Chern-Simons and Born-Infeld gravity theories and Maxwell algebras type

Recently was shown that standard odd and even-dimensional General Relativity can be obtained from a $(2n+1)$-dimensional Chern-Simons Lagrangian invariant under the $B_{2n+1}$ algebra and from a $(2n)$-dimensional Born-Infeld Lagrangian invariant under a subalgebra $\cal{L}^{B_{2n+1}}$ respectively. Very Recently, it was shown that the generalized In\"on\"u-Wigner contraction of the generalized AdS-Maxwell algebras provides Maxwell algebras types $\cal{M}_{m}$ which correspond to the so called $B_{m}$ Lie algebras. In this article we report on a simple model that suggests a mechanism by which standard odd-dimensional General Relativity may emerge as a weak coupling constant limit of a $(2p+1)$-dimensional Chern-Simons Lagrangian invariant under the Maxwell algebra type $\cal{M}_{2m+1}$, if and only if $m\geq p$. Similarly, we show that standard even-dimensional General Relativity emerges as a weak coupling constant limit of a $(2p)$-dimensional Born-Infeld type Lagrangian invariant under a subalgebra $\cal{L}^{\cal{M}_{2m}}$ of the Maxwell algebra type, if and only if $m\geq p$. It is shown that when $m<p$ this is not possible for a $(2p+1)$-dimensional Chern-Simons Lagrangian invariant under the $\cal{M}_{2m+1}$ and for a $(2p)$-dimensional Born-Infeld type Lagrangian invariant under $\cal{L}^{\cal{M}_{2m}}$ algebra.


Introduction
The most general action for the metric satisfying the criteria of general covariance and second-order field equations for d > 4 is a polynomial of degree [d/2] in the curvature known as the Lanczos-Lovelock gravity theory (LL) [1], [2]. The LL lagrangian in a d-dimensional Riemannian manifold can be defined as a linear combination of the dimensional continuation of all the Euler classes of dimension 2p < d [3], [4]: where α p are arbitrary constants and L p = ε a1a2······a d R a1a2 · · · ·R a2p−1a2p e a2p+1 · · · ·e a d (2) with R ab = dω ab + ω a c ω cb . The expression (1) can be used both for even and for odd dimensions.
The large number of dimensionful constants in the LL theory α p , p = 0, 1, · · ·, [d/2] , which are not fixed from first principles, contrast with the two constants of the Einstein-Hilbert action.
In ref. [5] it was found that these parameters can be fixed in terms of the gravitational and the cosmological constants, and that the action in odd dimensions can be formulated as a Chern-Simons theory of the AdS group.
The closest one can get to a Chern-Simons theory in even dimensions is with the so-called Born-Infeld theories [5] [6], [7], [8]. The Born-Infeld lagrangian is obtained by a particular choice of the parameters in the Lovelock series, so that the lagrangian is invariant only under local Lorentz rotations in the same way as is the Einstein-Hilbert action.
If Chern-Simons theory is the appropriate odd-dimensional gauge theory and if Born-Infeld theory is the appropriate even-dimensional theories to provide a framework for the gravitational interaction, then these theories must satisfy the correspondence principle, namely they must be related to General Relativity.
In Ref. [10] was shown that the standard, odd-dimensional General Relativity (without a cosmological constant) can be obtained from Chern-Simons gravity theory for a certain Lie algebra B and recently was found that standard, even-dimensional General Relativity (without a cosmological constant) emerges as a limit of a Born-Infeld theory invariant under a certain subalgebra of the Lie algebra B [11].
Very recently was found that the so called B m Lie algebra of Ref. [10] correspond to Maxwell algebras type M m [12]. In fact, it was shown that the generalized Inönü-Wigner contraction of the generalized AdS-Maxwell algebras provides maxwell algebras types M m which correspond to B m Lie algebra. These Maxwell algebras type M m algebras can be obtained by S-expansion resonant reduced of the AdS Lie algebra when we use S α=0 as semigroup.
It is the purpose of this paper to show that standard odd General Relativity emerges as a weak coupling constant limit of a (2p + 1)-dimensional Chern-Simons Lagrangian invariant under the M 2m+1 algebra, if and only if m ≥ p. Similarly, we show that standard even General Relativity emerges as a weak coupling constant limit of a (2p)-dimensional Born-Infeld type Lagrangian invariant under the L M2m algebra, if and only if m ≥ p. It is shown that when m < p this is not possible for a (2p + 1)-dimensional Chern-Simons Lagrangian invariant under the M 2m+1 and for a (2p)-dimensional Born-Infeld type Lagrangian invariant under L M2m . This paper is organized as follows: In Sec. II we briefly review some aspect of: (i) Lovelock gravity theory, (ii) the construction of the so called M 2n+1 algebra and (iii) obtaining odd and even dimensional general relativity from Chern-Simons gravity theory and from Born-Infeld theory respectively. In Section III it is shown that the odd-dimensional Einstein-Hilbert Lagrangian can be obtained from a Chern-Simons Lagrangian in (2p+1)-dimensions invariant under the algebra M 2m+1 , if and only if m ≥ p. However, this is not possible for Chern-Simons Lagrangian in (2p + 1)-dimension invariant under the M 2m+1 algebra when m < p.
In Section IV it is shown that the even-dimensional Einstein-Hilbert Lagrangian can be obtained from a Born-Infeld type Lagrangian in (2p)-dimensions invariant under the L M2m subalgebra of the M 2m algebra , if and only if m ≥ p. However, this is not possible for Born-Infeld type Lagrangians in (2p)dimensions invariant under the L M2m subalgebra when m < p.
Sec.V concludes the work with a comment about possible developments.
2 The Lovelock action, The M 2n+1 algebra and general relativity In this section we shall review some aspects of higher dimensional gravity, the construction of the so called Maxwell algebra types, and obtaining odd and even dimensional general relativity from Chern-Simons gravity theory and from Born-Infeld theory respectively. The main point of this section is to display the differences between the invariances of Lovelock action when odd and even dimensions are considered.

