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

TL;DR

This paper explores K-theoretic Donaldson-Thomas invariants of toric Calabi-Yau threefolds, deriving explicit formulas and dualities that connect to the Hilbert scheme of points on surfaces, advancing enumerative geometry techniques.

Contribution

It introduces new K-theoretic invariants and explicit combinatorial formulas related to the refined topological vertex, revealing dualities in tautological bundle generating functions.

Findings

Explicit formulas for limits of K-theoretic Donaldson-Thomas partition functions.

Dualities in generating functions for tautological bundles.

Connections between Donaldson-Thomas invariants and the Hilbert scheme of points.

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, Kos\c{c}az 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.