Brane nucleation in supersymmetric models

JHEP10(2023)061 Published for SISSA by Springer Received:July 10, 2023 Revised:September 20, 2023 Accepted:October 3, 2023 Published:October 11, 2023 Brane nucleation in supersymmetric models Igor Bandos,a,b,c Jose J. Blanco-Pillado,a,b,c Kepa Sousadand Mikel A. Urkiolab,e aDepartment of Physics, University of the Basque Country UPV/EHU, Barrio Sarriena s/n, 48940 Leioa, Spain bEHU Quantum Center, University of the Basque Country UPV/EHU, Barrio Sarriena s/n, 48940 Leioa, Spain cIKERBASQUE, Basque Foundation for Science, 48011, Bilbao, Spain dDepartment of Physics and Mathematics, University of Alcalá, 28805 Alcalá de Henares (Madrid), Spain eDepartment of Applied Mathematics, University of the Basque Country UPV/EHU, Plaza Ingeniero Torres Quevedo 1, 48013 Bilbao, Spain E-mail: [email protected],[email protected], [email protected],[email protected] Abstract: This paper explores the process of vacuum decay in supersymmetric models related to flux compactifications. In particular, we describe these instabilities within supersymmetric Lagrangians for a single three-form multiplet. This multiplet combines scalar fields, representing the moduli fields in four dimensions, with 3-form fields that influence the potential for these moduli via the integer flux of their associated 4-form field strength. Furthermore, using supersymmetry as a guide we obtain the form of the couplings of these fields to the membranes that act as sources to the 3-form potentials. Adding small supersymmetry breaking terms to these Lagrangians one can obtain instanton solutions describing the decay of the vacua in these models by the formation of a membrane bubble. These instantons combine the usual Coleman-de Luccia and the Brown-Teitelboim formalisms in a single unified model. We study simple numerical examples of theories with and without gravity in this new framework and generalize known Euclidean methods to accomodate the simulataneous inclusion of scalar fields and charged membranes to these instanton solutions. Moreover, we show explicitly in these examples how one recovers the static supersymmetric solutions in the limiting case where the supersymmetry breaking terms vanish. In this limit, the bubble becomes infinite and flat and represents a hybrid between the usual supersymmetric domain walls of field theory models and the brane solutions interpolating between the supersymmetric vacua; a sort of dressed supermembrane BPS solution. Finally, we briefly comment on the implications of these solutions in cosmological models based on the String Theory Landscape where these type of 4d effective theories could be relevant in inflationary scenarios. Keywords: Cosmological models, String and Brane Phenomenology, Supergravity Models ArXiv ePrint: 2306.09412 Open Access,c The Authors. Article funded by SCOAP3.https://doi.org/10.1007/JHEP10(2023)061 JHEP10(2023)061 Contents 1 Introduction 1 2 Rigid supersymmetric scalar field theory models 4 2.1 Field theory supersymmetric domain walls 4 2.1.1 Example. Double well potential 5 2.2 Stability and vacuum decay 6 3 Rigid supersymmetric models with 3-form potentials 8 3.1 Supersymmetric membranes coupled to 3-form potentials 9 3.2 Example with quadratic superpotential 11 3.3 Membrane nucleation 12 3.3.1 Numerical example 14 4 Supergravity models 16 4.1 Flat membrane solutions in supergravity models 18 4.2 Example: quartic superpotential 20 5 Membrane nucleation in supergravity 21 5.1 Example: quartic superpotential 22 5.1.1 AdS/Minkowski to AdS transitions 23 5.1.2 dS to AdS transitions 26 6 Conclusions 27 A Three-form multiplets in supersymmetry 29 A.1 Including 3-forms into the chiral superfields: special chiral superfields 30 B Rigid supersymmetry, bosonic membranes, and BPS equations 33 C Three-form multiplets in supergravity 36 C.1 Supergravity interacting with 3-form multiplets 38 C.1.1 Super-Weyl transformations 39 1 Introduction Many high energy extensions of the Standard Model make use in some way or another of supersymmetry. Furthermore, the presence of multiple vacua is quite generic in these models. Some of these vacua preserve part of the original supersymmetry while others – 1 – JHEP10(2023)061 break it completely. Understanding the stability of these vacua and their possible decay processes is, therefore, an essential aspect of the low energy description of these theories. Some of the most interesting examples of this type of scenarios are the effective four dimensional models obtained from string theory compactifications. Compactifying 10d string theory to four dimensions leaves us with a low energy effective theory with a collection of scalar fields (the moduli fields) that parametrize the possible deformations of the internal compact manifold. One can further impose that the compactification mechanism preserves some supersymmetry so that we end up with a low energy theory that can be classified as a supersymmetric scalar field theory.1This led the authors in [1] to consider the non-perturbative stability of a N=1 model of supergravity. They demonstrated that supersymmetric vacua are stable with respect to the usual process of bubble nucleation. Furthermore, they also showed that this stability is due to the restriction imposed by supersymmetry on the tension of the wall that interpolates between the two vacua. In fact, this quenching phenomenon is nothing more than the Coleman-deLuccia [2] suppression. In this limit the bubble radius would be infinite and the solution would be a planar domain wall that preserves part of the supersymmetry [3]. On the other hand, recent developments in models of string compactification based on the use of fluxes along the internal dimensions has led us to the idea of an extremely rich Landscape of possible 4dEffective Field Theories (EFTs), also referred to as flux vacua [4]. Each of these vacua is characterized by the presence of a set of integer fluxes that thread some cycles in the internal manifold. The stabilizing potential for the moduli in each of these sectors of the theory is also quite complicated and could easily have many local minima itself. One could therefore use the conclusions discussed earlier for the scalar field potential in each of these sectors to study their stability and find the bounce solutions using the techniques derived by Coleman and collaborators [5,6]. One can then ask whether there is an analogous process that would take us from one sector with some set of fluxes to another one where one or several of those fluxes have been changed. Naively this would seem impossible since the flux is quantized and therefore cannot be continuously changed. However, similarly to what happens in the Schwinger process [7], one could reduce the flux by the creation of sources charged with respect to the same field that produces it.2In fact, this type of process had been already discussed in a purely four dimensional context a number of years ago by Brown and Teitelboim (BT) [9,10]. In this model the presence of a 4-form field strength in four dimensions induces an effective cosmological constant that can only be changed by the nucleation of membranes charged with respect to its 3-form potential. The instanton solutions describing this type of instability of the model were studied in detail in connection to the possible self-tuning mechanism of the cosmological constant. 1More rigorously, one should speak about supermultiplets involving scalar fields; in the case of N=1 supersymmetry, these include not only the scalar supermultiplet but also the so-called three form supermultiplets (see below). 