Randall-Sundrum brane universe as a ground state for Chern-Simons gravity
Fabrizio Cordonier-Tello, Fernando Izaurieta, Patricio Mella, Eduardo Rodríguez
RRandall–Sundrum brane universe as a ground state for Chern–Simons gravity
Fabrizio Cordonier-Tello, ∗ Fernando Izaurieta, † Patricio Mella, ‡ and Eduardo Rodr´ıguez § Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universit¨atM¨unchen. Theresienstraße 37, 80333 M¨unchen, Germany Departamento de F´ısica, Universidad de Concepci´on, Casilla 160-C, Concepci´on, Chile Centro de Docencia de Ciencias B´asicas para Ingenier´ıa, Universidad Austral de Chile, Casilla 567, Valdivia, Chile Departamento de F´ısica, Universidad Nacional de Colombia, Bogot´a, Colombia (Dated: October 25, 2016)In stark contrast with the three-dimensional case, higher-dimensional Chern–Simons theories canhave non-topological, propagating degrees of freedom. Finding those vacua that allow for the prop-agation of linear perturbations, however, proves to be surprisingly challenging. The simplest solu-tions are somehow “hyper-stable,” preventing the construction of realistic, four-dimensional physicalmodels.Here, we show that a Randall–Sundrum brane universe can be regarded as a vacuum solution ofChern–Simons gravity in five-dimensional spacetime, with non vanishing torsion along the dimen-sion perpendicular to the brane. Linearized perturbations around this solution not only exist, butbehave as standard gravitational waves on a four-dimensional Minkowski background. In the non-perturbative regime, the solution leads to a four-dimensional “cosmological function” Λ ( x ) whichdepends on the Euler density of the brane.Interestingly, the fact that the solution admits nontrivial linear perturbations seems to be relatedto an often neglected property of the Randall–Sundrum spacetime: that it is a group manifold, or,more precisely, two identical group manifolds glued together along the brane. The gravitationaltheory is then built around this fact, adding the Lorentz generators and one scalar generator neededto close the algebra. In this way, a conjecture emerges: a spacetime that is also a group manifoldcan be regarded as the ground state of a Chern–Simons theory for an appropriate Lie algebra. PACS numbers: 04.50.+h, 04.50.KdKeywords: Chern–Simons Gravity, Randall–Sundrum Brane, Non-Vanishing Torsion
I. INTRODUCTION
Higher-dimensional Chern–Simons (CS) theories enjoya host of properties that make them interesting theoreti-cal laboratories. For a recent review of their relevance togravitation theory, the reader is advised to check Ref. [1].We would like to focus here on a generic problem thatappears in five and higher dimensions, namely, that thesimplest solutions are “hyper-stable,” meaning that nolinearized perturbations can propagate on them. In par-ticular, we want to showcase how this problem can beovercome in a particular setting, and what we can learnfrom this.The setting is a five-dimensional CS theory built uponthe Weyl subalgebra of the conformal algebra, which for-goes special conformal transformations to include onlyrotations, translations, and dilations. This is a theoryof gravity (in its first-order guise, as usual when deal-ing with CS theories) non-minimally coupled to a one-form field. The details of the theory are given in sec-tion III, where the Lagrangian, gauge transformations,and field equations are derived from the general proper-ties of CS theories. It must be noted, however, that the ∗ [email protected] † fi[email protected] ‡ [email protected] § [email protected] gravitational Lagrangian is not Einstein–Hilbert’s, butthe dimensional continuation of the Euler density. Thisterm is a total derivative in four dimensions, but not infive, where it leads to nontrivial dynamics [2]. In sec-tion IV, we show that a Randall–Sundrum (RS) braneuniverse [3, 4], with nonzero torsion along the directionperpendicular to the brane, solves the CS field equa-tions and admits linearized perturbations that propagateas standard gravitational waves on a four-dimensionalMinkowski background.Why should this be so? An attempt at answering thisquestion begins in section II, where we show that the RSbrane universe can be regarded as two copies of half a cer-tain group manifold, with the brane acting as a mirrorbetween the two. The commutation relations for the as-sociated Lie algebra are explicitly displayed. A CS theorybuilt upon this algebra would not be a theory of gravity,however, since there is no Lorentz symmetry and henceno spin connection or curvature. As the most economicaloption, we embed the algebra in the Weyl subalgebra ofthe conformal algebra, which includes the original gen-erators, plus the Lorentz symmetry and a single extragenerator. It then comes as less of a surprise that the RSbrane universe is indeed a solution of a CS theory basedupon the Weyl algebra.While the resulting system may not quite turn out tobe particularly realistic (we discuss some of its cosmo-logical implications in section IV, where we show that itcan lead to unphysical consequences), and should there- a r X i v : . [ h e p - t h ] O c t fore be regarded as a toy model, we believe that its maininterest lies in the lesson that is suggests: to constructan admissible ground state for a CS theory, look for asolution that admits a group structure.Our method can be briefly described as follows.Let A be a Lie algebra-valued one-form gauge connec-tion and let F = d A + [ A , A ] be its associated gaugecurvature. In d = 2 n + 1 dimensions, the CS field equa-tions can be written as (cid:104) F n G A (cid:105) = 0, where the G A generators span a basis for the gauge algebra, and (cid:104)· · · (cid:105) stands for a multilinear symmetric form of rank n + 1 in-variant under the algebra, such as the symmetrized tracein a suitable matrix representation. The simplest solu-tions satisfy F = 0, which, for n ≥
