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arXiv:gr-qc/9304047v1 2 May 1993UB-ECM-PF-93/10

Invariant connections with torsion on group manifolds and theirapplication in Kaluza-Klein theoriesKubyshin Yu.A.∗†Departament d’Estructura i Constituents de la Mat´eriaUniversitat de BarcelonaAv. Diagonal 647, 08028 Barcelona, SpainMalyshenko V.O.

and Mar´ın Ricoy D.Nuclear Physics Institute, Moscow State University,Moscow 119899, Russia20 April 1993AbstractInvariant connections with torsion on simple group manifolds S are studied andan explicit formula describing them is presented. This result is used for the dimen-sional reduction in a theory of multidimensional gravity with curvature squared termson M4 × S. We calculate the potential of scalar fields, emerging from extra compo-nents of the metric and torsion, and analyze the role of the torsion for the stability ofspontaneous compactification.∗On sabbatical leave from Nuclear Physics Institute, Moscow State University, 119899 Moscow, Russia†E-mail address: kubyshin@ebubecm1.bitnet1

1IntroductionThe ideas of Kaluza and Klein about possible multidimensional nature of our space-time wereformulated in the twenties [1] and nowdays are regarded as an important ingredient of manysupergravity and superstrings theories [2] and other schemes of unification of interactions.In the framework of this approach it is common to assume that the multidimensionalspace-time is of the form M4 × I, where M4 is the macroscopic four-dimensional part ofthe space-time and I is a compact space of extra dimensions often called internal space.The compactification of extra dimensions should occur spontaneously, as a solution of theequations of motion, which also determine the size L of I after compactification. In most ofthe schemes L ∼LP l ∼10−33 cm.In the standard Kaluza-Klein approach the bosonic sector of the multidimensional theoryincludes only pure Einstein gravity, and the electromagnetic and/or non-abelian gauge fieldsand scalar fields emerge from the extra components of the multidimensional metric afterreduction of the theory to the four-dimensional space-time [3].

However, there are somedifficulties in this approach.One of them is due to the fact that there are no vacuumsolutions with the structure of the space-time M4 × K/H, where M4 is the Minkowskispace-time and K/H is a homogeneous space with non-abelian isometry group K, which isnecessary for obtaining non-abelian gauge fields after the dimensional reduction. Anotherdifficulty is related to impossibility to obtain chiral fermions in the dimensionally reducedtheory.

It was found [4] that in order to solve these problems one can generalize the standardKaluza-Klein approach by adding gauge fields to the multidimensional Lagrangian (see [5],[6] and [7] for reviews). Later other types of generalizations were proposed.

One of themis to remain whithin the pure gravity but to add terms quadratic in the curvature tensorcomponents.An interesting possibility in the framework of the latter generalzation is to consider amodel with non-zero torsion. This option was investigated in a series of papers [8] - [10].There is a big literature on the theory of gravity in the Riemann-Cartan space-time withtorsion (see [11] for a review and an extensive list of references on the subject).

It is wellknown that torsion is a non-dynamical variable in the pure Einstein - Cartan theory [12],but it becomes dynamical, if one adds quadratic in curvature terms to the standard EinsteinLagrangian. Such terms are also motivated by the quantum field theory limit of strings [2],[13]-[15].

Problem of spontaneous compactification in multidimensional theories with torsionwas investigated in [10], [16] - [18].The aim of the present paper is to give a description of the class of invariant connectionswith torsion compatible with the invariant metric on group manifolds S, where S is a simpleLie group, and then apply this result to investigation of spontaneous compactification inmultidimensional gravity with R2-terms on the space-time M4 × I with I = S. We shouldnotice that description of invariant connections with torsion, in the case when the internalspace I is a homogeneous space K/H from a certain class, was obtained in [19] and we willborrow some of the methods from this paper for our analysis here. But due to some specificfeatures of group manifolds the case of I = S is not covered by the results of [19] and needsa special treatment.The paper is organized as follows.

In section 2 we give a brief description of K-invariantmetric compatible linear connections on homogeneous spaces K/H. In section 3 we consider2

the case of group manifolds S. We prove a theorem about decomposition of antisymmetricsquare of the adjoint representation of a simple Lie algebra, which will enable us to constructthe invariant connections on S explicitly and to calculate the components of the multidimen-sional curvature tensor. In section 4 we consider multidimensional gravity with R2- termsand derive the potential of scalar fields of the dimensionally reduced theory.

