Oberdieck's quasi-modularity conjecture for primitive abelian-surface Gromov–Witten classes

Let $E_1$ and $E_2$ be nonsingular elliptic curves, let $\mathsf{A}=E_1\times E_2$ be fibered over $E_1$, and let ${\mathcal A}_{g,n}(\gamma_1,\ldots,\gamma_n)$ denote the reduced primitive Gromov–Witten potential for insertions $\gamma_i\in H^*(\mathsf{A})$. Quasi-modularity conjecture.

$${\mathcal A}_{g,n}(\gamma_1,\ldots,\gamma_n)\in\operatorname{QMod}\otimes H_*({\overline M}_{g,n}(E_1,1)).$$

The conjecture concerns quasi-modularity of reduced Gromov–Witten classes in primitive curve classes. The paper states that the holomorphic anomaly equation is proved numerically in primitive classes, while quasi-modularity had been established previously; the displayed conjecture is therefore a solved result.