Poincare invariant gravity with local supersymmetry as a gauge theory for the M-algebra

Here we consider a gravitational action having local Poincare invariance which is given by the dimensional continuation of the Euler density in ten dimensions. It is shown that the local supersymmetric extension of this action requires the algebra to be the maximal extension of the N=1 super-Poincare algebra. The resulting action is shown to describe a gauge theory for the M-algebra, and is not the eleven-dimensional supergravity theory of Cremmer-Julia-Scherk. The theory admits a class of vacuum solutions of the form S^{10-d} x Y_{d+1}, where Y_{d+1} is a warped product of R with a d-dimensional spacetime. It is shown that a nontrivial propagator for the graviton exists only for d=4 and positive cosmological constant. Perturbations of the metric around this solution reproduce linearized General Relativity around four-dimensional de Sitter spacetime.


Introduction
A consensus has emerged in the high energy community that a consistent unified theory of all interactions and matter should be formulated in some dimension higher than four. Strong theoretical evidence, both in supergravity and in string theory, leads to conjecture the existence of an underlying fundamental theory in eleven dimensions [1], [2,3]. This is nowadays called M-Theory (see, e.g., [4]). The standard procedure to link the higher dimensional theory with four-dimensional physics has been either to compactify the extra dimensions by the Kaluza-Klein reduction (see, e.g., [5]), or through some more recent alternatives [6].
In these frameworks, however, the physical spacetime dimension is an input rather than a prediction of the theory. In fact, in standard theories whose gravitational sector is described by the Einstein-Hilbert action, there is no obstruction to perform dimensional reductions to spacetimes of dimensions d = 4. Then the question arises, since eleven-dimensional Minkowski space is a maximally (super)symmetric state, and the theory is well-behaved around it, why the theory does not select this configuration as the vacuum, but instead, it chooses a particular compactified space with less symmetry. An ideal situation, instead, would be that the elevendimensional theory dynamically predicted a low energy regime which could only be a fourdimensional effective theory. In such a scenario, a background solution with an effective spacetime dimension d > 4 should be expected to be a false vacuum where the propagators for the dynamical fields are ill-defined, lest a low energy effective theory could exist in dimensions higher than four.
In this paper, a new eleven-dimensional theory sharing some of these features is constructed. Indeed, for this theory, eleven-dimensional Minkowski spacetime is a maximally supersymmetric solution that would be a natural candidate for the vacuum. However, propagators around this background are ill-defined and hence it is a sort of false vacuum. On the other hand, the theory admits vacuum geometries of the form S 10−d × X d+1 , where X d+1 is a domain wall whose worldsheet is a d-dimensional constant curvature spacetime M d . These solutions exist only if M d has a non-negative cosmological constant, and the graviton can only propagate provided M d is a four-dimensional de-Sitter space. Moreover, the gravitational perturbations reproduce linearized General Relativity in four dimensions. Thus, the resulting four-dimensional effective theory is indistinguishable from gravity with positive cosmological constant in perturbation theory. Our motivation to choose eleven dimensions is to explore new geometrical and dynamical structures that are expected to exist in d = 11, and could be regarded as new "cusps" of M-theory (see e. g. [7]). The theory presented here is not equivalent to the Cremmer-Julia-Scherk supergravity in eleven dimensions [1].
The gravitational action we propose is selected by requiring local Poincaré invariance and is given by the dimensional continuation of the Euler density in ten dimensions. Its local supersymmetric extension requires the algebra to be the maximal extension of the N = 1 super-Poincaré algebra in eleven dimensions, commonly known as the M-algebra. This algebra is spanned by the set G A = {J ab , P a , Q α , Z ab , Z abcde }, where J ab and P a are the generators of the Poincaré group and Q α is a Majorana spinor supercharge with anticommutator [8] {Q α , Q β } = (CΓ a ) αβ P a + (CΓ ab ) αβ Z ab + (CΓ abcde ) αβ Z abcde . (1.1) The charge conjugation matrix C is antisymmetric, and the "central charges" Z ab and Z abcde are tensors under Lorentz rotations but otherwise Abelian generators 1 . As shown below, the algebra fixes the field content to include, apart from the graviton e a µ , the spin connection ω ab µ and the gravitino ψ µ , two one-form fields b ab µ , b abcde µ , which are rank two and five antisymmetric tensors under the Lorentz group, respectively. The local supersymmetry transformations close off-shell without requiring auxiliary fields. As will be seen below, the supersymmetric Lagrangian can be explicitly written as a Chern-Simons form. It is known that for Chern-Simons theories bosonic and fermionic degrees of freedom do not necessarily match, since there exists an alternative to the introduction of auxiliary fields (see e. g. [9]). Indeed, the matching may not occur when the dynamical fields are assumed to belong to a connection instead of a multiplet for the supergroup [10].

