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

TL;DR

This paper reviews a new method for computing the special K"ahler geometry of Calabi--Yau moduli spaces, especially for hypersurfaces in weighted projective spaces, using Frobenius manifolds and Milnor rings.

Contribution

It introduces an efficient procedure to compute the K"ahler potential of CY moduli spaces via Frobenius manifold structures and demonstrates its application on the quintic threefold.

Findings

Successfully computed the special geometry of the 101-dimensional moduli space.

Validated the method's efficiency through explicit calculations.

Provided insights into the structure of CY moduli spaces near orbifold points.

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}.