Higher Genus Gromov-Witten Theory of C^n/Z_n I: Holomorphic Anomaly Equations

TL;DR

This paper establishes holomorphic anomaly equations for higher genus Gromov-Witten theory of the quotient stack [C^n/Z_n], extending previous specific cases to arbitrary n≥3, thus advancing the understanding of the mathematical structure of these theories.

Contribution

It generalizes holomorphic anomaly equations from specific cases to all n≥3 for the Gromov-Witten theory of [C^n/Z_n], providing a unifying framework.

Findings

Proved holomorphic anomaly equations for [C^n/Z_n] for all n≥3.

Extended previous results from specific cases to a general n.

Enhanced understanding of the structure of higher genus Gromov-Witten invariants.

Abstract

We study the structure of higher genus Gromov-Witten theory of the quotient stack $[\mathbb{C}^n/\mathbb{Z}_n]$. We prove holomorphic anomaly equations for $[\mathbb{C}^n/\mathbb{Z}_n]$, generalizing previous results of Lho-Pandharipande arXiv:1804.03168 for the case of $[\mathbb{C}^3/\mathbb{Z}_3]$ and ours arXiv:2211.15878 for the case $[\mathbb{C}^5/\mathbb{Z}_5]$ to arbitrary $n\geq{3}$.