2, means that they arehyper-stable: no linear perturbations propagate aroundthis background. This happens because perturbationsmust satisfy (cid:10) F n − δ F G A (cid:11) = 0, which becomes emptywhen F = 0. For n = 1, on the other hand, pertur-bations are not affected by the fact that F = 0, andindeed there exist interesting cases of propagating fieldsin three-dimensional CS theory (see, e.g., Ref. [5]). Thisis even more remarkable given the fact that, providedthe algebra is semisimple (so that the bilinear form (cid:104)· · · (cid:105) is invertible), the CS field equations in three dimensionsare actually equivalent to F = 0. A higher-dimensionalvacuum that admits linear perturbations must necessar-ily have a non-vanishing gauge curvature, all the whilesatisfying (cid:104) F n G A (cid:105) = 0.Pick now a spacetime that is also a group manifold,meaning that its vielbein E A can be regarded as theMaurer–Cartan (MC) forms, θ = l E A Z A , of a cer-tain Lie algebra spanned by the Z A generators. TheseMC forms satisfy d θ + [ θ , θ ] = 0. If A is given by A = θ + W + · · · , where W is the Lorentz algebra-valuedspin connection, and the dots stand for any additionalfields, then the gauge curvature reads F = d W + 12 [ W , W ] + d θ + 12 [ θ , θ ] + [ W , θ ] + · · · . (1)Here, we identify R = d W + [ W , W ] and T =d θ + [ W , θ ] with the Lorentz curvature and torsion, re-spectively.The fact that spacetime is a group manifold now hasseveral consequences. First of all, since d θ + [ θ , θ ] = 0,some components of the gauge curvature vanish. On theother hand, there’s no a priori reason why torsion shouldvanish, so these models in general feature nonzero tor-sion. The torsional degrees of freedom can be param-eterized as T A = κ AB E B , where κ AB is the cotorsionone-form. The cotorsion then determines the spin con-nection via W AB = ˚ W AB + κ AB , where the torsion-less component ˚ W AB is found by solving the equationd E A + ˚ W AB E B = 0. In turn, the torsionless part of the This is sometimes referred to as the “linearization instability”problem in the literature. spin connection is used to define the Riemann curvature,˚ R AB = d ˚ W AB + ˚ W AC ˚ W CB . Finally, the Lorentz curva-ture is computed as R AB = ˚ R AB + ˚D κ AB + κ AC κ CB .To summarize, the metric structure encoded in thevielbein determines the torsionless component of the spinconnection and the Riemann curvature, while the cotor-sion is needed to compute the full Lorentz curvature. Thefact that spacetime has a group structure means that theproblem of finding a general background that allows forlinear perturbations to propagate has been reduced tofinding a one-form cotorsion with this property, whichproves to be more tractable.We close with some final thoughts and an outlook forfuture work in section V. II. RANDALL–SUNDRUM AND THE BIANCHICLASSIFICATION
Although far from self-evident, the RS metric is in-trinsically related to the Weyl subgroup of the conformalgroup in any dimension. The easiest way to see this isby means of the MC approach to Lie groups [6].Let M be a spacetime manifold endowed with a notionof metricity g codified through a basis of vielbein one-forms E A = E Aµ d x µ , g = η AB E A ⊗ E B = g µν d x µ ⊗ d x ν . (2)Consider now a constant vector k A and a basis m A = m Aµ d x µ of vielbein one-forms for the Minkowski space-time. Then it is straightforward to show that, when E A ( x ) = − k B m Bλ x λ m A , (3)then M is the higher-dimensional analogue of a three-dimensional Type V Bianchi spatial section [7].As matter of fact, by defining the dimensionless one-form θ A = 1 l E A , (4)where l is a constant parameter with units of length, one can prove that θ A satisfiesd θ C = − l (cid:0) δ CA k B − δ CB k A (cid:1) θ A ∧ θ B , (5) We prefer the convention where the structure constants of aLie algebra are always dimensionless, and therefore the MCone-forms θ A must be dimensionless too. Since the metric g = g µν d x µ ⊗ d x ν has units of length squared, the vielbeinone-form E A has units of length. That is why it is necessaryto introduce a constant parameter l with units of length in or-der to make everything dimensionally consistent. The same hasto be done in eq. (31). From a physical point of view it mayseem natural to identify l with the Planck length. The presenceof l can partially determine the structure of a theory; see, e.g.,Ref. [8]. and can therefore be regarded as the MC one-form forthe associated Lie algebra[ Z A , Z B ] = l (cid:0) δ CA k B − δ CB k A (cid:1) Z C . (6)Eq. (6) is of course the generalization to an arbitrary di-mension of the Bianchi Type V algebra, implying that M is in this case also a group manifold. This is not strange;the Bianchi classification of three-dimensional spaces hasbeen very well studied because of its importance in cos-mology, but the general concept can be defined in anydimension.It is straightforward to check that the vielbein (3) in-duces a warped metric g = η AB E A ⊗ E B . For clarity,let us choose a five-dimensional spacetime and a range ofindices A, B, . . . = 0 , , . . . ,
