The root-stack determination conjecture for Gromov–Witten theory

Let $\mathcal{X}$ be a Deligne–Mumford stack, let $\mathcal{D}\subset\mathcal{X}$ be a smooth irreducible divisor, and let $\mathcal{X}_r$ be the stack of $r$-th roots of $\mathcal{D}$ for a positive integer $r$. Let $H^*_{CR}(\mathcal{X})\to H^*_{CR}(\mathcal{D})$ be the restriction map in Chen–Ruan cohomology. Root-stack conjecture. The Gromov–Witten theory of $\mathcal{X}_r$ is determined by the Gromov–Witten theories of $\mathcal{X}$ and $\mathcal{D}$, together with the restriction map $H^*_{CR}(\mathcal{X})\to H^*_{CR}(\mathcal{D})$. Root constructions are essentially the only way stack structures arise in codimension $1$, and proving this determination statement is presented as the next goal for the codimension-$1$ setting.