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5.2 Dynamics in the Cartan subalgebra

We have just learned that, in some cases, the group of reflections that describe the (possibly chaotic) dynamics in the BKL-limit is a Lorentzian Coxeter group ℭ. In this section we fully exploit this algebraic fact and show that whenever ℭ is crystallographic, the dynamics takes place in the Cartan subalgebra 𝔥 of the Lorentzian Kac–Moody algebra 𝔤, for which ℭ is the Weyl group. Moreover, we show that the “billiard table” can be identified with the fundamental Weyl chamber in 𝔥.

5.2.1 Billiard dynamics in the Cartan subalgebra

Scale factor space and the wall system

Let us first briefly review some of the salient features encountered so far in the analysis. In the following we denote by ℳ β the Lorentzian “scale factor”-space (or β-space) in which the billiard dynamics takes place. Recall that the metric in ℳ β, induced by the Einstein–Hilbert action, is a flat Lorentzian metric, whose explicit form in terms of the (logarithmic) scale factors reads

d ( d ) ( d ) μ ν ∑ i i ∑ i ∑ j Gμν dβ dβ = dβ dβ − dβ dβ + dφ dφ, (5.12 ) i=1 i=1 j=1
where d counts the number of physical spatial dimensions (see Section 2.5). The role of all other “off-diagonal” variables in the theory is to interrupt the free-flight motion of the particle, by adding walls in ℳ β that confine the motion to a limited region of scale factor space, namely a convex cone bounded by timelike hyperplanes. When projected onto the unit hyperboloid, this region defines a simplex in hyperbolic space which we refer to as the “billiard table”.

One has, in fact, more than just the walls. The theory provides these walls with a specific normalization through the Lagrangian, which is crucial for the connection to Kac–Moody algebras. Let us therefore discuss in somewhat more detail the geometric properties of the wall system. The metric, Equation (5.12View Equation), in scale factor space can be seen as an extension of a flat Euclidean metric in Cartesian coordinates, and reflects the Lorentzian nature of the vector space ℳ β. In this space we may identify a pair of coordinates (βi,φ) with the components of a vector β ∈ ℳ β, with respect to a basis { ¯uμ} of ℳ β, such that

¯uμ ⋅ ¯uν = G μν. (5.13 )
The walls themselves are then defined by hyperplanes in this linear space, i.e., as linear forms μ ω = ωμσ-, for which ω = 0, where μ {σ-} is the basis dual to μ {¯u }. The pairing ω (β) between a vector β ∈ ℳ β and a form ω ∈ ℳ ⋆β is sometimes also denoted by ⟨ω, β⟩, and for the two dual bases we have, of course,
⟨σ-μ, ¯uν⟩ = δμν. (5.14 )
We therefore find that the walls can be written as linear forms in the scale factors:
∑ ∑ ∑d ω (β) = ωμβν ⟨σμ, ¯uν⟩ = ωμβ μ = ωiβi + ω φφ. (5.15 ) μ,ν μ i=1
We call ω (β) wall forms. With this interpretation they belong to the dual space ℳ ⋆β, i.e.,
⋆ ℳ β ∋ ω : ℳ β − → ℝ, β ↦− → ω(β ). (5.16 )
From Equation (5.16View Equation) we may conclude that the walls bounding the billiard are the hyperplanes ω = 0 through the origin in ℳ β which are orthogonal to the vector with components ωμ = G μνω ν.

It is important to note that it is the wall forms that the theory provides, as arguments of the exponentials in the potential, and not just the hyperplanes on which these forms ω vanish. The scalar products between the wall forms are computed using the metric in the dual space ⋆ ℳ β, whose explicit form was given in Section 2.5,

( ) ( ) ∑d 1 ∑d ∑d (ω|ω′) ≡ Gμνω μων = ωiω′i − ------ ωi ω′j + ωφω ′φ, ω, ω′ ∈ ℳ β. (5.17 ) i=1 d − 1 i=1 j=1

Scale factor space and the Cartan subalgebra

The crucial additional observation is that (for the “interesting” theories) the matrix A associated with the relevant walls ωA,

(ω |ω ) AAB = 2---A--B- (5.18 ) (ωA |ωA)
is a Cartan matrix, i.e., besides having 2’s on its diagonal, which is rather obvious, it has as off-diagonal entries non-positive integers (with the property AAB ⁄= 0 ⇒ ABA ⁄= 0). This Cartan matrix is of course symmetrizable since it derives from a scalar product.

