly referred as localization.

To better understand the mechanics of localization we need to study equivariant cohomology. This is particularly well understood for finite dimensional integrals. A good mathematical reference is [49] while [50] is more convenient for a physicist point of view.

Without entering in too many mathematical details we will explain briefly localization of finite dimensional integrals. The idea behind localization is that is possible to define on a manifold an operator which has the property that it squares to an isometry of the space. In other words

(14)

where is the Lie derivative. On the space of forms the operator , also called twisted de Rham differential, has the form

(15)

where is the de Rham differential and is the contraction operator by the vector . Since the vector generates an isometry of the manifold we have , that is, is a Killing vector. This operator then allows to define a cohomology on the space of forms which are left invariant under the isometry generated by , that is, on the space of forms for which . These forms are also called equivariant forms.

It turns out that the integral over of a closed form , that is, , localizes on the fixed points of action of . To show localization we consider the auxiliary integral parametrized by

(16)

with an equivariant differential form. Since both and are closed under we can show by integration by parts that

(17)

or in other words is an exact deformation. A clever choice for is to take

(18)

with a metric on the manifold . The property that ensures that is an equivariant form. That is a clever choice because the ”deformation” has a term which is positive everywhere on

(19)

This can be used to show that in the limit the integral collapses onto the fixed points of , rendering the saddle point approximation exact. This is the equivariant localization principle.

The same idea can be applied to infinite dimensional integrals. The idea is to extend the properties of the operator to the space of fields. Since it mixes forms of even and odd degrees it behaves much like a supercharge in supersymmetric field theories that sends bosonic to fermionic fields and vice-versa. The twisted de Rham operator becomes a functional and can be identified with the action of a real supersymmetric transformation while the analog of a closed equivariant form is given by a supersymmetric functional. By the same token we can deform the integral by an exact equivariant functional and show localization of the theory.

To make things simple consider the case of one-dimensional supersymmetric quantum mechanics on a circle with period [50].

There exists a supersymmetry that takes a boson to a fermion and vice-versa

(20)

with the coordinate on the circle. These transformations can be used to define the functional equivariant operator

(21)

with

(22)

It is an easy exercise to show that this operator squares to translations as expected from the supersymmetry algebra , with , the hamiltonian,

(23)

The space of equivariant functionals is determined by functionals that vanish under the action of . This immediately gives the condition

(24)

Since this should be valid for any and the condition is satisfied if we impose periodic conditions on both the scalars and fermions . In other words the space of equivariant functionals is the space of functionals with both and fields periodically indentified on the circle. The localization principle follows analogously. We deform the original integral by adding an exact deformation to the action

(25)

Using the fact that the deformed integrand is equivariantly closed we can show as in (17) that the integral

(26)

does not depend on the parameter and consequently the limit can be used to prove localization of the theory on the space of configurations for which

(27)

That is, the theory localizes on constant fields . Further corrections, which include the contribution from the Kaluza-Klein modes, are one-loop exact.

Without much effort the same idea can be applied to higher dimensional supersymmetric theories. In general, localization in rigid supersymmetric gauge theories is quite straightforward as long as there is a fermionic symmetry that squares to a compact Killing symmetry of the background. More generally there is an odd symmetry with the property that

(28)

where is a Killing vector and denotes a gauge transformation with parameter . With a set of fields that respect this algebra we can easily construct an exact deformation of the physical action. For this reason any deformation of the form with gauge invariant and , will be an exact deformation if the fields respect periodic boundary conditions along the compact direction. In other words

(29)

Pestun in his seminal work [25] gives a beautiful application of this formalism in the computation of Wilson loops in SYM defined on . In this case he uses a fermionic symmetry which is a combination of a conventional Q-supersymmetry and a special S-supersymmetry. This fermionic symmetry squares to an antiself-dual rotation of the sphere plus -symmetry and gauge transformations.

