Generalized cosmological term in D=4 supergravity from a new AdS-Lorentz superalgebra

A new supersymmetrization of the so-called AdS-Lorentz algebra is presented. It involves two fermionic generators and is obtained by performing an abelian semigroup expansion of the superalgebra osp(4|1). The peculiar properties of the aforesaid expansion method are then exploited to construct a D=4 supergravity action involving a generalized supersymmetric cosmological term in a geometric way, only from the curvatures of the novel superalgebra. The action obtained with this procedure is a MacDowell-Mansouri like action. Gauge invariance and supersymmetry of the action are also analyzed.


Introduction
It is well known that a good candidate for describing dark energy corresponds to the cosmological constant (see, for example, [1,2]). Then, it becomes interesting to analyze the ways in which cosmological constant terms can be introduced in gravity theories. In this context, for instance, a cosmological term can be introduced in a gravity theory in four dimensions by using the Anti de Sitter (AdS) algebra. Concerning the supersymmetric extension of gravity including a cosmological term, such a theory can be obtained in a geometric formulation. In particular, in this framework, supergravity is built from the curvatures of osp(4|1) and the action is known as the MacDowell-Mansouri action [3]. A further motivation in the construction of MacDowell-Mansouri like actions lies in the fact that, as shown in [4,5], the D = 4 renormalized action for AdS gravity, which corresponds to the bosonic MacDowell-Mansouri action, is equivalent on-shell to the square of the Weyl tensor describing conformal gravity. Consequently, the supergravity actionà la MacDowell-Mansouri suggests a superconformal structure.
Considering, instead, Minkowski spacetime, it is commonly accepted that its symmetries are described by the Poincaré algebra, which is generated by {J ab , P a }, being J ab and P a the Lorentz and spacetime translations generators, respectively. Minkowski spacetime was then generalized extending its symmetries from the Poincaré to the Maxwell symmetries [6][7][8][9][10][11][12][13][14] (see also the more recent work [15] for an alternative way of closing Maxwell-like algebras). The Maxwell algebra (M) is generated by the set {J ab , P a , Z ab } and its commutation relations read as follows: [P a , P b ] = ΛZ ab , [J ab , J cd ] = η bc J ad − η ac J bd − η bd J ac + η ad J bc , [J ab , P c ] = η bc P a − η ac P b , [J ab , Z cd ] = η bc Z ad − η ac Z bd − η bd Z ac + η ad Z bc , [Z ab , Z cd ] = 0, [Z ab , P c ] = 0. (1.1) The bosonic generators Z ab (Z ba = −Z ab ) correspond to tensorial abelian charges. The constant Λ can be related to the cosmological constant when [Λ] = M 2 . If we put Λ = e, where e is the electromagnetic coupling constant, we get the description of an enlarged spacetime in the presence of a constant electromagnetic background field. Indeed, in order to interpret the Maxwell algebra (and the corresponding Maxwell group), a Maxwell group-invariant particle model on the extended spacetime (x µ , φ µν ), with the translations of φ µν generated by Z µν (that is Z ab in the components language), was studied [10][11][12][13][14]: The interaction term described by a Maxwell-invariant 1-form introduces new tensor degrees of freedom f µν , momenta conjugate to φ µν , that, in the equations of motion, play the role of a background electromagnetic field which is constant on-shell and leads to a closed, Maxwell-invariant 2-form. Interestingly, in [16] the authors presented an alternative way of introducing the generalized cosmological constant term using the Maxwell algebra.
On the other hand, it was shown that deformations of the Maxwell symmetries lead to the so(D − 1, 2) ⊕ so(D − 1, 1) or so(D, 1) ⊕ so(D − 1, 1) algebras [17,18]. In this cases, the Z ab generators are non-abelian. Then, if spacetime symmetries are considered as local symmetries, it is possible to construct Chern-Simons gravity actions where dark energy can be interpreted as part of the metric of spacetime.