Gravitational sector
In dimensions higher than four, under the same assumptions of General Relativity in four dimensions (i.e., general covariance, second order field equations for the metric), the socalled Lovelock actions are obtained [11], which include the Einstein-Hilbert lagrangian as a particular case. In general, these lagrangians are linear combinations of the dimensional continuations of the Euler densities from all lower dimensions [12] and therefore contain higher powers of the curvature. Since the action can be expressed in terms of differential forms without using the Hodge dual, it is easy to see why these theories do not yield higher derivative field equations. In the first order formalism (analogous to Palatini's) the field equations can only involve first order derivatives of the dynamical fields (for more on this, see [13]). Furthermore, if one then imposes the torsion to vanish the field equations become at most second order. In the vanishing torsion sector, the theory has the same degrees of freedom as General Relativity [14].
Without imposing the torsion constraint, the field equations remain first order, even if one couples this theory to other p-form fields without involving the Hodge. In fact, in this way it is impossible to generate higher derivative terms in this theory.
An action containing (1.1) as a local symmetry must be, in particular, invariant under local translations, The only gravitational action in eleven dimensions constructed out of the vielbein e a and the spin connection ω ab , leading to second order field equations for the metric, invariant under diffeomorphisms and local Poincaré transformations is given by [15,13] I G [e, ω] = M 11 ǫ a 1 ···a 11 R a 1 a 2 · · · R a 9 a 10 e a 11 . (2.2) Here R ab = dω ab + ω a c ω cb is the curvature two-form, and wedge product between forms is understood [16]. For the reason given above, we take I G as the gravitational sector of our theory rather than the Einstein-Hilbert action which, is not invariant under (2.1) [17]. The Lagrangian in (2.2) is the ten-dimensional Euler density continued to eleven dimensions and contains the degrees of freedom of eleven dimensional gravity [14].
A local Poincaré transformation acting on the dynamical fields is a gauge transformation δ λ A = dλ+[A, λ], with parameter λ = λ a P a + 1 2 λ ab J ab , provided e a and ω ab are the components of a single connection for the Poincaré group, A = e a P a + 1 2 ω ab J ab . This observation will be the guiding principle for the construction of a locally supersymmetric extension of I G .

Supersymmetric extension
A natural way to construct a locally supersymmetric extension of (2.2) without breaking local Poincaré invariance is that the extra fields required by supersymmetry enter on a similar footing with the original fields. In other words, all dynamical fields will be assumed to belong to a connection for a supersymmetric extension of the Poincaré group. This approach strongly deviates from the standard assumption in supergravity, where the fields are assumed to belong to a multiplet. As we shall see now, the M-algebra emerges naturally from our approach. The simplest tentative option would be to consider the N = 1 super Poincaré algebra without central extensions. However, this possibility must be ruled out. Indeed, in this case, the connection would be extended by the addition of a gravitino as A → A+ψQ/ √ 2, and the gauge generator would change as λ → λ + ǫQ/ √ 2, where ǫ is a zero-form Majorana spinor. This would fix the supersymmetric transformations to be δe a =ǭΓ a ψ/2, δψ = Dǫ and δω ab = 0. Then the variation of (2.2) under supersymmetry can be cancelled by a kinetic term for the gravitino of the form where R abc := ǫ abca 1 ···a 8 R a 1 a 2 · · · R a 7 a 8 . However, the variation of I ψ produces, in turn, an extra piece which cannot be cancelled by a local Lagrangian for e a , ω ab , and ψ, and hence the super Poincaré algebra is not rich enough to ensure the off-shell supersymmetry of the action. Nevertheless, following the Noether procedure, it can be seen that supersymmetry can be achieved introducing additional bosonic fields. These fields can only be either a second-rank or a fifth-rank tensor one-forms b ab , and b abcde , that transform likeǭΓ ab ψ and ǫΓ abcde ψ, respectively. Assuming that the dynamical fields belong to a single connection for a supersymmetric extension of the Poincaré group, the only option that brings in these extra bosonic fields is to consider the M-algebra (1.1), which also prescribes their supersymmetry transformations in the expected form. This means that the field content is given by the components of a single fundamental field, the M-algebra connection,

2)
and hence, the required local supersymmetry transformations are obtained from a gauge transformation of the M-connection (3.2) with parameter λ = 1/ √ 2ǫ α Q α , Thus, the supersymmetric extension of (2.

