The polynomiality conjecture for descendent invariants of K3 surfaces

Let $S$ be a K3 surface, let $\beta\in H_2(S,\mathbb Z)$ be primitive and effective with $\beta^2\ne0$, and let $\delta_1,\ldots,\delta_t\in H^2(S)$ satisfy $\delta_i\cdot\beta=0$. Consider the normalized invariant

\left\llangle \prod_{i=1}^{r}\tau_{k_i}(1)\prod_{i=1}^{s}\tau_{\ell_i}(\beta)\prod_{i=1}^{t}\tau_{m_i}(\delta_i)\prod_{i=1}^{u}\tau_{n_i}({\mathsf p})\right\rrangle^S_\beta,

with $k_i,m_i\geq1$. Polynomiality conjecture. There exists a polynomial $p(x_1,\ldots,x_{r+s+t+u})$ of degree $\beta^2+2-2u-t+r$ such that, whenever

$$k_i\geq\beta^2/2+3-(u+t/2),\qquad \ell_i,m_i,n_i\geq\beta^2/2+1-(u+t/2),$$

the normalized invariant equals $p(k_1,\ldots,k_r,\ell_1,\ldots,\ell_s,m_1,\ldots,m_t,n_1,\ldots,n_u)$. This conjecture predicts a strong eventual polynomial structure in the descendent indices, potentially making broad classes of K3 Gromov–Witten invariants effectively computable.