10.3 Cosmological solutions with electric flux
Let us now make use of these considerations to construct some explicit supergravity solutions. We begin
by analyzing the simplest configuration
, of three points and one line. It is displayed in Figure 50.
This case is the only possible configuration for
.
This example also exhibits some subtleties associated with the Hamiltonian constraint and the ensuing
need to extend
when the algebra dual to the geometric configuration is finite-dimensional. We will come
back to this issue below.
10.3.1 General discussion
In light of our discussion, considering the geometric configuration
is equivalent to turning on only
the component
of the 3-form that parametrizes the generator
in the coset representative
. Moreover, in order to have the full coset description, we must also turn on the
diagonal metric components corresponding to the Cartan generator
. The algebra has thus
basis
with
where the form of
followed directly from the general commutator between
and
in
Section 8. The Cartan matrix is just
and is nondegenerate. It defines an
regular subalgebra. The Chevalley–Serre relations, which are guaranteed to hold according to
the general argument, are easily verified. The configuration
is thus dual to
,
This
algebra is simply the
-algebra associated with the simple root
. Because the
Killing form of
restricted to the Cartan subalgebra
is positive definite, one cannot find a
solution of the Hamiltonian constraint if one turns on only the fields corresponding to
. One needs to
enlarge
(at least) by a one-dimensional subalgebra
of
that is timelike. As will be discussed
further below, we take for
the Cartan element
, which
ensures isotropy in the directions not supporting the electric field. Thus, the appropriate regular subalgebra
of
in this case is
.
The need to enlarge the algebra
was discussed in the paper [127
] where a group theoretical
interpretation of some cosmological solutions of eleven-dimensional supergravity was given. In that paper, it
was also observed that
can be viewed as the Cartan subalgebra of the (non-regularly embedded)
subalgebra
associated with an imaginary root at level 21, but since the corresponding field is not
excited, the relevant subalgebra is really
.
10.3.2 The solution
In order to make the above discussion a little less abstract we now show how to obtain the relevant
supergravity solution by solving the
-sigma model equations of motion and then translating these,
using the dictionary from Section 9, to supergravity solutions. For this particular example the analysis was
done in [127
].
In order to better understand the role of the timelike generator
we begin the
analysis by omitting it. The truncation then amounts to considering the coset representative
The projection
onto the coset becomes
where the exponent is the linear form
representing the exceptional simple root
of
.
More precisely, it is the linear form
acting on the Cartan generator
, as follows:
The Lagrangian becomes
For convenience we have chosen the gauge
of the free parameter in the
-Lagrangian
(see Section 9). Recall that for the level one fields we have
, which is why only the
partial derivative of
appears in the Lagrangian.
The reason why this simple looking model contains information about eleven-dimensional supergravity is
that the
subalgebra represented by
is embedded in
through the level 1-generator
, and hence this Lagrangian corresponds to a consistent subgroup truncation of the
- sigma
model.
