Mirror symmetry and quantum cohomology of projective bundles

TL;DR

This paper proves a conjecture relating quantum D-modules of a variety and its projective bundle, demonstrating how quantum cohomology relations lift and deform in this context, especially for complete intersections in toric varieties.

Contribution

It proves a conjecture connecting quantum D-modules of a variety and its projectivization for complete intersections in toric varieties, and shows how quantum cohomology relations extend.

Findings

Quantum D-modules of $X$ relate to those of projective bundles.

Relations in small quantum cohomology lift to projective bundles.

Basic cohomology relations deform into quantum cohomology relations.

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.