2See [8] for a simple description of several field theory models in different dimensions of spacetime that exhibit a similar behaviour to the one described in this paper. – 2 – JHEP10(2023)061 In models of string theory compactification the situation is quite similar, and one can assume that each of the forms present in the 4d theory would have an associated brane charged with respect to that specific form. Indeed one can identify 4dmembrane objects by wrapping higher dimensional branes along some internal cycles. Similarly, one can also find the effective 4-form field that couples to the 4-dimensional membrane and understand its higher dimensional origin. Taking this point of view, these models of flux compactification seem to lead to a model closely related to the Brown-Teitelboim idea [4]. There is however an important difference between these two models. In the original Brown-Teitelboim’s model the nucleation of the membranes would lead to a jump in the value of the 4dcosmological constant. In models with higher dimensions this change in the flux would lead us to a different sector with a different moduli potential. This is interesting since it allows us to avoid the so called “empty universe problem” in the purely 4dBT model by postulating a period of inflation after the bubble nucleation driven by the compactification potential (see for example [11]). The previous arguments suggest the idea of combining both models into a unifying picture where we can describe within the same theory the presence of the scalar field moduli and the 4-form fields. Moreover, following what was done before in the purely scalar field model, we will look for guidance in supersymmetric models that include both types of degrees of freedom, the 4-form field as well as the moduli fields. The inclusion of form fields in supersymmetric and supergravity multiplets has been done before in [12–18]. Furthermore, their interaction with supersymmetric membranes (supermembranes) was studied in [14,19–21] and their role in the low energy description of flux compactifications has been recently discussed in [18,21–23]. In this paper we will study the non-perturbative stability of these models and their modified versions once we introduce small supersymmetry-breaking terms. More concretely, we will deform the supersymmetric field-theoretic part of the action by including soft supersymmetry breaking terms [24], while keeping the (super)membrane part of the action untouched. This paper is organized as follows. In section 2we will review the results of global supersymmetric scalar field theories (more concretely, self-interacting scalar supermultiplet models which can be described in terms of a generic chiral superfield), and the existence of supersymmetric domain wall solutions in these models. We will show with explicit examples how these solutions appear as limiting cases of a bubble decay process from a non-supersymmetric vacua. In section 3we will explain how to introduce 3-form gauge fields in our supersymmetric theory, as well as membranes that naturally couple to these forms. This will lead us to discuss the so-called 3-form supermultiplets described by a special type of chiral superfields. Furthermore, we will obtain the instanton solutions describing the decay processes in these theories, all the while explicitly showing how they connect in a proper limit with the supersymmetric solutions found in the previous section. Finally, in sections 4and 5we will describe a similar situation in the case of supergravity and find explicit solutions of the equations of motion describing the bubble nucleation. We end with some conclusions in section 6. – 3 – JHEP10(2023)061 2 Rigid supersymmetric scalar field theory models As a warm up exercise, in this section, we will consider the stability of vacua in a scalar field Lagrangian of an N=1 four dimensional globally supersymmetric field theory of a single chiral superfield, which describes the so-called scalar supermultiplet.3Many of the results presented in this section can be found in the literature, in particular in [1,3]. In the following sections we will discuss more complicated models following a similar reasoning to the one presented here. 2.1 Field theory supersymmetric domain walls The model we will be considering here is given by the most general Lagrangian for a complex scalar field (ϕ)which is given by the leading component of a generic chiral superfield (which describes the so-called scalar multiplet). This Lagrangian reads L=−Kϕ¯ ϕ∂µϕ∂µ¯ ϕ−Kϕ¯ ϕ|Wϕ|2(2.1) where we have introduced the two functions that define the model: the real Kahler potential K(ϕ, ¯ ϕ)and the complex holomorphic superpotential W(ϕ). We will denote their derivatives as ∂ϕ∂¯ ϕK=Kϕ¯ ϕ= 1/Kϕ¯ ϕand Wϕ=∂ϕW(ϕ), thus following the conventions of [25].4 The equations of motion for this theory are Kϕ¯ ϕ∂µ∂µϕ−Kϕϕ¯ ϕ∂µϕ∂µ¯ ϕ+Kϕ¯ ϕ¯ ϕ(Kϕ¯ ϕ)2|Wϕ|2−Kϕ¯ ϕWϕ¯ W¯ ϕ¯ ϕ= 0 (2.2) and its complex conjugate. These reduce to the usual Klein-Gordon equation for a complex scalar field with a scalar potential given by V(ϕ, ¯ ϕ) = |Wϕ|2if the kinetic term is canonical (which particularly requires K=ϕ¯ ϕ). We are looking for a domain wall solution in this model that interpolates between two supersymmetric minima, in other words, between two points whose superpotential satisfies Wϕ(ϕ±) = 0. Recall that all supersymmetric minima have a vanishing potential, and therefore degenerate in energy. For that matter, let us consider a flat domain wall whose transverse direction is given by the coordinate z. One can then show [3] that the static solution preserving half of the N=1 supersymmetry solves the first-order equation ∂zϕ(z) = eiθKϕ¯ ϕ¯ W¯ ϕ(¯ ϕ(z)),(2.3) known as the BPS equation, where the phase θis given by eiθ =∆W |∆W|,with ∆W≡W(ϕ(z=∞)) −W(ϕ(z=−∞)).(2.4) Of course, given appropriate boundary conditions, both the first-order and secondorder equations should yield the same static solution for the supersymmetric domain wall. 3In the main part of this paper we will only deal with the bosonic components of the supermultiplets. See, e.g., [25] for a more complete treatment of the simplest cases. 4In this paper we will use the (−+ ++) signature convention. – 4 – JHEP10(2023)061 The tension of the domain wall in this model can be computed writing the energy per unit area as follows: σ=Z∞ −∞ dz Kϕ¯ ϕ∂zϕ(z)−eiθKϕ¯ ϕ¯ W¯ ϕ(ϕ(z)) 2+ 2 Re[e−iθ∆W](2.5) One can readily see that in the case of supersymmetric bosonic solutions, where (2.3) holds, the tension becomes σBPS = 2|∆W|.(2.6) 2.1.1 Example. Double well potential Let us illustrate all of the above by considering the model defined by: K(ϕ, ¯ ϕ) = ϕ¯ ϕ, W(ϕ) = 1 3ϕ3−a2ϕ(2.7) where we are assuming that a > 0. For this particular model, the Lagrangian (2.1) simplifies to L=−∂µϕ∂µ¯ ϕ−|ϕ2−a2|2.(2.8) The potential of the theory, restricted to the real part of ϕ, has been plotted in figure 1(a) (dashed line) where we see the double well potential form with the two supersymmetric minima located at ϕ±=±a. The second-order equations of motion (2.2) read, in this case, ∂2 zϕ(z)−2¯ ϕ(ϕ2−a2)=0.(2.9) On the other hand, the first-order BPS equation (2.3) reads ∂zϕ(z) = −(¯ ϕ2−a2),(2.10) where we have chosen W(ϕ(z=∞)) < W(ϕ(z=−∞), which implies5eiθ =−1. The solution of this equation is given by the real field configuration, ϕ(z) = atanh(az),(2.11) while the tension of this domain wall is given by σBPS = 2|∆W|=8 3a3.(2.12) As we mentioned earlier, any solution to the BPS equation preserves some supersymmetry by construction, so it is clear that it cannot represent the decay of the vacuum. We can also see this noting that both vacua are supersymmetric, degenerate in energy, and the wall is flat and infinite, so there is no way these vacua can decay. On the other hand, the solution we found here is purely real. This is consistent with the potential we have constructed since its form is such that perturbations around the 5On the other hand, eiθ = +1 would yield the mirrored profile, often denoted as the anti-domain wall. This simply corresponds to flipping the boundary conditions imposed at ±∞. – 5 – JHEP10(2023)061 solution in the imaginary field directions are stabilized. We can check this by expanding the potential in the real and imaginary parts of the field, namely ϕ(z) = ψ(z) + i s(z),(2.13) so the potential reads V(ψ, s)=(ψ2−a2)2+ 2(ψ2+a2)s2+s4.(2.14) This is why we can concentrate on the solution along the s= 0 line. In all of the examples we show here, we have checked that this is indeed the case; therefore, in all of our illustrations we will simply draw the results concerning the real part of the fields. 