Higher genus Gromov-Witten theory of C^n/Z_n: Holomorphic anomaly equations and crepant resolution

Abstract: In this talk, we present certain results regarding the higher genus Gromov-Witten theory of $\mathbb{C}^n/\mathbb{Z}_n$ obtained by studying its cohomological field theory structure in detail. Holomorphic anomaly equations are certain recursive partial differential equations predicted by physicists for the Gromov-Witten potential of a Calabi-Yau threefold. We prove holomorphic anomaly equations for $\mathbb{C}^n/\mathbb{Z}_n$ for any $n\ge3$. In other words, we present a phenomenon of holomorphic anomaly equations in arbitrary dimensions, a result beyond the consideration of physicists. The proof of this fact relies on showing that the Gromov-Witten potential of $\mathbb{C}^n/\mathbb{Z}_n$ lies in a certain polynomial ring. Moreover, we prove an arbitrary genera crepant resolution correspondence for $\mathbb{C}^n/\mathbb{Z}_n$ by showing that its cohomological field theory matches with that of $K\mathbb{P}^{n-1}$, where $K\mathbb{P}^{n-1}$ is the total space of the canonical bundle of $\mathbb{P}^{n-1}$. More precisely, we show that the Gromov-Witten potential of $K\mathbb{P}^{n-1}$ also lies in a similar polynomial ring, and we show that it matches with the Gromov-Witten potential of $\mathbb{C}^n/\mathbb{Z}_n$ under an isomorphism of these polynomial rings. This talk is based on the joint works arXiv:2301.08389 and arXiv:2308.00780 with Hsian-Hua Tseng.

algebraic geometrycombinatorics

Audience: researchers in the topic

Online Nottingham algebraic geometry seminar

Series comments: Online geometry seminar, typically held on Thursday. This seminar takes place online via Microsoft Teams on the Nottingham University "Algebraic Geometry" team.

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