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

TL;DR

This work develops a contour-integral framework for the K-theoretic equivariant 3-vertex in gauge origami, showing that, with an appropriate JK reference vector, both the equivariant DT3 and PT3 vertices emerge from the same integrand. It systematically analyzes DT3, PT3, and their higher-rank generalizations, establishing a DT/PT correspondence and mapping PT3 to Macdonald-refined and refined topological vertices in several limits. A central advance is the introduction of Pandharipande–Thomas $qq$-characters as operator lifts of PT3 data, with free-field realizations and screening charges tying the counting problems to quantum toroidal $rak{gl}_1$ structure. The results illuminate wall-crossing phenomena in this setting and provide a robust framework to study higher-rank and supergroup generalizations, including tetrahedron instantons. Collectively, the paper bridges gauge-theory partition functions, refined topological string vertices, and BPS/CFT operators in a unified JK-residue/dyson–Schwinger context with potential implications for enumerative geometry and 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 $\mathfrak{gl}_{1}$.

Figures (26)

• Figure 1: Left: Minimal plane partition with three boundary Young diagrams $\lambda=\{3,2,1\}$, $\mu=\{2,1\}$, and $\nu=\{3,1\}$. Right: The corresponding 2d lattice of the minimal plane partition.
• Figure 2: Plane partitions with boundary conditions.
• Figure 3: PT configurations for $\lambda=\mu=\varnothing$ and $\nu=\{3,1\}$ with one and two boxes. The coordinates of the set of orange boxes are $(\epsilon_{1}-\epsilon_{3})$, $(2\epsilon_{2}-\epsilon_{3})$, $(\epsilon_{1}-\epsilon_{3},2\epsilon_{2}-\epsilon_{3})$, $(\epsilon_{1}-\epsilon_{3},\epsilon_{1}-2\epsilon_{3})$, $(\epsilon_{2}-\epsilon_{3},2\epsilon_{2}-\epsilon_{3})$, and $(2\epsilon_{2}-\epsilon_{3},2\epsilon_{2}-2\epsilon_{3})$, respectively.
• Figure 4: Construction of the PT box-counting with two nontrivial legs: $\lambda=\{3,2,1\}$, $\mu=\{2,1\}$, and $\nu=\varnothing$. Left: Configuration with asymptotic cylinders extended in both directions obtained from the minimal plane partition. Right: Configuration with boxes in the positive directions removed and boxes in the intersection colored. The blue boxes are the boxes belonging to the intersection of the two legs and they are kept.
• Figure 5: Construction of the PT box-counting with three trivial legs: $\lambda=\{3,2,1\}$, $\mu=\{2,1\}$, and $\nu=\{3,1\}$. Left: Configuration with asymptotic cylinders extended in both directions obtained from the minimal plane partition. Right: Configuration with boxes in the positive directions removed and boxes in the intersection colored. Red boxes are boxes belonging to the three legs.

Theorems & Definitions (57)

• Theorem 2.1
• Theorem 2.2
• Definition 3.1
• Theorem 3.2
• Definition 3.3
• Proposition 3.4
• Theorem 3.5
• Theorem 3.6
• Proposition 3.7
• Theorem 3.8