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arxiv: 1407.7659 · v2 · pith:OERZFCLSnew · submitted 2014-07-29 · 🧮 math.AG · math.DG

Donaldson-Thomas theory for Calabi-Yau 4-folds

Yalong Cao , Naichung Conan Leung This is my paper
classification 🧮 math.AG math.DG
keywords invariantscalabi-yaudonaldson-thomasfoldfoldscertaindefinemoduli
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Let $X$ be a compact complex Calabi-Yau 4-fold. Under certain assumptions, we define Donaldson-Thomas type deformation invariants ($DT_{4}$ 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 $DT_{4}$ 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 $DT_{4}$ invariants. We also discuss the non-commutative version of $DT_{4}$ invariants for quivers with relations. Finally, we compute $DT_{4}$ invariants for certain Calabi-Yau 4-folds when moduli spaces are smooth and find a $DT_{4}/GW$ correspondence for $X$. Examples of wall-crossing phenomenon in $DT_{4}$ theory are also given.

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