Double ramification cycle conjecture for Gromov–Witten invariants of K3 surfaces

Let $S$ be a K3 surface, let $\beta\in H^2(S,\mathbb{Z})$ be primitive and effective, and let $\mathsf{a}=(\mathsf{a}_1,\ldots,\mathsf{a}_n)$ be integers with $\sum_i\mathsf{a}_i=0$. For insertions $\gamma_i\in H^*(S)$, write $\operatorname{DR}_g(\mathsf{a})\in H^{2g}(\overline{M}_{g,n})$ for the double ramification cycle, let $\Theta(z,\tau)$ be the odd renormalized Jacobi theta function, and let $\Delta(\tau)$ be the discriminant. The Mukai pairing on $H^*(S)$ is denoted by $(\,\ )$, and $\deg(\gamma)$ is defined by $\gamma\in H^{2\deg(\gamma)}(S)$.

Double ramification cycle conjecture. There exist quasi-Jacobi forms $\varphi_m(z,\tau)$ and $\varphi_{m,n}(z,\tau)$ such that

$$\sum_{g=0}^{\infty}\left\langle \operatorname{DR}_g(\mathsf{a});\gamma_1,\ldots,\gamma_n\right\rangle^S_{g,\beta}(-1)^{g+n}z^{2g-2+n}=\frac{1}{\prod_i\mathsf{a}_i^{\deg(\gamma_i)}}\operatorname{Coeff}_{q^{\frac12\beta^2}}\left(\sum_{\{(a_j,b_j)\}_j,\{c_j\}_j}\frac{1}{\Theta^2\Delta}\prod_j(\gamma_{a_j},\gamma_{b_j})\varphi_{a_jb_j}\cdot\prod_j(\gamma_{c_j},\beta)\varphi_{c_j}\right).$$

Here the sum is over all partitions of $\{(\mathsf{a}_i,\gamma_i)\}_{i=1}^n$ into parts of size at most $2$, with singleton parts labeled by $(c_j,\gamma_{c_j})$ and pairs labeled by $\{(a_j,\gamma_{a_j}),(b_j,\gamma_{b_j})\}$. This generalizes the Katz–Klemm–Vafa formula for the case with no marked points, while the functions $\varphi_m$ and $\varphi_{m,n}$ were initially left indeterminate; further evidence is provided by related results for K3 surfaces and products with $\mathbb{P}^1$.