Special geometry on Calabi--Yau moduli spaces and $Q$--invariant Milnor rings

Abstract

The moduli spaces of Calabi--Yau (CY) manifolds are the special K\"ahler manifolds. The special K\"ahler geometry determines the low-energy effective theory which arises in Superstring theory after the compactification on a CY manifold. For the cases, where the CY manifold is given as a hypersurface in the weighted projective space, a new procedure for computing the K\"ahler potential of the moduli space has been proposed in \cite {AKBA1,AKBA2, AKBA3}. The method is based on the fact that the moduli space of CY manifolds is a marginal subspace of the Frobenius manifold which arises on the deformation space of the corresponding Landau--Ginzburg superpotential. I review this approach and demonstrate its efficiency by computing the Special geometry of the 101-dimensional moduli space of the quintic threefold around the orbifold point \cite {AKBA3}.