Hyperbolic billiards of pure D=4 supergravities

TL;DR

This paper analyzes the chaotic billiard dynamics in pure D=4 supergravities and their relation to hyperbolic Kac-Moody algebras, revealing new twisted algebra structures for certain supersymmetry cases.

Contribution

It identifies the billiard regions with fundamental Weyl chambers of hyperbolic Kac-Moody algebras, including new twisted affine types, and links these structures to the symmetry data of supergravity theories.

Findings

Billiards correspond to Weyl chambers of hyperbolic Kac-Moody algebras.

Chaotic dynamics are confirmed in the BKL limit for all cases studied.

Twisted affine Kac-Moody algebras appear in certain supergravity cases.

Abstract

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, N=3 and N=2. 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 summarize: split symmetry controls chaos.