Hyperbolic billiards of pureD= 4 supergravities

We compute the billiards that emerge in the Belinskii-Khalatnikov-Lifshitz(BKL) limit for all pure supergravities in D=4 spacetime dimensions, as well asfor D=4, N=4 supergravities coupled to k (N=4) Maxwell supermultiplets. We findthat just as for the cases N=0 and N=8 investigated previously, these billiardscan be identified with the fundamental Weyl chambers of hyperbolic Kac-Moodyalgebras. Hence, the dynamics is chaotic in the BKL limit. A new featurearises, however, which is that the relevant Kac-Moody algebra can be theLorentzian extension of a twisted affine Kac-Moody algebra, while the N=0 andN=8 cases are untwisted. This occurs for N=5, N=3 and N=2. An understanding ofthis property is provided by showing that the data relevant for determining thebilliards are the restricted root system and the maximal split subalgebra ofthe finite-dimensional real symmetry algebra characterizing the toroidalreduction to D=3 spacetime dimensions. To summarize: split symmetry controlschaos.Comment: 21 page