Multigravity from a discrete extra dimension

Multigravity theories are constructed from the discretization of the extra dimension of five dimensional gravity. Using an ADM decomposition, the discretization is performed while maintaining the four dimensional diffeomorphism invariance on each site. We relate the Goldstone bosons used to realize nonlinearly general covariance in discretized gravity to the shift fields of the higher dimensional metric. We investigate the scalar excitations of the resulting theory and show the absence of ghosts and massive modes; this is due to a local symmetry inherited from the reparametrization invariance along the fifth dimension.


Introduction
Since the work of Pauli and Fierz [1], it is known that the consistency of theories with spin 2 fields is highly non trivial. A manifestly covariant lagrangian formulation relies on a symmetric rank 2 tensor which contains a priori ten degrees of freedom. Using kinetic terms with four local invariances results in constraints which diminish the number of degrees of freedom to six. The additional scalar degree of freedom turns out to be a ghost whose elimination, at the linear level, dictates the Pauli-Fierz quadratic combination. Boulware and Deser [2] have argued that the incorporation of self interaction generically results in the reappearance of the pathologic scalar degree of freedom. The same reasons render the consistency of multigravity theories [3,4,5,6], with many interacting metrics, highly non trivial. Higher dimensional Einstein gravity when compactified to four dimensions results in a finite number of massless modes among whom is a graviton and an infinite tower of massive modes. The truncation to a finite number of modes is however inconsistent [7,8]. On the other hand a procedure has been recently proposed to obtain from five dimensional Yang-Mills theories a four dimensional theory which approach the five dimensional one in the infrared and which has a finite number of modes. The key point in the approach is to replace the extra component of the vector field by bifundamental scalars which can be viewed as arising from the latticized Wilson line along the fifth dimension. The goal of the present letter is to present such a construction for five dimensional gravity and to see to which extend the consistency of the higher dimensional theory descends to the discretized version.
Our starting point will be the 4+1 ADM decomposition [9] of the five dimensional metric, which will be much more convenient than the Kaluza-Klein splitting. In particular we will show the analogy between the shift vector in the gravity side and the fifth component of the gauge field in the Yang-Mills side. The bifundamental scalars in the Yang-Mills side can be seen as providing a mapping from fundamentals of a gauge group on a site to the fundamentals of the gauge group on the neighbouring site. We will show that the corresponding object in the gravity side is a mapping from one four dimensional manifold located on a site in the discrete extra dimension to the neighbouring one. The map reduces in the continuum limit to the shift vector of the ADM decomposition.
The paper is organized as follows. The second Section is devoted to the introduction of the (4+1) ADM decomposition of the 5D metric and the transformations of the various resulting fields under the 5D diffeomorphisms. In the third Section we show how to discretize the 5D Einstein-Hilbert action while maintaining 4D diffeomorphism invariance on each site. We relate the link fields [10] used to realize non linearly 4D general covariance to the shift vector field of the ADM formalism. In the fourth Section we discuss the spectrum of the resulting action. We exhibit, at the quadratic level, an additional local symmetry inherited from the reparametrization invariance along the fifth dimension. This symmetry is crucial to show the absence of ghosts and massive scalar modes. Our analysis suggests to reconsider previous work [11] on strong coupling effects in these theories. Figure 1: The points P and P ′ have 4D coordinates x µ , Q is on the normal to Σ y at P .

