{'fraction_non_alphanumeric': 0.04493498322147651, 'fraction_numerical': 0.016516359060402684, 'mean_word_length': 4.287773773218741, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 4, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Ferrara-Porrati-Sagnotti approach and the one-dimensional supersymmetric model with PBGS

We apply Ferrara-Porrati-Sagnotti approach to the case of one-dimensional supersymmetric model with N=2 supersymmetry spontaneously broken to the N=1 one. We explicitly demonstrate that only one superfield can be treated as the Goldstone one, while the second one has the meaning of the matter superfield. The general action for such a system is constructed and also two relevant particular cases are considered.


Introduction
In [1] J. Bagger and A. Galperin (BG) proposed the approach to construct the N = 2, D = 4 supersymmetric Born-Infeld theory. Combining two N = 1 supermultiplets, namely, N = 1 vector and N = 1 chiral ones, they realized the transformations of additional N = 1 supersymmetry with parameters η α ,ηα as Due to the presence of the constant term in this transformation law, this additional supersymmetry is spontaneously broken. Keeping in mind, that from the vector superfield W α Bianchi identity, for the N = 1 vector supermultiplet it follows that Once again, due to the presence of the constant in (1.4), this constraint can be explicitly solved by expressing X in terms of the superfields W α , Wα. The resulting action is just the N = 2 Born Infeld one.
In the recent paper [2] S. Ferrara, M. Porrati and A. Sagnotti (FPS) proposed the generalization of the BG construction to the cases of several N = 1 vector supermultiplets. The FPS approach includes two basic ingredients.
Firstly, the nonlinear constraint (1.4) is generalized to be where W aα and X a are n-copies of N = 1 vector and chiral multiplets respectively, and the totally symmetric tensor d abc and vector m a are a set of constants. The realization of hidden N = 1 supersymmetry is similar to the BG case and reads δW aα = m a − 1 4 D 2 X a η α − i∂ αα X aηα , δX a = −2W α a η α . (1.6) The invariance of (1.5) with respect to (1.6) involves the new additional constraint which was automatically satisfied in the BG case. The constraints (1.5), (1.7) can be solved to express the bosonic N = 1 chiral superfields X a through the N = 1 vector supermultiplets W aα . The next nontrivial step of the FPS approach yields the structure of the corresponding action, which proved to be more involved where e a = const, C ab = C ba = const. In fact, the additional term with C ab is invariant with respect to S-transformations (1.6). Such a term does not exist in the case of one supermultiplet, but it proved to be essential in the cases of several supermultiplets. The action (1.8) with X a being the solution of the constraints (1.5), (1.7) is treated as the many-field extension of N = 2 Born-Infeld theory.
In [2] the detailed analysis of the n = 2 case was presented, which can be divided into two subcases: where I 4 is a quartic invariant. Then, the analysis of the next n = 3 case has been given in [3]. The above, quite short sketch of the FPS approach is enough to raise two interesting questions: • whether the supermultiplets with different constants d abc are really different?
• whether the action (1.8) is unique as it happened in the n = 1 case?
The reasons for raising the first question are the following.
After resolving the constraints (1.5), (1.7), we will have in the theory n copies of N = 1 vector multiplets W aα with highly nonlinear transformation properties under hidden N = 1 supersymmetry (1.6). The presence of the constants m a in (1.6) at first sight means that all of them have Goldstone nature. But it cannot be so, because the spontaneous breaking of N = 2 to N = 1 supersymmetries is accompanied by only one Goldstone superfield [4,5]. Therefore, it should exist a special basis in which only one superfield is a Goldstone one, while the remaining superfields have to be matter superfields with trivial transformation properties with respect to the broken S-supersymmetry. Due to close relations between linear and nonlinear realizations of partially broken supersymmetries [6], the same property has to be visible within the linear realizations method, the FPS approach is dealing with. Thus, the claim that different choices of the constants d abc in the basic constraints (1.5), (1.7) must be equivalent, should be carefully analyzed.
The second question about the uniqueness of the action we are raising, is also related with the existence of the special basis in which we have one Goldstone superfield and an arbitrary number of matter superfields. In such a basis the general action will have a functional freedom. Thus, it has to be quite interesting to analyze this general action and understand the reasons that just select the FPS actions.
Unfortunately, the analysis of N = 2 extended Born-Infeld theory is quite involved, and the ideological part is hidden behind complicated calculations. Fortunately enough, the FPS approach, as we demonstrated in [7], is not limited by application to the Born-Infeld theory only, and the questions we raised above may be answered through the analysis of a much simpler system, which the FPS approach can be applied to.
In our paper [8] we explored the generalization of the FPS construction for the simplest model of super-particle which realized the partial breaking of N = 2 to N = 1 supersymmetry in one dimension. We explicitly established the transformations that relate two different n = 2 cases with the special basis discussed above, as well as the relations between them. We also constructed the general action for such a system containing two arbitrary functions and provided the explicit form of these functions reducing the general action to FPS-like ones.
In this contribution we review our results.
2 Partial breaking in one dimension and the superparticle action We begin with a simple example of a superparticle model describing N = 2 → N = 1 partial breaking of global supersymmetry in d = 1, which was considered in paper [9] The anticommutation relation of N = 1, d = 1 Poincaré superalgebra reads and the coordinates (t, θ) of N = 1 superspace satisfy the following rules of complex conjugation: t † = −t, θ † = θ. Now we define the bosonic and fermionic superfields v(t, θ) and ψ(t, θ) related as where the spinor derivative D satisfies the relation 2.1 Linear realization: one-particle case In full analogy with BG approach [1], we introduce two spinor superfields ψ(t, θ) and ν(t, θ), assuming that an additional spontaneously broken N = 1 supersymmetry is realized on them as The presence of the constant shift in the first relation suggests the interpretation of ψ(t, θ) as the Goldstone fermion accompanying the N = 2 → N = 1 breaking. The superfield ν(t, θ), due to its transformation properties under the broken N = 1 supersymmetry, may be chosen as a Lagrangian density, since the integral over the superspace is invariant with respect to both broken and unbroken supersymmetries.
To be meaningful, the action (2.5) should be accompanied by an additional constraint, which is invariant under transformations (2.4). One may easily check, that the corresponding invariant constraint looks quite similar to (1.4) ψDψ with evident solution given by It is easy to find that from (2.6) it follows that The bosonic limit of the action (2.5) with (2.7) corresponds to the action of a particle in d = 1 Thus, this simplest system demonstrate a close analogy with the BG construction of the supersymmetric Born-Infeld action.

