Euler Chern Simons Gravity from Lovelock Born Infeld Gravity

In the context of a gauge theoretical formulation, higher dimensional gravity invariant under the AdS group is dimensionally reduced to Euler-Chern-Simons gravity. The dimensional reduction procedure of Grignani-Nardelli [Phys. Lett. B 300, 38 (1993)] is generalized so as to permit reducing D-dimensional Lanczos Lovelock gravity to d=D-1 dimensions.


I. INTRODUCTION
Odd-dimensional gravity may be cast as a gauge theory for the (A)dS groups [1]. The lagrangian is the Euler-Chern-Simons form in D = 2n − 1 dimensions [2,3] p , whose exterior derivative is Euler's topological invariant in 2n dimensions. The constants κ and l are related to Newton's constant G and to the cosmological constant Λ through 1/κ = 2 (D − 2)!Ω D−2 G (where Ω D−2 is the area of the (D − 2) unit sphere) and Λ = ± (D − 1) (D − 2) /2l 2 . The Chern-Simons lagrangian remains invariant under local Lorentz rotations in tangent space δe A = λ A B e B , δω AB = −Dλ AB , and changes by a total derivative under an infinitesimal (A)dS boost δe A = −Dλ A , δω AB = 1 ℓ 2 λ A e B − λ B e A . With appropriate boundary conditions, this means that the action is left invariant by the (A)dS gauge transformations. Furthermore, the vielbein and the spin connection correspond to the gauge fields associated with (A)dS boosts and Lorentz rotations, respectively. Thus, odd-dimensional Chern-Simons gravity is a good gauge theory for the (A)dS group, but its usefulness is limited to odd dimensions. This is related to the fact that no topological invariants have been found in even dimensions, and therefore the derivative of an evendimensional lagrangian cannot be made equal to any of them [3]. * Electronic address: fizaurie@udec.cl † Electronic address: edurodriguez@udec.cl ‡ Electronic address: pasalgad@udec.cl The requirement that the equations of motion fully determine the dynamics for as many components of the independent fields as possible may also be used in the even-dimensional case, leading to the so-called Lovelock-Born-Infeld action [2,3] whereR AB ≡ R AB + 1 l 2 e A e B and l is a length. For D = 2n = 4, (1) reduces to the EH action with a cosmological constant Λ = ±3/l 2 plus Euler's topological invariant with a fixed weight factor.
When one considers the spin connection ω AB and the vielbein e A as components of a connection for the (A)dS group, one finds that the action (1) where λ A is the infinitesimal parameter of the transformation. It simply makes no sense to use the equations of motion associated with the action (1) to enforce the invariance; any action is on-shell invariant under any infinitesimal transformation, just by the definition of the equations of motion. On the other hand, one could try to set the torsion equal to zero by fiat, and impose T A = 0 as an off-shell identity. This is unsatisfactory in the sense that, in the (A)dS gauge picture, torsion and curvature stand on a similar footing as fields strengths of the connection whose components are e A and ω AB . It would seem rather odd to have some components of the field strength set arbitrarily to zero while the others remain untouched. The gauge interpretation of even-dimensional Lovelock-Born-Infeld gravity is thus spoiled by the lack of invariance of the action (1) under infinitesimal (A)dS boosts.
A truly (A)dS-invariant action for even as well as for odd dimensions was constructed in ref. [4] using the Stelle-West formalism [5] for non-linear gauge theories. The action is Here ζ A corresponds to the so-called AdS coordinate, which parametrizes the coset space SO (D + 1) /SO (D), and z = ζ/l. The method devised in ref. [4] allows for the evendimensional action to become (A)dS gauge invariant. The same construction can be applied in odd dimensions, where its only outcome is the addition of a boundary term to the Chern-Simons action.
(2n − 1)-dimensional gravity has attracted a growing attention in recent years, both as a good theoretical laboratory for a possible quantum theory of gravity and as a limit of the so called M theory. In this context it is then interesting to establish a clear link between D = 2n y D = 2n − 1 gravities by a dimensional reduction. This is the aim of the present letter and it is achieved in the framework of a gauge theoretical formulation of both theories. In fact, as is shown in [3,4], gravity in 2n − 1 and 2n dimensions can be formulated as a gauge theory of the AdS group. In 2n − 1 dimensions this formulation is especially attractive as the Lanczos-Lovelock action becomes the Chern-Simons term of the AdS group. Such a Chern-Simons action with the correct AdS gauge transformations can then be derived by dimensionally reducing the 2n dimensional Lanczos-Lovelock action in its gauge theoretical formulation.
In [6] was proved, in the context of a Poincaré gauge theoretical formulation, that pure gravity in 3 + 1dimensions can be dimensionally reduced to gravity in 2 + 1 dimensions. However, the mechanism of Grignani-Nardelli is not applicable in the context of an AdS gauge theoretical formulation. One of the goals of this paper is to find a generalization of the procedure of Grignani-Nardelli that permits, in the context of an AdS gauge theoretical formulation, to reduce D-dimensional LL gravity to d ≡ D − 1 dimensions.

