Hyperbolic billiards of pure D=4 supergravities -: art. no. 047

We compute the billiards that emerge in the Belinskii-Khalatnikov-Lifshitz (BKL) limit for all pure supergravities in D = 4 spacetime dimensions, as well as for D = 4, N = 4 supergravities coupled to k (N = 4) Maxwell supermultiplets. We find that just as for the cases N = 0 and N = 8 investigated previously, these billiards can be identified with the fundamental Weyl chambers of hyperbolic Kac-Moody algebras. Hence, the dynamics is chaotic in the BKL limit. A new feature arises, however, which is that the relevant Kac-Moody algebra can be the lorentzian extension of a twisted affine Kac-Moody algebra, while the N = 0 and N = 8 cases are untwisted. This occurs for N = 5, where one gets A(4)((2)boolean AND), and for N = 3 and 2, for which one gets A(2)((2)boolean AND). An understanding of this property is provided by showing that the data relevant for determining the billiards are the restricted root system and the maximal split subalgebra of the finite-dimensional real symmetry algebra characterizing the toroidal reduction to D = 3 spacetime dimensions. To summarise: split symmetry controls chaos.