LBNL 44356

UCB-PTH-99/49

McGill 99-32

Saclay t99/112

hep-th/9910081

Supergravity Inspired

Warped Compactifications

and

Effective Cosmological Constants

C. Grojean , J. Cline , and G. Servant

Department of Physics, University of California, Berkeley, CA 94720

Theoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, CA 94720

Physics Department, McGill University, Montréal, Québec, Canada H3A 2T8

CEA-SACLAY, Service de Physique Théorique, F-91191 Gif-sur-Yvette, France

We propose a supergravity inspired derivation of a Randall–Sundrum’s type action as an effective description of the dynamics of a brane coupled to the bulk through gravity only. The cosmological constants in the bulk and on the brane appear at the classical level when solving the equations of motion describing the bosonic sector of supergravities in ten and eleven dimensions coupled to the brane. They are related to physical quantities like the brane electric charge and thus inherit some of their physical properties. The most appealing property is their quantization: in extra dimensions, goes like and like . This dynamical origin also explains the apparent fine-tuning required in the Randall–Sundrum scenario. In our approach, the cosmological constants are derived parameters and cannot be chosen arbitrarily; instead they are determined by the underlying Lagrangian. Some of the branes we construct that support cosmological constant in the bulk have supersymmetric properties: D3-branes of type IIB superstring theory provide an explicit example.

## 1 Introduction

The coexistence of two hierarchical scales in particle physics is probably the most challenging puzzle to solve before hoping to construct a quantum theory of gravity. When the Schwarzchild radius () of a system of mass becomes of the same order as its Compton length (), a quantum mechanical extension of general relativity is surely needed. Therefore the natural scale of quantum gravity is the Planck mass, GeV. Understanding how, in such a theory, the tiny electroweak scale observed in experimental particle physics can arise and be stabilized against radiative corrections constitutes the so-called ‘gauge hierarchy problem’. In low energy supersymmetry [1], this vast disparity in scales can be protected from quantum destabilization. However a more fundamental explanation is certainly to be found in string theory and its latest developments. String theory relates the string scale to two other fundamental scales, namely the GUT scale connected to gauge interactions, and the Planck scale connected to the gravitational interaction. The link between these two is the geometry of extra dimensions, which can lower both scales [2] down to the TeV range [3] and thus partially answer the gauge hierarchy problem, or at least translate it into geometrical terms.

Subsequent to studies of thin shells in general relativity [4] and their revival in a -theory context [5, 6, 7], Randall and Sundrum (RS) have recently proposed [8] a new phenomenological mechanism for solving the gauge hierarchy problem, without requiring the extra dimension to be particularly large or small–in fact it could be noncompact. An exponential hierarchy is generated by the localization of gravity near a self-gravitating brane with positive tension, obtained by solving Einstein equations. The solution is a nonfactorizable metric, i.e., a metric with an exponentially decaying warp factor [9] along the single extra dimension. Restricting the Standard Model to a second parallel brane with negative tension at some distance in this transverse dimension, the electroweak scale in our world then follows from a redshifting of the Planck scale on the second brane. Since the exponential suppression by the redshift factor does not require an unnaturally large interbrane separation, the hierarchy problem can be explained without fine tuning, and without requiring any special size for the extra dimensions.

The cosmological implications of this scenario have been studied [10, 11], with emphasis on the danger of placing the Standard Model on a brane with negative tension since, for instance, the Friedmann equation governing the expansion of the universe appears with a wrong sign. A similar difficulty is also faced [12] when trying to reproduce the unification of gauge couplings. The original scenario can be modified [8, 12, 13] by maximizing the warp factor on the Standard Model brane, which can be achieved if its tension is taken positive. The two former problems are overcome but the electroweak scale seems now difficult to accommodate. More recently it has been shown that the correct cosmological expansion can be obtained if the second brane tension is negative, but not too much so [14]. Thus the RS scenario remains attractive, especially with regard to the possibility of an infinite extra dimension probed only by gravity. It is appealing that, despite a continuous Kaluza–Klein spectrum without any mass gap, Newton’s law of gravity is still reproduced [8, 13, 15] within the current experimental precision. Ref. [16] also proposed explicit models where a mass gap separates the ‘massless graviton’ from its KK excitations while the Yukawa type deviations from the 4D Newton law remain compatible with experimental bounds.

