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Donaldson-Thomas theory associates integers (which are virtual counts of sheaves) to a Calabi-Yau threefold X. The simplest example is that of C^3, when the Donaldson-Thomas (DT) invariant of sheaves of zero dimensional support and length d is p(d), the number of plane partitions of d. The DT invariants can be recovered as the Euler characteristic of a collection of vector spaces. It is natural to ask whether for a given Calabi-Yau threefold X, there exists a categorification of these vector spaces (and thus of DT invariants), for example a category whose periodic cyclic homology recovers (a Z/2-periodic version of) these vector spaces, and thus recovers the DT invariants of X. There exist such categorifications, constructed using matrix factorizations, when X is the total space of the canonical bundle over a surface S.

I will focus on the categorification C of DT invariants for C^3, in particular we will construct semi-orthogonal decompositions of C and study the K-theory of C, and explain how these computations relate to p(d).

This is joint work with Yukinobu Toda.