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Fermionic Kac-Moody Billiards and Supergravity

We study the"fermionic billiards", i.e. the chaotic dynamics of the gravitino, that arise in the near-spacelike-singularity limit of eleven-dimensional supergravity and of its dimensional truncations (notably four-dimensional simple supergravity). By exploiting the gravity-coset correspondence, we show that the billiard dynamics of the gravitino is described by a `spin extension' of the E(10) Weyl group, namely as a product of 90 degree `vector-spinor rotations' along certain simple-root-related generators of the maximal compact subalgebra K(E(10)) of the hyperbolic Kac--Moody algebra E(10). The `super-billiard' that combines the bosonic and fermionic billiards is found to have a remarkably simple structure, which exhibits a striking analogy with a polarized photon propagating in the ten-dimensional Lorentzian Weyl chamber of E(10).


Introduction
Some time ago, the studyà la Belinski, Khalatnikov, Lifshitz (BKL) [1] of the chaotic behaviour [2] of the general solution of the bosonic sector of 11dimensional supergravity (SUGRA 11 ) near a spacelike singularity revealed a hidden connection with the hyperbolic Kac-Moody algebra E 10 [3]. More precisely, it was found that, in the near-singularity limit, most bosonic degrees of freedom "freeze" (i.e. have some finite limit), except for the 10 'diagonal' components of the spatial metric (parametrized by their logarithms; β a ∼ − ln g aa ) which undergo, at each spatial point, a chaotic billiard motion within a conical polyhedron in an auxiliary 10-dimensional Lorentzian space, which can be identified with the Weyl chamber of E 10 . Further work has shown that this E 10related billiard motion was the lowest approximation ('height 1') of a hidden correspondence, which was checked up to height 29, between the SUGRA 11 dynamics and the dynamics of a massless particle on the infinite dimensional coset space E 10 /K(E 10 ) [4]. [Here, K(E 10 ) denotes the maximal compact subgroup of E 10 , see below.] More evidence for the presence of a hidden E 10 symmetry in SUGRA 11 came from the consideration of the fermionic sector of supergravity which could also be put into correspondence (up to some approximation roughly equivalent to the height 29 of the bosonic sector) with a fermionic extension of the E 10 /K(E 10 ) coset model [5,6,7]. For more work on the correspondences between gravity theories and Lorentzian Kac-Moody algebras see [8,9,10,11,12]. For a recent review, see [13]. For other works suggesting a hidden role of Lorentzian Kac-Moody algebras in supergravity or string theory, see [14,15,16,17].
The aim of the present paper is to study the fermionic side of the billiard dynamics taking place in the near-singularity limit. Instead of testing the conjectural gravity-coset correspondence to higher levels of approximation, we shall here go back to the lowest level of approximation ('height 1') and consider the Kac-Moody-related mathematical structure behind the billiard dynamics of the SUGRA 11 -gravitino. We shall find that this structure exhibits a remarkably simple interplay between E 10 (hyperbolic) and K(E 10 ) (elliptic) aspects.
In this work, we shall study the fermionic side of cosmological billiards within the standard algebraic framework where the 'classical' supergravity dynamics makes mathematical sense. Namely, we assume that the fermionic variables (ψ) take values in the odd-part of a Grassmann algebra, while the bosonic variables (g, A) take values in the even part of the same algebra. Then, with respect to the grading defined by a basis in the underlying Grassmann algebra, one has ψ = ψ 1 + ψ 3 + . . . , while g = g 0 + g 2 + . . . , A = A 0 + A 2 + . . . . The 'body' (g 0 , A 0 ) then satisfies the Einstein-3-form equations without any 'back-reaction' of ψ (which only enters the 'soul' parts g 2 , A 2 , . . . ), while the lowest component ψ 1 of ψ follows a dynamics driven by g 0 and A 0 . We are aware that [18,19] have shown that, in the case of an Einstein-Dirac system, the replacement of the O(ψ 2 ) source terms for g 2 by classical VEV's (of grading zero instead of two) strongly modifies the bosonic dynamics (then taking entirely place at grading zero), and, in particular, destroys the chaotic nature of the near-singularity limit. However, we do not think that such a replacement is mathemetically or physically justified. Pending a truly quantum analysis (in Hilbert space) of the back reaction of fermions, we think our analysis below is mathematically sound.
We shall also consider the truncation of this fermionic billiard to 4-dimensional N = 1 supergravity (SUGRA 4 ). In this case, the relevant (hidden) hyperbolic Kac-Moody symmetry is AE 3 (or A ++ 1 , the hyperbolic extension of A 1 ≃ sl 2 ), and the relevant coset model involves AE 3 /K(AE 3 ) [14]. In both cases (E 10 /K(E 10 ) and AE 3 /K(AE 3 )), we shall find that the 'fermionic billiards' exhibit a remarkable factorized structure involving both Lorentzian (SO (9,1) or SO(2, 1)) and Euclidean (SO (10) or SO(3)) structures. These results may suggest new ways of thinking about the fermionic extension of the E 10 /K(E 10 ) coset model. Some of our results may also be of interest for mathematics as they involve what we shall call "spin extensions of the Weyl groups" of hyperbolic Kac-Moody algebras.

Cosmological billiards and the conjectured gravity/coset correspondence
In this section, we shall present a brief survey of the so-called "cosmological billiards" [14] and describe some elements of the bosonic side of the gravitycoset correspondence. This will allow us to fix the conventions and notations used in this article.

Maximal supergravity in the BKL-limit
We start with a short review of the billiard limit of the bosonic sector of D = 11 supergravity. 1 In other words, we consider the behaviour of a general solution close to a spacelike singularity in the leading BKL-like 'gradient approximation', where one keeps only time derivatives and the dominant terms involving spatial gradients. We will adopt a space-time slicing such that the singularity "occurs" on the coordinate time slice t = +∞. This slicing is built by using pseudo-Gaussian coordinates defined by a vanishing shift leading to a metric of the form Here, ξ m (x) = 10 i=1 ξ m i (x)dx i could be an arbitrary spatial frame. For simplicity, we will use a coordinate frame ξ m (x) = dx m . Note that, in order to avoid ambiguities in later formulae that involve repeated indices but no sum, we shall often dispense with the Einstein summation convention and explicitly indicate the needed sums. Furthermore, we choose the time coordinate 2 t so that the lapse function is linked to the spatial volume density via N (t, x) = det g mn (t, x) (2.2) and we adopt a generalized temporal gauge for the three-form potential, i.e. A 0mn = 0. Without loss of generality, we describe the independent degrees of freedom of the spatial metric g mn by an Iwasawa decomposition of the associated vielbein e a m (g mn = 10 a=1 e a m e a n ), e a m =: e −β a N a m (2.3) implying g mn = 10 a=1 e −2β a N a m N a n . (2.4) Here the N a m 's are upper triangular matrices which have 1 as diagonal entries. The logarithmic scale factors β a describing the 'diagonal' components of the metric (see Eq. (2.4)) play a crucial role in the 'near-spacelike-singularity limit' (which corresponds to T → 0, t → +∞ and 10 a=1 β a → +∞, so that the local volume This defines a flat Lorentzian metric G in the d = D − 1 = 10 dimensional space of scale factors β a with signature (− + · · · +). This Lorentzian metric G plays a crucial role in the gravity-Kac-Moody-coset correspondence. 3 It will also be of prime importance for the discussion of the gravitino in section 3. If the dynamics were completely described by the Lagrangian Lβ (2.5), the scale factors β would follow a geodesic in the flat Lorentzian space (R 10 , G) (or β-space): (2. 6) In addition, the leadingβ-terms in the Hamiltonian constraint impose the condition 0 = 10 a,b=1 which means that the geodesic (2.6) is a null geodesic.
The null geodesic (2.6, 2.7) is the β-space description of a Kasner solution. However, such a Kasner-like solution is profoundly altered when one takes into account the additional contributions to the Lagrangian (beyond (2.5)) coming from the off-diagonal components N a m (2.4) of the metric, the electric and magnetic energy of the three-form A mnp (t, x), and from the spatial gradients of the metric. The detailed analysis in [14] shows, after having changed to the Hamiltonian formalism, that all the other degrees of freedom of supergravity result in adding to Lβ (2.5) several potential densities V A (labelled by an index A) which have an exponential dependence upon the β's: (2.8) Here, the coefficients c A depend on other degrees of freedom (N a m , A mnp ,. . . and their conjugate momenta) and the α A (β) are linear functionals α A (β) = 10 a=1 α A, a β a . The sum of (2.5) and (2.8) leads to a kind of Toda model for the dynamics of the β's. In the near-singularity limit 10 a=1 β a → +∞, one can order the various potentials in terms of their importance for altering the monotonic zeroth-order geodesic Kasner motion (2.6) into a chaotic dynamics. There are 10 dominant Toda terms with exponents α i (β) with i = 0, . . . , 9 (see Eq. (2.10) below), which all have positive coefficients c i . The ten terms ∝ e −2αi(β) conventionally define the 'height 1' in an approximation scheme where the higher order terms are made of various products of the ten dominant terms (e.g. e −2α1 e −2α2 = e −2(α1+α2) is of 'height 2', e −2(α1+α2+α3) of 'height 3', etc.). 3 G is part of the deWitt supermetric.
Nine of the dominant Toda terms come from terms (2.8) in the Hamiltonian where the coefficients are quadratic in the conjugate momenta of N a m (2.4). They are called 'symmetry walls'. The choice of parametrizing the spacetime metric g by upper-triangular matrices N (2.4) implies that the functionals α A corresponding to symmetry walls have the form [14] α s (ab) (β) = β b − β a with a < b = 1, . . . , 10. (2.9) Among the (10 × 9)/2 = 45 symmetry walls, the dominant ones are = β i+1 − β i for i = 1, . . . , 9. (2.10a) The tenth dominant Toda term comes from the potential (2.8) in the Hamiltonian which arises from the 'electric' energy density of the 3-form A mnp (t, x). The antisymmetry of A mnp implies that the corresponding 'electric walls' have the form α el abc (β) = β a + β b + β c with a, b, c ∈ {1, . . . , 10} being all different. Among these, the dominant 'electric wall' is α 0 (β) := α el 123 (β) = β 1 + β 2 + β 3 . (2.10b) All the other potential terms, which include the subdominant symmetry and electric walls, the 'magnetic walls' α mag a1...a6 (β) = β a1 + · · · + β a6 , and the 'gravitational walls' α (g) (see e.g. (2.18) below) can be written as linear combinations with positive integer coefficients of the 10 dominant walls α i (2.10). It has been shown in [4,8] that, up to height 29 in an expansion in dominant walls, the SUGRA 11 -dynamics (for all the bosonic degrees of freedom) following from the sum of (2.5) and all the other terms (2.8) can be identified with the geodesic dynamics of a particle on the infinite-dimensional coset space E 10 /K(E 10 ) (which can be written as the sum of (2.5) and of an infinite number of potential terms (2.8) where the linear forms α A (β) now label the positive roots of E 10 ).
In the following, we shall focus on the 'height 1' approximation (which is common to SUGRA 11 and to the E 10 /K(E 10 ) coset) that consists of keeping only the 10 dominant Toda walls, i.e. of adding to (2.5) a potential term of the form (with c i > 0) c i e −2αi(β) . (2.11) One can even further approximate the 10 Toda walls (2.11) in the BKL-limit 10 a=1 β a → +∞ by a sum of 'sharp walls' Thus, the motion of the scale factors is restricted to the polywedge α i (β) ≥ 0 for i = 0, . . . , 9. In addition, the instantaneous 'velocity' v a =β a (2.6) of the β-motion is still constrained by the BKL-limit of the Hamiltonian constraint to satisfy Eq. (2.7), i.e. to be lightlike. The effect of each collision on a particular wall, say α i (2.13) 4 is to transform the incoming Kasner velocity v (2.6) into the outgoing one v ′ = r i (v) by the usual formula for a geometric reflection r i in the hyperplane (2.13) in β-space (R 10 , G) [7]: (2.14) Here α # i denotes the contravariant version of the covariant vector (or linear where G ab denotes the inverse of the basic β-space metric (2.5). The round brackets (·|·) appearing in the denominator in Eq. (2.14) denote the scalar product defined by the metric G ab , or its inverse (depending on whether one considers covariant or contravariant vectors). For instance, (α i |α j ) = 10 a,b=1 G ab α i ,a α j ,b , which is equal to ( From the scalar products between the dominant walls α i (2.10), one defines the following matrix: It is found that this matrix is integer valued: with A ii = 2 and −A ij ∈ N when i = j. In addition, for SUGRA 11 (i.e. E 10 ) as well as for SUGRA 4 (i.e. AE 3 ), all the dominant walls have the same 'squared length' (α i |α i ) = 2, so that A ij is symmetric. One can associate to A a Dynkin diagram by drawing a node for the 10 dominant walls α i (2.10) and link them by −A ij lines, which results in: This is the Dynkin diagram of the hyperbolic Kac-Moody algebra E 10 [3]. The matrix A (2.16) is the corresponding Cartan matrix [21] of E 10 for the present case of D = 11 supergravity. In addition, the restriction of the E 10 invariant metric to the Cartan subalgebra coincides (when using suitable coordinates, also denoted β a in the Cartan subalgebra) with the basic β-space metric G ab introduced in Eq. (2.5).
Summarizing so far: the SUGRA 11 billiard picture in β-space, arising as leading approximation in the sharp wall BKL-limit, can be identified with a 'Kac-Moody billiard' via a dictionary that maps: 1. the ten gravity scale factors to coordinates β a in the Cartan subalgebra of E 10 2. the ten SUGRA 11 dominant walls α i to the ten simple roots of E 10 .
In other words, the billiard table, defined by the ten inequalities α i (β) ≥ 0 is identified with the Weyl chamber of E 10 , and the bosonic billiard dynamics, comprising successive reflections (2.14) in the dominant walls, is mapped into a product of Weyl reflections r i k , i.e. an element w of the Weyl group W E10 of E 10 , say where the length of the Weyl 'word' w ∈ W E10 (describing the billiard dynamics towards the singularity) grows indefinitely as t → +∞. It is this aspect of the bosonic gravity-coset correspondence that we will generalize to the fermionic sector in section 3.

