Our main objective is to study how braneworld models of higher codimension differ from the 5D case and traditional Kaluza-Klein compactifications. We first derive the classical dynamics describing the physical fluctuations in a wide class of models incorporating gravity, non-Abelian gauge fields, the dilaton and two-form potential, as well as 3-brane sources. Next, we use these results to study braneworld compactifications in 6D supergravity, focusing on the bosonic fields in the minimal model; composed of the supergravity-tensor multiplet and the gauge multiplet whose flux supports the compactification. For unwarped models sourced by positive tension branes, a harmonic analysis allows us to solve the large, coupled, differential system completely and obtain the full 4D spin-2,1 and 0 particle spectra, establishing (marginal) stability and a qualitative behaviour similar to the smooth sphere compactification. We also find interesting results for models with negative tension branes; extra massless Kaluza-Klein vector fields can appear in the spectra, beyond those expected from the isometries in the internal space. These fields imply an enhanced gauge symmetry in the low energy 4D effective theory obtained by truncating to the massless sector, which is explicitly broken as higher modes are excited, until the full 6D symmetries are restored far above the Kaluza-Klein scale. Remarkably, the low energy effective theory does not seem to distinguish between a compactification on a smooth sphere and these singular, deformed spheres.

DESY 09-009


General Perturbations for Braneworld Compactifications

and the Six Dimensional Case

S. L. Parameswaran111Email:, S. Randjbar-Daemi222Email: and A. Salvio333Email:

II Institute for Theoretical Physics,

University of Hamburg, DESY Theory Group, Notkestrasse 85, Bldg. 2a, D-22603 Hamburg, Germany

International Center for Theoretical Physics,

Strada Costiera 11, 34014 Trieste, Italy

Institut de Théorie des Phénomènes Physiques,

EPFL, CH-1015 Lausanne, Switzerland


IFAE, Universitat Autònoma de Barcelona,

08193 Bellaterra, Barcelona, Spain

1 Introduction

Almost two decades on, branes are evermore ubiquitous in the models constructed to understand particle physics and cosmology, with all their How?’s and Why?’s. As fundamental objects, they are the D-branes and NS-branes (or M-branes) of string (or M) theory, but within a low-energy effective field theory description, they are introduced as braneworlds. Often these braneworlds are considered as infinitely thin but finite tension objects, like for their more fundamental cousins, although sometimes it proves necessary to resolve their structure by adding some thickness.

A codimension one brane necessarily forms a boundary in the bulk space, since there is no path which can lead from one side to the other without traversing the brane. The gravitational backreaction of these objects is well understood; whilst the metric is continuous across the brane, its first derivative can have finite discontinuities. Branes with more than one transverse dimension are qualitatively different, and much harder, due to their sourcing of singularities in the transverse space. Still, codimension two branes can also be handled with some control; they backreact on the geometry in such a way as to produce relatively mild conical singularities.

The construction of solutions sourced by branes, with up to two codimensions, in various field theory models is by now a well-developed art. In 5D, the archetype is of course the construction of Randall and Sundrum [1, 2]. In 6D, we take the general warped braneworld compactifications (“conical-GGP solutions”) of 6D N=1 gauged supergravity [3] found in [4, 5] as representative. These solutions additionally invoke fluxes, which are also playing a dominant role in string compactifications today, and indeed models with two extra dimensions are the simplest in which flux compactifications can be studied. Having established the solutions, we can begin to ask about their physics: Are they stable to small perturbations? What are the symmetries and particle content of the low energy effective field theory? Is it chiral? What are the modifications to 4D Einsteinian gravity? What would be the effective vacuum energy measured by a 4D observer? What role do the branes play in these and other phenomena? And so on.

The first step towards answering these questions is to analyze the classical spectra of small fluctuations around the solution. A number of such studies have been made recently for the conical-GGP solutions. In [6] we worked out the spectra for certain 4D gauge fields and fermions present in the model and no tachyons or ghosts were found amongst them. A similar (marginal) stability was found in [7], where the axially symmetric modes for some of the scalar perturbations were calculated. The spectrum for the gravitino has also been analyzed in [8]. In [9], meanwhile, we studied the tachyonic instabilities that can arise from the non-axially symmetric, 4D scalar fluctuations descending from 6D gauge fields, and charged under the background fluxes444The end point of this instability is studied in [10].. Whether a given model with a given flux suffers from this instability turns out to depend on the tensions of the branes present.

