A conformally invariant generalization of string theory to higher-dimensional objects. Hierarchy of coupling constants
aa r X i v : . [ g r- q c ] O c t A CONFORMALLY INVARIANT GENERALIZATION OF STRING THEORYTO HIGHER-DIMENSIONAL OBJECTS.HIERARCHY OF COUPLING CONSTANTS
F.Sh. Zaripov
Department of Mathematics, Kazan State Humanitarian and Pedagogical University, 1 Mezhlauk St., Kazan 420021,Russia; e-mail: [email protected]
We suggest a conformally invariant generalization of string theory toi higher-dimensional objects. As such a model,we consider a conformally invariant σ model. For this theory, the Hamiltonian formalism is constructed, and the fullset of constraints is found. The equations obtained are studied under a fixed gauge. It is shown that special cases ofthe model are string theory and Einstein’s theory of gravity. Cosmological application of the suggested theory arestudied. It is shown that Friedmann-like models can be described in this framework. Our models make it possibleto interpret the Universe evolution as evolution of three-dimensional objects embedded in a higher-dimensional flatspace-time. The remarkable achievements of string and superstringtheories are well known: evaluation of the space-timedimension, fixing a particular gauge group, inclusion ofgravity into a unified scheme etc. [1]. These achieve-ments stimulate an interest in studies of geometricobjects of higher dimension, such as membranes orp-branes. It is known, however, that [5] in standardmembrane theories the absence of conformal invari-ance precludes the usage of string-theoretical methods.For instance, the requirement that conformal invarianceshould be preserved at the quantum level leads, in stringtheory, to fixing the space-time dimension [2]. Thereare also other arguments [3] in favour of the require-ment that a physical field theory should be conformallyinvariant, at least at the classical level.On this basis, we have previouly suggested a confor-mally invariant generalization of string theory to higher-dimensional objects [4]. This paper, aimed at furtherrealization of this approach, is devoted to obtaining andinvestigation of Hamiltonian equations and constraintequations of the theory under consideration. This ideawas originally suggested as a quantum theory by anal-ogy with string theory. However, a further analysis hasshown the necessity of an initial classical analysis ofthis theory. It has turned out that even the classicallevel of the theory contains results of interest related togravitation theory and p-brane theory. The action thatserves as a basis for the suggested theory, being a gener-alization of string and p-brane theory, is simultaneouslya certain generalization of Einstein’s general relativity.We suggest that general relativity should be consideredas a special case of a conformally invariant sigma model,appearing as a result of conformal symmetry violation.This paper is devoted to foundation and analysis ofthe above ideas in the classical case.The recent development of multidimensional theo-ries have been, to a large extent, related to the so-called branes. In this theory [6–9], the observable Universe isconsidered as a surface (brane) embedded in a higher-dimensional space-time. It is hoped that this approachcan lead to a success in solving the fundamental prob-lem of the hierarchy of physical coupling constants andthe cosmological constant problem. The hierarchy prob-lem lies in the existence of a huge difference between theelecroweak energy scale of about 1 TeV and the gravi-tational energy scale of the order of 10 GeV. Besides,the energy density related to the cosmological constantshould be about 120 orders of magnitude smaller thanthe possible energy density values for the known modelsof quantum theory of the weak and strong interactions.In the theory suggested, the gravitational constantis related to the dynamic characteristics of the model,and it is obtained in multidimensional space-time dueto localization of solutions to nonlinear equations, byanalogy with the Higgs effect in gauge field theory.
