The fact that the billiard structure is preserved under reduction turns out to be very useful for understanding the emergence of “overextended” algebras in the BKL-limit. By computing the billiard in three spacetime dimensions instead of in maximal dimension, the relation to U-duality groups becomes particularly transparent and the computation of the billiard follows a similar pattern for all cases. We will see that if is the algebra representing the internal symmetry of the non-gravitational degrees of freedom in three dimensions, then the billiard is controlled by the Weyl group of the overextended algebra . The analysis is somewhat more involved when is non-split, and we postpone a discussion of this until Section 7.
It was shown in  that the structure of the billiard for any given theory is completely unaffected by dimensional reduction on a torus. In this section we illustrate this by an explicit example rather than in full generality. We consider the case of reduction of eleven-dimensional supergravity on a circle.
The compactification ansatz in the conventions of [35, 41] is16. We will refer to the new lower-dimensional fields and as the dilaton and the Kaluza–Klein (KK) vector, respectively. Quite generally, hatted fields are low-dimensional fields. The ten-dimensional Lagrangian becomes
Examining the new form of the metric reveals that the role of the scale factor , associated to the compactified dimension, is now instead played by the ten-dimensional dilaton, . Explicitly we have
We have thus shown that the structure is sufficiently rigid to withstand compactification on a circle with the new simple roots explicitly given by
For non-toroidal reductions the above analysis is drastically modified [166, 165]. The topology of the internal manifold affects the dominant wall system, and hence the algebraic structure in the lower-dimensional theory is modified. In many cases, the billiard of the effective compactified theory is described by a (non-hyperbolic) regular Lorentzian subalgebra of the original hyperbolic Kac–Moody algebra .
The walls are also invariant under dualization of a -form into a -form; this simply exchanges magnetic and electric walls.
We will now exploit the invariance of the billiard under dimensional reduction, by considering theories that – when compactified on a torus to three dimensions – exhibit “hidden” internal global symmetries . By this we mean that the three-dimensional reduced theory is described, after dualization of all vectors to scalars, by the sum of the Einstein–Hilbert Lagrangian coupled to the Lagrangian for the nonlinear sigma model . Here, is the maximal compact subgroup defining the “local symmetries”. In order to understand the connection between the U-duality group and the Kac–Moody algebras appearing in the BKL-limit, we must first discuss some important features of the Lie algebra .
Let be a split real form, meaning that it can be defined in terms of the Chevalley–Serre presentation of the complexified Lie algebra by simply restricting all linear combinations of generators to the real numbers . Let be the Cartan subalgebra of appearing in the Chevalley–Serre presentation, spanned by the generators . It is maximally noncompact (see Section 6). An Iwasawa decomposition of is a direct sum of vector spaces of the following form,
The corresponding Iwasawa decomposition at the group level enables one to write uniquely any group element as a product of an element of the maximally compact subgroup times an element in the subgroup whose Lie algebra is times an element in the subgroup whose Lie algebra is . An arbitrary element of the coset is defined as the set of equivalence classes of elements of the group modulo elements in the maximally compact subgroup. Using the Iwasawa decomposition, one can go to the “Borel gauge”, where the elements in the coset are obtained by exponentiating only generators belonging to the Borel subalgebra,dilatons and the fields as axions. There is one axion field for each positive root and one dilaton field for each Cartan generator .
The Lagrangian (5.45) coupled to the pure three-dimensional Einstein–Hilbert term is the key to understanding the appearance of the Lorentzian Coxeter group in the BKL-limit.
To make the point explicit, we will again limit our analysis to the example of eleven-dimensional supergravity. Our starting point is then the Lagrangian for this theory compactified on an 8-torus, , to spacetime dimensions (after all form fields have been dualized into scalars),[33, 134, 35]. The 8 dilatons and the 120 axions are coordinates on the coset space17. Furthermore, the are linear forms on the elements of the Cartan subalgebra and they correspond to the positive roots of 18. As before, we do not write explicitly the corrections to the curvatures that appear in the compactification process. The entire set of positive roots can be obtained by taking linear combinations of the seven simple roots of (we omit the “hatted” notation on the roots since there is no longer any risk of confusion), , except for the additional factor of needed to compensate for the fact that the aforementioned reference has an additional factor of in the Killing form. Hence, using the Euclidean metric one may check that the roots defined above indeed reproduce the Cartan matrix of .
Next, we want to determine the billiard structure for this Lagrangian. As was briefly mentioned before, in the reduction from eleven to three dimensions all the non-gravity walls associated to the eleven-dimensional 3-form have been transformed, in the same spirit as for the example given above, into electric and magnetic walls of the axionic scalars . Since the terms involving the electric fields possess no spatial indices, the corresponding wall forms do not contain any of the remaining scale factors , and are simply linear forms on the dilatons only. In fact the dominant electric wall forms are just the simple roots of ,
The structure of the corresponding Lorentzian Kac–Moody algebra is now easy to establish in view of our discussion of overextensions in Section 4.9. The relevant walls listed above are the simple roots of the (untwisted) overextension . Indeed, the relevant electric roots are the simple roots of , the magnetic root of Equation (5.50) provides the affine extension, while the gravitational root of Equation (5.52) is the overextended root.
What we have found here in the case of eleven-dimensional supergravity also holds for the other theories with U-duality algebra in 3 dimensions when is a split real form. The Coxeter group and the corresponding Kac–Moody algebra are given by the untwisted overextension . This overextension emerges as follows :
Thus we see that the appearance of overextended algebras in the BKL-analysis of supergravity theories is a generic phenomenon closely linked to hidden symmetries.
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