Subsequently, in [19] it was shown that a generalized cosmological constant term can also be included in a Born-Infeld like action built from the curvatures of the so-called AdS-Lorentz algebra, being Q α a four-components Majorana spinor charge. The presence of the generators Z ab implies the introduction of a new bosonic "matter" field k ab which modifies the definition of the curvatures when building a (super)gravity theory for the AdS-Lorentz (super)algebra (s)AdS − L 4 . In (s)AdS − L 4 , unlike the case of the (minimal supersymmetrization of) the Maxwell algebra (s)M, the new generators Z ab are non-abelian and behave as Lorentz generators; furthermore, sAdS−L 4 contains only one spinor charge, while, as shown in [37], the minimal supersymmetrization of the Maxwell algebra requires two Majorana spinor charges.
Actually, as we have already mentioned, in [29] the authors also presented the generalized minimal AdS-Lorentz superalgebra, which involves two fermionic charges. However, it also includes two extra bosonic generators (Z ab andZ a ), and an Inönü-Wigner contraction of the generalized minimal AdS-Lorentz superalgebra provides the generalized minimal Maxwell superalgebra sM 4 [28], which involves an extra bosonic generatorZ ab with respect to the minimal super-Maxwell algebra sM of [37]. Then, in order to end up with the Maxwell algebra M, one should further contract the bosonic subalgebra of sM 4 in such a way to remove the extra bosonic generatorZ ab .
On the other hand, the AdS-Lorentz type (super)algebras also possesses the non-commutativity [P a , P b ] = Z ab , which is also present in the Maxwell (super)symmetries. Nevertheless, unlike the MacDowell-Mansouri Lagrangians for osp(4|1) and for sAdS − L 4 , it was shown in [28] that the supergravity actionà la MacDowell-Mansouri based on the generalized minimal Maxwell superalgebra sM 4 does not reproduce the supersymmetric cosmological constant term in the action. This is a direct consequence of the S-expansion procedure. The result of [28] corresponds to a generalization of the one previously presented in [46]. Let us mention that, in both cases, the super-Maxwell fields only appears in boundary terms of the resulting Lagrangian. Thus, introducing a generalized supersymmetric cosmological term through super-Maxwell symmetries in the context of supergravity seems to be a hard task. 1 Conversely, the AdS-Lorentz type superalgebras seem to be better candidates in order to introduce the cosmological term in supergravity, in the presence of bosonic generators Z ab . Let us also mention that, in [29], unlike the case of Maxwell-type superalgebras, the bosonic fields associated to the Lorentz-like generators Z ab appear not only in the boundary terms, but also in the bulk Lagrangian of the model.
With this motivation, the aim of the present paper it to construct a MacDowell-Mansouri like action based on a new AdS-Lorentz type superalgebra involving two fermionic charges and possessing a bosonic subalgebra that does not contains any additional generator with respect to the Maxwell algebra M, in such a way that, when contracted, it directly reproduces M. In this sense, such a superalgebra could also be viewed as the minimal supersymmetrization of a minimal Maxwell-like algebra in which the bosonic generator Z ab is non-abelian (in particular, [Z ab , Z cd ] = η bc Z ad − η ac Z bd − η bd Z ac + η ad Z bc ) and where [Z ab , P c ] = η bc P a − η ac P b , even if, as we will see, the generators Z ab in this case will not behave as Lorentz generators when considering the supersymmetric extension and the corresponding commutation relations with the fermionic charges.
Thus, in this paper we first present the aforesaid new supersymmetrization of AdS − L 4 (we will call itsAdS − L 4 , for short) as an abelian semigroup expansion of osp(4|1). Then, we show thatsAdS − L 4 allows to construct in a geometric way a D = 4 supergravity action containing a generalized supersymmetric cosmological term. The action we end up with corresponds to a MacDowell-Mansouri like action written in terms of thesAdS − L 4 curvatures. Our result is a new supersymmetric extension of [19] involving two fermionic generators. In our model, thesAdS − L 4 fields will appear not only in the boundary terms, but also in the bulk Lagrangian (analogously to what happened in [29] and differently from what happened in [28,46]). Referring to the recent works [47][48][49], we conjecture that the presence of thesAdS − L 4 fields in the bulk and in the boundary would allow to recover the supersymmetry invariance of a supergravity theory based on the superalgebrasAdS − L 4 in the presence of a non-trivial boundary of spacetime in the so-called rheonomic (i.e. geometric) approach.
