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7.2 “Split symmetry controls chaos”

The main point of this section is to illustrate and explain the statement “split symmetry controls chaos” [95Jump To The Next Citation Point]. To this end, we will now extend the analysis of Section 5 to non-split real forms, using the Iwasawa decomposition. As we have seen, there are two main cases to be considered:

In the first case, the billiard is governed by the overextended algebra 𝔣++, where 𝔣 is the “maximal split subalgebra” of 𝔲3. Indeed, the coupling to gravity of the coset Lagrangian of Equation (7.8View Equation) will introduce, besides the simple roots of 𝔣 (electric walls) the affine root of 𝔣 (dominant magnetic wall) and the overextended root (symmetry wall), just as in the split case (but for 𝔣 instead of 𝔲3). This is therefore a straightforward generalization of the analysis in Section 5.

The second case, however, introduces a new phenomenon, the twisted overextensions of Section 4. This is because the highest root of the (BC ) n system differs from the highest root of the B n system. Hence, the dominant magnetic wall will provide a twisted affine root, to which the symmetry wall will attach itself as usual [95Jump To The Next Citation Point].

We illustrate the two possible cases in terms of explicit examples. The first one is the simplest case for which a twisted overextension appears, namely the case of pure four-dimensional gravity coupled to a Maxwell field. This is the bosonic sector of 𝒩 = 2 supergravity in four dimensions, which has the non-split real form 𝔰𝔲(2,1) as its U-duality symmetry. The restricted root system of 𝔰𝔲(2,1 ) is the non-reduced (BC )1-system, and, consequently, as we shall see explicitly, the billiard is governed by the twisted overextension A (2)+ 2.

The second example is that of heterotic supergravity, which exhibits an SO (8,24)βˆ•(SO (8) ×SO (24)) coset symmetry in three dimensions. The U-duality algebra is thus 𝔰𝔬(8,24 ), which is non-split. In this example, however, the restricted root system is B8, which is reduced, and so the billiard is governed by a standard overextension of the maximal split subalgebra 𝔰𝔬(8,9) ⊂ 𝔰𝔬 (8, 24).

7.2.1 (BC )1 and 𝓝 = 2, D = 4 pure supergravity

We consider 𝒩 = 2 supergravity in four dimensions where the bosonic sector consists of gravity coupled to a Maxwell field. It is illuminating to compare the construction of the billiard in the two limiting dimensions, D = 4 and D = 3.

In maximal dimension the metric contains three scale factors, 1 2 β ,β and 3 β, which give rise to three symmetry wall forms,

s (β ) = β2 − β1, s (β) = β3 − β2, s (β) = β3 − β1, (7.9 ) 21 32 31
where only s21 and s32 are dominant. In four dimensions the curvature walls read
1 2 3 c123(β) ≡ c1(β) = 2β , c231(β) ≡ c2(β) = 2β , c312(β) ≡ c3(β ) = 2 β . (7.10 )
Finally we have the electric and magnetic wall forms of the Maxwell field. These are equal because there is no dilaton. Hence, the wall forms are
e (β) = m (β) = β1, e (β) = m (β) = β2, e (β) = m (β) = β3. (7.11 ) 1 1 2 2 3 3
The billiard region ℬ β„³ β is defined by the set of dominant wall forms,
ℬ = {β ∈ β„³ |e (β),s (β),s (β) > 0}. (7.12 ) β„³ β β 1 21 32
The first dominant wall form, e1(β), is twice degenerate because it occurs once as an electric wall form and once as a magnetic wall form. Because of the existence of the curvature wall, c1(β) = 2β1, we see that 2α 1 is also a root.

The same billiard emerges after reduction to three spacetime dimensions, where the algebraic structure is easier to exhibit. As before, we perform the reduction along the first spatial direction. The associated scale factor is then replaced by the Kaluza–Klein dilaton ˆΟ• such that

1 -1-- β = √2-Ο•ˆ. (7.13 )
The remaining scale factors change accordingly,
2 2 1 3 3 1 β = ˆβ − √--Ο•ˆ, β = ˆβ − √--ˆΟ•, (7.14 ) 2 2
and the two symmetry walls become
2 √ -- 3 2 s21(ˆβ,Ο•ˆ) = ˆβ − 2ˆΟ•, ˆs32(βˆ) = ˆβ − ˆβ . (7.15 )
In addition to the dilaton ˆΟ•, there are three axions: one (ˆχ) arising from the dualization of the Kaluza–Klein vector, one (E χˆ) coming from the component A1 of the Maxwell vector potential and one (C ˆχ) coming from dualization of the Maxwell vector potential in 3 dimensions (see, e.g., [35Jump To The Next Citation Point] for a review). There are then a total of four scalars. These parametrize the coset space SU (2, 1)βˆ•S (U (2) × U (1)) [113Jump To The Next Citation Point].

