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
Abstract

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