II. GRIGNANI NARDELLI PROCEDURE AND ADS INVARIANCE
Latin letters from the beginning of the alphabet will be used for tangent space indices; A, B, C = 1, 2, . . . , D, and a, b, c = 1, 2, . . . , d. Greek letters and latin letters from the middle of the alphabet will denote space-time indices; λ, µ, ν = 1, 2, . . . , D, and i, j, k = 1, 2, . . . , d. Fields belonging to d = D − 1 dimensions will be distinguished by underlining, as in ω ab . Exterior derivatives will be denoted as d = dx µ ∂ µ and d = dx i ∂ i . First, we consider the dimensional reduction from (3 + 1) to (2 + 1) dimensions. The Lovelock Born Infeld lagrangian in D = 4 is given by where is the curvature tensor. This lagrangian can be written in the form where we have used ε abc4 = ε abc . Method I: The table A of [6] can be written as:

This means
By substituting these results in L BI one gets which is very different of the Chern-Simons Lagrangian in 2 + 1 dimensions. Method II: The table B of [6] can be written as This means with z = mγ. By substituting these results in L BI one gets which it is again different from the Chern-Simons Lagrangian in 2+1 dimensions. Similar results are obtained for higher dimensions.

III. GENERALIZATION
We now consider a generalization of the mechanism of Grignani Nardelli. The basic idea is the following. Let where Consider a surface σ (x) = const and a vector field n not lying on the surface; i.e., I n (dσ) = 0, where I ξ is the contraction operator. It will prove convenient to normalize the vector n to make it fulfill the condition I n (dσ) = 1.
A p-form field ψ living in D dimensions can always be decomposed according to where we have definedψ ≡ dσI n ψ,ψ ≡ I n (dσψ) . The two fieldsψ andψ together carry the same information as the original field ψ, as can be seen from (5) The action is written as where M d is a d-dimensional manifold that belongs to the equivalence class of manifolds induced by σ (an example is shown below). It is perhaps interesting to note that so far there is no need to assume that the integrated direction has any especial feature such as being compact or extremely curved; the reduction procedure remains well-defined whether we make these assumptions or not. Now we consider the simple case σ (x) = x D ; that is, we deal with slices of constant x D across D-dimensional space-time. A natural choice for n is thus n = ∂ D ≡ ∂/∂x D , which satisfies I n dσ = I ∂D dx D = 1. With these choices, the d-form lagrangian L d may be written as and M d is simply any x D = const manifold. For all of them, the integral has the same value. However, the relabeling of the fields that remains to be done in (9) may be more natural on one specific surface. There is much freedom in the way this field-relabeling process is performed, as the only strong constraints come from symmetries. In general, the original fields in L D transform locally under a group G over M D , leaving L D invariant. On the other hand, the relabeled fields that enter L d must transform locally under a group G ′ over M d , leaving L d invariant. However, the lagrangian L d is still invariant under the G group, which gets realized now in a different way. Thus, the relabeling process must be carried out in such a way that this requirement is satisfied. A good example is provided by the spin connection ω AB . Under a local, infinitesimal Lorentz transformation Λ = 1 + 1 2 λ AB J AB defined over M D , it changes by δω AB = −Dλ AB . Our question now is, what components of this SO (D) connection may be relabeled as the SO (d) connection ω ab ? To find the answer, perform an M dlocal SO (d) transformation on ω AB , i.e., demand that the SO (D) infinitesimal parameters λ AB satisfy the conditions ∂ D λ AB = 0, λ A,D = 0 . These conditions turn the remaining λ ab into the right parameters for a SO (d) infinitesimal transformation. It is straightforward to show that, when this is the case, we have δω ab = −Ďλ ab , δω ab = λ a cω cb + λ b cω ac , whereĎ is the exterior covariant derivative in the connectionω ab . These last eqs. express thatω ab transforms as a SO (d)-connection, whileω ab behaves as a SO (d)-tensor. Therefore, an identification such asω ab → ω ab + [SO (d) -tensor] ab (10) seems quite reasonable. In general, one simple way to respect the relevant symmetries is to identify the components of a D-dimensional field with those of the corresponding d-dimensional one.
Here we shall perform this kind of identifications in their simplest possible form. Many components of the fields will be frozen to zero; this corresponds to our early assertion that we are only interested (by now) in showing the possibility of getting d-dimensional gravity from its higher-dimensional version. This is no longer true, of course, when we face the full dimensional reduction procedure, for in this case freezing some components of the fields yields a reduced gauge group as well.

