The folklore conjecture on quasi-modularity for Calabi–Yau 1-folds

Let $\mathcal{X}_r$ be a Calabi–Yau 1-fold, and let $\phi_1,\ldots,\phi_n$ be insertions with nonnegative integers $k_1,\ldots,k_n$. Define the ancestor Gromov–Witten correlation function by

$$\langle\!\langle \phi_1\psi_1^{k_1},\ldots,\phi_n\psi_n^{k_n}\rangle\!\rangle_{g,n}:=\sum_{d\geq 0}\langle \phi_1\psi_1^{k_1},\ldots,\phi_n\psi_n^{k_n}\rangle_{g,n,d}q^d.$$

Calabi–Yau 1-fold quasi-modularity conjecture. The ancestor Gromov–Witten correlation functions defined above are quasi-modular forms. This is the one-dimensional formulation of the broader folklore expectation; the paper studies cases in which such quasi-modularity leads to Ramanujan-type differential identities, while the general assertion remains open.