Donaldson-Thomas theory for Calabi-Yau 4-folds

Let $X$ be a compact complex Calabi-Yau 4-fold. Under certain assumptions, we define Donaldson-Thomas type deformation invariants ($D{T}_4$ invariants) by studying moduli spaces of solutions to the Donaldson-Thomas equations on $X$. We also study sheaves counting problems on local Calabi-Yau 4-folds. We relate $D{T}_4$ invariants of $K_Y$ to the Donaldson-Thomas invariants of the associated Fano 3-fold $Y$. When the Calabi-Yau 4-fold is toric, we adapt the virtual localization formula to define the corresponding equivariant $D{T}_4$ invariants. We also discuss the non-commutative version of $D{T}_4$ invariants for quivers with relations. Finally, we compute $D{T}_4$ invariants for certain Calabi-Yau 4-folds when moduli spaces are smooth and find a $D{T}_4\mathrm{/}GW$ correspondence for $X$. Examples of wall-crossing phenomenon in $D{T}_4$ theory are also given.