Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the 1-leg Donaldson-Thomas $\mathbb{Z}_2\times\mathbb{Z}_2$-vertex

Yijie Lin

TL;DR

This work addresses the open problem of an explicit formula for the 1-leg Donaldson-Thomas vertex of the orbifold $\mathbb{Z}_2\times\mathbb{Z}_2$. It introduces restricted pyramid configurations to encode leg-induced interlacing and identifies a unique symmetric interlacing class for staircase leg partitions $\nu=(m,m-1,\dots,1)$, enabling an explicit 1-leg $\mathbb{Z}_2\times\mathbb{Z}_2$-vertex formula via a connection to the 1-leg $\mathbb{Z}_4$-vertex using vertex-operator methods. The paper develops a detailed operator framework, proves the equality of two RPC families in the staircase case, and derives an explicit staircase formula that depends on $m\bmod 4$. It also clarifies the relation between the $\mathbb{Z}_2\times\mathbb{Z}_2$-vertex and the $\mathbb{Z}_4$-vertex, thereby establishing a concrete bridge between different orbifold DT vertices and providing a pathway for extending to more general leg partitions and higher-leg cases.

Abstract

We introduce a notion of restricted pyramid configurations for computing the 1-leg Donaldson-Thomas $\mathbb{Z}_2\times\mathbb{Z}_2$-vertex. We study a special type of restricted pyramid configurations with the prescribed 1-leg partitions, and find one unique class of them satisfying the symmetric interlacing property. This leads us to obtain an explicit formula for a class of 1-leg Donaldson-Thomas $\mathbb{Z}_2\times\mathbb{Z}_2$-vertex through establishing its connection with 1-leg Donaldson-Thomas $\mathbb{Z}_4$-vertex using the vertex operator methods of Okounkov-Reshetikhin-Vafa and Bryan-Young.

On the 1-leg Donaldson-Thomas $\mathbb{Z}_2\times\mathbb{Z}_2$-vertex

TL;DR

This work addresses the open problem of an explicit formula for the 1-leg Donaldson-Thomas vertex of the orbifold . It introduces restricted pyramid configurations to encode leg-induced interlacing and identifies a unique symmetric interlacing class for staircase leg partitions , enabling an explicit 1-leg -vertex formula via a connection to the 1-leg -vertex using vertex-operator methods. The paper develops a detailed operator framework, proves the equality of two RPC families in the staircase case, and derives an explicit staircase formula that depends on . It also clarifies the relation between the -vertex and the -vertex, thereby establishing a concrete bridge between different orbifold DT vertices and providing a pathway for extending to more general leg partitions and higher-leg cases.

Abstract

We introduce a notion of restricted pyramid configurations for computing the 1-leg Donaldson-Thomas -vertex. We study a special type of restricted pyramid configurations with the prescribed 1-leg partitions, and find one unique class of them satisfying the symmetric interlacing property. This leads us to obtain an explicit formula for a class of 1-leg Donaldson-Thomas -vertex through establishing its connection with 1-leg Donaldson-Thomas -vertex using the vertex operator methods of Okounkov-Reshetikhin-Vafa and Bryan-Young.
Paper Structure (16 sections, 56 theorems, 274 equations)

This paper contains 16 sections, 56 theorems, 274 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Vertex operators
  4. Pyramid partitions
  5. The orbifold topological vertex
  6. Restricted pyramid configurations
  7. Restricted pyramid configurations
  8. Restricted pyramid configurations with symmetric interlacing property
  9. Other restricted pyramid configurations with symmetric interlacing property
  10. The 1-leg DT $\mathbb{Z}_2\times\mathbb{Z}_2$-vertex
  11. Vertex operator products, topological vertices and restricted pyramid configurations
  12. 1-leg DT $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$-vertex and restricted pyramid configurations
  13. 1-leg DT $\mathbb{Z}_{4}$-vertex and restricted pyramid configurations
  14. 1-leg DT $\mathbb{Z}_2\times\mathbb{Z}_2$-vertex and 1-leg DT $\mathbb{Z}_{4}$-vertex
  15. An explicit formula for a class of 1-leg DT $\mathbb{Z}_2\times\mathbb{Z}_2$-vertex
  16. ...and 1 more sections

Key Result

Theorem 1.1

(see Theorem Z2Z2-Z4) Assume $\nu=(m,m-1,\cdots,2,1)$ with $m\in\mathbb{Z}_{\geq1}$. Let $q=q_aq_bq_cq_0$. Then we have (i) if $m\equiv0\; \hbox{(mod 4)}$ or $m\equiv3\; \hbox{(mod 4)}$, (ii) if $m\equiv1\; \hbox{(mod 4)}$ or $m\equiv2\; \hbox{(mod 4)}$, where

Theorems & Definitions (136)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Definition 2.2
  • Lemma 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Lemma 2.8
  • ...and 126 more