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Hyperbolic Kac-Moody Algebras and Chaos in Kaluza-Klein Models


AEI-2001-025 IHES/P/01/12 LPT-ENS/01-14 ULB-TH/01-05

arXiv:hep-th/0103094v1 13 Mar 2001

Hyperbolic Kac-Moody Algebras and Chaos in Kaluza-Klein Models
Thibault Damour1, Marc Henneaux2,3, Bernard Julia4 and Hermann Nicolai5
1

Institut des Hautes Etudes Scienti?ques, 35, route de Chartres, F-91440 Bures-sur-Yvette, France
2

Physique Th? eorique et Math? ematique, Universit? e Libre de Bruxelles, C.P. 231, B-1050, Bruxelles, Belgium
3 4

Centro de Estudios Cient? ??cos, Casilla 1469, Valdivia, Chile

Laboratoire de Physique Th? eorique de l’Ecole Normale Sup? erieure, 24, rue Lhomond, F-75231 Paris CEDEX 05

5

Max-Planck-Institut f¨ ur Gravitationsphysik, Albert-Einstein-Institut, M¨ uhlenberg 1, D-14476 Golm, Germany

Abstract Some time ago, it was found that the never-ending oscillatory chaotic behaviour discovered by Belinsky, Khalatnikov and Lifshitz (BKL) for the generic solution of the vacuum Einstein equations in the vicinity of a spacelike (“cosmological”) singularity disappears in spacetime dimensions D ≡ d + 1 > 10. Recently, a study of the generalization of the BKL chaotic behaviour to the superstring e?ective Lagrangians has revealed that this chaos is rooted in the structure of the fundamental Weyl chamber of some underlying hyperbolic Kac-Moody algebra. In this letter, we show that the same connection applies to pure gravity in any spacetime dimension ≥ 4, where the relevant algebras are AEd . In this way the disappearance of chaos in pure gravity models in D ≥ 11 dimensions becomes linked to the fact that the Kac-Moody algebras AEd are no longer hyperbolic for d ≥ 10.

1

Introduction

A remarkable result in theoretical cosmology has been the construction, by Belinsky, Khalatnikov and Lifshitz (BKL), of a generic solution to the 4-dimensional vacuum Einstein equations in the vicinity of a spacelike (“cosmological”) singularity [1]. This solution exhibits a never-ending oscillatory behaviour of the mixmaster type [2, 3] with strong chaotic properties. Some time ago, it was found that the BKL analysis for pure gravity leads to completely di?erent qualitative features in spacetime dimensions D ≥ 11 [4, 5]. Namely, for those dimensions, the generic solution to the vacuum Einstein equations ceases to exhibit chaotic features, but is instead asymptotically characterized by a monotonic Kasner-like solution (for a review, see [6]). The critical dimension D = 11 was discovered by a straightforward but lengthy procedure, with no direct interpretation. Another system for which chaos is known to disappear is the pure gravity-dilaton system in all spacetime dimensions [7, 8]. More recently [9, 10, 11], the BKL analysis was extended to the supergravity Lagrangians in 10 [12, 13] and 11 dimensions [14] that emerge as the low energy limits of the superstring theories (IIA,IIB, I, HO, HE) and M Theory, respectively. Contrary to what happens for the gravity-dilaton system in 10 dimensions or pure gravity in 11 dimensions, the chaotic oscillatory behaviour was found to be generic in all superstring and M -theory models thanks to the p-forms present in the ?eld spectrum [9]. It was furthermore proved that this chaos was rooted in the structure of the fundamental Weyl chamber of some Kac-Moody algebra [11]. More precisely, reformulating the asymptotic analysis of the dynamics as a billiard problem a ` la Chitre-Misner [15, 16], it was shown that the never ending oscillatory BKL behaviour could be described as a relativistic billiard within a simplex in 9-dimensional hyperbolic space. The re?ections on the faces of this billiard were shown to generate a Coxeter group, which was then identi?ed with the Weyl group of the hyperbolic Kac-Moody algebras E10 for the type IIA,IIB,and M theories, and BE10 for the type I, HO, HE theories (for background on Kac-Moody algebras and notations, see the textbooks [17, 18]). In this way, a relation was established between the fact that the billiard has ?nite volume, and hence chaotic dynamics, and the hyperbolicity of the underlying inde?nite Kac-Moody algebras E10 and BE10 . In this letter, we re-examine the case of pure gravity in arbitrary spacetime dimension D ≡ d + 1 in the light of these results. We demonstrate that the asymptotic dynamics (for t → 0, at any point in space) can again be viewed as a billiard in the fundamental Weyl chamber of an inde?nite H Kac-Moody algebra, which is now AEd ≡ A∧∧ d?2 ≡ Ad?2 . This algebra is the “overextended” [19] or “canonical hyperbolic” extension [17] of the (?nite dimensional) Lie algebra Ad?2 ; its associated Dynkin diagram is obtained by attaching, at the a?ne node, one more node to the Dynkin diagram of 1

