Generalized Poincare algebras and Lovelock-Cartan gravity theory

We show that the Lagrangian for Lovelock-Cartan gravity theory can be re-formulated as an action which leads to General Relativity in a certain limit. In odd dimensions the Lagrangian leads to a Chern-Simons theory invariant under the generalized Poincar\'{e} algebra $\mathfrak{B}_{2n+1},$ while in even dimensions the Lagrangian leads to a Born-Infeld theory invariant under a subalgebra of the $\mathfrak{B}_{2n+1}$ algebra. It is also shown that torsion may occur explicitly in the Lagrangian leading to new torsional Lagrangians, which are related to the Chern-Pontryagin character for the $B_{2n+1}$ group.


Introduction
The most general metric theory of gravity satisfying the criteria of general covariance and yielding to second-order field equations is a polynomial of degree [d/2] in the curvature known as the Lanczos-Lovelock gravity theory (LL) [1,2]. The LL action can be written as the most general d-form invariant under local Lorentz transformations, constructed with the spin connection, the vielbein and their exterior derivatives, without the Hodge dual [3,4], where R ab = dω ab + ω a c ω cb is the Lorentz curvature, e a corresponds to the oneform vielbein and the coefficientesα p , p = 0, 1, . . . , [d/2] , are arbitrary constants and they are not fixed from first principles.
It is an accepted fact that requiring the LL theory to have the maximum possible number of degrees of freedom, fixes the parametersα p 's in terms of the gravitational and the cosmological constants [5]. As a consequence, the action in odd dimensions can be formulated as a Chern-Simons (ChS) theory of the AdS group, while in even dimensions the action has a Born-Infeld (BI) form invariant only under local Lorentz rotations in the same way as the Einstein-Hilbert action [5,6,7,8].
Although the Einstein-Hilbert term is contained in the LL action, the ChS gravity for the AdS group and the BI gravity for the Lorentz group are dynamically very different from standard General Relativity.
In Ref. [9] it was shown that the standard, odd-dimensional General Relativity can be obtained from a Chern-Simons gravity theory for a certain B m Lie algebra, which will be called generalized Poincaré algebra 1 (where the particular case B 4 corresponds to the so-called Maxwell algebra [10]). The generalized Poincaré algebras can be obtained by a resonant reduced S-expansion of the AdS Lie algebra using S α=0 as semigroup [9]. The S-expansion method has been introduced in Ref. [13] (see also [14], [15], [16] ) and consists in a powerfull tool in order to obtain new Lie algebras from original ones. The method is based on combining the structure constants of a Lie algebra g with the inner multiplication law of a semigroup S. The new Lie algebra G = S × g is called the S-expanded algebra. Interestingly, when a decomposition of the semigroup S = p∈I S p (where I is a set of indices) satisfies the same structure that the subspaces V p of the original algebra g = p∈I V p , we say that G R = p∈I S p ×V p is a resonant subalgebra of G =S ×g.
In particular, when the semigroup has a zero element 0 S , the reduced algebra is obtained imposing 0 S × g = 0.
Subsequently, in Ref. [11] it was found that standard even-dimensional General Relativity emerges as a limit of a Born-Infeld theory invariant under a certain subalgebra L Bm of the B m Lie algebra. These odd-and even-dimensional theories are described by the so called Einstein-Chern-Simons (EChS) and the Einstein-Born-Infeld (EBI) actions, respectively.
Very recently it was found in Ref. [12] that standard odd and even-dimensional General Relativity emerges as a weak coupling constant limit of a (2p + 1)dimensional Chern-Simons Lagrangian and of a 2p-dimensional Born-Infeld Lagrangian invariant under B 2m+1 and L B2m , respectively, if and only if m ≥ p.