For this reason, one can usefully identify the space of the scale factors with the Cartan subalgebra 𝔥 of the Kac–Moody algebra 𝔤 (A ) defined by A. In that identification, the wall forms become the simple roots, which span the vector space 𝔥⋆ = span{α ,⋅⋅⋅ ,α } 1 r dual to the Cartan subalgebra. The rank r of the algebra is equal to the number of scale factors μ β, including the dilaton(s) if any ((βμ) ≡ (βi,φ )). This number is also equal to the number of walls since we assume the billiard to be a simplex. So, both A and μ run from 1 to r. The metric in ℳ β, Equation (5.12View Equation), can be identified with the invariant bilinear form of 𝔤, restricted to the Cartan subalgebra 𝔥 ⊂ 𝔤. The scale factors βμ of ℳ β become then coordinates hμ on the Cartan subalgebra 𝔥 ⊂ 𝔤 (A).

The Weyl group of a Kac–Moody algebra has been defined first in the space 𝔥⋆ as the group of reflections in the walls orthogonal to the simple roots. Since the metric is non degenerate, one can equivalently define by duality the Weyl group in the Cartan algebra 𝔥 itself (see Section 4.7). For each reflection ri on ⋆ 𝔥 we associate a dual reflection ∨ ri as follows,

∨ ∨ ∨ ri (β ) = β − ⟨αi,β⟩ αi , β,αi ∈ 𝔥, (5.19 )
which is the reflection relative to the hyperplane αi(β ) = ⟨αi,β ⟩ = 0. This expression can be rewritten (see Equation (4.59View Equation)),
2(β|α∨) r∨i (β) = β − --∨--i∨--α∨i , (5.20 ) (αi |αi )
or, in terms of the scale factor coordinates βμ,
2 (β |ω ∨) β μ −→ βμ′ = βμ − ---∨--∨-ω ∨μ. (5.21 ) (ω |ω )
This is precisely the billiard reflection Equation (2.45View Equation) found in Section 2.4.

Thus, we have the following correspondence:

ℳ β ≡ 𝔥, ⋆ ⋆ ℳ β ≡ 𝔥 , ωA (β) ≡ αA (h), (5.22 ) billiard wall reflections ≡ fundamental Weyl reflections.
As we have also seen, the Kac–Moody algebra 𝔤(A ) is Lorentzian since the signature of the metric Equation (5.12View Equation) is Lorentzian. This fact will be crucial in the analysis of subsequent sections and is due to the presence of gravity, where conformal rescalings of the metric define timelike directions in scale factor space.

We thereby arrive at the following important result [45Jump To The Next Citation Point46Jump To The Next Citation Point48Jump To The Next Citation Point]:

The dynamics of (a restricted set of) theories coupled to gravity can in the BKL-limit be
mapped to a billiard motion in the Cartan subalgebra 𝔥 of a Lorentzian Kac–Moody algebra 𝔤.


5.2.2 The fundamental Weyl chamber and the billiard table

Let ℬ ℳ β denote the region in scale factor space to which the billiard motion is confined,

ℬ ℳ β = {β ∈ ℳ β |ωA(β ) ≥ 0}, (5.23 )
where the index A runs over all relevant walls. On the algebraic side, the fundamental Weyl chamber in 𝔥 is the closed convex (half) cone given by
𝒲 𝔥 = {h ∈ 𝔥 |αA (h ) ≥ 0; A = 1,⋅⋅⋅ ,rank 𝔤}. (5.24 )
We see that the conditions αA (h) ≥ 0 defining 𝒲 𝔥 are equivalent, upon examination of Equation (5.22View Equation), to the conditions ωA(β ) ≥ 0 defining the billiard table ℬ ℳ β.

