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

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 ◦mar of the so-called marginal deformations of f. For each point σ ∈ M ◦mar we introduce the notion of an opposite subspace in the twisted de Rham cohomology of the corresponding singularity f σ and prove that opposite subspaces are in one-to-one correspondence with the splittings of the Hodge structure in the vanishing cohomology of f σ . Therefore, according to M. Saito, an opposite subspace gives rise to a semi-simple Frobenius structure on the space of miniversal deformations of f σ . Using Givental's higher genus reconstruction we define a total ancestor potential A σ ( , q) whose properties can be described quite elegantly in terms of the properties of the corresponding opposite subspace. For example, if the opposite subspace corresponds to the splitting of the Hodge structure given by complex conjugation, then the total ancestor potential is monodromy invariant and it satisfies the BCOV holomorphic anomaly equations. The coefficients of the total ancestor potential could be viewed as quasi-modular forms on M ◦mar in a certain generalized sense.