DT4 correspondence conjectures for Calabi–Yau fourfold invariants

Let $X$ be a projective Calabi–Yau $4$-fold. Write $\operatorname{Hilb}^n(X)$ for the Hilbert scheme of $n$ points on $X$, let $L\to X$ be a line bundle, and distinguish Gromov–Witten, Donaldson–Thomas theory in dimension four (DT4), and Pandharipande–Thomas style invariants, including their reduced versions for holomorphic symplectic $4$-folds. DT4 correspondence conjectures. The literature proposes relations between conventional invariants of $X$ and DT4 invariants, including: an explicit generating function for $\int_{\operatorname{Hilb}^n(X)}c_n(L^{[n]})$; formulae for integrals of Segre classes, Verlinde classes, and Nekrasov genera over $\operatorname{Hilb}^n(X)$; correspondences between genus $0$ Gromov–Witten invariants and $1$-dimensional DT4 invariants; correspondences between genus $0,1$ Gromov–Witten invariants and Pandharipande–Thomas style DT4 invariants; correspondences between genus $0,1$ Gromov–Witten invariants and rank $1$ DT4 invariants; and, for holomorphic symplectic $4$-folds, correspondences between reduced genus $0,1,2$ Gromov–Witten invariants and reduced DT4 invariants counting $1$-dimensional sheaves, as well as reduced Pandharipande–Thomas style DT4 invariants. These proposals seek to relate conventional enumerative invariants, possibly after a change of variables in generating functions, to DT4 invariants. The source summarizes conjectures made in the cited literature rather than asserting a single new formula; their general status is not resolved here.