Three-dimensional Maxwellian Extended Newtonian gravity and flat limit

In the present work we find novel Newtonian gravity models in three spacetime dimensions. We first present a Maxwellian version of the extended Newtonian gravity, which is obtained as the non-relativistic limit of a particular $U(1)$-enlargement of an enhanced Maxwell Chern-Simons gravity. We show that the extended Newtonian gravity appears as a particular sub-case. Then, the introduction of a cosmological constant to the Maxwellian extended Newtonian theory is also explored. To this purpose, we consider the non-relativistic limit of an enlarged symmetry. An alternative method to obtain our results is presented by applying the semigroup expansion method to the enhanced Nappi-Witten algebra. The advantages of considering the Lie algebra expansion procedure is also discussed.

An action principle of the Newtonian gravity has recently been presented in [28]. Such construction has required to extend the so-called extended Bargmann algebra [29][30][31][32][33][34][35] with three additional generators. Subsequently, a three-dimensional Chern-Simons (CS) formulation based on this particular extension of the Bargmann algebra has been introduced in [36]. Such NR gauge invariant theory has been denoted as extended Newtonian gravity and has been obtained as a contraction of a bi-metric model being the sum of the Einstein gravity in the Lorentzian and Euclidean signatures. The incorporation of a cosmological constant to the extended Newtonian gravity has then been presented in [37] by introducing by hand a new NR symmetry called exotic Newtonian algebra. The respective relativistic counterparts of the extended Newtonian gravity and its exotic version have been recently studied in [38]. In particular, the authors of [38] showed that the extended Newtonian gravity model and the exotic one appear as NR limits of relativistic CS gravity theories based on the co-adjoint Poincaré and co-adjoint AdS algebra, respectively.
Another extension of the Bargmann algebra is given by the recently introduced Maxwellian extended Bargmann (MEB) algebra allowing to construct a NR CS gravity action in presence of a covariantly constant electromagnetic field [20] without cosmological constant. Such NR model has been obtained as a NR limit of the [Maxwell]⊕ u(1) ⊕ u(1) ⊕ u(1) CS gravity. As in the extended Bargmann case [35], the U (1) enlargement of the relativistic counterpart is crucial to define a proper NR limit leading to a finite NR gravity action based on a non-degenerate bilinear form. The relativistic Maxwell algebra has been introduced to describe a Minkowski background in presence of a constant electromagnetic field [39][40][41]. Such symmetry and its generalizations have been useful to recover standard General Relativity without cosmological constant as a limit of CS and Born-Infeld (BI) gravity theories [42][43][44][45][46]. Let us also mention that supersymmetric extensions of the Maxwell algebra found relevant applications in the development and study of supergravity models (see, for instance, [47][48][49][50][51] and references therein). More recently, the dual version of the Maxwell symmetry, denoted as Hietarinta-Maxwell [52], allowed to recover the topological and minimal massive gravity theories. In particular, both of these massive gravity theories appear as particular cases of a more general massive gravity arising from a spontaneous breaking of a local symmetry in a CS gravity theory invariant under the Hietarinta-Maxwell algebra [53].
One may then ask if the MEB gravity can admit a cosmological constant in a similar way to the one based on the Newton-Hooke symmetry [54][55][56][57][58][59][60][61]. The answer to the question is affirmative. Indeed, there is the enlarged extended Bargmann gravity (EEB) which reproduces the MEB one in the vanishing cosmological constant limit → ∞ [23]. The EEB has been obtained as an Inönü-Wigner (IW) contraction [62,63] of the [AdS-Lorentz] ⊕ u(1) ⊕ u(1) ⊕ u(1) algebra in three spacetime dimensions. The AdS-Lorentz (AdS-L, for short) algebra has been first introduced in arbitrary spacetime dimensions as a semi-simple extension of the Poincaré algebra in [64,65] and has been useful to recover pure Lovelock theory [66,67] as a particular limit of CS and BI gravity theories [68][69][70]. Moreover, supersymmetric extentions of the AdS-L algebra proved useful in the context of supergravity (see [71][72][73][74][75] and references therein).
It is natural to address the question whether the extended Newtonian and its exotic version admit a Maxwell generalization in three spacetime dimensions. The motivation to consider threedimensional CS gravity theories is twofold. First, CS formalism offers us a simple framework to construct gauge invariant gravity actions. Second, three-dimensional geometry can be seen as an interesting toy model to approach more difficult theories in higher dimensions. In this paper we will show that a three-dimensional Maxwellian extended Newtonian (MENt) gravity theory can be defined by generalizing the results obtained in [38]. In particular, the MENt algebra appears as a NR limit of a particular enhancement of the Maxwell gravity enlarged with three u(1) gauge generators. The introduction of a cosmological constant to the MENt theory is also explored by defining a new NR gravity theory denoted as enlarged extended Newtonian (EEN) gravity which is the NR version of an enhanced AdS-L algebra enlarged with three u(1) gauge generators.
Interestingly, we show that the MENt and EEN symmetries contain the MEB and EEB algebras as sub-cases, respectively. Nevertheless, the respective novel NR CS actions do not contain the MEB and EEB gravity actions as particular cases. This is mainly due to the problematic to define a unique rescaling for the relativistic parameters of the invariant tensor. Such difficulty can be overcame by considering the semigroup expansion (S-expansion) method [76]. In particular, we show that the new NR symmetries presented here can alternatively been obtained as S-expansion of the enhanced Nappi-Witten algebra introduced in [38]. Remarkably, the S-expansion procedure not only allows us to obtain the MENt and EEN algebras but also provides us with the most general non-degenerate invariant tensor for the respective NR algebras.
The organization of the paper is as follows: in Section 2, we briefly review the relativistic Maxwell and AdS-L gravity theories in three spacetime dimensions. In Section 3, we present new relativistic algebras enlarged with three u(1) generators which can be seen as enhancements of the Maxwell and AdS-L symmetries. The respective CS actions based on the enhanced algebras are also constructed. In Section 4, we present the Maxwellian version of the extended Newtonian gravity. In particular, we introduce the MENt algebra as a NR limit of the [enhanced Maxwell] ⊕ u(1) ⊕ u(1) ⊕ u(1) algebra. The NR limit of the relativistic CS action is also considered. The generalization of our results to the presence of a cosmological constant is explored in Section 5. A novel NR symmetry denoted as EEN algebra is introduced by considering the NR limit of the [enhanced AdS-L] ⊕ u(1) ⊕ u(1) ⊕ u(1) algebra. We work out that the MENt gravity theory appears as a flat limit → ∞ of the EEN one. In Section 6, we show that our results can alternatively be obtained by considering the S-expansion method. Section 7 concludes our work with some discussions about future developments.

