CERN-TH/99-270

hep–th/9909041

States and Curves of Five-Dimensional Gauged Supergravity

Ioannis Bakas and Konstadinos Sfetsos Department of Physics, University of Patras

GR-26500 Patras, Greece

, ajax.physics.upatras.gr

Theory Division, CERN

CH-1211 Geneva 23, Switzerland

Abstract

We consider the sector of five-dimensional gauged supergravity with non-trivial scalar fields in the coset space , plus the metric. We find that the most general supersymmetric solution is parametrized by six real moduli and analyze its properties using the theory of algebraic curves. In the generic case, where no continuous subgroup of the original symmetry remains unbroken, the algebraic curve of the corresponding solution is a Riemann surface of genus seven. When some cycles shrink to zero size the symmetry group is enhanced, whereas the genus of the Riemann surface is lowered accordingly. The uniformization of the curves is carried out explicitly and yields various supersymmetric configurations in terms of elliptic functions. We also analyze the ten-dimensional type-IIB supergravity origin of our solutions and show that they represent the gravitational field of a large number of D3-branes continuously distributed on hyper-surfaces embedded in the six-dimensional space transverse to the branes. The spectra of massless scalar and graviton excitations are also studied on these backgrounds by casting the associated differential equations into Schrödinger equations with non-trivial potentials. The potentials are found to be of Calogero type, rational or elliptic, depending on the background configuration that is used.

CERN-TH/99-270

September 1999

## 1 Introduction

Ungauged and gauged supergravities in five dimensions were constructed several years ago in [1] and [2, 3], following the analogous construction made in four dimensions in [4] and [5]. More recently it has become clear that solutions of five-dimensional gauged supergravity play an important rôle in the context of the AdS/CFT correspondence [6, 7, 8]. In particular the maximum supersymmetric vacuum state in five-dimensional gauged supergravity with geometry, originates from the solution in ten-dimensional type-IIB supergravity. The latter solution arises as the near horizon geometry of the solution representing the gravitational field of a large number of coincident D3-branes and has been conjectured to provide the correct framework for analyzing supersymmetric Yang–Mills for large and ’t Hooft coupling constant at the conformal point of the Coulomb branch.

The supergravity approach to gauge theories at strong coupling is applicable not only at conformality, but also away from it. In particular, when the six scalar fields of the supersymmetric Yang–Mills theory acquire Higgs expectation values we move away from the origin of the Coulomb branch and the appropriate supergravity solution corresponds to a multicenter distribution of D3-branes with the centers, where the branes are located, associated with the scalar Higgs expectation values in the gauge theory side. A prototype example of such D3-brane distributions is the two-center solution that has been studied in [6, 9, 10], whereas examples of continuous D3-brane distributions arise naturally in the supersymmetric limit of rotating D3-brane solutions [11, 12]. Concentrating on the case of continuous distributions, note that from a ten-dimensional type-IIB supergravity view point the symmetry, associated with the round -sphere, is broken because this sphere is deformed. On the other hand, from the point of view of five-dimensional gauged supergravity the deformation of the sphere is associated with the fact that some of the scalar fields in the theory are turned on. Hence, finding solutions of five-dimensional gauged supersgravity might shed more light into the AdS/CFT correspondence as far as the Coulomb branch is concerned. Using such solutions, investigations of the spectrum of massless scalars excitations and of the quark-antiquark potential have already been carried out with sometimes suprising results [13, 14, 15]. Solutions of the five-dimensional theory are also important in a non-perturbative treatment of the renormalization group flow in gauge theories at strong coupling [16, 17, 18, 19].

An additional motivation for studying solutions of five-dimensional gauged supergravity is the fact that for a class of such configurations, four-dimensional Poincaré invariance is preserved. It turns out that our four-dimensional space-time can be viewed as being embedded non-trivially in the five-dimensional solution with a warp factor. This particular idea of our space-time as a membrane in higher dimensions is quite old [20] and has been recently revived with interesting phenomenological consequences on the mass hierarchy problem [21]. In that work, in particular, our four-dimensional world was embedded into the space from which a slice was cut out; it results into a normalizable graviton zero mode, but also to a continuum spectrum of massive ones above it with no mass gap separating them. The use of more general solutions of five-dimensional gauged supergravity certainly creates more possibilities and in fact there are solutions with a mass gap that separates the massless mode from the massive ones [22].

