An example of crepant resolution conjecture in two steps

TL;DR

This paper investigates the relationships between genus 0 Gromov-Witten theories of three related spaces connected via crepant resolutions and blowups, providing evidence for a conjecture in this context.

Contribution

It formulates and verifies a version of the Crepant Resolution Conjecture for a specific sequence of spaces involving orbifolds and resolutions.

Findings

Confirmed the conjectural relationship among Gromov-Witten theories

Established a two-step crepant resolution correspondence

Extended the conjecture to a new class of orbifold and resolution spaces

Abstract

We study the relation among the genus 0 Gromov-Witten theories of the three spaces $\mathcal{X}\leftarrow\mathcal{Z}\leftarrow Y$, where $\mathcal{X}=[\c^2/\z_3]$, $\mathcal{Z}$ is obtained by a weighted blowup at the stacky point of $\mathcal{X}$, and $Y$ is the crepant resolution of the $A_2$ singularity. We formulate and verify a statement similar to the Crepant Resolution Conjecture of Bryan and Graber.b