2), invariant under (3.3) is found to be
where R 2 := R ab R ba and (R 3 ) ab := R ac R cd R db . Here α is a dimensionless constant whose meaning will be discussed below. This action is invariant under (2.1), (3.3), local Lorentz rotations, and also under the local Abelian transformations δb ab = Dθ ab , δb abcde = Dθ abcde . (3.5) Invariance under general coordinate transformations is guaranteed by the use of forms. It is simple to see that the local invariances of the action, including Poincaré transformations, supersymmetry (3.3) together with (3.5), are a gauge transformation for the M-connection (3.2) with parameter λ = λ a P a + 1 2 λ ab J ab +θ ab Z ab +θ abcde Z abcde +1/ √ 2ǫ α Q α . As a consequence, the invariance of the action under the supersymmetry algebra is ensured by construction without invoking field equations or requiring auxiliary fields.

Manifest M-Covariance
The action (3.4) describes a gauge theory for the M-algebra with fiber bundle structure, which can be seen explicitly by writing the Lagrangian as a Chern-Simons form [18] for the M-connection (3.2). Indeed, the Lagrangian satisfies dL = F 6 , where the curvature F = dA + A 2 is given by withT a = De a − (1/4)ψΓ a ψ andF [k] = Db [k] − (1/4)ψΓ [k] ψ for k = 2, 5. The bracket ... stands for a multilinear form of the M-algebra generators G A whose only nonvanishing components are given by J a 1 a 2 , · · · , J a 9 a 10 , P a 11 = 16 3 ǫ a 1 ···a 11 , J a 1 a 2 , · · · , J a 9 a 10 , Z abcde = −α 4 9 ǫ a 1 ···a 8 abc η [a 9 a 10 ][de] , J a 1 a 2 , J a 3 a 4 , J a 5 a 6 , J a 7 a 8 , J a 9 a 10 , Z ab = (1 − α) 16 3 δ a 7 ···a 10 ab a 1 ······a 6 − δ a 9 a 10 ab a 1 ···a 4 δ a 7 a 8 a 5 a 6 Q, J a 1 a 2 , J a 3 a 4 , J a 5 a 6 , J a 7 a 8 , Q = 32 15 CΓ a 1 a 2 ab CΓ a 7 a 8 ab + 2CΓ a 3 ···a 6 δ a 7 a 8 where (anti-)symmetrization under permutations of each pair of generators is understood when all the indices are lowered. The existence of this bracket allows writing the field equations in a manifestly covariant form as In addition, if the eleven-dimensional spacetime is the boundary of a twelve-dimensional manifold, ∂Ω 12 = M 11 , the action (3.4) can also be written as I = Ω 12 F 6 , which describes a topological theory in twelve dimensions. In spite of its topological origin, the action does possess propagating degrees of freedom and hence it should not be thought of as a topological field theory.