The paper is organized as follows: In Section 2, we present the new superalgebrasAdS − L 4 as an S-expansion of osp(4|1). Subsequently, in Section 3 we construct in a geometric way a D = 4 supergravity model containing a generalized supersymmetric cosmological term only from the curvatures ofsAdS − L 4 . The action corresponds to a MacDowell-Mansouri like action. Then, we analyze the supersymmetry invariance of the theory. Finally, Section 4 contains the conclusions and possible future developments. In Appendix 6 we collect our conventions and some useful formulas.
2 AdS-Lorentz superalgebrasAdS −L 4 as an S-expansion of osp(4|1) In the following, we apply the S-expansion method to osp(4|1) by using a particular abelian semigroup and obtain a new supersymmetrization of AdS − L 4 . We will name this novel superal-gebrasAdS − L 4 , for short.
Let us first recall that the generators {J ab ,P a ,Q α } (with a = 0, 1, 2, 3, α = 1, 2, 3, 4) of osp(4|1) satisfy the following (anti)commutation relations: where γ ab , γ a are Dirac gamma matrices in four dimensions and C is the charge conjugation matrix;J ab are the Lorentz generators,P a the spacetime translations generators, andQ α is a four-components Majorana spinor charge. Then, let us consider, on the same lines of [29], the following decomposition of the original superalgebra g = osp(4|1) in subspaces V p , p = 0, 1, 2: Then, the subspace structure can be written as Now, let us consider the abelian semigroupŜ = {λ 0 , λ 1 , λ 2 , λ 3 } described by the following multiplication table: Then, consider the subset decompositionŜ =Ŝ 0 ∪Ŝ 1 ∪Ŝ 2 witĥ The decomposition (2.4) is said to be resonant [21], since it satisfieŝ which has the same form of (2.3). Then, according to Theorem IV.2 of [21], the subalgebra is a resonant subalgebra ofŜ × g. We can then perform the following identification: being {J ab , P a , Z ab , Q α , Σ α } the set of generators of the new superalgebra obtained after a resonant S-expansion of osp(4|1). This superalgebra corresponds to a new supersymmetrization of the AdS-Lorentz algebra AdS − L 4 of [19]. Let us call itsAdS − L 4 . Its (anti)commutation relations can be obtained by using the multiplication rules of the semigroupŜ (see Table 1) together with the commutation relations (2.1) of the original superalgebra, namely osp(4|1); they read: (2.9) One can see that a new Majorana spinor charge Σ α has been introduced as a direct consequence of theŜ-expansion procedure.
The new AdS-Lorentz superalgebra (2.9) contains the so-called AdS − L 4 algebra generated by {J ab , P a , Z ab } as a bosonic subalgebra. The AdS − L 4 algebra and its generalizations have been largely studied in [19]. In particular, it was proven that AdS − L 4 allows to include a generalized cosmological term in a Born-Infeld gravity action. Furthermore, performing the rescaling and taking the limit µ → ∞ (Inönü-Wigner contraction) in AdS − L 4 , one obtains the minimal Maxwell algebra (1.1).
On the other hand, AdS − L 4 is also a bosonic subalgebra of the AdS-Lorentz superalgebra of [29]. In other words, the new AdS-Lorentz superalgebra (2.9) and the AdS-Lorentz superalgebra of [29] share the same bosonic subalgebra AdS − L 4 .
Nevertheless, as we have already mentioned in the introduction, the AdS-Lorentz superalgebra of [29] has just one fermionic generator. Differently,sAdS − L 4 , given by (2.9), possesses two fermionic charges Q α and Σ α . Then, since it also contains AdS − L 4 as a bosonic subalgebra, in this sense (2.9) could also be viewed as the minimal supersymmetrization of a minimal Maxwelllike algebra in which the bosonic generator Z ab is non-abelian ([Z ab , Z cd ] = η bc Z ad − η ac Z bd − η bd Z ac + η ad Z bc ) and where [Z ab , P c ] = η bc P a − η ac P b , even if the generators Z ab , in this case, do not behave as Lorentz generators when considering the supersymmetric extension and the corresponding commutation relations with the fermionic charges. In this context, we observe that the behavior of the generators Z ab insAdS − L 4 is also different from the behavior of the Z ab 's in (a contraction of) the generalized minimal AdS-Lorentz superalgebra of [29].
