Kavli Affiliate: Yukinobu Toda
| First 5 Authors: Tudor Pădurariu, Yukinobu Toda, , ,
| Summary:
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. We also study the topological K-theory of DT categories and
prove similar results as above.
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