Abstract
We generalize the framework of Higgsed networks of intertwiners to the quantum toroidal algebra associated to Lie algebra \( \mathfrak{gl} \)N. Using our formalism we obtain a systems of screening operators corresponding to W-algebras associated to toric strip geometries and reproduce partition functions of 3d theories on orbifolded backgrounds.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J.-t. Ding and K. Iohara, Generalization and deformation of Drinfeld quantum affine algebras, Lett. Math. Phys. 41 (1997) 181 [INSPIRE].
K. Miki, A (q, γ)-analog of the W1+∞ algebra, J. Math. Phys. 48 (2007) 1.
L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville Correlation Functions from Four-dimensional Gauge Theories, Lett. Math. Phys. 91 (2010) 167 [arXiv:0906.3219] [INSPIRE].
N. Wyllard, A(N-1) conformal Toda field theory correlation functions from conformal N = 2 SU(N) quiver gauge theories, JHEP 11 (2009) 002 [arXiv:0907.2189] [INSPIRE].
A. Mironov and A. Morozov, On AGT relation in the case of U(3), Nucl. Phys. B 825 (2010) 1 [arXiv:0908.2569] [INSPIRE].
H. Nakajima, Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras, Duke Math. J. 76 (1994) 365 [INSPIRE].
I. Grojnowski, Instantons and affine algebras I: The Hilbert scheme and vertex operators, alg-geom/9506020 [INSPIRE].
E. Carlsson, N. Nekrasov and A. Okounkov, Five dimensional gauge theories and vertex operators, Moscow Math. J. 14 (2014) 39 [arXiv:1308.2465] [INSPIRE].
Y. Zenkevich, Higgsed network calculus, JHEP 08 (2021) 149 [arXiv:1812.11961] [INSPIRE].
H. Awata, B. Feigin and J. Shiraishi, Quantum Algebraic Approach to Refined Topological Vertex, JHEP 03 (2012) 041 [arXiv:1112.6074] [INSPIRE].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].
H. Awata, H. Kanno, A. Mironov, A. Morozov, K. Suetake and Y. Zenkevich, (q, t)-KZ equations for quantum toroidal algebra and Nekrasov partition functions on ALE spaces, JHEP 03 (2018) 192 [arXiv:1712.08016] [INSPIRE].
J.-E. Bourgine and S. Jeong, New quantum toroidal algebras from 5D \( \mathcal{N} \) = 1 instantons on orbifolds, JHEP 05 (2020) 127 [arXiv:1906.01625] [INSPIRE].
Y. Saito, Quantum toroidal algebras and their vertex representations, Publ. Res. Inst. Math. Sci. 34 (1998) 155 [q-alg/9611030].
Y. Saito, K. Takemura and D. Uglov, Transform. Groups 3 (1998) 75 [q-alg/9702024].
B. Feigin, M. Jimbo, T. Miwa and E. Mukhin, Branching rules for quantum toroidal gln, Adv. Math. 300 (2016) 229 [arXiv:1309.2147] [INSPIRE].
A. Tsymbaliuk, Several realizations of Fock modules for quantum toroidal algebras of sl(n), Algebr. Represent. Theory 22 (2019) 177 [arXiv:1603.08915] [INSPIRE].
B. Feigin, M. Jimbo, T. Miwa and E. Mukhin, Representations of quantum toroidal gln J. Algebra 380 (2013) 78 [arXiv:1204.5378].
B. Feigin, M. Jimbo and E. Mukhin, The (\( \mathfrak{gl} \)m, \( \mathfrak{gl} \)n) Duality in the Quantum Toroidal Setting, Commun. Math. Phys. 367 (2019) 455 [arXiv:1801.08433] [INSPIRE].
I.B. Frenkel and V.G. Kac, Basic representations of affine Lie algebras and dual resonance models, Invent. Math. 62 (1980) 23.
G. Segal, Unitarity Representations of Some Infinite Dimensional Groups, Commun. Math. Phys. 80 (1981) 301 [INSPIRE].
K. Nagao, Quiver varieties and Frenkel-Kac construction, J. Algebra 321 (2009) 3764 [math/0703107].
M. Wakimoto, Fock representations of the affine lie algebra A1(1), Commun. Math. Phys. 104 (1986) 605 [INSPIRE].
L. Bezerra and E. Mukhin, Quantum toroidal algebra associated with \( \mathfrak{gl} \)m|n, Algebr. Represent. Theory 24 (2021) 541 [arXiv:1904.07297] [INSPIRE].
T. Procházka and M. Rapčák, Webs of W-algebras, JHEP 11 (2018) 109 [arXiv:1711.06888] [INSPIRE].
T. Procházka and M. Rapčák, \( \mathcal{W} \)-algebra modules, free fields, and Gukov-Witten defects, JHEP 05 (2019) 159 [arXiv:1808.08837] [INSPIRE].
B. Feigin and S. Gukov, Voa[m4], J. Math. Phys. 61 (2020) 012302 [arXiv:1806.02470] [INSPIRE].
M. Rapčák, On extensions of \( \mathfrak{gl} \)(\( \hat{m\mid n} \)) Kac-Moody algebras and Calabi-Yau singularities, JHEP 01 (2020) 042 [arXiv:1910.00031] [INSPIRE].
J. Zhao, Orbifold Vortex and Super Liouville Theory, arXiv:1111.7095 [INSPIRE].
T. Fujimori, T. Kimura, M. Nitta and K. Ohashi, Vortex counting from field theory, JHEP 06 (2012) 028 [arXiv:1204.1968] [INSPIRE].
Y. Yoshida, Localization of Vortex Partition Functions in \( \mathcal{N} \) = (2, 2) Super Yang-Mills theory, arXiv:1101.0872 [INSPIRE].
G. Bonelli, A. Tanzini and J. Zhao, Vertices, Vortices and Interacting Surface Operators, JHEP 06 (2012) 178 [arXiv:1102.0184] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 1912.13372
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Zenkevich, Y. \( \mathfrak{gl} \)N Higgsed networks. J. High Energ. Phys. 2021, 34 (2021). https://doi.org/10.1007/JHEP12(2021)034
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2021)034