2.2 Stability and vacuum decay In the previous section we showed how one can find supersymmetric domain wall configurations that interpolate between supersymmetric vacua in our model. The fact that these vacua are stable is not surprising since supersymmetry imposes them to be degenerate global minima of the potential. Let us now consider the case where there is a small supersymmetry breaking term in our potential and study the stability of the resultant vacua. For simplicity let us assume that we introduce in the theory a couple of soft supersymmetry breaking terms [24] of the form Ssoft =−Zd4x√−ghµ2ϕ¯ ϕ+bϕ3+¯ ϕ3i,(2.15) which break supersymmetry explicitly. In the following, we will consider the coefficients to be small enough so that many of the properties of the solution found earlier will still hold. This means we will consider the case where µ2≪a2as well as b≪a, see the coloured curves of 1(a), where for simplicity we have plotted some curves for the cases of b=µ. In this regime we can see that the theory still has two minima given by ϕ±=±a+δ±(µ2, b),(2.16) where the solutions have only shifted slightly, i.e. |δ±/a| ≪ 1. The interesting point now is that both of these minima break supersymmetry. If one takes b > 0, the potential at ϕ+becomes slightly higher than the other minimum; this means that this vacuum will be unstable with respect to the nucleation of bubbles of the true vacuum at ϕ−. Furthermore, the form of the supersymmetry-breaking terms allows for the tunneling to happen along the real direction of the field, see figure 1(a). In order to compute the probability of the decay and its bubble profile in terms of the scalar field, we will resort to the usual methods developed by Coleman and collaborators [5,6] in the context of False Vacuum Decay. Thus, we only have to extremize the Euclidean action associated to the Lagrangian (2.1) completed by the supersymmetrybreaking terms. Assuming a standard kinetic term (K=ϕ¯ ϕ) and an O(4) symmetry in Euclidean space, the equations of motion for the field are given by ϕ′′ +3 ρϕ′=∂V ∂¯ ϕ(2.17) – 6 – JHEP10(2023)061 -1.5 -1.0 -0.5 0.5 1.0 1.5 -1 1 2 (a) -3-2-1 1 2 3 -1.0 -0.5 0.5 1.0 (b) Figure 1. (a) Potential (2.19) with a= 1, for several supersymmetry-breaking parameters. (b) Solutions to (2.17), centered around the inflection point corresponding to each profile, labeled as ρ∗. The BPS limit is shown with a dashed line, representing the domain wall solution arising from the dashed potential in (a). where ρ=√τ2+x2,τbeing the Euclidean time, and primes denote derivatives with respect to ρ. This equation is to be solved considering the boundary conditions lim ρ→∞ ϕ(ρ) = ϕ+, ϕ′(0) = 0.(2.18) Note that the Euclidean O(4) symmetry will turn to O(1,3) when transporting the solution back to Lorentzian spacetime. Among other things, this will mean that the profile ϕ(ρ)obtained via (2.17) will correspond to the emergent scalar profile for the bubble at formation. Furthermore, this symmetry implies that the radius of the bubble so formed will expand outward following a constant acceleration trajectory. Following the example presented in the previous subsection, we will work with the following potential V(ϕ, ¯ ϕ)=(ϕ2−a2)(¯ ϕ2−a2) + µ2ϕ¯ ϕ+b(ϕ3+¯ ϕ3)(2.19) which already includes a contribution from supersymmetry-breaking terms within its definition. In this one-dimensional setup, the profile of the scalar field can be easily found in Euclidean radial coordinates using an undershoot/overshoot algorithm [5]. Essentially, since the boundary conditions (2.18) do not specify the initial starting point ϕ(0), we can first obtain a couple of points where the field either undershoots or overshoots the false vacuum at ϕ+. Iteratively reducing this field range, we will eventually find a starting point ϕ(0) which stays sufficiently close to the true vacuum at ϕ−for large values of ρ. The solutions found this way have been contrasted with ones obtained using the software AnyBubble [26], which applies an alternative multiple shooting method, and have been shown to be essentially identical. Using this numerical algorithm, we can check that indeed the form of the domain wall forming the bubble does not change qualitatively in comparison to the supersymmetric case given above. In order to visually make this comparison we have plotted in figure 1(b) the resultant domain wall profiles for the bubble nucleation centered on the same point. It – 7 – JHEP10(2023)061 is clear that dialling back to zero the supersymmetry breaking coefficients b, µ one recovers the profiles obtained with the first-order BPS equation (2.10), i.e., the supersymmetric domain wall profile. Of course, this is consistent since in this case the bubble radius is infinite and the tunnelling rate would be zero, as this limit takes us back to the stable supersymmetric vacua presented in the previous section. 3 Rigid supersymmetric models with 3-form potentials As we described in the introduction, we are interested in studying supersymmetric models with some new degrees of freedom beyond the complex scalar field presented in the previous section. In particular, we would like to introduce a 3-form potential into our model. Actually, using supersymmetry as a guiding principle, we can describe both the scalar field and the 3-form gauge field into a single multiplet described in terms of a special type of chiral superfield. We show in appendix Ahow one can construct such a special chiral superfield describing the so-called single 3-form supermultiplet from unconstrained real scalar superfields. In this case, the bosonic content of the multiplet is simply given by complex scalar fields and auxiliary fields, the real part of which will be related to a field strength of the 3-form potential. Following this prescription we find that the simplest bulk action for such fields is given by Sbulk =Zd4x−Kϕ¯ ϕ∂µϕ∂µ¯ ϕ−1 4·4!Kϕ¯ ϕFµνσρFµνσρ +1 2·4! Wϕ+¯ W¯ ϕϵµνσρFµνσρ +1 4Kϕ¯ ϕWϕ−¯ W¯ ϕ2,(3.1) where Fµνρσ = 4∂[µAνρσ]and as before Kand Wdenote the Kähler potential and superpotential which define the model for the complex scalar field ϕ(x).6 As is usually the case in models involving gauge fields, this action must be supplemented with some boundary terms that are required to make the variation of the action well posed [10,27]. In our present case, the boundary term is given by Sbd =1 2·3! Zd4x ∂µhAνρσ Kϕ¯ ϕFµνρσ −ϵµνρσ Wϕ+¯ W¯ ϕi,(3.2) see appendix Afor a derivation of this result. The equation of motion for the 4-form field strength can be written as ∂µhKϕ¯ ϕFµνρσ −ϵµνρσ Wϕ+¯ W¯ ϕi= 0 ,(3.3) which can be integrated to give, Fµνρσ =Kϕ¯ ϕϵµνρσ Wϕ+¯ W¯ ϕ−2n,(3.4) where n∈Ris a real integration constant. This expression allows us to integrate out the 3-form potential from the original action to obtain a new effective theory written in terms 6For the sake of simplicity, will study models consisting of a single scalar field here. See [18] for a study of similar systems with an arbitrary number of scalar fields. – 8 – JHEP10(2023)061 01234 1 2 3 4 5 6 7 Figure 3. Scalar potential (3.29) for n= 1,q= 2 and some values of the supersymmetry-breaking parameters. The darker curve shows the potential inside the membrane (ρ < R), while the lighter one represents the potential outside it (ρ>R). ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ 2 4 6 8 10 0 0.05 0.1 0.15 ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ 5 10 15 20 25 1 2 3 4 5 ▼ ▲ ◆ ■ ● (a) 5 10 15 20 0 1 2 3 4 (b) Figure 4. (a) Euclidean action for solutions of the equation of motion (3.27), with n= 1,q= 2 and several values for R,band µ. (b) Energy and Euclidean action difference with respect to the false vacuum state for several radii, in the case where µ=b= 0.03. The maximum of Bcoincides with the root of ∆E, as expected. of the radius R. We take this to be the correct value of the radius of the instanton that mediates the vacuum decay of interest to us. We also checked the variation in the total energy of the profile with respect to the background at the time of nucleation. Indeed, there should be no energy loss or gain for the instanton solution at the time of its emergence. Thus, the correct bubble profile should correspond to the roots of the energy density difference with respect to the false vacuum background. This quantity is defined by ∆E= 4πZ∞ 0 dr r2"Kϕ¯ ϕ dϕ dr  2 +V(ϕ, ¯ ϕ)+2|qϕ|δ(r−R)−Vfv#t=0 .