Joyce–Song wall-crossing formula for twisted Vafa–Witten invariants

Let $S$ be a smooth projective surface, let ${\mathfrak S}\to S$ be a $\mu_r$-gerbe, and let $\alpha=(\operatorname{rk},L,c_2)$. Let ${\mathcal P}^{\perp,\operatorname{tw}}_\alpha(m)$ be the virtual localization invariant of the moduli stack of twisted stable pairs with fixed determinant and trace-free Higgs field. Joyce–Song twisted wall-crossing conjecture. If $H^{0,1}(S)=H^{0,2}(S)=0$, there exist rational numbers $\operatorname{VW}^{\operatorname{tw}}_{\alpha_i}({\mathfrak S})$ such that, for $m\gg0$,

$${\mathcal P}^{\perp,\operatorname{tw}}_{\alpha}(m)=\sum_{\substack{\ell\geq1,\ (\alpha_i=\delta_i\alpha)_{i=1}^{\ell}:\\ \delta_i>0,\ \sum_{i=1}^{\ell}\delta_i=1}}\frac{(-1)^\ell}{\ell!}\prod_{i=1}^{\ell}(-1)^{\chi(\alpha_i(m))}\cdot\chi(\alpha_i(m))\cdot\operatorname{VW}^{\operatorname{tw}}_{\alpha_i}({\mathfrak S}).$$

When either $H^{0,1}(S)$ or $H^{0,2}(S)$ is nonzero, only the first term is taken:

$${\mathcal P}^{\perp,\operatorname{tw}}_{\alpha}(m)=(-1)^{\chi(\alpha(m))-1}\cdot\chi(\alpha(m))\cdot\operatorname{VW}^{\operatorname{tw}}_{\alpha}({\mathfrak S}).$$

This is proposed by analogy with the Joyce–Song/Tanaka–Thomas treatment of Vafa–Witten invariants. The paper uses the first-term version in relevant calculations, while the full wall-crossing assertion is not established in general.