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 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.

Higher rank K-theoretic Donaldson-Thomas theory of points

“…Remark 4.3. As remarked in [46,1], choices of square roots of K vir differ by a 2-torsion element in the Picard group, which implies that χ (Quot 3 ( ⊕r , n), vir ) does not depend on such choices of square roots. Thus there is no ambiguity in (4.4).…”

“…Here, 0] (X ) defined by Illusie [26]. A perfect obstruction theory is called symmetric (see [8]) if there exists an isomorphism θ : → ∨ [1] such that θ = θ ∨ [1].…”

“…For the latter, see some recent developments after Nekrasov-Okounkov, e.g. [46,49,1,62] and [45,47,13] for a generalisation to Calabi-Yau 4-folds. In this paper we deal with K-theoretic DT theory.…”

Hyperbolic localization of the Donaldson-Thomas sheaf

“…The equation (1.3) in the non-projective case explains the origin of this ambiguity: one computes by toric localization only the virtual cohomology of the attracting variety, which is not the whole moduli space and depends on the chosen C * -action. This dependency on the slope was studied explicitly in [27] for the moduli space of framed representations of a toric quiver, and this was related to the ambiguity in the refined topological vertex of [28]. In this case, there is a two dimensional torus invariant acting on the moduli space of framed representations, scaling the arrows of the quiver by leaving the potential invariant, hence the space of slopes is P 1 R .…”

“…The fixed points can be described as molten crystals from [29]. In [27,Prop 3.3], it was established that there is a wall and chamber structure on the space of slopes, with the generating functions of framed invariants being constant in a chamber and jumping at a wall, the wall corresponding to slopes where the weight of an elementary cycle of the quiver becomes attracting or repelling. This is quite strange at first sight, because inside a given chamber the cohomological weight of a given molten crystal does changes on many walls, but the final result does not change: those walls are 'invisible'.…”