# Quasi-modularity and holomorphic anomaly equation for the twisted Gromov-Witten theory: $\mathcal{O}(3)$ over $\mathbb{P}^2$

## TL;DR

This paper establishes the quasi-modularity and derives the holomorphic anomaly equation for the twisted Gromov-Witten theory of the line bundle 3 over the projective plane, advancing understanding in enumerative geometry.

## Contribution

It proves the quasi-modularity property and formulates the holomorphic anomaly equation for this specific twisted Gromov-Witten theory.

## Key findings

- Proved quasi-modularity of the theory.
- Derived the holomorphic anomaly equation.
- Enhances understanding of enumerative invariants.

## Abstract

In this paper, we prove quasi-modularity property for the twisted Gromov-Witten theory of $\mathcal{O}(3)$ over $\mathbb{P}^2$. Meanwhile, we derive its holomorphic anomaly equation.