UG0009
CERNTH/2000192
KULTF2000/20
hepth/0007044
Supersymmetry in Singular Spaces
Eric Bergshoeff , Renata Kallosh,, and Antoine Van Proeyen
Institute for Theoretical Physics, Nijenborgh 4, 9747 AG Groningen, The Netherlands
[3mm] Theory Division, CERN, CH1211 Genève 23, Switzerland
[3mm] Instituut voor Theoretische Fysica, Katholieke Universiteit Leuven,
Celestijnenlaan 200D B3001 Leuven, Belgium
Abstract
We develop the concept of supersymmetry in singular spaces, apply it in an example for 3branes in and comment on 8branes in . The new construction has an interpretation that the brane is a sink for the flux and requires adding to the standard supergravity a form field and a supersymmetry singlet field. This allows a consistent definition of supersymmetry on a orbifold, the bulk and the brane actions being separately supersymmetric.
Randall–Sundrum braneworlds can be reproduced in this framework without fine tuning. For fixed scalars, the doubling of unbroken supersymmetries takes place and the negative tension brane can be pushed to infinity. In more general BPS domain walls with 1/2 of unbroken supersymmetries, the distance between branes in some cases may be restricted by the collapsing cycles of the Calabi–Yau manifold.
The energy of any static dependent bosonic configuration vanishes,
, in analogy with the vanishing of the Hamiltonian in a closed universe.
On leave of
absence from Stanford University until 1 September 2000
Onderzoeksdirecteur, FWO, Belgium
Contents
1 Introduction
Usually, supersymmetry is associated with nonnegative energy. In asymptotically flat spaces this is related to the time translation operator which is a square of fermionic operators . In curved space the positivity of energy theorem is proved via Nestor–Israel–Witten construction where it is usually assumed that the space is nonsingular. In case of black holes with the singularity covered by the horizon, the argument about the positivity of energy was also extended [1].
Recently there were number of reasons to reconsider the issues of supersymmetry, in general. In the brane world scenarios with fine tuning where singular branes are introduced [2, 3], the status of the supersymmetric embedding was not clear. The first attempts to relax the fine tuning was to find smooth solutions of supergravity where domain walls are build from some scalar fields [4, 5, 6, 7, 8]. With respect to the ‘alternative to compactification scenario’ [3], nogo theorems were established [9, 10] for the BPS smooth solutions of a certain class of supergravity theories ( with vector [11, 12] and tensor multiplets [13]) on the basis of the negative definiteness of the derivative of the function in the relevant renormalization group behavior (UV behavior).
More recently, the complete supergravity theory was constructed in [14] where also the hypermultiplets are included. The status of hypermultiplets in the adS brane world is not yet settled^{1}^{1}1J. Louis, talk at SUSY2K, July 2000., more work is required. Even if the solutions with the IR fixed point will be found, there will still be a problem to find a configuration in a smooth supergravity which will connect two such IR fixed points. This may be also explained following a suggestive argument^{2}^{2}2K. Stelle, talks at Fradkin and Gürsey conferences, June 2000.: BPS solutions are expected to be given by harmonic functions, in codimension 1 they must have a kink. A kink can not appear as a smooth solution. The complimentary argument was given in [15]: the fermion mass does not change sign when passing through the wall, which may explain the absence of smooth solutions with the required properties. The version of a nogo theorem for smooth brane worlds from compactifications, under some assumptions about the potential, was recently proposed in [16]. None of these arguments seems to give an unconditional final proof that there are no smooth domain walls for a brane world scenario. However, they suggest to find out whether the clear framework is available for the nonsmooth supersymmetric solutions.
Thus the purpose of this paper is to generalize supersymmetry for singular spaces, where the curvature may have some function singularities. In singular spaces there was no consistent and complete definition of supersymmetry so far^{3}^{3}3We will comment on the available information [17, 18, 19, 20, 21, 22] below. and the related issue of nonnegative energy was not clearly addressed.
An important reason to reformulate supersymmetry in singular spaces is related to supersymmetric domain walls: objects of codimension , i.e. branes in dimensional space. They may be associated with form fluxes which are dual to a scalar and piecewise constant. Such forms do not fall off with distance. In an infinite volume, they may lead to an infinite energy and would be unphysical. Therefore, objects like the brane in and 3branes in may not exist as independent objects. They may need some planes that serve as sinks for the fluxes for supersymmetric configurations of codimension 1 [23]. Note that the fluxes in the branes of higher codimension do vanish at infinity and this problem can be avoided.
