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.

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 .

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 setWe 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.

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.

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^{39}.
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.

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