JETP Letters: issues online

Ten-dimensional Superstring theory unifies the Standard Model of the strong, electromagnetic, and weak interactions with quantum gravity. Starting with 10-dimensional superstring theory, we can get a 4-dimensional theory with spacetime supersymmetry following the Kaluza-Klein idea by compactifying six of the ten dimensions. For phenomenological reasons we need to do this while maintaining N=1 Supersymmetry of 4-dimensional spacetime. To achieve this, as Candelas, Horowitz, Strominger and Witten [Candelas:1985en] have shown, we must compactify six of the ten dimensions of to the so called Calabi-Yau manifolds. Another equivalent approach developed by D. Gepner [Gepner1, Gepner2] is the compactification of 6 dimensions onto some N=2 Superconformal Field theory with the central charge c=9. Each of these two equivalent approaches has its own merits. Say, using exactly solvability of the Minimal models of N=2 Superconformal Field Theory, it is possible to obtain the explicit solution of the considered models. In this article we review a series of our works on Calabi-Yau manifolds, their mirror symmetry, special geometry on the moduli space of Calabi-Yau, the connection of Calabi-Yau manifolds with N=2 superconformal minimal models and with supersymmetric gauged linear sigma models.