3 Generalized supersymmetric cosmological term in D = 4 from sAdS − L 4 In order to construct an action based onsAdS − L 4 we start, on the same lines of [28,29], from the following 1-form connection: where the 1-form gauge fields are given by in terms of the components of the osp(4|1) connection. Note that, in order to properly interpret the gauge fields, it is necessary to introduce a length scale ℓ. The associated 2-form curvature F = dA + A ∧ A is given by: where 2 As recalled in [28,29], the general form of the MacDowell-Mansouri action [3] constructed with the osp(4|1) 2-form curvatureF is given by with the following choice of the invariant tensor: Observe that if one chooses the whole T ATB osp(4|1) as an invariant tensor (which satisfies the Bianchi identities) for the OSp(4|1) supergroup, then the action (3.6) is a topological invariant and gives no equations of motion. Nevertheless, with the choice (3.7) of the invariant tensor (which breaks the OSp(4|1) supergroup to its Lorentz subgroup), (3.6) becomes a dynamical action that corresponds to the MacDowell-Mansouri action for the osp(4|1) superalgebra [3,51]. Writing the explicit form of the action (3.6) with the choice (3.7) and omitting the boundary terms, the result is the N = 1 supergravity action in four dimensions, given by the Einstein-Hilbert and Rarita-Schwinger terms plus the usual supersymmetric cosmological terms; the aforementioned action is not invariant under the osp(4|1) gauge transformations. However, the invariance of the action under supersymmetry transformation can be obtained by modifying the supersymmetry transformation of the spin connectionω ab [52]. Now, in order to construct a MacDowell-Mansouri like action forsAdS − L 4 , we consider thê S-expansion of T ATB osp(4|1) and the 2-form curvature F in (3.3). In particular, the action for sAdS − L 4 can be written as where T A T B sAdS−L 4 can be derived from the original components of the invariant tensor written in (3.7), using Theorem VII.1 of [21]. One can then show that the non-vanishing components of where C 0 and C 2 are dimensionless arbitrary independent constants. This choice of the invariant tensor breaks the AdS-Lorentz supergroup to its Lorentz-like subgroup. Thus, considering (3.9) and the 2-form curvature (3.3), it is possible to write an action of the form (3.8) as can be written explicitly as where we have also isolated the boundary terms and defined The action (3.11) has been intentionally separated in different pieces: The term proportional to C 0 corresponds to the Gauss-Bonnet term; the second piece is an Euler invariant term which can be seen as a Gauss-Bonnet like term (it does not contribute to the dynamics and can be written as a boundary term) that involves the newsAdS − L 4 field k ab ; the third term contains the Einstein-Hilbert and Rarita-Schwinger Lagrangian, describing pure supergravity, plus three additional terms involving the new spinor 1-form field ξ and the bosonic field k ab ; the fourth term corresponds to a generalized supersymmetric cosmological term which contains, besides the usual supersymmetric cosmological term, also two additional terms involving thesAdS − L 4 spinor field ξ and one additional term involving k ab ; the last piece is a boundary term. Thus, we have shown that the MacDowell-Mansouri like action constructed applying the properties of the S-expansion procedure in the case of a resonantŜ-expansion (see Table 1 for the multiplication table of the semigroupŜ) of osp(4|1) describes a supergravity model with a generalized supersymmetric cosmological term. In other words, we have introduced in alternative way the supersymmetric cosmological term in a supergravity model, building a (deformed) D = 4 supergravity action which includes a generalized supersymmetric cosmological constant term from the new AdS-Lorentz superalgebrasAdS − L 4 . Our result corresponds to a new supersymmetric extension of [19] involving two fermionic generators.
Let us observe that, if we consider k ab = 0 and ξ = 0 in the action (3.11), we obtain the MacDowell-Mansouri action for the supergroup OSp(4|1). On the other hand, setting only ξ = 0 in (3.11), we obtain the action for the AdS-Lorentz superalgebra found in [29].
