Gromov-Witten theory of product stacks
Elena Andreini, Yunfeng Jiang, Hsian-Hua Tseng
TL;DR
The paper extends Behrend’s product formula to orbifold Gromov-Witten theory by establishing a product formula for invariants of the product stack $\mathcal{X}_1\times\mathcal{X}_2$ in terms of the invariants of the factors, using a careful analysis of virtual fundamental classes via log geometry. Central to the approach is the construction of the stack $\mathfrak{D}^{tw}(\widetilde{\tau})$ and a comparison of relative obstruction theories, which yields a precise identity for virtual classes (Theorems virtual_class_formula and weighted_virtual_class_formula) and thus for Gromov-Witten classes. The framework also provides an application to trivial gerbes, verifying a Gromov-Witten decomposition conjecture in this basic case (complementing Jarvis–Kimura results). Overall, the work furnishes a robust toolset for decomposing orbifold GW theories of product stacks and enables explicit computations in the presence of stack structures and gerbes.
Abstract
Let $\mathcal{X}_1$ and $\mathcal{X}_2$ be smooth proper Deligne-Mumford stacks with projective coarse moduli spaces. We prove a formula for orbifold Gromov-Witten invariants of the product stack $\mathcal{X}_1\times \mathcal{X}_2$ in terms of Gromov-Witten invariants of the factors $\mathcal{X}_1$ and $\mathcal{X}_2$. As an application, we deduce a decomposition result for Gromov-Witten theory of trivial gerbes.