Relative Gromov–Witten quasi-Jacobi-form and holomorphic anomaly conjecture

Let $\pi:X\to B$ be an elliptic fibration with section, let $D\subset X$ be a smooth divisor inducing an elliptic fibration over $A\subset B$, and let $\mathbb C_{g,\mathsf k}^{\pi/D,\bullet}(\gamma_1,\ldots,\gamma_n;\underline\eta)$ denote the associated cycle-valued relative disconnected Gromov–Witten series. Here $N_\iota$ is the normal bundle of the section, $\Delta(q)$ is the discriminant, and $\mathsf{QJac}_{Q_k/2}$ denotes the relevant space of quasi-Jacobi forms.

Relative holomorphic anomaly conjecture. The series lies in

$$H_{\ast}(\overline M_{g,n}^{\bullet}(B/A,\mathsf k;\eta))\otimes \Delta(q)^{\frac12 c_1(N_\iota)\cdot\mathsf k}\mathsf{QJac}_{\frac12 Q_k},$$

and satisfies a holomorphic anomaly equation with respect to $G_2$ of the stated four-term form. This conjecture is attributed to the cited work [RES], whose precise four terms are not reproduced in the source. The source uses it to deduce an elliptic holomorphic anomaly equation; its general validity remains conjectural.