2 1 1 0

?1

0

?1

d?3

d?2

Figure 1: Dynkin diagram of AEd (with AE3 on the left)

the a?ne algebra Ad?2 ≡ A∧ d?2 and is displayed in Figure 1. The algebra H has been particularly studied in [20] and was related in AE3 ≡ A∧∧ ≡ A 1 1 [21] to D = 4 (super-)gravity. Note that the double-line (in the conventions of [17]) in its Dynkin diagram can be viewed as the formal limit of the loop of AEd as d → 3. [It is interesting to remark that the Weyl group of AE3 is P GL2 (Z), which is arithmetic [22, 20].] Furthermore we show very explicitly how the occurrence of chaotic behaviour is correlated to the hyperbolicity of the underlying Kac-Moody algebra. More speci?cally, the algebras AEd are hyperbolic (in the sense de?ned in section 3 below) for d < 10, whence pure gravity in dimensions 4 ≤ D ≤ 10 is chaotic, whereas chaos disappears in dimensions D ≥ 11 in accord with the fact that the algebras AEd are no longer hyperbolic for d ≥ 10. The very existence of a connection between the BKL dynamics and inde?nite Kac-Moody algebras is already remarkable in itself. For the generic Einstein system with matter couplings, one can always de?ne a billiard that describes the asymptotic dynamics, but in general, this billiard will not exhibit any noticeable regularity properties. In particular, the faces of this billiard need not intersect at angles which are submultiples of π , and consequently the associated re?ections will not generate a Coxeter (discrete) re?ection group in general; a fortiori, the billiard need not be the fundamental Weyl chamber of any Kac-Moody algebra. The hyperbolic Kac Moody algebra E10 (and DE10 ) was already conjectured in [23, 19] to be a hidden symmetry of maximal supergravity reduced to one dimension. The results of [11] and of this letter indeed support the idea that hyperbolic Dynkin diagrams play a key r? ole in the massless bosonic sectors of supergravity and superstring theory. But we should emphasize that the Kac-Moody algebras do not appear in the present BKL analysis as symmetry algebras with associated Noether charges. They underlie nevertherless the dynamics through their Weyl group, in the sense that the dynamics can be described in terms of “Weyl words” Wi1 Wi2 . . . made out of the “letters” Wi generating the 2

(1)

Weyl re?ections. It is amazing to see the chaos being controlled by the U-duality group G of the toroidal compacti?cation to 3 dimensions via its overextension G∧∧ . Recently, it has been shown [24] that both G = SO(8, 8) and G = SO(8, 9) are the U-duality groups of anomaly-free string models; in fact, other SO(8, 8 + n) groups can be realised beyond the heterotic SO(8, 24). A possible explanation for the universality of BE10 will be given there as well.

