Donaldson-Thomas theory for Calabi-Yau 4-folds

Abstract

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 $D{T}_4/GW$ correspondence for $X$. Examples of wall-crossing phenomenon in $D{T}_4$ theory are also given.

v2: 47 pages, some corrections and improvements

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• Calabi-Yau algebras
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Cited by in corpus (32)

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• $K$-theoretic DT/PT correspondence for toric Calabi-Yau 4-folds
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• Curve counting and DT/PT correspondence for Calabi-Yau 4-folds
• Orientability for gauge theories on Calabi-Yau manifolds
• On orientations for gauge-theoretic moduli spaces
• Stable maps to Looijenga pairs
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• Canonical vertex formalism in DT theory of toric Calabi-Yau 4-folds
• Genus zero Gopakumar-Vafa type invariants for Calabi-Yau 4-folds II: Fano 3-folds
• Torus localization and wall crossing for cosection localized virtual cycles
• Counting perverse coherent systems on Calabi-Yau 4-folds
• Dimensional reduction in cohomological Donaldson-Thomas theory
• Homotopy L-infinity spaces
• Counting conics on sextic 4-folds
• Counting zero-dimensional subschemes in higher dimensions
• The homology of moduli stacks of complexes
• Enumerative invariants and wall-crossing formulae in abelian categories
• On Framed Quivers, BPS Invariants and Defects
• Crystals and Double Quiver Algebras from Jeffrey-Kirwan Residues
• Universal structures in $\mathbb{C}$-linear enumerative invariant theories
• Cosection Localization and Vanishing for Virtual Fundamental Classes of D-Manifolds
• Virtual pullbacks in Donaldson-Thomas theory of Calabi-Yau 4-folds
• Remarks on mirror symmetry of Donaldson-Thomas theory for Calabi-Yau 4-folds
• Virtual classes via vanishing cycles
• Correspondence of Donaldson-Thomas and Gopakumar-Vafa invariants on local Calabi-Yau 4-folds over V_5 and V_22
• Mukai flops and Plücker type formulas for hyper-Kähler manifolds
• Wall-crossing for punctual Quot-schemes