Abstract
We explore the quantum algebraic formalism of the gauge origami system in ℂ4, 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.
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Acknowledgments
We would like to thank Yalong Cao, Hiroaki Kanno, Hitoshi Konno, Norton Lee, and Hiraku Nakajima for fruitful discussions on the project. The work of TK was supported by CNRS through MITI interdisciplinary programs, EIPHI Graduate School (No. ANR-17-EURE-0002) and Bourgogne-Franche-Comte region. A part of the results in this paper has been presented by TK at XIII Workshop on Geometric Correspondences of Gauge Theories, June 2023, Trieste, Italy, and The 16th MSJ-SI: Elliptic Integrable Systems, Representation Theory and Hypergeometric Functions, July 2023, Tokyo, Japan. He is grateful to the organizers for the invitation and hospitality. GN is supported by JSPS KAKENHI Grant-in-Aid for JSPS fellows Grant No. JP22J20944, JSR Fellowship, and FoPM (WINGS Program), the University of Tokyo. GN is also grateful for the hospitality during his stay in Universite de Bourgogne where a part of this work was initiated.
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Kimura, T., Noshita, G. Gauge origami and quiver W-algebras. J. High Energ. Phys. 2024, 208 (2024). https://doi.org/10.1007/JHEP05(2024)208
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DOI: https://doi.org/10.1007/JHEP05(2024)208