2

Gravitational billiard in d + 1 dimensions

We ?rst review how Einstein’s theory gives rise to a “gravitational billiard” as one approaches a cosmological singularity; for more details, see [11]. As usual, we assume that the singularity is at t → 0+ , where t is the proper time in a Gaussian coordinate system adapted to the singularity. In fact, it is convenient to use a time coordinate τ ? ? log t such that τ → +∞ as t → 0+ [1, 2]. In the asymptotic limit, the metric takes the form √ ds2 = ?(N gdτ )2 +
d ?=1

exp [?2β ? (τ, xi )] (ω ? )2 ,

(2.1)

j i where the time dependence of the spatial one-forms ω ? ≡ e? i (x , τ )dx (i = 1, · · · , d) can be neglected with respect to the time-dependence of the scale functions β ? . In (2.1), N is the (rescaled) lapse ?g00 /g , where ? g = exp (?2 d ?=1 β ) is the determinant of the spatial metric in the frame {ω ? }. We assume d ≥ 3 (i.e. D ≥ 4) since pure gravity in D = 3 spacetime dimensions has no local degrees of freedom. The central feature that enables one to investigate the equations of motion in the vicinity of a spacelike singularity is the asymptotic decoupling of the dynamics at the di?erent spatial points [1]. The remaining e?ect of the spatial gradients can be accounted for by potential terms for the local scale factors β ? . Therefore, we focus from now on a speci?c spatial point and drop reference to the spatial coordinates xi . In the limit τ → +∞, the dynamics for the scale factors β ? is governed by the action

S [β ? (τ ), N (τ )] =



G?ν dβ ? dβ ν ? N V (β ? ) N dτ dτ

(2.2)

where G?ν is the metric de?ned by the Einstein-Hilbert action in a ddimensional auxiliary space Md spanned by the “coordinates” β ? , which must not be confused with physical space-time. This metric is ?at and of Minkowskian signature (?, +, +, · · · , +); explicitly, it reads
d d d

G?ν V ? W ν =
?=1

V ?W ? ? (

V ? )(
?=1 ν =1

W ν ),

(2.3)

3

We shall also need the inverse metric G?ν
d

G?ν θ? ψν =
?=1

θ? ψ? ?

1 ψν ). θ? )( ( d ? 1 ?=1 ν =1

d

d

(2.4)

In (2.2), the potential V is a sum of sharp wall potentials, V =
i

Vi , Vi = Θ∞ ( ? 2wi (β ))

(2.5)

where Θ∞ vanishes for negative argument and is (positive) in?nite for positive argument1 . The functions wi (β ) are homogeneous linear forms, viz. wi (β ) = wi? β ? where the covectors wi? will be given explicitly below. Varying the rescaled lapse N yields the Hamiltonian constraint G?ν dβ ? dβ ν +V =0 dτ dτ (2.7) (2.6)

√ where we have set N = 1 (i.e., dt = ? gdτ ) after taking the variation, since this gauge choice simpli?es the formulas (note that this implies indeed √ τ ? ? log t since g ? t [1, 2]). The dynamics is also subject to the spatial di?eomorphism (momentum) constraints, but these a?ect the spatial gradients of the initial data and need not concern us here. We stress that the action (2.2) is not obtained by making a dimensional reduction to one dimension of the D -dimensional Einstein-Hilbert action assuming some internal d-dimensional group manifold. Rather, the action (2.2), or, more precisely, the sum over all spatial points of copies of (2.2), supplemented by the momentum constraints, is supposed to yield the asymptotic dynamics in the limit t → 0+ for generic inhomogeneous solutions [1]. We should mention that the derivation of (2.2) from the Einstein-Hilbert action involves a number of steps that have not been rigorously justi?ed so far. Nevertheless, there is now a wealth of supporting evidence for the BKL analysis, both of analytical and of numerical type [26, 27]. Let us study the dynamics of the billiard ball whose motion is described by the functions β ? = β ? (τ ). From (2.5) we immediately see that the interior region of the billiard is de?ned by the inequalities wi (β ) ≥ 0, and that its walls are coincident with the hyperplanes wi (β ) = 0. Away from the walls, the Hamiltonian constraint becomes G?ν dβ ? dβ ν = 0. dτ dτ (2.8)

1 Of course, the factor 2 in the argument of Θ∞ in (2.5) could be dropped (Θ∞ (λx) = Θ∞ (x) for λ > 0), but we keep it in order to emphasize that the walls come with a natural normalization linked to the fact that they initially appear as Toda walls ? exp(?2wi (β )) [11]. These exponential walls become sharp in the Chitre-Misner limit [15, 16], generalized to higher dimensions [25, 11].

