Refined Joyce–Song pair conjecture for Vafa–Witten invariants

Let $(S,\mathcal O_S(1))$ be a polarised surface, let $X=K_S$, and let $\alpha=(r,c_1,c_2)\in H^{\operatorname{ev}}(S)$ be a charge. Let $P^\perp_\alpha(n,t)$ be the refined invariant of stable $SU(r)$ Joyce–Song pairs, and let $[m]_t=(t^{m/2}-t^{-m/2})/(t^{1/2}-t^{-1/2})$. Assume that $\mathcal O_S(1)$ is generic for $\alpha$ and that $H^{0,1}(S)=H^{0,2}(S)=0$. Refined Joyce–Song pair conjecture. There exist $\mathsf{VW}_{\alpha_i}(t)\in\mathbb Q(t^{1/2})$ such that, for $n\gg0$,

$$P^\perp_{\alpha}(n,t)=\sum_{\substack{\ell\ge1,\ (\alpha_i=\delta_i\alpha)_{i=1}^{\ell}\\ \delta_i>0,\ \sum_i\delta_i=1}}\frac{(-1)^\ell}{\ell!}\prod_{i=1}^{\ell}(-1)^{\chi(\alpha_i(n))}[\chi(\alpha_i(n))]_t\mathsf{VW}_{\alpha_i}(t).$$

If either $H^{0,1}(S)$ or $H^{0,2}(S)$ is nonzero, the asserted formula is instead

$$P^\perp_\alpha(n,t)=(-1)^{\chi(\alpha(n))-1}[\chi(\alpha(n))]_t\mathsf{VW}_\alpha(t).$$

This refines the numerical Vafa–Witten wall-crossing formula, whose specialization at $t=1$ is known in many cases; the refined statement remains conjectural in general.