Lovelock gravities from Born-Infeld gravity theory

We present a Born-Infeld gravity theory based on generalizations of Maxwell symmetries denoted as $\mathfrak{C}_{m}$. We analyze different configuration limits allowing to recover diverse Lovelock gravity actions in six dimensions. Further, the generalization to higher even dimensions is also considered.


Introduction
It is a common assumption in theoretical physics that the spacetime may have more than four dimensions. This requires a generalization of General Relativity (GR) theory of gravity that includes general covariance and second order field equations for the metric. Although the Einstein-Hilbert (EH) action can be generalized to higher dimensions, the most general metric theory of gravity, satisfying the criteria of general covariance and giving second order field equations, is given by the Lanczos-Lovelock theory (LL) [1,2]. The LL action is constructed as a polynomial of degree [D/2] in the Riemann curvature tensor R α βµν , with and where α k are arbitrary constants and δ µ 1 ...µ 2k ν 1 ...ν 2k is the generalized Kronecker delta. Although the EH action is contained in the LL action, the action with higher powers of the curvature in D > 4 are dynamically different from GR and are not perturbatively related.
Using first order formulation, where the affin connection Γ λ µν is suppose to be independent from the metric g µν , the LL theory acquires, in general, torsional degrees of freedom. This can be easily seen using Riemann-Cartan formulation of gravity in terms of the vielbein and spin connection one-forms e a ,ω ab , where a, b = 0, 1, . . . , D are the local Lorentz indices. In that case the LL action can be regarded as the most general D-form invariant under local Lorentz transformations, constructed out of the vielbein e a , the spin connection ω ab and their exterior derivatives without using the Hodge dual [3,4], where α k are arbitrary constants which are not fixed from first principles and L (k) = ǫ a 1 ···a D R a 1 a 2 · · · R a 2k−1 a 2k e a 2k+1 · · · e a D .
and where the Riemann curvature and torsion 2-forms are defined as R ab = dω ab + ω a c ω cb and T a = de a + ω a b e b , respectively. Then it is direct to note that a torsional dynamical field equation will arise for k ≥ 2.
It is worth to notice that the relation between the Riemann-Cartan action (3) with the tensorial first order formalism, this latter formulated in terms of the metric and affin connection g µν , Γ λ µν , is given through g µν = η ab e a µ e b ν and Γ λ µν = ω ab ν e λ a e aµ + e λ a ∂ ν e a µ . The last expression which is related with the metricity condition ∇ Γ g µν = 0, assures that the Riemann curvature and torsion expressed as 2-forms and tensors are essentially the same objects in both languages, i.e., R ab µν = e a α e b β R αβ µν , T a µν = e a α T α µν .
In the present work we will deal with the language of differential forms, since it makes calculation much more compact and because it is more suitable to describe the gauge structure that the theory possesses.
As shown in Ref. [5], requiring the theory to have the maximum possible number of degrees of freedom fixes, in the first order formalism, the α k constants in (3). In odd dimensions, the Lagrangian becomes a Chern-Simons (CS) form [6,7], which is a functional L CS [A] of a gauge connection one-form A containing the vielbein and spin connection. The corresponding CS action is invariant, up to a boundary term, under a bigger symmetry (dS, AdS or Poincaré groups). In even dimensions the same requirement leads to a Born-Infeld (BI) action which is also constructed in terms of the curvature associated with the gauge connection, but it is locally invariant only under the Lorentz subgroup.
Another interesting family, called Pure Lovelock (PL) gravity, has been recently proposed in [8,9,10] as another way of fixing the α's. It consists of only two terms, the cosmological one and a single p-power in the curvature, with p = 1, . . . , N = [(D − 1) /2]. Remarkably, their black holes solutions behave asymptotically like the ones in GR and like the dimensionally continued black holes [11,12] near the horizon.
In Ref. [4] it was suggested that the metric LL theory should have the same degrees of freedom (dof) as the higher-dimensional EH gravity, i.e., D (D − 3) /2. However, the non-linearity of the theory makes the symplectic matrix to change the rank with the backgrounds [13] generating extra local symmetries and decreasing degrees of freedom in some of them. This behavior, typical for LL theories, has also been found in Lovelock-Chern-Simons gravities [14,15] and recently in PL gravity [16] where the number of dof changes with the backgrounds between 0 and D (D − 3) /2. Besides, this property is not only intrinsic of the Riemann sector, but also happens when torsional degrees of freedom are considered. This is the case when one looks for charged black hole solutions in CS supergravity [17].