ADM decomposition and invariances
Let us briefly recall the ADM splitting of the five dimensional metricg and the corresponding expression for the Einstein-Hilbert action Above, and in the following, we use expressions with a tilde for denoting quantities of the 5D continuum theory, like the 5D metricg AB , upper case Latin letters from the beginning of the alphabet, A, B, C... to denote 5D indices and lower case Greek letters, µ, ν, α, ... will be denoting 4D indices. We consider a foliation of the five dimensional manifold by four dimensional ones, Σ y , located at given y. Each slice has its four dimensional metric g µν (x, y). The distance between the manifolds Σ y and Σ y+δy is denoted by N δy defining the lapse field N . The normal to Σ y at a point with 4D coordinates x µ hits Σ y+δy at a point with coordinates which in general differ from x µ by a vector N µ δy which defines the shift field (see figure 1). In brief, the fields N , N µ and g µν are related to components of the 5D metricg AB bỹ After an integration by part, the 5D Einstein-Hilbert action (1) can be written as where K µν is the extrinsic curvature of surfaces Σ y : where D µ is the covariant derivative associated with the induced metric g µν and a prime denotes an ordinary derivative with respect to y. The equations of motions derived from action (5) varying with respect to N , N µ and g µν yield respectively where G µν is the Einstein tensor for four dimensional metric g µν , and K is defined by K ≡ K µν g µν . We now turn to determine the transformation properties of the 4D fields under the various diffeomorphisms. These are generated by the vector fields and act on the metric as where Lξ is the Lie derivative with respect toξ. It will be very useful to write the corresponding transformations for the 4D metric, the lapse and the shift fields. If we define the shift vector byN and the covariant derivative, D y , by then, under y dependent four dimensional diffeomorphisms, that is for ξ 5 = 0, we have The transformation of g µν and N is as expected the one of a 4D metric and scalar respectively. The transformation ofN has however an additional term with respect to the usual one characterizing the transformation of a vector. This new term is reminiscent of the inhomogeneous term contributing to the transformation of a gauge field. Indeed this analogy justifies the covariant derivative name we gave to D y : suppose for example that φ is a 5D scalar and consider ∂ y φ, it is not a scalar under a diffeomorphism generated by ξ whereas D y φ is indeed a scalar where we used the transformation rule ofN (15). Similarly if T is a tensor then L Dy T is also a tensor under 4D y-dependent diffeomorphisms. One can thus view the role ofN as rendering possible the formulation of an action invariant under y dependent 4D diffeomorphisms. Incidentally, this remark tells us how the shift fields enter into the equations of motion: it suffices to replace everywhere ∂ y by D y . Notice also that, using D y , the extrinsic curvature can be simply expressed as K µν = (L Dy g µν )/(2N ). We next consider diffeomorphisms along the fifth dimension. In fact it is more convenient to consider diffeomorphisms generated by D y , that isξ = ζD y , with ζ depending on y as well as on x. A short calculation gives the following rules To end this section, let us also mention that it is also convenient to write the action (5) in the "Einstein frame" (from the point of view of the 4D metric g µν ). This can be achieved by performing the Weyl rescaling with

Discretization
We first review the deconstruction of gauge theories [12]. Consider a 5D non-abelian gauge field A = A a A t a dx A ≡ A µ dx µ + A 5 dy with gauge group, e.g., SO(M). Under a y dependent gauge transformation the transformation rules are where u is an element of SO(M). These reduce to the 4D y-dependent transformations for A µ dx µ and A 5 , which is a scalar viewed from 4D, has the following transformation which gives for an infinitesimal transformation Note the formal analogy between (26) and (15): the Lie bracket in (15) is replaced in (26) by a matrix commutator. It is very convenient when discretizing the y dimension to replace A 5 by the Wilson line [13,14] which transforms as The lattice version of the gauge theory is now easily obtained: one has N sites, on each site a gauge field with a corresponding gauge group SO(M) i and on each link between neighbouring sites a scalar W (y i , y i+1 ) transforming in the bifundamental of the gauge groups SO(M) i × SO(M) i+1 . One can then write an effective action for the gauge fields and the scalars. The continuum limit is recovered when the number of sites goes to infinity and a goes to zero; the vacuum expectation value of the scalars is then the identity: where a is the lattice spacing. In the broken phase one has a massless gauge boson corresponding to the diagonal subgroup and a collection of massive spin one particles with masses These reproduce, in the infrared, the Kaluza-Klein spectrum of the first modes when the radius is given by aN.