2.2
Truncating SUGRA 11 to SUGRA 4 , and correlatively reducing E 10 to AE 3 Before doing so, let us explain a result that we shall need below: how the inclusion of N = 1 SUGRA 4 within SUGRA 11 corresponds to an embedding of AE 3 within E 10 . Here, we shall indicate how a sub-billiard of the (full) SUGRA 11 billiard gives rise to the Cartan matrix of AE 3 in the BKL-limit.
To obtain N = 1 supergravity in D = 4 from SUGRA 11 , we have to perform two steps. First, we reduce the theory down to 4 dimensions by compactifying on a flat seven torus T 7 . Second, we truncate the degrees of freedom of SUGRA 11 down to those of N = 1 SUGRA 4 . In other words, we discard the three-form and in our pseudo-Gaussian gauge (2.1), we can restrict the indices m, n to the values 1, 2, 3 only. The β-space now becomes 3-dimensional, and endowed with a 'reduced' Lorentzian metric G r (of signature (− + +)) defined by restricting the sums in Eq. (2.5) to a, b = 1, 2, 3. Among the 9 symmetry walls (2.10a), only two survive: α 1 := α s (12) and α 2 := α s (23) . The previously tenth dominant wall (2.10b) disappears due to the truncation of the 3-form. However, we must now take into account the gravitational walls, which we could neglect above, because they were hidden 'behind' other walls (i.e. subdominant), but which now will affect the dynamics of the β-motion near a singularity.
In D = 11 supergravity, the gravitational walls are given by [14] α g The dimensional reduction on T 7 to D = 3 + 1 dimensions with the gauge fixing of the metric as in (2.1) implies that only the three spatial one forms θ 1 Iwa , θ 2 Iwa , θ 3 Iwa have a non-trivial coordinate dependence. Therefore, all structure functions C c de apart from the ones with c, d, e ∈ {1, 2, 3} are zero. This entails that only the gravitational walls (2.18) with these indices are present, because the corresponding potential (2.8) vanishes for zero structure functions C c de [14]. Among the surviving gravitational walls, one easily finds (making use of the dominant symmetry walls α s (12) (β) = β 2 − β 1 > 0 and α s (23) (β) = β 3 − β 2 > 0) that the dominant surviving gravitational wall is: 6 Then following the prescription presented in the previous section, one computes the scalar products between the three dominant walls α s (12) , α s (23) , α g 123 , with respect to the inverse of the reduced metric G r , thus obtaining a 3 × 3 matrix A ij . One finds that this matrix is the Cartan matrix of AE 3 , whose Dynkin diagram is:d 2.3 Kac-Moody coset model and the role of the connection Q After having briefly reviewed how the BKL-limit of the bosonic sector of supergravity gives rise to a billiard motion taking place in a Lorentzian polywedge, which can be identified with the Weyl chamber of a Kac-Moody algebra, let us review some of the elements of the definition of the E 10 /K(E 10 ) (respectively AE 3 /K(AE 3 )) coset model which has been conjectured to capture the hidden E 10 (resp. AE 3 ) symmetry of SUGRA 11 (resp. SUGRA 4 ). First, we recall that the relevant Kac-Moody algebra is inductively constructed by its Chevalley-Serre presentation from the set of generators (e i , f i , h i ) each corresponding to a simple root or to a node in the Dynkin diagrams of E 10 or AE 3 . The elements h i span the Cartan subalgebra h ([h i , h j ] = 0). The basic Lie brackets among with Jacobi identities and Serre relations [21]. The maximal compact subalgebra K(g) of an algebra g is defined as the fixed point set of the Chevalley involution defined by ω(

20)
6 Note that this dominant surviving gravitational wall in D = 4 differs from the gravitational wall that would dominate the other gravitational walls in a direct analysis of SUGRA 11 , namely α (g) 1 9 10 . 7 Note that for g = gl d , the automorphism −ω corresponds to the transposition in the standard matrix representation of GL(d) and hence the corresponding subgroup K(G) = SO(d) indeed is the maximally compact subgroup of the group G = GL(d) in the topological sense.
where E α,s is a general 'raising operator', i.e. a multiple commutator of the simple "raising operators" e i , and E −α,s := −ω(E α,s ). Each general raising operator is labelled by: (1) a positive root 8 α ∈ ∆ + (i.e. [h, E α,s ] = α(h)E α,s for h ∈ h) and (2) a degeneracy index s = 1, . . . , mult(α) labelling the different elements of g having the same root α. Due to the similarity of the coset descriptions for hyperbolic Kac-Moody algebras, in particular for E 10 and AE 3 , we will for clarity focus on the former. It is straightforward to generalize the construction to other algebras, which will also be addressed in Appendix B.
Assuming the existence of an Iwasawa decomposition, we parametrize the coset V ∈ E 10 /K(E 10 ) for the Lie algebra g = E 10 by 9 Here H a (a = 1, . . . , 10) is a general basis in the Cartan subalgebra h that was identified with β-space and is hence endowed with the metric G ab (2.5) via (H a |H b ) = G ab . The link between the general basis H a of h and the specific ten Cartan elements h i entering the Serre-Chevalley presentation involves the simple roots α i (identified with the ten simple walls (2.9, 2.10b) via Note that the 'co-root'α i ≡ 10 a=1α a i H a ≡ h i (which belongs to the Cartan subalgebra, i.e. the dual of the root space to which α i belongs) enters both in the (Weyl) reflections (2.14) in the hyperplanes defined by the roots, which can be written as and in the definition of the Cartan matrix (2.16) tantamount to Let us also recall that a Kac-Moody algebra is called 'simply laced' if the Cartan matrix A ij is symmetric. This implies that all the simple roots (and more generally all the real roots) have the same length, which is conventionally normalized to (α i |α i ) = 2. In that case, which applies to the two specific Kac-Moody algebras E 10 and AE 3 that we shall consider here, the co-root h i =α i becomes equal to the contravariant version of the corresponding root α # i (2.15) and the metric in Cartan space simply yields (α i |α j ) = (h i |h j ) = (α i |α j ) = α j (α i ) = A ij . For simplicity, we shall assume in the main text that we are in this simply-laced case. The general case of a non-symmetric Cartan matrix will be discussed in Appendix B. 8 We recall that a root α is a linear form on Cartan space h and that a positive root α is a sum P i n i α i where n i ∈ N and α i is a simple root. 9 We use the same symbol for groups and algebras. It should be clear from the context what is meant.
The decomposition of the Lie algebra valued 'Maurer-Cartan velocity' into an "antisymmetric" part Q ∈ K(E 10 ) and a "symmetric" one P ∈ E 10 ⊖ K(E 10 ) allows us to define the coset Lagrangian The bilinear form (·|·) entering the coset Lagrangian (2.24) is the (unique) E 10invariant bilinear form [21]. In the simply-laced case, it is such that (H a |H b ) = G ab , (h i |h j ) = A ij and (E α,s |E α ′ ,s ′ ) = δ α+α ′ δ ss ′ . See Appendix B for the modifications that are necessary for the non-simply laced case.
The bosonic coset equations of motion derived from the Lagrangian (2.24) read (in the gauge n(t) = 1 that we shall henceforth use for simplicity) ∂ t P(t) = Q(t), P(t) . (2.25) In addition, the variation of the coset 'lapse' n(t) in (2.24) yields the constraint 0 ≈ (P|P) . (2.26) In these equations, P ∈ E 10 ⊖ K(E 10 ) represents the 'velocity' of the coset 'particle' moving on the (infinite dimensional) coset space E 10 /K(E 10 ). See [8] for the explicit proof that the equations of motion (2.25) can be identified (up to height 29) with the equations of motion of SUGRA 11 via a 'dictionary' which relates the (first few rungs of the infinite ladder of) coset variables (β a , ν α ) to the bosonic gravity variables (β a , N a m , A mnp , ∂ [m A npq] , C a bc ).
The fermionic extension of the coset model [5,6,7] has similarly shown that the bosonic gravity-coset 'dictionary' could be extended (up to an approximation that corresponds to the bosonic one) to a map relating the gravitino field ψ (11) M (t, x) of SUGRA 11 to a coset fermionic variable ψ(t) which belongs to the so-called 'vector-spinor' vs representation of K(E 10 ). Then, the fermionic coset equation of motion simply reads ∂ t ψ(t) = Q vs (t)ψ(t) (2.27) where Q vs denotes the 'vector-spinor' representation of the object Q ∈ K(E 10 ) that entered the basic decomposition (2.23).
When comparing Eq. (2.25) to (2.27), we see that the 'antisymmetric' part Q ∈ K(E 10 ) of the Lie algebra valued velocityVV −1 ∈ E 10 (2.23) plays the role of a K(E 10 )-connection, realizing the 'parallel transport' of both the bosonic (coset-valued) 'velocity' P ∈ E 10 ⊖ K(E 10 ) and the fermionic coset variable ψ ∈ S vs K(E10) (where S vs K(E10) denotes the K(E 10 ) 'vector-spinor' representation space). In more intuitive terms, we can think of Q(t) ∈ K(E 10 ) as being a continuous rotational angular velocity under which both P(t) and ψ(t) 'turn' within their respective linear representation spaces of K(E 10 ). The common occurrence of the same Q(t) in the bosonic and fermionic dynamics will play a central role in the present work.

'
Spiky' structure of the K(E 10 ) 'angular velocity' Q(t) in the near-singularity limit Having reviewed the crucial role of the K(E 10 ) 'angular velocity' Q(t) in both the bosonic and the fermionic equations of motion (2.25, 2.27), let us study the structure of Q(t) ∈ K(E 10 ) in the BKL-type, near-singularity limit. In that limit, previous work on cosmological billiards has shown that, as a function of the coset time t (which tends to +∞ rougly proportionally to minus the logarithm of the proper time) Q(t) was exponentially small most of the time, except around the time of a 'collision' with a dominant wall, when Q(t) rose up to a finite value [14,11].
To study the precise structure of these 'spikes' in Q(t), it is enough to consider the 'collision' with a single wall. [Indeed, as t → +∞, the coset time intervalls between wall collisions grow linearly in t. This 'one-wall collision' process is anyway interesting in its own right ("S-brane solution", see below).] For simplicity, as this will be the case for the simply-laced algebras E 10 and AE 3 that we focus on here, we consider a wall form α(β) where α is a real (positive) root α ∈ ∆ + with norm (α|α) = 2. 10 In this one-wall approximation, the Iwasawa form of the coset element V (2.21) only contains one term, ν α (t)E α , in the second exponent of (2.21). 11 Then, one can easily evaluate the 'Maurer-Cartan velocity' (2.23) by using the basic commutation relation [h, E α ] = α(h)E α involving any Cartan subalgebra element h ∈ h:   Its splitting into a 'symmetric' part P ∈ E 10 ⊖ K(E 10 ) and an 'antisymmetric' one Q ∈ K(E 10 ) as in (2.23) (2.20) allows one to evaluate the Lagrangian (2.24), using the gauge n(t) = 1 for simplicity again: Note that the normalization of the kinetic term of ν α has come from the normalization (E α |E −α ) = 1 of the generators entering the K(E 10 ) generator J α .
The 'cyclicity' of the variable ν α implies that its conjugate momentum is constant. This conservation law will prove important for analyzing the time evolution of the Cartan variables β a and of the K(E 10 )-connection Q (2.29b). The analysis of the β-dynamics is greatly simplified by adapting the notation to the geometry. This is achieved by decomposing β into a component parallel (β a || ) and a component orthogonal (β a ⊥ ) to the considered wall α.
with α(β) = 10 a=1 α a β a and the metric (·|·) as used in (2.14). Note that this decomposition uses the Lorentzian metric G (2.15). As the wall potential in (2.30) only depends on the one-dimensional orthogonal component β ⊥ ∝ α(β), it is easily seen that the motion parallel to the wall proceeds with a constant velocityβ a || as in (2.6). By contrast, the motion orthogonal to the wall undergoes a non-trivial 'Toda collision'. The corresponding dynamics can be exactly solved by using the conservation of energy, which reads || denotes minus the conserved parallel kinetic energy (which must be positive in view of the null constraint (2.7) valid far from the wall). Note that the normalization factor 1 4 in the kinetic term of α(β) (2.33) has made use of (α|α) = 2. The conservation law (2.33) allows one to explicitly find the time evolution of the 'orthogonal' distance α(β). Besides the constants Π α and E || , it contains a time shift t c ∈ R as a constant of integration: Note that, when inserting the definition of Π α (2.31) and the result (2.34) into the expression of Q (2.29b), we obtain where ε α = Πα |Πα| = ±1 is the sign of Π α . Equation (2.35) exhibits the precise structure of the 'spike' of the 'angular velocity' Q ∈ K(E 10 ), which occurs at each collision with a Toda wall: it is centered at t = t c and given by an inverse hyperbolic cosine multiplying the particular K(E 10 ) generator J α = E α − E −α (2.20) associated to the considered root.
It will be useful in the following to also consider the integrated effect of Q(t) through the collision. For this, we define a finite 'angle of rotation' θ(t) about the fixed K(E 10 ) 'rotation axis' J α such that the instantaneous value of the 'angular velocity' Q(t) reads (2.36) In other words, comparing with (2.35) we havė (2.37) Integrating this differential equation yields the following expression for the time evolution of the 'angle of rotation': We clearly see on (2.38) the nature of the K(E 10 )-'rotation' that takes place at each wall: it consists of an arctan 'kink' (or 'antikink'), localized around the central time t c of the collision, and taking place on a finite time scale ∆t ≈ 1/ E || , during which most of the rotation occurs. It is remarkable that the 'total K(E 10 )-rotation' linking the initial state at t = −∞ to the final one 12 at t = +∞ is universally given by and is essentially independent of the initial conditions and of the characteristics of the considered wall. [We show in Appendix B that, if one normalizes the generators E α in a standard Kac-Moody way [21], the result (2.39) is also independent of the norm of the root α.] The only trace of the initial data is the sign ε α of the momentum Π α conjugate to ν α . In other words, the sign of Π α determines whether θ(t) undergoes a kink or an antikink.
We have checked that the universal result (2.38), that we have derived here from the coset model, can also be obtained by using gravity variables directly. For instance, in the case where the 'collision' occurs on an electric wall, we can derive (2.38) from the electric-wall bounce solution written in [22] (using a "Freund-Rubin ansatz"), while in the case of a symmetry wall, Eq. (2.38) follows from the formulae given in [14]. We have here another example where the coset model provides a common basis for expressing, in a uniform way, a priori unconnected special solutions of maximal supergravity that are sometimes referred to as "S-brane solutions" [23].