The action for a membrane (or p-brane) does not ad-mit conformal transformations, and these models do notpossess a natural candidate for the role of an anoma-lous symmetry like conformal symmetry in string the-ory. To circumvent this difficulty without abandoningthe string-theoretical ideology, we suggest the followinggeneralization of string theory: S = 1 w Z (cid:26) −
12 ( ∇ ν X, ∇ ν X ) + ˜ ξ n R ( X, X )+ Λ(
X, X ) ρ (cid:27) √− g ˆ d p +1 σ, (1)where we use the notations:( X, X ) = X A X B η AB , ( ∇ ν X, ∇ ν X ) = ∇ ν X A ∇ µ X B g νµ η AB ,ρ = ( p + 1) / ( p − . In the action (1), the functions X A = X A ( σ µ ), with A, B = 1 , , . . . , D ; µ, ν = 0 , , . . . , p , map the n = p + 1 -dimensional manifold Π, described by the met-ric g µν , into D -dimensional space-time M with themetric η AB , where the space M is determined by theMinkowski metric with the signature ( − , + , . . . , +).However, it turns out that, in a detailed study, it ismore convenient to leave the signature of M arbitrary.The flat space signature is here understood as the setof signs of the elements along the main diagonal (+1and − n R is thescalar curvature of the manifold Π, the operator ∇ ν means a covariant derivative in the manifold Π, wherethe Christoffel symbols are connected with the metricin the standard manner. We will assume that the spaceΠ is parametrized by the coordinates σ µ , where σ = t is the temporal coordinate while the components σ i ( i = 1 , , . . . , p ) describe a certain p -dimensional ob-ject, to be designated as Γ. The quantities w , ˜ ξ and Λare constants. The models like that with the action (1)are also often called nonlinear σ models.The action (1) is conformally invariant if˜ ξ = ξ ≡ − p − p . (2)This invariance is expressed in the fact that the equa-tions obtained by varying the action (1) with respect tothe fields ˆ g and ˆ X are invariant under the local Weylscale changes g µν e φ g µν , X A e ξpφ X A , (3)for an arbitraryf φ = φ ( σ µ ).After varying the action (1), the field equations forˆ X and ˆ g have the following form: Y A ≡ (cid:3) X A + 2 ξ n R X A + 2Λ ρ ( X, X ) ρ − X A = 0 , (4) T αβ ≡ T αβ + 2 ξw [ − n R αβ + 12 n R g αβ + ∇ α ∇ β − g αβ (cid:3) ]( XX ) = 0 , (5)where T αβ = 1 w [( ∇ α X, ∇ β X ) + L g αβ ] (6)are the terms appearing due to variation of the La-grangian density L = − w g µν ( ∇ µ X, ∇ ν X ) + 1 w Λ( X, X ) ρ . (7)If the action is supplemented by Lagrange functionsof other matter fields, then Eq. (5) is replaced by theequation T αβ + T eαβ = 0 , (8)where T eαβ is the energy-momentum tensor of the otherfields. In case ( X, X ) = const, as follows from (8) and(5), the equations are similar to Einstein’s, with thecanonical energy-momentum tensor T αβ and the effec-tive gravitational constant G e = − w πξ ( X, X ) , w = w π . (9) Let us point out the important fact that for strings( p = 1 ) the general solution to Eqs. (5) has the form Bg µν = ( ∇ µ X, ∇ ν X ) , µ, ν = 0 , p, (10)where B is an arbitrary function. Thus the originalmetric g µν is connected by a conformal transformationwith the induced metric ( ∇ µ X, ∇ ν X ). Unfortunately,in the general case p > g with the metricinduced by the solutions X A = X A ( σ µ ), as well as thatof a physical interpretation of this connection, have notbeen solved for an arbitrary dimension.In what follows, we will consider some special solu-tions to Eqs. (4)–(8), being of interest for physics. To pass over to the Hamiltonian formalism, we make,in the action (1), a ( p + 1)-partition. Employing theresults of Refs. [16, 17], we introduce the parameters N and N i , the “lapse” or “shift” functions, and the metricfunctions of the p -dimensional geometry h ij , where i =1 , p : g = N s N s − N , g i = N i ,g ij = h ij , √− g = N √ h. Then, taking into account the results of Ref. [16], wecan present the scalar curvature in the form n R = R − (Sp ˆ K ) + Sp( ˆ K ) − N √ h ∂ α [ N √ h ( n α Sp ˆ K + ∇ β