Gromov--Witten invariants of root stacks with mid-ages and the loop axiom

Abstract: We study orbifold Gromov--Witten invariants of the $r$-th root stack $X_{D,r}$ with a pair of mid-ages when $r$ is sufficiently large. We prove that genus $g$ invariants with a pair of mid-ages $k_a/r$ and $1-k_a/r$ are polynomials in $k_a$ and the $k_a^i$-coefficients are polynomials in $r$ with degree bounded by $2g$. Moreover, genus zero invariants with a pair of mid-ages are independent of the choice of mid-ages. As an application, we obtain an identity for relative Gromov--Witten theory which can be viewed as a modified version of the usual loop axiom.