Although the gravity localization mechanism seems to be specific to
codimension one branes, several works [15, 17] have managed
to extend it by considering many intersecting codimension one
branes.^{1}^{1}1See also ref. [18] for a recent construction of
warped compactification in two transverse dimensions. Oda and Hatanaka
et al. [19] also obtain solutions with a more involved
content of branes with a single one extra-dimension. In this context also,
ref. [14] finds a cosmological solution
for the bulk and the branes inflating at the same rate with a time
dependant Planck mass. In that case, the hierarchy between the weak and
the Planck scales is fixed at the end of inflation.
.

Undoubtedly, the localization of gravity by the RS mechanism has rich phenomenological and cosmological consequences [10, 11, 13, 12, 20, 21, 22, 14, 19, 15]; but at the present stage it seems lacking in generality, and it suffers from apparently ad hoc fine-tunings required between the cosmological constants in the bulk and on the branes, in order to obtain a solution to Einstein equations. Verlinde [23] has reexamined the RS scenario in superstring language and shown that the warp factor can be interpreted as a renormalization group scaling. In the context of the AdS/CFT correspondence, the extra dimension plays the role of the energy scale.

In this paper, we offer a derivation of the effective action used by
RS, starting from a more fundamental, string-inspired origin. Recent
works [5, 7, 24, 25] have studied the dynamics of a
brane-universe; here we propose an explicit embedding of
the RS model in supergravity theories and examine its physical
implications, following refs. [16, 26], which
have previously addressed this question at a more formal level.
Our starting point will be the bosonic action of
supergravity theories in ten or eleven dimensions. We emphasize that,
instead of neglecting various fields specific to these actions like
the dilaton and some -differential forms, taking them into account
can lead to an effective description in terms of cosmological
constants. Using -brane solutions^{2}^{2}2
The branes we construct are solution to the bosonic equations of motion
and thus, as we will see later, they are not necessarily supersymmetric even if
they are embedded in a supergravity theory., we construct such a description
for codimension one branes, which allows us to identify the effective
cosmological constants with physical quantities like the electric
charge carried by the brane and its mass density on the worldvolume.
Since the electric charge of a -brane obeys a generalized Dirac
quantization rule, we are led to the interesting conclusion that the
cosmological constants are also quantized.

The advantage of this approach is that we derive the stress-energy tensor , which is needed to solve the Einstein equations, starting from an action for fundamental fields, rather than putting it in by hand. Thus our is on the same footing as the Einstein tensor itself, from the point of view of fundamentality. Moreover we are able to generalize the procedure to higher codimension brane-universes (e.g., 3-branes embedded in more than one extra dimension), providing some of the first such solutions. In this case the bulk energy is no longer a cosmological “constant,” but depends on the distance from the brane.

## 2 Brane cosmological constant as a warp in an anti-de Sitter bulk

We begin with a review of the model studied by Randall and Sundrum [8]. This model is a particular case of the ones proposed by Chamblin and Reall [6], in which a scalar field was coupling a dynamical brane to an embedding bulk. Here we consider the restricted scenario of a static brane embedded in a spacetime curved by a bulk cosmological constant . The physics of this model is governed by the following action:

(1) |

where is the location of the brane in the transverse (extra dimensional) subspace and is the determinant of the metric, assumed to be factorizable, in this subspace. The Einstein equations derived from (1) when the transverse space is flat are (Greek indices denote longitudinal coordinates, and Latin indices are coordinates transverse to the brane, ):

(2) | |||||

(3) |

Randall and Sundrum solved these equations in the case of a codimension one brane. With the ansatz

