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

Abstract

We derive two multivariate generating functions for three-dimensional Young diagrams (also called plane partitions). The variables correspond to a colouring of the boxes according to a finite Abelian subgroup G of SO(3). We use the vertex operator methods of Okounkov--Reshetikhin--Vafa for the easy case G = Z/n; to handle the considerably more difficult case G=Z/2 x Z/2, we will 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 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.

38 pages, 10 figures. v2:fixed errors in defn 1.1, defn of e_n and h_n, the first displayed equation in the appendix (by Jim Bryan), and the sign for orbifold DT invariants of [C3/Z_3] in Remark A.5. Added example in section 3, updated bibliography, and made slight clarifications and changes in wording throughout