4

Thus the ball travels freely at the speed of light on straight lines until it ˙? hits one of the walls and gets re?ected. The change of the velocity v ? ≡ β after a collision on the wall wi (β ) = 0 is given by a geometric re?ection in the corresponding wall hyperplane [5, 11] v ? → v ′ = (Wi (v ))? ≡ v ? ? 2
?

wiν v ν ? ρ wi wiρ wi

(no sum over i)

(2.9)

? where wi ≡ G?ν wiν are the contravariant components of wi . For a timelike wall (whose normal vector is spacelike), the re?ection is an orthochronous Lorentz transformation; hence the velocity remains null and future-oriented. Let C + denote the future light cone with vertex at the origin (β ? = 0) where the walls intersect. In the asymptotic regime under study, the initial point from which one starts the motion has positive value of the timelike ? combination d ?=1 β of the coordinates; therefore, since the walls wi (β ) = 0 are all timelike – see below –, the ball wordline remains within C + [11]. The con?nement of the billard motion to the forward light cone enables one to project, if one so wishes, the piecewise linear motion of the ball in the Minkowski space Md onto the upper sheet Hd?1 of the unit hyperboloid: d

Hd?1 : G?ν β ? β ν = ?1,

β ? > 0.
?=1

(2.10)

A projection is in fact physically necessary in order to take into account the gauge redundancy (time-reparametrization invariance) and its associated Hamiltonian constraint. One of the β ? ’s does not correspond to an independent degree of freedom. The projection to the upper hyperboloid Hd?1 corresponds to viewing the d ? 1 coordinates of Hd?1 as the physical d ? degrees of freedom and ?=1 β (or a function of it) as the “time” (see e.g. [28]). For practical purposes, however, it is also convenient to keep the redundant description in terms of which the evolution is piecewise linear. We shall switch back and forth between the two descriptions. Note that the linear motion of β ? projects to a geodesic motion on hyperbolic space Hd?1 , so the problem is equivalent, in the limit under consideration, to a billiard in hyperbolic space. We now wish to describe in more detail the convex (half) cone W + de?ned by the simultaneous ful?llment of all the conditions wi (β ) ≥ 0, to which the motion of the billiard ball is also con?ned. There are altogether two types of walls. Setting n ≡ d ? 2, they are 1. Symmetry walls [11] wi (β ) = β i ? β i?1
d?1 d

w0 (β ) = β

w?1 (β ) = β ? β



d?2

(i = 2, · · · , n ≡ d ? 2), ,

(2.11) (2.12) (2.13)

d?1

5

2. Gravitational wall [4]
d?2

w1 (β ) = 2β 1 +
i=2

β i (d ≥ 4)

(2.14)

(for d = 3, w?1 = β 3 ? β 2 , w0 = β 2 ? β 1 and w1 = 2β 1 ). There is a total of d walls, which are all timelike since the associated wall forms (normal vectors) wi (i = ?1, 0, 1, · · · n) are spacelike in any spacetime dimension: G?ν wi? wiν = 2 (i ?xed). (2.15)