(4) |

the Einstein equations reduce to

(5) | |||

(6) |

where primes denote derivatives with respect to the transverse coordinate . For this system of equations to admit a solution that matches the singular terms, a fine-tuning between and is necessary:

(7) |

A general solution then takes the form:

(8) |

where is a regular function and the constant is related to the brane cosmological constant by: , being the sign of . A particular class of solutions that will play an important role in our analysis corresponds to:

(9) |

where and are two positive constants. An appropriate change of coordinates brings this solution to the form proposed by Randall and Sundrum [8]: defining and , the metric reads:

(10) |

If the brane located at the origin is identified as the “Planck brane”
of Lykken–Randall [13], an electroweak scale will be generated on
the “TeV brane” if and only if the power is negative, which
corresponds to a positive cosmological constant on the Planck
brane.^{3}^{3}3This connection between the signs of and
is specific to one transverse dimension. In section 4, we will see that we
can have whereas . In any case, the discussion about
the hierarchy problem deals with the sign of only. Another
motivation for requiring comes from computing the four-dimensional
effective Planck mass, , which is finite
for but diverges for .

Figure 1. The boundary of an anti-de Sitter of dimension space is topologically . In the system of coordinates and , this boundary is located at and : the piece at infinity is a -dimensional Minkowskian space, while the horizon at corresponds to the union of a point and . A codimension one brane embedded in this space acts as a warp in the sense that it cuts a part of the bulk: a brane with a positive cosmological constant cuts the vicinity of the boundary located at the infinity, while a brane with a negative cosmological constant removes the horizon at the origin.

We can make another diffeomorphism that clarifies the geometry of the solution. Defining , with , we now obtain:

(11) |

where we see that the geometry of the bulk corresponds to an anti-de Sitter space of radius , or at least a slice of an anti-de Sitter space, since the variable ranges only over a part of . Indeed, for , the range of variation of is restricted to , while for this range becomes . Although in both cases the whole space is covered in the limit , it is interesting to note which part is cut when . As we will argue in the appendix, the boundary of an anti-de Sitter space of dimension space is topologically , and in the system of coordinates and , this boundary is located at and : the piece at infinity is a -dimensional Minkowskian space, while the horizon at corresponds to the union of a point and . So the case, which corresponds to a positive cosmological constant on the brane, removes the part at infinity, while the case, i.e. , cuts the horizon at the origin. Note that in the AdS/CFT correspondence [27], a superconformal theory describes the dynamics of a brane near the horizon of an space while this dynamics should become free near infinity [28].

As presented, the model studied by Randall and Sundrum leaves one wondering whether it can be derived from some more fundamental starting point. In particular, the ad hoc fine-tuning between the cosmological constants is rather mysterious and begs for a better understanding. One suggestion is that this relation might arise from the requirement that tadpole amplitudes are zero in the underlying string theory [11]. (See also ref. [29] for recent progress about this question). Here we will see the cosmological constants as effective parameters which cannot be chosen arbitrarily, so the fine-tuning problem is ameliorated. The aim of this work is to motivate the RS model from a supersymmetry/superstring framework.

## 3 Effective cosmological constants from dynamics of codimension one branes

In this section, we would like to show that the theory derived from the action (1) can be seen as an effective description of a brane of codimension one, i.e., of an extended object with spatial dimensions embedded in a () dimensional spacetime.

The dynamics of an object extended in spatial directions is
governed by the generalization of the Nambu--Goto action^{4}^{4}4Concerning the
indices, our conventions will be the following:
hatted Greek indices are spacetime indices ()
while Latin indices are worldvolume indices(). [30]:

(12) |

where are the coordinates in the embedding spacetime of a point on the brane characterized by its worldvolume coordinates ; is the scale mass in so-called “-brane units” which is simply related to the Planck scale, , in the embedding spacetime; see below eq. (16). This action is known [31] to be equivalent to:

(13) |

where is an auxiliary field that gives the metric on the worldvolume.

