Supermembrane interaction with dynamical D = 4 N = 1 supergravity. Superfield Lagrangian description and spacetime equations of motion

We obtain the complete set of equations of motion for the interacting system of supermembrane and dynamical D = 4 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \mathcal{N} $\end{document} = 1 supergravity by varying its complete superfield action and writing the resulting superfield equations in the special \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ ``{\text{W}}{{\text{Z}}_{{\widehat{\theta } = 0}}}'' $\end{document} gauge where the supermembrane Goldstone field is set to zero \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \left( {\widehat{\theta } = 0} \right) $\end{document}. We solve the equations for auxiliary fields and discuss the effect of dynamical generation of cosmological constant in the Einstein equation of interacting system and its renormalization due to some regular contributions from supermembrane. These two effects (discussed in late 70th and 80th, in the bosonic perspective and in the supergravity literature) result in that, generically, the cosmological constant has different values in the branches of the spacetime separated by the supermembrane worldvolume.


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tem of D = 4 N = 1 supermembrane interacting with supergravity and matter multiplets was developed in [23]. Thus the interacting system of the dynamical supergravity and super-p-brane can be studied in the frame of the covariant but gauge fixed description even when the superfield action for supergravity is unknown (see [22] for some results on the D = 11 supergravitysupermembrane interaction). However, the study of the complete superfield description, when it is known or can be obtained, in particular in D = 4, is also interesting as a basis of possible phenomenological (model building) applications. It could also bring some new insights on the nature of supergravity-supermembrane interaction. In particular, as we will see below, this allows to understand the role of the auxiliary fields of supergravity in some supergravity-super-p-brane interacting systems.
The basic examples studied in [20,21] were D = 4 N = 1 supersymmetric systems of massless superparticle plus minimal supergravity [20] and superstring interacting with the minimal supergravity and tensorial supermultiplet [21]. The study of the superfield Lagrangian description of the D = 4 N = 1 supergravity-supermembrane interacting system was started in [24] (see [25] for the study of interacting system of supermembrane and D = 4 N = 1 scalar multiplet), where the Wess-Zumino type approach to the Grisaru-Siegel-Gates-Ovrut-Waldram special minimal supergravity [26][27][28] 4 was developed and the explicit expressions for supermembrane current superfields, entering the r.h.s. of the superfield supergravity equations, were obtained.
In this paper we obtain and study the equations of motion for the D = 4 N = 1 supergravity-supermembrane interacting system. The superfield supergravity equations with supermembrane contributions are quite complex, but simplify essentially in a special gauge which we call WZθ =0 gauge. It is reached by fixing the usual Wess-Zumino (WZ) gauge for supergravity and then by using a half of the local spacetime supersymmetry on the supermembrane worldvolume to fix the gauge where the supermembrane Goldstone fermion vanishes,θβ(ξ) = 0. In such a gauge we solve the auxiliary field equations and show that there are three types of supermembrane contributions to the Einstein equation of the interacting system. Besides the expected singular terms with the support on the (super)membrane worldvolume W 3 , the supermembrane produces two types of regular contributions which can be considered as contributions to the cosmological constant on the pieces of space-time separated by the supermembrane. The first, known from the study in [28] (and before in [30][31][32][33], see [24] for more references and discussion on them), is the cosmological constant generated dynamically. 5 The second nonsingular contribution from 4 For our best knowledge, this formulation of supergravity was discussed for the first time in [26] and [27], where it was noticed as a formulation with a simplified structure of the ghost sector of quantum supergravity. Ovrut and Waldram discovered it independently under the name of three form supergravity in [28] and elaborated it using an elegant mixture of the spacetime (component) and superfield formalism (much in the spirit of [29]). 5 Although it is not proportional to the supermembrane tension and is a result of the structure of the auxiliary field sector of special minimal supergravity, this dynamically generated cosmological constant should be considered as a contribution from supermembrane as far as the requirement that supergravity should have such auxiliary field, and not the ones of the generic minimal supergravity, comes from selfconsistency of the dynamical supergravity interaction with supermembrane.

