Seminar

Integrals of characteristic classes of tautological sheaves on the Hilbert scheme of points on a surface often arise in enumerative problems. We explain how to use the K-theoretic Donaldson-Thomas theory of toric Calabi-Yau threefolds to study K-theoretic versions of such expressions.

Namely, we explicate a precise relationship between K-theoretic Donaldson-Thomas theory and the refined topological vertex of Iqbal, Kosçaz and Vafa. Applying such results to specific toric threefolds, we deduce dualities satisfied by generating functions built from tautological bundles on the Hilbert scheme of points on the complex plane. We then explain how to use these dualities to evaluate certain Euler characteristics of tautological bundles on the Hilbert scheme of points on a general surface.