We analyze ofthis potential and make conclusions about stability of spontaneous compactification in thetheory.2General properties of invariant connections on ho-mogeneous spacesIn the present section we present the main results from the theory of invariant connectionson homogeneous spaces I = K/H. They will be used in the next section for explicit con-struction of such connections in the case when I has the structure of a simple Lie group.Our considerations are based on [20].Let us consider the principle fibre bundle O(M) with the structure group SO(d), whered = dim M, of orthonormal frames over M = K/H with reductive decomposition of the Liealgebra K = Lie(K) of the group K: K = H⊕M, adH(M) ⊂M, where H = Lie(H).

Thegroup K acts transitively on the base M and induces a natural automorphism on the bundleL(M). We will be interested in metrics g on M and metric compatible connections ω on thebundle L(M), which are invariant under the action of the group K. The construction of K- invariant connections on homogeneous spaces is based on the Wang theorem [20].

It statesthat there is 1 - 1 correspondence between K - invariant connections on the bundle L(M)and linear mappings Λ : M →so(d) = Lie(SO(d)), which satisfy the following conditionΛ(Adh(X)) = Ad(λ(h))(Λ(X)), X ∈M, h ∈H,(1)where Ad denotes the adjoint representation of H and λ is the homomorphism λ : H →SO(d) induced by the action of H on the tangent space To(K/H) (the corresponding homo-morphism of algebras is also denoted by λ). In terms of the mapping Λ the formulas for theinvariant torsion and curvature tensors at the point o = [H] acquire simple form, namelyTo(X, Y )=Λ(X)Y −Λ(Y )X −[X, Y ]M,(2)Ro(X, Y )=[Λ(X), Λ(Y )] −Λ([X, Y ]M) −λ([X, Y ]H),X, Y ∈M.

(3)Here we identified the tangent space To(K/H), M and Rd.Notice also that so(d) ∼=Rd ∧Rd ∼= M ∧M. Let us introduce the mapping β : M ⊗M →M by the formulaβ(X, Y ) = Λ(X)Y .

The connection form ω can be decomposed into the sum of the Levi-Civita connection form0ω with zero torsion and the contorsion form ¯ω. This leads to thecorresponding decomposition for Λ =0Λ +¯Λ and β =0β +¯β.

For reductive homogeneous spacesthe expression for0Λ was obtained by Nomizu (see [20]). It is given by the formula0β (X, Y ) ≡0Λ (X)Y = 12[X, Y ]M+0U (X, Y ),X, Y ∈M,(4)3

where0U is a symmetric bilinear mapping,0U: M ⊗M →M. We will return to it shortly.K - invariant metrics g on K/H are known to be in 1 - 1 correspondence with adH -invariant bilinear forms B on M, namely g0(X, Y ) = B(X, Y ),X, Y ∈M ∼= T0(K/H).The invariance of B with respect to adH meansB([A, X], Y ) + B(X, [A, Y ]) = 0,X, Y ∈M, A ∈H.

(5)It is easy to verify that the condition of compatibility of the invariant connection form ωwith the invariant metric g can be written as:B(β(X, Y ), Z) + B(Y, β(X, Z)) = 0. (6)We wish to construct the form ¯β, which describes nonzero torsion.

We can represent it asthe sum of the symmetric and antisymmetric parts: ¯β = ¯βs + ¯βas with ¯βs(X, Y ) = ¯βs(Y, X)and ¯βas(X, Y ) = −¯βas(Y, X). Combining the condition of metric compatibility (6) with twoother formulas obtained from (6) by the cyclic permutation of X, Y and Z it is easy to derivethe following relation between the symmetric part βs(X, Y ) =0U (X, Y ) + ¯βs(X, Y ) and theantisymmetric part βas(X, Y ) = 12[X, Y ]M + ¯βas(X, Y ) of the full mapping β(X, Y ):B(βs(X, Y ), Z) = B(X, βas(Y, Z)) + B(Y, βas(Z, X)),X, Y, Z ∈M.

(7)Thus, if we construct all mappings ¯βas, we will be able to find all invariant connections onK/H using eq. (7).