Relativistic Maxwell and AdS-Lorentz gravity theories
In this section, we briefly review the three-dimensional relativistic Maxwell CS gravity [77][78][79][80][81] and its generalization to the so-called AdS-Lorentz gravity [82][83][84]. The latter allows to include a cosmological constant into the Maxwell CS gravity action.
The Maxwell algebra can be seen as an extension and deformation of the Poincaré algebra and it is spanned by the set of generators {J A , P A , Z A } which satisfy the following non-vanishing commutation relations where A, B, C = 0, 1, 2 are Lorentz indices which are lowered and raised with the Minkowski metric η AB = (−1, 1, 1) and ABC is the three-dimensional Levi Civita tensor which satisfy 012 = − 012 = 1. Here, J A correspond to the spacetime rotations, P A are the spacetime translations and Z A are the so-called gravitational Maxwell generators.
Such relativistic algebra admits the following non-vanishing components of the invariant tensor of rank 2, where σ 0 , σ 1 , and σ 2 are arbitrary constants.
On the other hand, the gauge connection one-form A for the Maxwell algebra reads where W A denotes the spin-connection, E A is the vielbein and K A is the gravitational Maxwell gauge field. The corresponding curvature two-form is given by Here as well as in the sequel, the index "A" of W A , E A , K A in the parenthesis on the left-hand side is understood in order to lighten the notation, meaning, in fact, R A (W ) = R A W B and so on.
Considering the gauge connection one-form (2.3) and the non-vanishing components of the invariant tensor (2.2) in the three-dimensional CS expression, with k being the CS level of the theory related to the gravitational constant G, we find the following relativistic CS gravity action for the Maxwell algebra [77][78][79]: where T A = R A (E) denotes the torsion two-form. One can see that the CS action contains three independent sectors proportional to σ 0 , σ 1 , and σ 2 . The gravitational Lagrangian [85] appears along the σ 0 constant, while the Poincaré Lagrangian given by the Einstein-Hilbert term is related to the σ 1 constant. On the other hand, the term proportional to σ 2 contains a torsional term and the gravitational Maxwell gauge field K A coupled to the gravitational gauge field. As mentioned in [79], the presence of the additional gauge field K A modifies not only the asymptotic sector but also the vacuum of the theory. Interestingly, an enlarged relativistic algebra allows us to include a cosmological constant into the Maxwell CS gravity theory. Such enlarged algebra is the so-called AdS-L algebra {J A , P A , Z A } whose generators satisfy (2.1) along with where is a length parameter related to the cosmological constant Λ. Naturally, the flat limit → ∞ reproduces the Maxwell algebra. One can see that the following redefinition of the generators allows to rewrite the algebra given by (2.1) and (2.8) as the direct sum of the so (2, 2) and so (2, 1) algebras.
The non-vanishing components of the invariant tensor for the AdS-L algebra are given by (2.2) along with [84] Then, considering the gauge connection one-form for the AdS-L algebra which coincides with the Maxwell one (2.3) and the invariant tensor (2.2) and (2.10) into the general expression of the CS action (2.6), we find the following three-dimensional relativistic CS action [84]: where we have defined the curvature F A (K) : One can see that such relativistic symmetry not only introduces a comological constant term to the Einstein-Hilbert term but also includes the gauge field K A along the σ 1 constant. Furthermore, the term along σ 2 is also modified by the presence of the new commutation relations (2.8). Naturally, the vanishing cosmological constant limit → ∞ reproduces the relativistic Maxwell CS gravity action (2.7). NR versions of the Maxwell and the AdS-L algebras have been considered in [20,23,27,86,87] and have required to consider U (1) enlargements in order to have well-defined finite NR CS actions with non-degenerate bilinear forms. In particular, as shown in [20], the NR limit of the [Maxwell] ⊕ u (1) 3 algebra reproduces the MEB algebra. On the other hand, the EEB appears as a NR limit of the [AdS-L] ⊕ u (1) 3 algebra [23]. Such NR algebras are, as their relativistic versions, related through a flat limit.
In what follows, we study new enhancements of the Maxwell and AdS-Lorentz algebras whose IW contraction will lead us to novel NR algebras.

