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5.1 More on Coxeter billiards

5.1.1 The Coxeter billiard of pure gravity in D spacetime dimensions

We start by providing other examples of theories leading to regular billiards, focusing first on pure gravity in any number of D (> 3) spacetime dimensions. In this case, there are d = D − 1 scale factors i β and the relevant walls are the symmetry walls, Equation (2.48View Equation),

si(β) ≡ βi+1 − βi = 0 (i = 1,2,⋅ ⋅⋅ ,d − 1), (5.1 )
and the curvature wall, Equation (2.49View Equation),
r(β ) ≡ 2β1 + β2 + ⋅⋅⋅ + βd −2 = 0. (5.2 )
There are thus d relevant walls, which define a simplex in (d − 1)-dimensional hyperbolic space ℋ d−1. The scalar products of the linear forms defining these walls are easily computed. One finds as non-vanishing products
(si|si) = 2 (i = 1,⋅⋅⋅ ,d − 1 ), (r|r) = 2, (si+1|si) = − 1 (i = 2,⋅⋅⋅ ,d − 1 ) (5.3 ) (r|s1) = − 1, (r|sd−2) = − 1.
The matrix of the scalar products of the wall forms is thus the Cartan matrix of the (simply-laced) Lorentzian Kac–Moody algebra ++ Ad−2 with Dynkin diagram as in Figure 24View Image. The roots of the underlying finite-dimensional algebra Ad −2 are given by si (i = 1,⋅⋅⋅ ,d − 3) and r. The affine root is sd− 2 and the overextended root is sd−1.
View Image

Figure 24: The Dynkin diagram of the hyperbolic Kac–Moody algebra ++ A d− 2 which controls the billiard dynamics of pure gravity in D = d + 1 dimensions. The nodes s1,⋅⋅⋅ ,sd− 1 represent the “symmetry walls” arising from the off-diagonal components of the spatial metric, and the node r corresponds to a “curvature wall” coming from the spatial curvature. The horizontal line is the Dynkin diagram of the underlying Ad−2-subalgebra and the two topmost nodes, sd−2 and sd−1, give the affine- and overextension, respectively.

Accordingly, in the case of pure gravity in any number of spacetime dimensions, one finds also that the billiard region is regular. This provides new examples of Coxeter billiards, with Coxeter groups A++d− 2, which are also Kac–Moody billiards since the Coxeter groups are the Weyl groups of the Kac–Moody algebras ++ A d− 2.

5.1.2 The Coxeter billiard for the coupled gravity-3-Form system

Coxeter polyhedra

Let us review the conditions that must be fulfilled in order to get a Kac–Moody billiard and let us emphasize how restrictive these conditions are. The billiard region computed from any theory coupled to gravity with n dilatons in D = d + 1 dimensions always defines a convex polyhedron in a (d + n − 1)-dimensional hyperbolic space ℋd+n − 1. In the general case, the dihedral angles between adjacent faces of ℋ d+n−1 can take arbitrary continuous values, which depend on the dilaton couplings, the spacetime dimensions and the ranks of the p-forms involved. However, only if the dihedral angles are integer submultiples of π (meaning of the form π ∕k for k ∈ ℤ≥2) do the reflections in the faces of ℋd+n − 1 define a Coxeter group. In this special case the polyhedron is called a Coxeter polyhedron. This Coxeter group is then a (discrete) subgroup of the isometry group of ℋ d+n− 1.

In order for the billiard region to be identifiable with the fundamental Weyl chamber of a Kac–Moody algebra, the Coxeter polyhedron should be a simplex, i.e., bounded by d + n walls in a d + n − 1-dimensional space. In general, the Coxeter polyhedron need not be a simplex.

There is one additional condition. The angle ϑ between two adjacent faces i and j is given, in terms of the Coxeter exponents, by

ϑ = -π--. (5.4 ) mij
Coxeter groups that correspond to Weyl groups of Kac–Moody algebras are the crystallographic Coxeter groups for which mij ∈ {2,3,4,6,∞ }. So, the requirement for a gravitational theory to have a Kac–Moody algebraic description is not just that the billiard region is a Coxeter simplex but also that the angles between adjacent walls are such that the group of reflections in these walls is crystallographic.

These conditions are very restrictive and hence gravitational theories which can be mapped to a Kac–Moody algebra in the BKL-limit are rare.

The Coxeter billiard of eleven-dimensional supergravity

Consider for instance the action (2.1View Equation) for gravity coupled to a single three-form in D = d + 1 spacetime dimensions. We assume D ≥ 6 since in lower dimensions the 3-form is equivalent to a scalar (D = 5) or has no degree of freedom (D < 5).

