Gauge origami and quiver W-algebras III: Donaldson--Thomas $qq$-characters

TL;DR

This paper advances the understanding of gauge origami and quiver W-algebras by introducing Donaldson--Thomas $qq$-characters, connecting them to partition functions of toric Calabi--Yau spaces, and exploring their algebraic properties.

Contribution

It introduces DT $qq$-characters for multi-dimensional partitions with boundary conditions and provides a quantum algebraic derivation of sign rules for the partition function.

Findings

DT $qq$-characters are operator versions of equivariant DT vertices.

D6 and D8 $qq$-characters commute under proper sign rules.

Revisits construction of D8 $qq$-characters and their algebraic properties.

Abstract

We further develop the BPS/CFT correspondence between quiver W-algebras/$qq$-characters and partition functions of gauge origami. We introduce $qq$-characters associated with multi-dimensional partitions with nontrivial boundary conditions which we call Donaldson--Thomas (DT) $qq$-characters. They are operator versions of the equivariant DT vertices of toric Calabi--Yau three and four-folds. Moreover, we revisit the construction of the D8 $qq$-characters with no boundary conditions and give a quantum algebraic derivation of the sign rules of the magnificent four partition function. We also show that under the proper sign rules, the D6 and D8 $qq$-characters with no boundary conditions all commute with each other and discuss its physical interpretation.