We now intend to complete the spectral analysis for the bosonic fluctuations about the braneworld solutions of 6D supergravity. Our particular focus in this paper is on the so-called Salam-Sezgin sector – that arising from the supergravity-tensor multiplet and the gauge multiplet in which the background monopole lies – which was partially treated in [11, 7]. The remaining sectors have been completed elsewhere [6, 9]. We will calculate the corresponding spectra for the 4D spin-2 and – for unwarped backgrounds – spin-1 and spin-0 fields. The model that we are studying is complicated, and technically difficult. However, this goes hand in hand with its advantage of generality, and indeed the results for several simpler scenarios can be extracted from our work at its various stages.

Our approach will be that established in [12], where a formalism was developed to analyze the spectra of small perturbations about arbitrary solutions of Einstein, Yang-Mills and scalar systems. The first part of this paper can be considered as a generalization of that work, where we now include the presence of thin source 3-branes and extra bulk fields that are generically present in supergravity theories; the dilaton and anti-symmetric two-form potential. With little extra cost, we actually keep the number of dimensions transverse to the brane general.

We first derive the general form of the bilinear action that describes the behaviour of small fluctuations. For codimension-two or higher, we include fluctuations of the brane positions in the transverse directions, the so-called “branons”. We then apply the light-cone gauge (for bulk fields) and static gauge (for branons) to restrict to physical degrees of freedom, and decouple the dynamics for the spin-2, -1 and -0 fluctuations. The gauge-fixed bilinear action thus obtained provides the starting point to calculate the Kaluza-Klein (KK) spectra for the conical-GGP solutions, as well as, for example, the 5D Randall-Sundrum models and the non-supersymmetric Einstein-Yang Mills(-dilaton) model in any dimension.

In the second part of this paper, we use these general equations to study the behaviour of braneworld models in 6D (along the way also recover some of well-known aspects of the 5D scenarios). Here, since we include the backreaction of the branes, the dynamics of the branons are not well-defined555Indeed, the behaviour of the branons is usually considered under the probe brane approximation, in which the brane tension is much smaller than the bulk gravitational scale, so that the backreaction can be neglected [13, 14].. Therefore, to study the spin-0 sector, we choose to truncate the branons by e.g. placing the branes at orbifold fixed points, or taking the brane tensions to be very large making the branes rigid within our range of validity. Meanwhile, the conical singularities in the curvature that are induced by the codimension two branes do not prevent us from understanding the behaviour of the bulk fluctuations.

We are able to derive the spectrum for the 4D spin-2 fields in the model’s full warped generality. The spin-1 and spin-0 sectors present large coupled differential systems, and by finding a set of harmonics on the 2D internal space ( the “rugbyball”), we are also able to solve these systems analytically for the unwarped case. In this way, we obtain all the 4D modes for unwarped compactifications with positive tension brane sources, and qualitatively, we observe the same behaviour as in the smooth sphere compactification without branes – including marginal stability.

In the presence of negative tension codimension-two branes, meanwhile, the physics can surprise. Here, despite the fact that brane sources clearly break the isometries of the sphere to , three massless spin-1 fields666In addition to any massless gauge fields arising from unbroken higher dimensional gauge symmetries. can be found amongst the KK spectra for special values of the conical deficit angle. These special deficit angles, , allow three Killing vectors to be well-defined everywhere outside the branes, although only one of them can be globally integrated to an isometry.

Whether or not the massless vectors are gauge bosons of an enhanced gauge symmetry in the 4D theory can be understood by going beyond bilinear order and considering the interaction terms. We find the presence of KK modes that are not in well-defined representations of the generated by the Killing vectors, and therefore the full 4D theory does not enjoy an gauge symmetry. For this reason, we do not expect the classical masslessness of the vector fields to survive quantum corrections. Meanwhile, all our bosonic massless modes do fall into well-defined representations, and therefore we argue that the classical low energy 4D effective field theory – obtained by truncating to the massless sector – does enjoy an enhanced KK gauge symmetry beyond the isometries! Moreover, it appears that the low energy theory does not distinguish between compactifications on the smooth sphere and these singular, deformed spheres.

