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 elevendimensional supergravity and of its dimensional truncations (notably fourdimensional 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 E10 Weyl group, namely as a product of 90 ◦ ‘vectorspinor rotations’ along certain simple-root-related generators of the maximal compact subalgebra K(E10) of the hyperbolic Kac–Moody algebra E10. 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.