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 $\left[{ℂ}^2∕{\mathbb{Z}}_{n+1}\right]\times {\mathbb{P}}^1$.

中文翻译：

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Gromov–Witten 理论 [ℂ2∕ℤn+1] × ℙ1

我们计算的相对轨道 Gromov-Witten 不变量$\left[{ℂ}^2∕{\mathbb{Z}}_{n+1}\right]\times {\mathbb{P}}^1$相对于垂直纤维。通过 Hurwitz-Hodge 束的消失特性，使用 Pixton 的双分支循环公式和轨道量子 Riemann-Roch 明确计算 2 点橡胶不变量。因此，与它的 crepant 分辨率对应物平行${\mathcal{𝒜}}_n$, GW/DT/Hilb/Sym 对应建立为$\left[{ℂ}^2∕{\mathbb{Z}}_{n+1}\right]$. 该计算还暗示了相对轨道 Gromov-Witten 理论的 crepant 分辨率猜想$\left[{ℂ}^2∕{\mathbb{Z}}_{n+1}\right]\times {\mathbb{P}}^1$.