Notice that if we omit the boundary terms in (3.11), we get: a γ a γ 5 ρ + 4ψV a γ a γ 5 σ + 4ξV a γ a γ 5 σ + 4ξV a γ a γ 5 ρ Then, if we consider k ab = 0 and ξ = 0 in the action (3.13), we are left with the Einstein-Hilbert and Rarita-Schwinger Lagrangian plus the usual supersymmetric cosmological term. Now, in order to obtain the field equations, we compute the variation of the Lagrangian with respect to the differentsAdS − L 4 fields. One can prove that, computing the variation of the Lagrangian with respect to the spin connection ω ab and imposing δ ω L = 0, we get the following field equation (modulo boundary terms) for thesAdS − L 4 supertorsion: (3.14) From the variation of the Lagrangian with respect to k ab , we obtain the same equation. On the other hand, computing the variation of the Lagrangian with respect to the vielbein V a and imposing δ V L = 0, we get and from the variation of the Lagrangian with respect to the gravitino field ψ, imposing δ ψ L = 0, we obtain (modulo boundary terms): 8V a γ a γ 5 (ρ + Ξ) + 4γ a γ 5 (ψ + ξ) R a = 0. (3.16) The variation of the Lagrangian with respect to the spinor 1-form field ξ leads to the same equation. Note that the field equations (3.14), (3.15), and (3.16) are similar to those of osp(4|1) supergravity, the only differences being related to the presence of the new fields k ab and ξ. Interestingly, we can define a new bosonic field aŝ together with its respective covariant derivativê and a new spinor 1-form field asψ = ψ + ξ. Then, exploiting the new definitions, the equations of motion (3.14), (3.15), and (3.16) become, respectively: 8V a γ a γ 5ρ + 4γ a γ 5ψR a = 0, (3.20c) where we have also definedR These new curvatures have the same form of the osp(4|1) ones, and the fields {ω ab , V a ,ψ} fulfill equations of motion that have the same form of those of the osp(4|1) supergravity theory. In fact, let us also observe that, at the price of introducing the 1-form fields k ab and ξ (and the corresponding Maurer-Cartan equations), the osp(4|1) superalgebra can be mapped intosAdS − L 4 , whereby the spin connection and the gravitino are respectively identified with the Lorentz connection and gravitino of a D = 4 Minkowski spacetime with vanishing Lorentz curvature and vanishing gravitino super field-strength, albeit with a modification of the supertorsion, the latter being non-vanishing in both cases. This point will be further analyzed in a future work.

3.1sAdS − L 4 gauge transformations and supersymmetry invariance
The gauge transformation of the connection A (see (3.1)) is where ρ is thesAdS − L 4 gauge parameter Then, using we obtain that thesAdS − L 4 gauge transformations are given by: Analogously, from the gauge variation of the curvature, we can write the following gauge transformations of the curvature F (see (3.3)): Although the MacDowell-Mansouri like action (3.11) is built fromsAdS − L 4 , one can prove that it is not invariant under thesAdS − L 4 gauge transformations. In fact, as we can see, the action does not correspond to a Yang-Mills action, nor to a topological invariant. Furthermore, if we consider the variation of the action under gauge supersymmetry, we find: Thus, as in the osp(4|1) and super-Poincaré cases, the action is invariant under gauge supersymmetry imposing the supertorsion constraint R a = 0. (3.29) However, this causes the spin connection ω ab to be expressed in terms of the other fields V a , k ab , ψ, ξ (second order formalism). Alternatively, on the same lines of [28,29], one can recover the supersymmetry invariance of the action in the first order formalism by modifying the supersymmetry transformation of ω ab . Indeed, if we consider the variation of the action under an arbitrary δω ab , we find which vanishes for arbitrary δω ab if R a = 0. Then, it is possible to modify δω ab by adding an extra piece to the gauge transformation such that the variation of the action can be written as the supersymmetry invariance being fulfilled when δ extra ω ab = 2ǫ abcd ζ ec γ d ǫ +ζ de γ c γ 5 ǫ −ζ cd γ e γ 5 ǫ V