Let us now give an outline for the remainder of the paper. The first part presents a rather general analysis that determines the dynamics of perturbations in braneworld compactifications. In the next section, we introduce the model (both theories and background solutions) and discuss the scenarios to which our analysis can be applied. In Section 3, we introduce the perturbations about the background, obtain the bilinear action that describes their dynamics, and discuss the local symmetries of this action. In Section 4, we use these symmetries to fix to the “light cone static gauge”, and give the bilinear action in this gauge, in which the different spin sectors decouple.

Then begins the second part, which uses the previous results to study the 4D fields that emerge in various scenarios. In Section 5, our main interest is in the braneworld solutions of 6D supergravity, but we also discuss a non-supersymmetric 6D model and the 5D Randall-Sundrum models. In the main text we present the KK spectra for spin-2 and spin-1 fields and identify the massless spin-0 fields; the complete spin-0 sector can be found in the appendices. Finally, we understand in detail the physical significance of the extra massless 4D vector modes that can appear in the spectra, and the gauge invariance that emerges in the 4D theory.

We summarise our results in Section 6, before concluding in Section 7.

2 The Model

We begin with the definition of our model. The main focus of the present paper will be a class of bosonic 6D field theories with thin codimension-two branes. In particular we are interested in the bosonic part of 6D N=1 gauged supergravity [3]. However, throughout the article we shall keep a general space-time dimension as far as possible, and certain truncations of the field content allow our analysis to be applied to several different scenarios, including the non-supersymmetric Einstein-Yang-Mills theory or the Randall-Sundrum Model.

2.1 Field content

The basic ingredients of our model are the higher dimensional metric , where the space-time indices run over , and the gauge field of a compact Lie group . These are bulk fields in the sense that they depend on all the space-time coordinates .

We also want to consider a certain number of 3-branes embedded in the -dimensional space time. To do so we introduce, following Ref. [15], functions , which represent the positions of the branes in the -dimensional space time. The represent the 4D coordinates on the brane, , where are the 4D indices. Not all the space-time components of are physical degrees of freedom: 4 space-time components for each can be gauged away by using the 4D (general) coordinate transformation invariance acting on [15], as we will explicitly do in Subsection 4.1. We consider to be a brane field because it depends only on a 4D world-volume coordinate. These fields are important to introduce the branes in a covariant way, and indeed we can construct the induced metrics on the branes by means of


In order to complete the bosonic part of the 6D supergravity, one should add other bulk fields in addition to and , that is a dilaton and a 2-form field , which emerge from the graviton multiplet and an antisymmetric tensor multiplet [3]. We will refer to as the Kalb-Ramond field. Moreover, concerning the 6D supergravity, we shall assume that is a product of simple groups that include a gauged R-symmetry. In general one can also add some hypermultiplets [3], which turn out to be important to cancel gauge and gravitational anomalies [16, 17]. In the bosonic sector this leads to additional scalar fields (hyperscalars) in some representation of ; however, from now on we set . We do so because we are interested in the linear perturbations which mix with the -dimensional gravitational fluctuations : indeed, for the class of backgrounds we are interested in (see Subsection 2.3), the decouple from . Their inclusion should be straightforward.

Therefore the bulk and the brane field contents that we consider are respectively:


The dots in the second set of (2.2) represent additional brane fields that we can always introduce, but which are not required by general covariance; for example they can be the fields of the Standard Model (SM).

2.2 The action

We split the action functional into the bulk action , which depends only on the bulk fields, and the brane action that is a functional of the brane fields as well.

The bulk action is777We choose signature , and define and .[3]


where is the determinant of and is the -dimensional Planck scale; also888A trace overall is understood when we write a product of Lie algebra valued objects: e.g. in Eq. (2.3) . and . The explicit expression for the gauge field strength is999We define the cross-product as , with the structure constants of : , where are the generators of .


where is the gauge coupling, which in fact represents a collection of independent gauge couplings including that of the subgroup, . is the Kalb-Ramond field strength, which contains a Chern-Simons coupling as follows [18]:


The function is the dilaton potential. In the supersymmetric model this is fixed to be .

Meanwhile, we consider the following 3-brane action


where is the determinant of (2.1) and are the tensions of the branes. From now on (unless otherwise stated) we suppress the index , as we have done on the right hand side of (2.6). The reader may have noticed that we have not introduced the Gibbons-Hawking boundary term, which is generically necessary to treat codimension one branes [19]. Indeed, we shall apply our analysis only to those codimension one models whose branes are placed on orbifold fixed points, in which case the Gibbons-Hawking boundary term is not present [20].