This paper is organized as follows: In section 2 we present a brief summary of some basic facts about five-dimensional gauged supergavity with gauge group . In particular, we restrict our attention to the sector of the theory where only the metric and the scalar fields associated with the coset space are turned one. In section 3 we find the most general supersymmetric configuration in this sector, which as it turns out, depends on six real moduli. Our solutions have a ten-dimensional origin within type-IIB supergravity and represent the gravitational field of continuous distributions of D3-branes in hyper-surfaces embedded in the transverse space to the branes. In section 4 we further analyze our solution using some concepts from the theory of algebraic curves and in particular Riemann surfaces. We find that our states correspond to Riemann surfaces with genus up to seven, depending on their symmetry groups, which are all subgroups of . In section 5 we provide details concerning the geometrical origin of the supersymmetric states in five dimensions from a ten-dimensional point of view using various distributions of D3-branes in type-IIB supergravity. This approach yields explicit expressions for the metric and the scalar fields, and it can be viewed as complementary to the algebro-geometric classification of section 4 in terms of Riemann surfaces. In section 6 we consider massless scalar and graviton fluctuations propagating on our backgrounds. We formulate the problem equivalently as a Schrödinger equation in one dimension and compute the potential in some cases of particular interest. We also note intriguing connections of these potentials to Calogero models and various elliptic generalizations thereof. Finally, we end the paper with section 7 where we present our conclusions and some directions for future work.

## 2 Elements of five-dimensional gauged supergravity

supergravity in five dimensions involves 42 scalar fields parametrizing the non-compact coset space that describes their couplings in the form of a non-linear -model [1]. In five-dimensional gauged supergravity the global symmetry group breaks into an subgroup which corresponds to the gauge symmetry group of the resulting theory, and a non-trivial potential develops [2, 3]. In the framework of the AdS/CFT correspondence [6, 7, 8] the supergravity scalars represent the couplings of the marginal and relevant chiral primary operators of the supersymmetric Yang–Mills theory in four dimensions. The invariance of the theory with respect to the gauge group, as well as the symmetry inherited from type-IIB supergravity in ten dimensions, restricts the scalar potential to depend on invariants of the above groups. However, it seems still practically impossible to deal with such a general potential. In this paper we restrict attention to the scalar subsector corresponding to the symmetric traceless representation of , which parametrizes the coset , and set all other fields (except the metric) equal to zero. In this sector we will be able to find explicitly the general solution of the classical equations of motion that preserves supersymmetry.

The Lagrangian for this particular coupled gravity-scalar sector includes the usual Einstein–Hilbert term, the usual kinetic term for the scalars as well as their potential

(2.1) |

A few explanations concerning the scalar-field part of this action are in order. It has been shown that in this subsector the scalar potential depends on the symmetric matrix only, where is an element of [3] (for a recent discussion see also [17, 13]). Diagonalization of this matrix yields a form that depends only on five scalar fields. It is convenient, nevertheless, to represent this sector in terms of six scalar fields , as [13]

(2.2) |

where

(2.3) |

Note that the matrix that relates the auxiliary scalars with the ’s is not unique; it only has to satisfy the condition . The choice in (2.3) is particularly useful for certain computational purposes. It also has the property that if the fields are canonically normalized, the five independent scalar fields will be canonically normalized as well, i.e. .

The form of the kinetic term for the scalars in (2.1) suggests that the metric in the corresponding coset space is taken to be . This was explicitly shown for the case of only one scalar field in [17] and the general result was quoted without detailed explanation in [13]. One can generally prove this statement by first realizing that the kinetic term of these scalars can depend on two type of terms, namely and . Since belongs to the algebra of the first term is zero because of the traceless condition. The second term gives a result proportional to , thus showing that the scalar kinetic term in (2.1) has indeed the above form. The equations of motion follow by varying the action (2.1) with respect to the five-dimensional metric and the scalar fields. Using the metric , we have

(2.4) |

There is a maximally supersymmetric solution of the above equations that preserves all 32 supercharges, in which all scalar fields are set zero and the metric is that of space. Then, the potential in (2.2) becomes and equals by definition to the negative cosmological constant of the theory. This defines the length scale that will be used in the following.

