Atiyah Class and sheaf counting on local Calabi Yau 4 folds

Emanuel Diaconescu Rutgers University Artan Sheshmani Harvard CMSA/QGM Shing-Tung Yau Harvard University

Algebraic Geometry mathscidoc:1810.01001

We discuss Donaldson-Thomas (DT) invariants of torsion sheaves with 2 dimensional support on a smooth projective surface in an ambient non-compact Calabi Yau fourfold given by the total space of a rank 2 bundle on the surface. We prove that in certain cases, when the rank 2 bundle is chosen appropriately, the universal truncated Atiyah class of these codimension 2 sheaves reduces to one, defined over the moduli space of such sheaves realized as torsion codimension 1 sheaves in a noncompact divisor (threefold) embedded in the ambient fourfold. Such reduction property of universal Atiyah class enables us to relate our fourfold DT theory to a reduced DT theory of a threefold and subsequently then to the moduli spaces of sheaves on the base surface using results in arXiv:1701.08899 and arXiv:1701.08902 and . We finally make predictions about modularity of such fourfold invariants when the base surface is an elliptic K3.
Atiyah class, Calabi Yau 4 fold, Coherent sheaf, Vafa-Witten invariant, Seiberg-Witten invariant
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  • 54 pages.
  title={Atiyah Class and sheaf counting on local Calabi Yau 4 folds},
  author={Emanuel Diaconescu, Artan Sheshmani, and Shing-Tung Yau},
Emanuel Diaconescu, Artan Sheshmani, and Shing-Tung Yau. Atiyah Class and sheaf counting on local Calabi Yau 4 folds. 2018.
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