The walls therefore intersect the upper light cone C + . The qualitative dynamics of the billiard can be understood in terms of the relative positions of C + and W + . Two cases are possible : 1. W + is contained within C + (i.e., all vectors of W + are timelike or null); 2. W + is not entirely contained within C + (i.e., there are not only timelike and null but also spacelike vectors in W + ). In the ?rst case, the walls de?ne a generalized, ?nite-volume simplex in hyperbolic space Hd?1 (generalized because some vertices can be at in?nity, which occurs when some edges of the cone W + are lightlike2 ). As the walls are timelike, the ball will undergo an in?nite number of collisions because, moving at the speed of light, it will always catch up with one of the walls. The only exception, of measure zero, occurs when the ball moves precisely parallel to a lightlike edge of the billiard (there is always at least one such edge). As we shall see in the next section, the dihedral angles of the wall are all submultiples of π , so that the re?ections on the sides of the billiard generate a discrete group of isometries of hyperbolic space. Similarly to what happens in the superstring case [11], the projected dynamics on Hd?1 is then chaotic (Anosov ?ow) according to general theorems on the geodesic motion on ?nite-volume manifolds with constant negative curvature. In the second case, some walls intersect outside C + and the billiard on Hd?1 has in?nite volume. The ball undergoes a ?nite number of collisions until its motion is directed toward a region of W + that lies outside C + . It then never catches a wall anymore because it cannot leave C + : no “cushion” impedes its motion. The dynamics on Hd?1 is non-chaotic and the spacetime metric asymptotically tends to a generalized Kasner metric, corresponding to an uninterrupted geodesic motion of the ball.
2

The edges of W + are the (one-dimensional) intersections of d ? 1 distinct faces of W + .

6

The question of chaos vs. regular motion is thereby reduced to determining whether it is case 1 or case 2 that is realized. We discuss this in the next section by relating the “wall cone” W + to the fundamental Weyl chamber of a certain inde?nite Kac-Moody algebra.

3

Hyperbolic Kac-Moody algebras and chaos

In this section, we show that the re?ections (2.9) can be identi?ed with the fundamental Weyl re?ections of the inde?nite Kac-Moody algebra AEd , and therefore that the cone W + can be identi?ed with the fundamental Weyl chamber of AEd . To do that, we need to compute the dihedral angles between the walls. A direct calculation shows that the Gram matrix Aij ≡ G?ν wi? wjν for i, j = ?1, 0, 1, · · · , n (3.1)

of the scalar products of the wall forms is given by ? ? 2 ?1 0 Aij = ? ?1 2 ?2 ? for d = 3 0 ?2 2 and 2 ?1 0 0 ? ?1 2 ?1 0 ? ? 0 ?1 2 ?1 ? . . Aij = ? ? . ? 0 0 0 0 ? ? 0 0 0 0 0 ?1 0 0 ? ··· ··· ··· 0 0 0 0 0 0

(3.2)

? 0 ?1 ? ? 0 ? ? ? for d > 3. (3.3) ? · · · 2 ?1 0 ? ? · · · ?1 2 ?1 ? · · · 0 ?1 2 √ In both cases, the wall forms have same length 2. As in [11], we identify them with the simple roots of a Kac-Moody algebra. To emphasize the identi?cations “wall forms = simple roots”, we shall henceforth switch to a new notation and denote the wall forms wi by ri . We shall also denote the Cartan subalgebra of the Kac-Moody algebra by H and its dual (space of linear forms on H, i.e., the “root space”) by H? . Thus, wi ≡ ri ∈ H? . (3.4)

We recall that the root space H? is endowed with a bilinear form, which we identify with the bilinear form de?ned by the (contravariant) metric G?ν given above, ri · rj ≡ G?ν ri? rjν (3.5)

7

Since the roots have all same length squared 2, the algebra is “simply-laced” and the Gram matrix Aij computed in (3.2) and (3.3) is also the Cartan matrix aij , aij ≡ 2ri · rj , ri · ri (3.6)