The coupled system of non-linear differential equations (2.4) is in general difficult to solve. In this paper we will be interested in solutions preserving four-dimensional Poincaré invariance . Hence, we make the following ansatz for the five-dimensional metric

(2.5) |

where is the four-dimensional Minkowski metric and the conformal factor , as well as the scalar fields , depend only on the variable . In addition, we demand that our solutions preserve supersymmetry. The corresponding Killing spinor equations, arising from the supersymmetry transformation rules for the 8 gravitinos and the 42 spin- fields, give rise to the first order equations [13]

(2.6) |

where

(2.7) |

and the derivative is taken with respect to the coordinate . It is straightforward to check that all supersymmetric solutions satisfying the first order equations (2.6) also satisfy the second order equations (2.4). In doing so, it is convenient to use the alternative expression for the potential, instead of (2.2),

(2.8) |

## 3 The general supersymmetric solution

We begin this section with the construction of the most general solution of the non-linear system of equations (2.6) and discuss some of the general properties of the corresponding supersymmetric configurations. We also show how our solution can be lifted to ten dimensions in the context of type-IIB supergravity.

### 3.1 Five-dimensional solutions

It might still seem difficult to find solutions of the coupled system of equations (2.6) at first sight, due to non-linearity. It turns out, however, that this is not the case, but instead it is possible to find the most general solution. In order to proceed further, we first compute the evolution of the auxiliary scalar fields . Using (2.3) and (2.6) we find

(3.1) |

where for the last equality we have used the first equation in (2.6). This substitution results into six decoupled first order equations for the ’s which can be easily integrated, as we will soon demonstrate. Of course, after deriving the explicit solution for the , we also have to check the self-consistency of this substitution.

Let us reparametrize the function in terms of an auxiliary function as follows

(3.2) |

We have included a minus sign in this definition since, according to the boundary conditions that we will later choose, will be a decreasing function of . Then, according to this ansatz, the general solution of (3.1) is given by

(3.3) |

where the prime denotes here the derivative with respect to the argument . The ’s are six constants of integration, which, sometimes is convenient to order as

(3.4) |

without loss of generality. Note that we may fix one combination of them to an arbitrary constant value because (3.2) determines the function up to an additive constant. Also, since the sum of the ’s is zero, we find that the function has to satisfy the differential equation

(3.5) |

which thus contains all the information about the supersymmetric configurations and provides a non-trivial algebraic constraint. Using (3.2), (3.3) and (3.5) one may easily check that the first equation in (2.6) is also satisfied. If we insist on presenting the solution in the conformally flat form (2.5) the differential equation (3.5) needs to be solved to obtain . This will be studied in detail in section 4, as it is a necessary step for investigating the massless scalar and graviton fluctuations in section 6.

At the moment we present our general solution in an alternative coordinate system, where is viewed as the independent variable. Indeed, using (3.5), we obtain for the metric

(3.6) |

whereas the expression for the scalar fields in (3.3) becomes

(3.7) |

When the constants are all equal, our solution becomes nothing but with all scalar fields turned off to zero. In the opposite case, when all constants are unequal from one another, there is no continuous subgroup of preserved by our solution. If we let some of the ’s to coincide we restore various continuous subgroups of accordingly. As for the five scalar fields , they can be found using (2.3)

(3.8) | |||||

Note that imposing the reality condition on the scalars in (3.7) restricts the values of to be larger that the maximum of the constants , which according to the ordering in (3.4) means that . For the scalars tend to zero and , in which case the metric in (3.6) approaches as expected; put differently, in this limit close to that is taken as the origin of the -coordinate. For intermediate values of we have a flow in the five-dimensional space spanned by all scalar fields . In general we may have , with , when is -fold degenerate. In this case, the solution preserves an subgroup of and the flow is actually taking place in dimensions. On the other hand, let us consider the case when approaches its lower value . Then, the scalars in (3.7) are approaching

