Towards $A + B$ theory in conifold transitions for Calabi-Yau threefolds

For projective conifold transitions between Calabi-Yau threefolds $X$ and $Y$, with $X$ close to $Y$ in the moduli, we show that the combined information provided by the $A$ model (Gromov--Witten theory in all genera) and $B$ model (variation of Hodge structures) on $X$, linked along the vanishing cycles, determines the corresponding combined information on $Y$. Similar result holds in the reverse direction when linked with the exceptional curves.
Comment: 47 pages, references updated. To appear in J. Diff. Geometry