The Implications of N =2 Supergravity Cosmology On the Topology of the Calabi-Yau Manifold

When N = 1 D = 11 supergravity is compactified on CY threefold to N = 2 D = 5 supergravity the action of the last is given in terms of the geometery of the CY manifold space, namely, in terms of the hypermultiplets. There are z i ( i = 1 , ..., h 2 , 1 ) complex structure moduli in the moduli space of the CY manifold which’s a special K¨ahler manifold with a metric G i ¯ j . We solve the field equations of the complex structure moduli with the solution of the Einstein field equations to the moduli velocity norm G i ¯ j z i z ¯ j in case of a 3- brane filled with radiation, dust and energy embedded in the bulk of D = 5 supergravity. We get the time dependence of the moduli and the metric. Then we can further deduce the geometry of the moduli space by getting the K¨ahler potential which directly relates to the volume of the CY manifold.