(3.9) |

where . Consequently, we have a one-dimensional flow in this limit since the scalar fields can be expressed in terms of a single (canonically normalized) scalar , as

(3.10) |

It is also useful to find the limiting form of the metric (3.6) when . Changing the variable to as

(3.11) |

the metric (3.6) becomes for

(3.12) |

Hence, at (or equivalently at ) there is a naked singularity which has an interpetation, as we will see later in the ten-dimensional context, as the location of a distribution of D3-branes. It is instructive to compare this with the singular behaviour of non-conformal non-supersymmetric solutions found in [23]. A similar naked singularity was found there, but the corresponding metric near the singularity had a power law behaviour in with exponent equal to , which coincides with the result in (3.12) only for .

### 3.2 Type-IIB supergravity origin

It is possible to lift our solution with metric and scalars given by (3.6) and (3.7) to a supersymmetric solution of type-IIB supergravity, where only the metric and the self-dual five-form are turned on. This proves that our five-dimensional solution is a true compactification of type-IIB supergravity on . This is not a priori obvious because unlike the case of the compactification of eleven-dimensional supergravity to four dimensions [24], there is no general proof that the full non-linear five-dimensional gauged supergravity action can be fully encoded into the action or equations of motion of the type-IIB supergravity for the compactification. However, there is a lot of evidence that this is indeed the case and our result gives further support in its favour.

We will show that the ten-dimensional metric corresponds to the gravitational field of a large number of D3-branes in the field theory limit with a special continuous distribution of branes in the transverse six-dimensional space. Namely, the metric has the form

(3.13) |

where is a harmonic function (yet to be determined) in the six-dimensional space transverse to the brane parametrized by the coordinates. However, instead of being asymptotically flat, the metric (3.13) will become asymptotically for large radial distances (or equivalently in the UV region using the terminology of the AdS/CFT correspondence). The ten-dimensional dilaton field is constant, i.e. and, as usual, the self-dual five-form is turned on. Under these conditions, the ten-dimensional solution breaks half of the maximum number of supersymmetries (see, for instance, [25]).

We proceed further by first performing the coordinate change in (3.13)

(3.14) |

where the ’s define a unit five-sphere, i.e. they obey . Various convenient bases for these unit vectors can be chosen, depending on the particular applications that will be presented later. It can be shown that the flat six-dimensional metric in the transverse part of the brane metric (3.13) can be written as

(3.15) |

where the line element defines the metric of a deformed five-sphere given by

(3.16) |

For later use, we have also written the expression for the determinant of the deformed five-sphere in (3.16). In computing this determinant we have used the fact that the sum of the ’s is zero. Note that a similar expression also holds for a general -sphere.

The harmonic function is determined by comparing the massless scalar equation for the ten-dimensional metric (3.13) with the equation arising using the five-dimensional metric (2.5), i.e. . In both cases one makes the ansatz that the solution does not depend on the sphere coordinates, i.e. . Since the solutions for the scalar should be the same in any consistent trancation of theory, the resulting second order ordinary differential equations should be identical. A comparison of terms proportional to determines the function as follows,

(3.17) |

where in the second equality the harmonic function has been expressed in terms of the transverse coordinates . Comparison of the terms proportional to the first and second derivative of yields, using the expression for in (3.16), an identity and provides no further information. The coordinate is determined in terms of the transverse coordinates as a solution of the algebraic equation

(3.18) |

This is a sixth order algebraic equation for general choices of the constants , and its solution cannot be written in closed form. However, this becomes possible when some of the ’s coincide in such a way that the degree of (3.18) is reduced to four or less. Even then, the resulting expressions are not very illuminating and we will refrain from presenting them except in the simplest case in section 5 below.

The corresponding D3-brane solution that is asymptotically flat is obtained by replacing in (3.13) by . Then, in this context, the length parameter has a microscopic interpretation using the string scale , the string coupling , and the (large) number of D3-branes , as .

In the rest of this section we demonstrate for completeness the proof that the function , as defined in (3.17), is indeed harmonic in the six-dimensional transverse space spanned by , . This is not a trivial check since that appears in (3.17) is itself a function of the transverse space coordinates due to the condition (3.18). For notational convenience we define the functions

(3.19) |

Then, using (3.18) we determine the derivative of the function

(3.20) |

Also, the first derivative of with respect to turns out to be

(3.21) |

Taking the derivative with respect to once more, summing over the free indices and after some algebraic manipulations, we obtain the desired result

where the terms appearing in the three different lines above arise from the three distinct terms of (3.21) respectively.

