Minisuperspace quantum supersymmetric cosmology (and its hidden hyperbolic Kac-Moody structures)

Abstract

This work summarises recent progress 1,2 obtained by the mini-superpspace quantization of N = 1, d = 4 supergravity, formulated in the framework of the Bianchi IX cosmological model. The emphasis is put on three main results: the completeness of the solution space obtained, the elements suggesting a hidden Kac-Moody structure of the theory and those leading to conjecture an avoidance of the cosmological singularity by some branches of the wave function of the Universe.Keywords: Quantum cosmology; supergravity; quantum gravity; Kac-Moody algebras. Some MotivationsIn the framework of Einstein gravity, under very general assumptions, several theorems 3 predict ineluctable occurrence of cosmological singularities. These singularities reflect the inadequacy of a classical description of the physics of the first instants of the Universe. The relevant theory -maybe a workable string field theory -is not yet known. However, we may think that the era since which physics is reasonably well described by a metric governed by general relativity is preceded by an epoch where a quantized version of it is relevant. Of course this is only a pis aller, comparable to the quantization of the hydrogen atom or to the Fermi model that can be seen as approximations to quantum electrodynamics or quantum non-abelian gauge theories.On the other hand, a study of classical cosmological solutions in the vicinity of their singularities 4,5 has shown that in a generic case the dynamics of the system becomes chaotic. Even homogeneous cosmologies like the Bianchi VIII or IX models approximately evolve as a succession of periods described by Kasner like metrics:separated by collisions. Here {θ k } is a set of SO(2, 1) or SU (2) invariant forms and {p k } the set of so-called Kasner exponents. These exponents are constants during each period, but randomly reshuffled by the collision processes. The evolution of the classical geometry of a general non-homogeneous model appears as an ensemble