(3.30) This last expression can be easily evaluated from the Euclidean solution, since the profile obtained in Euclidean space corresponds to the profile of the emerging bubble in Lorentzian space at t= 0. In all of the profiles that we have computed, the value of R corresponding to no energy loss or gain with respect to the background has been found to correspond with the maximum of the Euclidean action, see figure 4(b) for an explicit example. – 15 – JHEP10(2023)061 0 5 10 15 20 25 30 1.0 1.5 2.0 2.5 (a) -2-1 1 2 3 4 5 1.0 1.5 2.0 2.5 3.0 (b) Figure 5. (a) Scalar field profiles corresponding to each maximum Euclidean action for some supersymmetry-breaking parameters. (b) The same profiles as before, centered around the membrane radius corresponding to each profile, denoted by ρ∗. The dashed line represents the BPS solution from eq. (3.22), with n= 1 and q= 2, which is clearly the asymptotic behaviour of the profiles as the supersymmetry-breaking parameters band µtend to 0. The profiles corresponding to the solutions which extremize SEhave been plotted in figure 5(a). As expected, as the supersymmetry-breaking parameters are made smaller, the radius of the emerging membrane increases and the scalar field profiles progressively tend towards the BPS solution derived above, as shown in figure 5(b). A curious feature of these profiles is that, when considered as a particle in the inverted potential −V, they first tend to get away from the false vacuum, only to then be projected in the r > R potential with enough velocity to asymptotically reach the false vacuum. 4 Supergravity models We now turn to study the same setup as in the previous section, with gravity taken into consideration. We will work in the context of N=1, D= 4 Supergravity coupled to chiral matter. We will be interested in generalizing the false vacuum decay discussed in the previous section including gravity. However, before analysing more generic situations, we will first study the supersymmetric limit of flat membrane solutions in supergravity. Thus, in this section, we will start by analyzing the action of the system composed by scalar fields, real three-forms and flat membranes in a spacetime of Lorentzian signature. The action we will be considering is once again given by the sum of the following terms, S=Sbulk +Smembrane +Sboundary terms.(4.1) As we review in appendix A, the bosonic part of the bulk action of the system, which includes gravity, scalar fields and 3-forms, is given by the following supergravity action Sbulk =Zd4x√−g1 2R−Kϕ¯ ϕ∂µϕ∂µ¯ ϕ−1 3e−K(M+K¯ ϕ¯ F) ( ¯ M+KϕF)−M¯ W−¯ MW +e−KKϕ¯ ϕF¯ F+FWϕ+¯ F¯ W¯ ϕ(4.2) – 16 – JHEP10(2023)061 where, as before, K(ϕ, ¯ ϕ)and W(ϕ)describe the real Kähler potential and the holomorphic superpotential functions of the scalar field and F=1 2(DµAµ+id) + 2 3¯ ϕM +1 3ϕ¯ M. (4.3) In the above equations we use MPl = 1 and denote the Ricci scalar by R. Furthermore, M is a complex scalar auxiliary field of the minimal supergravity multiplet, d is a real scalar auxiliary field and Aµis the Hodge dual of the three-form. In our setting this latter field belongs to the matter supermultiplet, which also includes the scalar field.8 Just as in the non-gravitational case, boundary terms must be included in the full action to ensure the variation of the action with respect to the form field is well posed. We leave their derivation for appendix C. Note that these boundary terms will also include the Gibbons-Hawking term [35] in order for the variational problem to be well defined also in the gravitational sector. The action of the membrane can be fixed by requiring the interacting system (4.1), with fermionic terms taken into account in the first two terms, to remain invariant under a half of local supersymmetry after the bosonic membrane is included [36,37]. It reads Smemb. =−ZM d3ξ√−h2eK/2|qϕ|+q 3! ZM d3ξAµνρ ∂xµ ∂ξa ∂xν ∂ξb ∂xρ ∂ξcϵabc (4.4) which as before describes the coupling between the membrane and the 3-form potential as well as the scalar field. Note however that in supergravity the Nambu-Goto term receives a correction from the exponential of the Kähler potential.9 It is easy to check that setting the form field and auxiliary fields on-shell and taking into account the contribution of the boundary terms, the action reads [21] S=Zd4x√−gR 2−Kϕ¯ ϕ∂µϕ∂µ¯ ϕ−V(ϕ, ¯ ϕ)+SGH −ZM d3ξ√−h2eK/2|qϕ|(4.5) where SGH represents the Gibbons-Hawking boundary term and V(ϕ, ¯ ϕ)is the N=1, D= 4 matter-coupled supergravity scalar potential of the form V(ϕ, ¯ ϕ) = eKhDϕˆ WKϕ¯ ϕD¯ ϕˆ ¯ W−3|ˆ W|2i(4.6) with the usual Kähler-covariant derivative denoted by Dϕ=∂ϕ+Kϕ. As in the global supersymmetric models the information about the 3-form flux is encapsulated in the form of the effective superpotential: ˆ W=W−(n+qH(x))ϕ , (4.7) where, just as in the non-gravitational case, H(x)is given by eq. (3.11). This modified superpotential will allow us to find a profile for the dressed membrane which interpolates between supersymmetric minima of different potentials. 8We note that a 3-form fields may also be included as an auxiliary field of the supergravity supermultiplet [14,28–34]. Even though this provides a more straightforward supersymmetric generalization of the Brown-Teitelboim construction [9,10], we will not consider it here since the advantage of inclusion of the 3-form in the matter supermultiplet is that this construction has a smooth and nontrivial flat space limit. 9The exponential factor eK/2arises due to the super-Weyl rescaling and field redefinitions required to bring the action to Einstein frame, see appendix Cfor more detail. – 17 – JHEP10(2023)061 4.1 Flat membrane solutions in supergravity models Similarly to what we did in the flat space case, one can find smooth domain wall solutions in the supergravity context. This has been extensively studied in the literature starting in [3] (see also a more recent discussion about this topic in [38] and references therein). Here we show how one can generalize these solutions to include the presence of a supermembrane as it was described in [21]. Let us begin with the case of a flat membrane interpolating between two supersymmetric vacua. Let us assume this static membrane sits at z= 0. In order to study the profile across such a membrane, we will assume the following ansatz for the metric10 [21,38]: ds2=e2D(z)(−dt2+dx2+dy2) + dz2(4.8) so that √−g=e3D(z). Let us turn to study the equation of motion for the scalar field, which is 1 √−g∂µ√−gKϕ¯ ϕgµν∂νϕ=Kϕ¯ ϕ¯ ϕ∂µϕ∂µ¯ ϕ+∂V ∂¯ ϕ+eK/2hK¯ ϕ|qϕ|−qeiηiδ(z),(4.9) where eiη is defined as eiη =−qϕ |qϕ|z=0 .(4.10) It is natural to assume that scalar field only depends on the transverse coordinate to the membrane, i.e., ϕ=ϕ(z). In that particular case, the field obeys ∂z(Kϕ¯ ϕ∂zϕ)+3Kϕ¯ ϕ∂zD ∂zϕ=Kϕ¯ ϕ¯ ϕ|∂zϕ|2+∂V ∂¯ ϕ+eK/2hK¯ ϕ|qϕ|−qeiηiδ(z).(4.11) On the other hand, the Einstein equations for the metric can be combined to give ∂2 zD+ 3(∂zD)2=−V−eK/2|qϕ|δ(z)(4.12) Note that the deltas at z= 0 will yield jumps in the first derivative of both ϕ, as in the non-gravitational case, and in the scale factor D. A supersymmetric and static domain wall may interpolate between non-degenerate minima, since essentially the gravitational contribution may compensate the difference in scalar potential between both vacua [3]. If supersymmetry is partly conserved across the profile of the domain wall, then the minima are bound to be either Minkowski or AdS vacua (note, however, that no supersymmetric domain wall may interpolate between two Minkowski vacua when gravity is included [3]). With these remarks at hand, the BPS equations acquire the form [3,21,38] ϕ′(z) = ∓eK/2eiarg( ˆ W)K¯ ϕϕD¯ ϕˆ W(4.13) D′(z) = ±eK/2|ˆ W|(4.14) 10Note that with this choice of metric and in the so-called static gauge xa(ξ) = ξa, for flat membrane, z(ξ) = 0, we will have √−h=√−g, where the r.h.s. is calculated at z= 0. – 18 – JHEP10(2023)061 where primes denote derivatives with respect to z, The second order equations are obeyed by the solutions of the first-order BPS equations when suitable boundary conditions are imposed. In particular, taking the derivative of eq. (4.14) and using (4.13), we find that the second order equation (4.12) is satisfied provided11 eiarg( ˆ W)z=0 =∓eiη .(4.15) It is convenient, following [38], to write these equations in terms of Z ≡ eK/2ˆ W. (4.16) Note that the value of the scalar potential at supersymmetric critical points is then given by Vsusy =−3|Z|2. In terms of Z, the BPS equations simplify to ϕ′(z) = ∓2K¯ ϕϕ∂¯ ϕ|Z| (4.17) D′(z) = ±|Z| .(4.18) The sign to use in the integration of the BPS equations is determined by the value of |Z| at z→ ±∞. In order to see this, let us firstly note that d|Z| dz = (ϕ′∂ϕ|Z|+¯ ϕ′∂¯ ϕ|Z|)∓eK/2|qϕ|δ(z) = ∓h4Kϕ¯ ϕ∂ϕ|Z|∂¯ ϕ|Z|+eK/2|qϕ|δ(z)i(4.19) where, in the second step, we have used (4.17). Then it is easy to observe that, if for example we choose the lower sign in eqs. (4.17) and (4.18), which then become ϕ′(z)=2K¯ ϕϕ∂¯ ϕ|Z| (4.20) D′(z) = −|Z| ,(4.21) we find that the derivative of |Z| is positive d|Z| dz = 4Kϕ¯ ϕ∂ϕ|Z|∂¯ ϕ|Z|+eK/2|qϕ|δ(z)>0.