Notice that ¯βas can be considered as a mapping from M ∧M intoM, ¯βas : M ∧M →M.The condition (1) can be rewritten for ¯βas in the infinitesimal form as follows:¯βas(adA ∧1 + 1 ∧adA)(ξ) = adA(¯βas(ξ)),ξ ∈M ∧M, A ∈H. (8)This enables us to consider ¯βas as an intertwining operator, which intertwines equivalentrepresentations of the algebra H in the linear spaces M ∧M and M. Thus, the generalscheme of construction of the operator ¯βas is the following [19].

We decompose linear spacesM∧M and M into subspaces carrying irreducible representations (irreps) of the algebra HM ∧M =XUk,M =XVn.According to Schur’s lemma, the operator ¯βas is equal to ¯βas =P fknβkn, where βkn is theunit operator establishing the isomorphism between the subspaces Uk and Vn if they carryequivalent irreps and βkn = 0 otherwise. Similar intertwining operators appear in the cosetspace dimensional reduction of multidimentional Yang-Mills theories.

See [6], [7] for thediscussion of the problem of the construction of such operators in gauge theories.In order to illustrate the general scheme of calculation of ¯βas let us consider two examples.1. K/H = G2/SU(3).

From the results in [21],[22] we have after complexificationadH(M)=3 ⊕3∗,adH(M ∧M)=8 ⊕3 ⊕3∗⊕1,where 8 is the adjoint representation of H. We see that there are only two irreps, 3 and3∗, which enter both decompositions. Therefore, the intertwining operator is of the form:4

βas = f33β33 + f3∗3∗β3∗3∗, wheref33 and f3∗3∗are arbitrary complex parameters. The realitycondition for βas implies (f33)∗= f3∗3∗.2.

K/H = (SU(3) × SU(3))/diag(SU(3) × SU(3)) ∼= SU(3). In this exampleadH(M ∧M)=adH ⊕10 ⊕10∗, adH = 8,(9)adH(M)=adH.There is only one irrep which enters both decompositions.

Therefore, the intertwining op-erator has the form ¯βas = fβ88, i.e. only one real contorsion field exists.

This exampleillustrates the case we are interested in, namely the case of group manifolds represented asa homogeneous space.More examples of construction of the contorsion form as the intertwining operator aregiven in [19].To conclude the section we note that the structure of the algebra K of reductive spacesadmits two natural intertwining operators: φ : M ∧M →M and ψ : M ∧M →H. Theyare given byφ(X ∧Y )=[X, Y ]M,(10)ψ(X ∧Y )=[X, Y ]H,X, Y ∈M.

(11)These operators will be used in the next section.3Construction of invariant connections on group ma-nifoldsOne of the aims of the present paper is to investigate in detail the case of the group manifoldsS, represented as a homogeneous space S = K/H with K = S × S and H = diag(S × S).There are three natural reductive decompositions for K [20], namely K = H ⊕M withM = M0, M+, M−,M0={(X/2, −X/2), X ∈S},M+={(0, −X), X ∈S},M−={(X, 0), X ∈S},(12)with H = Lie(H), S = Lie(S). Obviously, M ∼= H ∼= S. The (0) - decomposition (the firstdecomposition in (12) ) corresponds to the case when S is represented as a symmetric space.The subspaces M and H carry the adjoint representation of S only, therefore, to constructthe mapping ¯βas in this case we must study the decomposition of adS ∧adS into irreps.

Thefact, that the adS is contained in adS ∧adS at least once is guaranteed by the existence ofthe non-trivial intertwining operator φ (see (10)) in the case of (±) - decomposition and theoperator j ◦ψ (see (11)), where j is the isomorphism H →M, for (0) - decomposition.In fact, we can prove the followingTheorem. For simple Lie algebras S the decomposition of adS ∧adS into irreps of Shas one of the following formsadS ∧adS=adS ⊕γ ⊕γ∗,for S = An,adS ∧adS=adS ⊕γ,for other simple Lie algebras,(13)5

where γ and γ∗are irreps different from adS. Namely,dim γ = n(n −1)(n + 2)(n + 3)/4 for S = An,dim γ = n(n −1)(2n −1)(2n + 3)/2 for S = Bn and Cn,dim γ = n(n + 1)(2n −1)(2n −3)/2 for S = Dn,dim γ = 77 for G2, 1274 for F4, 2925 for E6, 8645 for E7 and 30380 for E8.Proof.

For the proof of the theorem it is convenient to complexify M and H and applyresults from the theory of Lie algebras [23],[24]. Here we will present the proof for the caseS = An, other cases can be proved in a similar way.