For non-rigid supersymmetric theories it is not known if the same idea can be applied. We do not know how to construct, if it exists, an exact deformation that is both gauge invariant and background independent. In general it is difficult to find an odd symmetry that satisfies the condition (28). The case is even worst when there is gravity. However in a certain region of the phase configuration space it is possible to realize linearly such an algebra. For instance the authors in [27] claim to have computed exactly the path integral of supergravity on . The results are quite astonishing. Assuming that the background remains fixed they localize the gauge theory sector and find that for each vector multiplet a normalizable fluctuation of the scalars is allowed if the corresponding auxiliary scalar also fluctuates. They have found that the theory localizes on the set of fluctuations of the form

(30)

with a scalar and the auxiliary scalar field, in the coordinates (8). Integration over the constants yields a finite dimensional integral which agrees with the microscopic predictions for BPS black holes in string theory [28].

Since in general the susy transformations of supergravity do not respect equivariant properties, the strategy that we pursue here is to find in which region of configuration space those properties are realized. This brings additional constraints on the fields. On this restricted subspace we can deform the path integral and show localization. We believe that in the full quantum gauge fixed theory such a restriction would follow naturally.

4 superconformal gravity and near horizon analysis

In this section we introduce the off-shell superconformal formalism for five dimensional supergravity. We present the various multiplets and respective supersymmetric transformations. We introduce the lagrangian with supersymmetric higher derivative corrections and present the BPS attractor equations for the near horizon geometry of the BMPV black hole.

4.1 Superconformal formalism

The superconformal calculus was originally constructed for supergravity in four dimensions [51, 52, 53] but only recently a formulation in five dimensions was developed [38, 39, 41, 40]. The idea is to construct a supersymmetric theory for the five dimensional conformal group by gauging the global generators and then imposing appropriate gauge fixing conditions. This is similar to the example of a scalar conformally coupled to the Einstein-Hilbert term. By gauge fixing the scalar to a constant we recover Poincaré gravity. One major distinction between four and five dimensional formulations is that while the first has R-symmetry, the five dimensional theory only has R-symmetry. This means, for instance, that the scalars in the vector multiplets are real.

In the following we give a summary of the content of the various supermultiplets, namely the Weyl multiplet, the vector multiplet, the linear multiplet and the hypermultiplet, and respective supertransformation rules. We follow closely the paper [54] where more details can be found.

We denote coordinate indices by greek letters , tangent space indices by roman letters and R-symmetry indices by .

Weyl multiplet:

the independent fields consist of the funfbein , the gravitino field , the dilatational gauge field , the R-symmetry gauge fields (anti-hermitian traceless matrix in the indices ), a real tensor field , a scalar and a spinor field . Both , , and are auxiliary fields. For the problem we want to solve we set and gauge the special conformal transformations parameters to zero. The conventional and special supersymmetry transformations, parametrized respectively by the spinors and , are as follows:

(31)

The derivatives are covariant derivatives.

Vector multiplet:

the vector multiplet consists of a real scalar , a gauge field , a triplet of auxiliary fields and a fermion field . The superconformal transformations are as follows:

(32)

with , and the supercovariant field strength is defined as,

(33)
Linear multiplet:

though they do not play any relevant role in our work we decided to include the supersymmetric transformations of the linear multiplet for congruence of the exposition. The linear multiplet consists of a triplet of scalars , a divergence-free vector , an auxiliary scalar and a fermion field . The superconformal transformations are as follows:

(34)

The divergence free condition of can be easily solved by considering the three-rank antisymmetric tensor via the equation .

Hypermultiplet:

hypermultiplets are usually associated with target spaces of dimension that are hyperkahler cones. The superconformal transformations are written in terms of local sections of an bundle as follows

(35)

The covariant derivative contains the connection associated with rotations of the fermions. Moreover the sections are pseudo-real in the sense that they obey the constraint , where is a covariantly constant skew-symmetric tensor with its complex conjugate satisfying . The information on the target space metric is contained in the hyperkahler potential

(36)

Note that the hypermultiplets do not exist as an off-shell supermultiplet. The superconformal transformations close only up to fermionic equations of motion.