The Chern-Simons gravity
The Lovelock action is a polynomial of degree [d/2] in curvature, which can be written in terms of the Riemann curvature and the vielbein e a in the form (1), (2). In first order formalism the Lovelock action is regarded as a functional of the vielbein and spin connection, and the corresponding field equations obtained by varying with respect to e a and ω ab read [5]: Here T a = de a + ω a b e b is the torsion 2-form. Using the Bianchi identity one finds [5] Dε a = Moreover From (6) and (7) one finds for d = 2n − 1 where for any dimensions l is a length parameter related to the cosmological constant by Λ = ±(d − 1)(d − 2)/2l 2 With these coefficients, the Lovelock action is a Chern-Simons (2n − 1)-form invariant not only under standard local Lorentz rotations δe a = κ a b e b , δω ab = −Dκ ab , but also under a local AdS boost [5].
With these coefficients the LL lagrangian takes the form [5] which is the Pfaffian of the 2-formR ab = R ab + 1 l 2 e a e b and can be formally written as the Born-Infeld like form [5], [8] . The corresponding action, known as Born-Infeld action is invariant only under local Lorentz rotations.
The corresponding Born-Infeld action is given by [5], [8] where e a corresponds to the 1-form vielbein, and R ab = dω ab + ω a c ω cb to the Riemann curvature in the first order formalism.
In order to interpret the gauge field as the vielbein, one is forced to introduce a length scale l in the theory. To see why this happens, consider the following argument: Given that (i) the exterior derivative operator d = dx µ ∂ µ is dimensionless, and (ii) one always chooses Lie algebra generators T A to be dimensionless as well, the one-form connection fields A = A A µ T A dx µ must also be dimensionless. However, the vielbein e a = e a µ dx µ must have dimensions of length if it is to be related to the spacetime metric g µν through the usual equation g µν = e a µ e b ν η ab . This means that the "true" gauge field must be of the form e a /l, with l a length parameter.
Therefore, following Refs. [15], [16], the one-form gauge field A of the Chern-Simons theory is given in this case by It is important to notice that once the length scale l is brought into the Born-Infeld theory, the lagrangian splits into several sectors, each one of them proportional to a different power of l, as we can see directly in eq. (14).