It is the purpose of this paper to show that: (i) it is possible to reformulate the Lagrangian for Lovelock-Cartan gravity theory, which we call "Lagrangian of Einstein-Lovelock-Cartan (ELC)", such that, in odd dimensions leads to the Einstein-Chern-Simons Lagrangian, and in even dimensions leads to the Einstein-Born-Infeld Lagrangian; (ii) the torsion may occur explicitly in the Lagrangian and that, following a procedure analogous to that of Ref. [5], it is possible to find new torsional Lagrangians, which are related to the Chern-Pontryagin character for the B 2n+1 group. This paper is organized as follows: In Section 2 we briefly review some aspects of the construction of the so called generalized Poincaré algebras and how it is possible to obtain General Relativity from the Chern-Simons and Born-Infeld formalism using these algebras.
In Section 3 the ELC-Lagrangian is constructed. It is shown that this 1 Alternatively known as the Maxwell algebra type.
Lagrangian leads in odd dimensions to the EChS Lagrangian and in even dimensions leads to the EBI Lagrangian.
In Section 5 we show that using the dual formulation of the S-expansion introduced in Ref. [17], it is possible to relate the Euler type invariant and the Pontryagin type invariant in d = 3 dimensions. Section 6 concludes the work with a comment and possible developments.
2 General Relativity and the generalized Poincaré algebras B 2n+1 In order to describe how the action for General Relativity can be obtained from the gravity actions invariant under generalized Poincaré algebras, let us review here the results obtained in Refs. [9,11,12]. Following the definitions of Ref. [13] let us consider the S-expansion of the Anti-de Sitter (AdS) Lie algebra using as a semigroup S (2n−1) E = {λ 0 , · · · , λ 2n } endowed with the multiplication law λ α λ β = λ α+β when α + β ≤ 2n; λ α λ β = λ 2n when α + β > 2n. TheJ ab , P a generators of the AdS algebra satisfy the following commutation relations where a, b = 0, . . . , 2n and η ab corresponds to the Minkowski metric. Let us consider the following subset decomposition S where λ 2n corresponds to the zero element of the semigroup (0 S = λ 2n ). After extracting a resonant subalgebra and performing its 0 S (= λ 2n )-reduction, one finds the generalized Poincaré algebra B 2n+1 , [J ab, where i, j = 1, · · · , n − 1. Let us note that the generators of the B 2n+1 algebra are related to the original ones through The generalized Poincaré algebra B 2n+1 is also known as the Maxwell algebra type which was introduced in Ref. [12]. We note that setting Z (i+1) ab and Z (i) a equal to zero, we obtain the B 4 algebra which coincides with the Maxwell algebra M [10]. In fact, every generalized Poincaré algebra B l can be obtained from B 2n+1 setting some generators equal to zero. Besides, one can see that the commutators (8), (12) and (13) form a Lorentz type subalgebra of the B 2n+1 algebra. This subalgebra denoted as L B2n+1 can be obtained as an S-expansion of the Lorentz algebra L using S (2n−1) 0 = {λ 0 , λ 2 , λ 4 , . . . , λ 2n } as the relevant semigroup [11]. The generalized Poincaré algebras are particularly interesting in the context of gravity since it was shown in [9] that standard odd-dimensional General Relativity may emerge as the weak coupling constant limit (l → 0) of a (2n + 1)dimensional Chern-Simons Lagrangian invariant under the B 2n+1 algebra, where and α j are arbitrary constants which appear as a consequence of the S-expansion process. Let us note that the S-expanded fields are related to the AdS fields ẽ a ,ω ab as follows, where j = 0, 1, . . . , n − 1. In a similar way, the S-expanded Lorentz curvature R (ab,2i) is related to the Lorentz curvatureR ab = dω ab +ω a cω cb as R (ab,2i) = λ 2iR ab . Similarly, it was shown in [11] that standard even-dimensional General Relativity emerges as the weak coupling constant limit (l → 0) of a (2n)-dimensional Born-Infeld type Lagrangian invariant under a subalgebra 2 L B2n of the B 2n+1 algebra, e (a 2k+2 ,q1) · · · e (a2n−1,p n−k ) e (a2n,q n−k ) .
These results have recently been generalized in Ref. [12] in which the autors have shown that L B2m+1 CS (2n+1) and L L B 2m BI (2n) lead to the Einstein-Hilbert Lagrangian in a weak coupling constant limit, if and only if m ≥ n.