Rederivation of the bosonic wall Weyl reflection from the connection 'kink'
Summarizing so far: We have shown that the K(E 10 )-valued 'angular velocity' Q(t) which entered (in different representations) both in the bosonic coset equation of motion (2.25) and in the fermionic one (2.27) was made of a sequence of well separated inverse-cosh 'spikes', proportional to the K(E 10 ) generators (2.20) associated to the wall α on which the considered collision takes place. Before discussing, in the next section, the effect of these spikes on the fermionic variables, let us see how their effect on the bosonic variables reproduces the well-known BKL-type result (2.14) saying that the β-space 'velocity' of the incoming billiard ball is Weyl-reflected in each wall.
To do this, we have to notice that, in view of Eq. (2.25), the action of the K(E 10 )-connection Q on the coset velocity P (2.23) is given by a commutator, i.e. by a (formal) adjoint action. 13 As the 'rotation axis' J α is fixed, one can formally integrate the equation of motion (2.25) in the one-wall approximation to obtain We see here how, at any intermediate time, the effect of a collision with one wall α on P can be understood as a continuous 'rotation' given by the adjoint action (within E 10 ) of the associated K(E 10 )-group element R α (t): where the superscript Ad indicates the adjoint group action of R α (t) ∈ K(E 10 ) on the coset element P and where the "rotation" R α (t) about the axis J α is obtained from a formal exponentiation of the algebra element θ(t)J α . One might worry whether the formal exponentiation (2.41) is a well-defined procedure. It is in the present case, as we are going to see. The basic reason is that, for a given real root α, the entire computation takes place in an SL(2, R) subgroup. 14 Let us evaluate the action of the total K(E 10 )-rotation linking the initial configuration P(−∞) to the final one P(+∞): In order to evaluate this 'total rotation' R α,εα on P(−∞), one starts by noticing that, far away from the wall (t → ±∞), theν-terms in (2.29a) become negligible due to e −α(β) → 0 (2.31, 2.34), which leads to a simple expression for the initial datum

45)
13 Mathematically speaking, one should reserve the name 'adjoint action' to the adjoint action of K(E 10 ) on itself. Here, P belongs to a different representation space: the coset one E 10 ⊖ K(E 10 ). The action of Q on P should therefore be called the 'coset action'. Note, however, that this coset action is canonically deduced from the adjoint E 10 -action, given the decomposition (2.23).
14 Note that the adjoint representation is integrable and that hence any sl 2 -subalgebra generated by the Chevalley triple (e i , f i , h i ) for a simple root α = α i can be integrated to the corresponding group [21]. As far as the action on the Cartan subalgebra h is concerned, this statement extends to any real root of E 10 , which can also be considered as a simple root of a subalgebra.
where v a =β a (2.6) is the 'incoming' β-velocity. Hence, the initial coset 'velocity' (2.45) belongs to the Cartan subalgebra. This drastically simplifies the calculation. The easiest way to evaluate the action (2.43) is to work within the SL(2, R) subgroup generated by the algebra elements E α , E −α and [E α , E −α ] = h α ≡α ∈ h. As in equation (2.32), we decompose the Cartansubalgebra valued initial datum P(−∞) (2.45) into two parts P ⊥ and P || where α(P || ) = 0. The component P || is found to commute with E α and E −α , while the action of the SO(2) rotation R α,εα (2.44) on P ⊥ ∝ h α is obtained by ex- α . This is easily done by noting that the commutation relations of h α , E α and E −α are those of an sl 2subalgebra. In the canonical basis of this subalgebra, we can represent E α ≈ into e θJα = 1l 2 cos θ + J α sin θ, so that the adjoint action (2.43) acting on the perpendicular component of P, We see that, while for intermediate angles of rotation, P ⊥ is mapped out of the Cartan subalgebra h and acquires a component ∝ sin(2θ)(E α + E −α ), the full rotation by θ = ± π 2 associated to a wall α transforms h α into −h α (i.e. back into h).
In order to keep the notation simple, we have confined the analysis to the simply-laced case from the start by assuming (α|α) = 2. It is explicitly checked in Appendix B that, actually, the norm of the (real) root α drops out of the final result and always yields a geometrical reflection (2.47) for the β-velocity. The fact that a general Weyl reflection of a finite-dimensional group G can be expressed by a π 2 -adjoint rotation about the 'axis' J α in the compact subgroup K ⊂ G was recently noticed in [24], too. We also note that a different exponential representation of a Weyl reflection (in the general Kac-Moody case) is used in [21] (p. 36). Its relation to (2.44) will be discussed below.
In this section, we have shown explicitly that the Kac-Moody billiard associated to the BKL-limit of D = 11 supergravity can be understood as a sequence of 90 • -K(E 10 )-rotations about the different axes J αi that are defined by the dominant walls α i (2.10, 2.20). Note that the ten 'axes' J α are generators in K(E 10 ) which are orthogonal to each other with respect to the E 10 -metric (·|·). We will see in the next section that the same picture holds for the fermions. Namely, the same formal rotation R α,εα (2.44), now evaluated in the spinor representation of K(E 10 ) [5,6,7], acts on the fermions of supergravity, which can be considered as the induced effect of the "collision of the scale factors β a with a wall α". This will lead us to introducing some 'spin extension' of the Weyl group.
3 Fermionic billiard of D = 11 supergravity and its K(E 10 ) structure

Gravitino dynamics: gravity versus coset description
As a prelude to our study of fermionic billiards, let us recall the link between the standard supergravity description of fermions and the coset one, which reveals its hidden K(E 10 )-structure.
The gravity form of the fermionic equations of motion (neglecting non-linear fermionic terms) derived from the Cremmer-Julia-Scherk action [20] reads (in the conventions recalled in Appendix A, using Einstein's summation convention and flat indices A, . . . = 0, . . . , 10, e.g. In this form, the coupling to the 3-form A (or rather F = dA) appears explicitly on the RHS, while the coupling to gravity is contained in the spin connection ω A[BC] appearing on the LHS of (3.1): In the coset formulation, all the F -dependent terms on the RHS can be merged with the spin connection to form a K( where the zero level term Q (0) involves the part of the spin connection which contains time derivatives of the metric g mn (t, x) (2.1), while the level 1 term Q (1) involves time derivatives of the three-form A mnp , the level 2 term Q (2) space derivatives of A mnp and the level 3 term Q (3) space derivatives of the metric (in the form of the structure constants C a bc ). To reveal this structure, one needs to gauge-fix both the metric and the gravitino, and to work with some redefined fermionic variables: namely the following 'coset' combinations of the original D = 11 gravitino field ψ (11) and of the supersymmetry transformation parameter ǫ (11) ψ 0 := det(g mn ) (3.2a) (3.2b) One fixes the supersymmetry gauge freedom by setting Note that we are not exhibiting the 32-valued spinor index of neither the supersymmetry parameter ǫ, nor the vector-spinor gravitino field ψ a at this stage. As already mentioned above, the gravitino equations of motion in the gauge (3.3) and up to 'level 3' in the sense indicated above, were found to be expressible as a parallel transport of an element of a 'vector-spinor' representation of K(E 10 ) by the same connection Q(t) ∈ K(E 10 ) that entered the bosonic equation of motion (2.27): Let us again emphasize the analogies between fermions and bosons. The time evolution of the gravitino (3.4) is completely analogous to the bosonic one (2.25): at each moment, ψ transforms by an infinitesimal K(E 10 )-'rotation' with the same 'angular velocity' Q(t) ∈ K(E [10 ) as for the bosonic case, except that now, Q is acting in a different representation. The reader should keep in mind that the 'fermionic' representation of Q(t) will have, modulo the fact that it lives in a different space, the same time-structure that we studied in section 2.4, namely a sequence of well-separated inverse-cosh 'spikes'. 15 In the near-singularity limit, we need only to consider the 'spikes' associated to collisions on the ten dominant walls α i (2.10) for i = 0, . . . , 9. We need now to recall the definition of the vector-spinor representation vs of K(E 10 ). It lives in a 320 dimensional vector space and was defined in [5,6,7]. Here, it will be enough for our purpose to know the action of the ten generators J αi (2.20) corresponding to the simple roots of E 10 . They act on ψ in the vector-spinor representation vs as 320 × 320-matrices. The explicit form of these matrices follows from the results of [7] (see. Eq. (2.26) there) or [6] and read Here, the Γ a 's, a = 1, . . . , 10, are SO(10) real gamma matrices ({Γ a , Γ b } = δ ab ). As a next step, recall from section 2.4 that the effects of the walls on the dynamics are obtained by considering, separately and successively, the effect of each 'spike' in the connection Q. We then start by integrating Eq. (3.4) over one spike of the form The only difference with the previous case (2.35) is that J vs α now acts in a 320 dimensional space, according to (3.5). As in the bosonic case (2.41), one can integrate the sequence of infinitesimal K(E 10 )-rotations (3.4) with angular velocityθJ vs α into Finally, the total effect of the collision on a wall α will be an integrated K(E 10 ) rotation between the incoming gravitino ψ and the outgoing one ψ ′ , given by Therefore, if we consider, as in Eq. (2.17) above, the billiard dynamics towards a singularity, its effect on the gravitino variable ψ will be represented by a growing 'word' made of the product of the induced Weyl vector-spinor rotation (3.7) corresponding to the Weyl reflections in the bosonic Weyl word w (2.17), i.e.
At this stage, our task to describe the motion of the gravitino is reduced to exponentiating the operators (3.5) within the 320 dimensional representation space.
For a further investigation of this action, we now introduce the following (second) redefinition of fermionic variables The definition (3.9) would seem to drastically violate the SO(10) symmetry (the unbroken subgroup of the space-time Lorentz symmetry SO(10, 1)) which was present in the general action of the vector-spinor generators J vs α (see their general definition in [7]). However, we shall see that this loss of explicit symmetry is compensated by the appearance of simple structures linked to the SO(9, 1) Lorentzian metric G ab (the restriction of the invariant form (·|·) to the Cartan subalgebra of E 10 ) rather than to the SO(10) Euclidean metric δ ab built in the zeroth level of K(E 10 ).

