We introduce an integral structure in orbifold quantum cohomology associated to the K-group and the b Γ-class. In the case of compact toric orbifolds, we show that this integral structure matches with the natural integral structure for the Landau-Ginzburg model under mirror symmetry. By assuming the existence of an integral structure, we give a natural explanation for the specialization to a root of unity in Y. Ruan's crepant resolution conjecture [66].Using the fact that L(τ, z) −1 is the adjoint of L(τ, −z) with respect to the orbifold Poincaré pairing (see (15)), we can calculate the embedding J τ = L(τ, z) −1 : (π * O(F )) τ ֒→

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 singularityˆC 2 /Zn˜. We also compute some genus-zero invariants ofˆC 3 /Z 3˜, verifying predictions of Aganagic-Bouchard-Klemm. In a self-contained Appendix, we determine the relationship between the quantum cohomology of the An surface singularity and that of its crepant resolution, thereby proving the Crepant Resolution Conjectures of Ruan and Bryan-Graber in this case.

Let X be a Gorenstein orbifold with projective coarse moduli space X and let Y be a crepant resolution of X . We state a conjecture relating the genus-zero GromovWitten invariants of X to those of Y , which differs in general from the Crepant Resolution Conjectures of Ruan and Bryan-Graber, and prove our conjecture when X D P .1; 1; 2/ and X D P .1; 1; 1; 3/. As a consequence, we see that the original form of the Bryan-Graber Conjecture holds for P .1; 1; 2/ but is probably false for P .1; 1; 1; 3/. Our methods are based on mirror symmetry for toric orbifolds.