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

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.