Induced Weyl group action on the redefined gravitino
The usefulness of the redefinition (3.9) will appear in streamlining the structure of the wall-induced 'rotation' R vs α,εα (3.7) acting on the gravitino. Adapting the notation to the geometry as for theβ a reflection (2.32), we split the 'vector' index 16 of the gravitino ϕ a (3.9) into components parallel and orthogonal to the wall α: and 0 = α(ϕ || ) 16 In doing so, the hidden spinor index of ϕ a is left untouched.
where α(ϕ) is a short-hand notation for 10 a=1 α a ϕ a , modelled on the canonical pairing α(β) that appeared in the bosonic case (2.32). A quick calculation then reveals that the action of the generators J αi (3.5) drastically simplifies to where J s αi denote the Dirac spinor actions [19] (see also Eq. (2.23) in [7]) defined by It is convenient to summarize the structure of the Clifford algebra valued generators (3.12) by writing them simply as where one introduces the short-hand notation electric wall: Note that these ten Γ αi 's all satisfy the relation and therefore can be considered as imaginary units i (with i 2 = −1). A remarkable feature of the result (3.11, 3.12, 3.15) besides their simplicity (compared to the original form (3.5)), is the symmetry they reveal between a symmetry wall α i , i = 1, . . . , 9, and an electric one, α 0 .
The simplicity of the structure (3.11, 3.12) now allows us to compute the needed exponentiated action of ε α π 2 J vs α , namely Note that equations (3.11) mean that the parallel and the orthogonal parts ϕ a || and ϕ a ⊥ of the gravitino ϕ a are 'eigendirections' of the vector part of the vectorspinor generator J vs αi . More precisely, it is interesting to remark that, with the notation (3.13), the action of the vector-spinor generators reads thereby exhibiting the fact that the gravitino is a vector-spinor (spin 1 ⊗ spin 1 2 ) that algebraically contains both a spin 1 2 and a spin 3 2 part. This eigencharacter, together with the structure of the Dirac spinor action (3.13) implies that ϕ || will rotate, under (3.16), by exp(ε α π 4 Γ α ) (where α is any of the ten simple roots α i (3.14)) while ϕ ⊥ will rotate by exp(−ε α 3π 4 Γ α ). Using the fact that each Γ α (or ε α Γ α ) can be thought of as being an imaginary unit i (3.15), so that we can write exp(− 3π 4 i) = − exp( π 4 i), we find that the exponentiated action (3.16) boils down to we recognize in these formulae the general definition of a geometric reflection in the 10-dimensional β a -vector space (which was identified with the Cartan subalgebra h): the orthogonal component changes sign, whereas the parallel one remains unaltered. In other words, if we recombine the two 'eigenspaces', we find that the exponentiated vector-spinor rotation associated to any dominant wall α is simply given by is a 32 × 32-matrix representing the K(E 10 ) action on a Dirac spinor. To be clear, we have made explicit all the indices on the redefined gravitino ϕ in Eq.  (10)).
The crucial point to note in Eq. (3.17) is that the induced action on the fermionic variable ϕ (3.9) of what was a (Weyl) vector reflection on the bosonic velocity v ∈ h is the tensor product of a Dirac spinor action R s α,εα (3.18) and a vector action r α (2.47). In other words, the redefinition of the gravitino (3.9) revealed that the vector-spinor action R vs α,εα (3.5, 3.17) in fact factorizes into a Dirac spinor and a vector action. As is clear from the explicit expression (3.17), this factorization implies that, when considering no longer one isolated collision, but a succession of several of them, we can write, say for a consecutive collision with two walls β, α: Applying this factorization to the word representing the fermionic dynamics (3.8), we see that the fermionic word w vs (3.8) similarly factorizes in the tensor product of the bosonic Weyl word (2.17) with a corresponding word w s in the Dirac-spinor representation: In the next section, we will show that the redefinition of the gravitino (3.9) also simplifies the analysis of its degrees of freedom, before investigating the Dirac spinor action R s α,εα in more detail in section 3.5.

Supersymmetry constraint as a tranversality constraint on a 'polarized' billiard particle
A further instance of the simplifications brought by the redefinition of the gravitino (3.9) occurs when considering the "supersymmetry constraint" S ≈ 0 [7]. This is a constraint on the initial data for the gravitino, which originates from the equation of motion of the gravitino component ψ 0 . Its explicit expression has been written down in equation (5.14) in [7]. The first observation is that it considerably simplifies in the BKL-limit when considered away from any wall. This is due to the fact that the higher level contributions P a1...a6 , . . . as well as the off-diagonal components of P (0) ab become exponentially negligible far away from the wall (i.e. they are proportional to positive powers of e −α(β) that vanishes in the BKL limit 10 a=1 β a → +∞ (2.34)). Thus, after a rescaling, we are left with a simplified supersymmetry constraint S BKL ≈ 0, with (3.21) If we now replace the original (coset) fermionic variables ψ a in terms of the redefined gravitino ϕ a (3.9), we find that S BKL acquires a remarkably simple form: Note that, contrary to the original form (3.21), this form is now factorized in the sense that the hidden spinor index on ϕ b is left untouched. The constraint S BKL now affects only the vector index of ϕ a . More precisely, we see that the constraint S BKL ≈ 0 says that the contravariant billiard velocity v a =β a (2.6) is orthogonal (with respect to the β-space metric G (2.5)) to the 'contravariant vector' ϕ a (3.9). The latter could be thought of as a 'polarization direction' orthogonal to the lightlike velocity v a (2.7) of the 'billiard particle' (as if we were talking about a Maxwell-like vector of helicity ∝ ϕ a ).
Note the remarkable fact that the (K(E 10 )-related) Euclidean SO(10) structure of the original, exact supersymmetry constraint (exhibited in Eq. (5.14) in [7]) has given rise to a constraint which formally looks SO(9, 1) invariant, in the sense that it now involves the (E 10 related) Lorentzian structure G ab (2.5).
[One should however keep in mind that the hidden, factorized spinor index on ϕ maintains its SO(10)-related nature.] Furthermore, notice that the constraint S transforms as a Dirac spinor under a collision with a wall α: This follows from combining the transformation properties of v a (2.47) and of ϕ a (3.17) and it proves in particular that the constraint S BKL ≈ 0 is preserved under the evolution in the BKL-limit.
Note finally how the billiard limit has analagously simplified the Hamiltonian constraint (2.7) and the supersymmetry constraint (3.22). They now seem to unite in expressing the 'gauge invariance' (in β-space) of an effective Maxwell-like particle: the billiard limit of the Hamiltonian constraint expresses the massless nature of this β-space 'photon', while the billiard limit of the supersymmetry constraint expresses the transversality constraint η µν p µ A ν = 0 linked to the Maxwell gauge invariance. As both constraints originally followed from other gauge invariances (diffeomorphisms and local supersymmetry), this intriguing transformation of gauge constraints in the billiard limit may be the tip of the iceberg of an improved understanding of the symmetries of supergravity, and of their possible E 10 /K(E 10 ) coset reformulation.

Conserved bilinear form and 'gauge transformations'
It was found in [5,6,7] that there existed a symmetric 17 , bilinear form (·|·) vs on the 320-dimensional vector-spinor representation space which was K(E 10 )invariant, and therefore invariant under the exact, coset fermionic equations of motion (3.4). This bilinear form was, in particular, invariant under the SO(10) subgroup of K(E 10 ) corresponding to SO(10) rotations of the vector index a of ψ a . Indeed, its original expression in terms of two gravitini ψ 1 , ψ 2 was [7]: Let us see how this form is transformed when substituting ψ a by its expression in terms of the redefined gravitino ϕ a := Γ * Γ a ψ a (3.9). An easy calculation (using Γ ab = Γ a Γ b − δ ab ) yields Note again how the use of the ϕ a variables has revealed a hidden 'Lorentzian' structure in the form (·|·) vs . More precisely, the K(E 10 )-covariant bilinear form (3.24) of two gravitini ϕ a 1 and ϕ b 2 amounts to contracting their 32-valued Diracspinor indices in the standard Euclidean way, whereas their vector indices are contracted with the Lorentzian metric G ab that was defined on the space of scale factors β (2.5). Let us also consider the corresponding norm (when evaluated on real, commuting variables) On the full ϕ-space, this norm has a hyper-Lorentzian signature with 9 × 32 pluses and 32 minuses. However, we note that, when the supersymmetry constraint S BKL = 0 is satisfied, the norm Q vs (ϕ) becomes positive semi-definite. Indeed the supersymmetry constraint S is saying that (when forgetting about the spinor index) the vector ϕ a is G ab -orthogonal to the billiard velocityβ b (3.22). As, by the Hamiltonian constraint (2.7),β a is a G ab -lightlike vector, ϕ a must then be either G ab -spacelike or parallel toβ a , so that Q vs (ϕ) ≥ 0 (when evaluated on real, commuting variables).
In addition to this positivity property, the supersymmetry constraint S (3.22) also guarantees that the norm Q vs (3.26) is not affected by the following Maxwell-like (longitudinal) 'gauge transformation' of ϕ a along the lightlike directionβ a (2.7) with any Dirac spinor ξ ∈ 32: for ϕ ′a = ϕ a +β a ξ.
As another example of the intriguing appearance of a Maxwell-type structure in the fermionic billiard, it is interesting to observe that supersymmetry is a particular case of this gauge transformation (3.27). To show this, we insert the redefinition (3.9) into the supersymmetry variation δϕ a (Eq. (3.9) in [7]). Then in the BKL-limit, due to simplifications similar to those that occurred in the evaluation of the supersymmetry constraint S (3.21), the only surviving term on the RHS of Eq. (3.9) in [7] comes from the term ∝ 1 2 N Ω 0(ab) Γ b Γ 0 ε which yields (when keeping the only billiard-surviving contribution to Ω 0(ab) , which comes from the time derivative of the diagonal part of the metric) Note how this result, together with the r α transformation law ofβ a upon collision on a wall (2.14), implies that the supersymmetry variation term δ ǫ ϕ a (3.28) varies, under a collision, in exactly the same factorized way as the full ϕ a (3.17). This is another check of the consistency of our findings. Due to the importance of the rotations R s α,εα for the dynamics of all fermions, we will continue their analysis in the following section. To conclude, let us remark that Eq. (3.28) shows that the effective Maxwell-like gauge transformations of the 'polarization vector' added to the β-particle by the fermionic degrees of freedom comes from supersymmetry transformations. This shows again how the billiard limit exhibits some intriguing metamorphoses of gauge symmetries, as well as of group symmetries (SO(10) ↔ SO(9, 1)).

Spin extension W spin of the Weyl group W of E 10
Summarizing the review of section 2.1, the bosonic dynamics reduce in the BKL-limit to the evolution of the scale factors β a (2.4) that follow a billiard motion: Hitting the cushion of the billiard table corresponds to a collision with a dominant wall α i (2.10) which interrupts the intermediate Kasner behaviour (2.6) and whose effect consists of a Lorentzian reflection of the Kasner velocity v a =β a (2.14). The link to the Kac-Moody picture is established by identifying the dominant walls α i (2.10) with the simple roots of the Kac-Moody algebra E 10 and the space of scale factors β a with its Cartan subalgebra, spanned by the co-rootsα i = h i (2.22). Any Lorentzian reflection due to a collision with a dominant wall α i then corresponds to a fundamental Weyl reflection r αi ∈ W [21]. Hence, any sequence of collisions during the chaotic evolution of the bosonic billiard corresponds to a word w in the Weyl group W of E 10 in Eq. (2.17), and the full evolution towards a spacelike singularity corresponds to a semi-infinite Weyl word: · · · w n · · · w 2 w 1 .
We have shown in section 3.1 that the gravitino can be understood as a (spinor valued) 'polarization vector' ϕ a associated to the scale factors β a . In particular, any collision induces a discrete K(E 10 )-rotation R vs α,εα (3.7) in the (unfaithful) vector-spinor representation. When acting on this 'polarization vector' ϕ a , the discrete K(E 10 )-rotation R vs α,εα factorizes as the tensor product of an ordinary Weyl reflection r α (acting on the vector index of the suitably redefined gravitino variable ϕ a (3.9)), and of a Dirac spinor rotation R s α,εα (3.17, 3.18) (acting on the hidden spinor index of ϕ a ).
These results suggest that to describe the general behaviour of fermions in the billiard limit, one should consider a certain abstract 'spin' extension of the Weyl group of E 10 (and more generally of any Kac-Moody algebra), say W spin . This group can be abstractly 'defined' as the 'discrete' subgroup of (a suitable covering of) the 'group' K(E 10 ), which is (multiplicatively) generated by the formal exponentials where i labels the simple roots (i = 1, . . . , 10 in the E 10 -case) and, where J α ≡ E α +ω(E α ) = E α −E −α (2.20) is an element of the K(E 10 ) Lie algebra. We have put some quotation marks in this definition, because the precise mathematical setting within which one could define, in full generality, the 'group' K(E 10 ), its needed 'spin' covering (analogous to the Spin(n) covering of SO(n)), and the formal exponentials (3.31), is unclear to us. 18 On the other hand, our study above has given us two concrete realizations of the abstract generators (3.31), namely R vs α,εα (3.16) and R s α,εα (3.18). We shall then focus in the following on the two corresponding matrix groups, say W vs and W s . We define W vs as the subgroup of GL(320) multiplicatively generated by (for i = 1, . . . , 10) Here, we have dispensed with the signs ε α = ± that entered our study above for two reasons: 1. We shall see that the matrix generators R vs αi and R s αi are idempotent (so that a suitable power of, say, R vs αi equals (R vs αi ) −1 = e − π 2 J vs α i ).
2. The sign of E αi and therefore of J αi = E αi − E −αi (2.20), is conventional and can therefore be changed to absorb any sequence of ε αi , without loss of generality. 20 With this notation, we have proven some mathematical results on the concrete, matrix groups W vs and W s that we can summarize as follows, before indicating the main elements entering the proofs (the mathematical details of which are given in Appendix C).

