Multiple cover formula for Gromov–Witten invariants of Hilbert schemes of points on K3 surfaces

Let $S$ be a K3 surface with effective curve class $\beta$, let $S_k$ and $\varphi_k$ be the K3 surface and real cohomological isometry associated to each divisor $k\mid\beta$, with $\varphi_k(\beta/k)$ primitive and effective, and extend $\varphi_k$ to $H^*(S^{[n]})$ factorwise through Nakajima operators. Let $A$ be the additional curve class and let $\operatorname{deg}(\mathrm{taut})$ denote the degree of the tautological insertion.

Multiple cover formula.

$$\left\langle\mathrm{taut};\gamma_1,\ldots,\gamma_N\right\rangle^{S^{[n]}}_{g,\beta+rA}=\sum_{k\mid(\beta,r)}k^{3g-3+N-\operatorname{deg}(\mathrm{taut})}(-1)^{r+r/k}\left\langle\mathrm{taut};\varphi_k(\gamma_1),\ldots,\varphi_k(\gamma_N)\right\rangle^{S^{[n]}}_{g,\varphi_k(\beta/k)+(r/k)A}.$$

This is presented as an equivalent form of the earlier Multiple Cover Conjecture, not as a separate claim.