## Computing genus-zero twisted Gromov-Witten invariants.(English)Zbl 1176.14009

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.

### MSC:

 14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) 14A20 Generalizations (algebraic spaces, stacks) 53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
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