Superbranes have been constructed [32] as classical solutions of supergravity theories in ten or eleven dimensions: they are BPS objects, since they preserve half of the supersymmetries; they have a Poincaré invariance on their worldvolume universe and also a rotational invariance in the transverse space. A -brane is therefore coupled to the low-energy effective theory of superstrings. Below the fundamental energy scale, identified as the energy of the first massive excitations of the string, the theory can be described by supergravity theories whose bosonic spectrum contains the metric, a scalar field (the dilaton) and numerous differential forms. The bosonic effective action, in supergravity units, takes the general form ():

(14) |

where is the field strength of the ()-differential form , whose coupling to the dilaton is measured by the coefficient . The coefficient is explicitly determined by a string computation: the coupling of the dilaton to differential forms from the Ramond-Ramond sector appears at one loop and thus in supergravity units, while the Neveu-Schwarz–Neveu-Schwarz two-form couples at tree level, so . In some cases, we can also add a Chern–Simons term () to the action, but it does not have any effect on the classical solutions.

The -brane couples to a ()-differential form, which results in the addition of a Wess–Zumino term to the free action (13):

(15) |

The functions and implicitly depend on worldvolume coordinates through their dependence in the embedding coordinates . The coefficient defines the “-brane units;” it is fixed [33] by requiring the same scaling behavior for and , which leads to

(16) |

The relation between and then follows from the value of this coupling to the dilaton: , being the vacuum expectation value of the dilaton.

To proceed, we must now relax some of the constraints imposed by supersymmetry, while still maintaining the form of the action. For example in string theories, the values of and are related to one another in order to have supersymmetry on the worldvolume universe [34]. Also, as just mentioned, the coupling to the dilaton is fixed. By relaxing these constraints, we give up any claim that the following construction is a direct consequence of string theory. On the other hand it might be hoped that our results will persist in a realistic low energy limit of string theory, which includes the effects of supersymmetry breaking. In what follows, we will elucidate how the various fields, which play a crucial role for the existence of branes in supergravity, can give rise to an effective stress-energy tensor which resembles the cosmological constant terms needed for the Randall–Sundrum scenario.

The equations of motion derived from are

(17) | |||

(18) | |||

(19) | |||

(20) | |||

(21) |

The stress-energy tensor of the brane is given by

(22) |

The electric current created by the brane is

(23) |

And the source current for the dilaton equation is

(24) |

We will solve these equations in the case of a codimension one brane and we will see in the next section how the analysis can be extended to higher codimension. First we choose a system of spacetime coordinates related to the brane:

in the physical gauge where .

We are looking for a solution with a Poincaré invariance in () dimensions, so that we can make the following ansatz for the metric:

(25) |

The nonvanishing components of the ()-differential form that couples to the -brane are

(26) |

where is the antisymmetric tensor normalized to .

It is well known that (see for instance [33] for a review), corresponding to the ansatz (25–26), the solutions of eqs (17–21) can be expressed in terms of a harmonic function :

(27) | |||

(28) | |||

(29) |

where the powers are given by

(30) |

The consistency of the whole set of equations of motion with our -brane ansatz requires to adjust the coefficient of the Wess–Zumino term to to the coupling to the dilaton by

(31) |

the whole set of equations of motion is now equivalent to Poisson’s equation,

(32) |

the solution of which reads

(33) |

where is an arbitrary positive constant that can be normalized to one if a flat Minkowski space in the vicinity of the brane is wanted. At this stage, it is worth noticing that the derivation follows directly from the bosonic equations (17)-(21) and no supersymmetric argument has been used. The full supergravity equations also include a Killing spinor equation that can be consistently solved, provided that the coupling of the differential form to the dilaton takes its stringy value. This promotes the bosonic solution to a BPS one.