4. When k A is spacelike, itpossible to rotate the f¨unfbein in such a way that the onlynon-vanishing component of k A is k = k . After makingthe standard transformation of coordinates x = e kz /k ,the associated metric becomes g = e − kz η µν d x µ ⊗ d x ν + d z ⊗ d z. (7)This means that the RS metric with a flat brane,¯ g = e − k | z | η µν d x µ ⊗ d x ν + d z ⊗ d z, (8)may be regarded as describing half the group manifold,glued with its reflection on the z = 0 brane. The directionorthogonal to the brane is that of the vector k A .Our purpose is to construct an off-shell invariant the-ory of gravity in d = 5 that admits the RS metric as asolution, with well-defined perturbations around it. Toachieve this goal, one may try a CS theory for the simpleone-form gauge connection A = 1 l E A Z A , (9)but this leads us nowhere; in order to describe gravity,it is imperative to include the generators J AB of theLorentz symmetry. The set { Z A , J AB } , however, doesnot span a Lie algebra. In order to close the algebra, thesmallest possible modification is to add a single extragenerator D , leading us to the Lie algebra[ Z A , Z B ] = l ( Z A k B − Z B k A ) , (10)[ J AB , Z C ] = η BC Z A − η AC Z B ++ l ( k A η BC − k B η AC ) D , (11)[ J AB , J CD ] = η BC J AD − η BD J AC + − η AC J BD + η AD J BC , (12)[ D , Z A ] = Z A + lk A D . (13)This algebra is not as unusual as it may seem at firstsight. It suffices to make the change of basis P A = Z A + lk A D (14) to see that it is equivalent to[ P A , P B ] = 0 , (15)[ J AB , P C ] = η CB P A − η CA P B , (16)[ J AB , J CD ] = η BC J AD − η BD J AC + − η AC J BD + η AD J BC , (17)[ D , P A ] = P A , (18)which is the Weyl subalgebra of the conformal algebra,[ P A , P B ] = 0 , (19)[ K A , K B ] = 0 , (20)[ K A , P B ] = η AB D − J AB , (21)[ J AB , P C ] = η CB P A − η CA P B , (22)[ J AB , K C ] = η CB K A − η CA K B , (23)[ J AB , J CD ] = η BC J AD − η BD J AC + − η AC J BD + η AD J BC , (24)[ D , P A ] = P A , (25)[ D , K A ] = − K A . (26)Given this intimate relation between the Weyl subal-gebra and the RS spacetime, one may expect that the RSsolution represents an interesting vacuum for a gauge the-ory invariant under this symmetry. The Weyl CS gravityin d = 5 is precisely this kind of gauge theory. The con-struction of the Lagrangian and the study of its dynamicsare treated in the following sections. III. CHERN–SIMONS GRAVITY WITH WEYLINVARIANCE IN FIVE DIMENSIONS
CS gravity theories have been studied with interest inthe last decades. The idea originated in the 1980s withthe work of Refs. [9–16], and it has been developed in thecontext of gravity and supergravity since then. A recentreview, with a focus on its application to gravity theory,can be found in Ref. [1].In odd dimensions, d = 2 n + 1, a CS Lagrangian isdefined as L (2 n +1)CS = κ n (cid:90) d τ (cid:42) A ∧ (cid:18) τ d A + 12 τ [ A , A ] (cid:19) ∧ n (cid:43) , (27)where A corresponds to a Lie algebra-valued connectionone-form A = A Aµ d x µ ⊗ G A , (28)and (cid:104)· · · (cid:105) : g n +1 → R (29) (cid:0) G A , . . . , G A n +1 (cid:1) (cid:55)→ g A ··· A n +1 = (cid:10) G A · · · G A n +1 (cid:11) , (30)stands for a multilinear symmetric form of rank n + 1invariant under the Lie algebra g .This kind of Lagrangian has several attractive featuresas a field theory. For our purposes, the most relevantare [1]:1. CS gravities are background-free (a backgroundvielbein is not necessary to construct the theory,as it is for instance in the case of the Yang–MillsLagrangian).2. CS gravities are off-shell invariant under the sym-metry group (up to boundary terms). This standsin strong contrast with the standard Einstein–Hilbert Lagrangian in d = 4 [17, 18].3. Despite its topological origin, this kind of La-grangian has propagating degrees of freedom when d ≥ d ≥ d = 3 corre-sponds to a CS three-form and therefore the Lagrangianis off-shell gauge invariant, in stark contrast with thefour-dimensional case. It has been conjectured [13, 19]that this invariance could be the underlying reason forthe renormalizability of three-dimensional gravity [20–22]. Trying to repeat this feat has been perhaps one thestrongest motivations to pursue this kind of theory inhigher dimensions.The main challenge here lies in the dynamics. In d = 3,the degrees of freedom are topological and not propa-gating; the equations of motion are simply F = 0. In d ≥
5, the equations of motion are highly nonlinear andthe phase space has a complicated structure. In fact, ingeneral the standard Dirac method for constraints cannotbe directly used [23–25], and this has been a big obstaclefor the quantization of this kind of theory in higher di-mensions. It is interesting to notice that F = 0 is still asolution for the equations of motion in higher dimensions,but it is somehow “hyper-stable,” meaning that pertur-bations on this background lead to the equation 0 = 0and do not propagate. In practice, higher-dimensionalCS theories can have solutions admitting propagating de-grees of freedom, but finding these states is nontrivial.In order to solve this problem, warped spacetime solu-tions have been successfully used in the past as vacuumstates in CS gravity theories in d = 11 (see Refs. [26, 27]).In the present work, we will also use this kind of vacuum,but from a slightly different point of view: the vanishingtorsion condition is not imposed in the bulk, and the RSspace corresponds to the group manifold for the trans-lational part of the algebra. This means that on thisvacuum the f¨unfbein corresponds to the MC form for thetranslational piece.The theory is constructed using the Lagrangian (27)for n = 2 (i.e., d = 5) when the one-form connectiontakes values in the Weyl subalgebra [cf. eqs. (10)–(13)] A = 12 W AB J AB + 1 l E A Z A + φ D , (31) where E A is the f¨unfbein one-form, W AB is the five-dimensional spin connection and φ = φ µ d x µ is the one-form field associated to dilations. The choice of the basis(10)–(13) for the following work stems from practical pur-poses, since aspects of the RS scenario are made explicitthrough the algebra and its commuting relations.Under an infinitesimal gauge transformation generatedby the local parameter λ = 12 λ AB J AB + 1 l λ A Z A + λ D , (32)the fields transform as components of the connection one-form, i.e., δ A = − (d λ + [ A , λ ]) . For Lorentz transformations we find δW AB = − D W λ AB , (33) δE A = λ AB E B , (34) δφ = k A λ AB E B . (35)The Z A -boosts, on the other hand, act on the fields as δW AB = 0 , (36) δE A = − D W λ A − k B λ B E A + ( E ⊥ − φ ) λ A , (37) δφ = − k A W AB λ B − φk A λ A . (38)Finally, dilations read δW AB = 0 , (39) δE A = λE A , (40) δφ = − d λ + λE ⊥ , (41)where D W stands for the usual Lorentz covariant deriva-tive, and E ⊥ = k A E A .It must be stressed that despite the nonstandardform of the commutator (11), the f¨unfbein behaves as aLorentz vector, and therefore the metric (2) remains in-variant under local Lorentz transformations, as expected.However, it is interesting to notice that, in our chosen ba-sis [cf. eqs. (10)–(13)], the dilation field φ is not a Lorentzscalar, since Lorentz transformations on planes orthogo-nal to the brane are going to change it [cf. eq. (35)].The fundamental ingredient in order to shape the La-grangian for the theory is the invariant tensor. In orderto construct a nontrivial invariant tensor, we will use arepresentation in terms of Dirac matrices in D = 6 forthe generators of the algebra (10)–(13). See Appendix Afor further details.Using this matrix representation, we find that the non-vanishing symmetric invariant tensor components read (cid:104) J AB J CD Z E (cid:105) = 18 √ (cid:15) ABCDE ++ α η AC η BD − η AD η BC ) lk E , (42) (cid:104) J AB J CD D (cid:105) = − α η AC η BD − η AD η BC ) , (43) (cid:104) Z A Z B Z C (cid:105) = − (cid:18) α + α (cid:19) l k A k B k C , (44) (cid:104) Z A Z B D (cid:105) = (cid:18) α + α (cid:19) l k A k B , (45) (cid:104) Z A DD (cid:105) = − (cid:18) α + α (cid:19) lk A , (46) (cid:104) DDD (cid:105) = 34 α + α , (47)where α is an arbitrary constant.Using the subspace separation method introduced inRefs. [28, 29], it is possible to write down an explicitLorentz-invariant expression for the CS Lagrangian in d = 5 as L (5)CS ( A ) = 332 √ l (cid:15) ABCDE R AB ∧ R CD ∧ E E + (48)+ 38 α ( φ − E ⊥ ) ∧ R AB ∧ R BA + (49)+ (cid:18) α + α (cid:19) ( φ − E ⊥ ) ∧ (d φ − d E ⊥ ) ∧ , (50)where R AB = d W AB + W AC ∧ W CB and T A = d E A + W AB ∧ E B stand for the five-dimensional Lorentz curva-ture and torsion.The field equations (obtained by regarding E A , W AB and φ as independent fields) can be reduced to the fol-lowing independent relations in d = 5: (cid:15) ABCDE R AB ∧ R CD = 0 , (51)1 l (cid:15) ABCDE R CD ∧ T E − αR AB ∧ d ( φ − E ⊥ ) = 0 , (52) α (cid:20) R AB ∧ R BA + (cid:18)
34 + α (cid:19) [d ( φ − E ⊥ )] ∧ (cid:21) = 0 . (53)Some comments on the field equations (51)–(53) arein order. In section IV A, a background solution with R AB = 0 and T A (cid:54) = 0 is considered, and in section IV B,the propagation of perturbations on this background isstudied.Of course, the “hyper-stability” problem (also referredto as “linearization instability”) on the R AB = 0 back-ground is still present in eq. (51), but not in eq. (52).That is why the linearized dynamics of perturbations insection IV B stems from eqs. (52) and (53). Eq. (51) can-not be simply dismissed, of course, because in principle it could give rise to nonlinear constraints which couldseriously hinder the propagation of perturbations. Thatthis is not the case for the vacuum proposed by us canbe shown by considering the non-perturbative dynamicsof the system. This is done in section IV C, where it isshown that in fact eq. (51) does not prevent the prop-agation of perturbations on the torsional background.However, eq. (51) has nontrivial dynamical consequences:the four-dimensional cosmological constant has to be re-placed by a “cosmological function” related to the four-dimensional Euler density of the brane. IV. DYNAMICS
The goal of this section is to find a vacuum for the CStheory constructed in section III that allows for linearperturbations of the fields to propagate, and to study itsdynamics. First, in section IV A a vacuum is constructed,and propagation of linear perturbations on the field equa-tions (52) and (53) is studied in section IV B. After this,the dynamics generated by the whole set of field equa-tions, including eq. (51), is analyzed non-perturbativelyin section IV C.There are several possible vacuum candidates for thefield equations (51)–(53). A warped space choice alongthe lines of Refs. [26, 27] is a good solution to the prob-lem. Here we will use also a warped geometry solution,but with a twist: torsion doesn’t have to vanish. As wewill show, in this context it is very natural to have tor-sion in the bulk, but a torsion-free brane universe. Here,torsion is regarded as a pure geometric quantity, alongthe lines of the Einstein–Cartan formalism. It shouldbe noted though, that torsion could emerge from othersources, as happens for instance when we add fermionsto a gravity theory in d = 4, and in String Theory,where torsion arises from the Kalb–Ramond field poten-tial. When torsion is present, the spin connection and thevielbein represent independent degrees of freedom. Thefull spin connection can be split as W AB = ˚ W AB + κ AB ,where ˚ W AB stands for the torsion-free part, defined im-plicitly by d E A + ˚ W AB ∧ E B = 0, and κ AB is the co-torsion one-form tensor, which is related to the torsiontwo-form through T A = κ AB ∧ E B . (54)In this case one must distinguish between the Lorentzcurvature two-form, R AB = d W AB + W AC ∧ W CB , builtfrom the full Lorentz connection W AB , and the Riemanncurvature two-form, ˚ R AB = d ˚ W AB + ˚ W AC ∧ ˚ W CB , builtfrom the torsion-free part only. Both are related throughthe equation R AB = ˚ R AB + ˚D κ AB + κ AC ∧ κ CB , (55)where ˚D stands for the covariant derivative in the torsion-free connection ˚ W AB .From now on, we will use the range of indices in up-percase as A, B, C = 0 , . . . , a, b, c = 0 , . . . ,
A. Torsional Randall–Sundrum backgroundsolution