e , (3.32) withρ +Ξ =ζ ab V a V b . Thus, we can conclude that the action in the first order formalism is invariant under the following supersymmetry transformations: 3 δω ab = 2ǫ abcd ζ ec γ d ǫ +ζ de γ c γ 5 ǫ −ζ cd γ e γ 5 ǫ V e , (3.33a) δV a =ǭγ a ψ +ǭγ a ξ, (3.33c) Observe that supersymmetry is not a gauge symmetry of the action. Finally, note that there is also another kind of supersymmetry (i.e. a supersymmetry-like symmetry), related to the fermionic generator Σ α . The new supersymmetry-like transformations read as follows: δV a =εγ a ψ +εγ a ξ, (3.34c) If we now consider the variation of the action under the new supersymmetry-like transformations, we get: which has the same form of (3.28), the only difference relying in the parameter ε. Then, one can repeat the procedure described above, obtaining that the action in the first order formalism is invariant under the following new supersymmetry-like transformations: 4 δω ab = 2ǫ abcd ζ ec γ d ǫ +ζ de γ c γ 5 ǫ −ζ cd γ e γ 5 ǫ V e , (3.36a) δV a =εγ a ψ +εγ a ξ, (3.36c) where, again,ρ +Ξ =ζ ab V a V b . Also this supersymmetry-like symmetry is not a gauge symmetry of the action.

Discussion
In this paper, driven by the fact that from AdS-Lorentz type (super)algebras one can introduce the cosmological term in (super)gravity in the presence of an extra bosonic generator Z ab [19,29], we have presented a new supersymmetrization of the AdS-Lorentz algebra AdS − L 4 of [19] involving two fermionic charges. Our new AdS-Lorentz superalgebra (that we calledsAdS − L 4 ) possesses, by construction, a bosonic subalgebra (that is AdS − L 4 ) that does not contains any additional generator with respect to the Maxwell algebra M, recalled in (1.1), in such a way that, when contracted, it directly reproduces the Maxwell algebra M. In this sense,sAdS − L 4 could also be viewed as the minimal supersymmetrization of a minimal Maxwell-like algebra in which the bosonic generator Z ab is non-abelian ([Z ab , Z cd ] = η bc Z ad − η ac Z bd − η bd Z ac + η ad Z bc ) and where [Z ab , P c ] = η bc P a −η ac P b , even if the generators Z ab , in this case, do not behave as Lorentz generators when considering the supersymmetric extension and the corresponding commutation relations with the fermionic charges.
In particular, we have obtainedsAdS − L 4 as an S-expansion of osp(4|1), using the semigroup described by Table 1, and we have shown that it allows to construct in a geometric way a D = 4 supergravity model containing a generalized supersymmetric cosmological term. For this purpose, we have exploited some peculiar and useful properties of the abelian semigroup expansion method, which allowed us to construct a MacDowell-Mansouri like action, on the same lines of [28,29]. Our result is a new supersymmetric extension of [19] involving two fermionic generators. Interestingly, in our model the bosonic fields k ab (associated to the generators Z ab ) and the spinor 1-form field ξ (associated to the fermionic charge Σ) appear not only in the boundary terms, but also in the bulk Lagrangian (analogously to what happened in [29] and differently from what happened in [28,46]). It would be interesting to study the possibility of obtaining the action (3.11) from the rheonomic (i.e. geometric) approach adopted in [47][48][49] in the presence of a non-trivial boundary of spacetime.
On the other hand, a possible future development could consist in analyzing N -extended versions ofsAdS − L 4 and constructing N -extended supergravity models (also higher-dimensional and matter-coupled ones) from the aforementioned N -extended superalgebras. In this context, the S-expansion procedure could play an important role. Furthermore, it would be interesting to carry on an analysis in 2 + 1 dimensions, on the same lines of [36], considering the restriction to three dimensions ofsAdS − L 4 .

Acknowledgments
D.M.P. acknowledges DI-VRIEA for financial support through Proyecto Postdoctorado 2018 VRIEAPUCV. The authors wish to thank P. Concha and E. Rodríguez for the illuminating discussions.