Usually local supersymmetry is realized in supergravity when the Lagrangian is integrated over a continuous space. Under supersymmetry, the Lagrangian transforms as a total derivative. The parameters of local supersymmetry are assumed to fall off at infinity and therefore the action is invariant. If in addition to the supergravity action one considers the symmetric worldvolume action of the positive tension brane of codimension , it provides the functional sources for the harmonic functions that describe the configuration, .
In the case of branes of codimension defined on an orbifold , the harmonic functions satisfy the equations . This can be seen e.g. in the Hor̆ava–Witten (HW) construction [17] developed for 5dimensional 3branes in [18]. The metric of such solutions depends on and has two kinks at the orbifold fixed points: one at and the other at where is identified with . The supersymmetry in HW construction includes the contribution from anomalies and requires quantum consistency. It has some problematic features related to higher order corrections and quartic fermionic terms.
To set up a new general point of view on supersymmetry in singular spaces, and to clarify the issue of energy of brane worlds, we introduce here a set of rules defining consistent supersymmetry on singular spaces.
The most important new steps include: i) some constants (masses or gauge couplings) have to become “odd” to make supersymmetry commuting with symmetry. This can be achieved by promoting such a constant to the status of a supersymmetry singlet field, as suggested in [24]; ii) moreover, one has to add to the theory the form potential^{4}^{4}4The importance of the form field was realized by Duff and van Nieuwenhuizen [25], who pointed out 20 years ago the quantum inequivalence of the theories with and without such field in the context of trace anomalies and 1loop counterterms in topologically nontrivial backgrounds. At about the same time Aurilia, Nicolai and Townsend [26] have found that the form, which propagates no physical particles, carries a surprising physics. They have looked at parameter in QCD and at the cosmological constant in supergravity. to compensate the variation of the Lagrangian proportional to the derivative of the new field; iii) one can find afterwards the new bulk and the brane actions, which are separately invariant under supersymmetry.
The unusual features of the new supersymmetry are the presence of the supersymmetry singlet field and form field in the bulk, and the fact that the purely bosonic action on the brane is supersymmetric due to the fact that its fermionic partner is a odd fermion which vanishes on the brane.
In absence of brane actions, the new fields become irrelevant: on shell for the 4form and supersymmetry singlet, the bulk action reduces to the standard supergravity action supersymmetric under the standard rules.
In general, when the brane actions are added, the new fields play an important role in understanding the energy issue. We will find that the total energy of supersymmetric configurations vanishes locally at each brane. The positive (negative) energy of the brane tensions are compensated separately by the terms with the derivative of the supersymmetry singlet field. The energy of any static dependent bosonic configuration vanishes locally, , in analogy with the vanishing of the Hamiltonian in a closed universe^{5}^{5}5We are grateful to A. Linde who suggested this analogy..
The strategy is applied in detail in the particular case of a 3brane in on the basis of gauged supergravity interacting with abelian vector multiplets. We expect that it will work in other cases as well: in one can try to include more general gauging and tensor and hypermultiplets. The gauged and supergravities in are also natural candidates for an analogous extension. We will give a brief discussion of the particularly interesting case of the 8brane in .
The paper has two main parts.
In Part I, The Supersymmetric Theory: Bosons And Fermions, we construct the supersymmetric actions in the bulk and in the brane. To do so, we first identify the operator in section 2. Then, we construct the supersymmetric theory with three steps in section 3. The final result is written down in section 4
In Part II, The Background: Vanishing Fermions, Bosons Solve Equation of Motion, we study bosonic solutions of the theory from part I and their unbroken supersymmetries. Section 5 discusses the vanishing of the energy. The BPS equations and preserved supersymmetries in these singular spaces are discussed in section 6. We consider two cases, the fixed scalars, with doubling of supersymmetry, and the general stabilization equations in very special geometry, with 1/2 preserved supersymmetry. We show the resulting formulas for the example of the STU wall, and for a particular Calabi–Yau wall in section 7.
Finally, in Part III, we discuss the similar mechanism for the 8brane in 10 dimensions.
The notations and use of indices are presented in appendix A. For those unfamiliar with the 5dimensional supergravity, and for establishing the related notations, we present its structure in appendix B. In appendix C we discuss the previous attempts to define supersymmetry on orbifolds.
