Categorical and K-theoretic Donaldson-Thomas theory of $\mathbb{C}^3$ (part I)

TL;DR

This paper develops a categorical framework for Donaldson-Thomas invariants of , constructing semiorthogonal decompositions, explicit basis objects, and establishing a K-theoretic analogue of the McKay correspondence and MacMahon's formula.

Contribution

It introduces DT categories for , constructs categorical wall-crossing formulas, and explicitly describes basis objects in K-theory, advancing the categorification of DT invariants.

Findings

Semiorthogonal decompositions of DT categories constructed.

Explicit basis objects in K-theory identified.

K-theoretic analogue of MacMahon's formula established.

Abstract

We begin the study of categorifications of Donaldson-Thomas invariants associated with Hilbert schemes of points on the three-dimensional affine space, which we call DT categories. The DT category is defined to be the category of matrix factorizations on the non-commutative Hilbert scheme with a super-potential whose critical locus is the Hilbert scheme of points. The first main result in this paper is the construction of semiorthogonal decompositions of DT categories, which can be regarded as categorical wall-crossing formulae of the framed triple loop quiver. Each summand is given by the categorical Hall product of some subcategories of matrix factorizations, called quasi-BPS categories. They are categories of matrix factorizations on twisted versions of noncommutative resolutions of singularities considered by \v{S}penko-Van den Bergh, and were used by the first author to prove a PBW theorem for K-theoretic Hall algebras. We next construct explicit objects of quasi-BPS categories via Koszul duality equivalences, and show that they form a basis in the torus localized K-theory. These computations may be regarded as a numerical K-theoretic analogue in dimension three of the McKay correspondence for Hilbert schemes of points. In particular, the torus localized K-theory of DT categories has a basis whose cardinality is the number of plane partitions, giving a K-theoretic analogue of MacMahon's formula.