On the other hand, the supersymmetric version of the LL theory is not known in general, except for few cases such as the EH and CS ones. The existence of a supersymmetric version in D = 5 for non vanishing constants α 1 and α 2 is discussed in Ref. [18], even though its explicit form is still unknown. It has also been suggested in [19] that a supersymmetric version of PL theory might be constructed using new symmetries obtained through expansion methods of Lie algebras [20,21,22]. Indeed, those methods have already been used to relate diverse gravity theories. For example, it has been found that even and odd-dimensional GR can be obtained as a special limit of BI and CS theories, constructed with expansions of the so (D − 1, 2) algebra [23,24,25,26,27].
Recently, in Ref. [19], it has been shown that the PL action in odd dimensions can be obtained as a limit from a CS action based on a special expansion of the so (D − 1, 2) algebra, denoted by C m . Those symmetries were introduced in Refs.c [28,29,30,31,32,33,34] and can be regarded as generalizations of the so called Maxwell algebra [35,36], which describes the symmetries of quantum fields in Minkowski space with the presence of a constant electromagnetic field. Thus, for completeness and also due to the growing interest in the effect of higher-curvature terms in the holographic context (see for example [37,38,39,40]), in this work we will show that different Lovelock gravity actions in even dimensions can be obtained from a BI action based on the C m algebra. We shall start by considering the six-dimensional spacetime since it allows us to obtain a bigger variety of gravity theories. Indeed, by applying in four dimensions the prescription presented here would lead only to the Einstein gravity with cosmological constant term.
The present work is organized as follows: in Section 2 we briefly review the BI gravity theory. Section 3 and 4 contain our main results. We present the explicit expression for the six-dimensional BI type gravity action based on the C 7 two-form curvature. The general setup in order to derive different Lovelock gravity action in a particular limit is given. We conclude our work by providing the generalization to higher even dimensions.
2 Brief review about Born-Infeld gravity theory As was previously pointed out the Lanczos-Lovelock theory refers to a family parametrized by a set of real coefficients α k , which are not fixed from first principles. To require the theory possess the largest possible number of degrees of freedom, fixes the α k parameters in terms of the gravitational and the cosmological constants [5]. As a result, in even dimensions the action has a Born-Infeld form invariant only under local Lorentz rotations, in the same way as the EH action. In this section, we review the main aspects of the BI gravity theory. As was shown in Ref. [5], choosing the coefficients as with 0 ≤ k ≤ n and the LL Lagrangian leads to the so-called Lovelock-Born-Infeld (LBI) Lagrangian [41,42] in D = 2n whereR ab = R ab + 1 l 2 e a e b corresponds to the AdS curvature. Here R ab is the usual Lorentz curvature and l is a length parameter. Let us notice that the Lagrangian (7) is the Pfaffian of the 2-formR ab and can be rewritten as, which remind us the Born-Infeld electrodynamics Lagrangian. The BI gravity Lagrangian, which is basically contructed under the requirement of having a unique maximally degenerate AdS vacuum, has the advantage of having well defined black holes configurations. The family of gravity actions constructed with the coefficients (6) up to a certain fixed k, with 0 ≤ k ≤ n, are characterized by the fact that they have a unique k-order degenerate AdS vacuum, with the BI Lagrangian case being described by k = n. As shown in [11,12], such family (which includes EH and EGB gravities for k = 1, 2) is free of degeneracies in the static sherically symmetric sector of the space of solutions, i.e., they have well defined black holes configurations. This not occurs for the Lovelock theory which, for arbitrary α k constants, the field equations do not determine completely the components of the curvature and torsion in the static sector. Besides, the BI gravity Lagrangian possess a large number of appealing features like cosmological models, black hole solutions, etc.
It is important to note that the Lagrangian (7) is invariant only under local Lorentz transformations and not under the AdS group. In this way, in D = 2n the Levi-Civita symbol ǫ a 1 ···a 2n in (7) can be regarded as the only invariant tensor under the Lorentz group SO(2n − 1, 1). This choice of the invariant tensor, which is necessary in order to reproduce a non-trivial action principle, breaks the full AdS symmetry to their Lorentz subgroup. In fact, the BI gravity action can be written as follows Here F is the AdS 2-form curvature and T A 1 · · · T An is chosen as an invariant tensor for the Lorentz group only, Then the action (9) can be expressed as In D = 4 the BI action is written as a particular linear combination of the standard Einstein-Hilbert action with cosmological constant and the Euler density, In the same way, the Lovelock coefficients α 0 , α 1 , α 2 and α 3 are chosen so that the D = 6 BI gravity action is given by