n α )] . (11)Here R is the scalar curvature calculated for the metric h ij , a α = ∇ β n α n β is the ( p + 1)-dimensional acceler-ation of an observer moving along a timelike normal ~n to consecutive sections. The space-time Π is assumedto be foliated into a one-parameter family of spacelikehypersurfaces with the parameter t . The quantity ˆ K isthe extrinsic curvature tensor of the spacelike sections: n α = {− N, } , ( ~n · ~n ) = − , (12)Sp ˆ K = h ij K ij , Sp ˆ K = K ij K ij ,K ij = 12 N ( D i N J + D j N i − ∂ t h ij ) , (13)where D i is a covariant derivative calculated with themetric h ij .Let us now pass over to a description in terms of thephase-space variables, i.e., the generalized coordinatesand momenta: { q I } = { X A , h lk , N, N i } , { p I } = { P A , Π lk , p N , p N i } , where p I = δLδ ( ∂ t q I ) , (14)where L is the Lagrangian corresponding to the action(1). We obtain as a result: P = √ hN w [ ˙ X − N i D i X + 4 ξXN Sp ˆ K ] , Π lk = ξ √ hw (cid:20) ( X, X )(Sp ˆ K · h lk − K lk ) − N h lk ( ∂ t ( X, X ) − N i D i ( X, X )) (cid:21) , (15)the remaining momenta are zero. In the conformaltransformations (3), the phase variables are transformedas follows: h lk e φ h lk , X e ξpφ X , N e φ N,N i e φ N i , P e − ξpφ P, Π lk e − φ Π lk . (16)As follows from (16), there is a constraint between P and Π lk . This constraint may be written as M ≡ ξp ( P, X ) = 0 . (17)Integrating by parts and rejecting terms with a full di-vergence, one can write the action (1) in terms of thecanonical variables as S = Z [ ˙ h lk Π lk + ( P, ˙ X ) − N H − N i H i − λ M M − ∂ i Q i ] ˆ d p +1 σ, (18)where H ≡ wξ √ h ( X, X ) [Sp( ˆΠ ˆΠ) − p ( SP ˆΠ) ] + w √ h P + √ hw (cid:20) − ξ ( X, X ) R + 12 ( D s X, D s X )+ 2 ξ ∆( X, X ) − Λ( X, X ) ρ (cid:21) , (19) H l ≡ ( P, D l X ) − D s Π sl , (20) Q i = 2 √ hξw [( X, X ) D i N − N D i ( X, X )]+ 2 N k Π ki + N i (cid:20) ξp Sp ˆΠ + 1 wN E (cid:21) . (21)Here we have used the notations E = ∂ t ( X, X ) − N s D s ( X, X ) , (22)∆ = h lk D l D k and ( D s X, D s X ) = ( D i X, D j X ) h ij .The function λ M is arbitrary. This is related to theimpossibility of resolving the velocity λ E = ∂ t ( X, X ) interms of the momenta. It can be shown that p E ≡ δLδλ E = M ξp ( X, X ) . The constraint (17) has appeared because of the invari-ance of the theory with respect to(16). To take this factinto account explicitly, let us transform the integrand in (18) according to (16) for φ = ψ and introduce the field ψ . Then the expression (18) takes the form S = Z [ ˙ h lk Π lk + ( P, ˙ X ) − N H − N i H i − ( λ M − ˙ ψ + N s D s ψ ) M − ∂ i ˜ Q i ] ˆ d p +1 σ. (23)That is, we could use, instead of λ M , the field ψ ( σ ).The corresponding momentum is p ψ = M . The diver-gence term in Eq. (23) does not affect the equations ofmotion but affects the boundary conditions. After thetransformations (16), the quantities Q i turn into ˜ Q i ,where˜ Q i = Q i + ( X, X ) √ hw × (cid:20) ξpN D i ψ + ξpN i N ( ˙ ψ − N s D ψ ) (cid:21) . (24)The latter expression implies that, taking into accountthe boundary effects, we shall obtain certain boundaryconditions applied to the function ψ , violating the in-variance of the theory with respect to (16). It is proba-bly reasonable to omit the divergence term (24) from theaction, replacing the original Lagrangian density with L + ∂ i Q i . An argument in favour of such a replacementis that for ( X, X ) = const and X i = const, the actionacquires the Einstein form. In the construction of theHamiltonian formalism for Einstein’s theory, such termsare omitted [17].The conditions that the primary constraints are con-served in time,Φ (1) I : p µ ≡ { p N , p N i } = 0 , p ψ − M = 0 , (25)with the Hamiltonian constructed in the standard way[17], H = ∂ i Q i + N H + N s H s + λ M (26)and the extended Hamiltonian H = H + λ I Φ (1) I , do not allow one to determine the functions λ I , butthere emerge secondary constraints:Φ (2) I : H v ≡ (cid:8) H , H l , M (cid:9) = 0 . (27)Consider the conservation conditions for the con-straints (27). To do so, it is necessary to calculate thePoisson brackets:[Φ K , Φ J ] ≡ δ Φ K δq I δ Φ J δp I − δ Φ K δp I δ Φ J δq I . After cumbersome calculations, it can be shown that ifthe appearing divergence terms, leading to surface in-tegrals, vanish, then the constraint conservation condi-tions do not allow determining the functions λ I and donot lead to new constraints. All constraints are thusfirst-class constraints. The equations of motion ˙ q = [ q, H ] have the follow-ing form:˙ X = N w √ h P + N s D s X + 4 ξpλ M X, (28)˙ h lk = 2 wNξ √ h ( X, X ) (cid:20) Π lk − p (Sp Π) h lk (cid:21) + D ( l N k ) + 2 λ M h lk , (29)˙ P = 2 wNξ √ h ( X, X ) (cid:20) Sp( ˆΠ ˆΠ) − p (Sp ˆΠ) (cid:21) X + 2 ξ √ hw [ RN − N ] X + √ hw ( N ∆ X + D s N D s X )+ 2 ρN √ hw Λ( X, X ) ρ − X + D s ( N s P ) − ξpλ M P, (30)˙Π lk = − wNξ √ h ( X, X ) (cid:20) Π lm Π mk − p Sp( ˆΠ)Π lk (cid:21) + N √ h h lk (cid:26) wξ ( X, X ) [Sp( ˆΠ ˆΠ) − p (Sp ˆΠ) ] + w P (cid:27) + ξ √ hw [ N D l D k ( X, X ) − N R lk + ( X, X ) D l D k N − h lk (cid:0) ( X, X )∆ N + D s ( X, X ) D s N (cid:1) ]+ N √ h w ( D l X, D k X ) − c lk + 12 P N ( l D k ) X − λ M Π lk − N w √ h ( X, X ) h lk H , (31)where c lk = √ hD s (cid:18) √ h ( N ( l Π k ) s − N s Π lk ) (cid:19) In what follows, we will put the constant w equal tounity. If, instead of the indefinite coefficient λ M , we in-troduce the field ψ , we should make the following sub-stitution in the equations of motion: λ M = λ − ˙ ψ + ψ s N s , (32)where λ is an arbitrary function of the phase variables.This function may be chosen to be equal to zero, whichsimply re-defines the function ψ . Then, using the sub-stitutions h lk = e − ψ ¯ h lk , X = e − ξpψ ¯ X, N = e − ψ ¯ N ,N i = e − ψ ¯ N i , P = e ξpψ ¯ P , Π lk = e ψ ¯Π lk , (33)one can exclude the field ψ from Eqs. (28)–(31) andpass over to the conformally invariant canonical vari-ables { ¯ q I , ¯ p I } , which is equivalent to putting λ M = 0in the equations. However, for studying different gaugeconditions, it is more convenient to preserve the arbi-trariness in choosing the function λ . To impose thecanonical gauge, it is necessary to impose 2 p + 4 sup-plementary conditions, according to the number of first-class constraints. We will consider as such constraints the class of additional conditions Φ G of the form N = ˜ N , N l = ˜ N l , λ = 0 ,χ µ = 0 , F = 0 , (34)where χ µ are p + 1 functions of the phase variables h lk and X A , while F is a function of the phase variables h lk , X A , P, Π lk . These functions are chosen in sucha way that det[ ˆΦ , ˆΦ] = 0 , where ˆΦ = { Φ , Φ G } . Let usdenote H v = { H , H l , M } and χ u = { χ µ , F } , then, inthe case under consideration,det[ ˆΦ , ˆΦ] = (det[ H v , χ u ]) ([ λ, p ψ ]) . (35)A gauge, related to a choice of the function F , violatesthe conformal symmetry and determines a “representa-tive” from each class of conformally equivalent metrics.To reach comprehension of the different kinds of gaugeconditions, let us consider some consequences of the con-straint equations (27) and the equations of motion (28),(31). Let us define the conformally invariant tensor Θ lk which is traceless on the surface of the constraints:Θ lk ≡ ξ √ hN (cid:20) D l D k ( X, X ) − h lk p ∆( X, X ) − ( X, X )( R lk − h lk p R ) (cid:21) + 2 ξ √ h (cid:20) ( X, X )( D l D k N − h lk p ∆ N ) (cid:21) + N √ h (cid:20) ( D l X, D k X ) − h lk p ( D s X, D s X ) (cid:21) + h lk p H . (36)Then, using Eqs. (28) and (31) as well as the definitions(19), (20) and (17), it is easy to prove the followingidentity: ∂ t (cid:20) Π lk − δ lk p Π ss (cid:21) − D s (cid:20) N s (Π lk − δ lk p Π ss ) (cid:21) = F lk + 12 Θ lk + 12 N ( l H s ) h sk − δ lk p N s H s , (37)where F lk = Π ls D k N s − Π sk D s N l . The latter equation is equivalent to Eq. (31) providedthe conditions (28)–(30) hold.
