Special geometry on the Moduli space of Calabi-Yau manifolds, Localization and Two-Sphere partition function of GLSM

The requirement the Space-time supersymmetry in the String theory is equivalent to the geometrical condition of the compactification 6 of 10 dimentions on Calabi-Yau(CY) threefold. The properties of the Effective Lagrangian of the model, which describes the massless sector, are defined in terms of the so-called Special Kahler geometry on CY moduli space. I describe a new approach for computing the Special Kahler geometry based on the relation of Landau-Ginzburg superpotential of the model with a Frobenius manifold structure on the CY moduli space. I'l show how apply this approach for computing the Kahler metric on moduli space of the Calabi-Yau threefolds of Fermat type. Also I show how the Kahler potentials are connected with the partition functions of the $N=(2,2)$ Gauged Linear Sigma-Models on Two-Sphere.