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the supermembrane changes the value of the cosmological constant on one side of the supermembrane worldvolume W 3 making the values of cosmological constant in two branches of spacetime M 4 + and M − 4 separated by W 3 generically different. This effect, which can be referred to as shift or renormalization of the cosmological constant by supermembrane contributions, was discussed in pure bosonic perspective in [30] and [34].
Thus generically one can expect that the ground state solution of our interacting system describes the supermembrane separating two branches of AdS spaces with different values of the cosmological constant. In the purely bosonic context the solutions of such a type were studied (besides [30] and [34]) in [35][36][37][38][39][40][41]. The solution for the particular case of coincident cosmological constants can be found in [28]. In section 5 of this paper we present a preliminary discussion on possible supersymmetric solutions of the interacting system equations with supermembrane separating the asymptotically-AdS spaces with different values of the cosmological constants.
The paper is organized as follows. We begin in section 2 by a brief review of the (generic) minimal formulation of the superfield supergravity (sections 2.1 and 2.2). In section 2.3 we present the supermembrane action in minimal supergravity background, discuss its κ-symmetry and obtain the supermembrane equations of motion in such background; these formally coincide with the supermembrane part of the superfield equations of the interacting system.
The supermembrane can propagate in a generic superspace of minimal supergravity because this allows for the existence of an invariant closed 4-form H 4 . The knowledge of such H 4 obeying dH 4 = 0 is completely sufficient for describing the supermembrane in a background superspace. However, the existence of 3-form potential C 3 is crucial when one would like to find supermembrane current or to obtain the superfield equations of motion for the interacting system. As we briefly discuss in section 2.4, referring to [24] for the detail, the requirement of the existence of the 3-form potential in curved superspace imposes a restriction on the prepotential structure of minimal supergravity. Namely its chiral compensator becomes special chiral superfield expressed in terms of real prepotential (rather than through the complex prepotential as in the case of generic chiral superfield) [24]. Such a special minimal supergravity was found in [26], discussed in [27] and found independently and elaborated by Ovrut and Waldram [28], under the name of 'three-form supergravity', as a supergravity formulation proper for interacting with D = 4 supermembrane. In section 2.4 we give a very brief review of the Wess-Zumino superfield approach to the special minimal supergravity which was elaborated in [24]; we present there the explicit expression for the variation of special minimal supergravity action which is needed to obtain the equations of the interacting system, and discuss the effect of dynamical generation of cosmological constant [28,31].
In section 3 we present the superfield action for the supergravity-supermembrane interacting system, vary it with respect to the supergravity prepotentials (more precisely, extract the coefficients for the independent variations of the special minimal supergravity in the interacting action variation [24]) and find the explicit form of the supermembrane current superfields which enter the r.h.s.-s of the superfield supergravity equations of the interacting system. Section 4 is devoted to the study of the spacetime component equations JHEP08(2012)140 of the interacting system which follow from the above mentioned superfield equations. The convenient WZθ =0 gauge is described in section 4.1. In section 4.2. the essential components of the supermembrane current superfields are calculated in this gauge and it is shown that the Rarita-Schwinger equation of the interacting system does not contain the supermembrane contributions explicitly (although contains the spin connection which does contain supermembrane contributions).
The Einstein equation of the supergravity-supermembrane interacting system is obtained in section 4.3. where the general solution of the auxiliary field equations is also given. This contains an arbitrary constant and also a contribution proportional to the covariant version of the Heaviside step function Θ(x, x 0 ;x) multiplied by the supermembrane tension T 2 . In section 4.4 we show that these result in the effect of dynamical generation of cosmological constant [28,[30][31][32][33] and its 'renormalization' [30,34] in such a way that the cosmological constant values in the branches of spacetime separated by the supermembrane worldvolume are generically different. Section 5 contains discussion on the possible supersymmetric solution of the interacting system equations characterized by different value of cosmological constant in two branches of spacetime separated by the supermembrane worldvolume. We conclude in section 6. Some details on the independent variations of the superfields of special minimal supergravity can be found in the appendix.
A reader who is not familiar with superfield formalism but is rather interested in physical consequences of the equations of the interacting system, may skip the technical details in sections 2, 3, 4.1 and 4.2 and pass directly to the analysis of the Einstein equations of the interacting system in section 4.3, 4.4 and discussion on its supersymmetric solutions in section 5.
Sometimes it is convenient to collect the supervielbein one forms in 3)