The Einstein-Lovelock-Cartan Lagrangian
We have seen that the S-expansion procedure allows the construction of Chern-Simons gravities in odd dimensions invariant under the B 2n+1 algebra and Born-Infeld type gravities in even dimensions invariant under the L B2n+1 algebra, leading to General Relativity in a certain limit. These gravities are called the Einstein-Chern-Simons theories [9] and the Einstein-Born-Infeld theories [11], respectively. These findings show that it could be possible to reformulate the Lagrangian for Lovelock-Cartan gravity theory such that, in a certain limit, it leads to the General Relativity theory.
In this section we show that it is possible to write a Lovelock-Cartan Lagrangian leading to the EChS Lagrangian in d = 2n − 1 invariant under the B 2n−1 algebra, and to the EBI Lagrangian in d = 2n invariant under the L B2n algebra. For this purpose we shall use the useful properties of the S-expansion procedure using S (d−2) E as the relevant semigroup. The expanded action is given by where α p and µ i , with i = 0, ..., d − 2, are arbitrary constants and L (p,i) ELC is given by The expanded fields e (a,2i+1) , ω (ab,2i) are related to the AdS fields ẽ a ,ω ab as follows where , which is a semigroup that obey the following multiplication law (see Ref. [13]), Following the same procedure of Ref. [5], we consider the variation of the action with respect to e (a,i) and ω (ab,i) . The variation of the action (21) leads to the following equations: where and where T (a,i) = de (a,i) + η dc ω (ad,j) e (c,k) δ i j+k is the expanded 2-form torsion. Using the covariant exterior derivative D = d + [A, ·] ( where A corresponds to the one-form gauge connection B 2n−1 -valued) and the Bianchi identity for the expanded 2-form curvature DR (ab,ij ) = 0, we have Since one finds From (31) and (33) we have lf p ′ = p + 1 we find which can be rewritten as which by consistency with ε (i) a = 0 must also vanish. Taking the product of ε which vanishes by consistency with ε

Chern-Simons gravity invariant under B 2n−1
Following the same procedure of Ref. [5] one can see that in the odddimensional case Eqs. (36), (37), lead to the coefficients given by where α 0 and γ are related to the gravitational and the cosmological constants, For any dimension d, l is a length parameter related to the cosmological constant by and the gravitational constant G is related to κ through With these coefficients the Lagrangian (21) may be written as the Chern-Simons form Let us note that this Lagrangian can be expressed equivalently as follows 3 e (a 2k+2 ,q1) · · · e (a2n−3,p n−1−k ) e (a2n−2,q n−1−k ) e (a2n−1,in) , where and This is the Einstein-Chern-Simons Lagrangian [compare with eq. (19)] found in Ref. [9].

Born-Infeld gravity invariant under L B 2n
In the even-dimensional case, following the same procedure of Ref. [5] one can see that eqs. (36), (37), lead to the following coefficients With these coefficients the Lagrangian (21) is given by or equivalently 4 , which corresponds to the Einstein-Born-Infeld Lagrangian found in Ref. [11]. It is important to note that the coefficients α i = κµ i are arbitrary constants. In this way we have shown that the S-expansion procedure does not modify the α p 's coefficients defined in Ref. [5]. Unlike the Lanczos-Lovelock action, the expanded action (21) called the Einstein-Lovelock action, has the property of leading to General Relativity in a certain limit of the coupling constant l both even and odd dimensions.