D=6 Lovelock gravity actions from Born-Infeld type theory
In this section, we show the explicit construction of a BI type theory based on enlarged symmetries and its relation to different six-dimensional Lovelock gravity actions.

Why C m algebras?
Our objective requires to find a symmetry which allows to separate each term of the original Lovelock Lagrangian in different sectors. Thus, under a specific limit, the unwanted sector can be avoided leading to interesting gravity actions. To this purpose, every Lovelock term should be originated by different components of an invariant tensor leading to a Born-Infeld type action. These desired properties have origin in the so (D − 1, 2) ⊕ so (D − 1, 1) Lie algebra 1 [28,29,30,31] which has been generalized, using the abelian semigroup expansion method (S-expansion) [21], to a family of Maxwell type algebras denoted as C m [32,33]. Their supersymmetric extensions have also been constructed in Refs. [43,44].
As was shown in Ref. [19] the C m algebras are obtained from AdS considering S (m−2) M = {λ 0 , λ 1 , . . . , λ m−2 } as the relevant semigroup, whose multiplication law is given by After extracting a resonant subalgebra, one finds the C m algebra whose generators satisfy the following commutation relations 1 Also known as AdS-Lorentz algebra.
The new generators J ab,(i) , P a,(k) are related to the so (5, 2) ones J ab ,P a through J ab,(i) = λ 2i ⊗J ab , P a,(k) = λ 2k+1 ⊗P a , with i = 0, 1, . . . , m−2 2 and k = 0, 1, . . . , m−3 2 . The two-form curvature F = dA + A ∧ A for the C m algebra is given by where Let us note that ω ab,(0) and e a,(0) correspond to the spin connection ω ab and the vielbein e a , respectively.
In order to build a six-dimensional BI type gravity action based on the C m two-form curvature, we require the explicit expression of the invariant tensor. Fortunately, the S-expansion method offers the possibility of deriving the invariant tensor for the expanded algebra from the original one. Indeed, following the Theorem VII.2 of Ref. [21], the non-vanishing components of an invariant tensor are given by where σ 2j are arbitrary constants. As in the original BI case, this choice of the invariant tensor breaks the C m group to a Lorentz type subgroup generated by J ab,(i) . Since the original Lovelock terms will arise in the BI type action through R ab,(0) and R ab, (1) , it is straightforward to know from (17) in which sector every Lovelock term will appear. The following table clarifies this point: As RRR = ǫ abcdef R ab R cd R ef corresponds to a boundary term, the minimal algebra which allows to separate each Lovelock term in different sectors of the BI type action corresponds to the C 7 algebra.

