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
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 [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,
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,
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),
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 [41
]:
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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