Linear realization: the multiparticle case
The above considered case can be easily extended to that of an arbitrary number of N = 1 superfields strictly following the FPS approach. Doing so, we firstly introduce a set of n superfields ψ a (t, θ) and ν a (t, θ) transforming under implicit N = 1 supersymmetry as where m a are arbitrary constants. Now, again in full analogy with FPS approach, we impose the following generalization of the constraint (2.6) The invariance of this constraint under the broken S-supersymmetry leads to an additional restriction which can be reformulated as assuming that the constant of integration is equal to zero. Finally, the FPS-like generalization of the action reads where e a and C ab = C ba are arbitrary real constants. After repeating the basic steps of the FPS approach, let us consider the system of two superparticles in the details.

Standard basis for n = 2 cases
In the standard approach of the Partial Breaking of Global Supersymmety, the two-particle model should contain only one Goldstone superfield (with the Goldstone fermion among their components). The rest of the fields must be matter ones, i.e. they should not transform with respect to the broken Ssupersymmetry. Now we demonstrate how such a splitting works in a two-particle system for special choices of the components of tensor d abc . We show that there is a special basis in which the systems in question, corresponding to a different choice of d abc , can be described in a unified way.

The case with
With such a choice of the symmetric tensor d abc , the basic constraints (2.11), (2.13) have a splitting form By rescaling of the variables ψ a and ν a in (3.1), one may always choose then the transformations of fermions ψ a and ν a in (2.10) take the form Similarly to the one particle case (2.7), the constraints (3.1) can be solved as The FPS-like action (1.8), invariant with respect to both S-and Q-supersymmetries, for this case reads Now we introduce two Goldstone spinor superfields ξ a that transform as It is easy to check that the tilded fieldsψ a andν ã transform under (3.3), (3.6) as Thus, it is a covariant operation to put the superfields (3.7) equal to zero, which is equivalent to the relations Moreover, the constraintsψ a =ν a = 0 contain some additional information: they express the superfields of the linear realization ψ a and ν a in terms of the Goldstone superfields ξ a ν a = ξ a Dξ a 1 + Dξ a Dξ a , ψ a = ξ a 1 + Dξ a Dξ a (no summation over a). (3.10) Until now we have two Goldstone superfields ξ a (t, θ), while we are expecting to have only one essential Goldstone fermionic superfield and one matter superfield. This may be achieved by passing to the new superfields η(t, θ) and λ(t, θ) which, in virtue of (3.6), transform as expected 1 δη = ǫ + ǫη∂ t η, δλ = ǫη∂ t λ.

