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
.
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
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
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
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
[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
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 [47
] one may then conjecture that the
dynamics of the heterotic supergravity should be equivalent to a null geodesic on the coset space
[42
].
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