Let us call the “partial embedding” condition the choiceof the supplementary conditions Φ G obtained from therequirement h lk = B ( D l X, D k X ) , l, k = 1 , p, (38)where B is a certain function. Thus the metric h lk ,entering into the original action, is connected with theinduced metric ( ∇ l X, ∇ k X ) by a conformal mapping.In the string case, the general solution to the con-straing equations has the form Bg µν = ( ∇ µ X, ∇ ν X ) , µ, ν = 0 , p, (39)where B is an arbitrary function. This solution followsfrom (36) and (37) if one puts the momenta Π lk and theparameter ξ equal to zero. For an arbitrary dimension,Eq. (39) (for B = 1 ) determines the condition of fullembedding of the mainfold Π into the space-time M .This equation, written in the p + 1 -partition formalism,is equivalent to Eq. (38) and the equations( ˙ X, ˙ X ) = B ( N s N s − N ) , ( D l X, ˙ X ) = BN l , B ≡ ( D l X, D l X ) /p = 1 . (40)Thus we can consider two kinds of solutions corre-sponding to “partial” or full embedding. In the firstcase, the validity of Eq. (38) (for B = 1 ), as well asfor strings, would permit one to interpret the fields X A as the conventional coordinates of a d -object Γ in thespace-time M . In other words, this means that, fromeach class Γ of conformally equivalent manifolds, it ispossible to choose at least one “representative” Γ , suchthat the functions X A perform embedding of Γ intothe surrounding space M . Here, conformally equiva-lent manifolds are understood as manifolds whose met-rics are connected with the reparametrization invarianceand the conformal invariance (16).An invalidity of the relations (40), if (38) is valid,leads to some difficulties in the physical interpretation.If, by analogy with string theory, the space-time M isconsidered as physical space-time, then the coincidencebetween the original metric h lk and the induced metric( D l X, D k X ) makes the theory transparent, making itpossible to interpret the solutions X A = X A ( t, σ ) as anembedding of a p -dimensional object Γ into the physi-cal space-time M . However, a non-coincidence betweenthe “lapse” or “shift” functions of the original manifoldΠ and the p + 1 -dimensional “world history” mainfoldof the object Γ poses a question on the physical mean-ing of the original functions N and N i . To answer thisquestion, one can try to invoke the ideas of the Kaluza-Klein theory. We, however, put forward a conjectureaccording to which it is possible, at the expense of achoice of the corresponding reference frame and confor-mal gauge, and maybe also the dimension D , to achievethe validity of the conditions of full embedding of thewhole manifold Π into the space-time M . Eqs. (36)and (37) determine p ( p +1)2 − k p equations, whilethe number of arbitrary functions, determining the con-straints Φ G , is equal to 2 p + 4 . It is necessary to specify p + 2 functions N, N l , λ M . Besides, according to thenumber of first-class conditions, we shoud impose p + 2supplementary conditions. One can try to impose thelatter relations by requiring the p + 2 conditions (40)to be valid. As follows from Eqs. (36) and (37), to fulfilthe “partial embedding” conditions (38), the followingequations should hold: ∂ t (cid:20) Π lk − δ lk p Π ss (cid:21) − D s (cid:20) N s (Π lk − δ lk p Π ss ) (cid:21) − F lk = ξ √ h (cid:26) N (cid:20) D l D k ( X, X ) − h lk p ∆( X, X ) − ( X, X )( R lk − h lk p R ) (cid:21) + (cid:20) ( X, X ) (cid:18) D l D k N − h lk p ∆ N (cid:19)(cid:21)(cid:27) . (41)If Eq. (41) holds, then (38) follows from (36) and (37)The p + 2 functions N, N l , λ M should be chosen insuch a way that, due to this choice, Eqs. (41) hold. Thenumber of these functions for the dimensions p = 2 and p = 3 is 4 and 5, respectively, while the number ofequations (41) (the number k p ) is 2 and 5, respectively.This simple counting of the degrees of freedom showsthat, in the cases of interest p = 2 and p = 3 , it ispossible to choose a full embedding gauge.In the most general case, it can be proved that, tosatisfy the full embedding conditions (39) (for B = 1 ),it is necessary that the following equations hold:[ ξ n R + ρ ( X, X ) ρ − Λ] ∂ µ ( X, X ) = 0 , (42)where µ = 0 , p . The simplest proof can be performedwith the aid of the generally covariant equations (4).They are equivalent to the Hamiltonian equations (28)–(30). Acting with the covariant derivative ∇ γ on therelation (39) and contracting different pairs of indices,we obtain the equations( ∇ ν X, (cid:3) X ) + ( ∇ µ X, ∇ µ ∇ ν X ) = ∇ ν B, (43)2( ∇ µ X, ∇ µ ∇ γ X ) = ∇ γ B ( p + 1) , (44)where µ, ν, γ = 0 , p .From the latter equations combined with (4), we ob-tain Eq. (42). In the derivation, we taking into accountthe covariant constancy of the metric tensor g µν andthat B = 1 . Thus, as follows from (42), we can use twokinds of supplementary gauge conditions agreeing withthe full embedding condition:(1) F ≡ ( X, X ) − C = 0 , C = const; (45)(2) F ≡ ξ n R + ρ ( X, X ) ρ − Λ = 0 . (46)As follows from (5) and (45), in the first case the con-straint equations are similar to Einstein’s equations withthe canonical energy-momentum tensor (6) and the ef-fective gravitational constant (9). For the second case,one cannot exclude solutions in which G e is variable andcoordinate-dependent. The equations obtained, like Einstein’s equations, arestrongly nonlinear and cannot be solved in a generalform. However, the existing additional conformal sym-metry simplifies the search for solutions of these equa-tions. In this section, we simplify the equations obtainedby restricting the class of metrics under consideration.Paying more attention to the dimension n = 4 , let usconsider a model problem with the metric tensor h lk chosen in the form h lk = b ω lk , (47)where b = b ( t, σ ) is an arbitrary function and ω lk issome fixed metric. It will be essential for what followsthat the functions ω lk are time-independent: ˙ ω lk = 0 .In the two-dimensional case, the following relations al-ways hold: ω R lk − ω Rp ω lk = 0 . (48)The index ω means that the corresponding quanti-ties are calculated for the metric coefficients ω lk . Forinstance, ω lk may be chosen to be the metric of aconstant-curvature space. Then, ω R lk = k ( p − ω lk = − ξk pω lk , (49) k = { , , − } for surfaces of zero, positive and negat-ice curvature, respectively.Here and henceforth, we leave the dimension arbi-trary, considering simultaneously two-dimensional Γ ob-jects and objects of an arbitrary dimension, however, forthe latter we restrict ourselves to spaces which are con-formal to constant-curvature spaces. Let us impose the following conditions on the “lapse”and “shift” functions: N = b , N i = 0 . (50)After taking the trace of Eq. (29), it follows: λ M = ˙ bb = − ˙ ψ = ⇒ Π lk = 1 p (Sp Π) h lk . (51)With the constraint M = 0 , we obtainΠ lk = − ξ ( P, X ) h lk . (52)Consider the gauge condition (45)( X, X ) − C = 0 , C = const = 0 . (53)Then, substituting X = gb ξp , from (28) and (30)we obtain the equations for the field g A components:¨ g − ω ∆ g − ξg ω R = 2 ρ Λ C ρ − b g. (54)The constraint equations H = 0 and H l = 0 maybe brought to the form( ˙ X, ˙ X ) = − (4 ξpλ M ) C + b (2 ξCR + 2Λ C ρ − Bd ) , (55)( ˙ X, D l X ) = 16 ξ pCD l ( λ M N ) . (56) Eqs. (41) are reduced to the following ones:8 ξpC (cid:18) − D l D k ψ + D l ψD k ψ − h lk p ( − ∆ ψ + D s ψD s ψ ) (cid:19) + C ( ω R lk − ω lk p ω R ) = 0 . (57)Using (55), (56) and the consequence of Eq. (53)( ˙ X, ˙ X ) = − ( ¨ X, X ) , we obtain an equation for finding the conformal factor:¨ ψ + 4 ξp ˙ ψ − p b (cid:0) R + 2∆ ψ − D s ψD s ψ − p + 1) q (cid:1) = 0 . (58)The scalar curvature may be expressed in terms ofthe function ψ : R = ω R b − + 8 ξp [ − ψ − ( p − ψ s ψ s ] . (59)If one requires that the first and the second relationsof the condition (40) hold, this leads to the equations − ξp ˙ ψ + b ( R + 16 ξpq ) = 0 , (60) D l ˙ ψ = − ˙ ψD l ψ, (61)where q = 14 ξC (cid:18) B + Λ C ρ ξp (cid:19) . It can be shown that (58) is a differential consequenceof (60) and (61). Then Eq. (58) can be brought to theform p ¨ ψ + b ( − q + ∆ ψ − D s ψD s ψ ) = 0 , (62)or, in terms of the metric ω , p ¨ ψ + − qb + ω ∆ ψ + 8 ξpD ˜ s ψD ˜ s ψ = 0 , (63)Thus the function ψ is found by solving Eqs. (60), (61)and (57).Then the functions X A are determined by Eq. (54).If we write Eq. (54) directly in terms of the variables X A , we obtain a linear equation with respect to X A .This equation, with (58), may be written as¨ X − ξp ˙ ψ ˙ X − ω ∆ X + 8 ξpD ˜ s XD ˜ s ψ − p + 1 C b X = 0 . (64)Solutions of the latter equations should satisfy the re-maining constraint equations which have the form( ˙ X, ˙ X ) = − b , ( X, X ) = C, (65)( ˙ X, D l X ) = 0 . ( D k X, D l X ) = h kl . (66)Let us present some special