PART I. The Supersymmetric Theory: Bosons And Fermions
2 Local supersymmetry and
Our setup has the following basic features.

Orbifold construction. Fields live on a circle in the direction , with an orbifold condition. The circle implies that , where is some arbitrary parameter setting the length of the circle. We use a concept of parity to split the fields in even and odd under a :
(2.1) This implies that the odd fields vanish at and at , where we will put the branes. The supersymmetries are also split in half even () and half are odd (). Both have 4 real components. The bulk action is even, and all transformation rules are consistent with the assignments.

The brane action is introduced. We place two branes, one at and another one at . The actions depend on the values of the bulk fields at and . As the odd fields are zero on the brane, the brane action only depends on the even fields. Only the supersymmetries act on these fields.
We analyse all possibilities to make parity assignments consistent with supersymmetry commuting with symmetry. We conclude that the consistent supersymmetry on orbifolds without brane actions does not fix the properties of the gauge coupling. However, after adding the brane action we must require the gauge coupling to be odd^{6}^{6}6The second possibility, even gauge coupling, as chosen in [19], will be discussed in Appendix C..
We have to treat the fact that is only piecewise constant. This approach is inspired by [24]. It consists of the following steps

Replace in the action the constant by a scalar function and keep this field a supersymmetry singlet. The supersymmetry singlet field was introduced in the context of symmetric brane actions in [27]. This field has some peculiar properties: its shift under translation vanishes on shell but not off shell.

Add source terms that are separately invariant, but may contain and and are dependent on , where these are the components in the 4 dimensions of the brane. After these additions the field equation says that , and thus . This provides a supersymmetric mechanism to change the sign of the coupling constant (and of the fermion mass) when passing through the wall.
We will first consider the possibilities for a in the 5dimensional action. Define the operator for any field, being or whether it is even or odd under . For the symplectic Majorana spinors, the splitting might involve some projection operators. Let us look for a of the form (for any symplectic Majorana spinor )
(2.2) 
where is thus a number or , while is so far an undetermined matrix. If this has to be a , the operation should square to . Therefore should square to . Notice that this is independent of whether we included in (2.2) or not. Next, this has to be consistent with the reality condition (symplectic Majorana condition, see appendix A). This implies that should satisfy , or thus, with , , and real. The condition implies that
(2.3) 
which means that is hermitian and traceless. If were not included in (2.2), the numbers would have to be pure imaginary, and the condition would imply , but then we would have no projection at all. Thus we conclude that (2.2) with (2.3) is the only one that is possible.
Lowering the two indices on , it becomes a symmetric matrix:
(2.4) 
The parity will be related to the fifth direction. We will therefore assign a negative parity to the coordinate, or . Consider now the supersymmetry transformation laws in the bulk, see (B.12). For one of the fermions we may arbitrary assign even parity, e.g. for the components of the gravitino in the directions excluding ‘5’, i.e. . Consider first the supersymmetry transformation laws that are independent of the gauge coupling . The consistency of the parity assignments determines
(2.5) 
Note that also the supersymmetry parameters got a parity projection and that the parameters of the gauge transformations, , have to be odd.
Now we consider the terms with in the gravitino transformation law (B.12). They depend on the constant matrix , see (B.6). Taking the parity transformation of both sides, we find that they are proportional to
(2.6) 
where we allowed a parity transformation of . We find that if and commute (which means that they are proportional) one needs , while if they anticommute (thus are taken in orthogonal directions in the space), .
Taking an anticommuting and brings us to the setup of [19]. We will see below, that the addition of brane actions forces us to take a matrix that commutes with . If and commute, one has two possibilities to implement the parity assignment . In the approach of [22], who take , this assignment is realized by replacing by . On the other hand, we will be able to make this assignment by promoting to a field.
To continue in the direction of [19, 20, 22], one has to provide the consistent definition of supersymmetry with step functions and delta functions present in supersymmetry rules and in the action. The construction of the higher order fermionic terms and the structure of the algebra of supersymmetry in singular spaces in these approaches are difficult as the HW theory shows. Instead of this, a new way to introduce supersymmetry in singular spaces will be developed below.