Let us now study the dynamics of the Lagrangian in Equation (10.19). The equations of motion for
are
where
is a constant. The equations for the
field
may then be written as
Integrating once yields
where
plays the role of the energy for the dynamics of
. This equation can be solved exactly with
the result [127
]
We must also take into account the Hamiltonian constraint
arising from the variation of
in the
-sigma model. The Hamiltonian becomes
It is therefore impossible to satisfy the Hamiltonian constraint unless
. This is the problem which
was discussed above, and the reason why we need to enlarge the choice of coset representative to include
the timelike generator
. We choose
such that it commutes with
and
,
and such that isotropy in the directions not supported by the electric field is ensured. Most
importantly, in order to solve the problem of the Hamiltonian constraint,
must be timelike,
where
is the scalar product in the Cartan subalgebra of
. The subalgebra to which we truncate
the sigma model is thus given by
and the corresponding coset representative is
The Lagrangian now splits into two disconnected parts, corresponding to the direct product
,
The solution for
is therefore simply linear in time,
The new Hamiltonian now gets a contribution also from the Cartan generator
,
This contribution depends on the norm of
and since
, it is possible to satisfy the Hamiltonian
constraint, provided that we set
We have now found a consistent truncation of the
-invariant sigma model which exhibits
-invariance. We want to translate the solution to this model, Equation (10.23), to a
solution of eleven-dimensional supergravity. The embedding of
in Equation (10.14) induces
a natural “Freund–Rubin” type (
) split of the coordinates in the physical metric, where the
3-form is supported in the three spatial directions
. We must also choose an embedding of the
timelike generator
. In order to ensure isotropy in the directions
, where the electric field
has no support, it is natural to let
be extended only in the “transverse” directions and we take [127
]
which has norm
To find the metric solution corresponding to our sigma model, we first analyze the coset representative at
,
In order to make use of the dictionary from Section 9.3.6 it is necessary to rewrite this in a way
more suitable for comparison, i.e., to express the Cartan generators
and
in terms of the
-generators
. We thus introduce parameters
and
representing, respectively,
and
in the
-basis. The level zero coset representative may then be written as
where in the second line we have split the sum in order to highlight the underlying spacetime structure, i.e.,
to emphasize that
has no non-vanishing components in the directions
. Comparing this to
Equation (10.14) and Equation (10.34) gives the diagonal components of
and
,
Now, the dictionary from Section 9 identifies the physical spatial metric as follows:
By observation of Equation (10.38) we find the components of the metric to be
This result shows clearly how the embedding of
and
into
is reflected in the coordinate split of
the metric. The gauge fixing
(or
) gives the
-component of the metric,
Next we consider the generator
. The dictionary tells us that the field strength of the 3-form
in eleven-dimensional supergravity at some fixed spatial point
should be identified as
It is possible to eliminate the
in favor of the Cartan field
using the first integral of its
equations of motion, Equation (10.20),
In this way we may write the field strength in terms of
and the solution for
,
Finally, we write down the solution for the spacetime metric explicitly:
where
This solution coincides with the cosmological solution first found in [61
] for the geometric configuration
, and it is intriguing that it can be exactly reproduced from a manifestly
-invariant
action, a priori unrelated to any physical model.
Note that in modern terminology, this solution is an
-brane solution (see, e.g., [143] for a review)
since it can be interpreted as a spacelike (i.e., time-dependent) version of the
-brane solution. From
this point of view the world volume of the
-brane is extended in the directions
and
,
and so is Euclidean.
In the BKL-limit this solution describes two asymptotic Kasner regimes, at
and at
.
These are separated by a collision against an electric wall, corresponding to the blow-up of the electric field
at
. In the billiard picture the dynamics in the BKL-limit is thus
given by free-flight motion interrupted by one geometric reflection against the electric wall,
which is the exceptional simple root of
. This indicates that in the strict BKL-limit, electric walls and
-branes are actually equivalent.
10.3.3 Intersecting spacelike branes from geometric configurations
Let us now examine a slightly more complicated example. We consider the configuration
, shown in
Figure 51. This configuration has four lines and six points. As such the associated supergravity model
describes a cosmological solution with four components of the electric field turned on, or, equivalently, it
describes a set of four intersecting
-branes [96
].
From the configuration we read off the Chevalley–Serre generators associated to the simple roots of the
dual algebra:
The first thing to note is that all generators have one index in common since in the graph any two lines
share one node. This implies that the four lines in
define four commuting
subalgebras,
One can make sure that the Chevalley–Serre relations are indeed fulfilled for this embedding. For instance,
the Cartan element
(no summation on the fixed, distinct indices
) reads
Hence, the commutator
vanishes whenever
has only one
-index,
Furthermore, multiple commutators of the step operators are immediately killed at level
whenever they
have one index or more in common, e.g.,
To fulfill the Hamiltonian constraint, one must extend the algebra by taking a direct sum with
,
. Accordingly, the final algebra is
.
Because there is no magnetic field, the momentum constraint and Gauss’ law are identically
satisfied.
