K-theoretic Donaldson-Thomas theory and the Hilbert scheme of points on a surface

Abstract

Integrals of characteristic classes of tautological sheaves on the Hilbert scheme of points on a surface frequently arise in enumerative problems. We use the K-theoretic Donaldson-Thomas theory of certain toric Calabi-Yau threefolds to study K-theoretic variants of such expressions. We study limits of the K-theoretic Donaldson-Thomas partition function of a toric Calabi-Yau threefold under certain one-parameter subgroups called slopes, and formulate a condition under which two such limits coincide. We then explicitly compute the limits of components of the partition function under so-called preferred slopes, obtaining explicit combinatorial expressions related to the refined topological vertex of Iqbal, Kosçaz and Vafa. Applying these results to specific Calabi-Yau threefolds, we deduce dualities satisfied by a generating function built from tautological bundles on the Hilbert scheme of points on ${\mathbb{C}}^2$. We then use this duality to study holomorphic Euler characteristics of exterior and symmetric powers of tautological bundles on the Hilbert scheme of points on a general surface.

56 pages, 10 figures, exposition expanded and section on rank two vector bundles added

References in corpus (2)

• The Refined Topological Vertex
• Equivariant K-theory and refined Vafa-Witten invariants

Cited by in corpus (9)

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• Higher rank K-theoretic Donaldson-Thomas theory of points
• Tetrahedron instantons in Donaldson-Thomas theory
• Double nested Hilbert schemes and the local stable pairs theory of curves
• Universal structures in $\mathbb{C}$-linear enumerative invariant theories
• K-Theoretic Descendent Series for Hilbert Schemes of Points on Surfaces
• A Representation-Theoretic Approach to $qq$-Characters
• Big and Nef Tautological Vector Bundles over the Hilbert Scheme of Points
• Equivariant Segre and Verlinde invariants for Quot schemes