1  This is done mostly for notational convenience. If there were other dilatons among the 0forms, these should be separated off from the forms because they play a distinct role. They would appear as additional scale factors and would increase the dimensions of the relevant hyperbolic billiard (they define additional spacelike directions in the space of scale factors).  
2  Note that we have for convenience chosen to work with a coordinate coframe , with the imposed constraint . In general, one may of course use an arbitrary spatial coframe, say , for which the associated gauge choice reads , with being a density of weight . Such a frame will be used in Section 2.3.1. This general kind of spatial coframe was also used extensively in the recent work [40].  
3  The Hamiltonian heuristic derivation of [48] shares many features in common with the work of [122, 109, 112, 123], extended to some higherdimensional models in [110, 111]. The central feature of [48] is the Iwasawa decomposition which enables one to clearly see the role of offdiagonal variables.  
4  This Hamiltonian exists if , as we shall assume from now on.  
5  In this article we will exclusively restrict ourselves to considerations involving the sharp wall limit. However, in recent work [40] it was argued that in order to have a rigorous treatment of the dynamics close to the singularity also in the chaotic case, it is necessary to go beyond the sharp wall limit. This implies that one should retain the exponential structure of the dominant walls.  
6  is the reflection with respect to the hyperplane defined by , because it preserves the scalar product, fixes the plane orthogonal to and maps on . Note that we are here being deliberately careless about notation in order not to obscure the main point, namely that the billiard reflections are elements of a Coxeter group. To be precise, the linear forms , really represent the values of the linear maps . The billiard ball moves in the space of scale factors, say (space), and hence the maps , which define the walls, belong to the dual space of linear forms acting on . In order to be compatible with the treatment in Section 2.4 (cf. Equation (2.45)), Equation (3.7) – even though written here as a reflection in the space – really corresponds to a geometric reflection in the space , in which the particle moves. This will be carefully explained in Section 5.2 (cf. Equations (5.20) and (5.21)), after the necessary mathematical background has been introduced.  
7  Note that the discussion in Footnote 6 applies also here.  
8  Note that in the case of the infinite dihedral group , for which is degenerate, the definition does not give anything of interest since . When is degenerate, the formalism developed here can nevertheless be carried through but one must go to the dual space [107].  
9  We are employing the convention of Kac [116] for the Cartan matrix. There exists an alternative definition of Kac–Moody algebras in the literature, in which the transposed matrix is used instead.  
10  We recall that an ideal is a subalgebra such that . A simple algebra has no nontrivial ideals.  
11  Imaginary roots may have arbitrarily negative length squared in general.  
12  The generalized Casimir operator is the only known polynomial element of the center of the universal enveloping algebra of an indefinite Kac–Moody algebra . However, Kac [115] has proven the existence of higher order nonpolynomial Casimir operators which are elements of the center of a suitable completion of the universal enveloping algebra of . Recently, an explicit physicsinspired construction was made, following [115], for affine in terms of Wilson loops for WZWmodels [1].  
13  We discuss in detail a different kind of level decomposition in Section 8.  
14  If they were not, one would have by the second point above , or . If the minus sign holds, then is automatically and there is nothing to be proven. So we only need to consider the cases , or . In the first case, , in the second case , and in the last case so these three cases are in fact excluded by the assumption. We can therefore assume and to be linearly independent.  
15  We thank Axel Kleinschmidt for an informative comment on this point.  
16  Taking the first spatial direction as compactification direction is convenient, for it does not change the conventions on the simple roots. More precisely, the Kaluza–Klein ansatz is compatible in that case with our Iwasawa decomposition (2.8) of the spatial metric with an upper triangular matrix. The (equivalent) choice of the tenth direction as compactification direction would correspond to a different (equivalent) choice of .  
17  This structure of can be understood as follows. The 248dimensional Lie algebra can be represented as
(direct sum of vector spaces), where constitutes a 128dimensional representation space of the group
, that transforms like Majorana–Weyl spinors. Using Dirac matrices , the commutation relations
read: For more information about see [134], and for a general discussion of real forms of Lie algebras see Section 6. 

18  In the following we write simply and it is understood that we refer to the split real form .  
19  Actually, the structure constants are integers and thus allows for defining the arithmetic subgroup .  
20  A conjugation on a complex Lie algebra is an antilinear involution, preserving the Lie algebra structure.  
21  This decomposition is just the “standard” decomposition of any Lorentz transformation, into the product of a rotation followed by a boost in a fixed direction and finally followed by yet another rotation.  
22  We say that an object is “unique” when it is unique up to an internal automorphism.  
23  For example, for the split form of , the 8 level 3elements and the 28 level 2elements form an Abelian subalgebra since there are no elements at levels (the level is defined in Section 8). This Abelian subalgebra has dimension 36, which is clearly much greater than the rank (8). We thank Bernard Julia for a discussion on this example. Note that for subgroups of the unitary group, diagonalizability is automatic.  
24  Quite generally, if is a vector in and is a vector in , then one has .  
25  An algebra is said to be compact if its group of internal automorphisms is compact in the topological sense. A classic theorem states that a semisimple algebra is compact if and only if its Killing form is negative definite.  
26  In the notation for a real form of the simple complex Lie algebra , with the integer referring to the signature.  
27  The coefficients are determined from the commutation relations as follows: . Moreover, because is an automorphism of the root lattice we have and so if and are equal to , then so is . But since this is true for the simple roots it remains true for all roots.  
28  A system is closed if implies that and .  
29  Geometrically, this results from the orthogonality of roots and such that and , or, equivalently, because .  
30  If , this basis consists only of noncompact generators.  
31  The decomposition of into representations of was done in [75].  
32  D.P. would like to thank Bengt E.W. Nilsson and Jakob Palmkvist for helpful discussions during the creation of Figure 47.  
33  Since we are, in fact, using conjugate Dynkin labels, these conventions are equivalent to the standard ones if one replaces covariant indices by contravariant ones, and viceversa.  
34  Strictly speaking, the coset space defined in this way should be written as . However, we follow what has become common practice in the literature and denote it by .  
35  As an example, consider the projection of a three index tensor onto the Young
tableaux


36  This does not exclude that other approaches would be successful. That , or perhaps , does encode a lot of information about Mtheory is a fact, but that this should be translated into a sigma model reformulation of the theory appears to be questionable.  
37  One may also consider a point incidence diagram defined as follows: The nodes of the point incidence diagram are the points of the geometric configuration. Two nodes are joined by a single bond if and only if there is no straight line connecting the corresponding points. The point incidence diagrams of the configurations are given in [105]. For these configurations, projective duality between lines and points lead to identical line and point incidence diagrams. Unless otherwise stated, the expression “incidence diagram” will mean “line incidence diagram”.  
38  A true secant is here defined as a line, say , distinct from and with a nonempty intersection with .  
39  This was also pointed out in [127].  
40  In [90] they were dealing with a hyperbolic internal space so there was an additional function in the transverse spacetime.  
41  We recall that a Hamiltonian path is defined as a path in an undirected graph which intersects each node once and only once. A Hamiltonian cycle is then a Hamiltonian path which also returns to its initial node.  
42  When no Coxeter exponent is equal to infinity, the Coxeter group is called 2spherical. 2spherical Coxeter subgroups of are rare [27].  
43  To convince oneself of the validity of this commutation relation, it suffices to check it in a basis where the (finitedimensional) matrix is diagonal, using the symmetry of the matrix .  
44  The uniqueness derives from the fact that the internal automorphism groups of and are identical. 
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