The orbifold DT/PT vertex correspondence

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 Znvertex in terms of loop Schur functions and prove the multi-regular orbifold DT/PT correspondence. Contents 1. Introduction 1 2. Preliminaries 4 2.1. Geometry of toric Calabi-Yau 3-orbifolds 5 2.2. Orbifold PT stable pairs 6 2.3. The DT partition function 8 3. The orbifold PT topological vertex 8 3.1. Partitions, labelled box configurations and the topological vertex 8 3.2. K-theory decomposition of orbifold PT stable pairs 12 3.3. The PT partition function 15 3.4. The PT topological vertex formalism 20 4. Orbifold DT theory and the graphical condensation recurrence 22 4.1. The dimer model for orbifold DT theory 22 4.2. The graphical condensation recurrence for orbifold DT theory 25 4.3. Weights for orbifold DT theory 28 5. Orbifold PT theory and the graphical condensation recurrence 37 5.1. Labelled box configurations and AB configurations for orbifold PT theory 37 5.2. The double-dimer model for orbifold PT theory 38 5.3. The graphical condensation recurrence for orbifold PT theory 40 5.4. Weights for orbifold PT theory 44 6. Orbifold DT/PT correspondence 52 6.1. A proof of orbifold DT/PT vertex correspondence 52 6.2. Remarks on orbifold GW/DT correspondence and DT crepant resolution conjecture 54 References 54