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 .
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 .
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,
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 for a review). There are then a total of four scalars. These parametrize the coset space .
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
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. 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
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 , withtwisted 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 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.
Pure supergravity in dimensions has a billiard description in terms of the hyperbolic Kac–Moody algebra . 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 . 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 .
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 . 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
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.,
Because of these properties of the Lagrangian for the coset
The Lagrangian constructed from the coset representative in Equation (7.30) becomes (again, neglecting corrections to the single derivative terms of the form “”)
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:
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  one may then conjecture that the dynamics of the heterotic supergravity should be equivalent to a null geodesic on the coset space .
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