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

Taro Kimura, Go Noshita

TL;DR

This work advances the BPS/CFT dictionary by introducing Donaldson--Thomas $qq$-characters as operator lifts of equivariant DT vertices and develops a comprehensive free-field realization framework for D4, D6, and D8 brane configurations with nontrivial boundary data. It constructs DT$qq$-characters across D4, D6, and D8 sectors, including legs, surfaces, and hypersurfaces, and provides explicit contour-integral and vertex-operator realizations that reproduce DT and magnificent four partition functions. A central achievement is the demonstration of commutativity and sign rules for D6 and D8 $qq$-characters, justified via infinite-product fusion and supported by low-instanton checks and plethystic interpretations, which underpins a consistent BPS/CFT correspondence. The results pave the way toward webs or 4G networks of BPS $qq$-characters, enabling gluing rules for toric Calabi--Yau 4-folds and connecting to DT/PT-type counts, with future work aimed at PT $qq$-characters and a full edge/face gluing formalism.

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.

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

TL;DR

This work advances the BPS/CFT dictionary by introducing Donaldson--Thomas -characters as operator lifts of equivariant DT vertices and develops a comprehensive free-field realization framework for D4, D6, and D8 brane configurations with nontrivial boundary data. It constructs DT-characters across D4, D6, and D8 sectors, including legs, surfaces, and hypersurfaces, and provides explicit contour-integral and vertex-operator realizations that reproduce DT and magnificent four partition functions. A central achievement is the demonstration of commutativity and sign rules for D6 and D8 -characters, justified via infinite-product fusion and supported by low-instanton checks and plethystic interpretations, which underpins a consistent BPS/CFT correspondence. The results pave the way toward webs or 4G networks of BPS -characters, enabling gluing rules for toric Calabi--Yau 4-folds and connecting to DT/PT-type counts, with future work aimed at PT -characters and a full edge/face gluing formalism.

Abstract

We further develop the BPS/CFT correspondence between quiver W-algebras/-characters and partition functions of gauge origami. We introduce -characters associated with multi-dimensional partitions with nontrivial boundary conditions which we call Donaldson--Thomas (DT) -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 -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 -characters with no boundary conditions all commute with each other and discuss its physical interpretation.

Paper Structure

This paper contains 88 sections, 49 theorems, 408 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Partition functions
  3. D4 partition functions
  4. Spiked instantons and finite Young diagrams
  5. Young diagrams with infinite size
  6. D4 partition functions with boundary conditions
  7. D6 partition functions
  8. Tetrahedron instantons and finite plane partitions
  9. Plane partitions with infinite size
  10. D6 partition functions with boundary conditions
  11. Surface boundary condition
  12. Leg boundary conditions
  13. Tetrahedron instantons
  14. D8 partition functions
  15. Magnificent four and finite solid partitions
  16. ...and 73 more sections

Key Result

Proposition 3.2

The operator products of the operators $\mathsf{A}(x),\mathsf{S}_{\bar{a}}(x),\mathsf{X}_{A}(x)\,(A\in\underline{\textbf{6}}),\mathsf{W}_{\bar{a}}(x)\,(a\in\underline{\textbf{4}}),\mathsf{Z}(K,x)$ are where the structure functions are given in eq:structure-funct.

Figures (2)

  • Figure 1: Decomposition of solid partitions. The solid partition extends in the 4-axis and for each layer orthogonal to the 4-axis, we have plane partitions. The plane partitions get smaller along the 4-axis which comes from the condition \ref{['eq:solidpartitionmelting']} of the solid partition.
  • Figure 2: Positions of boxes

Theorems & Definitions (87)

  • Remark 2.1
  • Remark 2.2
  • Definition 3.1: Kimura:2023bxy
  • Proposition 3.2: Kimura:2023bxy
  • Proposition 3.3: Kimura:2023bxy
  • Proposition 3.4: D4 two-legs
  • Proposition 3.5
  • Proposition 3.6: D6 one-leg
  • Proposition 3.7: D6 two-legs
  • Proposition 3.8: D6 three-legs
  • ...and 77 more