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

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 $n\ge 3$.

to appear in Advances in Mathematics