The case with
For these values of the constants the constraints (2.11) read while those in (2.13) take form Now we introduce the complex superfields ψ, ν and complex parameter m as So, we have a full analogy with the previously considered splitting case. Let us introduce two Goldstone spinor superfields ξ(t, θ) andξ(t, θ) which transform as Then the tilded fieldsψ,ψ andν,ν, which are defined by the following expressions Thus, one may again to put these superfields equal to zero, that means It is obvious that the constraints (3.23) follow from (3.27). The constraints (3.27) can be solved as In full analogy with the previously considered case, we introduce the superfields η(t, θ) and λ(t, θ) by and find that they have the same transformation properties as in (3.12) δη = ǫ + ǫη∂ t η, δλ = ǫη∂ t λ. (3.30) The superfields ξ(t, θ),ξ(t, θ) as well as the genuine superfields ψ(t, θ), ν(t, θ) can be expressed in terms of the Goldstone superfield η(t, θ) and matter fermionic superfield λ(t, θ):

34)
Thus, at least in one dimension, the two cases are completely equivalent, because they are related by invertible fields redefinitions (3.33), (3.34). The only difference between these cases lies in the diverse definition of the invariant actions (3.5), (3.20). In fact, these two actions are not the most general ones. We will demonstrate this by constructing the most general action for the Goldstone fermion η(t, θ) and the matter fermionic superfield λ(t, θ).

Nonlinear realizations approach
In this section we provide an alternative description of the system with a partial breaking of the global N = 2, d = 1 supersymmetry within the nonlinear realizations approach. It turns out that in the present case this approach is more suitable for the construction of the most general superfield action. and calculating the expression g −1 dg = ω P P + ω Q Q + ω Z Z + ω S S, one finds the explicit structure for the Cartan forms

NLR approach in one dimension
The covariant derivatives can be found in a standard way, and now they read where the explicit forms of the einbein E and its inverse E −1 are These derivatives satisfy the following (anti)commutation relations Acting on the coset element g (4.2) from the left by different elements g 0 of the N = 2, d = 1 Poincaré supergroup, one can find the transformation properties of the coordinates (t, θ) and the Goldstone superfield η(t, θ). In particular, under both unbroken Q-and broken S-supersymmetries one gets: • Unbroken supersymmetry (g 0 = e ǫQ ): • Broken supersymmetry (g 0 = e εS ): The final step is to express the Goldstone superfield η(t, θ) in terms of the bosonic superfield q(t, θ) by imposing the constraint [10] In order to describe the matter spinor superfield λ(t, θ) within the nonlinear realizations approach, it is necessary to postulate its transformation properties under the S-supersymmetry δ S λ = 0. (4.10) It is obvious that, due to the transformation of the coordinate t under the S-supersymmetry in (4.8), the variation of λ(t, θ) reads δλ = εη∂ t λ. (4.11) One may also define the bosonic superfield φ(t, θ) with the transformation property which is related with λ(t, θ) in the same way as in (4.9)