Enhanced Maxwell and AdS-Lorentz algebras and U (1) enlargements
In this section, we generalize the results presented in [38] to enlarged symmetries. In particular, we present new enhancements of the [Maxwell] ⊕ u (1) 3 and [AdS-L] ⊕ u (1) 3 algebras whose NR limits shall be explored in the next section. We expect to find a Maxwellian version of the Extended Newtonian gravity [36] and a generalization including a cosmological constant. The explicit CS action for the relativistic enhancement symmetries is also constructed.

Enhanced Maxwell
Such algebra can be seen as an extension and deformation of the coadjoint Poincaré algebra studied in [38,88,89]. Interestingly, the commutation relations (3.1) can also be recovered as an semigroup expansion (S-expansion) [76] of the Poincaré algebra iso(2, 1) = Ĵ A ,P A , whose generators satisfy the non-vanishing commutation relations Indeed, let us consider S (2) E = {λ 0 , λ 1 , λ 2 , λ 3 } as the relevant semigroup whose elements satisfy where λ 3 = 0 s is the zero element of the semigroup such that 0 s λ α = 0 s . Then, one can show that the enhanced Maxwell algebra (3.1) appears as a 0 s -reduction of the expanded algebra S E ×iso(2, 1), where the expanded generators are expressed in terms of the Poincaré ones through the semigroup elements as Let us note that the possibility to obtain generalizations of the coadjoint Poincaré algebra by considering S

(N )
E -expansion of the iso(2, 1) algebra has first been discussed in [38]. In the present analysis, we have shown that considering N = 2 allows to define an enhanced Maxwell algebra. As we shall see in the next section, by considering a different semigroup, a new relativistic symmetry will be obtained.
An interesting feature of the S-expansion method is that it immediately provides us with the non-vanishing components of the invariant tensor of the expanded algebra in terms of the original ones [76]. In particular, the non-vanishing components of the invariant tensor for the Poincaré algebra are given by Thus, one can show that the enhanced Maxwell algebra admits the following invariant tensor: where µ's are arbitrary constants and are related to the Poincaré constant as Similarly to what happens in the MEB algebra, one can generalize the relativistic algebra by including three U (1) generators given by Y 1 , Y 2 , and Y 3 . The inclusion of these U (1) generators in the case of the original Maxwell algebra assures to have a well-defined non-degenerate invariant tensor after applying the NR limit. Although these additional U (1) generators act as central charges in the relativistic algebra, their presence implies additional components of the invariant tensor given by [20] The non-vanishing components of the invariant tensor (3.6) and (3.8) define an invariant tensor for the [enhanced Maxwell] ⊕ u (1) 3 algebra. The gauge connection one-form for the [enhanced Maxwell] ⊕ u (1) 3 algebra now includes not only three additional bosonic gauge fields, Σ A , L A and Γ A in (2.3) but also three extra U (1) gauge field one-forms as The corresponding curvature two-form is given by where R A (W ), R A (E) and R A (K) are given by (2.5) and Then, considering the gauge connection one-form (3.9) and the non-vanishing components of the invariant tensor (3.6) and (3.8) in the general expression of the CS action (2.6), we find the following relativistic CS action for the [enhanced Maxwell] ⊕ u (1) 3 algebra: where It is interesting to note that the terms proportional to µ 2 and µ 3 reproduce the CS action for the coadjoint Poincaré algebra presented in [38]. In particular, the Einstein-Hilbert term appears along µ 2 . On the other hand, the term µ 0 contains the usual exotic Lagrangian [85]. The terms proportional to µ 1 , µ 3 , and µ 5 contains new contributions to the Maxwell CS gravity action given by the additional set of gauge fields. The non-degeneracy of the invariant tensor implies that the gauge fields are dynamically determined by the vanishing of every curvature (2.5) and (3.10) provided µ 5 = 0. In particular, the vanishing of R A (K) can be seen as the constancy of the background electromagnetic field in flat spacetime.
As was discussed in [20], the inclusion of three extra U (1) gauge fields to the Maxwell algebra allows to reproduce a non-degenerate invariant tensor at the NR level. Here, we shall see that the addition of those U (1) gauge fields in the enhanced Maxwell algebra (3.1) will be crucial to obtain a well-defined Maxwellian version of the extended Newtonian gravity.

