DT4/Gopakumar–Vafa correspondence for holomorphic symplectic 4-folds

Let $X$ be a holomorphic symplectic $4$-fold and let $M_\beta$ be the moduli scheme of one-dimensional stable sheaves $F$ on $X$ with $\operatorname{ch}_3(F)=\beta$ and $\chi(F)=1$. Choose a reduced virtual class $[M_\beta]^{\mathrm{vir}}\in A_2(M_\beta,\mathbb{Q})$ and an orientation, and define $\operatorname{DT}_4$ descendent invariants by

$$\left\langle\tau_{k_1}(\gamma_1),\ldots,\tau_{k_n}(\gamma_n)\right\rangle^{\operatorname{DT}_4}_\beta=\int_{[M_\beta]^{\mathrm{vir}}}\prod_{i=1}^n\tau_{k_i}(\gamma_i).$$

Let $n_{0,\beta}(\gamma_1,\ldots,\gamma_n)$, $n_{1,\beta}(\gamma)$, and $n_{2,\beta}$ denote the Gopakumar–Vafa invariants defined in the paper. DT4/Gopakumar–Vafa correspondence. For a certain choice of orientation, if $\beta$ is an effective curve class, then

$$\left\langle\tau_0(\gamma_1),\ldots,\tau_0(\gamma_n)\right\rangle^{\operatorname{DT}_4}_\beta=n_{0,\beta}(\gamma_1,\ldots,\gamma_n).$$

If $\beta$ is a primitive curve class, then

$$\left\langle\tau_1(\gamma)\right\rangle^{\operatorname{DT}_4}_\beta=-\frac{1}{2}\left\langle\tau_1(\gamma)\right\rangle^{\operatorname{GW}}_{0,\beta}-n_{1,\beta}(\gamma),$$

and

$$-\left\langle\tau_3(1)\right\rangle^{\operatorname{DT}_4}_\beta-\frac{1}{12}\left\langle\tau_1(c_2(X))\right\rangle^{\operatorname{DT}_4}_\beta=n_{2,\beta}.$$

The conjecture proposes a sheaf-theoretic interpretation of all genus Gopakumar–Vafa invariants through reduced $4$-dimensional Donaldson–Thomas invariants. The source gives no resolution status.