## 4 Riemann surfaces in gauged supergravity

In this section we will present the basic mathematical aspects of our general ansatz for the supersymmetric conditions of five-dimensional gauged supergravity and find the means to obtain explicit solutions in several cases by appealing to methods of algebraic geometry. In fact, we will classify all possible solutions according to symmetry groups (subgroups of ) and use the uniformization of algebraic curves that result in this approach for deriving the corresponding expressions. To simplify matters the parameter will be set equal to 1, but it can be easily reinstated by appropriate scaling in .

### 4.1 Schwarz–Christoffel transform

A useful way to think about the differential equation for the unknown function is in the context of complex analysis. Suppose that and are extended in the complex domain and let us consider a closed polygon in the -plane, including its interior, and map it via a Schwarz–Christoffel transformation onto the upper half -plane. This provides a one-to-one conformal transformation and it is assumed that is analytic in the polygon and is continuous in the closed region consisting of the polygon together with its interior. Considering the behaviour of and as the polygon is transversed in the counter-clockwise direction, we know that the transformation is described as

(4.1) |

where is some constant that changes by rescaling . The vertices of the polygon are mapped to the points on the real axis of the upper complex -plane and the exponents that appear in the transformation are the exterior (deflection) angles of the polygon at the corresponding vertices. When the polygon is closed their sum is . Of course, without loss of generality, we may take one point (say ) to infinity. Letting we see that as the Schwarz–Christoffel transformation becomes

(4.2) |

where is another constant factor. To make contact with our problem we choose and let the angles . Then, we arrive at the differential equation

(4.3) |

which is the same as the one implied by our ansatz for the general solution of gauged supergravity (with ).

The solutions of this equation are difficult to obtain in practice for generic values of the moduli . We will investigate this problem in connection with the theory of algebraic curves in and we will see that in many cases explicit solutions can be given using the theory of elliptic functions. Before proceeding further we note that in our formulation we are looking for the map from the interior of the polygon onto the upper half-plane, , and not for the inverse transformation.

### 4.2 Symmetries and algebraic curves

If we extend the variable to the complex domain, as before, and set

(4.4) |

the Schwarz–Christoffel differential equation will become an algebraic curve in ,

(4.5) |

This is a convenient formulation for finding solutions of the supersymmetry equations, but at the end we have to restrict to real values of and demand that the resulting supergravity fields are also real. For generic values of the parameters , so that they are all unequal and hence there is no symmetry in the solution of five-dimensional gauged supergravity, the genus of the curve can be easily determined (like in any other case) via the Riemann–Hurwitz relation. Recall that for any curve of the form

(4.6) |

which is reduced, i.e. the integers and have no common factors, and all ’s are unequal, the genus can be found by first writing the ratios

(4.7) |

in terms of relatively prime numbers and then using the relation

(4.8) |

According to this the genus of our surface turns out to be when all are unequal, and so it is difficult to determine explicitly the solution in the general case. However, by imposing some isometries in the solution of gauged supergravity the genus becomes smaller and hence the problem becomes more tractable. The presence of isometries manifests by allowing for multiple branch points in the general form of the algebraic curve, which in turn degenerates along certain cycles that effectively reduce its genus.

Note for completeness that if we had not taken to infinity in our discussion of the Schwarz–Christoffel transformation, we would have had an additional factor in the equation of the algebraic curve because instead of that was chosen for the remaining ’s. It can be easily verified that this does not affect the genus of the curve, as expected on general grounds.

Next, we enumerate all possible cases with a certain degree of symmetry that correspond to various subgroups of ; this amounts to various deformations of the round five-sphere, , which is used for the compactification of the theory from 10 to 5 dimensions. Consequently, this will in principle determine the solution for the scalar fields in the remaining 5 dimensions as we will see later in detail. If all the branch points are different, the isometry of will be completely broken, whereas if all of them coalesce to the same point the maximal isometry will be manifestly present. The classification is presented below in an order of increasing symmetry or else in decreasing values of .