Theorem: Whenever a p-form (p ≥ 1) is present, the curvature wall is subdominant as it can be expressed as a linear combination with positive coefficients of the electric and magnetic walls of the p-forms. (These walls are all listed in Section 2.5.)

Proof: The dominant electric wall is (assuming the presence of a dilaton)

1 2 p λp- e1⋅⋅⋅p(β) ≡ β + β + ⋅⋅⋅ + β − 2 φ = 0, (5.5 )
while one of the magnetic wall reads
m1,p+1,⋅⋅⋅,d−2(β) ≡ β1 + βp+1 + ⋅⋅⋅ + βd−2 + λpφ = 0, (5.6 ) 2
so that the dominant curvature wall is just the sum e1⋅⋅⋅p(β) + m1,p+1,⋅⋅⋅,d−2(β ).

It follows that in the case of gravity coupled to a single three-form in D = d + 1 spacetime dimensions, the relevant walls are the symmetry walls, Equation (2.48View Equation),

i+1 i si(β) ≡ β − β = 0, i = 1,2,⋅ ⋅⋅ ,d − 1 (5.7 )
(as always) and the electric wall
e123(β ) ≡ β1 + β2 + β3 = 0 (5.8 )
(D ≥ 8) or the magnetic wall
1 2 D −5 m1 ⋅⋅⋅D− 5(β) ≡ β + β + ⋅⋅⋅β = 0 (5.9 )
(D ≤ 8). Indeed, one can express the magnetic walls as linear combinations with (in general non-integer) positive coefficients of the electric walls for D ≥ 8 and vice versa for D ≤ 8. Hence the billiard table is always a simplex (this would not be true had one a dilaton and various forms with different dilaton couplings).

However, it is only for D = 11 that the billiard is a Coxeter billiard. In all the other spacetime dimensions, the angle between the relevant p-form wall and the symmetry wall that does not intersect it orthogonally is not an integer submultiple of π. More precisely, the angle between

is easily verified to be an integer submultiple of π only for D = 11, for which it is equal to π ∕3.

From the point of view of the regularity of the billiard, the spacetime dimension D = 11 is thus privileged. This is of course also the dimension privileged by supersymmetry. It is quite intriguing that considerations a priori quite different (billiard regularity on the one hand, supersymmetry on the other hand) lead to the same conclusion that the gravity-3-form system is quite special in D = 11 spacetime dimensions.

For completeness, we here present the wall system relevant for the special case of D = 11. We obtain ten dominant wall forms, which we rename α1,⋅⋅⋅ ,α10,

αm (β) = βm+1 − βm (m = 1,⋅⋅⋅ ,10), 1 2 3 (5.10 ) α10(β) = β + β + β .
Then, defining a new collective index i = (m, 10), we see that the scalar products between these wall forms can be organized into the matrix
( 2 − 1 0 0 0 0 0 0 0 0 ) | | | − 1 2 − 1 0 0 0 0 0 0 0 | || 0 − 1 2 − 1 0 0 0 0 0 − 1 || || 0 0 − 1 2 − 1 0 0 0 0 0 || (αi|αj) | 0 0 0 − 1 2 − 1 0 0 0 0 | Aij = 2 -------= || 0 0 0 0 − 1 2 − 1 0 0 0 || , (5.11 ) (αi |αi) || || | 0 0 0 0 0 − 1 2 − 1 0 0 | || 0 0 0 0 0 0 − 1 2 − 1 0 || ( 0 0 0 0 0 0 0 − 1 2 0 ) 0 0 − 1 0 0 0 0 0 0 2
which can be identified with the Cartan matrix of the hyperbolic Kac–Moody algebra E10 that we have encountered in Section 4.10.2. We again display the corresponding Dynkin diagram in Figure 25View Image, where we point out the explicit relation between the simple roots and the walls of the Einstein–3-form theory. It is clear that the nine dominant symmetry wall forms correspond to the simple roots αm of the subalgebra 𝔰𝔩(10,ℝ ). The enlargement to E10 is due to the tenth exceptional root realized here through the dominant electric wall form e123.
View Image

Figure 25: The Dynkin diagram of E10. Labels m = 1,⋅⋅⋅ ,9 enumerate the nodes corresponding to simple roots, αm, of the 𝔰𝔩(10,ℝ ) subalgebra and the exceptional node, labeled “10”, is associated to the electric wall α10 = e123.

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