The main idea is rather simple. As ithas been mentioned above the adjoint representation adS is contained in adS ∧adS at leastonce.

We will find another irrep γ contained in adS ∧adS, calculate its dimension dim γand show thatdim(adAn ∧adAn) = dim adAn + dim γ + dim γ∗. (14)We will also see that dim γ ̸= dim An, and this will complete the proof of the theorem forS = An.

The important tool in the proof is Weil’s formula for the dimension of an irrep[23]. It is known that any irrep γ of Lie algebra S is characterisized by its highest weightΩ= (Ω1, .

. .

, Ωn), where Ωi are the Dynkin coefficients of Ωwith respect to a system of thesimple roots {αi, i = 1, · · · , n = rank S} of S, i.e. Ω= Pni=1 Ωiαi.

Weil’s formula statesthat the dimension of the irrep γ(Ω) is equal todim γ(Ω) =Xα>0Pi Ki(α)(1 + Ωi)⟨⟨αi, αi⟩⟩Pi Ki(α)⟨⟨αi, αi⟩⟩,where the first sum goes over all positive roots αi of S and Ki(α) are the coefficients of theroot α with respect to {αi}, α = Pni=1 Ki(α)αi. Here ⟨⟨·, ·⟩⟩is the canonical scalar productin the space dual to the Cartan subalgebra of S induced by the non-degenerate invariantbilinear form ⟨·, ·⟩in S ( proportional to the Killing form ).

Nonzero weights of the adjointrepresentation adS are roots of the algebra S. Our first step is to find any irrep γ in thedecomposition of adS ∧adS. There is a general procedure of finding the so called highestirrep in the antisymmetric tensor product (see for example ref.[22]).

Let us denote by Ωadthe heighest weight of adS and by ˜Ωad one of the next to the heighest weights, i.e. theweight which is obtained from Ωad by subtraction of one of the simple roots αi of S. Thenthe highest weight Ωof the highest irrep in adS ∧adS is given by the formula Ω= Ωad + ˜Ωad.For S = An there are two next to the highest weights.

Therefore, two highest irrepsγ1 and γ2 in adS ∧adS exist. Their highest weights are Ω1 = (0, 1, 0, · · ·, 0, 2) and Ω2 =(2, 0, · · ·, 0, 1, 0).

It is known for S = An that if two irreps have the highest weights Ω1 =(a1, · · · , an) and Ω2 = (an, · · · , a1), then they are conjugate to each other. Thus, γ2 = γ∗1.Weil’s formula givesdimγ1 = n(n −1)(n + 2)(n + 3)4Now we calculate dim S = dim(adS) and dim(adS ∧adS) :dim(adS) = n(n + 2),dim(adS ∧adS) = n(n + 2)[n(n + 2) −1]2.We see immediately that dim γ1 ̸= dim(adS), therefore γ1 and γ∗1 are not equal to the adjointrepresentation, and can check easily that the formula (14) is true.

This finishes the prooffor the case S = An ✷.6

This theorem is the development of the known result stating that adS is always containedin the tensor product adS ⊗adS [22], [24], [25]. Using the theorem proved above we canconstruct the mapping ¯βas, satisfying the intertwining condition (8), explicitly.Indeed,Schur’s lemma implies that ¯βas must be proportional to the intertwining operator φ or j ◦ψ(see (10), (11)) and the result (13) guarantees that there are no other intertwining operatorsmapping from M ∧M into M. Thus, we have¯βas( ˜X ∧˜Y )=f2[ ˜X, ˜Y ]Mfor (±) - decomposition,¯βas( ˜X ∧˜Y )=2f ◦j([ ˜X, ˜Y ]H)for (0) - decomposition,(15)where ˜X, ˜Y ∈M and f is an arbitrary real parameter.As for the symmetric part βs, we can show using eq.

(7) that βs is identically zero. Todo this we take an adK - invariant bilinear form B( ˜X, ˜Y ) = ⟨X1, Y1⟩+ ⟨X2, Y2⟩on the Liealgebra K of K. Here ˜X = (X1, X2),˜Y = (Y1, Y2),˜X, ˜Y ∈M, Xi, Yi ∈S, and as in theproof of the theorem ⟨·, ·⟩is an adS - invariant bilinear symmetric form on S = LieS, whichin our case is proportional to the bi - invariant metric g on S and to the Killing form.

