Week 13.12.2009 – 19.12.2009

We want to know all reductive subgroups of the five exceptional algebraic groups. In low characteristic some unusual subgroups appear which are contained inside parabolic subgroups. I will show how to find these by an archaeological dig.

The mini-superspace quantization of D=11 Supergravity is equivalent to the quantization of a E10-K(E10) coset space sigma model when the latter is restricted to the E10 Cartan subalgebra. As a consequence, the wavefunctions solving the relevant mini-superspace Wheeler-DeWitt equation involve automorphic (Maass wave) forms under the modular group W+(E10) cong PSL2(O). Using Dirichlet boundary conditions on the billiard domain a general inequality for the Laplace eigenvalues of these automorphic forms can be derived, which implies that the wave function of the universe is generically complex and always tends to zero when approaching the initial singularity. The potential significance of these properties for the question of singularity resolution in quantum cosmology is discussed, also in comparison with other approaches. The present approach offers interesting new perspectives on some long-standing issues in canonical quantum gravity.