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arXiv · 1612.00652

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

Zijun Zhou·Zhengyu Zong
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Abstract

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

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Zijun Zhou, Zhengyu Zong. 2016-12-02. Gromov--Witten theory of $[\mathbb{C}^2/\mathbb{Z}_{n+1}]\times \mathbb{P}^1$. https://doi.org/10.2140/ant.2022.16.1

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