Generating functions for colored 3D Young diagrams and the Donaldson-Thomas invariants of orbifolds

Abstract

We derive two multivariate generating functions for three-dimensional (3D) Young diagrams (also called plane partitions). The variables correspond to a coloring of the boxes according to a finite Abelian subgroup $G$ of $\mathrm{SO}(3)$. These generating functions turn out to be orbifold Donaldson-Thomas partition functions for the orbifold $[{\mathbb{C}}^3/G]$. We need only the vertex operator methods of Okounkov, Reshetikhin, and Vafa for the easy case $G={\mathbb{Z}}_n$; to handle the considerably more difficult case $G={\mathbb{Z}}_2\times {\mathbb{Z}}_2$, we also use a refinement of the author's recent $q$-enumeration of pyramid partitions.

In the appendix, we relate the diagram generating functions to the Donaldson-Thomas partition functions of the orbifold $[{\mathbb{C}}^3/G]$. We find a relationship between the Donaldson-Thomas partition functions of the orbifold and its $G$-Hilbert scheme resolution. We formulate a crepant resolution conjecture for the Donaldson-Thomas theory of local orbifolds satisfying the hard Lefschetz condition