Gauge origami and quiver W-algebras
Taro Kimura, Go Noshita
Abstract
A bstract We explore the quantum algebraic formalism of the gauge origami system in 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 $$ {\mathfrak{gl}}_1 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>gl</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> . 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.