Gromov-Witten Theory of Quotients of Fermat Calabi-Yau Varieties

Gromov-Witten Theory of Quotients of Fermat Calabi-Yau Varieties Synopsis

"We construct a global B-model for any quasi-homogeneous polynomial f that has properties similar to the properties of the physic's B-model on a Calabi-Yau manifold. The main ingredients in our construction are K. Saito's theory of primitive forms and Givental's higher genus reconstruction. More precisely, we consider the moduli space M[unfilled bullet]mar of the so-called marginal deformations of f. For each point [sigma] [is an element of] M[unfilled bullet]mar we introduce the notion of an opposite subspace in the twisted de Rham cohomology of the corresponding singularity f[sigma] and prove that opposite subspaces are in one-to-one correspondence with the splittings of the Hodge structure in the vanishing cohomology of f[sigma]. Therefore, according to M. Saito, an opposite subspace gives rise to a semi-simple Frobenius structure on the space of miniversal deformations of f[sigma]. Using Givental's higher genus reconstruction we def