Proposition 1
• The vector-spinor realization W vs of W spin , i.e. the group multiplicatively generated by the ten 320 × 320 matrices R vs αi (3.32), is infinite. to each SO(2) group generated by Jα i for any simple root α i . This interpretation will be supported by the observation in Eq. (3.34a) below that a rotation by 2π, i.e. R 4 α i = e 2πJα i (3.31), indeed corresponds to a change of sign acting in a spinorial representation of K(E 10 ). 19 We recall that the J vs α i preserve the Euclidean norm Q s (ǫ) = ǫ T ǫ [7]. 20 Note in this respect that εα i is the sign of a conserved quantity in the BKL-limit. Hence, the sign εα i for any dominant wall α i cannot flip during the asymptotic Billiard dynamics.
• The squares of the matrices (R vs αi ) 2 generate a normal subgroup D vs of W vs , which is non-abelian and whose cardinality is 2048.
• The Weyl group W of E 10 is isomorphic to the quotient group W vs /D vs : The corresponding statement concerning the Dirac-spinor realization of W spin looks similar, but differs in a very important aspect: W s is a finite group, while W vs was infinite (like the Weyl group of E 10 ): It is interesting to contrast these results on what we called representations of the 'spin' extension W spin of the Weyl group, to the results of Kac [21] (p.36) on some representations of the usual Weyl group W of a Kac-Moody algebra, namely:

Proposition 3 [Kac]
Let ρ be an integrable 21  These ten elements r ρ αi generate a group W ρ ⊂ End(V ). Furthermore, their squares form an abelian normal subgroup D ρ of W ρ D ρ := (r ρ αi ) 2 |i = 0, . . . , 9 (3.38) such that the quotient group is isomorphic to the Weyl group W of E 10 Referring to Appendix C for details of the proofs of proposition 1 and 2, let us comment on some of their salient aspects: • The fact that for W vs and W s , we get the specific generalized Coxeter relations of propositions 1 and 2 (i.e. (R vs α ) 4 = −1 and therefore (R vs α ) 8 = 1 instead of r 2 α = 1 for the usual Weyl group) comes from the spinorial nature of the representations vs and s. Technically, this is due to the factor 1 2 in J s α = 1 2 Γ α which leads to factors e ± iπ 4 in R vs α and R s α . By contrast, the action of R α = e π 2 Jα on a Cartan valued P in the coset representation led (as in the setting of Kac's proposition 3) to phase factors e ±iπ in R coset α (2.39, 2.46). More generally, note that an eigenvalue ±is of J α (that we shall refer to as being 'spin s') entails a phase factor e ± iπ 2 s . In this language, the phase factors e ±iπ in R coset α can be thought of as coming from the fact that P (0) ab was a symmetric SO(10)-tensor, hence 'spin 2' versus 'spin 1 2 ' of the basic Dirac-spinor representation J s [Note in this respect that the type of generalized Coxeter relations, and associated extensions of the Weyl group that have been studied in [21,25] would, in the present language, be connected to integer 'spin s'. The integral nature of s guarantees for these cases that the Weyl group is extended in such a way that the generators e π 2 Jα i (i = 0, . . . , 9) are represented by fourth roots of unity.] • The finiteness of W s was far from being a priori evident. This finite cardinality does not follow from the unfaithfulness of the spinor representation because the vector spinor representation vs is also unfaithful (it leads to a finite-dimensional representation space), but W vs is infinite. This finite cardinality does not either follow alone from the fact that W s preserves a positive definite norm, namely Q s (ǫ) = ǫ T ǫ, because the action of the infinite Weyl group on the compact unit spere ǫ T ǫ = 1 in R 32 could have led to a 'chaotic' non-periodic action with many accumulation points.
• The technical difficulty in proving the finiteness of W s came from the fact that (using (Γ α ) 2 = −1, like i) contains √ 2 in the denominator. Therefore, a product of n exponentials or, as we shall say, a word of length n, R s α1 R s α2 · · · R s αn a priori contains a factor (1/ √ 2) n which could decrease without bound as n → ∞ and generate an infinite number of group elements in W s . Appendix C explains in detail how we surmounted this difficulty. Let us here only say that we used a four-pronged approach: (1) We crucially use the fact that we are working within an algebra, which follows from the representation in Eq. (3.40). The anticommutation properties of Clifford matrices Γ a then imply some definite (anti-) commutation relations among the R s α which involve coefficients of the form a + b √ 2 with a, b ∈ Z. At this first step, the use of the algebraic relations among elementary products R s α1 R s α2 · · · has allowed us to push the dangerous √ 2's up to the numerator.
(2) Then using repeatedly, as in the proof of Wick's theorem, the (anti-) commutation relations among the R s α , we can reorder any element of W s and express it as a finite linear combination of ordered R s αproducts with Z[ √ 2]-valued coeffcients. [Here we use the fact that the Galois extension Z[ It might be interesting to add that our proof of proposition 2 of the generalized Coxeter properties and of the finiteness of the spinor representation W s of the abstract W spin on the one hand can be straightforwardly generalized to other Kac-Moody algebras, and, on the other hand, can be reformulated within a more abstract setting which does not make use of the gamma-matrix realization (3.13, 3.14) of the 'compact' generators J s αi . Indeed, it seems that the main structure behind the proof is the following abstract, Kac-Moody related 'graded Clifford algebra'. First, independently of any explicit representation, we can always define some abstract generators Γ αi (associated to each node of a Dynkin diagram) as Γ αi := 2J s αi , i.e. such that Eq. (3.13) holds by definition. An abstract 'Dirac-spinor' realization can then be defined by requiring that: (1) all the 'spin eigenvalues' of the basic ('Hermitian') rotation generators iJ s αi are equal to j s αi = ± 1 2 . This implies that (Γ αi ) 2 = −1l; (2) the Γ αi 's corresponding to different nodes of the Dynkin diagram satisfy either anticommutation or commutation relations [Γ αi , Γ αi ] ± = −δ ij . E.g. for the E 10 case, we have anticommutation for adjoint nodes (when A ij = −1), and commutation for disconnected nodes (A ij = 0).
The same relation between this 'graded' Clifford structure 22 (anticommutation versus commutation) and the Dynkin diagram (A ij = −1 or A ij = 0) holds (in the generic case D > 4) for the AE D−1 Kac-Moody algebra associated to pure gravity in space-time dimension D [12]. [The case D = 4, which will be treated below, is non-generic in that the special 'gravity-wall' generator Γ α * commutes with the symmetry-wall generators.] Our proof applies in such a general setting (provided the strictly ordered products of Γ αi 's define a linearly independent basis), even if the representation space were infinite (though necessarily made of finite, reducible components).
Concerning the links between the content of the propositions 1 and 2, and Kac's proposition 3, we think that the definition of r ρ αi (3.37), though it looks very different from that of R − αi = e − π 2 Jα i (3.31), will be formally equivalent to it in any E 10 -representation ρ in which the exponential e − π 2 J ρ α i is well defined. 23 Indeed, a reshuffling (à la Baker-Campbell-Hausdorff), needed to pass from one form to the other, only involves commutators of the sl 2 subalgebra generated by  [25]. However, Ref. [25] introduces an extension of the Weyl group by (Z 2 ) r (where r is the rank), which means that the generalized Coxeter generators are fourth roots of unity (contrary to our 'spinorial' extension that involves eighth roots of unity).
It is clear that there are similarities between the statements made in the propositions 1-3. In particular, the group W ρ of proposition 3 also is an extension of the Weyl group. We wish however to emphasize that our results in the propositions 1 and 2 can in no way be deduced from proposition 3 for several deep reasons. First, our propositions concern K(E 10 ) representations and not E 10 representations. Although any E 10 representation also is a K(E 10 ) representation by restricting the action of E 10 to its subalgebra K(E 10 ), the reverse is clearly false in the case we consider in our propositions, because both the vector-spinor representation vs (3.5) and the spinor representation s (3.12) of K(E 10 ) being unfaithful (and finite-dimensional) can certainly not be lifted to representations of E 10 [indeed, E 10 , being simple, does not admit unfaithful representations (apart from the trivial one)]. Second, the structure of the normal subgroups that are factored out is different: while in our propositions the normal subgroups D vs , D s are non-abelian, D ρ is abelian in the case of proposition 3. 22 It seems that this structure, i.e. the properties (1),(2) above, follow from the requirement that the eigenvalues of a general K(E 10 )-normalized ( P i n 2 α i = 1 in the simply-laced case) linear combination P i nα i J α i has eigenvalues ± i 2 . 23 Any E 10 representation ρ, defined by the representation of the Chevalley generators (e ρ i , f ρ i , h ρ i ), canonically induces a representation of its subalgebra K(E 10 ), generated by Note, however, that Kac has used the setting of his proposition 3 to prove a result which is closely related to the one we found above for the bosonic billiard. Indeed, by applying the proposition to the adjoint representation ρ = ad, he showed that the adjoint action of r ad αi (3.37) to the Cartan subalgebra h ⊂ E 10 reproduces the action of the fundamental Weyl reflection r αi [21]. Given, for E 10 representations ρ, the link r ρ α ∼ e ± π 2 J ρ α mentioned above, this agrees with our observation in section 2.4 in which we evaluated the K(E 10 )-rotation R Ad αi,εα i (2.44) on the element of the Cartan subalgebra that corresponded to an incoming Kasner geodesic (2.45, 2.47).
In the next section, we will investigate the truncation of SUGRA 11 to D = 4 N = 1 supergravity that can be linked to the Kac-Moody algebra AE 3 in the BKL-limit as mentioned in section 2.2. The analysis of this simpler case will provide a useful 'toy setting' in which some aspects of our E 10 results become easier to analyze in detail. In particular, we shall be able to solve explicitly the supersymmetry constraint S (3.22) and to explicitly exhibit the remaining degrees of freedom of the gravitino in terms of two Dirac spinors. In order to facilitate the link to the E 10 -description of maximal supergravity, we want to describe D = 4 N = 1 supergravity as a truncation of the former theory reduced on a seven-torus T 7 . The analysis of section 2.2 reviewed the known fact that the hyperbolic Kac-Moody algebra AE 3 naturally appears in a description of the BKL-limit of d = 4 gravity [1,14]. We start this section with a review of the bosonic dynamics. Since the gravitational walls are linked to the structure functions C a bc introduced in section 2.2, it is sufficient to focus on their dominant contribution to the supergravity Lagrangian in the BKL-limit, which is where the gravitational wall α g abc was defined in Eq. (2.18) and where we used as usual the gauge choice N = det(g mn ) (2.2) [14] with m, n = 1, 2, 3. It was shown in [4,14,11] that the (Iwasawa frame) structure functions C a bc have limits on the singularity, and that they can be identified with the conserved conjugate momenta Π α (2.31) of coset variables ν α parametrizing a level-1 coset variable Φ ab = Φ ba . Here, we are interested in extracting from all the gravitational walls the dominant one near the singularity. In view of the ordering among the β's imposed by the symmetry walls (α s (12) (β) > 0 and α s (23) (β) > 0 implying β 1 < β 2 < β 3 ), it is easily seen that the dominant gravitational wall will be α g 123 , which is connected to C 1 23 . Then, either by using the cosetgravity dictionary given in [4,14], or by identifying (in the gauge n = 1) the one wall potential e −2α * (β) Π 2 α * of Eq. (2.33) to the dominant gravitational potential 1 4 e −2α g 123 (β) (C 1 23 ) 2 (4.1), we find that the dominant gravitational wall α * ≡ α g 123 contributes the following term to the coset connection Q| α (2.36): This equation will be crucial to obtain the correct normalization of the K(AE 3 )generator J α * in the next section.