It is interesting to substitute this solution back into the Einstein equations (17) to obtain:

(34) | |||||

(35) |

In the limit of decoupling between the brane and the dilaton, i.e., , which also corresponds to using the constraint (31), the Einstein tensor involves two constants and :

(36) | |||||

(37) |

If we keep the factors fixed (since could go to infinity as ), these constants are given by

(38) |

They can be interpreted as effective cosmological constants since the metric (27) is a solution to the Einstein equations derived from the RS action (1).

The expression of the cosmological constants in terms of fundamental Planck mass in dimensions may give some insight into the origin of the apparently ad hoc fine-tuning (7) of the RS mechanism: here the cosmological constants are no longer fundamental parameters and the fine-tuning problem appears in a different way; in the present language it is a consequence of taking the limit where the dilaton decouples from the brane. Of course this represents just one point in the full parameter space. The more general solution, when the dilaton does not decouple, is a bulk energy density which depends on , rather than a cosmological constant term. Regardless of this difference, one can still obtain a rapidly decaying warp factor, as long as remains small enough – the warp factor follows a power law whose exponent is inversely proportional to . The new insight, then, is that the original RS solution is only the simplest possibility within a whole class of solutions which can solve the hierarchy problem.

Furthermore, our approach links the energy densities of the brane and bulk to physical quantities like the charge, , associated to the electric current (23):

(39) |

Not only is such a charge
conserved, but it also obeys Dirac’s quantization rule
[35]: solutions exist where the fiducial value of the
electric charge is multiplied by an integer and these can be
interpreted as a superposition of parallel branes.
From the multiplication of the source (15) by a factor ,
we easily deduce the following scaling when disentangling the contribution
from the source and from the bulk
in (34)–(35)^{5}^{5}5It is important
to notice that the singular part of the Einstein tensor depends in the source
not only through the powers of but also intrinsically.:

(40) |

which assures that and are quantized like and respectively.

A serious shortcoming with the above solution is that the dilaton decoupling regime requires a purely imaginary Wess–Zumino term (see eq. (31)), which implies an imaginary hence unphysical value for the electric charge. Let us consider what happens if we insist that be real-valued, which would happen if the dilaton coupling, , was sufficiently large, instead of zero as we previously assumed. In this case eq. (33) implies that the harmonic form vanishes at some value of , , where . The exponents and are positive, so the metric coefficients vanish at ; this indicates the presence of a horizon surrounding the brane. The nontrivial dependence on of the brane metric coefficient, , means that the compactification is still warped; however it is not an exponential warp factor as in the solution of Randall and Sundrum. This can be seen by transforming to the coordinate , which measures physical distance in the bulk. In this coordinate, .

If we now imagine placing a visible sector brane (with such small tension that it has negligible effect on the background geometry [13]) very close to the horizon, physical masses on that brane will be suppressed relative to the string scale by the small factor . However one must fine-tune the closeness of the brane to the horizon to get particles of weak-scale masses, so this does not provide a natural solution to the weak-scale hierarchy problem.

Because of the imaginary value of required for the
solution which corresponds to that of Randall and Sundrum, our
construction is still just a tantalizing hint at a stringy origin for
their proposal.
To be more convincing, it is essential to overcome this problem.
In the next section, we will adress this issue by going to a higher
number of (still non copact) extra dimensions, in the space transverse
to the brane. However, it may happen that the problem of the imaginary
Wess--Zumino coupling also disappears when considering the compactification
of some of these extra dimensions,
requiring a more complete analysis involving some interacting moduli
fields in gauged supergravity theories^{6}^{6}6This question has been
recently addressed by Behrndt and Cvetič [25]. See also
ref. [5] for an earlier discussion.. The problem should also
be reconsidered in a more complicated version [36] of ten
dimensional IIA supergravity including mass terms since a
codimension one supersymmetric object, the D-8 brane, has been
constructed by Bergshoeff et al. [37]. This subject was
partially addressed in the recent references [38].

