Time-dependent toroidal compactification proposals and the Bianchi type II model: Classical and quantum solutions

In this work we construct an effective four-dimensional model by compactifying a ten-dimensional theory of gravity coupled with a real scalar dilaton field on a time-dependent torus without the contributions of fluxes as first approximation. This approach is applied to anisotropic cosmological Bianchi type II model for which we study the classical coupling of the anisotropic scale factors with the two real scalar moduli produced by the compactification process. Also, we present some solutions to the corresponding Wheeler-DeWitt (WDW) equation in the context of Standard Quantum Cosmology and we claim that these quantum solution are generic in the moduli scalar field for all Bianchi Class A models. Also we give the relation to these solutions for asymptotic behavior to large argument in the corresponding quantum solution in the gravitational variables and compare with Bohm's solutions, finding that this corresponds to the lowest-order WKB approximation.