The Einstein–Maxwell Lagrangian in four dimensions yields indeed in three dimensions the Einstein–scalar Lagrangian, where the Lagrangian for the scalar fields is given by

β„’ = ∂ Ο•ˆ∂μˆΟ• + e2e1(Ο•ˆ)( ∂ ˆχE ∂μˆχE + ∂ ˆχC∂ μˆχC ) + e4e1(Ο•ˆ) (∂ ˆχ∂μχˆ) + ⋅⋅ ⋅ (7.16 ) SU(2,1)βˆ•S(U (2)×U(1)) μ μ μ μ
-1-- e1(ˆΟ•) = √2-Ο•ˆ.

Here, the ellipses denotes terms that are not relevant for understanding the billiard structure. The U-duality algebra of 𝒩 = 2 supergravity compactified to three dimensions is therefore

𝔲3 = 𝔰𝔲 (2,1), (7.17 )
which is a non-split real form of the complex Lie algebra 𝔰𝔩(3,β„‚ ). This is in agreement with Table 1 of [113Jump To The Next Citation Point]. The restricted root system of 𝔰𝔲 (2, 1) is of (BC )1-type (see Table 28 in Section 6.8) and has four roots: α1, 2α1, − α1 and − 2α1. One may take α1 to be the simple root, in which case Σ = {α ,2α } + 1 1 and 2α 1 is the highest root. The short root α 1 is degenerate twice while the long root 2α1 is nondegenerate. The Lagrangian (7.16View Equation) coincides with the Lagrangian (7.8View Equation) for 𝔰𝔲(2,1) with the identification
ˆα1 ≡ e1. (7.18 )
We clearly see from the Lagrangian that the simple root ˆα 1 has multiplicity 2 in the restricted root system, since the corresponding wall appears twice. The maximal split subalgebra may be taken to be A1 ≡ 𝔰𝔲(1,1) with root system {αˆ1, − ˆα1}.

Let us now see how one goes from 𝔰𝔲(2,1) described by the scalar Lagrangian to the full algebra, by including the gravitational scale factors. Let us examine in particular how the twist arises. For the standard root system of A1 the highest root is just αˆ1. However, as we have seen, for the (BC )1 root system the highest root is θ(BC) = 2ˆα1 1, with

(θ(BC) |θ(BC )) = 4(ˆα1|ˆα1) = 2. (7.19 ) 1 1
So we see that because of (ˆα1|ˆα1) = 12, the highest root θ(BC )1 already comes with the desired normalization. The affine root is therefore
√ -- ˆα2(ˆΟ•, ˆβ ) = mˆχ2ˆ(βˆ, ˆΟ•) = ˆβ2 − θ(BC)1 = ˆβ2 − 2ˆΟ•, (7.20 )
whose norm is
(ˆα2|ˆα2) = 2. (7.21 )
The scalar product between ˆα1 and ˆα2 is (ˆα1|ˆα2) = − 1 and the Cartan matrix at this stage becomes (i,j = 1,2)
(ˆα |αˆ ) ( 2 − 4 ) Aij[A (22)] = 2--i--j-= , (7.22 ) (ˆαi|ˆαi) − 1 2
which may be identified not with the affine extension of A1 but with the Cartan matrix of the twisted affine Kac–Moody algebra A (2) 2. It is the underlying (BC ) 1 root system that is solely responsible for the appearance of the twist. Because of the fact that θ(BC )1 = 2αˆ1 the two simple roots of the affine extension come with different length and hence the asymmetric Cartan matrix in Equation (7.22View Equation). It remains to include the overextended root
ˆα (βˆ) = ˆs (βˆ) = ˆβ3 − ˆβ2, (7.23 ) 3 32
which has non-vanishing scalar product only with ˆα2, (ˆα2|ˆα3) = − 1, and so its node in the Dynkin diagram is attached to the second node by a single link. The complete Cartan matrix is
( ) 2 − 4 0 A [A(2)+ ] = ( − 1 2 − 1) , (7.24 ) 2 0 − 1 2
which is the Cartan matrix of the Lorentzian extension A (2)+ 2 of A(2) 2 henceforth referred to as the twisted overextension of A1. Its Dynkin diagram is displayed in Figure 41View Image.

The algebra (2)+ A 2 was already analyzed in Section 4, where it was shown that its Weyl group coincides with the Weyl group of the algebra A+1+. Thus, in the BKL-limit the dynamics of the coupled Einstein–Maxwell system in four-dimensions is equivalent to that of pure four-dimensional gravity, although the set of dominant walls are different. Both theories are chaotic.

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Figure 41: The Dynkin diagram of (2)+ A2. Label 1 denotes the simple root ˆα (1) of the restricted root system of 𝔲3 = 𝔰𝔲(2,1). Labels 2 and 3 correspond to the affine and overextended roots, respectively. The arrow points towards the short root which is normalized such that (ˆα1|ˆα1) = 1 2.