The S-expansion procedure
In this subsection we shall review the main aspects of the S-expansion procedure and their properties introduced in Ref. [13].
Let S = {λ α } be an abelian semigroup with 2-selector K γ αβ defined by and g a Lie (super)algebra with basis {T A } and structure constant C C AB , Then it may be shown that the product G = S × g is also a Lie (super)algebra with structure constants C The proof is direct and may be found in Ref. [13].
Definition 1 Let S be an abelian semigroup and g a Lie algebra. The Lie algebra G defined by G = S × g is called S-Expanded algebra of g.
When the semigroup has a zero element 0 S ∈ S, it plays a somewhat peculiar role in the S-expanded algebra. The above considerations motivate the following definition: Definition 2 Let S be an abelian semigroup with a zero element 0 S ∈ S, and let G = S × g be an S-expanded algebra. The algebra obtained by imposing the condition 0 S T A = 0 on G (or a subalgebra of it) is called 0 S -reduced algebra of G (or of the subalgebra).
An S-expanded algebra has a fairly simple structure. Interestingly, there are at least two ways of extracting smaller algebras from S × g. The first one gives rise to a resonant subalgebra, while the second produces reduced algebras. In particular, a resonant subalgebra can be obtained as follow.
Let g = p∈I V p be a decomposition of g in subspaces V p , where I is a set of indices. For each p, q ∈ I it is always possible to define i (p,q) ⊂ I such that Now, let S = p∈I S p be a subset decomposition of the abelian semigroup S such that S p · S q ⊂ r∈i (p,q) S p .
When such subset decomposition S = p∈I S p exists, then we say that this decomposition is in resonance with the subspace decomposition of g, g = p∈I V p . The resonant subset decomposition is crucial in order to systematically extract subalgebras from the S-expanded algebra G = S × g, as is proven in the following Theorem IV.2 of Ref. [13]: Let g = p∈I V p be a subspace decomposition of g, with a structure described by eq. (20) , and let S = p∈I S p be a resonant subset decomposition of the abelian semigroup S, with the structure given in eq. (21). Define the subspaces of G = S × g, Then, is a subalgebra of G = S × g. Proof: the proof may be found in Ref. [13].

Definition 3
The algebra G R = p∈I W p obtained is called a Resonant Subalgebra of the S-expanded algebra G = S × g.
A useful property of the S-expansion procedure is that it provides us with an invariant tensor for the S-expanded algebra G = S × g in terms of an invariant tensor for g. As shown in Ref. [13] the theorem VII.2 provide a general expression for an invariant tensor for a 0 S -reduced algebra.
Theorem VII.2 of Ref. [13]: Let S be an abelian semigroup with nonzero elements λ i , i = 0, · · · , N and λ N +1 = 0 S . Let g be a Lie (super)algebra of basis {T A }, and let T An · · · T An be an invariant tensor for g. The expression where α j are arbitrary constants, corresponds to an invariant tensor for the 0 S -reduced algebra obtained from G = S × g.
Proof: the proof may be found in section 4.5 of Ref. [13].
We note that commutation relations (29), (33), (34) form a subalgebra of the M 2n+1 algebra which we will denote as L M2n+1 . This subalgebra can be obtained from S-expansion of the Lorentz-Lie algebra using as a semigroup the sub-semigroup S After extracting a resonant subalgebra and perfoming its 0 S (= λ 2n )-reduction, one finds the L M2n+1 algebra, which is a subalgebra of the M 2n+1 algebra, whose generators J ab = λ 0J ab , Z