(4.22) Therefore, |Z| must increase monotonically, which implies |Z|−∞ <|Z|+∞. On the other hand, if |Z|−∞ >|Z|+∞, the lower sign of the BPS equations eqs. (4.17)–(4.18) will apply. Finally, it should be noted that in case |Z| has some root along z, the situation becomes a bit more complicated. Indeed, the signs must be swapped after crossing a root of the superpotential in order to have a positive overall tension of the domain wall [21,38]. However, as noted in [3], such cases do not correspond to the limiting case of a bubble instability. In fact the spacetime induced by these solutions is asymptotically quite different and resembles the one in the Randall-Sundrum scenario [40] with a single positive tension brane.12 11Actually this identification can be obtained straightforwardly as it implies that on the worldvolume of the membrane the supersymmetry preserved by the solution coincides with κ-symmetry of the supermembrane action [39]. 12We will defer the exploration of these types of solutions in our context for a future publication. – 19 – JHEP10(2023)061 0.2 0.4 0.6 0.8 1.0 1.2 -0.4 -0.3 -0.2 -0.1 0.1 0.2 Figure 6. Scalar potential for the model described in this section, for two different values of the flux value n. -4-2 0 2 4 0.95 1.00 1.05 1.10 (a) -4-2 2 4 -1.0 -0.5 0.5 (b) Figure 7. Solution of the BPS equations to the model defined in (4.23), for (a) the scalar field (b) the scale factor, for the metric defined in (4.8). The potential at z < 0corresponds to the case n= 3, while z > 0corresponds to n= 2; thus, the membrane is charged with q=−1. 4.2 Example: quartic superpotential The above equations can be used to find the profiles of a scalar field interpolating between different minima of the model defined by K(ϕ, ¯ ϕ) = ϕ¯ ϕ, W(ϕ) = (10MPl)−1ϕ4.(4.23) where we have restored the Planck mass momentarily in order to show explicitly the energy scales involved in this example. The scalar potential defined by this Kähler potential and superpotential is shown in figure 6, once the 3-form has been integrated out. The minimum featured by each branch can be shown to be supersymmetric, i.e., it satisfies Dϕˆ W= 0. Going back to units where MPl = 1, the numerical BPS profile arising from this potential for both the scalar field and the scale factor D is shown in figure 7, where we have placed the lower minimum to the left (for easier comparison later on). We have also checked that the second-order equations (4.11) and (4.12), which explicitly incorporate the first-derivative jumps as Dirac deltas, yield exactly the same profiles. Note that for this particular solution the supersymmetric minima on both sides of the wall describe an anti-deSitter vacua with different values of their cosmological constant. – 20 – JHEP10(2023)061 5 Membrane nucleation in supergravity As we discussed in the introduction membrane nucleation in a model with a 4-form flux has been studied in the literature in the context of models similar to Brown-Teitelboim [9,10]. Our supergravity model includes a 3-form field and a brane charged with respect to it, however, as we have shown in the previous section it can also be casted exclusively in terms of a scalar field by integrating out the 3-form potential. We can then ask whether there will be bounce solutions similar to the flat space ones where the field interpolates between two minima of the flux-dependent potential for the scalar field as one crosses the wall. In order to illustrate these vacuum decay processes in our model, we will follow a similar procedure to the one we presented in the flat space limit in section 3.3 where we introduced small corrections to the supersymmetric Lagrangian. These terms will allow the possibility of having supersymmetry breaking vacua that could be susceptible to decay. In particular, we will study the Euclidean action given by13 SE=Zd4x√g−R 2+Kϕ¯ ϕgab∂aϕ∂b¯ ϕ+˜ V(ϕ, ¯ ϕ)+ 2 ZM d3ξ√heK/2|qϕ|+SGH (5.1) where, ˜ V(ϕ, ¯ ϕ) = eKKϕ¯ ϕDϕˆ W 2−3ˆ W 2+µ2ϕ¯ ϕ, ˆ W≡W−(n+qH(x))ϕ. (5.2) As we will see afterwards, the Gibbons-Hawking boundary term will become important once we evaluate the actual value of the Euclidean action. Furthermore, assuming an O(4) symmetric Euclidean solution for the instanton, we introduce the following ansatz for the metric ds2=dχ2+ρ(χ)2dΩ2 3.(5.3) Using these coordinates we search for a solution featuring a spherical membrane sitting at a fixed value of the radial coordinate, which we call χ=R. With this setting, one arrives to the following equations of motion for the metric function and the scalar field, ∂χKϕ¯ ϕ∂χϕ+3∂χρ ρKϕ¯ ϕ∂χϕ=Kϕ¯ ϕ¯ ϕ|∂χϕ|2+∂˜ V ∂¯ ϕ+eK/2hK¯ ϕ|qϕ|−qeiηiδ(χ−R)(5.4) ρ′′ =−1 3ρ2Kϕ¯ ϕ|ϕ′|2+˜ V−ρeK/2|qϕ|δ(χ−R),(5.5) which in the case of a canonical kinetic term for the field ϕ, that is, K(ϕ, ¯ ϕ) = ϕ¯ ϕ, reduce to ϕ′′ +3ρ′ ρϕ′=∂˜ V ∂¯ ϕ+e|ϕ|2/2hϕ|qϕ|−qeiηiδ(χ−R)(5.6) ρ′′ =−1 3ρ2|ϕ′|2+˜ V−ρe|ϕ|2/2|qϕ|δ(χ−R),(5.7) 13We have omitted the supersymmetry-breaking cubic term for simplicity, as its effect (in this model, at least) was identical to turning on the quadratic term. – 21 – JHEP10(2023)061 which are very similar to the set of equations to be solved in the case of a purely scalar field bounce solution in curved space first found by Coleman and de Luccia in [2] with the important difference that now both first derivatives of the functions ρand ϕpresent a jump at the position of the membrane. This is to be expected since locally the behaviour of the functions should be the same as the flat membrane case in these models. Just like in the non-gravitational case of section 3.3, one may integrate eqs. (5.4)–(5.5) around a small neighborhood of χ=Rin order to explicitly find the jump of ϕ′and ρ′ across the membrane: [∂χϕ]|χ=R=eK/2hK¯ ϕ|qϕ|−qeiηi,[∂χρ]|χ=R=−ρeK/2|qϕ|.(5.8) Finally, in order to find the radius of the membrane R, we will differentiate between transitions starting from AdS and Minkowski, which we will cover in section 5.1.1, and transitions starting from dS, which we will analyze in section 5.1.2. For the former cases, we will take a similar approach as the one we took in the non-gravitational case, namely, we will solve eqs. (5.4)–(5.4) for several choices of Rand find which one extremizes the Euclidean action (5.1). On the other hand, decays involving a dS false vacuum will require special attention due to the compactness of the instanton. One should note, however, that another possibility exists in order to find the radius of the membrane for all of these cases, which relies on the use of the χχ-component of the Einstein equations associated to our system: 1−(ρ′)2+ρ2 3Kϕ¯ ϕ|ϕ′|2−˜ V= 0 (5.9) Indeed, this equation will be satisfied across any profile only for the particular membrane radius Rwhich extremizes the action. Therefore, an alternative method to find such Ris to fine-tune it in order for the l.h.s. of (5.9) to be as close to zero as possible across the full profile. In what follows, we have relied on the explicit evaluation of the Euclidean action in order to find R. Furthermore, eq. (5.9) has been used to test the accuracy of the numerical profiles. 5.1 Example: quartic superpotential In this subsection, we will apply the machinery described above to the simple model of eq. (4.23). Taking into account the different values of the supersymmetry breaking parameter µone can encounter several situations depending on the nature of the false vacuum. In particular, the geometry of the Euclidean space will depend on the sign of the potential at the false vacuum, i.e., while for Vfv ≤0, the background will be a non-compact space, the case with Vfv >0will yield a compact de Sitter space for the background. This different properties of the background will become important for the way we handle our numerical solutions. All the examples below were solved numerically using a simple overshoot-undershoot algorithm to find the correct initial condition for the scalar field. Note that, as far as the boundary conditions are concerned, requiring the spacetime to be regular at the center of the bubble implies that ρ(χ) = χ+O(χ3)for χ→0, see [2] for further details. – 22 – JHEP10(2023)061 5.1.1 AdS/Minkowski to AdS transitions As we mentioned above, this tunnelling event occurs within a non-compact space. Therefore, the integral of the Euclidean action corresponding to the instanton with a membrane fixed at a particular value of the coordinate radius Ris given by, SE= 2π2Z∞ 0 dχ hρ3|ϕ′|2+˜ V(ϕ, ¯ ϕ)+ 3 ρ2ρ′′ +ρ(ρ′)2−ρi+ 