Gromov–Witten theory of [ℂ2/ℤn+1]×ℙ1

Abstract

We compute the relative orbifold Gromov–Witten invariants of $\left[{ℂ}^2∕{\mathbb{Z}}_{n+1}\right]\times {\mathbb{P}}^1$ 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 $\left[{ℂ}^2∕{\mathbb{Z}}_{n+1}\right]$. The computation also implies the crepant resolution conjecture for the relative orbifold Gromov–Witten theory of $\left[{ℂ}^2∕{\mathbb{Z}}_{n+1}\right]\times {\mathbb{P}}^1$.