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 [41] 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] is

where , i.e., the compactification is performed along the first spatial directionExamining 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

The nine remaining eleven-dimensional scale factors, , may in turn be written in terms of the new scale factors, , associated to the ten-dimensional metric, , and the dilaton in the following way: We are interested in finding the dominant wall forms in terms of the new scale factors and . It is clear that we will have eight ten-dimensional symmetry walls, which correspond to the eight simple roots of . Using Equation (5.35) and Equation (5.36) one may also check that the symmetry wall , that was associated with the compactified direction, gives rise to an electric wall of the Kaluza–Klein vector, The metric in the dual space gets modified in a natural way, i.e., the dilaton contributes with a flat spatial direction. Using this metric it is clear that has non-vanishing scalar product only with the second symmetry wall , , and it follows that the electric wall of the Kaluza–Klein vector plays the role of the first simple root of , . The final wall form that completes the set will correspond to the exceptional node labeled “10” in Figure 25 and is now given by one of the electric walls of the NS-NS 2-form , namely It is then easy to verify that this wall form has non-vanishing scalar product only with the third simple root , , as desired.We have thus shown that the structure is sufficiently rigid to withstand compactification on a circle with the new simple roots explicitly given by

This result is in fact true also for the general case of compactification on tori, . When reaching the limiting case of three dimensions, all the non-gravity wall forms correspond to the electric and magnetic walls of the axionic scalars. We will discuss this case in detail below.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 [98].

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,

where is the “maximal compact subalgebra” of , and is the nilpotent subalgebra spanned by the positive root generators .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,

In that gauge we have where and are (sets of) coordinates on the coset space . A Lagrangian based on this coset will then take the generic form (see Section 9) where denotes coordinates in spacetime and the “ellipses” denote correction terms that are of no relevance for our present purposes. We refer to the fields collectively as 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),

The second two terms in this Lagrangian correspond to the coset model , where denotes the group obtained by exponentiation of the split form of the complex Lie algebra and is the maximal compact subgroup of [33, 134, 35]. The 8 dilatons and the 120 axions are coordinates on the coset spaceNext, 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 magnetic wall forms naturally come with one factor of since the magnetic field strength carries one spatial index. The dominant magnetic wall form is then given by where denotes the highest root of which takes the following form in terms of the simple roots, Since we are in three dimensions there is no curvature wall and hence the only wall associated to the Einstein–Hilbert term is the symmetry wall coming from the three-dimensional metric (). We have thus found all the dominant wall forms in terms of the lower-dimensional variables.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 [41]:

- The dominant electric wall forms for the supergravity theory in question are in one-to-one correspondence with the simple roots of the associated U-duality algebra .
- Adding the dominant magnetic wall form corresponds to an affine extension of .
- Finally, completing the set of dominant wall forms with the symmetry wall , which is the only gravitational wall form existing in three dimensions, is equivalent to an overextension of .

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.

http://www.livingreviews.org/lrr-2008-1 |
This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 2.0 Germany License. Problems/comments to |