Orbifold quantum Riemann–Roch, Lefschetz and Serre

Abstract

Given a vector bundle $F$ on a smooth Deligne–Mumford stack $\mathcal{X}$ and an invertible multiplicative characteristic class $c$, we define orbifold Gromov–Witten invariants of $\mathcal{X}$ twisted by $F$ and $c$. We prove a “quantum Riemann–Roch theorem” 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 $\mathcal{X}$ and that of a complete intersection, under additional assumptions. This provides a way to verify mirror symmetry predictions for some complete intersection orbifolds.