4π2hρ3e|ϕ|2/2|qϕ|iχ=R+SGH = 2π2Z∞ 0 dχ hρ3|ϕ′|2+˜ V(ϕ, ¯ ϕ)−3ρ(ρ′)2+ρi+ 4π2hρ3e|ϕ|2/2|qϕ|iχ=R(5.10) where, in the last step, the term we have integrated out cancels the contribution from the GH term (see [41,42]). On the other hand, the integral corresponding to the background is given by, SE,bg = 2π2Z∞ 0 dχ hρ3 fvVfv −3ρfv(ρ′ fv)2+ρfvi,(5.11) where ρfv(χ) = H−1sinh (Hχ)(5.12) is the expression for the scale factor of Euclidean anti-deSitter space in the particular slicing given by eq. (5.3) and where we have introduced the Hubble parameter H=q|Vfv| 3. We can take the Minkowski limit of this expression to find that in the case of Vfv = 0, the scale factor becomes, ρfv(χ) = χ. It is easy to see that these background Euclidean actions as well as the ones obtained from the instanton solutions are, in fact, divergent. However, the physically relevant quantity is the difference between the instanton action and the background one. This action is finite. The key to computing this difference correctly lies in performing the integrations up to a certain ρmax, such that its corresponding radial coordinate χmax satisfies χmax ≫R. See [42] for more detail and an explicit proof of the convergence of this difference. In figure 8we show the result of computing this difference for several membrane radii and supersymmetry-breaking parameter values. We can clearly see that the difference between actions is finite and reaches a maximum at a certain R, depending on µ. Furthermore, as the potential tends towards its original and supersymmetric form, the radius of the membrane interpolating between both branches of the scalar potential gets bigger, which is consistent with the fact that at the supersymmetric limit the tunneling transitions becomes completely suppressed. The profiles for both the scalar field and the scale factor corresponding to the radii with maximum Euclidean action difference for each µare shown in figure 9. The scalar field profiles are all quite similar in shape, with the only differences resting on the positions of the true and false vacuum, and in the radius where the jump happens. On the other hand, the scale factor shows very clearly where the jump happens as well, and the exponential behaviour seems to pick up quite fast once the membrane has been crossed. – 23 – JHEP10(2023)061 0.7 0.8 0.9 1.0 1.1 1.2 -0.4 -0.3 -0.2 -0.1 (a) ● ●●●●● ● ● ■ ■■■■■■■■■■■■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ▲ ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ ▲ ▲ ▲ ▼ ▼▼▼▼▼▼▼▼▼▼▼▼▼ ▼ ▼ ▼ ▼ 6 8 10 12 0.1 0.2 0.3 0.4 ● ● ● ● ● ● ● ● ● ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ ▼ ▼ 5 10 15 20 0 5 10 15 20 25 ▼ ▲ ◆ ■ ● (b) Figure 8. (a) Scalar potential for n= 2 (dashed) and n= 3 (solid). (b) Euclidean action for different fixed membrane radii Rfor a list of supersymmetry breaking parameters µ. The color corresponding to each µis the same for both figures. Note that the radius Rwhich extremizes the action increases as we diminish the supersymmetry-breaking parameter µ. 0 5 10 15 20 25 30 0.90 0.95 1.00 1.05 (a) 0 5 10 15 20 25 30 2 4 6 8 10 12 14 (b) Figure 9. Evolution of (a) scalar field and (b) scale factor, for different supersymmetry parameters µ. Each case corresponds to the radius of maximum Euclidean action obtained in figure 8. In order to compare all these profiles with the limiting BPS case, we show in figure 10 a close-up plot around the membrane for all of them. Essentially, we see that the BPS profile is actually a limiting case for the scalar field, which concurs with our results in the non-gravitational case. Tuning the supersymmetry-breaking parameter µ, we can also analyze an almost Minkowskian false vacuum (µ= 0.33), see figure 11(a). Note that Minkowski false vacua are still non-compact spaces, and thus they should be analyzed in exactly the same fashion as AdS vacua. After computing the Euclidean action difference for several radii, we found that the radius with maximum Euclidean action was R= 5.2in the Minkowskian case. The profiles corresponding to a setting with such a membrane are shown in figure 12. Finally, as explicitly shown in [10,42], the requirement of energy conservation can be generalized to tunnelings where gravitational effects are considered and the false vacuum is Minkowskian, by requiring the ADM mass vanishes. After switching to Lorentzian – 24 – JHEP10(2023)061 uncostrained real superfield V=¯ V. It is convenient to define its θ-expansion as follows: V(x, θ, ¯ θ) = C+iθχ −i¯ θ¯χ+iθθ ¯ ϕ−i¯ θ¯ θϕ −θσµ¯ θvµ +iθθ¯ θ¯ λ+i 2¯σµ∂µχ−i¯ θ¯ θθ λ+i 2σµ∂µ¯χ+1 2θθ¯ θ¯ θD−1 2□C.(A.12) Here u(x)and D(x)are real scalar fields, ϕ(x)is a complex scalar field, vµ(x)is a real vector field and χ(x)and λ(x)are Weyl spinors. A special chiral superfield Ycan be obtained from the generic chiral superfield Φ = ¯ D¯ DP by expressing the complex prepotential by P=−i 4Vthe real superfield V, so that17 Y:= −i 4¯ D2V(A.13) (the prefactor is chosen for later convenience). It is easy to check that it fulfills the condition ¯ D˙αY= 0 from its definition. Its components can be projected using Y|=−i 4¯ D2V=ϕ(A.14) DαY|=−i 4Dα¯ D2V=λα(A.15) −1 4D2Y=i 16 D2¯ D2V=1 2(∂µvµ+iD)(A.16) Note that, component-wise, Yis almost identical to the original chiral field Φin that it contains a complex scalar field, a complex Weyl fermion and a complex auxiliary field, albeit in this case we are mostly interested in the real part of the latter, which is given by the devergence of a 4-vector field vµ. For our purposes, it will be convenient to consider vµas the one-form associated through Hodge duality to a three-form. Indeed, the Hodge dual of the three-form is given by the following vector field18 (∗A3)µ≡Aµ=1 3!ϵµνρσAνρσ (A.17) where the indices have been raised using the flat spacetime metric. The divergence of this vector field is related to the Hodge dual of the 4-form field strength F4, associated to Aνρσ through19 ∗F4=1 4!ϵµνρσFµνρσ =∂µAµ(A.18) 17Note that this definition allows for some freedom in choosing V, since (A.13) is invariant under the gauge superspace symmetry V→V+L, where Lis the so-called real linear superfield which satisfies ¯ D2L= 0 = D2L. Furthermore, the definition of Ymay be generalized to the case where the real superfield Vis not independent but composed from a so-called complex linear superfield Σ, which obeys D2Σ = 0, and its complex conjugate ¯ Σ, which obys ¯ D2¯ Σ = 0. This last case may be used to construct special chiral superfields which describe a double 3-form supermultiplet. See [18,21,23] for further detail. 18In our conventions, ϵ0123 =−ϵ0123 = 1 for Lorentzian signature (which we will use throughout this appendix), while ϵ0123 =ϵ0123 = 1 for Euclidean one. 19We have omitted the contribution of the metric determinant in these expressions to clarify the definitions. In the case of a curved spacetime, these expressions are (∗A3)µ≡Aµ=1 3! √−gϵµνρσAνρσ and ∗F4=1 √−g∂µ(√−gAµ). – 31 – JHEP10(2023)061 With these expressions in hand, we find that if vµis identified with Aµ, i.e, with the Hodge dual of the 3-form field, the composite auxiliary field of the special chiral superfield Yreads FY=−1 4D2Y=1 2(∗F4+iD).(A.19) The advantage of treating the vector vµas dual to 3-form is that a membrane can be coupled ’electrically’ (or minially) to the 3-form field (see below) and thus to the composite auxiliary field in the special chiral superfield Y. Since Ddoes not enter the membrane part of the full action, we will be able to remove it from the action by solving its algebraic equations early on, leaving ∗F4untouched for its interplay with the scalar field and the membrane. We can easily apply these expressions to a single three-form multiplet Y, with a certain Kähler potential K(Y, ¯ Y)and superpotential W(Y). Plugging the component fields of Y into the bosonic Lagrangian (A.10), we find L|bos. =−Kϕ¯ ϕ∂µϕ∂µ¯ ϕ+1 4Kϕ¯ ϕ(∗F4+iD) (∗F4−iD) + 1 2(∗F4+iD)Wϕ+1 2(∗F4−iD)¯ W¯ ϕ =−Kϕ¯ ϕ∂µϕ∂µ¯ ϕ+1 4Kϕ¯ ϕ(∗F4)2+1 2(∗F4)Wϕ+¯ W¯ ϕ+1 4Kϕ¯ ϕD2+i 2DWϕ−¯ W¯ ϕ (A.20) The equation of motion for the auxiliary field Dreads D=−iKϕ¯ ϕWϕ−¯ W¯ ϕ(A.21) which is completely algebraic, as expected. Plugging this into (A.20) yields L|bos. =−Kϕ¯ ϕ∂µϕ∂µ¯ ϕ+1 4Kϕ¯ ϕ(∗F4)2+1 2(∗F4)Wϕ+¯ W¯ ϕ+1 4Kϕ¯ ϕWϕ−¯ W¯ ϕ2(A.22) =−Kϕ¯ ϕ∂µϕ∂µ¯ ϕ−1 4·4!Kϕ¯ ϕFµνσρFµνσρ +1 2·4! Wϕ+¯ W¯ ϕϵµνσρFµνσρ +1 4Kϕ¯ ϕWϕ−¯ W¯ ϕ2(A.23) where, in the second step, we have rewritten the Hodge duals in terms of their original tensor fields for clarity. In order to proceed, recall that the physical field we wish to extremise in the action is not the field strength, but rather its antisymmetric tensor potential Aµνρ. In the following, it will be more convenient to work with the Hodge-dual Aµ, which is related to ∗F4by eq. (A.18). The variation of (A.23) with respect to Aµis 1 2Kϕ¯ ϕ(∗F4) (∂µδAµ) + 1 2(∂µδAµ)Wϕ+¯ W¯ ϕ =δAµ1 2∂µ−Kϕ¯ ϕ(∗F4) + Wϕ+¯ W¯ ϕ+∂µδAµ 1 2Kϕ¯ ϕ(∗F4)−Wϕ−¯ W¯ ϕ.