7.2.2 Heterotic supergravity and π–˜π–” (8,24 )

Pure 𝒩 = 1 supergravity in D = 10 dimensions has a billiard description in terms of the hyperbolic Kac–Moody algebra ++ DE10 = D 8 [45Jump To The Next Citation Point]. This algebra is the overextension of the U-duality algebra, 𝔲3 = D8 ≡ 𝔰𝔬(8,8), appearing upon compactification to three dimensions. In this case, 𝔰𝔬(8,8) is the split form of the complex Lie algebra D8, so we have 𝔣 = 𝔲3.

By adding one Maxwell field to the theory we modify the billiard to the hyperbolic Kac–Moody algebra BE = B++ 10 8, which is the overextension of the split form 𝔰𝔬(8,9) of B 8 [45Jump To The Next Citation Point]. This is the case relevant for (the bosonic sector of) Type I supergravity in ten dimensions. In both these cases the relevant Kac–Moody algebra is the overextension of a split real form and so falls under the classification given in Section 5.

Let us now consider an interesting example for which the relevant U-duality algebra is non-split. For the heterotic string, the bosonic field content of the corresponding supergravity is given by pure gravity coupled to a dilaton, a 2-form and an E8 × E8 Yang–Mills gauge field. Assuming the gauge field to be in the Cartan subalgebra, this amounts to adding 16 𝒩 = 1 vector multiplets in the bosonic sector, i.e, to adding 16 Maxwell fields to the ten-dimensional theory discussed above. Geometrically, these 16 Maxwell fields correspond to the Kaluza–Klein vectors arising from the compactification on 16 T of the 26-dimensional bosonic left-moving sector of the heterotic string [89].

Consequently, the relevant U-duality algebra is 𝔰𝔬(8,8 + 16) = 𝔰𝔬(8,24 ) which is a non-split real form. But we know that the billiard for the heterotic string is governed by the same Kac–Moody algebra as for the Type I case mentioned above, namely ++ BE10 ≡ 𝔰𝔬(8,9), and not ++ 𝔰𝔬 (8, 24) as one might have expected [45Jump To The Next Citation Point]. The only difference is that the walls associated with the one-forms are degenerate 16 times. We now want to understand this apparent discrepancy using the machinery of non-split real forms exhibited in previous sections. The same discussion applies to the SO (32)-superstring.

In three dimensions the heterotic supergravity Lagrangian is given by a pure three-dimensional Einstein–Hilbert term coupled to a nonlinear sigma model for the coset SO (8,24)βˆ•(SO (8) × SO (24)). This Lagrangian can be understood by analyzing the Iwasawa decomposition of 𝔰 𝔬(8, 24) = Lie[SO (8,24)]. The maximal compact subalgebra is

𝔨 = 𝔰𝔬(8) ⊕ 𝔰𝔬(24). (7.25 ) 3
This subalgebra does not appear in the sigma model since it is divided out in the coset construction (see Equation (7.7View Equation)) and hence we only need to investigate the Borel subalgebra π”ž3 ⊕ 𝔫3 of 𝔰𝔬 (8, 24) in more detail.

As was emphasized in Section 7.1, an important feature of the Iwasawa decomposition is that the full Cartan subalgebra π”₯3 does not appear explicitly but only the maximal noncompact Cartan subalgebra π”ž3, associated with the restricted root system. This is the maximal Abelian subalgebra of 𝔲3 = 𝔰𝔬(8,24 ), whose adjoint action can be diagonalized over the reals. The remaining Cartan generators of π”₯ 3 are compact and so their adjoint actions have imaginary eigenvalues. The general case of 𝔰𝔬 (2q, 2p) was analyzed in detail in Section 6.7 where it was found that if q < p, the restricted root system is of type B2q. For the case at hand we have q = 4 and p = 12 which implies that the restricted root system of 𝔰𝔬(8,24 ) is (modulo multiplicities) Σ π”°π”¬(8,24) = B8.

The root system of B8 is eight-dimensional and hence there are eight Cartan generators that may be simultaneously diagonalized over the real numbers. Therefore the real rank of 𝔰𝔬(8,24) is eight, i.e.,

rankℝ 𝔲3 = dim π”ž3 = 8. (7.26 )
Moreover, it was shown in Section 6.7 that the restricted root system of 𝔰𝔬(2q,2p) has 4q(2q − 1) long roots which are nondegenerate, i.e., with multiplicity one, and 4q long roots with multiplicities 2 (p − q). In the example under consideration this corresponds to seven nondegenerate simple roots α1,⋅⋅⋅ ,α7 and one short simple root α8 with multiplicity 16. The Dynkin diagram for the restricted root system Σ π”°π”¬(8,24) is displayed in Figure 42View Image with the multiplicity indicated in brackets over the short root. It is important to note that the restricted root system Σ𝔰𝔬(8,24) differs from the standard root system of 𝔰𝔬(8,9) precisely because of the multiplicity 16 of the simple root α8.
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Figure 42: The Dynkin diagram representing the restricted root system Σ𝔰𝔬(8,24) of 𝔰𝔬(8,24). Labels 1,⋅⋅ ⋅ ,7 denote the long simple roots that are nondegenerate while the eighth simple root is short and has multiplicity 16.