Odd-dimensional general relativity
In Ref. [10], it was shown that the standard, odd-dimensional general relativity (without a cosmological constant) can be obtained from Chern-Simons gravity theory for the algebra M 2n+1 . The Chern-Simons Lagrangian is built from a M 2n+1 -valued, one-form gauge connection A which depends on a scale parameter l which can be interpreted as a coupling constant that characterizes diferent regimes whithin the theory. The field content induced by M 2n+1 includes the vielbein e a , the spin connection ω ab , and extra bosonic fields h a(i) and k ab(j) . The odd-dimensional Chern-Simons Lagrangian invariant under the M 2n+1 algebra is given by [10] where In the l −→ 0 limit, the only nonzero term in (40) corresponds to the case k = 1, whose only non-vanishing component occurs for p = q 1 = ···· = q 2n−1 = 0 and is proportional to the odd-dimensional Einstein-Hilbert Lagrangian [10]

Even-dimensional general relativity
In Ref. [11], it was recently shown that standard, even-dimensional General Relativity (without a cosmological constant) emerges as a limit of a Born-Infeld theory invariant under the subalgebra L M2n+1 of the Lie algebra M 2n+1 . The Born-Infeld Lagrangian is built from the two-form curvature S where which depends on a scale parameter l which can be interpreted as a coupling constant that characterizes different regimes within the theory. The field content induced by L M2n+1 includes the vielbein e a , the spin connection ω ab and extra bosonic fields h a(i) = e (a,2i+1) and k ab(i) = ω (ab,2i) , with i = 1, ..., n − 1. The even-dimensional Born-Infeld Gravity Lagrangian invariant under the L M2n+1 algebra is given by [11] where we can see that in the limit l = 0 the only nonzero term corresponds to the case k = 1, whose only nonzero component (corresponding to the case p = q 1 = · · · = q 2n−2 = 0) [11] is proportional to the even-dimensional Einstein-Hilbert Lagrangian

Chern-Simons Lagrangians invariant under the Maxwell algebra type
In this section it is shown that the Einstein-Hilbert Lagrangian for odd-dimensions can be obtained from a Chern-Simons Lagrangian in (2p + 1) dimensions invariant under the M 2m+1 algebra , if and only if m ≥ p. However, this is not possible when m < p for Chern-Simons Lagrangians in (2p + 1)-dimensions invariant under the M 2m+1 algebra . The 1-form gauge connection A M 2n+1 -valued, is given by where It is interesting to note that the Maxwell algebra type M 2m+1 can be used to construct different odd-dimensional Chern-Simons Lagrangians. For example, if we consider a S E -expansion of the AdS algebra SO (6, 2) and after extracting a resonant subalgebra and performing its 0 S -reduction, one fnds M 5 algebra in D = 7 dimensions. In this way, the CS Lagrangians L M5 CS (5) and L M5 CS (7) are invariant under the same M 5 algebra, however the indices of the generators T a runs over 5 and 7 values, respectively.
These considerations allow the construction of gravitational theories in every odd-dimension. Nevertheless, as discussed below, only in some dimensions it is possible to obtain General Relativity as a weak coupling constant limit of a Chern-Simons theory.