We noted previously thatN has a transformation law which is similar to the fifth component of the gauge potential, the gauge group being replaced by the diffeomorphism group. While the gauge field can be represented by a matrix this is no longer true forN which is rather to be thought of as an operator acting on functions on a four dimensional manifold. The gauge Wilson line (27) and the analogy betweenN and A 5 motivates the consideration of or more explicitly Now W (y ′ , y) defines a mapping from functions (scalar fields) on Σ y to functions (scalar fields) on Σ y ′ . Explicitly, let φ(x) be a scalar field defined on the hypersurface Σ y 0 and consider Then φ y verifies the equation ∂ y φ y =N y (φ y ) and is subject to the boundary condition φ y 0 = φ. Let ξ(y) generate a y-dependent 4D diffeomorphism then from the transformation ofN given in (15) we get δW (y ′ , y) = ξ(y ′ )W (y ′ , y) − W (y ′ , y)ξ(y), which implies that indeed φ y defined in (33) transforms under diffeomorphisms as δφ y = ξ(y)(φ y ) if φ transforms as δφ = ξ(y 0 )(φ). A convenient and useful way of writing (33) is where X(y, y 0 ) is a mapping from the manifold Σ y to Σ y 0 generated byN , that is which can be written as the right hand side of the first line being understood as the action of the W on the function x µ . It is possible to extend W so that it maps tensors of arbitrary rank on Σ y to tensors on Σ y ′ . This is done with the help of the Lie derivative as follows It reduces to the previous expression (31) when acting on scalars. The Leibniz rule for the Lie derivative results in a simple action of W on the direct product of tensor: where T 1 and T 2 are arbitrary tensors on Σ y . The commutation of the Lie derivative and the exterior derivative when acting on forms translates also to the simple property where ω y is an arbitrary form defined on Σ y . The geometric interpretation of the map W is clear from figure 1: when y and y ′ are infinitesimally close W maps the point P with coordinates x on Σ y to the point Q with coordinates x µ + δyN µ on Σ y+δy . We are now in a position of performing the discretization of the Einstein-Hilbert action along the y direction. We replace y by ia with i an integer and a the lattice spacing. The fields are thus the metric on each site g (i) µν , the lapse fields N (i) and the Wilson line W (i, i + 1) which, as in the gauge theory, replaces the shift vector. The y derivative, as we noted in the previous section, appears in the continuum in the combination D y . The Lie derivative of a tensor field with respect to D y can be written as ¿From this we see that the simplest discrete counterpart of the Lie derivative along D y is It is now immediate to get the discretized Einstein-Hilbert action from (5) The action is invariant under the product of all diffeomorphism groups associated to the points of the lattice. Under such a transformation generated by ξ i , the different fields transform as These reduce in the continuum limit to (14), (16) and (34). The explicit expression of the components of W (i, i + 1)T i+1 can be easily written down with the help of X µ (i, i + 1; x), a mapping between the manifolds at i and i + 1 which is the discrete counterpart of X µ (y, y 0 ; x) defined in (38). In fact, we have The variation of the action with respect to W (i, i + 1) amounts to a variation with respect to the mappings X µ (i, i + 1; x). Notice that the action is not the most general action with the symmetries (45) since it descends from a 5D theory which had also a reparametrization invariance along the y direction. This will have very important consequences as we will show in the next section.
