Three-dimensional Maxwellian extended Bargmann supergravity

We present a novel three-dimensional non-relativistic Chern-Simons supergravity theory invariant under a Maxwellian extended Bargmann superalgebra. We first study the non-relativistic limits of the minimal and the N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 2 Maxwell superalgebras. We show that a well-defined Maxwellian extended Bargmann supergravity requires to construct by hand a supersymmetric extension of the Maxwellian extended Bargmann algebra by introducing additional fermionic and bosonic generators. The new non-relativistic supergravity action presented here contains the extended Bargmann supergravity as a sub-case.


Introduction
Three-dimensional non-relativistic (NR) and ultra-relativistic (UR) versions of supergravity theory have only been explored recently in [1][2][3][4][5][6][7]. Although several generalizations and applications of supergravity have been developed by diverse authors these last four decades, its NR construction remains challenging and has only been approached in three spacetime dimensions. In particular, the formulation of a well-defined NR supergravity action has required the introduction of additional fermionic generators. As in NR bosonic cases, the addition of new generators allows to construct a non-degenerate invariant bilinear form which assures the proper construction of a Chern-Simons (CS) action.
The NR version of the Maxwell CS gravity theory has only been presented recently [74] (see also [75], where the related algebra has been recovered through Lie algebra expansion). Interestingly, the relativistic theory required the presence of three U(1) gauge fields in order to establish a well-defined NR limit and to avoid degeneracy. In the presence of supersymmetry, the NR version of the Maxwell CS supergravity was unknown till now. In this work, we explore the NR limit of the Maxwell superalgebra for N = 1 and N = 2. In particular, we show that a well-defined NR Maxwellian CS supergravity action requires to introduce by hand additional fermionic and bosonic generators. Our model is not only a novel NR supergravity theory without cosmological constant but contains the extended Bargmann supergravity as a sub-case.
The paper is organized as follows: in section 2, we briefly review the Maxwellian extended Bargmann gravity introduced in [74]. Sections 3 and 4 contain our main results. In section 3, we introduce the NR limits of the minimal and the N = 2 Maxwell superalgebras. In section 4, we present the Maxwellian extendad Bargmann superalgebra and the NR CS supergravity action. Section 5 is devoted to discussion and possible future developments. Some large formulas are collected in the appendix.

Maxwellian extended Bargmann gravity
In this section, we briefly review the Maxwellian extended Bargmann algebra introduced in [74] and the associated CS gravity theory developed in the same paper in three (2+1) dimensions. In [74] the authors proved that an alternative way to circumvent the degeneracy of the bilinear form in the [Maxwell] ⊕ u(1) ⊕ u(1) system analyzed in the same paper, is to add one more u(1) gauge field.
The non-vanishing commutation relations of the Maxwell algebra are given by where J A are the spacetime rotations, P A the spacetime translations, and Z A are new generators characterized and introduced in [39,40] (A = 0, 1, 2 and η AB = diag(−, +, +)). A gauge-invariant CS gravity action in three (meaning 2+1, here as well as in the sequel) dimensions based on the above written Maxwell algebra has been constructed in [47][48][49]74]. The CS action is constructed using the connection one-form A = A A T A taking values in JHEP04(2020)051 the Maxwell algebra generated by {J A , P A , Z A }, that is where E B , W B , and K B are one-form fields.
The CS form constructed with the invariant bilinear form defines an action for the relativistic gauge theory for the symmetry under consideration as In the specific case we are now reviewing, when the Maxwell algebra is supplemented with the three additional U(1) generators (Y 1 , Y 2 , and Y 3 ), the connection one-form involved in the construction reads where M , S, and T are the additional bosonic gauge fields. Also the bilinear form acquires further non-zero entries due to the presence of the new generators (see [74] for details). In particular, a non-degenerate bilinear form can be obtained from the aforesaid relativistic bilinear form, allowing for a well-defined and finite NR CS action. Specifically, in [74], the contraction leading to the NR generators is defined through the identifications and by subsequently taking ξ → ∞. Let us note that the index A = 0, 1, 2 has previously been decomposed as A → {0, a}, with a = 1, 2. Furthermore, Y 1 , Y 2 , and Y 3 are the three U(1) generators introduced at the relativistic level.
In terms of the NR generators and fields, the gauge connection one-form of [74],Ã = A AT A , is given bỹ A = τH + e aP a + ωJ + ω aG a + kZ + k aZ a + mM + sS + tT . (2.6) The NR version of the Maxwell algebra presented in [74] was called by the authors Maxwellian Exotic Bargmann (MEB) algebra, and its non-trivial commutations relations read