solutions to the equa-tions obtained for the case of conformally flat manifolds.Let us first consider a flat p -dimensional model: ω R = 0 . Let the metric matrix ( ω lk ) be a unit matrix. A solutionto Eqs. (60), (61) and (57) has the form b − ≡ e ψ = (cid:20) u r − t ) + mt + n i σ i + l ) (cid:21) , (67)where r = p X i =1 ( σ i ) ,u , m, n i , l being integration constants satisfying thecondition m + 2 u l − n i n i = 2 q/p . Here t is the timecoordinate, and the p -dimensional coordinates may beinterpreted as the conventional Cartesian coordinates.It can be verified by a direct inspection that the func-tions f = a ˙ ψ and f l = a l D l ψ (where a , a l = const)are special solutions to Eq. (54). Using this, let us buildsolutions which also satisfy Eqs. (65)–(66). Probably,there can be many such solutions. But we will here seeksolutions with a minimal set of fields X A . With thisapproach, we consider solutions which describe an em-bedding of the manifold Π into the 5-dimensional space M . Calculations show that the solutions linear in thefunctions f and f l satisfy Eqs. (65)–(66) with the fol-lowing values of the constants: q = p C = ⇒ C ρ = ( p − p + 3)8Λ , (68) a = a l = 1 u = Cc , (69)where c = 2 l /u . Without losing generality in thesolution (67), we put m = n i = 0 , which simply cor-responds to a parallel transport. Then the scale factor(67) is rewritten in the form b = 4 a ( r − t + c ) . (70)(69) it follows that all solutions split into two types:(1) C > , c > C < , c < . To embed the manifold Π into M , it turns out to beconvenient (see [18]) that, for the second type of solu-tions, the metric signature in M be ( − , + , + , + , − ). Forthe first type it should be ( − , + , + , + , +). If we define | c | = g , then the solution have the following form: X = t √ C g t − r − g ,X l = σ l √ C g t − r − g ,X = √ C t − r + g t − r − g (71)for the first type and X = t p | C | g t − r + g ) ,X l = σ l p | C | g t − r + g ,X = p | C | t − r − g t − r + g (72) for the second type.To study the global properties of the manifold, letus study its boundaries. For the second type of solu-tions, consider a range W o specified by the followingconstraints on the coordinate variables: −| g | ≤ ( t + r ) ≥ | g | , −| g | ≤ ( t − r ) ≥ | g | . (73)Let us introduce the new coordinate ( η, χ ) instead of( t, r ): t + r = g tanh (cid:18)
12 ( η + χ ) (cid:19) ,t − r = g tanh (cid:18)
12 ( η − χ ) (cid:19) . (74)In the new coordinates, using a conformal transforma-tion, the metric may be brought to a form exactly coin-ciding with the open anti-de Sitter space metric ds = a ( η )[ dη − ( dχ ) − K ( χ ) d Ω ] , (75)where a ( η ) = C cosh η , K ( χ ) = sinh χ, and d Ω is the metric form of a ( p − W c l : −∞ < t < + ∞ , −∞ < r < + ∞ may be mapped into a part of the compact (closed) deSitter space.To this end, we introduce new coordinates by therelations t + r = g tan (cid:18)
12 ( η + η + χ ) (cid:19) ,t − r = g tan (cid:18)
12 ( η + η − χ ) (cid:19) , (76)with η = const. The metric has the form (75), where a ( η ) = C cos ( η + η ) , K ( χ ) = sin χ, (77)The functions X A are scalars with respect to theabove coordinate transformations and may be rewrittenin the new coordinates. For instance, for de Sitter space, X = √ C tan ( η + η ) , X a = √ C cos ( η + η ) k a , (78)where k a are the embedding functions of a p -dimensionalsphere. For dimension p = 3 , these functions are k = sin χ sin θ cos φ, k = sin χ sin θ sin φ,k = sin χ cos θ, k = cos χ. (79)In this case, the metric form is ds = a ( η )[ dη − dχ − sin χ ( dθ + sin θdφ ] , (80)which corresponds to the Robertson-Walker metric de-scribing the Friedmann cosmological models.The solutions for anti-de Sitter space are obtainedfrom the above equations if one makes there the follow-ing substitution:sin χ sinh χ, cos χ cosh χ, cos η cosh η. In addition to considering different gauge conditions, letus note that the field equations and constraint equationsmay also be studiedly the canonical gauge. To do so, us-ing the substitution X = b ξp g , without imposing thesupplementary condition (53), one can entirely excludethe field λ M in the Hamiltonian equations if the condi-tion (50) is valid. In this case, the following equationsare obtained:¨ g = ω ∆ g + 2 ξg ω R +2 ρ Λ gZ ρ − , Z ≡ ( g, g ) , (81)12 ξ [( D l g, D k g ) − B g ω lk ] + D l D k Z − p ω ∆ Zω lk − [ ω R lk − p ω R