Superfield supergravity action, superspace constraints and equations of motion
The superfield action of the minimal off-shell formulation of D = 4, N = 1 supergravity [43] is given by the superdeterminant (or Berezinian) of the matrix of supervielbein coefficients, 3), which obey the set of supergravity constraints. These can be collected together with their consequences in the following expressions for the superspace torsion 2-forms (2.4), (2.5) (see [20] and refs. therein) and ∧ is the exterior product of the differential forms, in particular, [20,24] for more detail.

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The main superfields, real vector G a = (G a ) * and complex scalar R = (R) * , entering (2.11) and (2.12), obey (2.14) These relations can be obtained by studying the Bianchi identities (2.8), which also allow to find the expression for superfield generalization of the gravitino field strength, T bc α (Z), involving one more main superfield, W αβγ = W (αβγ) =: (Wαβγ) * . This obeys Studying the Bianchi identities with the constraints (2.11), (2.12) one also finds that the superfield generalization of the left hand side (l.h.s.) of the supergravity Rarita-Schwinger equation reads 18) and the superfield generalization of the Ricci tensor is

19) This suggests that superfield supergravity equation should have the form
See [20,24] for more detail on the superfield description of minimal supergravity in the present notation.
Eqs. (2.20) and (2.21) can be obtained by varying the action (2.10) with respect to supervielbein obeying the supergravity constraints (2.11), (2.12) [43]. Such admissible variations are expressed through a vector parameter δH a and complex scalar parameter δU = (δŪ ) * which enter the variation of the supervielbein and spin connection under the symbol of the chiral projector (D α D α −R) (see [20,24] for more detail). They correspond to the variations of the so-called prepotentials, unconstrained superfields which appear in the general solution of the supergravity constraints. The minimal supergravity constraints are solved in terms of the axial vector superfield H µ [44][45][46] and chiral compensator Φ [47]. This latter obeysDαΦ = 0 and, hence, can be expressed as Φ = (DαDα − R)U with a complex unconstrained superfield U .
Thus the set of minimal supergravity prepotentials includes H µ , U andŪ = (U ) * which are in one to one correspondence with the set of three independent variations δH a , δU and δŪ = (δU ) * of the Wess-Zumino approach to supergravity [43,48] producing the three superfield equations (2.20) and (2.21). In short, as it had been known already from [43], JHEP08(2012)140

Supermembrane action in minimal supergravity background
As it is well known, the supermembrane action [9,10,13] is given by the sum of the Dirac-Nambu-Goto and the Wess-Zumino term, The former is given by the volume of W 3 defined as integral of the determinant of the induced metric, g = det(g mn ), Here ξ m = (τ, σ 1 , σ 2 ) are local coordinates on W 3 andẐ M (ξ) are coordinates functions which determine the embedding of W 3 as a surface in target superspace Σ (4|4) , In the second equality of (2.23) the Dirac-Nambu-Goto term is written as an integral of the wedge product of the pull-back of the Σ (4|4) bosonic supervielbein form E a to W 3 , and of its Hodge dual two form defined with the use of the induced metric (2.24) and its inverse g mn , * Ê a := 1 2 dξ m ∧ dξ n √ gǫ mnk g klÊa l . (2.27) The second, Wess-Zumino term of the supermembrane action (2.23) describes the supermembrane coupling to a 3-form gauge potential C 3 defined on Σ (4|4) , Thus, to write a supermembrane action, one has to construct the 3-form gauge potential C 3 in the target superspace Σ (4|4) and take its pull-back to the supermembrane worldvolumê Actually, to study supermembrane in supergravity background, it is sufficient to know the field strength of the above 3-form potential, H 4 = dC 3 . This should be closed, dH 4 = 0, and supersymmetric invariant 4-form. In flat superspace such a form exists and represents a nontrivial Chevalley-Eilenberg cohomology of the N = 1 supersymmetry algebra [49][50][51].
The minimal supergravity superspace allows for existence of two closed 4-forms

