Maulik–Pandharipande–Thomas stable-pair/GV correspondence for Calabi–Yau fourfolds

Let $(X,\omega)$ be a smooth projective Calabi–Yau $4$-fold, let $P_{n,\beta}(\gamma)$ be its stable-pair invariants with $\gamma\in H^4(X,\mathbb Z)$, and let $n_{0,\beta}(\gamma)$ and $n_{1,\beta}$ be the genus $0$ and genus $1$ Gopakumar–Vafa type invariants. Let

$$M(q)=\prod_{k\geqslant1}(1-q^k)^{-k}$$

be the MacMahon function. Maulik–Pandharipande–Thomas conjecture. For certain choices of orientations,

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

and

$$\sum_{\beta\geqslant0}P_{0,\beta}q^\beta=\prod_{\beta>0}M(q^\beta)^{n_{1,\beta}}.$$

This conjecture proposes a PT/GV correspondence on Calabi–Yau fourfolds, expressing stable-pair invariants through genus $0$ and genus $1$ GV-type invariants. The source presents it as conjectural and records verification in several examples.