Let us start by considering a RS geometry describedby the f¨unfbein E A = (cid:26) E a = e − k | z | m a ,E = d z, (56)where m a corresponds to the vierbein of four-dimensionalMinkowski spacetime.The independent degrees of freedom of the spin con-nection described by the cotorsion can be chosen as κ AB = − sgn ( z ) (cid:0) k A E B − k B E A (cid:1) . (57)This is a very particular state. Calling ˚ ω ab the torsion-free spin connection for the brane (defined by d m a +˚ ω ab ∧ m b = 0), then the spin connection generated bythe cotorsion (57) is given by W AB = (cid:26) W ab = ˚ ω ab ,W a = 0 , (58)It may seem paradoxical, but this connection creates anon-vanishing torsion in the bulk given by T A = d E A + W AB ∧ E B = sgn ( z ) E A ∧ E ⊥ . (59)This is a state with nonzero torsion in the bulk, but withvanishing torsion in the brane. In a similar way, the stan-dard Riemann curvature two-form corresponds as usualto ˚ R ab = − sgn ( z ) k E a ∧ E b , (60)˚ R a = (cid:2) δ ( z ) − k sgn ( z ) (cid:3) E a ∧ E ⊥ , (61)but the spacetime is Lorentz-flat, i.e., R AB = 0, andtherefore the torsion is covariantly constant, D W T A =0. Finally, the dilation one-form field is given by φ = u ( x ) d v ( x ) , (62)with u ( x ) and v ( x ) two arbitrary scalar functions de-pending on the brane coordinates. This is a consequence of the Bianchi identities, D R AB = 0,D T A = R AB E B . B. Perturbations
The state described by eqs. (56), (57) and (62) satisfiesthe field equations (51)–(53) and proves to be an inter-esting vacuum for the CS theory. As a matter of fact, letus consider a perturbation on the solution parameterizedby E A → E A + 12 h A , (63) W AB → W AB + U AB + Ξ AB , (64) φ → φ + ϕ. (65)The perturbations of the spin connection are split inthe mode U AB ( h, ∂h ), which doesn’t change the torsion,and the mode Ξ AB , which does. In particular, this meansthat 12 D W h A + U AB ∧ E B = 0 , (66)and δT A = Ξ AB ∧ E B .The vacuum described by eqs. (56), (57), and (62)shows some interesting features. First of all, since R AB =0, eq. (51) doesn’t provide us with any information at thelinear level (the “0 = 0” problem), and therefore it will bestudied non-perturbatively in section IV C. In contrast,eq. (52) does allow for linear perturbations to propagateon this background. A particularly interesting case froma four-dimensional point of view occurs when the one-form h A is given by the warped ansatz h A = (cid:26) h = 0 ,h a = e − k | z | ˇ h ab ( x ) e b , (67)with d φ = 0 and Ξ AB = 0. In this case, linear perturba-tions on the field equation (52) take the form12 (cid:15) abcd D ˚ ω U ab ∧ e c ∧ E ⊥ = 0 , (68)which correspond to standard gravitational waves ˇ h + µν = ( h µν + h νµ ) on a four-dimensional Minkowski back-ground. The general modes with Ξ AB (cid:54) = 0 describe a“torsional wave,” which also depends on the antisym-metric component ˇ h − µν = ( h µν − h νµ ) of the one-form h A .This nontrivial dynamics for the propagation of linearperturbations seems highly satisfactory from a physicalpoint of view. However, its consistency with eq. (51)at the non-perturbative level still has to be checked; seesection IV C.A more rigorous analysis of the symplectic matrix ofthe system, Ω ijKL = − g KLM (cid:15) ijkl ¯ F M kl , (69)following the lines of Ref. [25], but considering dilationsinstead of U (1), has been performed in the particularcase when φ = φ ( x brane ) E . The maximal rank of thematrix is 64 − R AB = 0, but it does indicate that we havean irregular and degenerate state, and the counting of de-grees of freedom cannot be performed na¨ıvely followingthe standard Dirac procedure. C. Non perturbative dynamics
Let us slightly modify our ansatz. First, let us con-sider a RS brane of arbitrary geometry, described by thef¨unfbein one-form E A = (cid:26) E a = e − k | z | e a ,E = d z, (70)where now e a ( x ) is an arbitrary vierbein one-form forthe brane instead of the Minkowskian vierbein one-form m a we have used before.In the same way, instead of eqs. (57) and (62) forthe cotorsion κ AB and the dilation one-form φ , let usparametrize the fields as κ AB = (cid:104) τ ( x ) e k | z | − sgn ( z ) (cid:105) (cid:0) k A E B − k B E A (cid:1) , (71) φ = − (cid:104) kzτ ( x ) + e − k | z | (cid:105) d v ( x ) , (72)where τ ( x ) and v ( x ) are arbitrary scalar functions.Under this ansatz, the components of the two-formsLorentz curvature, torsion and dilation field strengthread R ab = ˚ r ab − k τ e a ∧ e b , (73) R a = − k d τ ∧ e a , (74) T A = (cid:104) sgn ( z ) − τ e k | z | (cid:105) E A ∧ E ⊥ , (75)d φ = − (cid:104) sgn ( z ) − τ e k | z | (cid:105) e − k | z | d v ∧ E ⊥ ++ 2 kz d v ∧ d τ, (76)where ˚ r ab = d˚ ω ab + ˚ ω ac ∧ ˚ ω cb corresponds to the Rie-mann two-form associated to the torsionless spin connec-tion ˚ ω ab on the brane (defined by d e a + ˚ ω ab ∧ e b = 0).Again, this is a torsionless configuration for the brane,but with non-vanishing torsion for the bulk. Using theBianchi identities it is straightforward to prove thatD W T A = − e k | z | d τ ∧ E A ∧ E ⊥ , (77)and therefore τ = const. corresponds to a covariantlyconstant torsion state, D W T a = 0.When the new ans¨atze for the fields, eqs. (70)–(72),are satisfied, it is possible to prove that the field equa-tions (51)–(53) reduce to the independent relations for the four-dimensional geometry (cid:15) abcd ˚ r ab ∧ ˚ r cd − k τ (cid:15) abcd e a ∧ e b ∧ e c ∧ e d = 0 , (78)12 (cid:15) abcd (cid:0) ˚ r ab − k τ e a ∧ e b (cid:1) ∧ e c − αkle d ∧ d v ∧ d τ = 0 , (79) α ˚ r ab ∧ ˚ r ba = 0 . (80)Eq. (80) implies that the four-dimensional brane geome-try is restricted to be Pontryagin-density-vanishing when α (cid:54) = 0. Eq. (79) corresponds to the standard four-dimensional Einstein–Hilbert equations, with an effectivefour-dimensional stress-energy tensor given by κ T ab ∗ e a = αkle b ∧ d v ∧ d τ (81)and a “cosmological function”Λ ( x ) = 3 k τ ( x ) . (82)Finally, eq. (78) indicates that this cosmological functionΛ ( x ) is not arbitrary, but related to the brane’s Eulerdensity, Λ ( x ) = (cid:115) ∗ (cid:18) − (cid:15) abcd ˚ r ab ∧ ˚ r cd (cid:19) . (83)Spherically symmetric solutions to this