$D=11$ supergravity near a spacelike singularity admits a cosmological billiard description based on the hyperbolic Kac-Moody group $E_{10}$. The quantization of this system via the supersymmetry constraint is shown to lead to wave functions involving automorphic (Maass wave) forms under the modular group $W^{+}(E_{10})\cong PS{L}_2(\mathtt{O})$ with Dirichlet boundary conditions on the billiard domain. A general inequality for the Laplace eigenvalues of these automorphic forms implies that the wave function of the Universe is generically complex and always tends to zero when approaching the initial singularity. We discuss possible implications of this result for the question of singularity resolution in quantum cosmology and comment on the differences with other approaches.

• Received 20 July 2009

DOI:https://doi.org/10.1103/PhysRevD.80.061701