Wesee now that for ¯βas, given by eq. (15), the r.h.s.

of (7) vanishes, thus βs = 0.Finally, the mappings Λ, corresponding to the invariant connection with torsion on thegroup manifold S, form a 1 - parameter family given byΛ( ˜X) ˜Y=1 + f2[ ˜X, ˜Y ]M,for (±) - decomposition,Λ( ˜X) ˜Y=2fj([ ˜X, ˜Y ])H,for (0) - decomposotion. (16)The mapping Λ with f = 0 corresponds to the Levi-Civita connection with zero torsionon S (see eq.(16)).

When f = −1 for (±) - decomposition and f = 0 for (0) - decompositionΛ describes the canonical connection. Notice that for (0) - decomposition cononical con-nection coincides with the Levi-Civita connection.

We would like to underline here that weconstructed non-trivial invariant connection for the case of the (0) - decomposition in (12)when the group manifold S is represented as a symmetric homogeneous space K/H. Thisdiffers from the case of simply connected compact irreducible symmetric spaces K/H whichare not group manifolds.

In the latter case, as it was shown in [19] ( Proposition 3.1), theLevi-Civita connection is the only K - symmetric metric compatible connection on K/H.Introducing the isomorphism i : S →M and using that ˜R( ˜Xk, ˜Xp) ˜Xj on K/H equalsi(R(Xk, Xp)Xj on S where i(Xk) = ˜Xk, i(Xp) = ˜Xp, etc., one gets from (3)R0(Xk, Xp)Xj = F(f)[[Xk, Xp], Xj],F(f) = f 2 −12,Xk, Xp, Xj ∈S,(17)which yields for the curvature tensor componentsRijkp = F(f)CakpCbajg(Xb, Xi),where g(·, ·) is the bi - invariant metric on S and Ckij are the structure constants of thealgebra S.Analogously, eq. (2) and (16) imply that the torsion tensor on S equalsT0(X, Y ) = f[X, Y ].

(18)7

4Dimensional reduction of multidimensional gravitywith torsionWe investigate the theory with the actionS =Zdˆxq−ˆg{ˆλ0 + ˆλ1R + ˆλ2R2},(19)where R2 = κRABCDRCDAB −4RABRBA + R2, on the space-time E = M4 × S, κ =0, 1, 2,A, B, C, D = 0, 1, 2, . .

. , d + 3.

As it has been pointed out in the introduction suchaction arises in the field theory limit of strings with κ = 0 for the superstring [13], κ = 1 forthe heterotic string [15] and κ = 2 for the bosonic strings [14]. We choose the metric tensorin the block diagonal formˆg = gαβ(x)00L2θma (ξ)θnb (ξ)δmn!,(20)where α, β = 0, 1, 2, 3, a, b = 4, .

. .

, d + 3, L is a constant of the dimension of lengthcharacterizing the size of the space S, x ∈M4, ξ ∈S, θma (ξ) are the vielbeins. Substituting(20) in (19) and taking the invariance of the metric and connections into account, we getS = vdZd4x Ld√−g{ ˆλ0 + ˆλ1( ¯R(4) + R(d)) + ˆλ2R2},(21)where Ldvd is the volume of the internal space, ¯R(4) and R(d) are the scalar curvatures of thespaces M4 and S respectively, and g = det gαβ.

To separate the term corresponging to thepure four-dimensional Einstein gravity we introduce the true physical metric η(x) on M4related to g(x) in the following way:gαβ(x) = ( LL0)−dηαβ(x),where L0 is the constant of the dimension of length to be fixed later on. Then the action(21) takes the formS =Zd4x√−η{ ¯λ1R(4)−W(L, f, d, κ, L0, ¯λ0, ¯λ2)},(22)W(L, f, d, κ, L0, ¯λ0, ¯λ2)=−{¯λ2( LL0)d(R2)(4) + 2¯λ2R(4)R(d)+¯λ0( LL0)−d+¯λ1( LL0)−dRd + ¯λ2( LL0)−d(R2)(d)},(23)where ¯λi = ˆλiLd0vd.

If we had considered the contorsion form (16) and the metric (20) withthe parameter f(x) and the size L(x) depending on the coordinates of M4, then after thedimensional reduction we would have obtained the Einstein gravity on M4 with the metrictensor ηαβ(x) coupled to the scalar fields ψ(x) = ln{L(x)/L0} and f(x) with kinetic terms,higher derivatives and the potential arising from (23). If L and f are constant, as in the caseconsidered in the present paper, we are left with (22).