Gravitons and four-dimensional spacetime
We now turn to the problem of identifying the true vacuum of the theory. Obviously, a configuration with a locally flat connection, F = 0, solves the field equations and would be a natural candidate for vacuum in a standard field theory. However, no local degrees of freedom can propagate on such background because all perturbations around it are zero modes. Note that eleven-dimensional Minkowski spacetime is maximally supersymmetric by virtue of (3.3), however as it obeys F = 0, the propagators on it are ill-defined, and hence it is a sort of false vacuum.
In a matter-free configuration, Eq. (3.6) is a set of quintic polynomials for the Riemann two-form R ab . The dynamical field equations take the form So, in order to have propagation for A j , the spatial components F ij cannot be small. Hence, a deviation around F = 0 that propagates cannot be infinitesimal and is therefore nonperturbative and non-local. A necessary condition to have well-defined perturbations is that the background solution be a simple zero of at least one of the polynomials. In particular, this requires the curvature to be nonvanishing on a submanifold of a large enough dimension. Let us consider a torsionless spacetime with a product geometry of the form X d+1 × S 10−d , where X d+1 is a domain wall whose worldsheet is a d-dimensional constant curvature spacetime M d . The line element is given by µν stands for the worldsheet metric with µ, ν = 0, ..., d − 1; γ is the metric of S 10−d of radius r 0 and a is a constant.
This Ansatz solves the vacuum field equations provided the projection of the Riemann tensor along the worldsheet, vanishes (hereẽ i andR ij stand for the vielbein and the Riemann curvature of the worldsheet, respectively). This means that M d is either locally de Sitter spacetime of radius a −1 , or locally Minkowski for a = 0. The requirement that the curvature of (4.2) be a simple zero, implies, after a straightforward computation, that d cannot be greater than four. Then, the condition of having well-defined propagators singles out the dimension of the worldsheet to be d = 4, and a 2 > 0. Indeed, for d = 4, the only relevant equation for the perturbations is the one that arises from the variation with respect toẽ i , Since for a = 0 this equation becomes empty, Minkowski spacetime must be ruled out. Thus, the existence of the propagator requires the four-dimensional cosmological constant to be strictly positive and given by Λ 4 = 3a 2 . Note that Eq. (4.3) has support only on the z = 0 plane. Perturbations along the worldsheet, δg µν = h µν (x) reproduce the linearized Einstein equations in four-dimensional de Sitter spacetime. The modes that depend on the coordinates transverse to the worldsheet fall into two classes. Those of the form δg µν = h µν (x, y) are massive Kaluza-Klein modes with a discrete spectrum, while δg µν = h µν (x, z) correspond to Randall-Sundrum-like massive modes whose spectrum is continuous and has a mass gap. The perturbations of the remaining metric components are zero modes, which is related to the fact that the equations are not deterministic for the compact space. A detailed analysis of this, as well as of the perturbations of matter fields will be presented elsewhere [19].

Discussion
We have presented a framework in which the spacetime dimension is dynamically selected to be four. The mechanism is based on a new eleven-dimensional action of the Chern-Simons type, which is a gauge theory for the M-algebra. The possibility of dynamical dimensional reduction arises because the theory has radically different spectra around backgrounds of different effective spacetime dimensions. Thus, in a family of product spaces of the form X d+1 ×S 10−d , the only option that yields a well defined low energy propagator for the graviton is d = 4 and Λ 4 > 0. It should be stressed that for all gravity theories of the type discussed here, possessing local Poincaré invariance in dimensions D = 2n + 1 ≥ 5, four-dimensional de Sitter spacetime is also uniquely selected by the same mechanism as the background for the low energy effective theory.
The action discussed in this paper has a free parameter α, which reflects the fact that the theory contains two natural limits which correspond to different subalgebras of (1.1). For α = 0, the action I 0 in Eq. (3.4) does not depend on b [5] and corresponds to a gauge theory for the supermembrane algebra, while for α = 1, the bosonic field b [2] decouples, and I 1 is a gauge theory for the super five-brane algebra as discussed in [20]. It is interesting to note that the linear combination of both limits, I α = I 0 + α(I 1 − I 0 ), is not only invariant under the intersection of both algebras, but under the entire M-algebra. As the term I 1 − I 0 does not couple to the vielbein and is invariant under supersymmetry by itself, α is an independent coupling constant. A similar situation occurs in nine dimensions where, in one limit, the theory corresponds to the super five-brane algebra, while for the other it is a gauge theory for the super-Poincaré algebra with a central extension [21].
In the presence of negative cosmological constant, the eleven-dimensional AdS supergravity presented in Ref. [10] can be written as a Chern-Simons theory for osp(32|1), which is the supersymmetric extension of AdS 11 . It is natural to ask whether there is a link between that theory in the vanishing cosmological constant limit, and the one discussed here. Since the M-algebra has 55 bosonic generators more than osp(32|1), these theories cannot be related through a Inönü-Wigner contraction for a generic value of α. However, it has been recently pointed out in [22], generalizing the procedure of [23], that it is possible to obtain the M-algebra from an expansion of osp(32|1). In this light, applying this procedure to the eleven-dimensional AdS supergravity theory, it should be expected that the action presented here will be recovered up to some additional terms decoupled from the vielbein, that are supersymmetric by themselves.