We can summarise by saying that our analysis will apply to the following two types of models:

  1. 6D N=1 gauged supergravity.

  2. Einstein-Yang-Mills theories, with a dilaton or cosmological constant , for a general space-time dimension.

The second case includes, for example, the RS models [1, 2] or the non-supersymmetric 6D Einstein-Yang-Mills- (EYM) model [13, 21]. They can be obtained by simply fixing the appropriate dimension and setting , and . Even if our main interest is in models of Type 1 we will also consider the second class for several reasons. In this way, we will see that our results can be applied in quite general contexts, and it will also provide interesting additional ways to check our formulae. Moreover, in the future it should help us to figure out the role of supersymmetry in the linear perturbations.

Finally, it is important to note that the actions and are invariant with respect to both the -dimensional and the 4D coordinate transformations (acting respectively on and ). We will discuss the local symmetries of the present model and an explicit gauge fixing for the linear perturbations in Subsections 3.2 and 4.1.

2.3 The equations of motion (EOMs) and solutions

The EOMs that follow from the variation of the action are:


where we have fixed , since our interest shall be in backgrounds that enjoy 4D Poincaré invariance. Moreover, in Eq. (2.9) and (2.10) we have introduced the notation , where is the covariant derivative, and . Recall also that we have suppressed the index on , which labels each of the branes. The last term in (2.7) represents the brane contribution to the Einstein equations, where is defined by


we note that the bulk quantity in (2.11) is computed at the position of the brane () because of the presence of the -dimensional delta function . Furthermore, since Eqs. (2.10) come from the variation of the brane action with respect to , there the bulk fields and are computed at the brane position ( and ).

In the present paper we will focus mainly on the following ansatz solution to (2.7)-(2.10):


where , , (we have ) and and are respectively the coordinate and the metric on the -dimensional space. Eq. (2.12) is not really an assumption because we can always use the 4D general coordinate invariance on the branes to set (2.12). Eq. (2.13) is instead a non trivial assumption. Moreover, in Eqs (2.14)-(2.17) we are assuming that the bulk field background has a 4D Poincaré invariance and that the functions , and depend only on the coordinate . We will also assume to lie in the Cartan subalgebra of Lie.

One of the simplest models that can be described by this set up is the Randall-Sundrum (RS) model [1], where we have , and and the internal space is with two branes on the fixed points of , say at and . The explicit form of the solution is given by


where is a positive constant. The object in (2.18) is equal to the absolute value of in the region and its value anywhere else is obtained by periodicity. In order for (2.18) to be a solution one needs and . In Section 5, we shall use this very well-known solution to check the result given in Section 4.

However, in this paper our main interest lies in the analysis of a class of solutions found by Gibbons, Güven and Pope (GGP) [4] to the 6D supergravity: the general set of warped solutions with 4D Poincaré symmetry, and axial symmetry in the transverse dimensions. Here we give only a subset of this general class, namely that which contains singularities no worse than conical and therefore can be sourced by brane terms of the form (2.6).

To give the explicit expression of the conical-GGP solutions, it turns out to be useful to introduce the following radial coordinate [6]


whose range is . In this frame the metric reads


The explicit conical-GGP solutions101010The coordinate is related to the coordinate in [4] by . are then the following particular case of the ansatz (2.12)-(2.17) [4]:


where and are generic real numbers and is a generator of a subgroup of a simple factor of , satisfying Tr. Also,


with and .

This solution is supported by two branes located at and . Indeed, as or , the metric tends to that of a cone, with respective deficit angles


and corresponding delta-function behaviours in the Ricci scalar. We will take without loss of generality. The tensions of the two branes and are related to the deficit angle as follows [22]:


Unlike the RS solution, here the warp factor is smooth on the brane positions and . In particular we have


By using (2.25), (2.12) and (2.13), it is also easy to check that the conical-GGP configuration satisfies the -equations (2.10) in addition to the bulk EOMs (2.7)-(2.9).