We may therefore make the crucial identification

𝒲 𝔥 ≡ ℬ ℳ β, (5.25 )
which means that the particle geodesic is confined to move within the fundamental Weyl chamber of 𝔥. From the billiard analysis in Section 2 we know that the piecewise motion in scale-factor space is controlled by geometric reflections with respect to the walls ωA(β ) = 0. By comparing with the dominant wall forms and using the correspondence in Equation (5.22View Equation) we may further conclude that the geometric reflections of the coordinates β μ(τ ) are controlled by the Weyl group in the Cartan subalgebra of 𝔤(A ).

5.2.3 Hyperbolicity implies chaos

We have learned that the BKL dynamics is chaotic if and only if the billiard table is of finite volume when projected onto the unit hyperboloid. From our discussion of hyperbolic Coxeter groups in Section 3.5, we see that this feature is equivalent to hyperbolicity of the corresponding Kac–Moody algebra. This leads to the crucial statement [45Jump To The Next Citation Point46Jump To The Next Citation Point48Jump To The Next Citation Point]:

If the billiard region of a gravitational system in the BKL-limit can be identified with the
fundamental Weyl chamber of a hyperbolic Kac–Moody algebra, then the dynamics is chaotic.


As we have also discussed above, hyperbolicity can be rephrased in terms of the fundamental weights Λi defined as

⟨Λ ,α ∨⟩ = 2(Λj-|αi) ≡ δ , α ∨∈ 𝔥, Λ ∈ 𝔥⋆. (5.26 ) j i (αi|αi ) ij i i
Just as the fundamental Weyl chamber in ⋆ 𝔥 can be expressed in terms of the fundamental weights (see Equation (3.40View Equation)), the fundamental Weyl chamber in 𝔥 can be expressed in a similar fashion in terms of the fundamental coweights:
∑ ∨ 𝒲 𝔥 = {β ∈ 𝔥 |β = aiΛi , ai ∈ ℝ≥0}. (5.27 ) i
As we have seen (Sections 3.5 and 4.8), hyperbolicity holds if and only if none of the fundamental weights are spacelike,
(Λi|Λi) ≤ 0, (5.28 )
for all i ∈ {1, ⋅⋅⋅ ,rank 𝔤}.

Example: Pure gravity in D = 3 + 1 and A++1

Let us return once more to the example of pure four-dimensional gravity, i.e., the original “BKL billiard”. We have already found in Section 3 that the three dominant wall forms give rise to the Cartan matrix of the hyperbolic Kac–Moody algebra ++ A 1 [46Jump To The Next Citation Point48Jump To The Next Citation Point]. Since the algebra is hyperbolic, this theory exhibits chaotic behavior. In this example, we verify that the Weyl chamber is indeed contained within the lightcone by computing explicitly the norms of the fundamental weights.

It is convenient to first write the simple roots in the β-basis as follows¿

α1 = (2,0,0) α2 = (− 1,1,0) (5.29 ) α3 = (0,− 1,1).

Since the Cartan matrix of A+1+ is symmetric, the relations defining the fundamental weights are

(αi |Λj ) ≡ δij. (5.30 )
By solving these equations for Λ i we deduce that the fundamental weights are
3 Λ1 = − --α1 − 2α2 − α3 = (− 1,− 1,− 1), 2 Λ2 = − 2α1 − 2α2 − 2 α3 = (0,1,− 1), (5.31 ) Λ3 = − α1 − α2 = (− 1,− 1,0),
where in the last step we have written the fundamental weights in the β-basis. The norms may now be computed with the metric in root space and become
3 (Λ1|Λ1 ) = − 2-, (Λ2 |Λ2 ) = − 2, (Λ3|Λ3) = 0. (5.32 )
We see that Λ1 and Λ2 are timelike and that Λ3 is lightlike. Thus, the Weyl chamber is indeed contained inside the lightcone, the algebra is hyperbolic and the billiard is of finite volume, in agreement with what we already found [46Jump To The Next Citation Point].
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