Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The orbifold DT/PT vertex correspondence

Yijie Lin

TL;DR

The work develops an orbifold topological vertex framework for PT and DT theories on toric Calabi–Yau 3-orbifolds with transverse $A_{n-1}$ singularities, culminating in a full 3-leg DT/PT vertex correspondence. It constructs both DT and PT dimer/double-dimer models on orbifold graphs, derives three graphical condensation recurrences on each side, and computes explicit weight factors and sign rulings that certify the correspondence. A key outcome is an explicit formula for the PT $Z_n$-vertex in terms of loop Schur data and the establishment of multi-regular DT/PT equivalence in this orbifold setting. The approach unifies combinatorial, geometric, and representation-theoretic ingredients, enabling explicit computations of orbifold DT/PT invariants and vertex functions. The results bear on broader GW/DT/PT correspondences and crepant-resolutions-type phenomena in orbifold Calabi–Yau geometries.

Abstract

We present an orbifold topological vertex formalism for PT invariants of toric Calabi-Yau 3-orbifolds with transverse $A_{n-1}$ singularities. We give a proof of the orbifold DT/PT Calabi-Yau topological vertex correspondence. As an application, we derive an explicit formula for the PT $\mathbb{Z}_{n}$-vertex in terms of loop Schur functions and prove the multi-regular orbifold DT/PT correspondence.

The orbifold DT/PT vertex correspondence

TL;DR

The work develops an orbifold topological vertex framework for PT and DT theories on toric Calabi–Yau 3-orbifolds with transverse singularities, culminating in a full 3-leg DT/PT vertex correspondence. It constructs both DT and PT dimer/double-dimer models on orbifold graphs, derives three graphical condensation recurrences on each side, and computes explicit weight factors and sign rulings that certify the correspondence. A key outcome is an explicit formula for the PT -vertex in terms of loop Schur data and the establishment of multi-regular DT/PT equivalence in this orbifold setting. The approach unifies combinatorial, geometric, and representation-theoretic ingredients, enabling explicit computations of orbifold DT/PT invariants and vertex functions. The results bear on broader GW/DT/PT correspondences and crepant-resolutions-type phenomena in orbifold Calabi–Yau geometries.

Abstract

We present an orbifold topological vertex formalism for PT invariants of toric Calabi-Yau 3-orbifolds with transverse singularities. We give a proof of the orbifold DT/PT Calabi-Yau topological vertex correspondence. As an application, we derive an explicit formula for the PT -vertex in terms of loop Schur functions and prove the multi-regular orbifold DT/PT correspondence.
Paper Structure (28 sections, 67 theorems, 295 equations, 3 figures)

This paper contains 28 sections, 67 theorems, 295 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Geometry of toric Calabi-Yau 3-orbifolds
  4. Orbifold PT stable pairs
  5. The DT partition function
  6. The orbifold PT topological vertex
  7. Partitions, labelled box configurations and topological vertices
  8. $K$-theory decomposition of orbifold PT stable pairs
  9. The PT partition function
  10. The PT topological vertex formalism
  11. Orbifold DT theory and the graphical condensation recurrence
  12. The dimer model for orbifold DT theory
  13. The graphical condensation recurrence for orbifold DT theory
  14. Weights for orbifold DT theory
  15. Weights for the graphical condensation recurrence with $\mathbf{G}=\mathbf{H}(N;\lambda^{rc},\mu^{rc},\nu)$
  16. ...and 13 more sections

Key Result

Theorem 1.1

(see Theorem and [Zhang, Theorem/Conjecture 4.19]) Let $\mathcal{X}$ be a toric CY 3-orbifold with transverse $A_{n-1}$ singularities. Then the PT partition function $PT(\mathcal{X})$ is where and with edges $(e,f_{v_{0}},f^\prime_{v_{0}},g_{v_{\infty}},g^\prime_{v_{\infty}})$ oriented as in Figure (see other notations in Section 3).

Figures (3)

  • Figure 1: Orientations of the edge $e$ and its adjacent edges.
  • Figure 2: The left graph $\mathbf{H}(4)$ is divided into 3 sectors with the labellings of outer vertices for DT case. The right graph is again $\mathbf{H}(4)$ but divided into another 3 sectors with the different labellings of outer vertices for PT case.
  • Figure 3: The graph $\mathbf{H}(4)$ with weights as assigned in Definition .

Theorems & Definitions (120)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Definition 3.1
  • Definition 3.2
  • Remark 3.3
  • Definition 3.4
  • Remark 3.5
  • Definition 3.6
  • ...and 110 more