i.e., Aij = aij . We then recognize the ?rst matrix as the Cartan matrix of the Kac-Moody algebra AE3 , while the second matrix is the Cartan matrix of the Kac-Moody algebra AEd (d > 3). This is what justi?es the identi?cations (3.4) and (3.5). The roots r0 , . . . rn form the closed ring of the Dynkin diagram, r0 is the (a?ne) root closing the ring, and r?1 is the overextended root connected to r0 . Once the wall forms are identi?ed with the simple roots of a Kac-Moody algebra, the space Md in which the dynamics of the scale factors takes place becomes identi?ed with the Cartan subalgebra H of AEd . The cone W + de?ning the billiard is given by the conditions ri , β ≥0 for all i = ?1, 0, 1, ..., n (3.7) where ri , β denotes the pairing between a form ri ∈ H? and a vector β ∈ H. The cone W + is then just the fundamental Weyl chamber [17, 18], as was anticipated by our notations. It is striking to note that the ?nite dimensional germ Ad?2 of the hyperbolic algebra AEd is nothing but the Ehlers symmetry of the toroidal compacti?cation of the original gravity from d + 1 to 3 dimensions [29]. The reduction to two dimensions brings the a?ne extension and the ?nal elimination of all spatial coordinates increases the rank further to d [23]. The above Cartan matrices are indecomposable. They are also of indefinite, Lorentzian type since the metric G?ν in H is of Lorentzian signature. A Cartan matrix with these properties is said to be of hyperbolic type if any subdiagram obtained by removing a node from its Dynkin diagram is either of ?nite or a?ne type [17]. The concept of hyperbolicity is particularly relevant here because it is a general result that the fundamental Weyl chamber W + of a hyperbolic Kac-Moody algebra is contained within the light cone C + ; the Weyl cell is then a (generalized) simplex of ?nite volume. Furthermore, for hyperbolic KM algebras the closure of the Tits cone, de?ned as the union of the fundamental Weyl chamber and all its images under the Weyl group, is just C + ([17], section 5.10). As already mentioned, the Kac-Moody algebras AEd are hyperbolic for d ≤ 9. We will now verify by explicit computation that their associated fundamental Weyl chambers are indeed contained in the forward light cone. The location of the fundamental Weyl chambers in the general case is most conveniently (and most easily) analyzed by means of the fundamental weights Λj ∈ H? . The latter are de?ned by ri · Λj ≡ G?ν ri? Λjν = δij i, j = ?1, 0, 1, . . . , n ≡ d ? 2. 8 (3.8)

Let us also introduce the coweights Λ∨ i ∈ H, i.e., the contravariant vectors ? ?ν associated with the forms Λi with components (Λ∨ i ) ≡ G Λiν . Because the fundamental Weyl chamber W + is de?ned by the conditions ri , β ≥ 0, we have
n

W

+

= β ∈ Md ≡ H | β =

i=?1

ai Λ∨ i , ai ∈ R, ai ≥ 0

(3.9)

The (one-dimensional) edges of W + are obtained by setting all aj except one to zero, which gives the vectors Λ∨ i . The question of determining whether the fundamental Weyl chamber is contained in the forward light cone or not is thus reduced to a simple computation of the norms of the fundamental weights. To get the fundamental weights, we observe that if the root r?1 is dropped, the associated Cartan matrix reduces to the Cartan matrix of a?ne sl(n + 1). The a?ne null root is given by δ = r0 + r1 + ...rn (3.10)

It obeys δ2 ≡ δ · δ = 0 = rj · δ for all j = 0, 1, ..., n (but r?1 · δ = ?1). The fundamental weights for the subalgebra An are de?ned by ri · λj = δij for i, j = 1, ..., n (3.11)

They are explicitly given by λj = n?j+1 r1 + 2r2 + . . . + jrj n+1 j + (n ? j )rj +1 + (n ? j ? 1)rj +2 + . . . + rn n+1 (3.12) (3.13)

with norm λ2 j = j (n ? j + 1) >0 n+1 (3.14)

(note that r0 ·λj = ?1 for all j = 1, ..., n). One then ?nds for the fundamental weights3 of AEd Λ?1 = ?δ , Λ0 = ?r?1 ? 2δ , Λj = Λ0 + λj for j = 1, ..., n (3.15)

Their norms (with Λ2 ≡ Λ · Λ ≡ G?ν Λ? Λν ) are easily computed: Λ2 ?1 = 0 ,
3

Λ2 0 = ?2 ,

Λ2 j = ?2 +

j (n ? j + 1) n+1
j

(3.16)

In the general case with highest root θ = fundamental weights are given by Λ?1 = ?δ ,

mj rj , we have r0 · λj = ?mj and the Λj = mj Λ0 + λj

Λ0 = ?r?1 ? 2δ ,
j (a ?1

An alternative representation is Λi = matrix.