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and its complex conjugate H 4R = (H 4L ) * (see [24]). Its real part, is also closed and provides the 4-form field strength associated to the Wess-Zumino (WZ) term of the supermembrane action in the minimal supergravity background [28], W 3 C 3 in (2.23). Indeed, the WZ term can also be defined as an integral of the closed 4 form H 4 , related to C 3 by H 4 = dC 3 , over some four dimensional space W 4 the boundary of which is given by the supermembrane worldvolume W 3 , The condition that the form H 4 is closed, dH 4 = 0, guaranties that the integral W 4 H 4 is independent on the choice of W 4 and, thus, is related to the supermembrane worldvolume W 3 . The fact that the knowledge of H 4 is completely sufficient for studying the properties of closed supermembrane in a supergravity background is related to that in this case the only dynamical variables are the supermembrane coordinate functionsẐ M (ξ), that the action is written in term of pull-back of differential forms to W 3 , and that the variation of the differential form with respect to the coordinates can be calculated with the use of the Lie derivative formula, in particular The supermembrane equations of motion in the minimal supergravity background, which are obtained by varying the action (2.23) with respect to the coordinate functions δẐ M (ξ), where the matrixγ βα is defined bȳ
(2.38) Some identities involving the above matrix arē They are useful, in particular, to show that the fermionic equations of motion obey the Noether identityΨ 3α = Ψ α 3γ αα reflecting the local fermionic κ-symmetry with the local fermionic "parameter" κ α (ξ) = (κα) * obeying The relation of the supermembrane κ-symmetry in curved superspace with the minimal supergravity constraints was discussed in [28]. The flat superspace limit of our equations reproduces the equations of the seminal paper [13].

3-form potential in the minimal supergravity superspace. Special minimal supergravity
Thus, as we have seen in the previous subsection, to find the equations of motion of supermembrane in supergravity background as well as to study its symmetries it is sufficient to know the closed 4-form H 4 = dC 3 in the background superspace. However, to calculate the supermembrane current(s) describing the supermembrane contribution(s) to the supergravity (super)field equations, one needs to vary the Wess-Zumino term Ĉ 3 of the supermembrane action with respect to the supergravity (super)fields. Thus one arrives at a separate problem of finding the variation such that dδC 3 = δH 4 reproduces the variation of H 4 from (2.31), (2.30), written in terms of the basic supergravity variations (we refer to [24] for the explicit expression of δH 4 ). Studying such a technical problem we have found [24] that it imposes a restriction on the independent variations of the supergravity prepotentials, or equivalently, on the independent parameters of the admissible supervielbein variations, thus transforming the generic minimal supergravity into a special minimal supergravity. This off-shell supergravity formulation had been described for the first time in [26], further discussed in [27] (see also latter [42]) and elaborated in [28] using the elegant combination of superfield results and the component 'tensor calculus' approach on the line of [29].