The same discretization procedure can be applied to action (21) yielding directly the action with canonical kinetic terms for the metric on each sites with Before closing this section let us note that a way to realize diffeomorphism invariance on every site has been proposed in ref. [10] with no a priori relation to an extra dimension. The procedure amounts to the incorporation of maps between interacting sites. This is the analog of our W (i, i + 1). Our construction shows that this map arises naturally in the discretization procedure and allows to identify the continuum limit of the map with the shift vector of the ADM decomposition. In fact in the continuum limit we have, from (38) Different approaches to the deconstruction of gravity theories relying mainly on the local Lorentz invariance in the Cartan moving basis formalism have been considered in [15]. The remaining degrees of freedom are expected to be shared by a number of zero modes (scalars and vectors), which does not depend on N but depends on the boundary conditions, as well as a number of massive scalars. The latter number depends on N as 2N + c, where c is a constant which depends on the boundary conditions but which does not depend on N. These scalars are potentially pathologic, they may lead to ghosts or tachyons. For a generic multigravity theory, ghosts and instabilities do indeed appear [3,2]. The higher dimensional theory we started with, before discretization, does not have these pathologies. It is thus possible that the action (44) inherited the consistency of the continuum action. The rest of this section is devoted to the proof that this is indeed the case at least to the quadratic order in the fluctuations. We will find that the massive scalar modes decouple at the quadratic level. This is due to an extra local symmetry which removes 2N − 2 degrees of freedom. In order to get the standard kinetic terms for the metrics, let us first perform a Weyl rescaling on the metric in (44). So we define the metrics γ (i) µν and the scalars φ (i) by g (i)

Spectrum of the action
The next step is to expand the action around the vacuum and to keep the quadratic fluctuations in the fields. In (50), M p is given by M 2 p = M 3 (5) a. We obtain where we have defined the finite difference operator ∆ acting on a field F (i) as two spacetime indices between a parenthesis indicates a symmetrization on these indices weighted by a factor two 5 , we have h αβ ≡ η αµ η βν h µν and h ≡ h αβ η αβ . The equations of motion derived from this action read .
They correspond respectively to linearization of equations (9), (8) and (7). In order to exhibit the spectrum encoded in the action (51) it is convenient to perform a discrete Fourier transformation. To each field F (i) with F (i+N ) = F (i) we defineF (k) bŷ The action (51) becomes The spin two and one content of the action is easily read from the action. We have one massless spin 2 particle given byĥ (0) µν , one massless spin 1 particlen (0) µ , one massless scalarφ (0) and a tower of massive spin two particles with a spectrum given by 5 That is to say F (µν) = 1 2 (F µν + F νµ ) The action has the local invariances which show that for k = 0, then (k) µ are Stuckelberg fields which are absorbed by the massive spin 2 fields and do not propagate. The invariances (58) are the linearized version of the invariance under 4D diffeomorphisms, they are expected by construction. Less expected is the invariance under the local transformations A generic multigravity theory with 4D diffeomorphism invariance on each site realized does not possess this symmetry, which is inherited from the diffeomorphism invariance under the y reparametrizations in the continuum theory. In fact the invariance under (59) eliminates, at the quadratic level, all except the massless, scalar modesφ (k) . It may also be used to eliminate the trace of theĥ (k) µν proving the absence of ghostlike excitations. Associated to this local invariance there is a constraint which removes one more set of scalars. At this point we note that while the Pauli-Fierz action removes the ghost by hand, the above action removes it with the aid of a local symmetry; in the gauge whereφ (k) andn (k) µ are zero the action reduces to the Pauli-Fierz form.
The scalar fluctuations are in fact potentially pathologic. This is mainly due to the fact that the conformal factor has a kinetic term with the wrong sign, that is for a metric of the form g = e 2σ η, the Einstein action reads 6 (∂e σ ) 2 , giving a ghostlike kinetic term for e σ In order to explicitly check the consistency of the action (51) it is instructive to isolate in (51) the scalar massive modes which in addition toφ (k) are given bŷ At quadratic order inψ (k) ,f (k) ,v (k) andφ (k) , the action (44) now reads On the above action one clearly sees the ghost like nature ofψ (k) as well as the fact that the discretization procedure is leading to tachyonic mass term forφ (k) . We saw however that both problems disappear once one carefully considers the invariances of the action (63) given in (58). The equations of motion derived from the above action (63) now read ✷φ (k) One can explicitly verify that these equations are obtained from the full equations of motion (53-55). The first three equations are respectively the equation of motion ofψ (k) ,f (k) ,φ (k) , while the last is the equation of motion for ∂ µv(k) considered as a dynamical variable. Note that had we considered (as in ref. [11]),v (k) , instead of ∂ µv(k) , as the dynamical variable, we would have obtained the equation of motion ✷ ψ (k) −φ (k) (exceptφ (0) ) decouple and thus have no propagators, we have a non-standard action which starts at a cubic or higher order level for these modes. This deserves a further study.