(A.24) We can clearly see that the vanishing of the first term in the r.h.s. will give us an equation of motion for the 3-form gauge field, while the second one will produce a boundary term. – 32 – JHEP10(2023)061 As originally discussed in [10], in order to deal with the second term in (A.24) we will need to add a boundary term to our original Lagrangian to convert this contribution into an expression containing the variations of gauge invariant quantities (F4and scalar fields). Indeed, otherwise the variational problem with respect to gauge potential would not be well posed (see [18] for recent discussion). This boundary term not only ensures the consistency of the variational problem but, as we will see shortly, it will have a noticeable effect on the final, on-shell result. The required boundary term is given by Lbd =−1 2∂µhAµKϕ¯ ϕ(∗F4)−Wϕ−¯ W¯ ϕi =1 2·3!∂µhAνρσ Kϕ¯ ϕFµνρσ +ϵµνρσ Wϕ+¯ W¯ ϕi (A.25) while the equation of motion for the form field is ∂µ1 2Kϕ¯ ϕ(∗F4) + Re Wϕ= 0 ⇒ ∗F4=−2Kϕ¯ ϕ(ReWϕ−n)(A.26) where n∈Ris an arbitrary real (integration) constant. As this constant appears in the expression for the 4-form field strength, it can be identified with the flux of the 3-form gauge field. Plugging this result into (A.23) and taking into account the non-vanishing contribution of the boundary term yields L|bos.,on-sh. =−Kϕ¯ ϕ∂µϕ∂µ¯ ϕ−Kϕ¯ ϕ(Wϕ−n)¯ W¯ ϕ−n.(A.27) From this final form of the Lagrangian we can conclude that the contribution of 3-forms in a supersymmetric setup results in a linear contribution to our original superpotential in which the field is multiplied by a constant associated with the flux of the 3-form gauge field. Thus, setting the 3-forms on shell reduces the original theory to a model of a scalar field described by the original Kähler potential and an effective superpotential given by ˆ W(ϕ)≡W(ϕ)−nϕ. (A.28) B Rigid supersymmetry, bosonic membranes, and BPS equations As it was shown in [36,37], the action of the interacting system composed of the supergravity multiplet and a bosonic membrane is invariant under a half of local supersymmetry, provided the membrane term of the interacting action is given by the bosonic ‘limit’ of a supermembrane. Furthermore, the preserved part of the local supersymmetry reflects the local fermionic κ-symmetry of the original supermembrane action. In the case of an interacting system composed of supersymmetric matter and a supermembrane, there is no rigorous way to find rigid supersymmetry invariance of the interacting system including the matter supermultiplet and a bosonic membrane. However, as we will show below, there exists a trick allowing us to see the κ-symmetry of the original – 33 – JHEP10(2023)061 supermembrane part of the action for the interacting system. It implies the manipulation of the supersymmetry transformation of the bosonic membrane action with the use of some ansatze both for the fields and for membrane configuration. Actually, such a possibility, when it exists, reflects the existence of purely bosonic supersymmetric solutions of the complete supersymmetric system.20 In the following we will show how using this arguments we obtained the simple form of the BPS equations for our brane in our model. Let us start by looking at the supersymmetry transformation in our model. Although we did not write those transformations of the 3-form supermultiplet in the main text, they can be easily restored from the association of the spacetime fields with the superfield components in eqs. (A.14), (A.15), (A.16) and the identification of supersymmetry as fermionic supertranslations in superspace. This leads to the following supersymmetric transformations for the fields, δϵϕ=ϵαDαY|=ϵαλα,(B.1) δϵλα=ϵβDβDαY|+ ¯ϵ˙ β¯ D˙ βDαY|= 2ϵαFY+ 2i(σµ¯ϵ)α∂µϕ =ϵα(∂µAµ+iD) + 2i(σµ¯ϵ)α∂µϕ , (B.2) and δϵFY=1 2(∂µδϵAµ+iδϵD) = −1 4¯ϵ˙ β¯ D˙ βD2Y|=i(∂µλσµ¯ϵ).(B.3) These transformations also imply δϵD = ∂µλσµ¯ϵ+ϵσµ∂µ¯ λ , δϵAµ=i(∂µλσµ¯ϵ−ϵσµ∂µ¯ λ),(B.4) as well as δϵAµνρ =iϵµνρσ(∂µλσσ¯ϵ−ϵσσ∂µ¯ λ),(B.5) for the dual 3-form Aµνρ =ϵµνρσAσ. Let us now perform the above supersymmetry transformation to the scalar and 3-form field in the bosonic membrane action (3.7): δϵSmembr.=−2|q|Zd3ξ√−hRe ϵαλα ¯ ϕ |¯ ϕ|!+ 2qZd3ξIm (λσµ¯ϵ)ϵµνρσ∂0xν∂1xρ∂0xσ. (B.6) Generically, this last expression does not vanish. However, if we consider a flat membrane lying at x3=z= 0 in the static gauge, i.e., xa(ξ) = ξa, x3(ξ) := z(ξ)=0,(B.7) then √−h= 1 and eq. (B.6) reduces to δϵSmembr.=−2|q|Zd3ξRe ϵαλα ¯ ϕ |¯ ϕ|−iq |q|(¯ϵ˜σ3λ)!.(B.8) 20The relation of the κ-symmetry of a super-p-brane with supersymmetry preserved by solutions of equations was described for the first time in [39]. – 34 – JHEP10(2023)061 This formally vanishes if we set ϵα=i(¯ϵ˜σ3)αqϕ |qϕ|z=0 .(B.9) This equation has nontrivial solutions for a constant fermionic spinor ϵαif qϕ |qϕ|z=0 is independent of ξa(i.e., as long as it represents a constant phase). This condition is clearly satisfied if we assume the scalar field to depend on the zcoordinate only, that is, ϕ(xa, z) = ϕ(z).(B.10) This and the restriction to a flat membrane given by the last equation in (B.7) define the ansatz for the supersymmetric bosonic solution of the interacting system of the 3-form multiplet and supermembrane. Let us repeat that these formal calculations reflect the κ-symmetry of the complete supermembrane action, which guarantees the existence of the purely bosonic supersymmetric solution of the equations for the interacting system of a supermembrane and the single 3-form matter supermultiplet. Let us look at the transformation of the fields in the bulk. For a purely bosonic solution obtained with the ansatz (B.10), the preservation of supersymmetry implies δϵλ= 0 which, after using (B.2) and the auxiliary field’s equation 2FY= (∗F4+iD) = −Kϕϕ ˆ ¯ W¯ ϕ(B.11) reduces to i(σ3¯ϵ)α∂zϕ=ϵαKϕϕ ˆ ¯ W¯ ϕ.(B.12) This equation has a nontrivial solution with the constant fermionic spinor obeying projecting condition ϵα=eiηi(σ3¯ϵ)α⇔ϵα=−eiηi(¯ϵ˜σ3)α(B.13) if the scalar field obeys the BPS equation ∂zϕ(z) = eiηKϕ¯ ϕˆ ¯ W¯ ϕ.(B.14) Furthermore, the supersymmetry preserved by the bosonic configuration will also preserve the supersymmetry of the static flat bosonic membrane confirguration given by (B.7) if (B.13) coincides with (B.9), i.e. when eiη =−qϕ |qϕ|z=0 .(B.15) Given the BPS equation (B.14), it is easy to check that the phase of ∂zˆ Wremains constant even across the membrane. Indeed, if we multiply both sides of the equation by ˆ Wϕ, we find (∂zϕ)ˆ Wϕ=eiη ˆ WϕKϕ¯ ϕˆ ¯ W¯ ϕ⇒∂zˆ W=eiη hV(ϕ, ¯ ϕ) + |qϕ|δ(z)i(B.16) – 35 – JHEP10(2023)061 Therefore, one finds that eiη may be written as eiη =−qϕ |qϕ|z=0 =∆ˆ W |∆ˆ W|(B.17) where ∆ˆ W≡ˆ Wz=+∞−ˆ Wz=−∞. Note that this last result exactly reproduces the phase obtained in [3], albeit in our case it includes the effect of the membrane localized at z= 0, which effectively changes the superpotential when crossing it. One may also use this result in order to find a closed expression for the tension of the domain wall solution in the presence of a supermembrane. As shown in [23], one may rewrite the action (3.13) as S=−Zd3xZdzKϕ¯ ϕh∂zϕ−eiβKϕ¯ ϕˆ ¯ W¯ ϕih∂z¯ ϕ−e−iβKϕ¯ ϕˆ Wϕi −Zd3x2|qϕ|z=0 + 2Re he−iβ(∆ ˆ W+qϕ|z=0)i.(B.18) where the phase eiβ is, at this stage, arbitrary. However, upon choosing eiβ =eiη one can see that the expression for the tension of any field configuration is maximized by the solution of the BPS equation (B.14). Taking this into account the tension becomes TDW+memb. =|2∆ ˆ W|.