Higher Genus Gromov-Witten Theory of C^n/Z_n II: Crepant Resolution Correspondence

TL;DR

This paper establishes a crepant resolution correspondence for higher genus Gromov-Witten theories of the total space of the canonical bundle of projective space and a related orbifold, generalizing previous specific cases.

Contribution

It proves the finite generation of the Gromov-Witten potential for $K ext{P}^{n-1}$ and establishes an isomorphism between polynomial rings, extending prior results to arbitrary $n \

Findings

Gromov-Witten potential of $K ext{P}^{n-1}$ is in an explicit polynomial ring.

Established a crepant resolution correspondence for higher genus Gromov-Witten theories.

Extended previous specific cases to general $n \

Abstract

We study the structure of the higher genus Gromov-Witten theory of the total space $K\mathbb{P}^{n-1}$ of the canonical bundle of the projective space $\mathbb{P}^{n-1}$. We prove the finite generation property for the Gromov-Witten potential of $K\mathbb{P}^{n-1}$ by working out the details of its cohomological field theory (CohFT). More precisely, we prove that the Gromov-Witten potential of $K\mathbb{P}^{n-1}$ lies in an explicit polynomial ring using the Givental-Teleman classification of the semisimple CohFTs. In arXiv:2301.08389, we carried out a parallel study for $[\mathbb{C}^n/\mathbb{Z}_n]$ and proved that the Gromov-Witten potential of $[\mathbb{C}^n/\mathbb{Z}_n]$ lies in a similar polynomial ring. The main result of this paper is a crepant resolution correspondence for higher genus Gromov-Witten theories of $K\mathbb{P}^{n-1}$ and $[\mathbb{C}^n/\mathbb{Z}_n]$, which is proved by establishing an isomorphism between the polynomial rings associated to $K\mathbb{P}^{n-1}$ and $[\mathbb{C}^n/\mathbb{Z}_n]$. This paper generalizes the works of Lho-Pandharipande arXiv:1804.03168 for the case of $[\mathbb{C}^3/\mathbb{Z}_3]$ and Lho arXiv:2211.15878 for the case $[\mathbb{C}^5/\mathbb{Z}_5]$ to arbitrary $n\geq 3$.