Vol. 16, No. 1, 2022

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

Zijun Zhou and Zhengyu Zong

Vol. 16 (2022), No. 1, 1–58

We compute the relative orbifold Gromov–Witten invariants of [2n+1] × 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 𝒜n, the GW/DT/Hilb/Sym correspondence is established for [2n+1]. The computation also implies the crepant resolution conjecture for the relative orbifold Gromov–Witten theory of [2n+1] × 1.

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relative orbifold GW theory, GW/DT correspondence, crepant resolution conjecture
Mathematical Subject Classification 2010
Primary: 14N35
Received: 1 February 2019
Revised: 1 March 2021
Accepted: 15 April 2021
Published: 22 February 2022
Zijun Zhou
Kavli Institute for the Physics and Mathematics of the Universe
The University of Tokyo
Zhengyu Zong
Department of Mathematical Sciences
Tsinghua University