Mirror symmetry and quantum cohomology of projective bundles

Abstract

In an earlier paper we conjectured a relation between the quantum $\mathcal{D}$-modules of a smooth variety $X$ and the projectivisation of a direct sum of line bundles over it. In this paper we prove the conjecture when $X$ is a complete intersection in a toric variety. We also use the conjecture to show that the relations of the small quantum cohomology ring of $X$ that come from differential operators lift to the projective bundle. The basic cohomology relation of the projective bundle deforms to a relation in the small quantum cohomology.

9 pages

Cited by in corpus (1)

• Borel $(\alpha ,\beta )$-multitransforms and Quantum Leray-Hirsch: integral representations of solutions of quantum differential equations for ${\mathbb{P}}^1$-bundles