Coproducts in the cohomological DT theory of 3-Calabi-Yau completions

26 October 2020
Ben Davison
Given a suitably friendly category D we can take the 3-Calabi Yau completion of D and obtain a 3-Calabi-Yau category E. The archetypal example has D as the category of coherent sheaves on a smooth quasiprojective surface, then E is the category of coherent sheaves on the total space of the canonical bundle - a quasiprojective 3CY variety. The moduli stack of semistable objects in the 3CY completion E supports a vanishing cycle-type sheaf, the hypercohomology of which is the basic object in the study of the DT theory of E. Something extra happens when our input category is itself 2CY: examples include the category of local systems on a Riemann surface, the category of coherent sheaves on a K3/Abelian surface, the category of Higgs bundles on a smooth complete curve, or the category of representations of a preprojective algebra. In these cases, the DT cohomology of E carries a cocommutative coproduct. I'll also explain how this interacts with older algebraic structures in cohomological DT theory to provide a geometric construction of both well-known and new quantum groups.

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  • Geometry and Analysis Seminar