The expression for the gauge field background in Eq. (2.21) is well-defined in the limit , but not as . We should therefore use a different patch to describe the brane, and this must be related to the patch including the brane by a single-valued gauge transformation. This leads to a Dirac quantization condition, which for a field interacting with through a charge gives


where is an integer that is called monopole number and is the gauge coupling constant corresponding to the background gauge field. For example, if lies in , then . The charge can be computed once we have selected the background gauge group, since it is an eigenvalue of the generator . Also, note that the internal space corresponding to Solutions (2.21) has an topology (its Euler number equals 2).

Finally, we observe that one can obtain the unwarped “rugbyball” compactification [21] simply by setting . In this case the metric is


where , and the background value of the dilaton is zero; therefore this is a solution also to the non-supersymmetric 6D EYM model. For the deficit angle is positive. The geometry is also well-defined when and the deficit angle is negative; we name these spaces “saddle-spheres” (see [9] for a detailed discussion on their properties). Moreover, we can smoothly retrieve the sphere compactification (with radius ) by taking in addition to .

3 General Perturbations

The main purpose of this paper is to study the linear perturbations in the above models. We therefore perturb the fields in (2.2) as follows:


The first terms in the right hand sides of (3.28) represent the background quantities of the corresponding fields. In fact, it is useful to introduce another 2-form field in order to describe the fluctuations of the Kalb-Ramond field. This can be done as follows. Since appears only quadratically in (2.3), and at the background level due to 4D Poincaré invariance, the linear approximation (which corresponds to the bilinear level in the action) involves only the linear perturbation of , that we denote with111111Since the background , and the background monopole, , lies in the Cartan subalgebra, we see that the exterior derivative acting on the background Kalb-Ramond potential must be zero. Also, . ,


where we have used the notation of p-forms and is the fluctuation in the Kalb-Ramond 2-form, and the background values of the gauge field and its field strength respectively and the perturbation of the gauge field. We now introduce the 2-form as follows:


whose components will be denoted by . can now be expressed in terms of and :


where we have introduced a new parameter ; for we recover the structure of required by the 6D supergravity, whereas for the fluctuations of are completely decoupled (at the linear level) from the rest. This will allow us to treat simultaneously the 6D supergravity and the EYM models.

Finally, we note that the fields describe the fluctuations of the brane positions, and as such they are 4D fields.

3.1 Bilinear action

Here we provide the linearized theory which corresponds to the bilinear approximation in the action. The bilinear action has been computed by considering the variation of under (3.28) and by keeping only terms up to the quadratic order121212The EOMs (2.7)-(2.10) guarantee that the linear terms vanish.. We split it into different contributions as follows:


where is the bilinear action that depends only on the fluctuations , represents the mixing term between and and so on. We have as a consequence of our background ansatz, for which . We give here the explicit expressions for the bilinear action that depend only on the bulk fields; the dynamics of the fields, are explicitly given in Appendix A. We find:


where the semicolon denotes the (background) gravitational covariant derivative, , is the Riemann tensor for the background metric and we have defined




The term proportional to in the last line of (3.33) is the contribution to coming from the brane action , whereas the term proportional to comes from the EOMs (2.7), which we have used to write in the form (3.33). Moreover,



We would like to remind the reader of the assumptions we have made to derive (3.33) and (3.36)-(3.42) (and (A.110)-(A.111) given in Appendix A):

  • If the Kalb-Ramond field and the term in (2.3) is not included, then the only assumption we made is that the background satisfies the EOMs (2.7)-(2.10).

  • If the Kalb-Ramond field and the term in (2.3) is instead included, we also assumed and the background gauge field to lie in the Cartan subalgebra.

We observe that if we want to focus on the -dimensional EYM system we can restrict ourselves to the terms , (for ), and the -dependent terms given in Appendix A. Instead, if we want to consider the 6D supergravity, we should put , and also take into account the terms (3.38)-(3.42). Finally, we note that our results reduce to those of Ref. [12] which studies a general non-supersymmetric class of thick brane models, once we take , and we neglect the fluctuations .

3.2 Local symmetries

As a consequence of the local symmetries of the complete model, the linearized theory also possesses a number of local symmetries:


Eqs. (3.43), (3.44) and (3.45) represent the effect of the local symmetries (descending from the -dimensional coordinate transformation invariance and gauge symmetry) on the metric, the gauge field and the dilaton fluctuations (see e.g. Ref. [12]). The bulk functions and are the gauge functions associated with the -dimensional coordinate invariance and gauge symmetry.