Enhanced AdS-Lorentz
It is interesting to note that such enhanced AdS-L algebra can be rewritten as three copies of the Poincaré algebra, that is to say by considering the following redefinition of the generators: (3.14) The relativistic enhanced AdS-L algebra can also be written as the direct sum of the coadjoint AdS algebra J A ,P A ,S A ,T A defined in [38] and the Poincaré algebra J A ,P A by considering the following redefinition: Although the relativistic algebra seems simpler in the form of three copies of the Poincaré algebra (3.13), we shall consider the basis {J A , P A , Z A , S A , T A , V A } since it allows to establish a welldefined and evident vanishing cosmological constant limit → ∞ reproducing the enhanced Maxwell algebra. As we shall see, the same behavior will be inherit to its NR version. An alternative procedure to obtain the enhanced AdS-L algebra ((3.1) and (3.12)) is given by the S-expansion method [76]. Indeed, let us consider S (2) M = {λ 0 , λ 1 , λ 2 } as the relevant semigroup and the Poincaré algebra iso(2, 1) as the original one. The semigroup elements satisfy the following multiplication law: (3.16) Then, one can show that the enhanced AdS-L algebra given by (3.1) and (3.12) appears as a S M -expansion of the Poincaré algebra where the expanded generators are expressed in terms of the Poincaré ones through the semigroup elements as Then, following theorem VII.2 of [76], one can show that the non-vanishing components of the enhanced AdS-L algebra can be expressed in terms of the Poincaré ones. Thus, the invariant tensor for the present relativistic algebra are given by (3.6) along with Naturally, the flat limit → ∞ reproduces the invariant tensor of the enhanced Maxwell one. Analogously to the procedure to obtain the EEB algebra [23], one can generalize the relativistic algebra by including three u (1) generators given by Y 1 , Y 2 , and Y 3 which provide the following non-vanishing components of the invariant tensor [23]: Then, considering the gauge connection one-form for the enhanced algebra which coincides with the Maxwell one (3.9) and the non-vanishing components of the invariant tensor (3.6), (3.18), and (3.8) in the general expression of the CS action (2.6), we find the following relativistic CS action for the [enhanced AdS-L] ⊕ u (1) 3 algebra: One can see that the terms proportional to µ 2 , µ 3 , µ 4 , and µ 5 contain new contributions from the new commutation relations (3.12) of the [enhanced AdS-L] ⊕ u (1) 3 algebra. In particular, a cosmological constant term is added to the µ 2 term together with the gauge field K A . Interestingly, the flat limit → ∞ reproduces the relativistic [enhanced Maxwell] ⊕ u (1) 3 gravity action.
As we shall see, the presence of the U (1) gauge fields will be essential to establish a well-defined NR limit allowing us to accommodate a cosmological constant into the Maxwellian version of the extended Newtonian gravity.

Maxwellian extended Newtonian gravity
In this section, we apply a NR limit to the previously introduced [enhanced Maxwell] ⊕ u (1) 3 gravity theory in three spacetime dimensions. We show that the new NR algebra corresponds to a Maxwellian version of the extended Newtonian algebra introduced in [36] and subsequently studied in [37,38]. The three-dimensional NR CS action based on this new symmetry is also discussed.

Maxwellian extended Newtonian algebra and non-relativistic limit
A NR version of the [enhanced Maxwell] ⊕ u(1) 3 algebra can be obtained through an IW contraction. To this end, we consider a dimensionless parameter ξ and we express the relativistic enhanced Maxwell generators as a linear combination of the NR ones (denoted with a tilde) as On the other hand, one can express the u (1) generators Y 1 , Y 2 and Y 3 in terms of the NR generators as After considering the contraction of the [enhanced Maxwell] ⊕ u(1) 3 algebra (3.1) and the limit ξ → ∞, we find that the generators of the novel NR algebra satisfy the MEB algebra [20], where a = 1, 2, ab ≡ 0ab , ab ≡ 0ab .
The new NR algebra corresponds to a Maxwellian version of the so-called extended Newtonian algebra [36] which we have denoted as MENt algebra. In particular, the MENt algebra contains the MEB generators J ,G a ,S,H,P a ,M ,Z,Z a ,T together with a set of additional generators T a ,B a ,Ṽ a ,B,Ỹ ,W . As we shall see, the three central chargesB,Ỹ , andW , appearing in the MENt algebra as a result of the NR limit, allows to have a non-degenerate bilinear form, thus assuring the proper construction of a three-dimensional NR CS gravity action.

