We study Hilbert schemes of points on a smooth projective Calabi-Yau 4-fold X. We define DT 4 invariants by integrating the Euler class of a tautological vector bundle L [n] against the virtual class. We conjecture a formula for their generating series, which we prove in certain cases when L corresponds to a smooth divisor on X. A parallel equivariant conjecture for toric Calabi-Yau 4-folds is proposed. This conjecture is proved for smooth toric divisors and verified for more general toric divisors in many examples.Combining the equivariant conjecture with a vertex calculation, we find explicit positive rational weights, which can be assigned to solid partitions. The weighted generating function of solid partitions is given by exp(M (q) − 1), where M (q) denotes the MacMahon function.The virtual class (1.1) depends on the choice of orientation o(L). On each connected component of Hilb n (X), there are two choices of orientations, which affects the corresponding contribution to the class (1.1) by a sign. We review facts about the DT 4 virtual class in Section 2.1.In order to define the invariants, we require insertions. Let L be a line bundle on X and denote by L [n] the tautological (rank n) vector bundle over Hilb n (X) with fibre H 0 (L| Z ) over Z ∈ Hilb n (X). Then it makes sense to define the following: Definition 1.1. Let X be a smooth projective Calabi-Yau 4-fold and let L be a line bundle on X. Let L be the determinant line bundle of Hilb n (X) with quadratic form Q induced from Serre duality. Suppose L is given an orientation o(L). We define DT 4 (X, L, n ; o(L)) := [Hilb n (X)] vir o(L) e(L [n] ) ∈ Z, if n 1,where e(−) denotes the Euler class. We also set DT 4 (X, L, 0 ; o(L)) := 1.We make the following conjecture for the generating series of these invariants: 1 2 YALONG CAO AND MARTIJN KOOL Conjecture 1.2 (Conjecture 2.2). Let X be a smooth projective Calabi-Yau 4-fold and L be a line bundle on X. There exist choices of orientation such that ∞ n=0 DT 4 (X, L, n ; o(L)) q n = M (−q) X c1(L)·c3(X) , where

As an analogy to Gopakumar-Vafa conjecture on CY 3-folds, Klemm-Pandharipande defined GV type invariants on CY 4-folds using GW theory and conjectured their integrality. In this paper, we define stable pair type invariants on CY 4-folds and use them to interpret these GV type invariants. Examples are computed for both compact and non-compact CY 4-folds to support our conjectures. 1 0.4. Verifications of the conjecture I: compact examples. We first prove our conjectures for some special compact Calabi-Yau 4-folds.Sextic 4-folds. Let X ⊆ P 5 be a degree six smooth hypersurface and [l] ∈ H 2 (X, Z) ∼ = H 2 (P 5 , Z) be the line class. We check our conjectures for β = [l] and 2[l]. Proposition 0.3. (Proposition 2.1, 2.2) Let X be a smooth sextic 4-fold and [l] ∈ H 2 (X, Z) be the line class. Then Conjecture 0.1 and 0.2 are true for β = [l] and 2[l].Elliptic fibrations. We consider a projective CY 4-fold X which admits an elliptic fibration π : X → P 3 ,given by a Weierstrass model (2.1). Let f be a general fiber of π and h be a hyperplane in P 3 , set B = π * h, E = ι(P 3 ) ∈ H 6 (X, Z), where ι is a section of π. Then we have

Recently, Cao-Maulik-Toda defined stable pair invariants of a compact Calabi-Yau 4-fold X. Their invariants are conjecturally related to the Gopakumar-Vafa type invariants of X defined using Gromov-Witten theory by Klemm-Pandharipande. In this paper, we consider curve counting invariants of X using Hilbert schemes of curves and conjecture a DT/PT correspondence which relates these to stable pair invariants of X. After providing evidence in the compact case, we define analogous invariants for toric Calabi-Yau 4-folds. We formulate a vertex formalism for both theories and conjecture a relation between the (fully equivariant) DT/PT vertex, which we check in several cases. This relation implies a DT/PT correspondence for toric Calabi-Yau 4-folds with primary insertions.