On degree zero elliptic orbifold Gromov-Witten invariants
Hsian-Hua Tseng
TL;DR
The paper computes genus $1$, degree $0$ orbifold Gromov–Witten invariants with non-stacky insertions for smooth proper DM stacks, expressing them in terms of integrals over the double inertia stack $II\mathcal{X}$ and validating conjectural dilaton/divisor relations.Two complementary methods are developed: a detailed moduli-stack analysis (via admissible $G$-covers and obstruction theory) and virtual localization in the presence of torus actions with isolated fixed points; both yield the same closed-form evaluations.For global quotients $\mathcal{X}=[M/G]$, the invariants are computed as $\langle\tau_1(1)\rangle_{1,1,0}'^{\mathcal{X}}=\frac{1}{24}\int_{II[M/G]}c_{top}(T_{II[M/G]})$ and $\langle\tau_0(D)\rangle_{1,1,0}'^{\mathcal{X}}=-\frac{1}{24}\int_{II[M/G]}\pi_{\mathcal{X}}^*D\cup c_{top-1}(T_{II[M/G]})$; these agree with the conjectures and extend to general stacks via étale descent.The framework is further extended to twisted genus $1$ invariants, and the techniques are shown to apply to arbitrary smooth DM stacks by local quotient descriptions, yielding a robust approach to orbifold Virasoro constraints in genus one.
Abstract
We compute, by two methods, the genus one degree zero orbifold Gromov-Witten invariants with non-stacky insertions which are exceptional cases of the dilaton and divisor equations. One method involves a detailed analysis of the relevant moduli spaces. The other method, valid in the presence of torus actions with isolated fixed points, is virtual localization. These computations verify the conjectural evaluations of these invariants. Some genus one twisted orbifold Gromov-Witten invariants are also computed.