Non-relativistic MENt Chern-Simons gravity action
The construction of a three-dimensional CS action based on this new NR symmetry requires the non-vanishing components of the invariant tensor for the MENt algebra together with the gauge connection one-form A = A µ dx µ .
The MENt algebra can be equipped with the most general extended Newtonian non-vanishing components of the invariant tensor [37], where the relativistic parameters µ's were rescaled as In particular, β 0 is related to an exotic sector of the extended Newtonian gravity [37]. Furthermore, the MENt algebra also admits the MEB non-vanishing components of the invariant tensor given by [20] (see also [23]) which can be obtained by setting µ 0 = µ 2 = µ 4 = 0 and considering the following rescaling of the relativistic parameters µ's: Nevertheless, at the level of the MENt algebra, the invariant tensor (4.8) alone results to be degenerate. On the other hand, putting (4.5), (4.6), and (4.8) all together one would obtain a non-degenerate bilinear form. However, we observe that (4.5), (4.6), and (4.8) pertain to two different NR limits. Thus, in the sequel we will construct a CS action based on the MENt algebra by exploiting the components given by (4.5) and (4.6) which describe non-degenerate invariant tensor for the MENt algebra. As we shall see in Section 6, the complete set of invariant tensor given by (4.5), (4.6), and (4.8) can alternatively be obtained through the S-expansion procedure allowing to write the most general CS action based on the MENt algebra. The gauge connection one-form A for the MENt algebra reads A = τH + e aP a + ωJ + ω aG a + kZ + k aZ a + mM + sS + tT + t aT a + b aB a +v aṼ a + bB + yỸ + W . (4.10) The curvature two-form F = dA + 1 2 [A, A] is given by where the explicit definition of every curvature can be found in Appendix A.
A CS gravity action based on the MENt algebra can be constructed by combining (4.5) and (4.6) with the gauge connection 1-form (4.10), and it reads, up to boundary terms, as follows: The CS gravity action (4.12) is invariant under the MENt algebra given by (4.3) and (4.4) by construction, and it is split into three different independent terms. In particular, the term proportional to β 0 corresponds to an exotic sector of the extended Newtonian gravity term introduced in [37] which can be obtained as a NR limit of an U (1)-extension of the exotic Einstein term [85]. The term proportional to β 1 is the standard extended Newtonian gravity action presented in [36].
On the other hand, the piece proportional to β 2 contains some terms appearing in the CS action invariant under the exotic Newtonian algebra 1 introduced in [37], together with some completely new terms.
It is interesting to note that an enhancement of the Maxwell algebra allows to construct a Maxwellian version of the extended Newtonian gravity containing known NR gravity theories as sub-cases. Although the field content is bigger than the extended Newtonian one, the present NR theory does not contain a cosmological constant term. As we shall see in the next section, in order to accommodate a cosmological constant to (4.12), it is necessary to consider a different NR algebra which can be obtained as an IW contraction of the relativistic [enhanced AdS-L] ⊕u(1) 3 algebra. The NR limit and a possible vanishing cosmological constant limit are not the only ways to obtain the present MENt gravity theory. Indeed, as we shall show, the MENt algebra and its most general invariant tensor can alternatively be found using the semigroup expansion method.
The non-degeneracy of the invariant tensor has allowed to achieve a well-defined NR CS gravity action whose equations of motion are given by the vanishing of all the curvatures (4.11).
The CS action (4.12) can equivalently be obtained as NR limit of (3.11). To see this, one shall express the relativistic enhanced Maxwell 1-form fields as a linear combination of the NR ones as (4.13) and the gauge fields dual to the u(1) generators Y 1 , Y 2 , and Y 3 (namely S, M , and T , respectively) as (4.14) Then, substituting back (4.13) and (4.14), together with (4.7), in (3.11), omitting boundary terms and taking the limit ξ → ∞, one precisely recovers (4.12). Finally, one can see that each independent term of the action (4.12) is invariant under the gauge transformation laws δA = dλ + [A, λ], being λ = ΛH + Λ aP a + ΩJ + Ω aG a + κZ + κ aZ a + χM + ςS + πT + π aT a + ρ aB a +ν aṼ a + ρB + γỸ + W (4.15) the gauge parameter. The gauge transformations of the theory can be found in Appendix B.

Enlarged extended Newtonian gravity and flat limit
In this section, we apply a NR limit to the [enhanced AdS-L] ⊕ u (1) 3 algebra previously introduced. We show that the new NR algebra can be seen as an enlargement of the extended Newtonian algebra [36] (which we have denoted as EEN algebra) and reproduces the MENt algebra (4.3)-(4.4) in the vanishing cosmological constant limit → ∞.

