Mirror symmetry for concavex vector bundles on projective spaces

Let $X\subset Y$ be smooth, projective manifolds. Assume that $X$ is the zero locus of a generic section of a direct sum $V+$ of positive line bundles on $\PP^n$. Furthermore assume that the normal bundle $N_{X/Y}$ is a direct sum $V-$ of negative line bundles. We show that a $V:=V+\oplus V-$-twisted Gromov-Witten theory of $\PP^n$ restricts to the Gromov-Witten theory of $X$ inherited form $Y$. The later one can be computed via a Mirror Theorem which we prove in this paper.