Computing Genus-Zero Twisted Gromov-Witten Invariants
Tom Coates, Alessio Corti, Hiroshi Iritani, Hsian-Hua Tseng
TL;DR
The paper addresses the challenge of computing genus-zero twisted Gromov-Witten invariants for orbifolds, which depend on a vector bundle $F$ and a multiplicative class ${\boldsymbol{c}}$. It extends Givental's symplectic formalism to twisted orbifold theories, proves a general orbifold quantum Riemann--Roch and a quantum Lefschetz-type result to express twisted genus-zero invariants in terms of untwisted data, and applies the machinery to compute invariants for local models such as $\left[\mathbb{C}^2/\mathbb{Z}_n\right]$ and $\left[\mathbb{C}^3/\mathbb{Z}_3\right]$, including a quintic hypersurface example. The results yield explicit generators for the twisted $J$-functions, establish a framework to derive genus-zero potentials for complete intersections, and prove Crepant Resolution Conjectures in the type $A$ surface singularity case via mirror symmetry. Overall, the work provides a comprehensive toolkit to translate twisted orbifold GW data into untwisted ambient data, with broad implications for mirror symmetry, local GW theory, and crepant-resolution phenomena.
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" which 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 C^2/Z_n. We also compute some genus-zero invariants of C^3/Z_3, verifying predictions of Aganagic-Bouchard-Klemm. 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 Bryan-Graber in this case.