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In [24] we described this special minimal supergravity in the complete Wess-Zumino superfield formalism. Referring to that paper for technical details, we only notice that the existence of the 3-form potential imposes a restriction on the prepotential structure of minimal supergravity which in our approach manifests itself in that the basic complex variations δU and δŪ = (δU ) * are expressed in terms of one real variation δV , essentially As a result, the variation of the special minimal supergravity action is essentially (see [24] and the appendix) The value of the cosmological constant is proportional to the square of the above arbitrary constant c, which has appeared as an integration constant, so that the special minimal supergravity is characterized by a cosmological constant generated dynamically.
The above mechanism of the dynamical generation of cosmological constant in special minimal supergravity is the same as was observed by Ogievetski and Sokatchev [31] in their theory of axial vector superfield. In the language of component spacetime approach to supergravity the dynamical generation of cosmological constant in the special minimal supergravity was described in [28] and before it, in purely bosonic perspective in [30,33,34] and in the context of spontaneously broken N = 8 supergravity in [32]. We refer to [24] for more references and discussion.

JHEP08(2012)140 3 Superspace action and superfield equations of motion for the interacting system of dynamical supergravity and supermembrane
The action for interacting system of dynamical supergravity and supermembrane reads where S p=2 is the same as in eq. (2.23), the supervielbein (2.3) and the 3-form potential (2.28) are assumed to be restricted by the minimal supergravity constraints (2.11), (2.12), (2.31), (2.30). Furthermore, as we have discussed in previous sections (and in more details in [24]; see also appendix), the existence of the 3-form potential imposes the restrictions (2.46) on the prepotentials of minimal supergravity or equivalently on the basic supergravity variations.
As a result, the superfield equations which appear as a result of variation of the interacting action (3.1) read [24] G a = T 2 J a , and where J a and X = (X ) * are supermembrane current superfields. Roughly speaking, they are obtained as a result of varying the supermembrane action with respect to the prepotentials of the special minimal supergravity, this is to say as δS p=2 /δH a and δS p=2 /δV [24], and have the form

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and Notice that, as a consequence of (2.14), the supermembrane current superfields obeȳ Although at first glance these relations look different from any of listed in [52][53][54], they can be reduced to the Ferrara-Zumino multiplet [55] if one takes into account eq.

(4.5)
A detailed discussion on this "WZθ =0 " gauge can be found in [17,[20][21][22]. We notice only few of its properties. Firstly, in the WZ gauge (4.1), (4.3) the leading component of supervielbein matrix has a triangular form, which implies, in particular, the following relation between the leading component of T ab α and the true gravitino field strength Secondly, we would like to comment on symmetries leaving eqs. (4.1)-(4.5) invariant. The WZ gauge (4.1), (4.3), (4.4) is preserved by spacetime diffeomorphisms, local Lorentz symmetry and supersymmetry. Fixing further the gauge (4.5) we break 1/2 of the local supersymmetry on the worldvolume of the supermembrane. The only restriction on the parameter of the local spacetime supersymmetry ǫ α (x) is the condition that its pull-back to W 3 ,ǫ α := ǫ α (x(ξ)), and its complex conjugateǫα :=ǭα(x(ξ)) are related bŷ whereγα α is the supermembrane κ-symmetry projector (2.37) calculated withθ(ξ) = 0. Eq. (4.8) is tantamount to saying that the pull-back of the local supersymmetry parameter to W 3 is expressed through the κ-symmetry parameter of the supermembrane. There are no restrictions on the local supersymmetry parameter outside the supermembrane worldvolume so that the equations (4.8) can be understood as the boundary condition imposed on the supersymmetry parameter on the domain wall W 3 .

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where θ ∧4 := θθθθ and θ ∧3 denotes terms proportional to either θθθ or θθθ (or both, which implies ∝ θθθθ). Using these relations and introducing the current pre-potential fields we find that the vector and scalar current superfields (3.4), (3.5) have the form and Using (4.15) and (4.16) one can easily check that eqs. (3.6) are satisfied at lowest order in θ.
One also sees that there is no explicit supermembrane contributions to the Rarita-Schwinger equations of the supergravity-supermembrane interacting system which thus reads ǫ µνρσ e a ν (x)D ρ ψ α σ (x) σ aαα = 0 . However, such a contribution is actually present in (4.17) implicitly, hidden inside the covariant derivative. Indeed, as indicated by Einstein equation, the bosonic vielbein and the spin connection do contain some contributions from supermembrane.

