In this last main section we shall show that the low level equivalence between the sigma model and eleven-dimensional supergravity can be put to practical use for finding exact solutions of eleven-dimensional supergravity. This is a satisfactory result because even in the cosmological context of homogeneous fields , that depend only on time (“Bianchi I cosmological models” ), the equations of motion of eleven-dimensional supergravity remain notoriously complicated, while the corresponding sigma model is, at least formally, integrable.
We will remain in the strictly cosmological sector where it is assumed that all spatial gradients can be neglected so that all fields depend only on time. Moreover, we impose diagonality of the spatial metric. These conditions must of course be compatible with the equations of motion; if the conditions are imposed initially, they should be preserved by the time evolution.
A large class of solutions to eleven-dimensional supergravity preserving these conditions were found in . These solutions have zero magnetic field but have a restricted number of electric field components turned on. Surprisingly, it was found that such solutions have an elegant interpretation in terms of so called geometric configurations, denoted , of points and lines (with ) drawn on a plane with certain pre-determined rules. That is, for each geometric configuration (whose definition is recalled below) one can associate a diagonal solution with some non-zero electric field components , determined by the configuration. In this section we re-examine this result from the point of view of the sigma model based on .
We show, following , that each configuration encodes information about a (regular) subalgebra of , and the supergravity solution associated to the configuration can be obtained by restricting the -sigma model to the subgroup whose Lie algebra is . Therefore, we will here make use of both the level truncation and the subgroup truncation simultaneously; first by truncating to a certain level and then by restricting to the relevant -algebra generated by a subset of the representations at this level. Large parts of this section are based on .
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