(2 + 1)-dimensional Chern-Simons Lagrangians invariant under M 7 -algebra
Before considering the Chern-Simons Lagrangian (2n + 1)-dimensional, we study the case of the M 7 algebra. The M 7 -algebra can be found by S-expansion of the AdS algebra using as semigroup S E . In fact, after extracting a resonant subalgebra and performing the 0 S reduction, one finds the M 7 -algebra whose generators satisfy the following commutation relations Consider the construction of a three-dimensional Chern-Simons Lagrangian invariant under M 7 . In fact, Using Theorem V II.2 of Ref. [13], it is possible to show that the only non-vanishing components of a invariant tensor for the M 7 algebra are given by where α 0 , α 1 , α 2 , α 3 , α 5 and α 5 are arbitrary independent constant dimensionless. The 1-form gauge connection A M 7 -valued, is given by, and the 2-form curvature is Using the dual procedure of S-expansion, we find that the 3-dimensional Chern-Simons Lagrangian invariant under the M 7 -algebra is given by where The Lagrangian (70) is split into six independent pieces, each one proportional to α 1 , α 3 , α 5 , α 0 , α 2 , α 4 . The term proportional to α 1 corresponds to the Chern-Simons Lagrangian for ISO (2, 1) which contains the Eintein-Hilbert term ε abc R ab e c .
which correspond to the equations of general relativity with cosmological constant in (2 + 1)-dimensions.

(4+1)-dimensional Chern-Simons Lagrangian invariant under M 7 -algebra
The only non-vanishing components of a invariant tensor for the M 7 algebra are given by In D = 5, the only non-vanishing components of a invariant tensor for the M 7 algebra are given by where α 1 , α 3 and α 5 are arbitrary independent constant of dimensions [length] −3 . Using the dual procedure of S-expansion, we find that the 5-dimensional Chern-Simons Lagrangian invariant under the M 7 -algebra is given by Varying the Lagrangian (81) we have When a solution without matter k (ab,1) = 0, k (ab,2) = 0, h (a,1) = 0, h (a,2) = 0 is singled out, we are left with δL M7 (4+1) = ε abcdf α 1 l 2 R ab R cd + 2α 3 R ab e c e d + 1 l 2 α 5 e a e b e c e d δe f + α 3 l 2 R ab R cd + 2α 5 R ab e c e d δh (f,1) + 2α 5 l 2 δk (ab,2) R cd T f + α 5 l 2 R ab R cd δh (f,2) +δk (ab,1) 2α 3 l 2 R cd T f + 2α 5 e c e d T f + δω ab 2α 1 l 2 R cd T f + 2α 3 e c e d T f So when α 1 and α 5 vanish we finally get Therefore, if we impose the torsionless condition, we see that the Chern-Simons Lagrangian in D = 5 invariant under M 7 leads to the same equations of motion than the Chern-Simons Lagrangian in D = 5 invariant under M 5 [10]. From (82), like in Ref. [10], we can see that in the limit where l = 0 the extra constraints just vanish, and δL CS = 0 leads us to the Einstein-Hilbert dynamics in vacuum, Similarly, when the cosmological constant is not considered and a solution without matter is singled out, the strict limit where the coupling constant l equals zero yields just to the Einstein Hilbert term in the Lagrangian

(6+1)-dimensional Chern-Simons Lagrangian invariant under M 5 -algebra
Now, consider a Chern-Simons action (6 + 1)-dimensional invariant under M 5algebra.The 1-form gauge connection A M 5 -valued, is given by and the 2-forma curvature is given by Using the dual procedure of S-expansion, we find that the 7-dimensional Chern-Simons Lagrangian invariant under the M 5 -algebra is given by where α 1 = λ 1 κ, α 3 = λ 3 κ. From here we see that the Einstein-Hilbert term is not present in the Lagrangian. This result holds for all D = p-dimensional Chern-Simons Lagrangian invariant under an algebra M m if p > m.
Varying the Lagrangian we have from which we can see it is not possible to obtain the Einstein-Hilbert dynamics.
In fact, imposing the torsionless condition and if we consider the case where k ab = 0 , h a = 0 with α 1 = 0 we find which obviously does not correspond to the dynamics of General Relativity.
Where we see that in the limit l → 0 we have that the Lagrangian leads to the Einstein Hilbert term . This is because to obtain the term Einstein-Hilbert, is necessary the presence of the J a1a2 Z a3a4 · · · Z a2p−1a2p P 2p+1 component of the invariant tensor, which is given by This observation leads to state the following theorem: Theorem 4 Let M 2m+1 be the Maxwell type algebra, which is obtained from AdS algebra by S The following table shows a set of Chern-Simons Lagrangian L M2m+1 CS (2p+1) , invariant under the Lie algebra M 2m+1 , that flow into the General Relativity Lagrangian in a certain limit: . . . . . .