modified the-ory of gravity include the Schwarzchild solution only inthe far-field limit. In order to avoid unphysical conse-quences, one might wish to turn to the supersymmetricextension of the theory. In this case, some components ofthe one-form gravitini in the super-connection can playthe role of dark matter fields non-minimally coupled tothe geometry. This may allow for a more realistic geom-etry solution, which will be treated elsewhere.Using the non-perturbative equations (78)–(80), it ispossible to check the consistency of the perturbative anal-ysis made in section IV B. In order to focus our atten-tion on the gravitational piece, we have chosen a non-perturbative solution which is much more general forthe curvature and torsion. However, we have chosen amore restrictive φ solution, in such a way that it satis-fies d φ ∧ d φ = 0 and kills many terms orthogonal to thebrane.After some algebraic manipulation, the “problematic”field equation (51) becomes eq. (78). The same happenswith eqs. (52) and (79), and eqs. (53) and (80), respec-tively.Now it is possible to understand why the linear per-turbations in section IV B were allowed to propagatein the form of standard gravitational waves on a four-dimensional Minkowskian background. The first term Here ∗ denotes the four-dimensional Hodge dual. in eq. (79) generates the standard four-dimensional tor-sionless Einstein–Hilbert dynamics. Eq. (78) does notfreeze the dynamics, but it requires the “cosmologicalfunction” given by eq. (83). The background consideredin section IV B is equivalent to ˚ r ab = τ = 0. In such acase, the cosmological function Λ = 3 k τ vanishes andit doesn’t change under linear perturbations. Therefore,torsion-preserving perturbations on such a backgroundmust reproduce standard four-dimensional gravitationalwaves on the brane, as was shown in eq. (68). D. Cosmological Models
So far, we have studied solutions for our d = 5 CSgravity theory when torsion is nonzero along the bulk.We have seen that torsion itself has an effect upon theEinstein equations for the brane [i.e., that torsion playsa role in the cosmological function (83)], so it would beinteresting to explore whether it somehow impacts thedynamics of certain systems, e.g., cosmological models.In order to probe the possible behavior when matter iscoupled to the theory (e.g., in a supersymmetric exten-sion), let us consider the cosmological toy model gener-ated when the cosmological constant is replaced by theexpression in eq. (83). To this end, we use the effectiveEinstein equations for the brane (79) with a spatial-flatFLRW metric, along with the effective four-dimensionalstress-energy tensor (81), taken to be that of a perfectfluid, suitable for a cosmological treatment.With the previous considerations, the cosmologicalfunction (83) becomesΛ ( t ) = 3 c ˙ aa (cid:114) ¨ aa = 3 c √− qH , (84)where q = − ¨ aa/ ˙ a is the cosmological deceleration pa-rameter. From eq. (84), we see that our model requires q < (cid:18) ˙ aa (cid:19) − ˙ aa (cid:114) ¨ aa = 13 c κ ρ, (85)2 ¨ aa + (cid:18) ˙ aa (cid:19) − aa (cid:114) ¨ aa = − c κ p. (86)We can see that eqs. (85)–(86) together imply the energybalance equation ˙ ρ + 3 ˙ aa ( ρ + p ) = 0 . (87)Now if we choose an equation of state in the form of abarotropic fluid p = wρ , plug it into the energy balanceequation, and solve the resulting system together with eq. (85), we obtain several families of solutions. One ofthem is a scale factor of the form a ( t ) = a exp (cid:32) c (cid:114) ¯Λ3 t (cid:33) , (88)where a and ¯Λ are constants. This alternative leads toa standard cosmology with a fixed cosmological constantΛ = ¯Λ and exponential growth, so we won’t pursuit itany further.More interesting solutions are of the form a ( t ) = a p (cid:18) tt p (cid:19) α w , (89)and a ( t ) = a (cid:18) H α w t (cid:19) α w , (90)where the α w exponent is given by α w = −
49 ( w + 1) (cid:0) w − (cid:1) . (91)Let us briefly consider both scenarios. When the α w exponent is positive, the solution describes a universewhere at t = 0 we have a (0) = 0 and a divergent Hubbleparameter, H ( t ) = α w t . (92)The interesting point is that we always have a negativeconstant deceleration parameter, q = − (cid:20) (cid:18) w + 13 (cid:19)(cid:21) , (93)implying an accelerating universe regardless of the valueof w .Fig. 1, drawn for the particular case of w = 0 (dust), α w = 4 /
3, is representative of the behavior of the cosmo-logical parameters.On the other hand, eq. (90) describes a scenario whereat t = 0 we have a (0) = a and H (0) = H . Again,we find exactly the same constant and always negativedeceleration parameter (93) from the former solution. Inthis particular model, it is possible to find some singu-larities for a ( t ) at finite future times depending in thevalues of w . However, the important point is that theidea of a cosmological function given by eq. (84) enforcesa negative deceleration parameter.It is worth observing that other “self-accelerating”models are known to arise from branes in higher-dimensional gravity (see Refs. [30–32]). Therefore, itseems natural to also find this kind of models in thecontext of the modified FLRW equations (85)–(86) orig-inated from a torsional RS state in five dimensional CSgravity. However, the model shown here has the partic-ular feature of always requiring acceleration in a naturalway, regardless of the details of the coupling to matter.In particular, it is possible to have accelerated universeseven in the case of only dust (dark matter). − t/t p a/a p H/t − p qρ/ρ p p/p p FIG. 1. Cosmological parameters a , H , q , ρ and p for thesolution in eq. (89), with w = 0. The curve for Λ is identicalto the curve for ρ (properly normalized). When w = 0, thepressure vanishes. V. CONCLUSIONS