Because of these properties of 𝔰𝔬(8,24) the Lagrangian for the coset

---SO--(8,-24)---- (7.27 ) SO (8) × SO (24)
takes a form very similar to the Lagrangian for the coset
----SO-(8,9)---. (7.28 ) SO (8) × SO (9)
The algebra constructed from the restricted root system B8 is the maximal split subalgebra
𝔣 = 𝔰𝔬(8,9). (7.29 )
Let us now take a closer look at the Lagrangian in three spacetime dimensions. We parametrize an element of the coset by
⌊ ⌋ [ 8 ] 𝒱(xμ) = Exp ∑ φ(i)(xμ)α∨ Exp ⌈ ∑ χ(γ)(xμ)E ⌉ ∈ ---SO--(8,24)---, (7.30 ) i γ SO (8) × SO (24) i=1 γ∈Δ+
where xμ(μ = 0,1,2) are the coordinates of the external three-dimensional spacetime, α ∨ i are the noncompact Cartan generators and Δ+ denotes the full set of positive roots of 𝔰𝔬(8,24 ).

The Lagrangian constructed from the coset representative in Equation (7.30View Equation) becomes (again, neglecting corrections to the single derivative terms of the form “∂xχ”)

8 7 ∑ (i) μ (i) ∑ αj(φ) (j) μ (j) β„’ 𝒰3βˆ•π’¦ (𝒰3) = ∂μφ (x) ∂ φ (x) + e ∂μχ (x)∂ χ (x) i=1 j=1 ( ∑16 ) ∑ mu∑lt(α) +e α8(φ) ∂μχ (8)(x)∂μχ (8)(x) + eα(φ) ∂μχ(α)(x)∂ μχ(α)(x), (7.31 ) [k] [k] &tidle;+ s =1 [sα] [sα] k=1 α∈ Σ α
where &tidle;Σ+ denotes all non-simple positive roots of Σ, i.e.,
&tidle; + + ¯ Σ = Σ βˆ•B (7.32 )
¯ B = {α1,⋅⋅⋅ ,α8}. (7.33 )
This Lagrangian is equivalent to the Lagrangian for SO (8,9)βˆ•(SO (8) × SO (9)) except for the existence of the non-trivial root multiplicities.

The billiard for this theory can now be computed with the same methods that were treated in detail in Section 5.3.3. In the BKL-limit, the simple roots α1,⋅⋅⋅ ,α8 become the non-gravitational dominant wall forms. In addition to this we get one magnetic and one gravitational dominant wall form:

α = β1 − θ (φ ), 0 (7.34 ) α−1 = β2 − β1,
where θ(φ) is the highest root of 𝔰𝔬(8,9):
θ = α1 + 2α2 + ⋅⋅⋅ + 2α7 + α8. (7.35 )
The affine root α0 attaches with a single link to the second simple root α2 in the Dynkin diagram of B 8. Similarly the overextended root α −1 attaches to α 0 with a single link so that the resulting Dynkin diagram corresponds to BE10 (see Figure 43View Image). It is important to note that the underlying root system is still an overextension of the restricted root system and hence the multiplicity of the simple short root α8 remains equal to 16. Of course, this does not affect the dynamics in the BKL-limit because the multiplicity of α8 simply translates to having multiple electric walls on top of each other and this does not alter the billiard motion.

This analysis again showed explicitly how it is always the split symmetry that controls the chaotic behavior in the BKL-limit. It is important to point out that when going beyond the strict BKL-limit, one expects more and more roots of the algebra to play a role. Then it is no longer sufficient to study only the maximal split subalgebra 𝔰𝔬(8,9)++ but instead the symmetry of the theory is believed to contain the full algebra 𝔰𝔬(8,24)++. In the spirit of [47Jump To The Next Citation Point] one may then conjecture that the dynamics of the heterotic supergravity should be equivalent to a null geodesic on the coset space ++ ++ SO (8, 24) βˆ•π’¦ (SO (8,24 ) ) [42Jump To The Next Citation Point].

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Figure 43: The Dynkin diagram representing the overextension B++8 of the restricted root system Σ = B8 of 𝔰𝔬(8,24). Labels − 1,0,1,⋅⋅⋅ ,7 denote the long simple roots that are nondegenerate while the eighth simple root is short and has multiplicity 16.

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