Born-Infeld lagrangian in D = 4 invariant under L M 5
Following the definitions of Ref. [13], let us consider the S-expansion of the Lie algebra SO (3, 1) using as a semigroup the sub-semigroup S After perfoming its 0 S (= λ 4 )-reduction, one finds a new Lie algebra, call it L M5 which is a subalgebra of the so called M 5 algebra, whose generators J ab = λ 0Jab , Z ab = λ 2Jab , satisfy the commutation relationships [Z ab, Z cd ] = 0.
In order to write down a Born-Infeld, we start from the two-form L M5 curvature F Using Theorem V II.2 of Ref. [13], it is possible to show that the only non-vanishing components of a invariant tensor for the L M5 algebra are given by where α 0 and α 2 are arbitrary independent constants of dimensions [length] −2 .
Using the dual procedure of S-expansion in terms of the Maurer-Cartan forms [14], we find that the 4-dimensional Born-Infeld Lagrangian invariant under the L M5 algebra is given by [11] Here we can see that the Lagrangian (100) is split into two independent pieces, one proportional to α 0 and the other to α 2 . The term proportional to α 0 corresponds to the Euler invariant. The piece proportional to α 2 contains the Einstein-Hilbert term ε abcd R ab e c e d plus a boundary term which contains, besides the usual curvature R ab , a bosonic matter field k ab .
Unlike the Born-Infeld Lagrangian the coupling constant l 2 does not appear explicitly in the Einstein Hilbert term but accompanies the remaining elements of the Lagrangian. This allows recover four dimensional the Einstein-Hilbert Lagrangian in the limit where l equals to zero.
The variation of the Lagrangian, modulo boundary terms, is given by from which we see that to recover the field equations of general relativity is not necessary to impose the limit l = 0. δL L M BI (4) = 0 leads to the dynamics of Relativity when considering the case of a solution without matter (k ab = 0). This is possible only in 4 dimensions. However, to recover the field equations of general relativity in dimensions greater than 4, is necessary to take a limit of the coupling constant l.

Born-Infeld lagrangian in D = 4 invariant under L M 7 algebra
Now, we consider the Born-Infeld lagrangian in D = 4 invariant under L M7 algebra whose generators satifsy the following commutation relations The two-form curvature S 0 -expanded reduced is ab .
Using theorem VII.2 of Ref. [13], it is posible to show that the only nonvanishing components of a invariant tensor for the L M7 algebra are given by where α 0 , α 2 and α 4 are arbitrary independent constants dimensionless.
Using the dual procedure of S-expansion in terms of the Maurer-Cartan forms [14], we find that the 4-dimensional Born-Infeld Lagrangian invariant under the L M7 algebra is given by where The Lagrangian (111) is split into three independent pieces, each one proportional to α 0 , α 2 , α 4 repectively. The term proportional to α 0 corresponds to the Euler invariant. The piece proportional to α 2 contains the Einstein-Hilbert term ε abcd R ab e c e d plus a boundary term which contains, besides the usual curvature R ab , a bosonic matter field k (ab,1) .
The variation of the Lagrangian, modulo boundary terms, is given by where α 2i+2j are arbitrary independent constants dimensionless and where we have defined Using the same procedure used in the previous cases, we found that the Lagrangian L L M 2n+1 BI (4) is given by Varying the Lagrangian and considering the case without matter, k (ab,i) = h (a,j) = 0 , we have It is also interesting to note that the L M2n algebra can be used to construct different even-dimensional Born-Infeld type lagrangians. For example, if we consider a reduced S These considerations allow the construction of gravitational theories in every even-dimension. However, as discussed below, only in some dimensions it is possible to obtain General Relativity as a weak coupling constant limit of a Born-Infeld theory.

