Exceptional symmetry structures appear quite unexpectedly in the study of the behavior of solutions to the gravitational field equations (Einstein's equations or their supersymmetric generalizations) in the vicinity of a space-like singularity (cosmological singularity or "big bang"). In this limit, where fields diverge, dynamics is controlled by hyperbolic infinite Coxeter groups. The most spectacular case is that of maximum supergravity, where the limit gives rise to the Coxeter group E(10), which possesses remarkable properties.
These Coxeter groups are closely related to the Kac-Moody extensions of the symmetry groups revealed by dimensional reduction and to the duality groups of string theory. The fact that they are infinite is a direct consequence of the presence of gravitation among dynamical fields.
These symmetries are unexpected, as they are not visible in the original Lagrangian formulation of the theory. These are known as "hidden symmetries".
This year's lecture will be devoted to studying the emergence of these remarkable symmetry structures, their properties and implications.
Questions addressed:
- Einstein equations in the vicinity of a cosmological singularity - Belinski-Lifshitz-Khalatnikov analysis
- Cosmological billiards - first contact with hyperbolic Coxeter groups
- Generalization to supergravities
- Dimensional reduction and hidden symmetries
- Gravitational duality
- Hyperbolic Coxeter groups
- Hyperbolic Kac-Moody algebras
- Recent developments