3 Supersymmetry in a singular space
3.1 Step 1: the bulk action
We consider 5dimensional supergravity coupled to vector multiplets. The coupling is determined by a 3index real symmetric constant tensor , where the indices run over values. We consider a gauging of the symmetry group determined by real constants . The direction in space is determined by a matrix . Obviously, the choice of that direction has no physical consequences. The full action, to which we will refer as the GST action [12], and transformation rules, are given in appendix B. Replacing the coupling constant by a scalar , the action is not any more invariant supersymmetry,
(3.1) 
and neither under the gauged symmetry,
(3.2) 
where is the parameter of the symmetry
(3.3) 
3.2 Step 2: the fourform
In the second step, we add the following Lagrange multiplier term:
(3.4) 
Now we can make the action invariant under supersymmetry. This is obtained 1) by taking invariant under supersymmetry, and 2) by defining the variation of such that all terms in the transformation of the rest of the action are cancelled. The fact that is invariant is consistent with the algebra because the translation of is a field equation ( is a supersymmetry singlet [27]). Thus the algebra is realized onshell. The resulting variation of under supersymmetry is
(3.5) 
Under gauge transformations the 4form transforms as follows
(3.6) 
We define the covariant flux as follows:
(3.7)  
The closure of the algebra on shell is due to a field equation
(3.8) 
where the bosonic part is related to the potential, given by
(3.9) 
This equation relates the flux to the potential via the singlet field (up to terms with fermions). We will see later that on shell the flux will change sign when passing through the wall. This will explain the role of the wall as a sink for the flux.
Remarks:

the action is invariant under 8 supersymmetries.

the Lagrangian is invariant up to a total derivative. This is sufficient if the fields either drop off at infinity (as it is supposed to be in the 4dimensional spacetime), or the space is cyclic and the fields are continuous (as it is supposed to be in the direction). To have at the end piecewise constant, but not everywhere the same, it is clear that we need at least 2 branes where it can jump.

The procedure outlined in step 2 is very general and does not depend on the details of the configuration.
We have explained in section 2 that the GSTaction allows two different parity assignments for . The action (3.4) respects both choices with the assignments
(3.10) 
3.3 Step 3: the brane as a sink for the flux
If we have only the bulk action, the field equation still implies that is constant everywhere. The flux is proportional to the potential and on shell for and we recover the standard supergravity.
We thus need sources to modify the field equation on . These we can choose according to a physical situation. For reproducing the scenario described above, we take two branes positioned at and at . For both branes we introduce a worldvolume action, which basically is the Dirac–Born–Infeld action in a curved background with all excitations of the worldvolume fields set equal to zero.
The brane action includes the pullback of the metric, the scalars and the 4form of the bulk action. The coefficient of the determinant of the metric is taken to be the function . It is related to the central charge, similar to what was obtained for black holes in , in [28]. The Wess–Zumino term describes the charge of the domain wall.
Remember that, as explained after (2.1), odd fields vanish on the branes. Therefore, if we want to use the pullback of the components on the brane, we need that their parity is even. This forces us to take the first choice in (3.10). This is consistent with the scenario of [22], but it is problematic to incorporate the approach of [19] in our framework of ‘consistent supersymmetry on singular spaces’.
The brane action is
(3.11) 
where is the determinant of the 4 by 4 vierbein , and is in the same way a 4density. The factor is a sign to be chosen later. The new field equation for is
(3.12) 
which has as solution
(3.13) 
for . One should understand as the function that is for , and for . Thus it has also a jump at , and
(3.14) 
Now we may look at the equations of motion for the flux with account of the value of the onshell field. The bosonic part of the flux (3.8) is onshell
(3.15) 
Clearly the flux is changing the sign when passing through the brane, which justifies the title of this section. This may be contrasted with the properties of the potential when the field is on shell:
(3.16) 
where the standard assumption is made that . The potential does not care about the existence of the brane.
The fermion mass terms on shell for the field also change the sign across the wall, as it follows from the terms in the action that are quadratic in fermions and linear in .
We now consider the invariance of the brane action. The fünfbein satisfies . This is due to the parity assignment and orbifold condition, which implies that only even parity fields are nonzero on the brane. The variation of the brane action is ( is zero on the brane, and therefore those contributions can be neglected)
(3.17)  
This vanishes if we apply the projections of section 2 with
(3.18) 
It thus implies that
(3.19) 
such that (3.17) vanishes.
