Computing genus-zero twisted Gromov-Witten invariants

Abstract

Twisted Gromov-Witten invariants are intersection numbers in moduli spaces of stable maps to a manifold or orbifold $X$ which depend in addition on a vector bundle over $X$ and an invertible multiplicative characteristic class. Special cases are closely related to local Gromov-Witten invariants of the bundle and to genus-zero one-point invariants of complete intersections in $X$. We develop tools for computing genus-zero twisted Gromov-Witten invariants of orbifolds and apply them to several examples. We prove a “quantum Lefschetz theorem” that expresses genus-zero one-point Gromov-Witten invariants of a complete intersection in terms of those of the ambient orbifold $X$. We determine the genus-zero Gromov-Witten potential of the type $A$ surface singularity $[{\mathbb{C}}^2/{\mathbb{Z}}_n]$. We also compute some genus-zero invariants of $[{\mathbb{C}}^3/{\mathbb{Z}}_3]$, verifying predictions of Aganagic, Bouchard, and Klemm [3]. In a self-contained appendix, we determine the relationship between the quantum cohomology of the $A_n$ surface singularity and that of its crepant resolution, thereby proving the Crepant Resolution Conjectures of Ruan and of Bryan and Graber [12] in this case