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

Benjamin Young, Jim Bryan

TL;DR

This work develops multivariate generating functions for colored 3D Young diagrams (plane partitions) under a finite Abelian group $G \subset SO(3)$, with variables assigned to group elements to refine MacMahon-type counts. Using vertex-operator methods, the authors derive explicit product formulas for $Z_{\mathbb{Z}_n}$ and $Z_{\mathbb{Z}_2\times\mathbb{Z}_2}$ and connect these to orbifold Donaldson–Thomas invariants of $[\mathbb{C}^3/G]$, as well as to DT theory on $G$-Hilbert scheme crepant resolutions. They introduce pyramid partitions as an auxiliary combinatorial model and prove a key relation $Z_{\mathbb{Z}_2\times\mathbb{Z}_2} = Z_{pyramid} \cdot \widetilde{M}(q_a q_b, q)$, enabling the $\mathbb{Z}_2\times\mathbb{Z}_2$ case and linking to orbifold DT. The paper also formulates a local crepant resolution conjecture for DT theory under the Hard Lefschetz condition and discusses broader extensions to other toric Calabi–Yau 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.

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

TL;DR

This work develops multivariate generating functions for colored 3D Young diagrams (plane partitions) under a finite Abelian group , with variables assigned to group elements to refine MacMahon-type counts. Using vertex-operator methods, the authors derive explicit product formulas for and and connect these to orbifold Donaldson–Thomas invariants of , as well as to DT theory on -Hilbert scheme crepant resolutions. They introduce pyramid partitions as an auxiliary combinatorial model and prove a key relation , enabling the case and linking to orbifold DT. The paper also formulates a local crepant resolution conjecture for DT theory under the Hard Lefschetz condition and discusses broader extensions to other toric Calabi–Yau 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.

Paper Structure

This paper contains 13 sections, 19 theorems, 119 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Review: the infinite wedge space
  3. The operators $\Gamma(x)$, $\Gamma'(x)$, and $Q_g$
  4. Counting with $Z_n$ colouring
  5. Pyramid partitions
  6. A generating function for pyramid partitions
  7. Counting $\mathbb{Z}_2 \times \mathbb{Z}_2$--coloured 3D Young diagrams
  8. Future work
  9. Don-ald-son-Thom-as theory of ${\mathbb C}^{3}/G$ and its crepant resolution (by Dr. Jim Bryan)
  10. Review of Don-ald-son-Thom-as theory
  11. Orbifold Don-ald-son-Thom-as theory of $[{\mathbb C} ^{3}/G]$
  12. The Don-ald-son-Thom-as crepant resolution conjecture
  13. Proof of Proposition \ref{['prop: DT CRC holds for Zn and\nZ2xZ2']} / Theorem \ref{['thm:resolution_dt_partition_functions']}:

Key Result

Theorem 1.4

Let $G = \mathbb{Z}_n$ and let the colouring $K_{Z_n}$ be given by Let $q=q_0\cdots q_{n-1}$ and for $a,b \in [1, n-1]$, let $q_{[a,b]} = q_aq_{a+1} \cdots q_b$. Then

Figures (10)

  • Figure 1: A partition coloured according to $K_{\mathbb{Z}_2 \times \mathbb{Z}_2}$ and to $K_{\mathbb{Z}_3}$
  • Figure 2: Applying $\alpha_{-3}$ to a partition
  • Figure 3: Slicing a $\mathbb{Z}_3$--coloured 3D diagram
  • Figure 4:
  • Figure 5:
  • ...and 5 more figures

Theorems & Definitions (49)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 2.1
  • Theorem 2.2
  • Corollary 2.3
  • ...and 39 more