The Abelian/non-Abelian Correspondence and Gromov-Witten Invariants of Blow-ups

TL;DR

This paper proves the Abelian/non-Abelian Correspondence for partial flag bundles and derives explicit formulas for how genus-zero Gromov-Witten invariants change under blow-ups, especially for Fano varieties.

Contribution

It generalizes the Abelian/non-Abelian Correspondence to bundles over partial flag varieties and provides explicit formulas for Gromov-Witten invariants after blow-ups.

Findings

Established the Abelian/non-Abelian Correspondence for partial flag bundles.

Derived formulas for Gromov-Witten invariants of blow-ups in complete intersections.

Provided closed-form expressions for invariants of Fano blow-ups.

Abstract

We prove the Abelian/non-Abelian Correspondence with bundles for target spaces that are partial flag bundles, combining and generalising results by Ciocan-Fontanine-Kim-Sabbah, Brown, and Oh. From this we deduce how genus-zero Gromov-Witten invariants change when a smooth projective variety X is blown up in a complete intersection defined by convex line bundles. In the case where the blow-up is Fano, our result gives closed-form expressions for certain genus-zero invariants of the blow-up in terms of invariants of X. We also give a reformulation of the Abelian/non-Abelian Correspondence in terms of Givental's formalism, which may be of independent interest.