The 4-fold Pandharipande--Thomas vertex

TL;DR

This paper proposes a detailed conjectural description of the K-theoretic equivariant vertex for Pandharipande--Thomas stable pairs on toric Calabi-Yau 4-folds, linking fixed loci to quiver Grassmannians and verifying the DT/PT correspondence computationally.

Contribution

It provides a full conjectural framework for the PT vertex on toric CY4s, connecting fixed loci to quiver Grassmannians and enabling computational verification.

Findings

Conjectural explicit description of the PT vertex.

Identification of fixed loci as quiver Grassmannians.

Verification of DT/PT correspondence in low degrees.

Abstract

We give a conjectural but full and explicit description of the (K-theoretic) equivariant vertex for Pandharipande--Thomas stable pairs on toric Calabi--Yau 4-folds, by identifying torus-fixed loci as certain quiver Grassmannians and prescribing a canonical half of the tangent-obstruction theory. For any number of non-trivial legs, the DT/PT vertex correspondence can then be verified by computer in low degrees.