Enlarged extended Newtonian algebra and non-relativistic limit
In the previous section we have constructed a Maxwellian version of the extended Newtonian algebra by applying an Inönü-Wigner contraction to the [enhanced Maxwell] ⊕ u (1) 3 algebra (3.1). In order to obtain a NR version of the [enhanced AdS-L] ⊕ u (1) 3 algebra (where the latter is given by (3.1) and (3.12)), we consider the same redefinitions of the relativistic generators as in (4.1)-(4.2) which provides us with a well-defined ξ → ∞ limit. The new NR algebra is then generated by the set of generators of the EEB algebra J ,G a ,S,H,P a ,M ,Z,Z a ,T , a set of additional generators T a ,B a ,Ṽ a , and three central charges given byB,Ỹ ,W . Such generators satisfy the commutation relations (4.3)-(4.4) along with Note that the commutation relations (4.3) and (5.1) define a subalgebra corresponding to the EEB algebra introduced in [23]. The novel non-relativistic algebra obtained here can be seen as an enlargement of the extended Newtonian algebra, and we will denote it as EEN algebra. An interesting feature of such NR algebra is given by the explicit presence of a scale which allows us to accommodate a cosmological constant into the Maxwellian version of the extended Newtonian gravity (4.3)-(4.4). Naturally, the vanishing cosmological constant limit → ∞ reproduces the MENt algebra. Let us note that the EEN algebra can be rewritten as three copies of the enhanced Nappi-Witten algebra defined in [38]. Indeed, the EEN algebra can be written as by considering the following redefinition of the generators: On the other hand, a different redefinition of the generators of the EEN algebra can be considered. In particular, the redefinitioñ allows to rewrite the EEN algebra (4.3), (4.4), (5.1), and (5.2) as the direct sum of the exotic Newtonian algebra [37], 4) and the enhanced Nappi-Witten algebra, The exotic Newtonian algebra has been recently introduced in [37] and subsequently studied in [38] and allows us to accommodate a cosmological constant into the extended Newtonian gravity theory. One can see that, as we have shown previously, the same behavior appears at the relativistic level. Indeed the enhanced AdS-Lorentz algebra can also be rewritten as three copies of the Poincaré algebra and as the direct sum of the coadjoint AdS and Poincaré algebra after an appropriate redefinition of the generators. In particular, the enhanced Nappi-Witten algebra (5.5) appears as an IW contraction of the Poincaré ⊕ u (1) algebra, while the exotic Newtonian algebra can be obtained as a NR limit of the coadjoint AdS algebra [38].
In what follows we present the NR CS action based on the EEN algebra (4.3), (4.4), (5.1), and (5.2). Such basis is preferred in order to make clear the flat limit leading to the MENt gravity. Furthermore, as we shall see, such basis present an alternative way to include a cosmological constant into a three-dimensional NR CS gravity action diverse to the one discussed in [37].

Non-relativistic EEN Chern-Simons gravity action and flat limit
A well-defined CS action requires invariant non-degenerate bilinear form. Interestingly, the presence of the central chargesB,Ỹ , andW assures to have a non-degenerate invariant tensor for the EEN algebra. In particular, the non-vanishing components of the invariant tensor for the EEN algebra are given by the MENt ones (4.5)-(4.6) along with where the relativistic parameters µ's have been rescaled as On the other hand, one can show that the EEN algebra can also admit the MEB invariant tensor (4.8) along with when we set µ 0 = µ 2 = µ 4 = 0 and consider the following rescaling of the relativistic parameters µ's: One can see that the components of the invariant tensor proportional to the α's are the respective components of the invariant tensor of the EEB algebra introduced in [23]. On the other hand, those related to the β's shall reproduce the enlarged extended Newtonian gravity action. As it was discussed in the MENt case, the NR limit does not allow to have both types of invariant tensor. In this section we shall focus on those proportional to the β's constants allowing to construct a novel NR CS action. In Section 6, we shall see that the complete set containing both the α's and the β's constants appearing in the full invariant tensor can be properly obtained considering the S-expansion method.
Although the gauge connection one-form A is the same than the MENt one, the corresponding curvature two-form F is different due to the presence of new commutation relations involving the parameter. Indeed, the curvature two-form F reads where the explicit expression of every curvature two-form is given in Appendix A.
A CS gravity action based on the EEN algebra can be constructed by combining (4.5)-(4.6) and (5.6) with the gauge connection 1-form (4.10), and it reads, up to boundary terms, as follows: The CS action (5.11) is invariant under the EEN algebra by construction. One can notice, looking also at (4.12), that there are three independent sectors proportional to the different coupling constants β 0 , β 1 , and β 2 . The term proportional to β 0 is exactly the same as in (4.12). On the other hand, the terms proportional to β 1 and β 2 involve, besides the MENt contributions, also new pieces. In the flat limit → ∞ of (5.11) we recover the MENt gravity action (4.12). Observe that our construction gives an alternative way to include a cosmological constant into the three-dimensional NR CS gravity action (4.12).
The non-degeneracy of the invariant tensor has allowed to achieve a well-defined CS gravity action whose equations of motion are given by the vanishing of all the curvatures (A.1).
The CS action (5.11) we have just constructed can equivalently be obtained as NR limit of the CS action (3.20) based on the [enhanced AdS-L] ⊕ u(1) 3 algebra. Indeed, expressing the relativistic enhanced AdS-L 1-form fields in (3.20) as a linear combination of the NR ones as in (4.13) and the gauge fields S, M , and T as in (4.14), together with (5.7), and taking the limit ξ → ∞, one precisely recovers the NR CS action (5.11) for the EEN algebra.
Concluding, we can see that each independent term of the action (5.11) is invariant under the gauge transformation laws δA = dλ + [A, λ] with gauge parameter λ given by (4.15). Specifically, the explicit gauge transformations of the theory are defined in Appendix B.