(B.19) Note that even though this expression is formally similar to the one in the scalar field model in [3] the result is now written in terms of the effective superpotential ˆ Wwhich includes the jump due to the different fluxes on both sides of the membrane. As described in [21] and references therein, similar arguments can be applied to obtain the counterpart of (B.14) for dynamical systems including supergravity, which lead to eqs. (4.13)–(4.14). C Three-form multiplets in supergravity In this section we will follow a similar reasoning as the one above, with gravity included. We will first present the action for scalar multiplets described by generic chiral superfields, which we will later on generalize to include special chiral superfields which have vector or 3-form components among their bosonic ingredients. All of the results we present here have also been derived in [18,23] using a super-Weyl invariant approach to matter-coupled supergravity, reaching the same conclusions. We recall that supergravity can be described in terms of the superspace supervielbein EA M(z)and the spin connection ωab M(z) = −ωba M(z)subject to a set of torsion constraints. After fixing the so-called Wess-Zumino gauge, the field content reduces to •eµ a, the vielbein, •ψµ α, the gravitino, •bµ, a real vector auxiliary field, •M, a complex scalar auxiliary field. – 36 – JHEP10(2023)061 The most general superspace action of interacting supergravity and scalar supermultiplets in terms of EA M(z)and generic chiral superfields (defined in curved supergravity superspace) can be found e.g. in [25]. The spacetime action for the component fields is then obtained upon fixing the Wess-Zumino gauge and integrating over the fermionic coordinates of superspace. In the conventions of [25], the spacetime Lagrangian of the bosonic sector of such an action reads 1 √−gL=1 2Re−1 3K+ Ωa¯ b∂µϕa∂µ¯ ϕ¯ b−1 3e−1 3K˜ M˜ ¯ M−˜ M¯ W−˜ ¯ MW +e−1 3KKa¯ bFa¯ F¯ b+Fa(Wa+KaW) + ¯ F¯ b(¯ W¯ b+K¯ b¯ W) −1 9Ωbµbµ−i 3bµ(∂µϕaΩa−∂µ¯ ϕ¯ bΩ¯ b),(C.1) where Ω(Φ,¯ Φ) = −3e−1 3K(Φ,¯ Φ),˜ M=M+K¯a¯ F¯a,˜ ¯ M=¯ M+KaFa.(C.2) This action is not written in Einstein frame. Therefore, it is customary to rescale the vielbein as follows: ea µ7→ ea µe1 6K,(C.3) and to supplement this with a suitable transformation of the spin connection. Furthermore, rescaling the auxiliary fields as Fi7→ Fie−1 6K, M 7→ M e−1 6K,(C.4) we arrive at the following action in Einstein frame 1 √−gL=1 2R−Ka¯ b∂µϕa∂µ¯ ϕ¯ b−1 3˜ M˜ ¯ M−e1 2K˜ M¯ W−e1 2K˜ ¯ MW +Ka¯ bFa¯ F¯ b+e1 2KFa(Wa+KaW) + e1 2K¯ F¯ b(¯ W¯ b+K¯ b¯ W).(C.5) Notice that in this last expression the auxiliary fields bµhave been integrated out using their equations of motion. If we are dealing with minimal supergravity and scalar multiplets, ˜ M and all Fiare independent, and the auxiliary field equations read ˜ M=−3e1 2KW , FaKa¯ b=−e1 2K(¯ W¯ b+K¯ b¯ W).(C.6) Substituting this into (C.5) we find the well known matter-coupled N=1, D= 4 supergravity Lagrangian 1 √−gL=1 2R−Ka¯ b∂µϕa∂µ¯ ϕ¯ b−V(ϕ, ¯ ϕ)(C.7) where the potential is given by V(ϕ, ¯ ϕ) = eKDaWKa¯ bD¯ b¯ W−3|W|2,(C.8) and Da=∂a+Kaare the so-called Kähler-covariant derivatives. – 37 – JHEP10(2023)061 C.1 Supergravity interacting with 3-form multiplets Just as in the non-gravitational case, we will implicitly introduce three-forms by passing from generic to special chiral superfields. In this case, special chiral superfields describing single three-form multiplets are given by S=−i 4(¯ D2−8R)P,P= (P)∗(C.9) where Ris the so-called main chiral superfield of minimal supergravity, whose leading component is proportional to the complex scalar auxiliary field, R| =−1 6M, and Pis an unconstrained real superfield. The latter is defined up to shift by a real linear superfield (C.9) P 7→ P +L, (¯ D2−8R)L= 0 = (D2−8¯ R)L. (C.10) This freedom is the manifestation of a gauge symmetry which can be used to fix the WessZumino gauge, where P| = 0 , DαP| = 0 ,¯ D˙αP| = 0 .(C.11) The remaining part of the Lsymmetry, preserving this gauge, coincides with the 2-form gauge symmetry for the 3-form dual to vector component of the prepotential superfield, σa α˙α[Dα,¯ D˙α]P| = 4Aa, Aa=∗(A3)a.(C.12) In the gauge (C.11), the highest component of the special chiral superfield of Ssimplifies to (in the notation of [25]) F ≡ FS=−1 4D2S|=i 16 D2¯ D2P|− i 2RD2P| =1 2(DµAµ+id) + 1 3(s¯ M+ 2¯sM).(C.13) where s=S|,Mand ¯ Mare the scalar auxiliary fields of minimal supergravity, dis a real auxiliary scalar field, and ∗F4=DµAµ=1 e∂µ(eAµ), e = det ea µ=√−g . (C.14) Fixing the WZ gauge and integrating over the fermionic coordinates of superspace in the superfield action we arrive at the following Lagrangian in the bosonic limit: 1 √−gL=1 2Re−1 3K+ Ωs¯s∂µs∂µ¯s−1 3e−1 3K(M+K¯s¯ F) ( ¯ M+KsF)−M¯ W−¯ MW +e−1 3KKs¯sF¯ F+FWs+¯ F¯ W¯s−1 9Ωbµbµ−i 3bµ(∂µsΩs−∂µ¯sΩ¯s) + 1 √−gLbd (C.15) This Lagrangian formally coincides with that of scalar multiplets interacting with supergravity (C.5) up to boundary term, and up to the composite nature of the F-component Fof the special chiral superfields (C.13). – 38 – JHEP10(2023)061 Before substituting the expression for the F-components of the special chiral superfields, (C.13), it is convenient to perform a Weyl rescaling of the fields with (C.3), which will bring our action to the Einstein frame. In the follwoing, will carefully consider the case of supergravity interacting with special chiral superfields describing 3-form multiplets, as this case has some peculiarities with respect to the case of generic chiral superfields describing scalar supermultiplets. C.1.1 Super-Weyl transformations At the beginning of this appendix, when considering the interaction of supergravity an chiral superfields Φ, we assumed that each superfield Φand its components ϕand Fare inert under Weyl transformations. This is a consistent assumption in the case of a generic chiral superfield.21 However, since the chiral superfield Shas been written in terms of a real superfield P, it is important to check how super-Weyl transformations, act on them. These are defined via the following transformations of supervielbein [25,68]: Ea7→ ˜ Ea=eΥ+¯ ΥEa,(C.16) Eα7→ ˜ Eα=e2¯ Υ−ΥEα−i 4Ea¯ D˙α¯ Υ˜σ˙αα a,(C.17) ¯ E˙α7→ ˜ ¯ E˙α=e2Υ−¯ Υ¯ E˙α+i 4Ea˜σ˙αα aDαΥ,(C.18) where Υis a chiral superfield ¯ D˙αΥ=0, Dα¯ Υ=0.(C.19) These transformations must be supplemented by a suitable transformations of the spin connection; however, their explicit form is not needed for our purposes (see [25,68] for more detail). It is important to note that the supergravity chiral projector transforms in an inhomogeneous way under super-Weyl transformations: (¯ D¯ D−8R)7→ e−4Υ(¯ D¯ D−R)e2¯ Υ,(DD −8¯ R)7→ e−4¯ Υ(DD −8¯ R)e2Υ.(C.20) In the case of a generic chiral superfield Φ=(¯ D¯ D−8R)P, this projector acts on the generic complex superfield potential P. Choosing the transformation of this superfield to be P7→ e+4 ¯ Υe−2ΥPwe can actually make Φinert under the super-Weyl tranformations. On the other hand, this is not possible in the case of a special chiral superfield S(C.9) constructed from the real superfield prepotential P= (P)∗. In this case, in the light of (C.20), the only way to obtain a covariant super-Weyl transformation of the special chiral superfields (C.9) is to attribute to its real prepotential the transformation rule P 7→ P e−2Υ−2¯ Υ(C.21) 21This can be checked by writing the chiral field in terms of an unconstrained complex superfield (with a similar definition as (C.9)). Choosing the Weyl weights of the transformation accordingly, it can be shown that a general chiral superfield Φis invariant under these rescalings. – 39 – JHEP10(2023)061 which results in S7→ S e−6Υ (C.22) and in the following transformations of its leading component22 s7→ s e−6Υ|.(C.23) We will be interested in the purely bosonic part of the super-Weyl transformations with Υ|=1 12K=¯ Υ|, DαΥ|= 0 , D2Υ|= 0 ,(C.24) since, in that case, ea µ7→ ea µe1 6K(C.25) as needed to write the Lagrangian in Einstein frame, just as in the case of supergravity interacting with chiral multiplets. However, in this scenario, both Mand Fwill be affected by this transformation. This can be seen from (D2−8¯ R)S|=−4F −8RS|=−4F+4 3s¯ M(C.26) whose transformation with the use of (C.23) and (C.24) results in s7→ e−1 2Ks(C.27) F 7→ e−2 3KF(C.28) M7→ e−1 6KM . (C.29) As far as the scalar field is concerned, it is convenient to combine the Weyl rescaling with the field redefinition ϕ:= e−1 2Ks(C.30) so that the kinetic term of ϕfield remains in a canonical form. Taking all of the above into account and integrating out the auxiliary field bµusing its algebraic equations of motion, we find that 1 √−gL=1 2R−Kϕ¯ ϕ∂µϕ∂µ¯ ϕJ−1 3(M+e−1 2KK¯ ϕ¯ F) ( ¯ M+e−1 2KKϕF) −e1 2KM¯ W−e1 2K¯ MW +e−KKϕ¯ ϕF¯ F+FWϕ+¯ F¯ W¯ ϕ+1 √−gLbd.(C.31) It is convenient to further redefine the supergravity auxiliary field Mas ˇ M≡Me1 2K(C.32) 22We do not write transformation of P|explicitly because it vanishes in the Wess-Zumino gauge (C.11). – 40 –