K(AE 3 ) and the fermions of D = 4 N = 1 supergravity
The dynamics of the gravitino ψ of D = 4 N = 1 supergravity can e.g. be taken from [26] (neglecting third order terms in fermions and with ∂ t = N ∂ 0 ) with the standard Levi-Civita connection ∇, the Clifford algebra {γ α , γ β } = η αβ and η = diag(− + ++). We use the same redefinitions (3.2) as for the E 10 -case (i.e. the same rescaling, but with indices restricted to a = 1, 2, 3) and adopt the supersymmetry gauge ψ 0 = 0 (3.3). This leads to the following explicit form for the time evolution of the (coset) fermionic variable ψ a (using Einstein's summation convention) Our task now is to single out on the RHS of this equation the dominant terms in the BKL-limit: i.e. the ones corresponding to the two symmetry walls α s (12) , α s (23) and the gravitational wall α * = α g 123 . First, we notice that the temporal derivatives of the metric coefficients completely follow the scheme used in the E 10 -case (3.4, 3.5): They give rise to the symmetry walls (2.9). One then easily finds that they lead to the same results (restricted to indices a = 1, 2, 3) for the corresponding 'compact' generators J α s (12) , J α s (23) . Denoting for simplicity α 1 := α s (12) and α 2 := α s (23) as in (2.9), one finds the following explicit expressions for the two putative K(AE 3 ) generators J α1 and J α2 corresponding to these symmetry walls: If we now extract from the RHS of Eq. (4.4) the contribution proportional to the dominant gravitational wall α * := α g 123 , we must be careful to collect several different connection terms. Indeed, C 1 23 is proportional to Ω 231 in the notation of [7]. Then the formula giving the 'spin connection' ω abc in terms of Ω abc (see Eq. (A.5)) shows that C 1 23 ∝ Ω 231 will enter in ω 123 , ω 231 and ω 312 . Finally, we get that the dominant gravitational wall contribution to the equation of motion reads after substituting (2.2, 2.3): By rewriting this equation in the universal coset-model form (3.4) we can fix (by comparison with our previous result (4.2), and remembering that the dominant gravitational wall α * (β) = 2β 1 (2.18) does indeed give a factor e −α * (β) = e −2β 1 ) the form of the putative K(AE 3 ) vector-spinor representation vs of the generator J α * in terms of γ-matrices: At this stage, we extracted from the SUGRA 4 fermionic equations of motion explicit γ-matrix expressions for the generators J vs α1 , J vs α2 and J vs α * that should give rise to a 'vector-spinor' representation of K(AE 3 ). The proof that these generators indeed form a representation of the compact subalgebra K(AE 3 ) of AE 3 crucially depends on the correct normalization of the 'gravitational wall' generator J vs α * (which is fixed by the precise coefficient in Eq. (4.2)). We start by recalling that K(AE 3 ) is defined by the free Lie algebra generated by the compact generators J αi (2.20) for i = * , 1, 2 modulo Serre-like relations [7] that are inherited from the Serre relations of AE 3 . In particular, the condition ad 3 Eα * (E α1 ) = 0 that is encoded in the Dynkin diagram (figure 2) gives rise to the following relation for J αi involving the exceptional node: Jα * (J α1 ) + 4ad Jα * (J α1 ) = 0. (4.8) As in the K(E 10 )-case treated in [7], the manifest SO(d) invariance of the gauge-fixed gravitino equations of motion (which follow from their original, local Lorentz covariance) ensures that the condition (4.8) is the only Serre-like relation that needs to be checked. By doing the explicit γ-algebra calculation involved in Eq. (4.8) when using the expressions (4.5, 4.7) for J vs α in terms of γ-matrices, we have verified that Eq. (4.8) is fulfilled and hence, that these generators J vs α ∈ R 12×12 indeed form a matrix representation of K(AE 3 ).
This result was expected because the fact that the equations of motion of SUGRA 4 admits a K(AE 3 )-covariant coset reformulation (in the BKL-like approximation where one keeps the leading effect of all the gravitational walls α g abc with a, b, c all different), actually follows from three known results: (1) The fermionic dynamics of SUGRA 11 admits a K(E 10 )-covariant reformulation up to a level of approximation ("l = 3 − ") which includes all the gravitational walls associated to real roots [4].
(2) AE 3 is a subalgebra of E 10 obtained by keeping only a fraction of the generators of E 10 [21,27]. This truncation, being compatible with the Chevalley involution, also implies that K(AE 3 ) is a subalgebra of K(E 10 ).
(3) The dynamics of SUGRA 4 can be obtained by reducing the dynamics of SUGRA 11 (truncating away, in particular, both the spatial dependence upon a seven-torus T 7 and the corresponding gravitino components).

Factorized structure of the K(AE 3 ) fermionic billiard
Let us now describe the SUGRA 4 fermionic dynamics in the billiard limit. The treatment is parallel to the SUGRA 11 case discussed in section 3.1 and consists in combining the effects on ψ coming from successive chaotic collisions on the three dominant walls α 1 , α 2 and α * of AE 3 . As for maximal supergravity in Eq. (3.7), we focus on a one-wall system and integrate the equation of motion (4.6) as before. The computations from section 2.4 still hold for the AE 3 -case, with (4.2) replacing (2.36) for the dominant gravitational wall α * . Hence, the net effect of a collision with a wall α is a K(AE 3 )-rotation about the axis J α with an angle ε α π 2 . This rotation turns the incoming gravitino ψ into an outgoing one ψ ′ : with R vs α,εα = e εα π 2 J vs α being a 12 × 12 matrix acting in the vector-spinor representation space of ψ aA (where a = 1, 2, 3, while A = 1, 2, 3, 4 is a real Majorana spinor index). Similarly to the SUGRA 11 treatment above, we found that the transformation law of the gravitino simplifies very much, if we replace ψ a by the following redefined fermionic variable: ϕ a := γ 0 γ a ψ a (no sum). (4.12) With this redefinition, we find again, in complete analogy to (3.17), that the transformation (4.11) in the vector-spinor representation factorizes as the tensor product of a Weyl reflection r α (now acting in a 3-dimensional vector space a = 1, 2, 3) with a Dirac-spinor rotation e εα π 2 J vs α (now acting in the 4-dimensional Majorana spinor space). In formulae, we have: and R s α,εα with J s α being Dirac-spinor generators acting in the 4-dimensional Majorana spinor space. For symmetry wall rotations associated to α 1 and α 2 , this result follows from a straightforward truncation of the K(E 10 )-results, with the Diracspinor representation (4.14) By contrast, the gravitational wall case needs a special treatment that we now sketch. We start with the observation that the redefinition (4.12) simplifies the Lie algebra transformation (4.7) (with J vs α * (ϕ) a = γ 0 γ a J vs α * (ψ) a ) to The orthogonal and parallel components ϕ a ⊥ and ϕ a || (3.10) of the gravitino ϕ a turn out again to be 'eigendirections' and hence, the J vs α * -action simplifies in the same way as in (3.11): where J s α * is the Dirac-spinor action now defined as In order to evaluate the group rotation R vs α,εα (4.11) on the eigenspaces spanned by ϕ ⊥ and ϕ || (4.16), we have to evaluate the exponential series defined in (4.13) for the gravitational wall α * : (4.18) Since the difference of the action on the eigenspaces (4.16) again is only a sign due to e − 3iπ 4 = −e iπ 4 , we finally obtain after recombining ϕ = ϕ ⊥ + ϕ || the factorized transformation of the gravitino that we quoted above (and that is completely similar to (3.17)). Let us also note (for future use) that the explicit form of the Weyl reflection r α * at the gravitational wall α * = α g 123 = 2β 1 reads: We will continue the investigation of this gravitino billiard in the next section, taking into account the supersymmetry constraint S.

'Longitudinal' and 'transversal polarizations' of the 'β-photon'
Concerning the supersymmetry constraint S, the AE 3 model, describing the BKL-limit of N = 1 d = 4 supergravity, exhibits a qualitative difference with respect to the E 10 -case of maximal supergravity: the constraint S ≈ 0 can be solved explicitly, which entails that the gravitino or 'polarization vector' ϕ a can be split into a 'longitudinal' and a 'transversal' part, both being Dirac-spinor valued. 29 As a first step, recall that the supersymmetry constraint in the billiard limit (far away from any wall) only depended on the velocitiesβ a , e.g. 29 Note however that these components do not transform as Dirac spinors under K(AE 3 ) as we shall see in Eq. (4.31) below.
As before, the formulation in terms of the redefined gravitino ϕ a (4.12) allows one to write S as an orthogonality constraint with respect to the Lorentzian β-space metric G r (2.5): From the point of view of the 3-dimensional Lorentzian geometry of β-space (forgetting about the hidden spinor index), the supersymmetry constraint S ≈ 0 (4.22) is a scalar constraint on the 3-dimensional vector ϕ a . It therefore restricts the gravitino ϕ a to live in a 2-dimensional hyperplane (actually, the 'null' hyperplane tangent alongβ to the β-light-cone). One can then explicitly parametrize ϕ a in terms of two (spinor-valued) scalars, say η and Φ, by using the existence of the cross product of vectors in three dimensions: Here, σ is the quantity (4.21), ε 123 = +1, and u a denotes the timelike vector in β-space which has unit (contravariant) components: u = (1, 1, 1). The vector u a plays a somewhat special role in the AE 3 billiard in that it naturally relates the Lorentzian metric G r (2.5) to the Euclidean metric, say v · w := 3 a=1 v a w a . Indeed, we have the identity: Keeping in mind thatβ a is lightlike (2.7), the parametrization of ϕ a in terms of Φ and η is easily seen to span the general solution of the supersymmetry constraint S BKL ≈ 0 (4.22). It is also easy to invert the parametrization (4.23) (constrained by (4.22)) and to express the two 'scalars' η and Φ in terms of ϕ a (using σ = u ·β (4.21) and σ 2 =β ·β (2.7)): Pursuing the analogy of considering the gravitino ϕ a as a polarization vector, η and Φ can be thought of as its longitudinal and transversal part, respectively. This point of view is reinforced by the fact that the bilinear form (·|·) (3.25) for any two 'polarizations' ϕ 1 , ϕ 2 , having the same null velocityβ a , only depends on their transversal parts Φ 1 , Φ 2 (4.23): involves a trace over the hidden spinor indices. Note also that the 'gauge transformation' of the gravitino ϕ a along the lightlike directionβ a preserves this bilinear form (3.27) and correspondingly only affects the longitudinal part η. At this point, we want to emphasize again that the decomposition of ϕ a into two scalar-like components (4.23) is a peculiarity of the three dimensions of the β-space of SUGRA 4 (or AE 3 ). In higher dimensions (i.e. higher rank ), one will still haveβ and u as special directions in β-space, but they will not provide enough structure for selecting special directions in the null hyperplane orthogonal toβ a (besides the 'longitudinal' directionβ a itself). Note also that instead of parametrizing ϕ a (in the AE 3 case) by two scalars, one could have also parametrized it by a lightlike vector λ a ( 3 a,b=1 G r ab λ a λ b = 0), with λ a =β a by writing ϕ =β × λ, i.e.
. We shall come back to this possibility below.
As a next step, we will investigate the transformation of the parameters η and Φ under a collision with a dominant wall. At first, recall that the dynamics of the velocitiesβ is described by the bosonic billiard, i.e. for any dominant wall α with (α|α) = 2, we obtain the Weyl reflection (2.14) For the dominant symmetry walls α 1 and α 2 (2.10), the reflection r αi (4.27) reduces to a mere permutation: On the other hand, a collision with the gravitational wall α * induces the Weyl reflection r α * (4.19)β (4.30) It implies σ ′ = σ + 2β 1 (4.21) and it allows one to deduce the induced transformations for Φ and η using the Hamiltonian constraintβ ·β = σ 2 (2.7): The gauge direction or longitudinal polarization η is changed in an inhomogeneous way, in contradistinction to the transversal polarization Φ. This implies that neither η nor Φ transform in the Dirac spinor representation of K(AE 3 ), the former due to the inhomogeneity and the latter due to the sign in (4.29b, 4.31b). 30 The homogeneous transformation behaviour of the transversal polarization Φ was expected due to the preservation of the Euclidean norm (ϕ|ϕ) = 2Φ T Φ (4.26). By contrast, as this norm does not contain η it does not provide any means to control the evolution of the longitudinal polarization η and is thereby compatible with the inhomogeneous character of (4.31a).

