Götsche–Kool virtual elliptic genus conjecture for rank 2 sheaf moduli spaces

Let $S$ be a smooth projective surface with $b_1(S)=0$, $p_g(S)>0$, $K_S\ne0$, and with only Seiberg–Witten basic classes $0$ and $K_S$. Let $H,c_1,c_2$ be chosen so that there are no rank $2$ strictly Gieseker $H$-semistable sheaves on $S$ with Chern classes $c_1,c_2$, and write $M:=M_S^H(2,c_1,c_2)$. Let $\operatorname{vd}(M)$ be its virtual dimension, and let $\operatorname{Ell}^{\mathrm{vir}}(M)$ denote its virtual elliptic genus. Götsche–Kool's virtual elliptic genus conjecture. $\operatorname{Ell}^{\mathrm{vir}}(M)$ is given by the coefficient of $p^{\operatorname{vd}(M)}$ in $\psi_S(p,q,y)$, with

$$\psi_S(p,q,y)=8\left(\frac{1}{2\mathsf L_2(\phi_{0,1})}\right)^{\chi(\mathcal O_S)}\left(\frac{2\mathsf L_4(2\phi_{0,\frac12}\phi_{0,\frac32})\mathsf L(-2\phi_{0,\frac12})}{\mathsf L_2\left(-2\phi_{0,\frac12}^{\operatorname{ev}}|_{(q^{\frac12},y)}-\phi_{0,\frac12}|_{(q^2,y^2)}+2\phi_{0,\frac12}^2\right)}\right)^{K_S^2},$$

where

$$\mathsf L_2(\phi_{0,1})=\mathsf L_2(\operatorname{Ell}(K3))^{\frac12}=\left(\frac{\chi_{10}(p^2,q,y)}{p^2\Delta(q)\phi_{-2,1}(q,y)}\right)^{\frac12}.$$

Here $\phi_{0,\frac12}$ and $\phi_{0,\frac32}$ are the quasi-Jacobi forms introduced in the paper, and $\mathsf L$, $\mathsf L_2$, and $\mathsf L_4$ are the corresponding lift operators. The formula is a rank $2$ analogue of the Dijkgraaf–Moore–Verlinde–Verlinde formula and is supported by the universal structure and calculations discussed in the paper, but remains conjectural.