The refined Vafa–Witten multiple-cover conjecture for Enriques surfaces

Let $Y$ be an Enriques surface and let $(r,\beta,n)\in H^{\ast}(Y,\mathbb{Z})$ be effective. For a smooth projective variety $X$, define the signed normalized $\chi_y$-genus and the quantum integer by

$$\widehat{\chi}_{-t}(X):=(-1)^{\dim(X)}t^{-\dim(X)/2}\chi_{-t}(X),\qquad [n]_t:=\frac{t^{n/2}-t^{-n/2}}{t^{1/2}-t^{-1/2}}.$$

The refined Vafa–Witten multiple-cover conjecture. The refined Vafa–Witten invariant satisfies

$$\mathsf{VW}(r,\beta,n)=2\sum_{\substack{k\mid(r,\beta,n)\\ k\geq1\ \operatorname{odd}}}\frac{1}{k[k]_t}\widehat{\chi}_{-t^k}\left(\mathsf{Hilb}^{\frac{\beta^2-2rn-r^2}{2k^2}+\frac12}(Y)\right).$$

This conjecture gives an explicit formula for refined Vafa–Witten invariants in arbitrary rank, generalizing the known unrefined formula and exhibiting multiple-cover behavior. The paper presents it as unproved; related formulas are known in low rank and for K3 surfaces.