Fermionic billiard dynamics and BKL-eras
Let us now briefly discuss the qualitative evolution of η and Φ near a singularity. We recall that, qualitatively speaking, the 3-dimensional BKL chaotic behaviour can be thought of as made of eras that collect together (possibly long) sequences of Kasner 'epochs' (where an epoch is an intervall between two successive wall collisions) [1]. To explain the meaning of these eras, we start by recalling the 'u-parametrization' of the 'Kasner circle' [1]. First, it is convenient to work with the Kasner exponents p a which are projective coordinates of the (homogeneous) Lorentzian β-space velocitiesβ a , namely By construction, these lie in the plane Furthermore, the Hamiltonian constraint (2.7) also restricts them to a sphere The intersection is a circle that is parametrized by u ∈ R ∪ {∞} following [1]: The transformation of the projective velocities p a (4.32) under a collision with a dominant wall directly follows from the action on the velocitiesβ a (4.28, 4.30). These induce the following transformations on the parameter u: 31 Next, we recall that the Weyl reflections r αi for i = * , 1, 2 fulfill the Coxeter relations that follow from the Dynkin diagram of AE 3 (figure 2). Apart from 30 Φ could, however, be described as a pseudo-Dirac spinor, in analogy to a pseudo-scalar in particle physics. 31 As an example, the symmetry wall α 1 interchanges p 1 and p 2 and leaves p 3 inert (4.28). This implies rα 1 (p 1 )(u) = p 2 (u) = p 1 (rα 1 (u)), rα 1 (p 2 )(u) = p 1 (u) = p 2 (rα 1 (u)) and rα 1 (p 3 )(u) = p 3 (u) = p 3 (rα 1 (u)).
The exponents k = 2, 3, ∞ correspond to the angles π k between the dominant walls that confine the billiard motion of the scale factors β a to a polyhedron, which intersects the unit hyperboloid G ab β a β b = −1 into a hyperbolic triangle of finite volume. The special geodesics with velocity u = +∞ correspond to the ones that end at the fixed point (p 1 , p 2 , p 3 ) = (1, 0, 0) (which corresponds to the flat Milne space-time), and hence do not exhibit a chaotic behaviour. Chaos is restored for generic values of u ∈ R, however. In particular note that this fixed point is unstable, because any finite, but large value u decreases in a sequence of consecutive collisions with the walls α * and α 1 (4.35) due to: This sequence is called a BKL-era of length n = [u] ∈ N [1]. 32 For long eras n >> 1, we can neglect contributions of order u −2 , which simplifies the parametrization of the projective velocities p a in terms of u (4.34) to Substituting this in the transformation for η (4.31a) yields for η u−1 = r α1 r α * (η) u : We iterate this transformation n = [u] times and obtain for large n an approximate asymptotic relation linking the values at the beginning η u to the ones at the end η u−n of the BKL-era: As for Φ, its evolution during the considered era is simply given by Qualitatively, we see that while the 'spinor indices' of both η and Φ continuously 'rotate' under the spinor actions R s α,εα 's, their 'scalar magnitudes' evolve in very different ways: the Φ one remains constant, while the η one is made of two parts: One part, 1 n 2 η u , decreases during an era, while the other part, 2nΦ, linearly increases during an era. When considering successive eras, this leads us to expect that the (gauge-like) 'longitudinal' part η will perform very large 'excursions', whose 'directions' (in spinor space) are restricted to the orbit of the Dirac spinor action R s α,εα on Φ. In a 'baby problem', where we would replace the 'spinor direction' by a direction in a complex plane, we conceive of η as performing a kind of Brownian motion in the complex plane under a randomly oriented force (∝ Φ) that grows linearly in time. In the next section, we shall study in detail the evolution of the Φ degrees of freedom, i.e. the rotations in spinor space generated by the product of many K(AE 3 ) spinor rotations R s α,εα .
But before doing so let us indicate that if one had used the parametrization of ϕ a by means of two lightlike vectors (β a , λ b ), the action of the fermionic billiard on the pair (β a , λ b ) (where only λ b has an extra spinor index) would be a "double BKL billiard", namelẏ with the same vector reflection r α (2.14) acting on the vector index of both lightlike vectors. We already know that the simple billiardβ ′ = r α (β) has interesting chaotic properties; we shall see in the next section that the R s α,εα -billiard is not chaotic, but rather periodic (finite multiplicative group). This means that the 'chaotic' part of the 'doubled billiard' (4.39) will essentially come from a double bosonic BKL billiard:β ′ = r α (β), λ ′ = r α (λ). We leave to further work the study of these dynamics.

Spinorial extension of the Weyl group W AE 3
An important link between the cosmological billiard dynamics and hyperbolic Kac-Moody algebras g consists of the fact that any sequence of billiard collisions corresponds to a word in the associated Weyl group W g . We have transferred this idea to the fermionic dynamics in section 3.5 by defining the spin extension W spin g of the Weyl group W g . For g = E 10 , we have found two concrete realizations W vs g and W s g in terms of matrix groups (generated by the matrices R vs αi (3.32) and R s αi,εα i (3.33) respectively) that were characterized by the propositions 1 and 2. For g = AE 3 , we can perform a similar analysis. Since AE 3 is a subalgebra of E 10 , the structure of the matrix realizations W vs AE3 and W s AE3 of the spin extension of the Weyl group shows great similarity to the E 10 -case, as exemplified by the following statements:
• The generators R vs αi of W vs AE3 satisfy the following generalized Coxeter relations: and R vs α * commutes with R vs α2 .
• The squares of the matrices (R vs αi,+ ) 2 generate a normal subgroup D vs

AE3
of W vs AE3 , which is non-abelian and whose cardinality is 2 × 8.
The discussion of the Dirac spinor action for the AE 3 -case can be made more precise than the corresponding one for E 10 in proposition 2: The proof of theorems 3 and 4 closely follows the one for the E 10 -case provided in Appendix C: the factorizability of the gravitino action R vs αi,εα i into a Weyl reflection r αi and a Dirac spinor action R s αi,εα i allows one to consider proposition 4 as a corollary of proposition 5 as before. The latter discusses the property of the group W s AE3 , whose generators can be written in the form (4.14, 4.17): In this form, it is clear that R s α * ,εα * commutes with both R s α1,εα 1 and R s α2,εα 2 and hence can be discussed separately from the other two. It is interesting to phrase this fact in terms of Dynkin diagrams: for AE 3 , it is straightfoward to check that all elements in the quotient group W s AE3 /D s AE3 fulfill the Coxeter relations corresponding to the disconnected diagram This is, however, not the Dynkin diagram of AE 3 ( figure 2), but the one of the finite dimensional product group SL(2) × SL (3). Its associated Weyl group is Z 2 × S 3 being isomorphic to the group of equivalence classes W s AE3 /D s AE3 . Note that this statement is in sharp contrast to the E 10 -case of section 3.5, for which there is no additional Coxeter relation for the quotient group. The associated Coxeter diagrams for E 10 and for the quotient group W s E10 /D s E10 are the same but nevertheless, the latter was of finite cardinality. We have checked that the 'disconnection' (figure 3) that occurs for AE 3 is non-generic in that the Diracspinor representations of the higher rank cases AE d lead to generalized Coxeter relations rooted in the full corresponding connected Dynkin diagram.
The statement that the kernel of the homomorphism (4.42) is a congruence subgroup (4.43) is related to the observation that the Weyl group of AE 3 is isomorphic to PGL(2, Z) [29]. We can make this precise within the context of the cosmological billiard dynamics. Using the parameter u (4.34) for their description, the transformations r αi of u (4.35) provide a natural link between the generators of the Weyl group r αi (for i = * , 1, 2) and 2 × 2 matrices A i := ai bi ci di with integer coefficients a i , b i , c i , d i and determinant ±1 tantamount to The three matrices A * , A 1 , A 2 generate the group PGL(2, Z) [29,13]. 33 As a next step, consider the two group homomorphisms 34 det : PGL(2, Z) → Z 2 and C : PGL(2, Z) → PGL(2, Z 2 ) = SL(2, Z 2 ).
The kernel of det is the modular group PSL(2, Z) that hence is a normal subgroup of PGL(2, Z). The same is true for the kernel of C. Furthermore, the intersection of two normal subgroups again is a normal subgroup and it is nontrivial in this case: On the one hand, we have r α * ∈ Ker(C) and r α * / ∈ Ker(det), whereas on the other hand r α * r α1 / ∈ Ker(C) and r α * r α1 ∈ Ker(det) (4.35). Therefore, there is a homomorphism with kernel Ker(C') = Ker(C) ∩ Ker(det) of the form C' : PGL(2, Z) → Z 2 × SL(2, Z 2 ). 33 The Z 2 -factor in PGL(2, Z) = GL(2, Z)/Z 2 corresponds to the subgroup (±1 l 2 ) that clearly corresponds to the same fractional transformation (4.44) keeping in mind that the matrix product agrees with the concatenation of two transformations (4.44). 34 The canonical homomorphism C amounts to mapping any element`a b c d´∈ PGL(2, Z) to the matrix of Z/2Z-equivalence classes`[ a] [b] [c] [d]´∈ PGL(2, Z 2 ).
We can also make this more precise in terms of 2 × 2-matrices. A comparison of (4.35) with (4.44) yields: It is an easy exercise to verify the Coxeter relations (4.36) in terms of these matrices. 35 The additional Coxeter relation from proposition 5, saying that the matrix R s α * commutes with R s α1 , corresponds to the fact that the images of the canonical homomorphism C(A * ) and C(A 1 ) commute in PSL(2, Z 2 ): It may also be interesting to investigate whether this pattern linking principal congruence subgroups to the description of the billiard dynamics of a Dirac spinor in terms of the finite group W s g allows for a generalization to other Kac-Moody algebras g. A promising result in this respect was recently provided in [30], where the Weyl group of E 10 was shown to be related to the 'modular group' PSL (

Conclusions
We have studied the "fermionic billiards", i.e. the chaotic dynamics of the gravitino, that arise in the near-spacelike-singularity limit (or Belinski-Khalatnikov-Lifshitz, BKL limit) of supergravity. We have focussed on eleven-dimensional supergravity (whose bosonic cosmological billiard takes place in the Weyl chamber of E 10 ), and have also considered in detail N = 1 D = 4 supergravity (whose bosonic billiard takes place in the Weyl chamber of AE 3 = A ++ 1 ). We have shown that a useful tool for studying the fermionic side of the near-singularity cosmological billiards is the Kac-Moody coset reformulation of the supergravity dynamics. In this reformulation, the bosonic coset velocity P, and the gravitino variable ψ, are both parallel-transported by the same abstract connection Q belonging to the maximal compact subalgebra of the relevant Kac-Moody algebra, namely K(E 10 ) for maximal supergravity and K(AE 3 ) for its D = 4 N = 1 truncation. The time evolution, near a spacelike singularity, of the, say, K(E 10 )-valued 'angular velocity' Q(t) was shown to consist of a succession of well-separated inverse-cosh 'spikes' associated to the collision on a 'Toda wall' (itself associated to a certain root α of E 10 ). Each such K(E 10 )-spike takes place along the 'rotational axis' J α = E α − E −α = E α + ω(E −α ) associated (within K(E 10 )) to the corresponding wall root α.
We found that the integral over the collision of each 'angular velocity' Q(t) generates a finite K(E 10 ) rotation given by the universal formula where the normalization of E α and E −α = −ω(E α ) (ω denoting the Chevalley involution) is such that E α , E −α and h α := [E α , E −α ] form a standardly normalized sl 2 -algebra (see Appendix B). When evaluated within the 'adjoint' representation which acts on the coset velocity P (and in the billiard limit where P ∈ h, the Cartan subalgebra of E 10 ), the K(E 10 )-rotation (5.1) reproduces the standard geometrical Weyl reflection r α , see Eq. (2.14). The effect of each collision on the fermionic variable ψ is obtained by evaluating the general abstract K(E 10 ) rotation (5.1) within the 'vector-spinor' representation of K(E 10 ) to which ψ belongs. As a consequence, the billiard dynamics of the gravitino can be described as a 'word', i.e. a product, of discrete vector-spinor K(E 10 ) rotations, say (as in (3.8)) where α i1 → α i2 → · · · → α iN denotes an unbounded sequence of collisions encountered by the billiard particle moving in the Weyl chamber of E 10 . Here, (α i1 , . . . , α iN ) is a sequence of simple roots (the signs ε αi = ± associated to each simple root are fixed in the BKL-limit and could here be all conventionally replaced by +). The vector-spinor word w vs (5.2) is the fermionic side of the corresponding bosonic billiard result, described by a word in the Weyl group, say w = r αi N · · · r αi 2 r αi 1 .
We found that the 'vector-spinor reflections' (5.2) happen to factorize as the tensor product of the usual vector Weyl reflections (5.3) and of a corresponding K(E 10 ) 'Dirac-spinor' reflection as in (3.20), induced by the same sequence of collisions α i1 → α i2 → · · · → α iN : We considered (5.2) and (5.4) as particular representations of an abstract 'spin' extension W spin of the Weyl group of E 10 (or more generally of any Kac-Moody algebra), formally defined as the discrete subgroup of a suitable covering of the 'group' K(E 10 ), which is multiplicatively generated by the formal exponentials (5.1). We found that the generators of the two representations (5.2) and (5.4) of W spin satisfy generalized Coxeter relations (see propositions 1 and 2 in section 3.5). We also found that its realization W vs in the vector-spinor representation is infinite and contains a quotient group W vs /D vs (where D vs is a normal subgroup of finite cardinality) that is isomorphic to the usual Weyl group W. Therefore, we can think of W vs as a finite-index extension of W. By contrast, we found that the Dirac-spinor realization W s of W spin is of finite cardinality. Physically, this means that a Dirac-billiard, described by the action of the words (5.4) on some initial 32-valued Dirac-spinor variable ǫ 0 , has only a finite orbit in R 32 (though, due to the chaotic nature of the underlying Weyl-chamber billiard, the sequence of Dirac variables ǫ 1 , ǫ 2 , . . . , ǫ n , . . . will run chaotically over the finite orbit W s ǫ 0 ).
Although our results have relied on the use of the only currently known spinorial representations of K(E 10 ) (which are both finite-dimensional and unfaithful), we think that these results could be more general, and will, in particular, apply to the representations of W spin G (where G is a 'Kac-Moody group') that are defined as the tensor product of a representation of the Weyl group W G and a 'pure spin 1 2 ' representation of K(G) (where each K(G) rotation generator has eigenvalue ± 1 2 ).
As a particular example of another Kac-Moody algebra of interest for physics we considered AE 3 , whose Weyl group describes the billiard of pure supergravity in D = 4 dimensions. We explicitly studied the fermionic billiard that arises in N = 1 D = 4 supergravity. It is a simpler version of the E 10 case, though it retains the main features mentioned above: e.g. factorizability, existence of generalized Coxeter relations, finite cardinality of W s AE3 . A special feature of the AE 3 case is, however, that W s AE3 can be decomposed into two commuting quotient groups, because the rotation generator J s α * ∈ K(AE 3 ) (associated to the dominant gravitational wall) happens to commute with the two other simple generators J s α1 and J s α2 . This situation is non-generic in that it does not occur for the AE d case (which is associated to pure gravity in D = d + 1 dimensions).
Finally, we found that the 'super-billiard' obtained by combining the bosonic and fermionic billiards exhibits (for E 10 as well as for AE 3 ) a striking analogy with the dynamics of a 'polarized photon' bouncing on the Lorentzian mirrors corresponding to the Weyl chamber of E 10 (or AE 3 ). It has a 'momentum' v a (given by the Cartan-space velocity v a =β a ), a 'polarization vector' ϕ a (linked to the vector-index of the gravitino, after a suitable redefinition (3.9) or (4.12)), two on-shell constraints ('masslessness', a,b G ab v a v b = 0, and 'transversality, a,b G ab v a ϕ b = 0, where G ab is the flat Lorentzioan metric in Cartan space), and a Maxwell-like gauge invariance (ϕ ′a = ϕ a + ξv a ). This analogy exhibits some intriguing metamorphoses of gauge symmetries (local supersymmetry ↔ Maxwell-like gauge symmetry), as well as of group symmetries (SO(d) ↔ SO(d − 1, 1)). It may hint at new ways of using the conjectural gravity-coset correspondence for illuminating the hidden symmetries of maximal supergravity.

