Computing genus-zero twisted Gromov-Witten invariants
The paper under review concerns twisted Gromov-Witten invariants of orbifolds. Fixing an orbifold \(\mathcal X\), we let \(\mathcal X_{g,n,d}\) be the moduli space of stable maps to \(\mathcal X\). Each vector bundle \(E\) over \(\mathcal X\) gives rise to a \(K\)-theory class \[ E_{g,n,d}=\pi_{!} f^{\star} E \] via pushforward from the universal family of the pullback of \(E\). The twisted Gromov-Witten invariants are integrals of the form \[ \int_{[\mathcal X_{g, n,d}]^{vir}} c(E_{g,n,d}) \prod_{i=1}^{n} {\text{ev}}_i^{\star} \alpha_i \psi_i^{k_i} \] for nonnegative integers \(k_i\) and classes \(\alpha_i\in H^{\star}_{orb}(\mathcal X).\) Here \(c\) denotes an invertible multiplicative characteristic class. The computation of twisted invariants is considered in this paper. In particular, the authors prove a ``quantum Lefschetz theorem'' expressing genus zero one point twisted invariants of complete intersections in terms of the invariants of the ambient orbifold. Furthermore, the genus zero potential of \(\left[\mathbb C^2/\mathbb Z_n\right]\) are computed. Calculations for \(\left[\mathbb C^3/\mathbb Z_3\right]\) also also presented. The appendix proves Ruan's Crepant Resolution Conjecture for type A surface singularity. Letting \(Y\) be the crepant resolution of the singularity \(\mathcal X=\left[\mathbb C^2/\mathbb Z_n\right]\), the Conjecture gives an isomorphism between the small quantum cohomology of \(Y\) and the quantum cohomology of \(\mathcal X\) after analytic continuation in the quantum parameters and specializiation of some quantum parameters to roots of unity. In fact, a stronger statement of Bryan-Graber is proved here, matching the big quantum cohomology after analytic continuation and specialization, via a linear isomorphism preserving the orbifold Poincaré pairing.
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