### 10.4 Cosmological solutions with magnetic flux

We will now briefly sketch how one can also obtain the -brane solutions from geometric
configurations and regular subalgebras of . In order to do this we consider “magnetic” subalgebras of
, constructed only from simple root generators at level two in the level decomposition of . To the
best of our knowldege, there is no theory of geometric configurations developed for the case of having
6 points on each line, which would be needed here. However, we may nevertheless continue
to investigate the simplest example of such a configuration, namely , displayed in
Figure 53.
The algebra dual to this configuration is an -subalgebra of with the following generators:

Although the embedding of this algebra is different from the electric cases considered previously, the sigma
model solution is still associated to an coset space and therefore the solutions for
and are the same as before. Because of the embedding, however, the sigma model translates to a
different type of supergravity solution, namely a spacelike five-brane whose world volume is extended in the
directions . The metric is given by
This solution coincides with the -brane found by Strominger and Gutperle
in [90].
Note that the correct power of for the five-brane arises here entirely due to the embedding of
into through Equation (10.65).
Because of the existence of electric-magnetic duality on the supergravity side, it is suggestive
to expect the existence of a duality between the two types of configurations and
, of which we have here seen the simplest realisation for the configurations and
.