Dynamics of multidimensional generalization of Bianchi type-IX cosmological models

We investigate the dynamics of an 11-dimensional homogeneous cosmological model. We assume that the t=const hypersurfaces are products of a 3-dimensional Bianchi type-IX space and a 7-dimensional torus. Most results of our investigation hold when the 7-dimensional torus is replaced by an m-dimensional torus $T^m$. We show that for a large class of vacuum solutions the physical space expands while the microspace contracts providing a natural mechanism of dimensional reduction. Matter satisfying a simple barotropic equation of state always breaks the process of dynamical dimensional reduction. With special attention we study the behavior of our model close to the initial singularity. In contrast with the 4-dimensional Bianchi type-IX cosmological model the Kasner solution always describes an approach to the initial singularity. We study the transition from the Kasner regime to the oscillatory regime. We show that matter does not significantly change this property. We have found some exact solutions; they describe isotropic physical space, and axially symmetric physical space. We also investigate high-temperature quantum effects and their effects on the dynamics of our model. We show that the high-temperature quantum corrections do not depend on the geometrical properties of space, provided it is compact. The high-temperature quantum corrections can be described by a macroscopic equation of state satisfied by radiation.