Asymptotic Hodge theory and quantum products

Assuming suitable convergence properties for the Gromov-Witten potential of a Calabi-Yau manifold X, one may construct a polarized variation of Hodge structure over the complexified Kähler cone of X. In this paper we show that, in the case of fourfolds, there is a correspondence between “quantum potentials” and polarized variations of Hodge structures that degenerate to a maximally unipotent boundary point. Under this correspondence, the WDVV equations are seen to be equivalent to the Griffiths’ trasversality property of a variation of Hodge structure.