IV. THE LANCZOS LOVELOCK ACTION
We shall start our dimensional reduction process with the D-dimensional LL action A more suitable version of action (11) is The first term in (12) shows that it is always possible to obtain a LL action in d dimensions starting from a LL action defined in D = d + 1.
The coefficients α p in (11) are selected according to the criterion that the equations of motion fully determine the dynamics for as many components of the independent fields as possible. This analysis leads to [4] A well-defined dynamics in D dimensions leads to a well-defined dynamics in d dimensions; however, we shall additionally demand that the purely gravitational terms in the reduced action produce well-defined dynamics as well. This means that the coefficients α p must be reduced accordingly; that is, the coefficients in the reduced gravitational lagrangian must correspond to α (d) p as given in (13), with D → d.
We consider the dimensional reduction from D =even to d = D − 1 =odd. First we note that the action (11) includes Euler's topological invariant for p = D/2, and we may write it as This action is decomposed as In this case the relation between the α p coefficients in D and d dimensions is given by [cf. eq. (13)] Plugging (15) into (14), we are led to where Now we show that it is possible to obtain an action for d = D − 1 dimensional gravity from the corresponding action in D dimensions. Any identification in the spirit of (10) will do the job; for example, Strictly speaking, one must perform a series expansion of the fields on the x D coordinate. The x D -independent term in this expansion leads, with the given identifications, to gravity in d = D − 1 dimensions. The last term, which does not contribute to the D-dimensional equations of motion, is the Chern-Simons action for ∂M D , i.e., The Chern-Simons term, S (G2) D , forces us to take as M d the boundary of M D , that is, M d = ∂M D . In this way the freedom we initially had to pick any manifold out of the equivalence class induced by σ is lost, and we are left with a precise choice for M d .
It is clear from the form of S (G1) D that, in order to obtain well-defined dynamics for the gravitational sector alone, one needs to integrate the fields over the x D coordinate before performing the identification process. This is due to the presence of the extra (d − 2p) factor, which precludes S (G1) D from leading to well-defined dynamics. Also, the zero mode hypothesis used in odd dimensions seems useless here, because we need some kind of dependence on the x D -coordinate to have a well-defined action for gravity in d dimensions. With this in mind, we shall take M d = ∂M D and parametrize the x D -coordinate in such a way that it ranges through −∞ < x D ≤ 0, with x D = 0 corresponding to the boundary of M D . We shall additionally assume that the vielbein may be written as where k is a real, positive constant with dimensions of [length] −1 and V a 0 and W ab are taken to be x D -independent. Clearly, V a 0 corresponds to V a 0 = V a x D = 0 .
With these assumptions, the action (17) takes the form Integration over x D from x D = −∞ to x D = 0 leads to The further addition of S where It is now clear that the identifications (20) lead to a well-defined action for gravity in d dimensions, since the coefficients in the action (22) correspond to those given in (13).