Refined K3 Vafa–Witten formula for prime rank

Let $S$ be a K3 surface with polarization $H$, first Chern class $c_1$, and prime rank $r$. Let $\mathsf Z_{K3,H,r,c_1}^{\mathrm{inst}}(q,y)$ and $\mathsf Z_{K3,H,r,c_1}^{\mathrm{mono}}(q,y)$ denote the instanton and monopole contributions. Refined K3 formula conjecture. These contributions are given by the two explicit modular expressions displayed in the source, namely the finite sum over $m=0,\ldots,r-1$ for the instanton branch and the delta-supported monopole expression. The formula is a proposed $y$-refinement of the known unrefined K3 formula; the source records supporting results in special cases but no general proof.