In summary, our study of codimension one branes suggests that the cosmological constants introduced by Randall and Sundrum are an effective description of the dynamics of a more complicated set of fields governing the physics of a brane that couples to the bulk through gravitational interactions only. Thus those effective cosmological constants inherit some physical properties of the brane, an intriguing one being their quantization. We point out that, for codimension one branes with no dilaton coupling, the solution (27) belongs to the general class of solutions (9). Since the exponent is negative, it follows from the general discussion of section 2 that this field configuration has an exponential decaying warp factor and thus can solve the gauge hierarchy problem in the manner proposed by Lykken and Randall [13]. Namely, physical particle masses will be exponentially suppressed on any test-brane (“TeV brane”) placed sufficiently far from the “Planck brane” featured in our solution.

## 4 Generalization to higher codimension brane-universe

We would now like to generalize the previous results to the case of a brane-universe of codimension greater than one. Requiring rotational invariance in the transverse space, the ansatz for the metric and for the ()-differential form will be a function only of the distance in the transverse space:

(41) |

The solutions (27–29) take the same form, but the powers are now given by:

(42) |

and the relation between the Wess–Zumino coupling and the dilaton coupling becomes:

(43) |

The function is harmonic in the transverse space:

(44) |

The rotational invariant solution is

(45) |

where is an arbitrary constant and is the volume of . (When the sphere degenerates into two points, giving .) The case of a brane of codimension two involves logarithmic behavior, and we will not specify it in the following. Whatever the value of is, (45) gives a solution to the equations of motion, however the solution associated to has enhanced symmetry properties. Moreover, as we will now demonstrate, when the dilaton decouples from the brane, the geometry of this solution can be derived from effective cosmological constants. Indeed the components of the Einstein tensor associated with the solution (45) are

(46) | |||||

(47) |

When the dilaton decouples, , implying and , the metric can then be written as:

(48) |

with

(49) |

This is the geometry of ; is the radius of the sphere and it is related to the radius of the space by . The expression of the Einstein tensor simplifies to:

(50) | |||||

(51) |

where the constants and are given by:

(52) |

What allows us to interpret them as effective cosmological constants is the fact that the metric (48) is actually a solution to the Einstein equations derived from a generalized RS action:

(53) |

where is defined by . It is noteworthy that when the metric in the transverse space is integrated out, i.e., fixed to the solution of its equation of motion, the action (53) reduces to the one introduced by RS.

In the expression (52), we notice that even if the power is positive, the cosmological constant on the brane is positive. Along the discussion of the section 2, this would not be the case with only one extra dimension, but when the extra transverse dimensions that live on the sphere also contribute to the singularity in the Einstein tensor and modify the singularity coming from the part of the space. Nevertheless, our discussion of the hierarchy problem is unaffected by the spherical extra dimensions and thus a positive power is undesirable as regards the gauge hierarchy problem, since it implies that the integral for the 4D effective Planck mass diverges. However a positive power naturally generates a gauge coupling unification along the lines of the scenario proposed in [12].

Just as in the case of codimension one, the effective cosmological constants are related to the charge associated to the electric current (23):

(54) |

which leads to their quantization in the multibrane configuration: goes to and goes to .

Not only does going to higher codimension brane-universes cure the problem of the imaginary Wess–Zumino term, but they can also be more easily embedded in a superstring framework. As we have seen, the theory defined by admits a -brane solution only when the two couplings and are related by eq. (31) that defines a line in the parameter space. On this line, one point, , has an effective description in terms of cosmological constants in the bulk and on the brane. On the other hand, there is another point where the -brane is supersymmetric, the couplings taking their stringy values. The two points may coincide as it is the case for the D-3 brane in type IIB theory or for the branes of -theory since they do not couple to the dilaton. At this stage, it would be interesting to incorporate in the field theoretical analysis of RS some stringy corrections to the supergravity action, like quadratic terms in curvarture, for instance, since they can modify the spectrum of the Kaluza–Klein graviton’s excitations.