The main result of the present article is the realiza-tion that a flat RS brane is a solution of five-dimensionalCS gravity, and more importantly, it is a vacuum whichadmits the propagation of linear perturbations. The CStheory is off-shell gauge invariant under the Weyl sub-group of Conf . Both, the solution and the symmetry,were carefully chosen in such a way that:1. The vacuum solution is a higher-dimensional gen-eralization of a Bianchi space.2. The symmetry of the CS theory contains theBianchi algebra as the translational subalgebra,and the f¨unfbein is the piece of the connection as-sociated to these generators.Together, both conditions imply that, on the vacuum, thef¨unfbein corresponds to the MC one-form for the trans-lational part of the algebra. In particular, this impliesthat the associated gauge curvature components vanish.Our solution requires that five-dimensional torsion benonzero. It then becomes necessary to distinguish be-tween Lorentzian and Riemannian curvature. The cho-sen vacuum is indeed Lorentz-flat but not Riemann-flat.In this sense, it is necessary to follow an approach verysimilar to the one of Refs. [33, 34] but in five insteadof three dimensions, and featuring a brane instead of ablack hole. And in the same way as in Refs. [33, 34],torsion is covariantly constant, D T A = 0.Despite all this, four-dimensional brane torsion van-ishes and the dynamics of the perturbations is thesame as standard gravitational waves in d = 4 on aMinkowskian background. However, far from the per-turbative regime it induces an effective four-dimensional “cosmological function” Λ ( x ) proportional to the squareroot of the Euler density of the brane,Λ ( x ) = (cid:115) ∗ (cid:18) − (cid:15) abcd ˚ r ab ∧ ˚ r cd (cid:19) (94)This turns out to be an interesting toy model from thepoint of view of cosmology, because the field equationsonly allow accelerated cosmologies with q <
0, regardlessof the details of matter interaction. The effect of torsionin braneworld scenarios on the effective four-dimensionalEinstein equations has also been studied in Refs. [35–39].There is a lot of room for further research from theideas presented here.First, it is necessary to remember that what we havepresented here is just a toy model, and it would be inter-esting to construct a more realistic version of it. Despiteof having found a nice solution to the vacuum problemfor this particular CS theory and having some interest-ing possibilities in cosmology, it is necessary to stress thathaving an effective “cosmological function” Λ ( x ) propor-tional to the square root of the Euler density also inducessome unphysical consequences (e.g., Schwarzschild solu-tion only in the far-field limit). However, the problemseems to be solvable by considering the supersymmetricextension of the CS theory here presented (i.e., CS super-gravity, see Refs. [15, 16]). In this case, it is necessary towork with a super-connection one-form, where the one-form gravitini fields are associated to the fermionic pieceof the superalgebra. This would relax some of the con-straints, and the gravitini should play the role of a darkmatter background for the brane geometry. On this back-ground, a more realistic dynamics is to be expected. Asomewhat similar system was studied in Ref. [40], where,however, the CS theory is based upon a supersymmetricversion of the anti-de Sitter algebra, instead of the Weylalgebra.Second, it is possible to conjecture that the underly-ing reason for the nice behavior of our vacuum choice isthat, on it, spacetime has the structure of a group man-ifold, with the f¨unfbein corresponding to its associatedMC one-form. Here we have studied just one single case(the generalized Type V space of the Bianchi classifica-tion), but many other possibilities remain to be studied. ACKNOWLEDGMENTS
The authors wish to thank Jorge Zanelli and PatricioSalgado for enlightening discussions, as well as the twoanonymous referees for their careful reading of the paperand their valuable suggestions. FI is grateful to Jos´e A.de Azc´arraga for his valuable suggestions regarding thetreatment of Lie algebras. This research was partiallyfunded by Fondecyt grants 1130653 and 1150719 (FI),and 3130444 (PM), and by Conicyt scholarship 22131299(FC-T), from the Government of Chile.0
Appendix A: Representation of the Weyl Subalgebra
In order to find a useful invariant tensor [cf. eqs. (42)–(47)] for the Weyl subalgebra (10)–(13), it is necessaryto find a suitable matrix representation.Such a representation can be constructed using theDirac representation in D = 6,Γ A Γ B + Γ B Γ A = 2 η AB , (A1)in terms of 2 × Dirac matrices. Here we use uppercaseindices with the range
A, B, C = 0 , , . . . , a, b, c = 0 , , . . . , { Γ A ··· A n } n =0 for all 2 × matrices asΓ A ··· A n = 1 n ! δ B ··· B n A ··· A n Γ B · · · Γ B n , where, with the exception of the identity matrix Γ = ,all Γ A ··· A n matrices are traceless.In particular, let us consider the matricesΓ AB = (cid:26) Γ ab , Γ a , Γ A = (cid:26) Γ a , Γ . In terms of them, a representation for the algebra (10)–(13) is given by J ab = 12 Γ ab , Z a = 12 √ a − Γ a ) − lk a (cid:18)
12 Γ + α (cid:19) , D = 12 Γ + α , with α an arbitrary constant.In order to construct the invariant tensor (42)–(47), itis necessary to use the symmetrized trace, some prop-erties of Dirac matrices in D = 6, and the algebra ofmatricesΓ M ··· M p Γ N ··· N q = min( p,q ) (cid:88) s =0 ( − s ( s − / ( s !) ( p − s )! ( q − s )! ×× η [ B ··· B s ][ C ··· C s ] δ A ··· A p − s B ··· B s M ··· ··· M p ×× δ C ··· C s A p − s +1 ··· A p + q − s N ··· ··· N q Γ A ··· A p + q − s , where η [ A ··· A s ][ B ··· B s ] = η A C · · · η A s C s δ C ··· C s B ··· B s . For more details on higher-dimensional Dirac matricesmanipulation, a valuable guide can be found in Ref. [41]. [1] J. Zanelli,
Modifications of Einstein’s Theory of Gravityat Large Distances , ch. Chern–Simons Forms andGravitation Theory, pp. 289–310. SpringerInternational Publishing, Switzerland, 2015.[2] D. Lovelock, “The Einstein tensor and itsgeneralizations,”