)ij rj where (a?1 )ij is the inverse Cartan

9

Note that Λ?1 is always lightlike, and Λ0 is timelike for all n. It is furthermore elementary to check that Λ2 j ≤0 for all j if n ≤ 7 (3.17)

with equality only for n = 7 and j = 4. For n ≥ 8 there is always at least one spacelike fundamental weight Λj ; e.g. for n = 8 we have
2 Λ2 4 = Λ5 =

2 >0 9

(3.18)

The above calculation then tells us that for n ≤ 7 (i.e. for AEd with d ≤ 9) the fundamental Weyl chamber is contained in the forward light cone with one edge touching the light cone (two edges for n = 7). For n ≥ 8 there is at least one spacelike edge, so the Weyl chamber contains timelike, lightlike and spacelike vectors. This is, then, the Kac-Moody theoretic understanding of the fact that the asymptotic solution of the vacuum Einstein equations in the vicinity of a spacelike singularity exhibits the never-ending oscillatory behaviour of the BKL type in spacetime dimensions ≤ 10, while this ceases to be the case for D ≥ 11 [4]. To conclude this letter we would like to stress once more that the emergence of a Kac-Moody algebra is not automatic for the gravitational systems under consideration. For instance, the billiard associated with the EinsteinMaxwell system in D spacetime dimensions has the same symmetry walls (2.11), (2.12), (2.13), but the gravitational wall (2.14) is replaced by the (asymptotically dominant) electric wall w1 (β ) = β 1 . This wall is orthogonal to all symmetry walls, except w2 (w0 for d = 3) with which it makes an angle α given by cos α = (d ? 1)/2(d ? 2). This dihedral angle is generically not a submultiple of π and the associated group of re?ections is not a discrete group, with two notable exceptions: (i) α is equal to zero for D = 4, where electric and gravitational walls coincide (though the wall forms are normalized di?erently), and (ii) the angle α is equal to π/6 for the case D = 5, whose study was advocated in [10] in the context of homogeneous models. Taking into account that the wall form w1 has norm squared equal to (d ? 2)/(d ? 1) = 2/3, one gets in that case the Cartan matrix ? ? 2 ?1 0 0 ? ?1 2 0 ?1 ? ? aij = ? (3.19) ? 0 0 2 ?3 ? 0 ?1 ?1 2 The underlying Kac-Moody algebra is the canonical hyperbolic extension of the exceptional Lie agebra G2 (hyperbolic algebra number 16 in table 2 of [30]). One Einstein-Maxwell theory in 5 dimensions is particularly interesting because it is the bosonic sector of simple supergravity in 5 dimensions, which shares many similarities with D = 11 supergravity, such as the cubic Chern-Simons term for the vector ?eld [31]. The relevance of the exceptional 10

group G2 to that theory was pointed out in [32, 29]. This system, as well as pure gravity or superstring models and M -theory, for which one does get the Weyl group of a Kac-Moody algebra, are thus rather exceptional [11].

Acknowledgements
T.D. thanks Victor Kac for informative communications. M.H. and H.N. are grateful to the Institut des Hautes Etudes Scienti?ques for its kind hospitality. The work of M.H. is partially supported by the “Actions de Recherche Concert? ees” of the “Direction de la Recherche Scienti?que - Communaut? e Fran? caise de Belgique”, by IISN - Belgium (convention 4.4505.86) and by the European Commission programme HPRN-CT-2000-00131 in which he is associated to K.U.Leuven. The work of H.N. was supported in part by the European Union under Contract No. HPRN-CT-2000-00122. The work of B.J. was supported in part by the European Union under Contract No. HPRN-CT-2000-00131.

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