Eq. (3.46) represents instead a local symmetry acting on , which descends from both the gauge symmetry and the Kalb-Ramond symmetry131313By Kalb-Ramond symmetry we mean the local invariance under of the action, where is a general 1-form.. For this reason and are independent (bulk) gauge functions. Let us explicitly check (3.46). To do so, it is enough to verify the invariance of the 3-form (3.31) under (3.44) and (3.46). We have


where we have used and represents the 1-form with components . Now, by using the 4D Poincaré invariance of the background and , which we always assume in the presence of the Kalb-Ramond field, we have and ; also, by remembering that is assumed to lie in the Cartan subalgebra, we have . These equations are sufficient to conclude .

Finally, Eq. (3.47) represents the local transformation of the perturbation of the brane position, descending from the -dimensional coordinate invariance and the 4D brane coordinate transformation invariance (respectively the first and the second term on the right hand side of (3.47)); the latter invariance is associated to (a function of ), which represents another independent gauge function.

4 Perturbations in the Light Cone Static Gauge

Having derived the general bilinear action, we now have to choose a gauge in order to study the physical spectrum. In this section we will discuss our gauge choice and give the corresponding bilinear action.

4.1 Gauge fixing

We have two types of local symmetries: the bulk local symmetries (which include the -dimensional coordinate transformation invariance, the gauge symmetry and the Kalb-Ramond symmetry) and the 4D coordinate transformation invariance on the brane. Let us start with the first group.

A very convenient gauge choice for the bulk local symmetry is the light cone gauge, as it ensures that the dynamics of sectors with different spin decouple at the bilinear level141414This has been observed in other studies, for example [24, 25, 26, 12].. Another advantage of the light cone gauge is that it does not involve gauge artifacts such as Faddeev-Popov ghosts, but contains only the physical spectrum [23, 24, 25]. To define this gauge, let us introduce and , for a general vector . Then the light cone gauge is defined by


It can be proved that, after imposing (4.49), the components of the different fields (i.e. , and ) are not independent, but can be expressed in terms of the other components by means of constraint equations [24, 25, 12]. We therefore end up with the following independent bulk fields: , , , , , , , and , where . In particular the field equation simply leads to the constraint


which brings a considerable amount of simplification.

Concerning the 4D coordinate transformation invariance, we instead impose the condition [15]


We will refer to (4.51) as to the static gauge. We observe that the light cone gauge and the static gauge are compatible because, once we fix the light cone gauge by choosing , and in a suitable way, we still have the freedom to perform the local transformations generated by . The static gauge is also free from Faddeev-Popov unphysical ghosts [15]. We observe that (4.51) does not remove completely the brane position fields , but we are left with their components along the extra dimensions . We will refer to them as branons. Even if the branons represent physical degrees of freedom, it can happen that they can be consistently truncated e.g. by imposing an orbifold symmetry, as in the RS models or in the conical-GGP compactification [9]. In the following we will confirm that the spin-0 fields do not have any mixing with the spin-2 and spin-1 sectors in the light cone gauge.

4.2 Bilinear action in the light cone static gauge

Here we provide the bilinear action in the light cone static gauge, that we have computed by imposing the gauge conditions (4.49) and (4.51) on the general bilinear action and by using the constraint equations for the components. In this section we assume the form given in (2.12)-(2.17) for the background solution, and give the part of the action that is independent of the branons. Those involving the branons are given in Appendix B.

The results that are presented here reduce to those for the non-supersymmetric model present in151515We do, however, correct some typos in that reference. [12] once we take , and we neglect the fluctuations ; they also correctly reduce (for and ) to the results of [27], where the linear perturbations of the sphere-monopole solution to the 6D supersymmetric model are analyzed.

4.2.1 Spin-2 action

The spin-2 action only contains the field and has the following simple expression in terms of :


We observe that (4.52) has exactly the same form as in [12] even if we have included the brane terms. Therefore, the brane sources do not explicitly contribute to the spin-2 dynamics. We shall use (4.52) to derive the 4D gravitational spectrum for the solutions described in Subsection 2.3.

4.2.2 Spin-1 action

The spin-1 action involves and . We have the following explicit expressions.


where . The last term in (4.53) is the brane contribution. We have introduced the notation and for the internal components of the coordinate and the brane position respectively, where the label stands for the codimension of the brane. The other non vanishing terms are the following.