Einstein equation of the supergravity-supermembrane interacting system
in the WZθ =0 gauge The Einstein equation with supermembrane current contributions can be obtained as leading term in the decomposition of eq. (2.19), i.e.

R bc
(4.18) The first two terms in the r.h.s. of eq. (4.18) can be easily calculated from eqs. (4.15), (4.16), while the last term, in the light of that the scalar superfield equation has the form of eq. (3.3), is expressed in terms of (R +R) 2 | θ=0 and requires a separate study. As an intermediate resume let us fix that

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The last term is the square of (R +R)| θ=0 which, as a result of (3.3), obeys the equation The solution of this equation can be written in the form where c is an arbitrary constant which corresponds to the value of (R +R) at the spacetime point x µ 0 providing the lower limit of the integral in the second term, c = (R(x 0 )+R(x 0 ))/8. One can easily check that entering the second term in the r.h.s. of (4.21), obeys Furthermore, using a convenient local frame in the neighborhood of the worldvolume, one can check that Θ(x, x 0 |x) vanishes if the points x µ and x µ 0 are on the same side of spacetime with respect to the domain wall provided by the supermembrane worldvolume while it is equal to ±1 if these points belongs to the different branches of the spacetime separated by this domain wall. This is to say that eq. (4.22) defines a counterpart of the Heaviside step function associated to the direction orthogonal to the supermembrane worldvolume. The last statement about association implies that Θ(x, x 0 |x) is a functional of the supermembrane coordinate functionx µ (ξ). Furthermore, as in the case of the standard Heaviside step function Θ(y), we can use 8 (Θ(x, x 0 |x)) 2 = Θ(x, x 0 |x).
Thus our solution of the auxiliary field equations (4.21) can be written as (cf. [30]) and implies that eq. (4.19) reads where P b a (x) is the singular contribution defined in (4.13).

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4.4 Cosmological constant generation in the interacting system and its "renormalization" due to supermembrane Let us analyze the supermembrane contribution to the Einstein equations. These can be separated in two classes, one containing singular contributions and the other containing regular contributions proportional to T 2 . Being a bit more provocative one can say about three classes, counting also the contribution proportional to square of the arbitrary integration constant c, as far as this comes from the auxiliary field sector of the special minimal supergravity, the off-shell formulation which is 'elected' by the supermembrane. As we have already commented in section 2.4. and in [24], the supermembrane can exist in a background of a generic minimal supergravity, however the supermembrane interaction with dynamical supergravity requires this to be special minimal supergravity. This in its turn, even in the absence of any matter (neither of the field theoretical type nor of branes), produces Einstein equations with a cosmological constant generated dynamically. Then this cosmological constant proportional to the square of the above arbitrary integration constant c should also be considered as a(n indirect) contribution of the supermembrane to the Einstein equation.
To be more concrete, eq. (4.25) can be written in the form R bc When T 2 is set to zero, it contains a nonvanishing cosmological constant contribution with Λ = −3c 2 . This (AdS-type) cosmological constant is generated dynamically as far as it is proportional to the (minus) square of the arbitrary integration constant c which is inevitable in the special minimal supergravity equations due to its auxiliary field structure (see [28] and [24] for references and more discussion). In its turn, special minimal supergravity, and not generic minimal supergravity can be included into the action of the supergravity-supermembrane interacting system. In this sense the cosmological constant generated dynamically is the first 'relict' contribution from the supermembrane to the Einstein equation of the interacting system. The second type of the supermembrane contributions to the r.h.s. of the Einstein equation are singular terms ∝ P c d (x) (4.13), which are expected when (super)gravity interact with supermembrane.
In the third type we collect the regular supermembrane contributions which are proportional to the supermembrane tension, (4.29) An evident observation is that, in the general case, the cosmological constants in different branches of spacetime separated by the worldvolume W 3 are different.
One also notices that the cosmological constants in M 4 + and M 4 − coincide if c = − T 2 256 . However, as far as c is an arbitrary integration constant, fixing its value is equivalent to imposing a kind of boundary conditions and we do not see any special reason to chose such boundary conditions in such a way that c = − T 2 256 . Rather we should allow a generic value of c and thus accept that the cosmological constant takes different values in the branches of spacetime separated by the supermembrane worldvolume.
Notice that the solution of the Einstein equation describing membranes separating two AdS 5 spaces with different values of cosmological constants were studied in [35], as a Brane world alternative to the dark matter, and [36][37][38] in relation with the hypothesis on possible change of signature in the Brane World models. See also [39][40][41] for the related studies. In the bosonic perspective the appearance of different cosmological constants on the different sides of a domain wall interacting with gravity and a 3-form gauge field was known from [30], where it was used as a basis for a bag model for hadrons, and from [34] where this effect was proposed as a mechanism for damping the cosmological constant. Our present study indicates that the result on the different values of cosmological constant on the different sides of the supermembrane domain wall is an imminent consequence of the dynamics of the supersymmetric interacting system of the supermembrane and dynamical D = 4 N = 1 supergravity.