4.2 The Lagrangian

We present the bosonic part of the Lagrangian.

The lagrangian is essentially the sum of three parcels, that is,

(37)

The first term is cubic in the vector multiplet fields,

(38)

where , symmetric in all its indices, are constants that encode the different couplings of the fields. The function is the contraction .

The term encodes the lagrangian for the hypermultiplets

(39)

while contains higher derivative corrections with couplings between vector and weyl multiplets fields

(40)

The constants encode the couplings of the higher derivative terms. The symbol denotes .

Note that and are respectively the Ricci scalar and tensor 121212In [54] the authors use a different convention for the spin connection. This results in a sign flip for the curvature tensors. while is the superconformal Weyl tensor. Other conventions can be found in the Appendix.

In the following we show how to obtain on-shell Poincaré supergravity by integrating out the auxiliary fields. The equation of motion for the auxiliary field is

(41)

which on the attractor background (44) reduces to

(42)

For simplicity consider the theory with a unique vector multiplet without higher derivative corrections, that is, . The function becomes . The gauge theory sector of the lagrangian, composed of a scalar , a vector and auxiliary fields becomes, after reintroducing the fermion fields, invariant under rigid superconformal transformations. Due to scale invariance we fix the scalar to a constant. If we further use the attractor equations and (44) we obtain

(43)

which upon including the gravitino field, is equal to the Lagrangian of pure five-dimensional supergravity. The Newton’s constant is identified with so that the Ricci scalar appears with the canonical prefactor .

4.3 BPS attractor equations and near horizon geometry

In this section we present the attractor field configuration that preserves full supersymmetry. The analysis is completely off-shell and therefore it does not depend on the specific higher derivative corrections the theory may contain. To fully determine the black hole attractor background these equations must be supplemented with the values of the charges which depend on details of the higher derivative corrections. For further details we refer the reader to [54].

Since ultimately we are interested in Poincaré supergravity we want to study the vanishing of the fermionic variations modded out by S-supersymmetry variations. This is achieved by constructing fermionic fields which are invariant under S-supersymmetry. This is basically the approach first outlined in [55]. The solutions are

(44)

where is the R-symmetry field strength .

The geometry has the form of a circle non-trivially fibered over

(45)

with fiber

(46)

The size of is determined via the condition

(47)

where are the only non-vanishing components of , where are the local Lorentz indices.

For the line element (45) can be rewritten for as

(48)

with

(49)

Up to the conformal factor , the second term in the line element (48) is diffeomorphic to flat space. However for we have a conical singularity at the origin. Requiring smoothness of the solution we fix by imposing the condition

(50)

Since the theory is scale invariant we set for convenience. The geometry is left with only one parameter defined via the equation (47) by setting

(51)

The line element (45) becomes

(52)

This is the near horizon geometry of a rotating black hole with angular momentum proportional to . The limiting case or has line element

(53)

The first three terms describe a local . So effectively, we have the space . If we insist on the identification we have the near horizon geometry of a black ring, while for noncompact we have an infinite black string. In this work we will be interested only in the case of a rotating black hole. The case, which is very interesting, will be postponed for a future work.

In summary, we have a one parameter family of geometries, which are locally , that interpolate between the non-rotating black hole with near horizon geometry and the black ring/string with near horizon geometry [56].

The euclidean version of this configuration follows from a standard Wick rotation together with . This is equivalent to the transformation . Definitions (51) become

(54)

and the line element (52) becomes

(55)

with . In the rest of the paper we will use hyperbolic coordinates which appear to be more convenient. The line element is now

(56)

with conformal boundary at , while the fiber becomes

(57)

From a four dimensional point of view this corresponds to a gauge fi