Non-relativistic algebras and semigroup expansion method
In this section, we present an alternative procedure to recover the new NR algebras previously introduced. In particular, we show that the MENt and EEN algebras can be alternatively obtained by considering the S-expansion procedure. The Lie algebra expansion method has first been introduced in [90] in the context of AdS superstring and then developed by considering the Maurer-Cartan equations in [91][92][93]. The expansion based on semigroups has then been introduced in [76] and subsequently developed in [94][95][96][97][98][99]. The S-expansion consists in defining a new expanded Lie algebra G = S × g, by combining the elements of the structure constants of a Lie algebra g with the semigroup S. Such procedure provides us not only with the commutation relations, but also with the complete set of non-vanishing components of the invariant tensor for the expanded algebra. At the NR level, the Lie algebra expansion method has been used by diverse authors leading to interesting results [100][101][102][103][104][105].
Here we shall consider the S-expansion of the enhanced Nappi-Witen algebra (5.5) formerly discussed. The motivation to consider the enhanced Nappi-Witten algebra as the original algebra is twofold. First, as was shown in [23,86], the S-expansion of the Nappi-Witten algebra allows us to recover diverse NR algebras. Second, since the iso(2, 1) ⊕ u(1) algebra can be used to obtain the respective relativistic counterparts, it seems natural to expect the same behavior for the NR version of the iso(2, 1) ⊕ u(1) algebra, which is given by the enhanced Nappi-Witten algebra.
Let us first recall the enhanced Nappi-Witten algebra g, whose generators satisfy This algebra appears as a particular IW contraction of the iso(2, 1) ⊕ u(1) algebra. Indeed, as was mentioned in [38], one can write the relativistic Poincaré and u (1) generators {J A , P A , Y 1 } in terms of the enhanced Nappi-Witten ones as Then, the aforesaid algebra (6.1) is revealed after applying the limit ξ → ∞. Naturally, the Nappi-Witten algebra [106,107] spanned by {J, G a , S} appears as a subalgebra.
Let us now consider S (2) E = {λ 0 , λ 1 , λ 2 , λ 3 } as the relevant semigroup whose elements satisfy the following multiplication law, where λ 3 = 0 s is the zero element of the semigroup such that 0 s λ α = 0 s . Then, after considering a 0 s -reduction of the expanded algebra S Using the multiplication law of the semigroup (6.3) and the enhanced Nappi-Witten commutators (6.1), one finds that the expanded generators satisfy the MENt algebra (4.3) and (4.4). Additionally, one can express the non-vanishing components of the invariant tensor of the S-expanded algebra in terms of the original ones. Interestingly, considering the Theorem VII. of [76], one can show that the non-vanishing components of the invariant tensor for the MENt algebra are given not only by (4.5)-(4.6) but also by the MEB ones (4.8). This can be seen directly from the invariant tensor of the enhanced Nappi-Witten algebra whose components are given by The expanded invariant tensor coming from JS and G a G b generate the non-vanishing components of the invariant tensor for the MEB algebra proportional to α's given by (4.8). In particular, the α's constants can be expressed in terms of γ 1 as The invariant tensor for the MENt algebra (4.5)-(4.6), proportional to the β's, are obtained from SS , SB , and G a B b with β 0 = λ 0 γ 2 , β 1 = λ 1 γ 2 , β 2 = λ 2 γ 2 . (6.7) As we have previously discussed, both classes of invariant tensor can be obtained separately by considering different NR limits. Although the complete set of invariant tensor (4.5)-(4.6) along with (4.8) are non-degenerate for the MENt algebra, they cannot be obtained simultaneously considering a unique NR limit. Here, the semigroup expansion procedure provides us with both components of the invariant tensor. In particular, the MEB ones are related to the γ 1 appearing in the invariant tensor of the enhanced Nappi-Witten algebra (6.5), while the MENt ones proportional to the β's have origin in γ 2 . Such feature is an additional advantage of considering the semigroup expansion method in the NR context. In particular, as the MENt algebra (4.3)-(4.4), the MENt CS gravity action constructed using the S-expansion procedure contains also the MEB gravity action [20] as a sub-case. It is interesting to note that the semigroup chosen is the same used to obtain the MEB algebra from the Nappi-Witten algebra [23,86]. Furthermore, as we have shown in Section 3, the same semigroup is used to obtain its respective relativistic algebra from the iso (2, 1) ⊕ u (1) algebra. The fact that the same semigroup can be used in two diverse regimes has already appeared in other contexts. Indeed, the same semigroup can be used to relate diverse finite-and infinitedimensional Lie (super)algebras [108][109][110] or higher-spin Lie algebras [111].
On the other hand, considering a different semigroup allows us to find a different NR algebra. One can show that the EEN algebra can alternatively be obtained by applying a S one obtains an S-expanded algebra whose generators are related to the enhanced Nappi-Witten ones as in (6.4) and satisfy the EEN algebra (4.3), (4.4), (5.1), and (5.2). Furthermore, considering the Theorem VII. of [76], one can show that the non-vanishing components of the invariant tensor for the EEN algebra are given not only by (5.6) but also by the EEB ones (5.8). As discussed in previous sections, both of the invariant tensors cannot be obtained from a unique NR limit. Here, the S-expansion gives us the complete set of invariant tensors considering only one semigroup, and allowing to write an EEN CS gravity action containing the EEB gravity [23] as a sub-case.
In particular, the arbitrary constants α's and β's are related to the original ones as in (6.6) and (6.7), respectively. As an ending remark, let us mention that one could obtain new NR algebras by expanding the enhanced Nappi-Witten algebra using the same semigroup used to obtain their respective relativistic algebras from the iso (2, 1) ⊕ u (1) algebra. It would be interesting to extend the present results to supersymmetric extensions of NR gravity theories.