J. Math. Phys. (1971) 498.[3] L. Randall and R. Sundrum, “A Large mass hierarchyfrom a small extra dimension,” Phys. Rev. Lett. (1999) 3370–3373, arXiv:hep-ph/9905221 .[4] L. Randall and R. Sundrum, “An Alternative tocompactification,” Phys. Rev. Lett. (1999)4690–4693, arXiv:hep-th/9906064 .[5] V. Cardoso and J. P. S. Lemos, “Scalar, electromagneticand Weyl perturbations of BTZ black holes:Quasinormal modes,” Phys. Rev. D (2001) 124015, arXiv:gr-qc/0101052 .[6] J. A. de Azc´arraga and J. M. Izquierdo, Lie groups, Liealgebras, Cohomology and some Applications in Physics .Cambridge University Press, Cambridge, UK, 1995.[7] L. Landau and E. Lifshitz,
Course of TheoreticalPhysics vol. 2: The Classical Theory of Fields .Butterworth-Heinemann, Oxford, UK, 1980.[8] F. Izaurieta and E. Rodr´ıguez, “On eleven-dimensionalsupergravity and Chern–Simons theory,”
Nucl. Phys. B (2012) 308–319, arXiv:1103.2182 [hep-th] .[9] S. Deser, R. Jackiw, and S. Templeton,“Three-dimensional massive gauge theories,”
Phys. Rev.Lett. (1982) 975–978. [10] S. Deser, R. Jackiw, and S. Templeton, “Topologicallymassive gauge theories,” Annals Phys. (1982)372–411.[11] A. Ach´ucarro and P. K. Townsend, “A Chern–Simonsaction for three-dimensional anti-de Sitter supergravitytheories,”
Phys. Lett. B (1986) 89–92.[12] A. Ach´ucarro and P. K. Townsend, “Extendedsupergravities in d = 2 + 1 as Chern–Simons theories,” Phys. Lett. B (1989) 383–387.[13] A. H. Chamseddine, “Topological gauge theory ofgravity in five and all odd dimensions,”
Phys. Lett. B (1989) 291–294.[14] A. H. Chamseddine, “Topological gravity andsupergravity in various dimensions,”
Nucl. Phys. B (1990) 213–234.[15] M. Ba˜nados, R. Troncoso, and J. Zanelli, “Higherdimensional Chern–Simons supergravity,”
Phys. Rev. D (1996) 2605–2611, arXiv:gr-qc/9601003 .[16] R. Troncoso and J. Zanelli, “New gauge supergravity inseven and eleven dimensions,” Phys. Rev. D (1998)101703, arXiv:hep-th/9710180 .[17] T. W. B. Kibble, “Lorentz invariance and thegravitational field,” J. Math. Phys. (1961) 212.[18] S. W. MacDowell and F. Mansouri, “Unified geometrictheory of gravity and supergravity,” Phys. Rev. Lett. (1977) 739–742.[19] J. Zanelli, “Chern–Simons forms in gravitationtheories,” Class. Quant. Grav. (2012) 133001, arXiv:1208.3353 [hep-th] .[20] E. Witten, “(2 + 1)-dimensional gravity as an exactlysoluble system,” Nucl. Phys. B (1988) 46–78.[21] E. Witten, “Topology changing amplitudes in(2 + 1)-dimensional gravity,”
Nucl. Phys. B (1989)113–140.[22] E. Witten, “Quantum Field Theory and the JonesPolynomial,”
Comm. Math. Phys. (1989) 351.[23] J. Saavedra, R. Troncoso, and J. Zanelli, “Degeneratedynamical systems,”
J. Math. Phys. (2001)4383–4390, hep-th/0011231 .[24] O. Miˇskovi´c and J. Zanelli, “Dynamical structure ofirregular constrained systems,” J. Math. Phys. (2003) 3876–3887, arXiv:hep-th/0302033 .[25] O. Miˇskovi´c, R. Troncoso, and J. Zanelli, “Canonicalsectors of five-dimensional Chern–Simons theories,” Phys. Lett. B (2005) 277–284, arXiv:hep-th/0504055 .[26] M. Hassa¨ıne, R. Troncoso, and J. Zanelli, “Poincar´einvariant gravity with local supersymmetry as a gaugetheory for the M-algebra,”
Phys. Lett. B (2004)132–137, arXiv:hep-th/0306258 .[27] M. Hassa¨ıne, R. Troncoso, and J. Zanelli, “11Dsupergravity as a gauge theory for the M-algebra,”
PoSWC (2005) 006, arXiv:hep-th/0503220 . http://pos.sissa.it/archive/conferences/013/006/WC2004_006.pdf .[28] F. Izaurieta, E. Rodr´ıguez, and P. Salgado, “Ontransgression forms and Chern–Simons (super)gravity,” arXiv:hep-th/0512014 .[29] F. Izaurieta, E. Rodr´ıguez, and P. Salgado, “Theextended Cartan homotopy formula and a subspaceseparation method for Chern–Simons theory,” Lett.Math. Phys. (2007) 127–138, arXiv:hep-th/0603061 .[30] C. Deffayet, “Cosmology on a Brane in MinkowskiBulk,” Phys. Lett. B (2001) 199–208, hep-th/0010186 .[31] C. Deffayet, G. Dvali, and G. Gabadadze, “Accelerateduniverse from gravity leaking to extra dimensions,”
Phys. Rev. D (2002) 044023, astro-ph/0105068 .[32] G. Dvali and M. S. Turner, “Dark Energy as aModification of the Friedmann Equation,” astro-ph/0301510 .[33] P. D. Alvarez, P. Pais, E. Rodr´ıguez,P. Salgado-Rebolledo, and J. Zanelli, “The BTZ blackhole as a Lorentz-flat geometry,” Phys. Lett. B (2014) 134–135, arXiv:1405.6657 [gr-qc] .[34] P. D. Alvarez, P. Pais, E. Rodr´ıguez,P. Salgado-Rebolledo, and J. Zanelli, “Supersymmetric3D model for gravity with SU (2) gauge symmetry, massgeneration and effective cosmological constant,” Class.Quant. Grav. (2015) 175014, arXiv:1505.03834[hep-th] .[35] J. M. Hoff da Silva and R. da Rocha, “Braneworldremarks in Riemann–Cartan manifolds,” Class. Quant.Grav. (2009) 055007, arXiv:0804.4261 [gr-qc] .[36] J. M. Hoff da Silva and R. da Rocha, “Braneworldremarks in Riemann–Cartan manifolds,” Class. Quant.Grav. (2009) 179801.[37] S. Khakshournia, “Comment on ‘Braneworld remarks inRiemann–Cartan manifolds’,” Class. Quant. Grav. (2009) 178001.[38] J. M. Hoff da Silva and R. da Rocha, “Reply to‘Comment on ‘Braneworld remarks in Riemann-Cartanmanifolds”,” Class. Quant. Grav. (2009) 178002.[39] J. M. Hoff da Silva and R. da Rocha, “Torsion Effectsin Braneworld Scenarios,” Phys. Rev. D (2010)024021, arXiv:0912.5186 [hep-th] .[40] R. S. Garavuso and F. Toppan, “Chern–Simons AdS supergravity in a Randall–Sundrum background,” Nucl.Phys. B (2008) 320–330, arXiv:0705.4082[hep-th] .[41] A. Van Proeyen, “Tools for supersymmetry,” in
Proceedings, Spring School on Quantum Field Theory.Supersymmetries and Superstrings: Calimanesti,Romania, April 24-30, 1998 , vol. 9, pp. 1–48. 1999. arXiv:hep-th/9910030 [hep-th] . http://cis01.central.ucv.ro/pauc/vol/1999_9_part1/1999partI-1-48.pdfhttp://cis01.central.ucv.ro/pauc/vol/1999_9_part1/1999partI-1-48.pdf