Adding torsion in the Lagrangian
The Lagrangian (21) can be interpreted as the most general d-form invariant under a Lorentz type subalgebra L B2n of the generalized Poincaré algebra. This Lagrangian is constructed from the expanded vielbein and the expanded spin connection e (a,2k+1) , ω (ab,2k) (k = 0, . . . , n − 1) and their exterior derivatives 5 .
One can see from the variation of the EL Lagrangian that eq. (30) does not imply in d > 4 the vanishing of the expanded torsion T (a,2k+1) . The condition T (a,2k+1) = 0 implies that the expanded spin connection ω (ab,2k) have a dependence on the expanded vielbein e (a,2k+1) . Thus the expanded fields ω (ab,2k) and e (a,2k+1) cannot be identified as the components of a connection 4 As in the odd-dimensional case p = 0 does not contribute to the sum because δ i i 1 +···+i 2n = 0 for any value of i and n. 5 When k = 0, e (a,1) and ω (ab,0) are identified with the usual vielbein e a and the spin connection ω ab , respectively. for the generalized Poincaré algebra. Therefore, impose T (a,2k+1) = 0 seems to be restrictive and arbitrary. In this section, we study the possibility of adding terms which contain the expanded torsion to the ELC Lagrangian.
The Einstein-Lovelock-Cartan Lagrangian can be generalized adding torsion explicitly following a procedure analogous to that of the Refs. [5,20].
The only terms invariant under L B2n that can be constructed out of e (a,2k+1) , ω (ab,2k) and their exterior derivatives, are R (ab,2k) , T (a,2k+1) , and products of them. Then the invariant combinations that can occur in the Lagrangian are: where i = 0, . . . , 2n − 2. So that, the Lagrangian can be written as a linear combination of products of these basic invariant combinations. In a similar way to Ref. [20], we find that the Lagrangian has to be of the form where the µ, α and β are constants, L Thus, the inclusion of the expanded torsion leads to a number of arbitrary coefficients βj. Interestingly, as in the AdS symmetry case, it is possible to choose the β's in order to enlarge the Lorentz type L B symmetry to the generalized Poincaré gauge symmetry.
In even dimensions, the B 2n+1 -invariant d-forms are given by where . . . denotes a symmetric invariant tensor for the B 2n+1 algebra. Here, F = dA + AA is the 2-form curvature for the generalized Poincaré algebra and it is given by with The ω (ab,2k) and e (a,2k+1) are the different components of the 1-form connection A, where J (ab,2k) and P (a,2k+1) are the generators of the generalized Poincaré algebra B 2n+1 . Naturally, one of the invariants present in even dimensions is the Euler type invariant which is obtained from the following components of an invariant tensor, However, there are other components of the invariant tensor which lead to a different invariant known as the Pontryagin invariant which exists only in 4p dimensions. This invariant corresponds to the B 2n+1 -invariant d-form built from e (a,2k+1) , R (ab,2k) , T (a,2k+1) and can be expressed as the exterior derivative of a Chern-Simons (4p − 1)-form, This implies that in odd dimensions there are two families of Lagrangians invariant under the generalized Poincaré algebra B 2n+1 : • The Euler-Chern-Simons form L B2n+1 E (2p+1) , in D = 2p + 1. Its exterior derivative is the Euler density in 2p + 2 dimensions and does not involve torsion explicitly.
These results generalize those obtained in Ref. [5] to our case. The similitude is not a surprise since the B 2n+1 algebra corresponds to an expansion of the AdS algebra. Nevertheless, unlike the AdS-invariant gravity theory, the locally B 2n+1 -invariant gravity theory leads to General Relativity in the weak coupling constant limit (l → 0) (see Ref. [9,11,12]).
Interestingly, in 4p dimensions, both families exist which allows us to write the most general Lagrangian for gravity in d = 4p − 1 invariant under the generalized Poincaré algebra, namely where i = 1, 3, 5, . . . , 2n − 1 and j = 0, 2, 4, . . . , 2n − 2. The α's are arbitrary and are a consequence of the S-expansion procedure. In the next subsection, we explore an example in d = 3 which clarifies this point.

Example for d = 3
Let us consider a (2 + 1)-dimensional Lagrangian invariant under the B 5 algebra. This algebra can be obtained from the AdS algebra, using the Sexpansion procedure of Ref. [13].
After extracting a resonant subalgebra and performing a 0 S -reduction, one finds the B 5 algebra, whose generators satisfy the following commutation relations In order to write down a Chern-Simons Lagrangian for the B 5 algebra, we start from the B 5 -valued one-form gauge connection and the associated two-form curvature Using Theorem VII.2 of Ref. [13], it is possible to show that the only nonvanishing components of an invariant tensor for the B 5 algebra are given by where α 0 , α 1 , α 2 and α 3 are arbitrary constants.
Using these components of the invariant tensor in the general expression for the ChS Lagrangian L ChS = AdA + 2 3 A 3 , we find that the ChS Lagrangian invariant under the B 5 algebra is given by The exterior derivative of this Lagrangian leads us to the following associated invariant where in addition to an Euler type density we can see that appears the usual Pontryagin density P (4) = R a b R b a , the Nieh-Yan N (4) = 2 l 2 T a T a − e a e b R ab and a Pontryagin type density P 4 (k) = 2R a b D ω k b a coming from the new fields. Note that these densities P (4) , N (4) and P (4) (k) are combined in a Pontryagin type invariant for the B 5 group which is written as follows (choosing α 0 = α 2 ) where In the next section we show that the B 5 -invariant Lagrangian (76) can be obtained directly from the Lorentz-invariant Lagrangian.

