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11 Conclusions

In this review, we have investigated the remarkable structures that emerge when studying gravitational theories in the BKL-limit, i.e., close to a spacelike singularity. Although it has been known for a long time that in this limit the dynamics can be described in terms of billiard motion in hyperbolic space, it is only recently that the connection between the billiards and Coxeter groups have been uncovered. Furthermore, the relevant Coxeter groups turn out to be the Weyl groups of the Lorentzian Kac–Moody algebra obtained by double extension (sometimes twisted) of the U-duality algebra appearing upon dimensional reduction to three dimensions.

These results, which in our opinion are solid and here to stay, necessitate some mathematical background which is not part of the average physicist’s working knowledge. For this reason, we have also devoted a few sections to the development of the necessary mathematical concepts.

We have then embarked on the exploration of more speculative territory. A natural question that arises is whether or not the emergence of Weyl groups of Kac–Moody algebras in the BKL-limit has a profound meaning independently of the BKL-limit (which would serve only as a “revelator”) and could indicate that the gravitational theories under investigation – possibly supplemented by additional degrees of freedom – possess these infinite Kac–Moody algebras as “hidden symmetries” (in any regime). The existence of these infinite-dimensional symmetries was also advocated in the pioneering work [113] and more recently [16715615710210374104158168] from a somewhat different point of view. It is also argued in those references that even bigger symmetries (E11 that contains E10, or Borcherds subalgebras) might actually be relevant. In order to make the conjectured E10-symmetry manifest (which is perhaps itself part of a bigger symmetry), we have investigated a nonlinear sigma model for the coset space ℰ ∕𝒦 (ℰ ) 10 10 using the level decomposition techniques introduced in [47]. Although very suggestive and partially successful, this approach exhibits limitations which, in spite of many efforts, have not yet been overcome. It is likely that new ideas are needed, or that the implementation of the symmetry must be made in a more subtle fashion, where duality will perhaps play a more central role.

Independently of the way they are actually implemented, it appears that infinite-dimensional Kac–Moody algebras (e.g, E10 or, perhaps, E11) do encode important features of gravitational theories, and the idea that they constitute essential elements of the final formulation will surely play an important role in future developments.

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