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

TL;DR

This paper develops generating functions for colored 3D Young diagrams linked to orbifold invariants, connecting combinatorics with Donaldson-Thomas theory and proposing a crepant resolution conjecture.

Contribution

It introduces new multivariate generating functions for colored 3D Young diagrams and relates them to Donaldson-Thomas invariants of orbifolds, extending vertex operator methods.

Findings

Derived generating functions for colored 3D Young diagrams.

Connected diagram generating functions to Donaldson-Thomas invariants.

Proposed a crepant resolution conjecture for orbifold DT theory.

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.