Gromov--Witten theory of $[{\mathbb{C}}^2\mathrm{/}{\mathbb{Z}}_{n+1}]\times {\mathbb{P}}^1$

We compute the relative orbifold Gromov-Witten invariants of $[{\mathbb{C}}^2\mathrm{/}{\mathbb{Z}}_{n+1}]\times {\mathbb{P}}^1$, with respect to vertical fibers. Via a vanishing property of the Hurwitz-Hodge bundle, 2-point rubber invariants are calculated explicitly using Pixton’s formula for the double ramification cycle, and the orbifold quantum Riemann-Roch. As a result parallel to its crepant resolution counterpart for ${\mathcal{A}}_n$, the GW/DT/Hilb/Sym correspondence is established for $[{\mathbb{C}}^2\mathrm{/}{\mathbb{Z}}_{n+1}]$. The computation also implies the crepant resolution conjecture for relative orbifold Gromov-Witten theory of $[$\mathbb{C}$^2/$\mathbb{Z}$_{n+1}]\times $\mathbb{P}$^1$.