Given a vector bundle F on a smooth Deligne-Mumford stack X and an invertible multiplicative characteristic class c, we define orbifold Gromov-Witten invariants of X twisted by F and c. We prove a "quantum Riemann-Roch theorem" (Theorem 4.2.1) which expresses the generating function of the twisted invariants in terms of the generating function of the untwisted invariants. A quantum Lefschetz hyperplane theorem is derived from this by specializing to genus zero. As an application, we determine the relationship between genus-0 orbifold Gromov-Witten invariants of X and that of a complete intersection, under additional assumptions. This provides a way to verify mirror symmetry predictions for some complete intersection orbifolds.
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.
We prove a Givental-style mirror theorem for toric Deligne-Mumford stacks X . This determines the genus-zero Gromov-Witten invariants of X in terms of an explicit hypergeometric function, called the I-function, that takes values in the Chen-Ruan orbifold cohomology of X .
Hodge classes on the moduli space of admissible covers with monodromy group G are associated to irreducible representations of G. We evaluate all linear Hodge integrals over moduli spaces of admissible covers with abelian monodromy in terms of multiplication in an associated wreath group algebra. In case G is cyclic and the representation is faithful, the evaluation is in terms of double Hurwitz numbers. In case G is trivial, the formula specializes to the well-known result of Ekedahl-Lando-Shapiro-Vainshtein for linear Hodge integrals over the moduli space of curves in terms of single Hurwitz numbers.
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