A Combinatorial Case of the Abelian-Nonabelian Correspondence

TL;DR

This paper proves a specific case of the abelian-nonabelian correspondence by explicitly relating genus zero Gromov-Witten invariants of Grassmannians and products of projective spaces using localization and moduli space comparison.

Contribution

It provides a concrete proof of a case of the abelian-nonabelian correspondence for genus zero invariants of Grassmannians and related spaces.

Findings

Established explicit relation between Gromov-Witten invariants of Grassmannians and products of projective spaces.

Used localization techniques to compute twisted Gromov-Witten invariants.

Compared moduli spaces of stable maps to complete the proof.

Abstract

The abelian-nonabelian correspondence outlined by Bertram, Ciocan-Fontanine, and Kim gives a broad conjectural relationship between (twisted) Gromov-Witten invariants of related GIT quotients. This paper proves a case of the correspondence explicitly relating genus zero m-pointed Gromov-Witten invariants of Grassmannians Gr(2,n) and products of projective space $\PP^{n-1} \times \PP^{n-1}$. Localization is used to compute twisted Gromov-Witten invariants of $\PP^{n-1} \times \PP^{n-1}$, and comparison of the moduli spaces of stable maps completes the proof.