Oberdieck–Pixton quasi-Jacobi form conjecture for relative elliptic fibrations

Let $\pi:X\to B$ be an elliptic fibration with section and integral fibers, let $D\subset X$ be a nonsingular divisor restricting to an elliptic fibration $\pi_D:D\to A$ over a nonsingular divisor $A\subset B$, and let $\underline{\eta}=((\eta_i,\delta_i))$ be an ordered cohomology-weighted partition. For $\gamma_1,\ldots,\gamma_n\in H^*(X)$ and $\mathsf{k}\in H_2(B,\mathbb{Z})$, let ${\mathcal C}_{g,\mathsf{k}}^{\pi/D}(\gamma_1,\ldots,\gamma_n;\underline{\eta})$ be the relative Gromov–Witten potential. Relative quasi-Jacobi form conjecture. The series is a cycle-valued quasi-Jacobi form of index $Q_{\mathsf{k}}/2$:

$${\mathcal C}_{g,\mathsf{k}}^{\pi/D}(\gamma_1,\ldots,\gamma_n;\underline{\eta})\in H_*({\overline M}_{g,n}(B/A,\mathsf{k};\eta))\otimes\frac{1}{\Delta(q)^m}\operatorname{QJac}_{Q_{\mathsf{k}}/2},$$

where $m=-\frac12c_1(N_{\iota})\cdot\mathsf{k}$. This extends the predicted modularity from absolute to relative elliptic-fibration geometries; the corresponding relative holomorphic anomaly equation is likewise conjectural in general.