Thus, W is the effective potential8

of scalar fields ψ and f of the four dimensional reduced theory. It determines vacua, i.e.constant with respect to four dimensional coordinates solutions of the equations of motion.We are going to analyze the form and properties of the potential and find its minima.Assuming that M4 is the Minkowski space-time and ηαβ is the Minkowski metric, thepotential W(L, f, .

. .) takes the formW(L, f, .

. .) = −( LL0)d{¯λ0 + ¯λ1R(d) + ¯λ2(R2)(d)}(24)Hereafter we will drop the symbol ”(d)” for the components corresponding to the spaceS.

The components of the curvature and Ricci tensors can be expressed in terms of theeigenvalue of the Casimir operator C2. Using the fact that in our case C2 = 1 for the adjointrepresentation [26] we find for R2 and RR2=F 2(f)L4d(d + κ −4),(25)R=−1L2 F(f)d.(26)Substituting (25), (26) into (24) and introducing the field ψ(x) = ln{L(x)/L0} we obtainthe following expression for the potential of the scalar fields ψ and f of the dimensionallyreduced theoryW(ψ, f) = e−ψd{λ0 + λ1Fe−2ψ + λ2F 2e−4ψ},(27)whereF(f) = f 2 −14, λ0 = −¯λ0, λ1 = ¯λ1dL20> 0, λ2 = −¯λ2d(d + κ −4)L40.Our next step is to investigate possible cases corresponding to the values of the parametersλ0, λ1, λ2, κ and d for which W(ψ, f) has a minimum.

We would like to underline here thatthe point (ψmin, fmin), where the potential has its minimum, is a constant solution of theequations of motion of the theory corresponding to spontaneous compactification of the extradimentions to the compact space S, so that the space-time has the form M4 × S (see [27]).We will use the notation ∆≡λ21/4λ0λ2.4.1Case 1: λ0, λ2 > 0, ∆= 1It can be checked that in this case the potential has degenerate minima at the points(ψmin, fmin) located on the curve ψmin(f) = 12 ln{λ2(1 −f 2)/2λ1}, | f |< 1, and its valuesat these points are equal to zero, i.e. W(ψmin, fmin) = 0 (see Fig.

1). The four-dimensionalcosmological constant Λ4, which is determined by the value of W at the point correspond-ing to the vacuum solution, vanishes.

We may fix the parameter L0 by the requirementψmin(0) = 0. This givesL0 =vuut−ˆλ2(d + κ −4)2ˆλ1,which is thus the size of the internal space in the vacuum corresponding to spontaneouscompactification with zero torsion.9

4.2Case 2: λ0, λ2 > 0, d(d+4)(d+2)2 < ∆< 1In this case the potential is positive for all (ψ, f) and vanishes when ψ →+∞. It can beverified that the potential has the minimum at the point(ψmin, fmin) = (12 lnλ2(d + 4)2λ1(d + 2)[1 +r1 + d(d+4)(d+2)2 ∆−1], 0).Since W(ψmin, 0) > 0 the minimum is local and the corresponding vacuum state is meta-stable.

We should note that the potential W(ψ, f) has two gutters in the region | f |< 1and ψ < 12 ln 2λ1/λ2, which join each other at the point ( 12 ln 2λ1/λ2, 0) and ascend whenψ →−∞. We fix the parameter L0 by the requirement that ψmin = 0.

This gives twovalues for L20 and we choose the one for which L20 = 0 when λ2 = 0 ( this corresponds to thecollapse of the internal space and absence of stable spontaneous compactification solutionfor the pure Einstein gravity, as it has been discussed in the Introduction):L20 = (d + 2) ˆλ12ˆλ0[−1 +vuut1 −ˆλ0 ˆλ2ˆλ21(d + κ −4)(d −4) ]. (28)The experimental bound for the four-dimensional cosmological constant Λ4 gives | 16πGΛ4 |<10−120.

By making the parameter λ0 approaching λ21/4λ2 from above we can obtain arbitrarysmall values for Λ4 = W(0, 0). Trying to make Λ4 = 0 it is easy to see that this is possibleif and only if ∆= 1, i.e.

in case 1.4.3Case 3: λ0, λ2 > 0, ∆> 1Now the potential has no minimum. As in the previous case for ψ < 12 ln 2λ1/λ2 and | f |< 1,the potential has two gutters which join each other at ( 12 ln 2λ1/λ2, 0), but contrary to thecase 2, lower to (−∞) when ψ →−∞.