Relation between the Pontryagin and Euler invariants
In this section we show that it is possible to relate the Lorentz invariant Lagrangian which depends only on the spin connection, and the Lagrangian obtained for the B 5 algebra . This means that by dual formulation of the Sexpansion [17] is possible to obtain both an Euler type invariant and a Pontryagin type invariant from the Pontryagin invariant. Consider first the Lorentz algebra L in (2 + 1)-dimensions, The one-form gauge connection A and the associated two-form curvature F are given by where R ab = dω ab + ω a c ω cb is the Lorentz curvature. The corresponding Chern-Simons Lagrangian invariant under the Lorentz algebra L is given by which can be written as where the invariant tensor · · · for the Lorentz algebra is Before starting the S-expansion of the Lorentz algebra is useful to define with Now let us consider the S E expansion of Lorentz algebra. The appropriate semigroup S (3) E = {λ 0 , λ 1 , λ 2 , λ 3 , λ 4 } is endowed with the following product: where λ 4 = 0 s is the zero element of the semigroup. In a similar way to Ref. [17], we define the spin connection and the 2-form curvature as where and Here we identify e a with the vielbein, R ab with the Lorentz curvature , T a with the torsion, and k ab and h a are identified as bosonic "matter" fields. Using Theorem VII.2 of Ref. [13], it is possible to show that the only non-vanishing components of an invariant tensor for the B 5 algebra are given by where α 0 , α 1 , α 2 and α 3 are arbitrary constants. Now if we use the components of the invariant tensor (94) − (99) in the general expression for a Chern-Simons Lagrangian we find the B 5 -invariant CS Lagrangian in (2 + 1) dimensions, The exterior derivative of this Lagrangian leads us to the following invariant polynomial, P B5 (4) = 1 l ǫ abc α 1 R ab T c + α 3 1 l 2 e a e b T c + R ab D ω h c + k c d e d + D ω k ab T c Thus we have shown that the S-expansion method allows us to relate the Pontryagin invariant of the Lorentz algebra with the invariants of the B 5 algebra studied in the previous section.
It is important to note that it is possible to generalize the previous result to the case of the B 2n+1 algebras. In fact, by considering the reduced S (2n−1) E expansion of the Lorentz algebra L and using the Theorem VII.2 of Ref. [13] we can find the non-vanishing components of an invariant tensor for the expanded algebra and thus build a (2 + 1)-dimensional Lagrangian invariant under B 2n+1 .

Comment and possible developments
In the present work we have shown that it is possible to construct an Einstein-Lovelock-Cartan Lagrangian that, in odd dimensions leads to the Einstein-Chern-Simons Lagrangian, and in even dimensions leads to the Einstein-Born-Infeld Lagrangian. The EChS and EBI theories are particularly interesting since it was shown in Refs. [9,11,12] that General Relativity can be obtained as a certain limit of these gravity theories. On the other hand we have shown that the Einstein-Lovelock-Cartan Lagrangian can be generalized adding torsional terms following a procedure analogous to that of Ref. [5]. Interestingly, the torsional terms appear explicitly in the Lagrangian only in 4p − 1 dimensions. Thus, the only 4p-forms invariant under the generalized Poincaré algebra B 2n+1 , constructed from e (a,2k+1) , R (ab,2k) and T (a,2k+1) (k = 0, · · · , n − 1), are the Pontryagin invariants P (4p) . Finally we have established a relation between the Pontryagin and the Euler invariants using the dual formulation of the Sexpansion method introduced in Ref. [17].
The 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 underlyng geometric structure of Supergravity could be (work in progress).