Gauge origami and quiver W-algebras

Abstract

We explore the quantum algebraic formalism of the gauge origami system in ${\mathbb{C}}^4$, where D2/D4/D6/D8-branes are present. We demonstrate that the contour integral formulas have free field interpretations, leading to the operator formalism of $qq$-characters associated with each D-brane. The $qq$-characters of D2 and D4-branes correspond to screening charges and generators of the affine quiver W-algebra, respectively. On the other hand, the $qq$-characters of D6 and D8-branes represent novel types of $qq$-characters, where monomial terms are characterized by plane partitions and solid partitions. The composition of these $qq$-characters yields the instanton partition functions of the gauge origami system, eventually establishing the BPS/CFT correspondence. Additionally, we demonstrate that the fusion of $qq$-characters of D-branes in lower dimensions results in higher-dimensional D-brane $qq$-characters. We also investigate quadratic relations among these $qq$-characters. Furthermore, we explore the relationship with the representations, $q$-characters, and the Bethe ansatz equations of the quantum toroidal ${\mathfrak{gl}}_1$. This connection provides insights into the Bethe/Gauge correspondence of the gauge origami system from both gauge-theoretic and quantum-algebraic perspectives. We finally present conjectures regarding generalizations to general toric Calabi-Yau four-folds. These generalizations imply the existence of an extensive class of $qq$-characters, which we call BPS $qq$-characters. These BPS $qq$-characters offer a new systematic approach to derive a broader range of BPS/CFT correspondence and Bethe/Gauge correspondence.

arXiv comment149+36 pages, v2: fixed typos and added references, published version

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• Gauge origami and quiver W-algebras III: Donaldson--Thomas $qq$-characters
• Crystals and Double Quiver Algebras from Jeffrey-Kirwan Residues
• An Overview of Crystals and Double Quiver Yangians
• Gauge origami and quiver W-algebras IV: Pandharipande--Thomas $qq$-characters
• Tunnels Under Geometries (or Instantons Know Their Algebras)
• The 4-fold Pandharipande--Thomas vertex and Jeffrey--Kirwan residue