## 5 Discussion

In this work we have presented solutions to the coupled equations for branes in extra dimensions and the low energy bosonic states of supergravity or superstring theories. The goal was to reproduce the effective stress-energy tensor needed for the Randall-Sundrum solution which uses gravitational trapping to solve the weak scale hierarchy problem. Let us summarize the results.

### Decoupled dilaton regime

Regardless of the dimensionality of the tranvserse space, we find that the stress-energy tensor takes a simple form only in the limit that the dilaton field decouples from the brane. Then there are three cases:

. It is necessary to go to an unphysical value of the Wess-Zumino coupling, , to obtain a solution, which does however then yield exactly the bulk and brane cosmological constants needed for the RS proposal.

. This appears to be an uninteresting case, because is forced to vanish, leading to trivial solutions.

. We now find solutions with positive and
physically acceptable values for the Wess-Zumino
coupling. Some of the solutions are supersymmetric and are identified
as the usual branes of string theories.
The bulk energy term looks conventional (constant) in the
brane components of , but it has a mild dependence on the
bulk coordinates in the tranvserse components, . The warp
factor goes like in coordinates where
represents the physical distance from the brane in the bulk (). Therefore the solution cannot be advocated to explain the
hierarchy between the Planck and electroweak scales. This is in
qualitative agreement with the solution recently found in
ref. [18]. It would therefore appear that the RS solution to
the hierarchy problem works only in the case of a single extra
dimension^{7}^{7}7Numerical solutions which we have found in the case
of also support this conclusion., or in the case of
several intersecting branes of codimension one. On the other hand, as
shown in ref. [12], despite infinitely large extra
dimensions, gauge coupling unification can naturally arise as a result
of the anomaly associated with the rescaling of the wave functions on
the brane. Moreover the presence of the spherical extra dimensions can
help to cure some phenomenological puzzles which occur when there is
only one transverse dimension, such as electroweak symmetry breaking
and obtaining small enough neutrino masses [12].

### Coupled dilaton regime

It is interesting to also consider the solutions where the dilaton does not decouple from the brane. The bulk energy is no longer constant in these solutions, so the resulting stress-energy tensor does not have the simple form proposed by RS. Nevertheless, these solutions are equally acceptable and may have interesting physical consequences.

. It is now possible to have a real-valued Wess-Zumino coupling and in this regime the metric develops a horizon at a finite distance from the brane.

. The solutions are no longer trivial, but have a logarithmic dependence on the bulk coordinate. We have not studied this special case in detail.

. The term in which looked like a bulk cosmological constant when the dilaton coupling vanished now has nontrivial spatial dependence in the bulk. Such behavior has recently been proposed as a condition for avoiding the generic problem of the incorrect Friedmann equation for the expansion of the brane [39]. In the latter, complicated and a priori unmotivated expressions for the dependence of on were derived using the requirement of correct cosmological expansion. Although we have not yet found inflationary solutions in the present context, it would be interesting to do so in order to check whether the dependence of advocated in ref. [39] can be justified by the presence of nontrivial dilaton fields.

## Appendix: the boundary of an anti-de Sitter space

An anti-de Sitter space of dimension can be seen as a hypersurface embedded in a flat space of signature (2,). Let , , be some coordinate system in this embedding space. The anti-de Sitter space of radius is defined by the equation:

(55) |

and the metric on is the embedding metric. In a convenient system of coordinates defined by

(56) |

the embedding metric factorizes:

(57) |

The boundary of is the set of points that satisfies equation (55) at the infinity of the flat space. More precisely, we can rescale the coordinates and consider the limit . The boundary is thus defined by the projective equations

(58) | |||

(59) |

which clearly describe . In the system of coordinates (56), the set of solutions to the boundary equations has two disconnected pieces: the first one is associated with , which is sent to by the rescaling, and it corresponds to a Minkowski space of dimension spanned by ; the second piece is associated with , i.e. , and corresponds to the union of a point and