Quantum Calabi-Yau manifolds and mirror symmetry

The primary topic of this work is the solution of string theories compactified on a Calabi-Yau manifold. The prepotentials and the geometry of the moduli spaces for a Calabi-Yau manifold and its mirror are computed. In this way all sigma model corrections to the Yukawa couplings and moduli space metric for the original manifold are obtained. The moduli space is found to be subject to the action of a modular group which, among other operations, exchanges large and small values of the radius though the action on the radius is not as simple as R → 1/R. It is shown also that the quantum corrections to the coupling decompose into a sum over instanton contributions and moreover that this sum converges so that there are no 'sub-instanton' corrections. This sum over instantons points to a deep connections between the modular group and the rational curves of the Calabi-Yau manifold. The burden of the present work is that a mirror pair of Calabi-Yau manifolds is an exactly soluble superconformal theory. They are also more general than exactly soluble models that have hitherto been discussed since here the theory for all points of the moduli space is solved. The other topic of this thesis is related also to the space of moduli of Calabi-Yau manifolds. It is shown that the topology changing paths on the moduli space between two Calabi-Yau manifolds are continuous even in the space of Ricci-flat Kaehler metrics