Gauge origami and quiver W-algebras IV: Pandharipande--Thomas $qq$-characters

TL;DR

This paper introduces a contour integral formalism for computing equivariant 3-vertices in K-theory, unifies DT and PT vertices via Jeffrey--Kirwan residues, and explores their limits, algebraic structures, and connections to quantum algebras.

Contribution

It develops a unified contour integral approach for DT and PT 3-vertices, analyzes their limits, and constructs the PT $qq$-character linked to quantum toroidal algebras.

Findings

Unified DT and PT 3-vertices via contour integrals

Recovered various topological vertices as limits

Constructed the PT $qq$-character and related it to quantum algebras

Abstract

We develop a contour integral formalism for computing the K-theoretic equivariant 3-vertex. Within the Jeffrey--Kirwan (JK) residue framework, we show that, by an appropriate choice of the reference vector, both the equivariant Donaldson--Thomas (DT) and Pandharipande--Thomas (PT) 3-vertices can be extracted from the same integrand. We analyze three distinct limits of the PT 3-vertex, recovering the unrefined topological vertex, the refined topological vertex, and the Macdonald refined topological vertex. Higher-rank extensions of PT counting and the DT/PT correspondence are also explored. From a quantum algebraic perspective, we construct an operator version of the equivariant PT 3-vertex and term it the Pandharipande--Thomas $qq$-character. We then discuss its connection with the quantum toroidal $\mathfrak{gl}_{1}$.