D=6 Lovelock gravity actions and C 7 algebra
Considering the C 7 algebra, we present different limits and conditions on the σ's leading to various Lovelock gravity actions in D = 6.
Let us first consider the C 7 -valuated connection one-form and the associated curvature two-form where Here D = d + ω denotes the Lorentz covariant exterior derivative and R ab is the usual Lorentz curvature. Following eq. (17), it is possible to show that the only non-vanishing components of an invariant tensor required to build a BI gravity action, are given by where σ 0 , σ 2 and σ 4 are dimensionless arbitrary constants. As was previously mentioned, this choice of invariant tensor breaks the C 7 algebra to its Lorentz type subalgebra L C 7 generated by J ab , Z ab ,Z ab . Then, considering the curvature two-form (19) and the non-vanishing components of the invariant tensor (20) in the general expression of a six-dimensional Born-Infeld type gravity action Separating the purely gravitational terms (ω, e) from those containing extra fields k,k, h,h the action (21) can be rewritten as Omitting the boundary term, each term of the Lovelock series appears in a different sector of the BI type gravity action. The same feature occurs in the seven-dimensional Chern-Simons case using the C 7 algebra [19]. Nevertheless, unlike the odd-dimensional case, the action is not gauge invariant of the entire gauge group but only under a Lorentz type subgroup. Let us note that the action (22) contains vielbein type fields h,h leading to products of different vielbeins. This action has an analogous structure to dRGT (de Rham, Gabadadze, Tolley) massive gravity theories in the vielbein formalism [45,46] suggesting a tri-metric theory in six dimensions, each of them with an EH type term. A smaller algebra (C 5 ) would lead only to two kind of vielbien (e, h) leading to Lagrangian terms analogous to the bi-gravity formalism [47,48]. It would be interesting to investigate under what conditions they might be connected to our results. However, important differences should be pointed out. Indeed such theories are in the second order formalism, whilst our description is a first order description. Additionally, we have introduced h's only as extra fields and it is not the original purpose to interpret them as new vielbeins to describe a n-gravity theory. Besides, since our purpose is to analyze the limits where those extra fields vanish, we postpone that discussion for future work.
Interestingly diverse gravity actions can be obtained following appropriate conditions on the σ's and applying suitable limits on the extra fields.

Pure Lovelock gravity action
The p = 1 PL action, which trivially corresponds to EH + Λ, emerges from the BI type gravity action (22) imposing σ 2 = 0, σ 0 = σ 4 and considering a matter-free configuration k =k = h =h = 0 where we have omitted the boundary term ǫ abcdef R ab R cd R ef . This feature is well desired since if BI type gravity theories are the appropriate theories to describe gravity then such theories should satisfy the correspondence principle. This property will be our principal requirement in a possible supersymmetric extension of PL theory. Not only the lowest order PL action can be recovered from the BI type action but also the maximal one (p = 2). Indeed, when the σ 4 constant vanishes and σ 0 = −σ 2 , the matter-free configuration limit k =k = h =h = 0 leads to As in the CS case, although we obtain the PL action, the dynamical limit requires a more subtle treatment. A detailed discussion about the dynamical issue in odd dimensions has been done in Ref. [19].

EH + LL gravity action
Another non-trivial election of the σ's leads to an alternative Lovelock gravity action. The EH + LL action, where LL is an arbitrary Lanczos-Lovelock term [18], can be derived from the BI type action (22) imposing σ 0 = 0, σ 2 = σ 4 , and considering a matter-free configuration limit k =k = h =h = 0 : Thus, the action contains the EH term and the Gauss-Bonnet (GB) term. Naturally, in six dimensions this is the only possibility to relate the Einstein-Hilbert term with another higher power in the curvature. Nevertheless, as we will see in the 2n-dimensional case, the presence of higher-curvature terms in the BI type action will allow to equip the EH term with an arbitrary p-order LL term (p = 0).
It is important to point out that the supersymmetric extension of the EH + LL gravity remains unsolved. Although some discussions can be found in Ref. [18], the explicit form of the supergravity action is still a mystery. The procedure presented here could be generalized to supergravity allowing to find a wider class of supergravity theories.
On the other hand, as it was mentioned in Ref. [49], the causality is violated for quadratic gravity theories when the GB coupling is finite. One could argue that considering other spin-2 fields could fix the causality violation, however this would lead to restrictive field equations where not even pp-wave could satisfy. Indeed, the situation is quite different to the one presented in Ref. [50] where no copy of the EH terms appears. In our present case, the C m symmetries imply to copy every original Lovelock terms avoiding the possibility to find Gauss-Bonnet equations when h orh is identified as the true vielbein. This does not occur in the PL case where a non-trivial identification of the extra fields allows to reproduce the appropriate dynamics [19].

