Fermionic Kac-Moody billiards and supergravity

We study the 'fermionic billiards', i.e. the chaotic dynamics of the gravitino, that arise in the near-spacelike-singularity limit of eleven-dimensional supergravity and of its dimensional truncations (notably four-dimensional simple supergravity). By exploiting the gravity-coset correspondence, we show that the billiard dynamics of the gravitino is described by a 'spin extension' of the Weyl group of the hyperbolic Kac-Moody algebra E10. This spin extension is a discrete subgroup of (a spin covering of) the maximal compact subgroup K(E10) of E10 that is generated by ten (simple-root-related) idempotent elements of order 8. The 'super-billiard' that combines the bosonic and fermionic billiards is found to have a remarkably simple structure, which exhibits a striking analogy with a polarized photon propagating in the ten-dimensional Lorentzian Weyl chamber of E10.