The 4-fold Pandharipande--Thomas vertex and Jeffrey--Kirwan residue

TL;DR

This paper develops a contour integral approach for computing the K-theoretic PT 4-vertex using Jeffrey--Kirwan residues, revealing connections to DT invariants and exploring higher rank generalizations.

Contribution

It introduces a novel contour integral formalism for the PT 4-vertex within the JK residue framework, linking it to DT invariants and extending to higher rank cases.

Findings

PT 4-vertex can be derived from the DT 4-vertex integrand via different reference vectors.

The formalism is demonstrated through examples involving curves and surfaces on 4-folds.

The work explores the DT/PT correspondence and its higher rank and supergroup-like generalizations.

Abstract

We present a contour integral formalism for computing the K-theoretic equivariant Pandharipande--Thomas (PT) 4-vertex. Within the Jeffrey--Kirwan (JK) residue framework, we show that the PT 4-vertex can be obtained from the same integrand as the Donaldson--Thomas (DT) 4-vertex by choosing a different reference vector. We illustrate the formalism through examples involving curves and surfaces on the 4-fold. Furthermore, we investigate the DT/PT correspondence for the 4-fold setting together with its higher rank and supergroup-like generalizations.