These features of the potential are of some importancefor understanding of the spontaneous compactification issue as will be discussed in the nextsection.5Discussion of the resultsLet us start our discussions with case 2. We have found that the minimum of the scalarpotential W(ψ, f) of the reduced theory is at (ψmin, 0) (for our choice of L0ψmin = 0)that corresponds to spontaneous compactification of extra dimensions to the group manifoldS with characteristic size L0 given by (28) and zero torsion.

This minimum is stable withrespect to fluctuation with non zero torsion and classicaly stable with respect to fluctuationin ψ - direction. Since W(0, 0) > 0 and W(∞, 0) = 0 the corresponding vacuum state ismetastable, and the system can pass to the region of large ψ via quantum tunnelling.

Thiscorresponds to decompactification of the space of extra dimensions. Analogous phenomenafor the multidimensional Einstein - Yang - Mills system without torsion were consideredin [28],[29].

But if the parameters of the lagrangian are tuned in such way that the four-dimensional cosmological constant Λ4 = W(0, 0) is small enough to satisfy the experimental10

bound | 16πGΛ4 |< 10−120, the lifetime of the metastable vacuum exceeds the lifetime of theUniverse (see estimations in [29]).Another interesting problem which can be addressed here is the dynamics of compactifi-cation of extra dimensions in the framework of the Kaluza - Klein cosmology (the analogousissue for the Einstein - Yang - Mills system without torsion was considered in [28]). The mainquestion is the following: if at the early stage the multidimensional Universe had started itsclassical evolution with large negative ψ (small size L) and large | ˙ψ |, would it have founditself in the minimum (ψ = 0, f = 0) corresponding to spontaneous compactification of extradimensions?

Since the height of the barrier separating the minimum from the region whereψ →+∞(decompactification of extra dimensions ) is finite, the system can have enoughenergy to overcome the barrier in spite of the loss of energy due to the friction terms whichare present in such sort of theory. In any case this question needs more detailed investigationfor which the explicit form of non-static terms in the reduced action must be known.

Thisis beyond the scope of the present paper.In case 1, the minimum of the potential is degenerate: W(ψmin(f), f) = 0 for 0 ≤| f |≤1.The vacuum with (ψmin(0), 0), corresponding to compactification with zero torsion, is notseparated by any barrier from another vacua with the same energy but non-zero torsion.The situation changes even more drastically in case 3. The potential W(ψ, f) does nothave any minimum at all.

But if we analyzed the same theory without torsion, we would see(as in case 1 also) that the potential W(ψ, 0) has minimum and expect to have spontaneouscompactification solution. However, taking torsion as additional degree of freedom in thetheory into account changes the situation.

The vacuum (ψmin(0), 0) is not stable and smallfluctuations of the fields and their time derivatives may initiate transitions of the system toanother state with the same (case 1) or less (case 3) energy. Non-zero torsion is developedin such transitions.We think that this example is rather instructive for deeper understanding of the sponta-neous compactification problem.

It illustrates some of the hidden difficulties that the Kaluza- Klein approach may face.In conclusion we would like to underline that the analysis of the Kaluza - Klein R2 -gravity with torsion was carried out for arbitrary S × S - invariant configurations of themetric and connection form.The mathematical results, obtained in Section 3, enabledus to describe all metric compatible invariant connections with non-zero torsion on groupmanifolds S, and thus the solution of the spontaneous compactification problem for this classof metrics and connections is complete.6AcknowledgementsIt is a pleasure for us to thank A.P. Demichev A.P., J.I.

P´erez Cadenas, G. Rudolph G.and I.P. Volobujev I.P.

for their useful comments and critical remarks and J. Mour˜ao forilluminating discussions and critical reading of the manuscript. Yu.

K. acknowledges supportfrom Direcci´on General De Investigaci´on Cient´ıfica y T´ecnica (sabbatical grant SAB 92 0267)during part of this work and thanks the Department d’ECM de la Universidad de Barcelonafor its warm hospitality.11

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Figure 1 Shape of the potential W(ψ, f) when λ0, λ0 > 0 and ∆= 1 (see Sect. 4.1).

Theminimum of the potential is degenerate and is located on the curve ψ −12 ln (1−f2)2λ1= 0in the (ψ, f)-plane.14

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