Born-Infeld Lagrangian in D = 6 invariant under L M4
The Born-Infeld lagrangian invariant under Lorentz algebra is given by (125) Following the definitions of Ref. [13], let us consider the S-expansion of the Lie algebra SO (5, 1) using S After performing its 0 S -reduction, one finds the L M4 algebra which corresponds to a subalgebra of M 4 algebra. The new algebra is generated by {J ab , Z ab }, where these new generators can be written as In this case,J ab corresponds to the original generator of SO (5, 1) and the λ α belong to a finite abelian semigroup S 0 . Using the invariant tensors we find that the six-dimensional Born-Infeld lagrangian invariant under L M4 algebra is given by Note that in this case the S-expansion procedure caused the Einstein-Hilbert term disappeared. This means that in the case of a six-dimensional Born-Infeld Lagrangian invariant under L M4 does not lead to General Relativity in no limit.

Born-Infeld Lagrangian in D = 6 invariant under L M6 algebra
In this case the 2-form curvature is given by [11] ab .
Using the invariant tensors we find that the six-dimensional Born-Infeld lagrangian invariant under L M6 algebra is given by The generators of the algebra L M2n satisfy the following commutation relations Theorem VII.2 of Ref. [10] allows us to see that the only nonzero components of the tensor invariant are given by where j = 0, · · · , 2n − 2 and α j are arbitrary independent constants of dimensions [length] 2−2n .
which is proportional to the Einstein-Hilbert lagrangian.
The results show that the 2p-dimensional Born-Infeld actions invariant under the algebra L M2m does not always lead to the action of General Relativity. Indeed, for certain values of m is impossible to obtain the Einstein-Hilbert term in the 2p-dimensional Born-Infeld type Lagrangian invariant under L M2m . This is because to obtain the term Hilbert-Einstein, is necessary the presence of the J a1a2 Z a3a4 · · · Z a2p−1a2p component of the invariant tensor, which is given by if m < p. (144) It is interesting to note that for each dimension D of space-time, we have the Lagrangian L BI (D) invariant under the L M2n algebra contains all other D-dimensional Lagrangian evaluated in an L M2m algebra with m < n. So it is always possible to obtain an action of a lower algebra off the appropriate fields.
It is also of interest to note that it was found that, analogously to what happens in the case of three-dimensiona Chern-Simons gravity, in four dimensions is not necessary to take the limit l = 0 to result in General Relativity.

Comments and Possible Developments
In the present work we have shown that: (i) standard odd-dimensional General Relativity (without a cosmological constant) emerges as a weak coupling constant limit of a (2p + 1)-dimensional Chern-Simons Lagrangian invariant under M 2m+1 algebra, if and only if m ≥ p.
(ii) when m < p, is impossible to obtain odd-dimensional General Relativity from a (2p + 1)-dimensional Chern-Simons Lagrangian invariant under the M 2m+1 algebra.
(iii) standard even-dimensional General Relativity (without a cosmological constant) emerges as a weak coupling constant limit of a (2p)-dimensional Born-Infeld type Lagrangian invariant under L M2m algebra, if and only if m ≥ p.
(iv) when m < p, is impossible to obtain even-dimensional General Relativity from a (2p)-dimensional Born-Infeld type Lagrangian invariant under L M2m algebra.
The toy model and procedure considered here could play an important role in the context of supergravity in higher dimensions. In fact, it seems likely that it is possible to recover the standard odd and even-dimensional supergravity from a Chern-Simons and Born-Infeld gravity theories, in a way very similar to the one shown here. In this way, the procedure sketched here could provide us with valuable information of what the underlying geometric structure of Supergravity could be (work in progress).