General action
The most general Ansatz for the N = 1 superfield action describing the Goldstone fermionic superfield η(t, θ) and the matter fermionic superfield λ(t, θ) and having no dimensional constants reads are arbitrary functions depending on ∇ θ η and ∇ θ λ only. The reason for such a form of Ansatz is quite understandable: • with respect to S-supersymmetry transformation, δ S t = −εη, δ S η = ε, δ S λ = 0, the "improved" measure dtdθ 1 + ηη is invariant; as well as ∇ t λ are also invariant with respect to the broken S-supersymmetry; have to be further restricted by imposing the invariance of the action (4.14) with respect to the broken S-supersymmetry.
The variation of the action (4.14) with respect to S-supersymmetry reads δ S S = dtdθ ε 1 + ηη F cov (4.15) Thus, the function F cov 2 may be chosen to be an arbitrary function of its arguments. To fix the functions F cov 1 , F cov 3 , F cov 4 from the condition δ S S = 0 one has, firstly, to integrate over θ in (4.15) and then to replace the arguments of these functions, ∇ θ η and ∇ θ λ, by ∇ θ η = 1 + ηη Dη ≡ 1 + ηη x, ∇ θ λ = Dλ + ηλDη ≡ y + ηλx. (4.16) Hence, the explicit expression for each of F cov [∇ θ η, ∇ θ λ] takes the form where, by definition, • performing the integration in (4.15) over θ, • expanding the functions F cov 1 , F cov 3 , F cov we will get the following three equations where a is an arbitrary constant. The equation (4.19b) may be integrated once, giving where and G[x] is an arbitrary function depending on x. Then the equation (4.19a) reads where b is a new constant. Finally, representing the function F 1 as we will finish with the equations which define the functions F 3 , F 4 in terms of an arbitrary function F 1 [x, y]. Thus, the general action (4.14) is invariant under the broken S-supersymmetry, if the functions F 1 , F 3 and F 4 are expressed through arbitrary function F 1 as (4.26) Thus, the most general action for the present system is defined up to two arbitrary functions F 1 and F 2 . Let us note that the constant b can be chosen equal to zero, because the action corresponding to the Lagrangian is trivial. Thus, the final result for the covariant functions F cov which enter the action (4.14) reads being arbitrary functions on ∇ θ η and ∇ θ λ

Interesting cases
Having at hands the most general expression for the action (4.14), (4.28) invariant with respect to both Q-and S-supersymmetries, it is interesting to visualize the functions F 1 , F 2 which reproduce the FPS actions (3.5), (3.20). Comparing (3.14) and (3.32) one may conclude, that these two FPS cases are related through the substitutions: λ → −iλ, y → −iy (where y = Dλ). Therefore, it is enough to consider the first case, with d 111 = d 222 = 1, only.
In order to simplify everything, it is useful to represent the integrand in (4.14) as follows where the functions F 1 and F 2 depend on x, y variables (4.18). Comparing the integrands in the actions (3.5) and (5.1), we will get for the two special choices of the constant parameters e a and C ab of the FPS action the following result (a) if e 2 = C ab = 0, then L F P S = ν 1 ⇒ , a = 1.
Unfortunately, the explicit form of the functions F 1 , F 2 which corresponds to FPS actions (5.3), (5.4) is not informative enough to understand why these actions were selected. Let us remind that the general action with the restrictions contains two arbitrary functions F cov 1 and F cov 2 . So, as we expected, the invariance with respect to additional, spontaneously broken N = 1 supersymmetry does not fix the action in the many-particles case, in contrast with the one-particle case.

Conclusion
In this work we analysed in details the application of the FPS approach in d = 1.
We demonstrated that the resulting system describes the N = 2, d = 1 supersymmetric action for two particles in which one of N = 1 supersymmetries is spontaneously broken. The final actions possess the same features as the FPS ones.
Using the nonlinear realization approach we reconsider the system in the basis where only one superfield has the Goldstone nature while the second superfield can be treated as the matter one. Having at hands the transformations relating the two selected FPS cases with our more generic one, we established the field redefinitions which relate these cases. Namely, in this basis, the two FPS supermultiplets are related by the redefinition of the matter superfield λ → −iλ. This analysis leads to the conclusion that the only difference between two FPS cases lies in a various choice of actions, while the supermultiplets specified by the FPS constraints are really the same.
Going further on with the nonlinear realization approach, we constructed the most general action for the system of two N = 1 superfields possessing one additional hidden spontaneously broken N = 1 supersymmetry. This action contains two arbitrary functions and reduces to the FPS actions upon specification of these functions.
Unfortunately, the exact form of these functions corresponding to FPS actions is not very informative and it gives no hint about the reason why the FPS cases were selected. Of course, our consideration was strictly one-dimensional and therefore we cannot argue that all features we discussed will appear in the generalized supersymmetric Born-Infeld theory constructed in [2,3]: Nevertheless, we believe that our results and the generality of the nonlinear realization approach are yield reasonable tools to reconsider the questions: • whether the supermultiplets with different constants d abc are really different?
• which additional properties select the FPS actions? in four dimensions.