Discussion
In this work we have presented a Maxwellian version of the three-dimensional extended Newtonian gravity theory introduced in [36]. We have shown that the MENt algebra appears as an IW contraction of the [enhanced Maxwell] ⊕ u(1) ⊕ u(1) ⊕ u(1) algebra. The additional U (1) gauge fields are necessary to have a finite and non-degenerate invariant tensor allowing the proper construction of a NR CS action. Furthermore, we have also applied the NR limit at the level of the CS action. We have then extended our results to accommodate a cosmological constant in the MENt gravity theory. To this end we have presented a new NR symmetry which can be seen as an enlargement of the extended Newtonian algebra and we have denoted it as EEN algebra. Such novel NR symmetry can be rewritten either as three copies of the enhanced Nappi-Witten algebra defined in [38] or as the direct sum of the exotic Newtonian algebra introduced in [37] with the enhanced Nappi-Witten one.
We have also explored an alternative procedure to recover our results by considering the semigroup expansion of the enhanced Nappi-Witten symmetry. In particular, we have shown that the same semigroup used to relate relativistic algebras can be adopted to reproduce their NR counterparts. In addition, we have pointed out that the vanishing cosmological constant limit appearing at the relativistic level is inherited in the NR version.

Figure 1: Expansions and limits
The S-expansion method has the advantage to provide not only the new Newtonian algebras but also to reproduce the most general invariant tensor for the respective NR algebras. Interestingly, the MENt and EEN gravity theories constructed with expanded invariant tensor contain known extended Bargmann gravity actions as sub-cases. One can notice that the scenario graphically displayed in Figure 1 also holds for the particular sub-cases. In particular, as was noticed in [23], the MEB and the EEB algebras can alternatively be obtained as S-expansions of the Nappi-Witten algebra.
Our results could be seen as a starting point for diverse studies. It would be interesting to explore the extensions of our analysis to supergravity. Although the large amount of NR gravity models, the supersymmetric extension of NR gravity theories has only been approached recently [35,36,87,[112][113][114] due to the difficulty to define a proper NR limit. One way to avoid such problem could be through the S-expansion method considered here. Indeed, one could conjecture that Figure 1 can be generalized to the presence of supersymmetry. The supersymmetric extension of the Maxwellian extended Newtonian gravity and its exotic version shall be approached in a future work.
It would be worth it to study the generalization of the results presented here to the ultra-relativistic regime. In particular, a Maxwellian version of the three-dimensional Carroll gravity theory [115,116] could be constructed analogously to the MEB one. Then, a cosmological constant could be accommodated in a very similar way to the one considered in the AdS-Caroll CS gravity [117][118][119]. Another natural extension of our results is the generalization of the obtained NR symmetries to four and higher spacetime dimensions. Another point that deserves further investigation is the physical implications of the additional gauge fields appearing in the enhanced Maxwell and AdS-L gravity theories introduced here. As it was discussed in [79], the additional gauge generator Z A appearing in the Maxwell algebra influences not only the asymptotic sector but also its vacuum energy and vacuum angular momentum. On the other hand, the boundary dynamics of the AdS-L CS gravity is described by three copies of the Virasoro algebra [84]. One may then ask how the conserved charges and solutions are modified by the presence of the additional set of relativistic generators {S A , T A , V A }.

Acknowledgments
This work was funded by the National Agency for Research and Development ANID (ex-CONICYT) -PAI grant No. 77190078 (P.C.) and the FONDECYT Project 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.

A Curvature two-forms
The curvature two-forms for the EEN algebra are given by The vanishing cosmological constant limit → ∞ naturally reproduces the curvature two-forms of the MENt algebra. In particular, in the flat limit, the curvature two-forms with an hat reduce to the MENt ones (4.11).

B Gauge transformations
The gauge transformations for the EEN gauge fields are given by Let us note that, in the limit → ∞, (B.1) reproduces the gauge transformations for the MENt gauge fields (4.10).