Appendices A D = 11 supergravity
Following the conventions in [7], we use a real representation of the Clifford algebra with η = diag(− + · · · +) and A, B = 0, . . . , 10. The Lagrangian of D = 11 supergravity [20] in the vielbein frame (2.3) reads modulo higher fermionic terms using Einstein's summation convention 37 Here The equations of motion following from the Lagrangian (A.2) are as in [7] to lowest order in fermions: We also use the decomposition of the spin connection ω ABC = ω A[BC] in terms of the anholonomy coefficients Ω ABC = Ω [AB]C that reads B General (non-simply laced) case for the one wall limit Let us indicate what are the modifications to bring to the reasonings of section 2.4 in the case of a non-simply laced algebra where the various real roots that one might have to consider do not all have the same norm. First, note that, in the context of the gravity-coset correspondence, the Cartan subalgebra is naturally endowed with the symmetric bilinear form G ab associated to Eq. (2.5). The metric G ab (together with its inverse G ab ) then defines the norm of any root: (α|α) := ab G ab α a α b . G ab also provides one with the general definition of the co-rootα ≡ h α ∈ h that is associated to a general (non-null) root α ∈ h * , namely (generalizing the simple root case (2.22)): The definition, and in particular the normalization (with the factor 2/(α|α)) of h α ≡α a H a ∈ h (where H a is the β-coordinate basis in h; G ab = (H a |H b )) is such that Kac [21] has shown that the symmetric bilinear form (·|·) defined on h (and h * ) admits a unique extension to the full Kac-Moody algebra g when requiring its invariance (([x, y]|z) = (x|[y, z]) for any x, y, z ∈ g). In particular, he has shown that two generators E α (with associated root α > 0) and Then, when decomposing the motion of β in parallel and orthogonal pieces, we also need to remember that the kinetic energy of the orthogonal part is no longer 1 4 α(β) 2 , but α(β) 2 2(α|α) = σα 4 α(β) 2 . The conservation of orthogonal energy (2.33) gets modified into Then, the solution (2.34) takes the form and the angular velocityθ (2.37) readṡ while its integral becomes with the sign ε α := Πα |Πα| ∈ {±1} as before. As we see, the final result is not affected by this generalization: the kink in θ(t) is ± π 2 independently of the initial data (apart from the sign of Π α ), and of the norm of α. The crucial point is that the 'rotation generator' J α = E α − E −α entering the final result (2.44) is normalized in the standard sl 2 way, see Eqs. (B.4, B.5). This is, e.g., what we had used in the explicit SL(2, R) calculation (2.46).

C Proof of the propositions of section 3.5
The factorizability of the vector-spinor action R vs α,εα into the tensor product of a Weyl reflection r α (2.14) and a Dirac spinor action (3.17) will allow us in fact to consider proposition 1 as a corollary of proposition 2. The proof of the latter will be divided into several parts. We start with proving the generalized Coxeter relations:

Lemma 1
The matrices R s αi ∈ R 32×32 (3.12, 3.33), associated to the dominant walls α i with i = 0, . . . , 9 (2.10), fulfill the following generalized Coxeter relations: If two nodes α i , α j in figure 1 are not linked, the associated matrices commute, otherwise these fulfill The ten elements R s αi generate a group W s . Note that this implies in particular that the generators R s αi,εα i (3.12, 3.18) would have generated the same group for all choices of signs ε αi .

Proof
We start with the observation that the matrices Γ αi = 2J s αi (3.14) with i = 0, . . . , 9 square to the negative identity matrix. This entails the relation (C.1) after an evaluation of a matrix exponential series (equivalent to e iπ = −1) with the unit matrix 1l: It also allows us to rewrite the group elements in a different manner: Given any two nodes α i , α j in figure 1 that are not linked, the identification of Γ αi with Γ-matrices (3.14) then immediately implies that the corresponding group elements commute, taking into account the Clifford algebra relation {Γ a , Γ b } = 2δ ab . For the case of linked nodes α i , α j , the product k of the anticommuting matrices i := Γ αi and j := Γ αj is a square root of −1l again. k := ij also anticommutes with i and j. The elements i, j, k generate a multiplicative group with 8 elements (namely {±1, ±i, ±j, ±k}). This allows for an easy evaluation of the product of the matrices R s αi (C.3): with mi + nj + pk =: m 2 + n 2 + p 2 I for m, n, p ∈ R, which is a standard formula for anticommuting quaternions i, j, k with I 2 = −1l. Since (C.4) is a third root of −1l, the relation (C.2) follows.
Another consequence of the relation (C.1) is that any matrix R s αi is an eighth root of unity. This in particular implies that its inverse also is contained in the set of words W s generated by arbitrary products of the matrices R s αi . Thus, W s is endowed with a group structure. Since the inverse of R s αi,εα i (3.18) is R s αi,−εα i , the group W s is not sensitive to the choices of ε αi ∈ {±1}.

Lemma 2
The group W s generated by the ten matrices R s αi (3.12, 3.33) is a finite subgroup of SO(32).

Proof
First, we recall from [7] that the algebra generators J s αi (3.12) are particular cases of K(E 10 )-generators acting unfaithfully on a 32 dimensional space endowed with an invariant norm Q s which reads for any ǫ ∈ R 32 : This implies J s αi ∈ so 32 and for the group generators R s αi = e π 2 J s α i (3.33) R s αi ∈ SO(32). Hence, the group W s is a subgroup of SO(32). We will show next that W s is finite. Squaring the equation (C.3) results for any i ∈ {0, . . . , 9} in the identity We have already shown in lemma 1 that two rotations R s αi commute, if the associated nodes in the Dynkin diagram of E 10 in figure 1 are not connected with a line. For all the remaining cases, it is a straightforward calculation with Γ-matrices to verify the following anticommutation relation: Hence, any matrix product of generators R s αi can be written as a sum of ordered matrix products, for which there only are a finite number of possibilities due to the relation (C.6). Furthermore, the coefficients of the ordered matrix products are contained in the Galois extension Z[ √ 2] of integers. In other words, the group action in W s stabilizes on a Z[ √ 2]-lattice, spanned by the ordered products of generators R s αi . In formulae, this means that any word g ∈ W s constructed from these generators can be associated to constants c * ∈ Z[ √ 2] such that ..i10 R s αi 1 · · · R s αi 10 .
(C. 8) In order to prove the finiteness of W s , it is hence sufficient to show that the constants c * ∈ Z[ √ 2] may only take a finite number of values. As a next step, we substitute the Γ-matrices for R s αi (3.14, C.3) in this expansion. This implies that a general element g ∈ W s has the form g = 1 32 d 0 1l + 10 a=1 d a Γ a + a1<a2 d a1a2 Γ a1a2 + · · · + a1<···<a10 d a1...a10 Γ a1...a10 .
The important observation is that there is a maximal denominator which is 32 = √ 2 10 and that the coefficients d * ∈ Z[ √ 2] are still Galois extended integers. Since the group W s is a subgroup of the special orthogonal group SO(32) (C.5), any g ∈ W s has to fulfill the orthogonality constraint g T g = 1l. Taking the trace of this constraint (i.e. considering the part ∝ 1l in the expansion of g T g along the ordered Γ-basis) leads us to the following diophantine equation: Since all terms on the RHS of (C.10) are positive and the coefficients d ′ * , d ′′ * are integer, the number of solutions of the equation (C.10) is finite. As the collective of integers d ′ * , d ′′ * parametrize a general word g ∈ W s , this proves that W s is of finite cardinality.

Lemma 3
The squares of the generating elements R s αi ∀g ∈ W s , ∀h ∈ D s g · h · g −1 ∈ D s (C.12) The subgroup D s is not abelian and its cardinality is 2048.
To prove that D s is a normal subgroup, it is sufficient to show that the relation (C.12) is fulfilled for the generators R s αi ∈ W s (C.3) (with i = 0, . . . , 9) and the generators h ∈ {±1l, ±Γ a } (for a = 1, . . . , 10) which can be easily checked by a direct computation.

Lemma 4
There is a homomorphism Φ between the Weyl group W of E 10 and the quotient group W s /D s : Φ : W → W s /D s (C.14)

Proof
As shown in lemma 3, D s is a normal subgroup of W s . Hence, the coset W s /D s in (C.14) is well-defined. Next, we define the mapping Φ (C.14) by linking the fundamental reflections r αi ∈ W to the D s -equivalence classes [R s αi ] of generating elements of W s : Φ : W → W s /D s (C. 15) r αi → [R s αi ] =:r αi The mapping Φ can then be extended to the entire Weyl group W by the usual relation for any two fundamental reflections r αi , r αj ∈ W with i, j ∈ {0, . . . , 9} Φ(r αi r αj ) := Φ(r αi )Φ(r αj ).
To complete the proof that Φ (C.15) defines a homomorphism, one has to check that the identity 1 ∈ W can be mapped to the D s -equivalence class [1l] of the identity in W s in a consistent way. This is equivalent to checking that the Coxeter relations that the Weyl group W is subject to and that are encoded in the Dynkin diagram of E 10 (figure 1), are preserved by the mapping Φ: The first equation is an immediate consequence of the definition of the normal subgroup D s , the second one follows from the relations (C.2, C.13) and the third one can be deduced from the first one (C.16a) taking into account that the matricesr αi andr αj commute for the case of non-adjacent nodes as shown in lemma 1.
The proof of theorem 2 is then completed by combining the four lemmas: Since the group W s is finite (lemma 2), the cardinality of the quotient group W s /D s also is finite. Combining this observation with the infinite cardinality of the Weyl group W of E 10 , any homomorphism Φ (C.14, lemma 4) must have a non-trivial kernel, corresponding to a normal subgroup of the Weyl group W of E 10 .
As mentioned above, the proof of proposition 1 follows from a combination of proposition 2 and the factorization of the vector-spinor action R vs α into the tensor product of a Weyl reflection r α (2.14) (acting on the vector part of the gravitino ϕ a ) and a Dirac spinor action R s αi (acting on its spinor part) as exhibited in Eq. (3.17). In formulae, for all dominant walls α i (2.10) with r αi ∈ W and R s αi ∈ W s we have R vs αi = r αi ⊗ R s αi . (C.17) The group structure of W vs is then inherited from the group structure of the Weyl group W and of the Dirac-spinor group W s . Since the fundamental reflections r αi of the Weyl group W are subject to the Coxeter relation r 2 αi = 1, the squares of the generators of W vs have the form (R vs αi,εα i This immediately implies that these matrices generate a normal subgroup D vs of W vs that is isomorphic to D s and hence shares all its properties. The generalized Coxeter relations stated in proposition 1 then also directly follow from the tensor product structure (keeping in mind lemma 1 and the standard Coxeter relations for the Weyl group W of E 10 (C.16)). Furthermore, a homomorphism Φ linking the Weyl group W of E 10 to the quotient group W vs /D vs can be constructed in a similar way as in (C.15). Hence, we are left with proving that the homomorphism Φ linking the Weyl group W of E 10 to the quotient group W vs /D vs is an isomorphism (3.36): Φ : W → W vs /D vs .
The tensor product structure of R vs αi = r αi ⊗ R s αi,εα i (C.17) together with the isomorphism of the quotient groups D vs ≃ D s and the Dirac-spinor mapping (C.15) implies that a general word w (with w = r α1 r α2 · · · r αn ) in the Weyl group W is mapped by Φ to: The issue is now to know whether one can uniquely reconstruct the pre-image w ∈ W, when one is given some equivalence class of matrices in the tensor product W vs . Clearly given a matrix R a b A B (3.17), it determines the factor w a b it comes from modulo a (real) factor, say λw a b . But we also know that w ∈ W is an orthochronous Lorentz transformation which preserves the Lorentzian norm G abβ aβb . This implies that there is a unique w within a "line" λw. [Even λ = −1 is impossible, because W preserves the future light cone in β-space.] This completes the proof of the propositions of section 3.5.