(1) : It corresponds to setting two of the equal to each other and the remaining are all unequal. The curve becomes

(4.9) |

and its genus turns out to be .

(2) : It corresponds to setting three of the equal and all other remain unequal. The curve becomes

(4.10) |

and the genus turns out to be .

(3) : In this case two pairs of are mutually equal and the remaining two parameters are unequal. The curve becomes

(4.11) |

and its genus is .

(4) : In this case three are equal and another two are also equal to each other. The curve becomes

(4.12) |

and its genus is . Therefore we know that it can be cast into a manifest hyper-elliptic form by introducing appropriate bi-rational transformations of the complex variables.

(5) : It corresponds to setting four equal to each other and the other two remain unequal. The curve becomes

(4.13) |

and its genus is . It can also be cast into a manifest (hyper)-elliptic form as we will see shortly.

(6) : It corresponds to three different pairs of mutually equal , but in this case the curve is not irreducible, since . The reduced form is

(4.14) |

and clearly has genus as it is written directly in (hyper)-elliptic form.

(7) : In this case we have two groups of triplets with equal values of . The curve becomes

(4.15) |

and its genus is . It can also be cast into a manifest (hyper)-elliptic form.

(8) : In this case five are equal to each other and the last remains different. The curve becomes

(4.16) |

and its genus is also as before.

(9) : It corresponds to separating the into four equal and another two equal parameters. The curve becomes , but it is not irreducible. The reduced form is

(4.17) |

and has genus , as it can also be obtained by degenerating a genus 1 surface along its cycles. Therefore, we expect the solution to be given in terms of elementary functions.

(10) : This is the case of maximal symmetry in which all are set equal to each other. The curve becomes , whose reduced form is

(4.18) |

and has genus as before.

Of course, when certain cycles contract by letting various branch points to coalesce, the higher genus surfaces reduce to lower genus and a bigger symmetry group emerges in the solutions corresponding to gauged supergravity. For genus one can always transform to a manifest hyper-elliptic form so that two sheets (instead of four) are needed for picturing the Riemann surface by gluing sheets together along their branch cuts. We will investigate in detail the cases corresponding to genus 0 and 1 surfaces since the solutions can be given explicitly in terms of elementary and elliptic functions respectively. Some results about the genus 2 case will also be presented. The other cases are more difficult to handle in detail even though the general form of the solution is known implicitly for all according to our ansatz.

### 4.3 Genus 0 surfaces

There are two genus 0 surfaces according to the previous discussion, namely the curve for the isometry group and the curve for the isometry group . According to algebraic geometry every irreducible curve with genus 0 is representable as a unicursal curve (straight line)

(4.19) |

by means of a bi-rational transformation , and conversely , . In our two examples the underlying transformations are summarized as follows:

(a) : We have

(4.20) |

and conversely

(4.21) |

(b) : We have

(4.22) |

and conversely

(4.23) |

Of course, the first curve arises as special case of the second for .

For general and we may use as a (trivial) uniformizing complex parameter for the unicursal curve, i.e. . Then, the expressions for and yield

(4.24) |

where we have taken into account the rescaling , , that was introduced earlier. So we can determine as a function of by simple integration since

(4.25) |

In fact there are three different cases for generic values of and . Choosing appropriately the integration constant, so that the resulting conformal factor will behave like as , we have:

(4.26) | |||||

(4.27) | |||||

(4.28) |

The first two cases correspond to the isometry and they are obtained by analytic continuation from one other, depending on the size of the ’s, whereas the last case has isometry. Here, we do not assume any given ordering among the ’s. As for the functions we have respectively

(4.29) |

Then, the expression for the conformal factor of the metric is

(4.30) | |||||

(4.31) | |||||

(4.32) |

which indeed behaves as in all three cases for .

The solution for the scalar fields of five-dimensional gauged supergravity follows by simple substitution into our ansatz. We have explicitly in each case

(4.34) | |||||

(4.35) |

Equally well we could have transformed the genus 0 curves into the quadratic form using the following transformation for the curve

(4.36) |

and conversely