### 7.2 “Split symmetry controls chaos”

The main point of this section is to illustrate and explain the statement “split symmetry controls chaos” [95]. 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:
• The restricted root system of is of reduced type, in which case it is one of the standard root systems for the Lie algebras or .
• The restricted root system, , of is of non-reduced type, in which case it is of -type.

In the first case, the billiard is governed by the overextended algebra , where is the “maximal split subalgebra” of . Indeed, the coupling to gravity of the coset Lagrangian of Equation (7.8) will introduce, besides the simple roots of (electric walls) the affine root of (dominant magnetic wall) and the overextended root (symmetry wall), just as in the split case (but for instead of ). 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 system differs from the highest root of the system. Hence, the dominant magnetic wall will provide a twisted affine root, to which the symmetry wall will attach itself as usual [95].

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 supergravity in four dimensions, which has the non-split real form as its U-duality symmetry. The restricted root system of is the non-reduced -system, and, consequently, as we shall see explicitly, the billiard is governed by the twisted overextension .

The second example is that of heterotic supergravity, which exhibits an coset symmetry in three dimensions. The U-duality algebra is thus , which is non-split. In this example, however, the restricted root system is , which is reduced, and so the billiard is governed by a standard overextension of the maximal split subalgebra .

#### 7.2.1 and , pure supergravity

We consider 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, and .

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

where only and are dominant. In four dimensions the curvature walls read
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
The billiard region is defined by the set of dominant wall forms,
The first dominant wall form, , 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, , we see that 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 such that

The remaining scale factors change accordingly,
and the two symmetry walls become
In addition to the dilaton , there are three axions: one () arising from the dualization of the Kaluza–Klein vector, one () coming from the component of the Maxwell vector potential and one () coming from dualization of the Maxwell vector potential in 3 dimensions (see, e.g., [35] for a review). There are then a total of four scalars. These parametrize the coset space  [113].

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

with

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

which is a non-split real form of the complex Lie algebra . This is in agreement with Table 1 of [113]. The restricted root system of is of -type (see Table 28 in Section 6.8) and has four roots: , , and . One may take to be the simple root, in which case and is the highest root. The short root is degenerate twice while the long root is nondegenerate. The Lagrangian (7.16) coincides with the Lagrangian (7.8) for with the identification
We clearly see from the Lagrangian that the simple root has multiplicity in the restricted root system, since the corresponding wall appears twice. The maximal split subalgebra may be taken to be with root system .

Let us now see how one goes from 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 the highest root is just . However, as we have seen, for the root system the highest root is , with

So we see that because of , the highest root already comes with the desired normalization. The affine root is therefore
whose norm is
The scalar product between and is and the Cartan matrix at this stage becomes
which may be identified not with the affine extension of but with the Cartan matrix of the twisted affine Kac–Moody algebra . It is the underlying root system that is solely responsible for the appearance of the twist. Because of the fact that the two simple roots of the affine extension come with different length and hence the asymmetric Cartan matrix in Equation (7.22). It remains to include the overextended root
which has non-vanishing scalar product only with , , and so its node in the Dynkin diagram is attached to the second node by a single link. The complete Cartan matrix is
which is the Cartan matrix of the Lorentzian extension of henceforth referred to as the twisted overextension of . Its Dynkin diagram is displayed in Figure 41.

The algebra was already analyzed in Section 4, where it was shown that its Weyl group coincides with the Weyl group of the algebra . 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.

#### 7.2.2 Heterotic supergravity and

Pure supergravity in dimensions has a billiard description in terms of the hyperbolic Kac–Moody algebra [45]. This algebra is the overextension of the U-duality algebra, , appearing upon compactification to three dimensions. In this case, is the split form of the complex Lie algebra , so we have .

By adding one Maxwell field to the theory we modify the billiard to the hyperbolic Kac–Moody algebra , which is the overextension of the split form of  [45]. 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 Yang–Mills gauge field. Assuming the gauge field to be in the Cartan subalgebra, this amounts to adding 16 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 of the 26-dimensional bosonic left-moving sector of the heterotic string [89].

Consequently, the relevant U-duality algebra is 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 , and not as one might have expected [45]. 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 -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 . This Lagrangian can be understood by analyzing the Iwasawa decomposition of . The maximal compact subalgebra is

This subalgebra does not appear in the sigma model since it is divided out in the coset construction (see Equation (7.7)) and hence we only need to investigate the Borel subalgebra of in more detail.

As was emphasized in Section 7.1, an important feature of the Iwasawa decomposition is that the full Cartan subalgebra does not appear explicitly but only the maximal noncompact Cartan subalgebra , associated with the restricted root system. This is the maximal Abelian subalgebra of , whose adjoint action can be diagonalized over the reals. The remaining Cartan generators of are compact and so their adjoint actions have imaginary eigenvalues. The general case of was analyzed in detail in Section 6.7 where it was found that if , the restricted root system is of type . For the case at hand we have and which implies that the restricted root system of is (modulo multiplicities) .

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

Moreover, it was shown in Section 6.7 that the restricted root system of has long roots which are nondegenerate, i.e., with multiplicity one, and long roots with multiplicities . In the example under consideration this corresponds to seven nondegenerate simple roots and one short simple root with multiplicity . The Dynkin diagram for the restricted root system is displayed in Figure 42 with the multiplicity indicated in brackets over the short root. It is important to note that the restricted root system differs from the standard root system of precisely because of the multiplicity 16 of the simple root .

Because of these properties of the Lagrangian for the coset

takes a form very similar to the Lagrangian for the coset
The algebra constructed from the restricted root system is the maximal split subalgebra
Let us now take a closer look at the Lagrangian in three spacetime dimensions. We parametrize an element of the coset by
where are the coordinates of the external three-dimensional spacetime, are the noncompact Cartan generators and denotes the full set of positive roots of .

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

where denotes all non-simple positive roots of , i.e.,
with
This Lagrangian is equivalent to the Lagrangian for 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 become the non-gravitational dominant wall forms. In addition to this we get one magnetic and one gravitational dominant wall form:

where is the highest root of :
The affine root attaches with a single link to the second simple root in the Dynkin diagram of . Similarly the overextended root attaches to with a single link so that the resulting Dynkin diagram corresponds to (see Figure 43). 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 remains equal to 16. Of course, this does not affect the dynamics in the BKL-limit because the multiplicity of 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 but instead the symmetry of the theory is believed to contain the full algebra . In the spirit of [47] one may then conjecture that the dynamics of the heterotic supergravity should be equivalent to a null geodesic on the coset space  [42].