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

TL;DR

This paper computes orbifold Gromov-Witten invariants for a specific class of orbifolds, establishing a correspondence with other enumerative theories and confirming the crepant resolution conjecture in this context.

Contribution

It explicitly calculates 2-point rubber invariants using Pixton's formula and proves the GW/DT/Hilb/Sym correspondence for $[C^2/Z_{n+1}]$, advancing understanding of orbifold Gromov-Witten theory.

Findings

Explicit computation of 2-point rubber invariants.

Establishment of GW/DT/Hilb/Sym correspondence for the orbifold.

Verification of the crepant resolution conjecture for the given orbifold.

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