Computing genus-zero twisted Gromov-Witten invariants

Abstract

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.

Effective Superpotentials for Compact D5-Brane Calabi-Yau Geometries

“…Note that the two-form sector, 19) which is closed with respect to the variations, ∂ z and ∂ u , and which will play an important role in deriving and solving the Picard-Fuchs differential equations of the relative periods (2.14).…”

“…[16,17,18], in the vicinity of orbifold points the superpotential encodes rational orbifold disk invariants [19,20,21].…”

“…That is to say we first solve the variational sub-system (2.19) governed by the two differential operators, L 1 and L 2 . The operator, L 1 , constraints a solution of the sub-system to have the form 19) whereas the second operator, L 2 , applied to this ansatz yields for the function, λ(u), the ordinary differential equation…”

Open Gromov-Witten Invariants from the Augmentation Polynomial

“…The right-hand side of Equation (A1) is sometimes referred to as a Gromov-Witten invariant "twisted" by F d . In [30], the authors have worked out a general procedure for obtaining the twisted J-function J tw M of a target space M, defined by…”

“…When c (F d ) is trivial, this reduces to the usual untwisted Gromov-Witten invariants. The next steps are to apply the twisted formula of [30] for X = O P 1 (−1, −1), and then localize the resulting expression.…”