4 Summary of supersymmetry on
In summary, the new Lagrangian in a singular space with the new supersymmetry is given by
(4.1) 
Here is a curl of the 4form and is the standard gauged supergravity of [12], see appendix B. The standard supergravity at is called ungauged supergravity. We will use the notation for its Lagrangian. Our new bulk theory has Lagrangian , there are terms linear in , which are proportional to the flux defined in (3.7), and terms quadratic in the field , see (3.9),
(4.2)  
In addition, we will denote by the part of supersymmetry that acts at and which forms the supersymmetry transformations of ungauged supergravity. For completeness we repeat here the brane actions.
(4.3) 
The new supersymmetry rules are
(4.4) 
The new gauge symmetry transformations are
(4.5) 
The action is defined by an integral over a product space of the 4dimensional manifold and an orbifold, .
(4.6) 
The 4d manifold is noncompact and the 5th dimension is compact but has no boundaries in . Therefore, all surface terms in the variation of the action vanish assuming as usual that the parameters of supersymmetry decrease at infinities of the 4d space. The bulk and the brane actions are separately invariant. The supersymmetry transformations form an onshell closed algebra.
PART II. The Background: Vanishing Fermions, Bosons Solve Equation of Motion
5 Vanishing energy
It is known that the Hamiltonian of the spatially closed universe vanishes since in absence of boundaries it is given by a diffeomorphism constraint [29]. The basic argument goes as follows. In 4d space when the ansatz for the metric is taken in the form one finds that the Hamiltonian of constraint is . Here has the contribution both from the gravity and from matter. Still one has to keep in mind that the definition of the energy of the closed universe is rather subtle.
In our new supersymmetric theory we also face the problem of how to define the energy, in general. In our case the space is not an asymptotically flat or an antide Sitter space. Moreover, since our space is singular, we can not easily apply the Nestor–Israel–Witten construction, which for asymptotically flat or antide Sitter spaces would predict a nonnegative energy.
Therefore, we would perform here only a partial analysis of the energy issue for supersymmetric theories in singular spaces, which has a clear conceptual basis. We hope, however, that a more general treatment of this problem is possible.
Here we limit ourselves to configurations which depend only on . For such configurations the natural definition of the energy functional was suggested and studied in [5, 7, 9]. We are using the warped metric in the form
(5.1) 
Starting with the new Lagrangian (4.1), we may present the energy functional for static dependent bosonic configurations as follows
(5.2)  
Here means . This expression in turn can be given in the BPStype form closely related to [5, 7, 9] but still different due to i) the presence of the 4form and the supersymmetry singlet off shell, ii) the presence of , which comes from the choice of the action on the fermions and is introduced in (3.18).
(5.3)  
The tension of the first brane, , is equal to . The tension of the second brane, , is given by . Even if any of them is negative, this causes no problem since we have a compensating contribution to the energy on each brane. In presence of the supersymmetry singlet field, there is an additional contribution to the energy at each brane due to the gradient of the supersymmetry singlet field . The term with cancels the tension contributions at each brane separately since due to the field equations for the 4form. With account of the and equations and no boundary condition, which allows to ignore , the energy functional takes the form
(5.4) 
Note that the energy functional is still not positive definite: one perfect square with the kinetic energy of the normal scalar is positive, however that with the kinetic energy of the conformal factor of the metric is negative, as it should be. One might have a concern about some configurations where the negative contribution will dominate over the positive one, which will lead to the instability of the theory. However, using the equations of motion for and (not the BPS equations), one finds that the energy vanishes for any dependent solution of the equations of motion. This can also be reinterpreted as derived using the canonical formulation of the gravitational theory where one starts from
(5.5) 
where is such that the resulting action has no second derivatives on the metric. The action of matter fields can be added and as in [29] one finds that an onshell energy , at least for timeindependent configurations, vanishes
(5.6) 
for any solution of the equations of motion.
One very important property of the vanishing of the energy of the closed universe is that it vanishes locally and not due to the compensation of the total energy, positive and negative, in different parts of the universe. As we explained this happens also here: the cancellation of energy on each brane takes case separately: the tension term is cancelled by the energy of the supersymmetry singlet field .
6 BPS construction in singular spaces
The squared terms in (5.3) suggest the BPS conditions:
(6.1) 
where we can use the on shell value of , which is equal to , so that we obtain
(6.2) 
From now on we will make the choice , i.e. pick up a particular property of fermions under parity. Physics depends on the sign of . Therefore, the change in the sign of can always be compensated by a change of the sign of .