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Such NR algebra admits the following non-vanishing components of the invariant tensor G aGb =α 0 δ ab , This bilinear form is non-degenerate ifα 2 = 0. The MEB curvature two-forms are given by The NR three-dimensional CS action obtained in [74] reads, up to boundary terms, as follows: As was noticed in [74], the NR CS action has three independent sectors proportional to three arbitrary constants,α 0 ,α 1 , andα 2 . The first term corresponds to the so-called exotic NR gravity. The second term is the CS action for the extended Bargmann algebra [76][77][78][79][80][81], while the last term reproduces the CS action for a new NR Maxwell algebra. Let us note that, since the bilinear form does not result to acquire degeneracy in the contraction process, the equations of motion from the NR action (2.10) are given by the vanishing of all the curvatures (2.9).

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3 On the supersymmetric extension of the Maxwellian extended Bargmann algebra In this section, we explore the supersymmetric extension of the NR Maxwell algebra by applying a NR limit to the N = 1 and N = 2 Maxwell superalgebra. Interestingly, we show that, in order to have a well-defined NR superalgebra, we have to consider the NR limit of a centrally extended N = 2 Maxwell superalgebra endowed with a so(2) generator. Indeed, a true supersymmetric extension of the MEB algebra in which the anti-commutator of two fermionic charges gives a time and a space translation requires, as in the Bargmann case, at least N = 2 supersymmetry. However, as we shall see in the next section, it is necessary to introduce by hand additional fermionic and bosonic generators in order to obtain a MEB superalgebra which allows the proper construction of a NR CS supergravity.
In three spacetime dimensions, the minimal Maxwell superalgebra is spanned by the set of generators , where, in particular, Q α are the supersymmetry generators. Besides, such supersymmetric extension of the Maxwell algebra is characterized by the presence of an additional Majorana fermionic generator Σ α whose presence assures the Jacobi identity (P a , Q α , Q β ). The introduction of a second spinorial charge is not new and have previously been considered in superstring theory [82] and D = 11 supergravity [83][84][85]. The (anti-)commutation relations of the minimal Maxwell superalgebra are given by where α, β = 1, 2 are spinorial indices, C is the charge conjugation matrix, and γ A are the Dirac matrices in three spacetime dimensions.
As was discussed in [74], it is necessary to include three additional U (1) generators given by Y 1 , Y 2 , and Y 3 in order to get the bosonic MEB algebra as a NR limit. At the supersymmetric level, a NR contraction can be applied by considering the rescaling of the bosonic generators as in (2.5) and the following rescaling, with a dimensionless parameter ξ, of the Majorana fermionic generators Q α and Σ α : Although the (anti-)commutation relations (3.3) are well-defined and satisfy the Jacobi identities, we cannot say that the N = 1 MEB superalgebra obtained here is a true supersymmetry algebra. Indeed, the anti-commutator of two supercharges leads to a central charge transformation instead of a time and space translation. This is analogous to the N = 1 Bargmann superalgebra case [1].
One way to circumvent such difficulty is to apply the NR contraction to a N = 2 relativistic Maxwell superalgebra. The N = 2 supersymmetric extension of the Maxwell algebra has been explored by diverse authors [62,64,71]. Here, we shall focus on the N = 2 centrally extended Maxwell superalgebra endowed with a so (2) internal symmetry generator introduced in [73]. Such N = 2 Maxwell superalgebra is spanned by the set of generators J A , P A , Z A , B, Z, Q i α , Σ i α , which satisfy the following non-vanishing (anti-)commutation relations: where i = 1, 2 is the number of supercharges. Let us note that the presence of a so (2) internal symmetry generator is crucial in order to admit a non-degenerate invariant inner product [73]. Then, following [86], let us consider the following definitions of the fermionic JHEP04(2020)051 A dimensionless parameter ξ can be introduced by considering the rescaling of the generators and central extension, Then, after taking the limit ξ → ∞, a particular N = 2 Maxwellian Bargmann superalgebra is obtained; its (anti-)commutation relations are given by the purely bosonic commutators Notice that, unlike the N = 1 superalgebra, the N = 2 Maxwellian Bargmann superalgebra obtained here can be seen as a true supersymmetry algebra. In particular, let us JHEP04(2020)051 note the presence of the non-vanishing commutator between theG a generator and supersymmetry generator. Nevertheless, this superalgebra does not contain the MEB algebra as a subalgebra. Indeed, the bosonic subalgebra (3.7) can be seen as a non-relativistic version of a [Maxwell]⊕u (1) ⊕ u (1) algebra.
Moreover, although the N = 2 NR Maxwell superalgebra (3.7)-(3.8) has the desired features of a true superalgebra, it is not a good candidate to construct a three-dimensional CS supergravity action. Indeed, in order to have a NR supergravity action based on a supersymmetric extension of the MEB algebra, we need a well-defined invariant tensor, which requires to introduce by hand additional fermionic generators. The explicit Maxwellian extended Bargmann superalgebra allowing to construct a NR supergravity action is presented in the next section.

Maxwellian extended Bargmann supergravity
Here, we present the explicit form of the Maxwellian extended Bargmann superalgebra allowing to construct a NR supergravity action. Consequently, we develop the aforementioned NR supergravity action by exploiting the CS construction in three dimensions.

Maxwellian extended Bargmann superalgebra
As we have discussed in the previous section, the N = 2 NR Maxwell superalgebra given by (3.7)-(3.8) does not allow for the proper construction of a NR CS supergravity action although its relativistic analogue is well-defined. In order to have a proper NR CS supergravity action based on a supersymmetric extension of the MEB algebra, one requires to find a NR superalgebra which not only contains the MEB algebra as a subalgebra but also admits a non-degenerate invariant supertrace. Indeed, when studying the NR limit of a theory, one has that the symplectic form of the NR model might become degenerate, making some fields not determined by the field equations, thus reducing the number of dynamical fields. In the case of a CS formulation in three dimensions, the non-degeneracy of the bilinear invariant trace of gauge generators implies the non-degeneracy of the symplectic form, which would ensure dynamically indeterminate fields in the NR theory. In particular, the non-degeneracy of the bilinear form is related to the physical requirement that the CS action involves a kinematical term for each field and the equation of motions imply that all curvatures vanish.
Here we construct by hand a supersymmetric extension of the MEB algebra by introducing six Majorana fermionic generatorsQ + α ,Q − α ,Σ + α ,Σ − α ,R α , andW α . Let us note that the presence of theR α andW α generators is similar to what happens in the extended Bargmann superalgebra presented in [3] and in the extended Newtonian superalgebra of [4], in which aR α generator is considered. Furthermore, we introduce six extra bosonic generators Y 1 , Y 2 , U 1 , U 2 , B 1 , and B 2 . Both B 1 and B 2 are central, while the others act non-trivially on the spinor generators, similarly to the extra bosonic generators introduced in the extended Newton-Hooke supergravity [5]. The proposed supersymmetric extension of the MEB algebra is generated by the set of bosonic and fermionic generators {J,G a ,S,H,P a ,M ,Z,Z a ,T ,Ỹ 1 ,Ỹ 2 ,Ũ 1 ,Ũ 2 ,B 1 ,

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Such generators satisfy the MEB algebra (2.7) along with the following non-vanishing (anti-)commutation relations: The superalgebra given by (2.7) and (4.2) will be denoted as the Maxwellian extended Bargmann superalgebra. One can note that theS generator is no longer a central charge in this supersymmetric extension of the MEB algebra but acts non-trivially on the spinor generatorsQ + α andΣ + α . It is important to emphasize that the MEB superalgebra obtained here has not been obtained through a NR limit of a relativistic superalgebra. Furthermore, the supersymmetric extension of the MEB algebra allowing a well-defined CS supergravity action could not be unique. Then, it would be interesting to study further supersymmetric extensions of the MEB algebra and the possibility of obtaining them by applying a NR limit to a relativistic theory.

