Oberdieck–Pixton holomorphic anomaly equation for elliptic fibrations

Let $\pi:X\to B$ be an elliptic fibration, let $\mathsf{k}\in H_2(B,\mathbb{Z})$, and let ${\mathcal C}^{\pi}_{g,\mathsf{k}}(\gamma_1,\ldots,\gamma_n)$ be the associated relative Gromov–Witten class. Let $\iota$ and $j$ be the gluing maps for the irreducible and separating boundary strata, respectively, and let $\Delta^!$ denote pullback along the diagonal of $B$. Let $\psi_i$ be the cotangent-line class at the $i$th marking. Holomorphic anomaly equation conjecture. On ${\overline M}_{g,n}(B,\mathsf{k})$,

$$\begin{aligned} \mathsf{T}_q {\mathcal C}^{\pi}_{g,\mathsf{k}}(\gamma_1,\ldots,\gamma_n)={}&\iota_*\Delta^!{\mathcal C}^{\pi}_{g-1,\mathsf{k}}(\gamma_1,\ldots,\gamma_n,\mathbf{1},\mathbf{1})\\ &+\sum j_*\Delta^!\left({\mathcal C}^{\pi}_{g_1,\mathsf{k}_1}(\gamma_{S_1},\mathbf{1})\boxtimes{\mathcal C}^{\pi}_{g_2,\mathsf{k}_2}(\gamma_{S_2},\mathbf{1})\right)\\ &-2\sum_{i=1}^n{\mathcal C}^{\pi}_{g,\mathsf{k}}(\gamma_1,\ldots,\gamma_{i-1},\pi^*\pi_*\gamma_i,\gamma_{i+1},\ldots,\gamma_n)\cdot\psi_i, \end{aligned}$$

with the separating sum over $g=g_1+g_2$, $\{1,\ldots,n\}=S_1\sqcup S_2$, and $\mathsf{k}=\mathsf{k}_1+\mathsf{k}_2$. This equation describes the failure of holomorphicity through boundary gluing and descendant terms; it is part of the general conjectural structure, with special geometries established in the paper.