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10 Cosmological Solutions from E10

In this last main section we shall show that the low level equivalence between the ℰ10∕ 𝒦(ℰ10) 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 G αβ(t), Fαβγδ(t) that depend only on time (“Bianchi I cosmological models” [61Jump To The Next Citation Point]), 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 [61Jump To The Next Citation Point]. 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 (nm, g3), of n points and g lines (with n ≤ 10) 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 Ftijk, determined by the configuration. In this section we re-examine this result from the point of view of the sigma model based on ℰ10∕𝒦 (ℰ10).

We show, following [96Jump To The Next Citation Point], that each configuration (nm,g3) encodes information about a (regular) subalgebra ¯𝔤 of E10, and the supergravity solution associated to the configuration (nmg3 ) can be obtained by restricting the ℰ10-sigma model to the subgroup ¯𝒢 whose Lie algebra is ¯𝔤. 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 ¯𝔤-algebra generated by a subset of the representations at this level. Large parts of this section are based on [96Jump To The Next Citation Point].

 10.1 Bianchi I models and eleven-dimensional supergravity
  10.1.1 Diagonal metrics and geometric configurations
 10.2 Geometric configurations and regular subalgebras of E10
  10.2.1 General considerations
  10.2.2 Incidence diagrams and Dynkin diagrams
 10.3 Cosmological solutions with electric flux
  10.3.1 General discussion
  10.3.2 The solution
  10.3.3 Intersecting spacelike branes from geometric configurations
  10.3.4 Intersection rules for spacelike branes
 10.4 Cosmological solutions with magnetic flux
 10.5 The Petersen algebra and the Desargues configuration
 10.6 Further comments

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