Non-relativistic Chern-Simons supergravity action
Let us construct a NR CS supergravity action based on the MEB superalgebra previously introduced.

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The non-vanishing components of the invariant tensor for the MEB superalgebra are given by (2.8) whereα 0 ,α 1 , andα 2 are arbitrary constants. The bilinear form associated with the MEB superalgebra is non-degenerate forα 2 = 0, analogously to the purely bosonic case [74]. On the other hand, the gauge connection one-formÃ for the MEB superalgebra reads 1 The corresponding curvature two-formF = dÃ +Ã ∧Ã = dÃ + 1 2 Ã ,Ã in terms of the generators is given bỹ In particular, the bosonic curvature two-forms are given by Here and in the sequel, we omit the spinor index α, in order to lighten the notation.

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together with , R (s), and R (t) correspond to the MEB curvatures already defined in (2.9). On the other hand, the covariant derivatives of the spinor 1-form fields read e a γ a ψ + + 1 2 ω a γ a ξ + (4.8) Let us note that the gauge fields related to the additional bosonic generators {Y 1 , Y 2 , U 1 , U 2 } appear explicitly on the spinorial curvature. One can see that b 1 and b 2 do not give any contribution to the fermionic curvatures since they are related to central charges.
A CS supergravity action based on the Maxwellian extended Bargmann superalgebra can be constructed by combining the non-zero invariant tensors (2.8) and (4.3) with the JHEP04(2020)051 gauge connection one-formÃ (4.4), and it reads, up to boundary terms, as follows: The CS action (4.9) obtained here describes the so-called Maxwellian extended Bargmann supergravity theory. Let us note that the NR CS supergravity action (4.9) contains three independent sectors proportional toα 0 ,α 1 , andα 2 . In particular, the term proportional toα 0 corresponds to a NR exotic Lagrangian, while the extended Bargmann supergravity introduced in [3] appears in theα 1 sector, endowed with some additional terms related to the presence of the bosonic 1-form fields y 1 , y 2 , u 1 , and u 2 . The term proportional toα 2 can be seen as the CS Lagrangian for a new NR Maxwell superalgebra. In addition, one can see that the bosonic part of the CS action (4.9) corresponds to the MEB gravity action presented in [74], supplemented with the bosonic 1-form fields y 1 , y 2 , b 1 , b 2 , u 1 , and u 2 . Note that, for α 2 = 0, the field equations from the NR CS supergravity action (4.9) reduce to the vanishing of the curvature two-forms (4.7), (4.6), and (4.8) associated with the MEB superalgebra. These curvatures transform covariantly with respect to the supersymmetry transformation laws given in (A.1).
The three-dimensional Maxwellian extended Bargmann supergravity theory obtained here corresponds to an alternative NR supergravity theory which contains the extended Bargmann supergravity [3] (supplemented with some additional bosonic 1-form fields) as a sub-case and which is distinct from the Newton-Cartan supergravity introduced in [1,2]. It is important to emphasize that, although our result generalizes the extended Bargmann supergravity [3], the new NR supergravity obtained here do not contain a cosmological constant. It would be then interesting to study the introduction of a cosmological constant in our model.

Discussion
In this work, we have studied the NR limit of the relativistic Maxwell superalgebra. A well-defined NR superalgebra with the desired features has been obtained by contracting the N = 2 Maxwell superalgebra introduced in [73]. Nevertheless, the construction of a proper NR supergravity action based on a NR version of the Maxwell superalgebra has required to introduce by hand new fermionic and bosonic generators. The new structure has been called as the Maxwellian extended Bargmann superalgebra and corresponds to a supersymmetric extension of the MEB algebra presented in [74]. In particular the MEB superalgebra admits a non-degenerate invariant bilinear form allowing to construct a proper NR CS supergravity action. Interestingly, the MEB CS supergravity theory presented here contains the extended Bargmann supergravity as a sub-case.