Lovelock gravity with generalized cosmological constant
A six-dimensional gravity action in presence of a generalized cosmological constant term [32,51] can be recovered in a particular limit. Indeed, whenk ab =h a = h a = 0, the BI type action reduces where we have omitted the boundary terms and where we have set σ 0 = σ 2 = σ 4 . The LΛ term includes the cosmological constant term plus additional pieces depending on the extra field k ab . This can be seen as a generalization of the result obtained for the Maxwell symmetry [51] to six dimensions. Interestingly, in this limit, the action corresponds to the six-dimensional BI type constructed out of the curvature two-form for the so (D − 1, 2) ⊕ so (D − 1, 1) algebra.
Let us observe that the same result can be obtained imposing k ab = 0 instead ofk ab = 0. Obviously, the additional terms appearing in the action would depend in such case onk ab . The presence of the extra fields k ab ork ab leads to an alternative extension of standard gravity allowing the introduction of a generalized cosmological term. At the supersymmetric level, an analogous result has been presented in Ref. [44] using a supersymmetric extension of the so (D − 1, 2) ⊕ so (D − 1, 1) algebra.

Extension to D = 2n gravities
In this section we present the 2n-dimensional BI type gravity action based on the C 2n+1 algebra. Additionally, we provide with the suitable limits on the extra fields and the general conditions on the σ's necessary to recover diverse gravity actions.
Following the same discussion of the previous section, one can see that the minimal symmetry allowing to separate each Lovelock term in different sectors corresponds to C 2n+1 . From Theorem VII.2 of Ref. [21], the non-vanishing components of an invariant tensor necessary to build a 2ndimensional BI type gravity action based on the C 2n+1 curvature two-form are given by J a 1 a 2 ,(i 1 ) · · · J a 2n−1 a 2n ,(in) = 2 n−1 n α 2j δ j (i 1 +···+in) mod[n] ǫ a 1 ···a 2n , with i = 0, 1, . . . , 2n−1 2 . This invariant tensor breaks the C 2n+1 symmetry to a Lorentz type subgroup L C 2n+1 generated by J ab,(i) . It is important to clarify that an invariant tensor of the whole C 2n+1 group would lead to a topological invariant.

Summary and discussions
In the present work, we have shown that deforming the BI gravity theory (based on the AdS curvature) using the abelian semigroup expansion allows to reproduce diverse Lovelock gravity theories considering appropriate matter-free configuration limit and imposing pertinent conditions on the σ constants.
The relations among the expanded BI gravity theories and the Lovelock one is motivated by the recent connection between even-dimensional standard GR and expanded BI gravity theories [24,25] where the properties of the S-expansion were used. Diverse results have been obtained, not only in the BI context, using the Lie Algebra expansion method. In fact, it was shown in Ref. [50], that EH Lagrangian can be derived from an expanded AdS CS gravity theory. Additionally the proper Einstein dynamics can also be obtained in an appropriate coupling constant limit using the S-expansion procedure [23].
The results obtained along this paper not only present some new explicit relations among BI and Lovelock theories but also could bring valuable information in order to derive other (super)gravities. Indeed, various supersymmetric extension of (pure)Lovelock gravity theories has not been explored yet and their standard constructions remain to be a highly difficult task. However, our results suggest that the super (pure)Lovelock could be related to supergravity theory with expanded Lie algebras. Besides, the same procedure could be applied to CS supergravity theories.