On supersymmetric solutions of the interacting system equations
When searching for purely bosonic supersymmetric solutions, setting ψ α µ = 0, one studies the Killing spinor equations, which appears as the conditions of supersymmetry preservation, δ ǫ ψ α µ = 0. When starting from superfield formulation of supergravity, δ ǫ ψ α µ can be calculated with the use of superspace Lie derivative, this is to say Hence, in a generic off-shell D = 4 N = 1 minimal supergravity background the Killing equations are

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We can split this on two Killing equations valid in two different branches of spacetime separated by the supermembrane worldvolume, The supersymmetry parameter should also obey the boundary conditions (4.8) on the worldvolume W 3 , which is the common boundary of M 4 + and M 4 − , The detailed study of these system of Killing spinor equations and the search for the supersymmetric solutions of the interacting system equations on their basis is an interesting subject for future. An intriguing question is whether the supersymmetric solutions of the equations of the interacting system exist in the generic case of arbitrary c corresponding to different values of cosmological constants on different sides of the supermembrane worldvolume, or supersymmetry selects some particular values of the constant c. Presently we can state that if obstructions existed, they would occur due to the singular terms with support on the worldvolume i.e. that M 4 ± are AdS spaces with apparently different cosmological constants. One can easily check that (5.8) and (5.9) solve our equations of motion (4.30) and (4.29) and thus describe the completely supersymmetric solution of the system of the supergravity equations of the interacting system (at least) when these are considered modulo singular terms with the support on W 3 .
Let us stress that such a system of equations does contain the supermembrane contributions: not only an indirect, which comes from an arbitrary cosmological constant

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The conditions of that δH 4 can be expressed in terms of the variation of the 3-form potential δC 3 , with δC 3 decomposed on the basic covariant 3-forms, as in eq. (2.45), restrict the set of independent variations by [24] (DD −R)δU = 1 12 This is equivalent to where δν α is an additional independent variation; this however does not contribute to (DD −R)δU and, hence, to the variations of supergravity potentials. Factoring out the gauge transformations, we can write the variation δC 3 , which produces (A.4) through (A.5), in the form (2.45) with [24] β αβγ (δ) = 0 = β αβγ (δ) , β αβa (δ) = iσ aαβ δV Notice that the variations δκ α and its complex conjugate (c.c.) result in equations D α R = 0 and its c.c., which are satisfied identically due the minimal (eq. (2.21)) or special minimal supergravity equations of motion (eq. (2.49)). In the WZθ =0 gauge (4.1)-(4.5) it is also relatively easy to check that δκ α does not produce any independent equation for the physical fields of the interacting system. This observation allowed us to simplify the discussion in the main text by neglecting the existence δκ α variation.