One can verify that the jump conditions on the branes^{7}^{7}7It was observed in [30] that in gauged supergravity the jump conditions on the branes may be satisfied if the tensions are related to the superpotential. However, putting the step functions in supersymmetry transformations by hand may in general cause problems with higher order corrections. Our work makes it plausible that gauged supergravity with addition of the flux and supersymmetry singlet may be constructed with the complete and consistent supersymmetry. This will generate the step functions in supersymmetry rules in presence of branes. , derived starting with the second order differential equations, are satisfied automatically due to the new supersymmetry (4.4), the onshell condition (3.12) and the presence of the 5form flux, changing the sign when passing through the wall.
Let us also consider the Killing spinors. In the background with only nonvanishing scalars and the warped metric (5.1), which has as only nonzero components of the spin connection , the transformations of the spinors are
(6.3) 
To solve these, we split
(6.4) 
Using the conditions (6.1), the last equation of (6.3) gives the dependence of the supersymmetries on . We obtain
(6.5) 
The second equation gives
(6.6) 
The solutions are thus
(6.7) 
where are constant spinors with each only 4 real components due to the projection (6.4). Remains the first Killing equation, which implies
(6.8) 
There are thus two possibilities to solve the Killing equations.

Maximal unbroken supersymmetry in the bulk ().
Here we require that the scalars are strictly constant, , and the superpotential is independent of the scalars, , at the solution. No constraints on Killing spinors arise from the gaugino.

1/2 of the maximal unbroken supersymmetry in the bulk ().
When the scalars are not constant, then there is the extra condition (6.8), leaving just one projected supersymmetry with 4 real components,
(6.9) where is constant. Note that this projection of the supersymmetries is on the brane consistent with (3.19). Thus, we remain in the bulk as well as on the brane with of the original supersymmetries. Vice versa, imposing the projection (6.9) one derives from the vanishing of (6.3) that the conditions (6.1) should be satisfied.
6.1 Fixed scalars, doubling of supersymmetries
and an alternative to compactification world brane
The field equation for the 4form and field Killing equations are solved if (6.1) are solved. In this section we look for the very particular solutions of these equations with maximal unbroken supersymmetry that have everywhere constant scalars^{8}^{8}8These solutions remind the so called doubleextreme black holes [32, 33], which have fixed scalars in their solutions [34]..
The ‘fixed scalar domain wall solution’ is given by
(6.10) 
The solution is given by the supersymmetric attractor equation [34, 33] in the form
(6.11) 
where
(6.12) 
and we have used charges normalized as
(6.13) 
Consistency implies
(6.14) 
In some cases the explicit solution of the attractor equation is known in the form
(6.15) 
(see e.g. [35, 33] where many examples are given). The metric is
(6.16) 
If we choose to be positive (which means that at we have a positive tension brane), the metric is that introduced in [2], where two branes are present at some finite distance from each other. We will refer to this scenario as RSI. The second brane has a negative tension and can be sent to infinity, in principle, which leads to an alternative to compactification, discussed in [3]. We will refer to this as RSII. Whether this limit is totally consistent is an independent issue, however the warp factor in the metric can be chosen to exponentially decrease away from the positive tension brane. This is not in a contradiction with the nogo theorem [9, 10]. For constant scalars at the critical point, the metric behaves differently from the case when scalars are not constant but approaching the critical point. This can also be explained by the doubling of unbroken supersymmetries in the bulk for these solutions. Indeed, the gaugino transformations are vanishing without any constraint on the Killing spinors and the gravitino transformations also have an 8dimensional zero mode as shown in (6.7). Note that the curvature is everywhere constant, except on the branes where the metric has a cusp. For example, the scalar curvature for this solution, is equal to
(6.17) 
The simplest example of fixed scalar domain wall (related to doubleextreme black holes) comes out from the M5brane compactified on so that and (see sec. 4.3 in [33]). Here are FI terms.
6.2 BPS equations in very special geometry (with vector multiplets)
In the context of our present work in the more general case with vector multiplets present and nonconstant scalars, we will first change coordinates, write down the energy functional in the new coordinate system and proceed by solving the BPS conditions following from the energy functional. We take
(6.18) 
so that for . The BPStype energy functional for static dependent bosonic configurations following from the new Lagrangian (4.1) is and we will use
The stabilization equations for the energy are the BPS conditions: