Cao–Maulik–Toda wall-crossing conjecture for stable pairs on Calabi–Yau fourfolds

Let $(X,\omega)$ be a smooth projective Calabi–Yau $4$-fold, let $\beta\in H_2(X,\mathbb Z)$ and $n\in\mathbb Z_{\geqslant0}$, and choose a generic $t\in\mathbb R_{>0}$. Let $P^t_{n,\beta}(\gamma)$ be the $Z_t$-stable-pair invariant, let $P_{0,\beta}$ be the degree-zero stable-pair invariant, and let $n_{0,\beta}(\gamma)$ be the genus $0$ Gopakumar–Vafa type invariant for $\gamma\in H^4(X,\mathbb Z)$. Cao–Maulik–Toda wall-crossing conjecture. For suitable choices of orientations,

$$P^t_{n,\beta}(\gamma)=\sum_{\substack{\beta_0+\beta_1+\cdots+\beta_n=\beta\\ \omega\cdot\beta_i>\frac1t,\ i=1,\ldots,n}}P_{0,\beta_0}\prod_{i=1}^n n_{0,\beta_i}(\gamma).$$

In particular, $P^t_{0,\beta}(\gamma)=P_{0,\beta}(\gamma)$ is independent of the choice of $t>0$.

This is the main conjecture of the paper and generalizes the PT/GV correspondence by describing wall crossing among $Z_t$-stable-pair invariants. The source states it as conjectural and verifies it in several geometric examples.