Quantum cohomology of projective bundles

TL;DR

This paper develops a new approach to quantum cohomology of projective bundles, showing how their quantum D-modules decompose into those of the base space, with implications for semisimplicity.

Contribution

It constructs an I-function for projective bundles associated with non-split vector bundles and proves the splitting of their quantum cohomology D-modules.

Findings

Constructed an I-function via Fourier transform of the S^1-equivariant J-function.

Proved the quantum cohomology D-module of P(V) splits into that of B.

Applied results to the semisimplicity of big quantum cohomology.

Abstract

We construct an I-function of the projective bundle P(V) associated with a not necessarily split vector bundle V\to B as a Fourier transform of the S^1-equivariant J-function of the total space of V and show that it lies on the Givental Lagrangian cone of P(V). Using this result, we show that the quantum cohomology D-module of P(V) splits into the direct sum of the quantum cohomology D-modules of the base space B. This has applications to the semisimplicity of big quantum cohomology.