ω lk ] Z = 0 , (82)( ˙ g, ˙ g ) + B g p + 4 ξ ω ∆ Z − ξZ ω R − Z ρ = 0 ,B g ≡ p ω lk ( D l g, D k g ) , (83) D l ˙ Z + 12 ξ ( ˙ g, D l g ) = 0 . (84)The latter two equations are equivalent to the constraintequations H , H l . As their consequence, we obtain anequation for the function Z :¨ Z = (1 − ξ ) ω ∆ Z + 8 ξZ ω R − ξ Λ Z ρ − dB g . (85)Eq. (41) has the form D l D k Z − p ω ∆ Zω lk − (cid:20) ω R lk − d ω R ω lk (cid:21) Z = 0 , (86)We will seek special solutions to Eqs. (81)–(85) forthe dimension p = 3 , when the metric ω lk is deter-mined by the 3-dimensional part of the linear elementof an open-type Robertson-Walker space-time. We seeksolutions in the form g = u ( η ) , g a = u ( η ) k a ( σ i ) , a = 1 , , , , (87)and, doing so, we do not require that the full embeddingconditions (40) should hold. Then, for the functions u ( η ) and u ( η ) we obtain˙ u + 4 ku − k Λ Z ( u − ku ) udu − H = 0 ,H = const , (88)˙ u + ku − k Λ Z ( u − ku ) u du + 2 H = 0 . (89)The last two equation with respect to the variables u ( η )and u ( η )) may be considered as a dynamic system withthe potential energy U ( u, u ) = − k Λ u ( u + 2 u ) + Λ u + k (2 u − u /
2) and zero total energy. Integrating by parts and sum-ming, we can obtain˙ u + ˙ u + 2 U = 0 . A further study shows that the previously found solu-tion, describing an open de Sitter space, is a stable ex-ceptional solution to Eqs. (88)–(89). In the present for-mulation, the following solution corresponds to the oneobtained above: u = r (cosh η ) , u = r tanh η cosh η , r = s | Λ | . (90)Using the terminology of the qualitative theory ofdifferential equations, the singular point u = 0 , u = 0is unstable. There are no other static points. Mean-while, the solution (90), being a separatrix in the phasespace of the variables u ( η ) and u ( η ), minimizes thetotal energy.From this point of view, it is of interest to invoke theHiggs mechanism to obtain the constraints (38). Thefields X A , being coordinates of the space M , may playthe role of Higgs’ fields in Grand Unification models.On the other hand, from the viewpoint of the Hamilto-nian formalism considered above, solutions with a bro-ken symmetry may be treated as a particular choice ofthe gauge. As has been already noted above, we here obtain equa-tions similar to the Einstein equations with an effec-tive gravitational constant, see (9). Indeed, in case(
X, X ) = C and if g µν = ( ∇ µ X, ∇ ν X ) , µ, ν = 0 , p, (91)Eqs. (8) take the form G αβ = 8 πG e T eαβ + Λ e g αβ , (92)where G e is given by Eq. (9), while the cosmologicalconstant isΛ e = − ξC ( − C ) . (93)From the solution (68) and (93), we find thatΛ = 32 C , Λ e = 3 C . (94)In a closed model, the constant C satisfies the equa-tion − ( X ) + ( X ) + ( X ) + ( X ) + ( X ) = C. (95)Then √ C characterizes the size of the observed part ofthe Universe. In the solution (77) (for η = π/ t and obtain: a ( t ) = √ C cosh( t/ √ C ) ,H ≡ ˙ aa = √ C tanh( t/ √ C ) . (96)Suppose that the Hubble “constant” H ∼ (3 · ) − c − (in the Planck units, ~ = 1 and c = 1 ) and that ourepoch corresponds to the time t ≃ √ C , then we obtainthe calue of C : √ C ≃ . · cm ∼ cm. Substi-tuting this value into (94), we find Λ e ∼ − cm − ,or, the same in energy units, Λ e ∼ − GeV . Thisresult confirms the existence of a nonzero cosmologicalconstant Λ , which is also in agreement with the obser-vational data, see, e.g., Ref. [19].Equating the expression (9) to 1 /M p ∼ − cm ,we find that w ∼ · − cm . The parameter w alsocorresponds to distances of the order l w = √ w ∼ . n > l < . /G ) R , which violates the conformal invarianceof the equations, then there emerges the effective grav-itational “constant” G e = wG ( w + 2 ξG ( X, X )) − . Asis shown in Ref. [14], this leads to an instability of cos-mological solutions for G e → References [1] .B. Green, J.H. Schwarz and E. Witten, “Superstringtheory”, Cambridge University Press, 1987.[2] L. Brink and . Henneaux, “Principles of String Theory”,Plenum Press, NY–London, 1988.[3] B. d Wit, “Introduction to Supergravity”, M., Mir,1985.[4] F.Sh. Zaripov, Proc. Int. School-Seminar “Foundationof the Theory of Gravity and Cosmology”, Odessa,1995, p. 35.[5] S.V. Ketov, “Introduction to Quantum Theory ofStrings and Superstrings”, Nauka, Novosibirsk, 1990 (inRussian).[6] L. Randall and R. Sundrum,
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