List of Footnotes

1 This is done mostly for notational convenience. If there were other dilatons among the 0-forms, these should be separated off from the p-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 i dx, with the imposed constraint √ - N = g. In general, one may of course use an arbitrary spatial coframe, say i θ (x), for which the associated gauge choice reads √ - N = w (x) g, with w (x ) being a density of weight − 1. 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 [40Jump To The Next Citation Point].
3 The Hamiltonian heuristic derivation of [48Jump To The Next Citation Point] shares many features in common with the work of [122, 109, 112, 123], extended to some higher-dimensional models in [110, 111Jump To The Next Citation Point]. The central feature of [48Jump To The Next Citation Point] is the Iwasawa decomposition which enables one to clearly see the role of off-diagonal variables.
4 This Hamiltonian exists if fiik = 0, 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 [40Jump To The Next Citation Point] 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 si is the reflection with respect to the hyperplane defined by αi = 0, because it preserves the scalar product, fixes the plane orthogonal to αi and maps αi on − αi. 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 αi(β ), i = 1,2,3, really represent the values of the linear maps αi : β → αi(β) ∈ ℝ. The billiard ball moves in the space of scale factors, say ℳ β (β-space), and hence the maps αi, which define the walls, belong to the dual space ⋆ ℳ β of linear forms acting on ℳ β. In order to be compatible with the treatment in Section 2.4 (cf. Equation (2.45View Equation)), Equation (3.7View Equation) – even though written here as a reflection in the space ⋆ ℳ β – really corresponds to a geometric reflection in the space ℳ β, in which the particle moves. This will be carefully explained in Section 5.2 (cf. Equations (5.20View Equation) and (5.21View Equation)), 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 I2(∞ ), for which B is degenerate, the definition does not give anything of interest since ℰ = ∅. When B is degenerate, the formalism developed here can nevertheless be carried through but one must go to the dual space ∗ V [107Jump To The Next Citation Point].
9 We are employing the convention of Kac [116Jump To The Next Citation Point] for the Cartan matrix. There exists an alternative definition of Kac–Moody algebras in the literature, in which the transposed matrix T A is used instead.
10 We recall that an ideal 𝔦 is a subalgebra such that [𝔦,𝔤] ⊂ 𝔦. A simple algebra has no non-trivial ideals.
11 Imaginary roots may have arbitrarily negative length squared in general.
12 The generalized Casimir operator Ω is the only known polynomial element of the center Z of the universal enveloping algebra U(𝔤) of an indefinite Kac–Moody algebra 𝔤. However, Kac [115Jump To The Next Citation Point] has proven the existence of higher order non-polynomial Casimir operators which are elements of the center Z𝔉 of a suitable completion U𝔉(𝔤) of the universal enveloping algebra of 𝔤. Recently, an explicit physics-inspired construction was made, following [115], for affine 𝔤 in terms of Wilson loops for WZW-models [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 β = ±12α, β = ± α or β = ±2α. If the minus sign holds, then (α|β) is automatically < 0 and there is nothing to be proven. So we only need to consider the cases β = +12α, β = + α or β = +2 α. In the first case, α− β = β ∈ Δ, in the second case α − β = 0, and in the last case α − β = − α ∈ Δ so these three cases are in fact excluded by the assumption. We can therefore assume α and β 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 (2.8View Equation) of the spatial metric with 𝒩 an upper triangular matrix. The (equivalent) choice of the tenth direction as compactification direction would correspond to a different (equivalent) choice of 𝒩.
17 This structure of ℰ8(8) can be understood as follows. The 248-dimensional Lie algebra E8(8) can be represented as 𝔰𝔬(16)⊕ 𝒮16 (direct sum of vector spaces), where 𝒮16 constitutes a 128-dimensional representation space of the group Spin(16), that transforms like Majorana–Weyl spinors. Using Dirac matrices Γ νa μ, the commutation relations read:
[Mab,Mcd ] = δacMbd + δbdMac − adMbc − δbcMad,
1 ν [Mab,Q μ] = 2Γ[ab]μQν,
[Qμ,Q ν] = Γ [ab]Mab. μν
For more information about E8 (8) see [134Jump To The Next Citation Point], and for a general discussion of real forms of Lie algebras see Section 6.
18 In the following we write simply E 8 and it is understood that we refer to the split real form E 8(8).
19 Actually, the structure constants are integers and thus allows for defining the arithmetic subgroup SL(2,ℤ) ⊂ SL (2,ℝ).
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 2+ 1 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 E8(8) of E8, the 8 level 3-elements and the 28 level 2-elements form an Abelian subalgebra since there are no elements at levels > 3 (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 Xα is a vector in gα and Y −α is a vector in g−α, then one has [Xα,Y−α] = B(Xα,Y− α)H α.
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 semi-simple algebra is compact if and only if its Killing form is negative definite.
26 In the notation Hr(σ) for a real form of the simple complex Lie algebra Hr, with the integer σ referring to the signature.
27 The coefficients aα are determined from the commutation relations as follows: N α,βaα+β = Nθ[α],θ[β]aαaβ. Moreover, because θ is an automorphism of the root lattice we have N2 = N 2 , α,β θ[α],θ[β] and so if aα and aβ are equal to ±1, then so is aα+β. But since this is true for the simple roots it remains true for all roots.
28 A system Δ is closed if α, β ∈ Δ implies that − α ∈ Δ and α +β ∈ Δ.
29 Geometrically, this results from the orthogonality of roots α and β such that Hα ∈ 𝔨 and Hβ ∈ 𝔭, or, equivalently, because α (H β) = θ[α](H β) = α(θ(Hβ)) = − α(Hβ).
30 If p = q+ 1, this basis consists only of noncompact generators.
31 The decomposition of AE3 into representations of A+ 1 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 47View Image.
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 vice-versa.
34 Strictly speaking, the coset space defined in this way should be written as 𝒦 (𝒢)∖𝒢. However, we follow what has become common practice in the literature and denote it by 𝒢∕𝒦 (𝒢 ).
35 As an example, consider the projection Pαβγ ≡ T⟨αβγ⟩ of a three index tensor Tαβγ onto the Young tableaux
|-|-| |-|-. ---
This projection is given by
1 Pαβγ = 3(Tαβγ + Tβαγ − Tγβα − Tβγα ),
which clearly satisfies
P αβγ = − P γβα, P[αβγ] = 0.
Note also that Pαβγ ⁄= Pβαγ.
36 This does not exclude that other approaches would be successful. That E10, or perhaps E11, does encode a lot of information about M-theory 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 (93,93) 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 Δ′, distinct from Δ and with a non-empty intersection with Δ.
39 This was also pointed out in [127].
40 In [90] they were dealing with a hyperbolic internal space so there was an additional sinh-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 m ij is equal to infinity, the Coxeter group is called 2-spherical. 2-spherical Coxeter subgroups of E 10 are rare [27].
43 To convince oneself of the validity of this commutation relation, it suffices to check it in a basis where the (finite-dimensional) matrix ρ is diagonal, using the symmetry of the matrix στ.
44 The uniqueness derives from the fact that the internal automorphism groups of 𝔤ℝ and 𝔤 are identical.