The multiple cover conjecture for descendent Gromov–Witten invariants of K3 surfaces

Let $S$ be a K3 surface, let $\beta\in H_2(S,\mathbb{Z})$ be an effective curve class, and let $\langle \tau_{k_1}(\gamma_1) \cdots \tau_{k_n}(\gamma_n) \rangle^S_{g,\beta}$ denote its reduced descendent Gromov–Witten invariant. Write ${\mathsf p}\in H^4(S,\mathbb{Z})$ for the class of a point. For every positive divisor $k\mid\beta$, let $S_k$ be a K3 surface and let $\varphi_k:H^2(S,\mathbb{R})\to H^2(S_k,\mathbb{R})$ be a real isometry such that $\varphi_k(\beta/k)$ is primitive and effective, extended by $\varphi_k(1)=1$ and $\varphi_k({\mathsf p})={\mathsf p}$. Multiple cover conjecture. Then

$$\left\langle \tau_{k_1}(\gamma_1) \cdots \tau_{k_n}(\gamma_n) \right\rangle^{S}_{g,\beta} =\sum_{k\mid\beta} k^{2g-3+\sum_{i=1}^{n}\deg_{\mathbb C}(\gamma_i)} \left\langle \tau_{k_1}(\varphi_k(\gamma_1)) \cdots \tau_{k_n}(\varphi_k(\gamma_n)) \right\rangle^{S_k}_{g,\varphi_k(\beta/k)}.$$

This conjecture predicts that all descendent invariants are determined by primitive invariants, extending the multiple-cover formula from the Gromov–Witten theory of K3 surfaces and $K3\times E$.