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The NR supergravity action constructed in this work, see (4.9), could serve as a starting point for diverse studies. In particular, our result is the first step to construct a new family of NR supergravity beyond the extended Bargmann supergravity. Furthermore, the MEB supergravity could be useful in the construction of a three-dimensional Horava-Lifshitz supergravity. Indeed, as was noticed in [3,33], the extended Bargmann gravity can be seen as a particular kinetic term of the Horava-Lifshitz gravity. In particular, it would be intriguing to explore the effects arising from the presence of the additional gauge field appearing in the Maxwell version of the extended Bargmann (super)algebra.
Moreover, the NR supergravity action (4.9) could have applications in the context of holography, as it already happens for NR gravity at the purely bosonic level [11,13], as well as in effective field theory, where gravitational fields are used as geometrical background response functions [16,98]. Also at this level, it would be interesting to examine the role played by the additional field involved in Maxwell version of the extended Bargmann (super)algebra. As a further remark, let us highlight that additional gauge fields appearing in (super)Maxwell algebras have already been proven to be of particular relevance in the study of the supersymmetry invariance of flat (relativistic) supergravity on a manifold with non-trivial boundary [68], where it has also been conjectured that the presence of the new gauge fields in the boundary would allow to regularize the supergravity action in the holographic renormalization language.
On the other hand, concerning, in particular, the purely bosonic restriction of our model, one could explore the phenomenological implications possibly related to Dark Matter at the non-relativistic level. Another aspect of the MEB gravity which deserves some investigation is the study of the asymptotic symmetries. As it was shown in [49], the asymptotic symmetry of the three-dimensional CS theory invariant under the Maxwell algebra is given by an extension and deformation of the BMS 3 algebra denoted as Max 3 [99]. One could then expect that, since the MEB gravity is the NR version of the Maxwell CS gravity, the corresponding asymptotic symmetry of the MEB gravity would be given by a NR version of the Max 3 algebra. Interestingly, the presence of the additional gauge field Z A modifies not only the asymptotic sector but also the vacuum of the theory. It would be then interesting to study the physical implications of the additional bosonic gauge field appearing in our model. One could go even further and extend this study to supersymmetric level. However, this would require first to establish the relativistic version of our NR model and then study its asymptotic structure by considering suitable boundary conditions. It would also be interesting to develop higher-dimensional extensions of our NR supergravity theory, since we argue that, due to the fact that the introduction of (at least) one second spinorial charge not only is required to construct the minimal supersymmetric extension of the Maxwell algebra but has also been considered in superstring [82] and in D = 11 supergravity [83][84][85], they could be relevant in the study and analysis of the NR limit of higher-dimensional supergravity models, and, more specifically, of D = 11 supergravity.
A future development could also consist in exploring the possibility to obtain the MEB superalgebra introduced here through a NR limit or contraction process from a relativistic superalgebra. An alternative limit which could be used to recover the MEB superalgebra JHEP04(2020)051 is the vanishing cosmological constant limit. In particular, it would be worth exploring the possibility to accommodate a cosmological constant to the MEB supergravity presented here. One could conjecture that a supersymmetric extension of the recent enlarged extended Bargmann gravity, introduced in [87], reproduces the present MEB supergravity in a flat limit [work in progress].
On the other hand, it would be interesting to apply the Lie algebra expansion method [88][89][90][91] to obtain the MEB superalgebra. One could follow the procedure used in [92][93][94] and consider the expansion of a relativistic Maxwell superalgebra. Alternatively, one might also extend the results obtained in [87,95] in which NR algebras appear as semigroup expansions of the so-called Nappi-Witten algebra.
Another aspect that deserves further investigation regards the development of a Maxwellian version of the extended Newtonian gravity [96] and its supersymmetric extension [4]. In particular, the result obtained here could correspond to a subcase of a Maxwellian generalization of the extended Newtonian supergravity. A cosmological constant has recently been accommodated in the extended Newtonian gravity action by including new generators to the Newton-Hooke algebra [97]. One could expect to obtain a Maxwellian Newtonian algebra by generalizing the MEB one in a similar way to [4,97]. It would be then compelling to explore possible matter couplings.

Acknowledgments
This work was supported by the CONICYT -PAI grant N • 77190078 (P.C.) and FONDE-CYT Projects N • 3170438 (E.R.). P.C. would like to thank to the Dirección de Investigación and Vice-rectoría de Investigación of the Universidad Católica de la Santísima Concepción, Chile, for their constant support.
Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.