By investigating the sigma model solution corresponding to the algebra
, augmented with the
timelike generator
,
we find a supergravity solution which generalizes the one found in [61
]. The solution describes a set of
four intersecting
-branes, with a five-dimensional transverse spacetime in the directions
.
Let us write down also this solution explicitly. The full set of generators for
is
The coset element for this configuration then becomes
We must further choose the timelike Cartan generator,
, appropriately. Examination of
Equation (10.54) reveals that the four electric fields are supported only in the spatial directions
so, again, in order to ensure isotropy in the directions transverse to the
-branes, we choose
the timelike Cartan generator as follows:
which implies
The Lagrangian for this system becomes
where
and
represent the
-invariant Lagrangians corresponding to the
four
-algebras. The solutions for
and
are separately identical
to the ones for
and
, respectively, in Section 10.3.2. From the embedding into
, provided in Equation (10.54), we may read off the solution for the spacetime metric,
As announced, this describes four intersecting
-branes with a
-dimensional transverse
spacetime. For example the brane that couples to the field associated with the first Cartan generator is
extended in the directions
. By restricting to the case
the metric
simplifies to
which coincides with the cosmological solution found in [61
] for the configuration
. We can
therefore conclude that the algebraic interpretation of the geometric configurations found in this paper
generalizes the solutions given in the aforementioned reference.
In a more general setting where we excite more roots of
, the solutions of course become more
complex. However, as long as we consider commuting subalgebras there will naturally be no
coupling in the Lagrangian between fields parametrizing different subalgebras. This implies that
if we excite a direct sum of
-algebras the total Lagrangian will split according to
where
is of the same form as Equation (10.19), and
is the Lagrangian for the timelike Cartan
element, needed in order to satisfy the Hamiltonian constraint. It follows that the associated solutions are
Furthermore, the resulting structure of the metric depends on the embedding of the
-algebras into
, i.e., which level 1-generators we choose to realize the step-operators and hence which Cartan
elements that are associated to the
’s. Each excited
-subalgebra will turn on an electric 3-form
that couples to an
-brane and hence the solution for the metric will describe a set of
intersecting
-branes.
As an additional nice example, we mention here the configuration
, also known as the Fano
plane, which consists of 7 lines and 7 points (see Figure 52). This configuration is well known for its relation
to the octonionic multiplication table [8]. For our purposes, it is interesting because none of the lines in the
configuration are parallell. Thus, the algebra dual to the Fano plane is a direct sum of seven
-algebras
and the supergravity solution derived from the sigma model describes a set of seven intersecting
-branes.
10.3.4 Intersection rules for spacelike branes
For multiple brane solutions, there are rules for how these branes may intersect in order to
describe allowed BPS-solutions [6]. These intersection rules also apply to spacelike branes [144]
and hence they apply to the solutions considered here. In this section we will show that the
intersection rules for multiple
-brane solutions are encoded in the associated geometric
configurations [96
].
For two spacelike
-branes,
and
, in
-theory the rules are
So, for example, if we have two
-branes the result is
which means that they are allowed to intersect on a 0-brane. Note that since we are dealing with spacelike
branes, a zero-brane is extended in one spatial direction, so the two
-branes may therefore intersect
in one spatial direction only. We see from Equation (10.59) that these rules are indeed fulfilled for the
configuration
.
In [72] it was found in the context of
-algebras that the intersection rules for
extremal branes are encoded in orthogonality conditions between the various roots from
which the branes arise. This is equivalent to saying that the subalgebras that we excite
are commuting, and hence the same result applies to
-algebras in the cosmological
context.
From this point of view, the intersection rules can also be read off from the geometric configurations
in the sense that the configurations encode information about whether or not the algebras
commute.
The next case of interest is the Fano plane,
. As mentioned above, this configuration
corresponds to the direct sum of 7 commuting
algebras and so the gravitational solution describes a
set of 7 intersecting
-branes. The intersection rules are guaranteed to be satisfied for the same reason
as before.